research article on relation between porosity of epitaxial...
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Research ArticleOn Relation between Porosity of Epitaxial Layer andQuantity of Radiation Defects Generated during RadiationProcessing in a Multilayer Structure
E L Pankratov1 and E A Bulaeva12
1Nizhny Novgorod State University 23 Gagarin Avenue Nizhny Novgorod 603950 Russia2Nizhny Novgorod State University of Architecture and Civil Engineering 65 Ilrsquoinsky Street Nizhny Novgorod 603950 Russia
Correspondence should be addressed to E L Pankratov elp2004mailru
Received 16 March 2016 Revised 5 May 2016 Accepted 25 May 2016
Academic Editor Oleg Lupan
Copyright copy 2016 E L Pankratov and E A Bulaeva This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We analyzed redistribution of radiation defects in a multilayer structure with porous epitaxial layer The radiation defects weregenerated during radiation processing It has been shown that porosity of epitaxial layer gives a possibility to decrease quantity ofradiation defects
1 Introduction
One of actual questions of solid state electronics is increasingtheir radiation resistance The increasing gives a possibilityto decrease influence of different types of irradiation on char-acteristics of solid state electronic devices Several methodshave been used to increase the radiation resistance of devicesof solid state electronics [1ndash5] One way to solve the problemis using an epitaxial layer over considered devices (overlayer)to organize the resistance Another way to increase theradiation resistance is using special epitaxial layers underconsidered devices to use these layers as drain of radiationdefects An alternative approach to the considered one isusing porous materials near device area to use the porosityas drain of radiation defects again In this paper we consideran approach to decreasing quantity of radiation defectsThe radiation defects have been generated during radiationprocessing of materials For the framework the approachwe consider a heterostructure The heterostructure consistsof a substrate and porous epitaxial layer (see Figure 1) Weassume that the substrate was under the influence of radiationprocessing (ion implantation cosmic radiation etc) throughthe epitaxial layer Radiation processing of materials leadsto generation of radiation defects Main aim of the presentpaper is analysis of influence of porosity of epitaxial layer
on distribution of concentration of radiation defects in theconsidered heterostructure
2 Method of Solution
To solve our aim we calculate distributions of concentrationsof radiation defects in considered heterostructure in spaceand time We determine the above distributions as solutionsof the following system of equations [6ndash12]
120597119868 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
Hindawi Publishing CorporationJournal of NanoscienceVolume 2016 Article ID 3491790 11 pageshttpdxdoiorg10115520163491790
2 Journal of Nanoscience
+120597
120597119909[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597119881 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+120597
120597119909[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(1)
Here 119868(119909 119910 119911 119905) and119881(119909 119910 119911 119905) are the distributions of con-centrations of radiation interstitials and vacancies in spaceand time respectively The first the second and the thirdterms of both equations describe diffusion of point defectswith the diffusion coefficients 119863
119868(119909 119910 119911 119879) for interstitials
and 119863119881(119909 119910 119911 119879) and vacancies respectively The fourth
terms of both equations describe recombination of pointdefects with the parameter of recombination 119896
119868119881(119909 119910 119911 119879)
The fifth sixth and seventh terms of both equations describecorrection to diffusion due to porosity of material Thefunctions 119863
119868119878(119909 119910 119911 119879) and 119863
119881119878(119909 119910 119911 119879) describe depen-
dencies of diffusion coefficients of defects due to porosityof materials on coordinate and temperature 119879 119896 is theBoltzmann constant119881 is themolar volume120583(119909 119910 119911 119879) = 119877sdot119879sdotln(119881
21198811) [10] is the chemical potential119881
1K1198812are the initial
and final volume of pores respectively119877 = 831 J(molesdotK) isthe molar gas constant Last terms of (1) with nonlinearity ofconcentrations of defects1198812(119909 119910 119911 119905) and 1198682(119909 119910 119911 119905) corre-spond to generation of divacancies and analogous complexesof interstitials (see eg [11] and appropriate references inthis work) The functions 119896
119868119868(119909 119910 119911 119879) and 119896
119881119881(119909 119910 119911 119879)
D(x)
D0
L0 a x0
x
D1
P1
D2 P2
C(x t = 0)
Figure 1 Heterostructure which consists of a substrate and anepitaxial layer The figure also shows distribution of concentrationof implanted dopant
describe dependencies of the parameters of generation ofcomplexes point defects
Boundary and initial conditions for (1) could be writtenas
120597119868 (119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597119868 (119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
120597119868 (119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597119868 (119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597119868 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597119868 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
120597119881 (119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597119881 (119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
120597119881 (119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597119881 (119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
Journal of Nanoscience 3
120597119881 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597119881 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
119868 (119909 119910 119911 0) = 119891119868(119909 119910 119911)
119881 (119909 119910 119911 0) = 119891119881(119909 119910 119911)
119881 (1199091+ 119881119899119905 1199101+ 119881119899119905 1199111+ 119881119899119905 119905)
= 119881lowast(1 +
2ℓ120596
119896119879radic11990921+ 11991021+ 11991121
)
(2)
Here 119868 and 119881lowast are the equilibrium distributions of concen-trations of interstitials and vacancies respectively 120596 = 119886
3119886 is the atomic spacing ℓ is the specific surface energy Theabove boundary conditions correspond to absence of flow ofpoint defects through external boundary of heterostructureand absorption of these defects by pores (last condition)The above initial conditions correspond to distributions ofconcentration of the above defects after finishing radiationprocessing To take into account porosity we assume thatporous are approximately cylindrical with average dimen-sions 119903 = radic1199092
1+ 11991021and 119911
1[13] With time small pores
decompose into vacancies and the vacancies are absorbedby large pores [10] The large pores take spherical formduring the absorption [10] Distribution of concentrationof vacancies which was formed due to porosity could bedetermined by summing all pores that is
119881 (119909 119910 119911 119905)
=
119897
sum
119894=0
119898
sum
119895=0
119899
sum
119896=0
119881119901(119909 + 119894120572 119910 + 119895120573 119911 + 119896120594 119905)
119877 = radic1199092 + 1199102 + 1199112
(3)
Here 120572 120573 and 120594 are averaged distances between centers ofpores in 119909 119910 and 119911 directions respectively 119897 119898 and 119899 arequantities of pores in the same directions
We determine distributions of concentrations of divacan-ciesΦ
119881(119909 119910 119911 119905) and diinterstitialsΦ
119868(119909 119910 119911 119905) in space and
time by solving the following system of equations [9 11 1214]
120597Φ119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
minus 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]120597Φ119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
minus 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
(4)
The first the second and the third terms of both equa-tions describe diffusion of point defects with the diffusioncoefficients 119863
Φ119868
(119909 119910 119911 119879) for diinterstitials and 119863Φ119881
(119909 119910
119911 119879) for divacancies The fourth terms of both equationscorrespond to generation of new diinterstitials and divacan-cies The fifth terms of the above equations correspond todecay of existing diinterstitials and divacanciesThe functions119896119868(119909 119910 119911 119879) and 119896
119881(119909 119910 119911 119879) describe the parameters of
decay of the above complexes on coordinate and tempera-ture The last terms of both equations describe correctionto diffusion due to porosity of material The functions119863Φ119868119878(119909 119910 119911 119879) and 119863
Φ119881119878(119909 119910 119911 119879) describe dependencies
of diffusion coefficients of defects due to porositymaterials oncoordinate and temperature Boundary and initial conditionsfor (4) could be written as
120597Φ119868(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
4 Journal of Nanoscience
120597Φ119868(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597Φ119868(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
120597Φ119881(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
120597Φ119881(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597Φ119881(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
Φ119868(119909 119910 119911 0) = 119891
Φ119868
(119909 119910 119911)
Φ119881(119909 119910 119911 0) = 119891
Φ119881
(119909 119910 119911)
(5)
The above boundary conditions correspond to absence offlow of point defects through external boundary of het-erostructure The above initial conditions correspond todistributions of concentration of the above defects afterfinishing radiation processing
Wedetermine distributions of concentrations of radiationdefects in space and time by method of averaging of functioncorrections [14ndash16] To use the approach we write (1) and (4)on account of initial distributions of defects that is
120597119868 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+ 119891119868(119909 119910 119911) 120575 (119905) +
120597
120597119909[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597119881 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881 (119909 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+ 119891119881(119909 119910 119911) 120575 (119905) +
120597
120597119909[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(1a)
120597Φ119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
minus 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
Journal of Nanoscience 5
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
minus 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
(3a)
Farther we replace the required concentrations in rightsides of (1a) and (3a) on their not yet known average values1205721120588The replacement gives us possibility to obtain the follow-
ing equations for determining the first-order approximationsof concentrations of radiation defects in the following form
1205971198681(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119868(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119868119896119868119868(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119881(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119881119896119881119881
(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
(1b)
120597Φ1119868(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597Φ1119881(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119881
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(3b)
Integration of the left and right sides of (1b) and (3b) ontime gives a possibility to obtain the first-order approxima-tions of concentrations of radiation defects in the final form
1198681(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119911int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119868int
119905
0
119896119868119868(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
6 Journal of Nanoscience
1198811(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119911int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119881int
119905
0
119896119881119881
(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
(1c)
Φ1119868(119909 119910 119911 119905)
= 119891Φ119868
(119909 119910 119911) + int
119905
0
119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
Φ1119881(119909 119910 119911 119905)
= 119891Φ119881
(119909 119910 119911) + int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
(3c)
Average values of the first-order approximations of therequired approximations could be calculated by the followingrelation [14ndash16]
1205721120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
1205881(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
(6)
Substitution of relations (1c) and (3c) into relation (6)gives a possibility to calculate the required average values inthe following form
1205721119868= radic
(1198863+ 119860)2
411988624
minus 4(119861 +Θ1198863119861 + Θ
21198711199091198711199101198711199111198861
1198864
)
minus1198863+ 119860
41198864
1205721119881
=1
11987811986811988100
[Θ
1205721119868
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 120572111986811987811986811986800
minus Θ119871119909119871119910119871119911]
1205721Φ119868
=1198771198681
Θ119871119909119871119910119871119911
+11987811986811986820
Θ119871119909119871119910119871119911
+1
119871119909119871119910119871119911
sdot int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119868
(119909 119910 119911) 119889119911 119889119910 119889119909
1205721Φ119881
=1
119871119909119871119910119871119911
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119881
(119909 119910 119911) 119889119911 119889119910 119889119909
+(1198771198811+ 11987811988111988120
)
Θ119871119909119871119910119871119911
(7)
Here
119878120588120588119894119895
= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896120588120588(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119881
119895
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
1198864= 11987811986811986800
(1198782
11986811988100minus 11987811986811986800
11987811988111988100
)
1198863= 11987811986811988100
11987811986811986800
+ 1198782
11986811988100minus 11987811986811986800
11987811988111988100
1198862= int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119881(119909 119910 119911) 119889119911 119889119910 119889119909119878
119868119881001198782
11986811988100+ 211987811988111988100
11987811986811986800
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 1198782
11986811988100int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909 + 119878
11986811988100Θ1198712
1199091198712
1199101198712
119911minus Θ1198712
1199091198712
1199101198712
11991111987811988111988100
Journal of Nanoscience 7
1198861= 11987811986811988100
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
119860 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
1198860= 11987811988111988100
[int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909]
2
119861 =Θ1198862
61198864
+3
radicradic1199022 + 1199013 minus 119902 minus3
radicradic1199022 + 1199013 + 119902
119902 =Θ31198862
2411988624
(41198860minus Θ119871119909119871119910119871119911
11988611198863
1198864
) minus Θ2 1198860
811988624
(4Θ1198862minus Θ21198862
3
1198864
) minusΘ31198863
2
5411988634
minus 1198712
1199091198712
1199101198712
119911
Θ41198862
1
811988624
119877120588119894= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896119868(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
119901 = Θ2 411988601198864
1211988624
minusΘ1198862
181198864
minusΘ11988611198863
1211988624
119871119909119871119910119871119911
(8)
We determine approximations with the second andhigher orders of concentrations of radiations defects frame-work standard iterative procedure of method of averagingof function corrections [14ndash16] For the framework of theprocedure we determine the approximation of 119899th orderby replacement of the concentrations of radiation defects119868(119909 119910 119911 119905) 119881(119909 119910 119911 119905) Φ
119868(119909 119910 119911 119905) and Φ
119881(119909 119910 119911 119905) in
right sides of (1b) and (3b) on the following sums 120572119899120588
+
120588119899minus1
(119909 119910 119911 119905) The replacement gives a possibility to obtainthe second-order approximations of concentrations of radia-tion defects
1205971198682(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119868119868(119909 119910 119911 119879)
sdot [1205721119868+ 1198681(119909 119910 119911 119905)]
2
1205971198812(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119881119881(119909 119910 119911 119879)
sdot [1205721119868+ 1198811(119909 119910 119911 119905)]
2
(1d)
120597Φ2119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119909]
8 Journal of Nanoscience
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905) + 119896
119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 119905) +120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ2119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905) + 119896
119881(119909 119910 119911 119879)
sdot 119881 (119909 119910 119911 119905) +120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119881
(119909 119910 119911) 120575 (119905)
(3d)
Integration of left and right sides of (1d) and (3d) gives apossibility to obtain relations for the second-order approxi-mations of the required concentrations of radiation defectsin the following form
1198682(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868(119909 119910 119911
119879)1205971198681(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119868119868(119909 119910 119911 119879) [120572
2119868+ 1198681(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [1205722119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119868119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
(1e)
1198812(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119881119881
(119909 119910 119911 119879) [1205722119881+ 1198811(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [120572119881119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119881119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
Φ2119868(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119868
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 120591)
120597119911119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 + int
119905
0
119896119868119868(119909 119910 119911 119879)
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
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BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
2 Journal of Nanoscience
+120597
120597119909[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597119881 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+120597
120597119909[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(1)
Here 119868(119909 119910 119911 119905) and119881(119909 119910 119911 119905) are the distributions of con-centrations of radiation interstitials and vacancies in spaceand time respectively The first the second and the thirdterms of both equations describe diffusion of point defectswith the diffusion coefficients 119863
119868(119909 119910 119911 119879) for interstitials
and 119863119881(119909 119910 119911 119879) and vacancies respectively The fourth
terms of both equations describe recombination of pointdefects with the parameter of recombination 119896
119868119881(119909 119910 119911 119879)
The fifth sixth and seventh terms of both equations describecorrection to diffusion due to porosity of material Thefunctions 119863
119868119878(119909 119910 119911 119879) and 119863
119881119878(119909 119910 119911 119879) describe depen-
dencies of diffusion coefficients of defects due to porosityof materials on coordinate and temperature 119879 119896 is theBoltzmann constant119881 is themolar volume120583(119909 119910 119911 119879) = 119877sdot119879sdotln(119881
21198811) [10] is the chemical potential119881
1K1198812are the initial
and final volume of pores respectively119877 = 831 J(molesdotK) isthe molar gas constant Last terms of (1) with nonlinearity ofconcentrations of defects1198812(119909 119910 119911 119905) and 1198682(119909 119910 119911 119905) corre-spond to generation of divacancies and analogous complexesof interstitials (see eg [11] and appropriate references inthis work) The functions 119896
119868119868(119909 119910 119911 119879) and 119896
119881119881(119909 119910 119911 119879)
D(x)
D0
L0 a x0
x
D1
P1
D2 P2
C(x t = 0)
Figure 1 Heterostructure which consists of a substrate and anepitaxial layer The figure also shows distribution of concentrationof implanted dopant
describe dependencies of the parameters of generation ofcomplexes point defects
Boundary and initial conditions for (1) could be writtenas
120597119868 (119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597119868 (119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
120597119868 (119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597119868 (119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597119868 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597119868 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
120597119881 (119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597119881 (119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
120597119881 (119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597119881 (119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
Journal of Nanoscience 3
120597119881 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597119881 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
119868 (119909 119910 119911 0) = 119891119868(119909 119910 119911)
119881 (119909 119910 119911 0) = 119891119881(119909 119910 119911)
119881 (1199091+ 119881119899119905 1199101+ 119881119899119905 1199111+ 119881119899119905 119905)
= 119881lowast(1 +
2ℓ120596
119896119879radic11990921+ 11991021+ 11991121
)
(2)
Here 119868 and 119881lowast are the equilibrium distributions of concen-trations of interstitials and vacancies respectively 120596 = 119886
3119886 is the atomic spacing ℓ is the specific surface energy Theabove boundary conditions correspond to absence of flow ofpoint defects through external boundary of heterostructureand absorption of these defects by pores (last condition)The above initial conditions correspond to distributions ofconcentration of the above defects after finishing radiationprocessing To take into account porosity we assume thatporous are approximately cylindrical with average dimen-sions 119903 = radic1199092
1+ 11991021and 119911
1[13] With time small pores
decompose into vacancies and the vacancies are absorbedby large pores [10] The large pores take spherical formduring the absorption [10] Distribution of concentrationof vacancies which was formed due to porosity could bedetermined by summing all pores that is
119881 (119909 119910 119911 119905)
=
119897
sum
119894=0
119898
sum
119895=0
119899
sum
119896=0
119881119901(119909 + 119894120572 119910 + 119895120573 119911 + 119896120594 119905)
119877 = radic1199092 + 1199102 + 1199112
(3)
Here 120572 120573 and 120594 are averaged distances between centers ofpores in 119909 119910 and 119911 directions respectively 119897 119898 and 119899 arequantities of pores in the same directions
We determine distributions of concentrations of divacan-ciesΦ
119881(119909 119910 119911 119905) and diinterstitialsΦ
119868(119909 119910 119911 119905) in space and
time by solving the following system of equations [9 11 1214]
120597Φ119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
minus 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]120597Φ119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
minus 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
(4)
The first the second and the third terms of both equa-tions describe diffusion of point defects with the diffusioncoefficients 119863
Φ119868
(119909 119910 119911 119879) for diinterstitials and 119863Φ119881
(119909 119910
119911 119879) for divacancies The fourth terms of both equationscorrespond to generation of new diinterstitials and divacan-cies The fifth terms of the above equations correspond todecay of existing diinterstitials and divacanciesThe functions119896119868(119909 119910 119911 119879) and 119896
119881(119909 119910 119911 119879) describe the parameters of
decay of the above complexes on coordinate and tempera-ture The last terms of both equations describe correctionto diffusion due to porosity of material The functions119863Φ119868119878(119909 119910 119911 119879) and 119863
Φ119881119878(119909 119910 119911 119879) describe dependencies
of diffusion coefficients of defects due to porositymaterials oncoordinate and temperature Boundary and initial conditionsfor (4) could be written as
120597Φ119868(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
4 Journal of Nanoscience
120597Φ119868(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597Φ119868(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
120597Φ119881(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
120597Φ119881(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597Φ119881(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
Φ119868(119909 119910 119911 0) = 119891
Φ119868
(119909 119910 119911)
Φ119881(119909 119910 119911 0) = 119891
Φ119881
(119909 119910 119911)
(5)
The above boundary conditions correspond to absence offlow of point defects through external boundary of het-erostructure The above initial conditions correspond todistributions of concentration of the above defects afterfinishing radiation processing
Wedetermine distributions of concentrations of radiationdefects in space and time by method of averaging of functioncorrections [14ndash16] To use the approach we write (1) and (4)on account of initial distributions of defects that is
120597119868 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+ 119891119868(119909 119910 119911) 120575 (119905) +
120597
120597119909[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597119881 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881 (119909 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+ 119891119881(119909 119910 119911) 120575 (119905) +
120597
120597119909[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(1a)
120597Φ119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
minus 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
Journal of Nanoscience 5
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
minus 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
(3a)
Farther we replace the required concentrations in rightsides of (1a) and (3a) on their not yet known average values1205721120588The replacement gives us possibility to obtain the follow-
ing equations for determining the first-order approximationsof concentrations of radiation defects in the following form
1205971198681(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119868(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119868119896119868119868(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119881(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119881119896119881119881
(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
(1b)
120597Φ1119868(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597Φ1119881(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119881
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(3b)
Integration of the left and right sides of (1b) and (3b) ontime gives a possibility to obtain the first-order approxima-tions of concentrations of radiation defects in the final form
1198681(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119911int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119868int
119905
0
119896119868119868(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
6 Journal of Nanoscience
1198811(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119911int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119881int
119905
0
119896119881119881
(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
(1c)
Φ1119868(119909 119910 119911 119905)
= 119891Φ119868
(119909 119910 119911) + int
119905
0
119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
Φ1119881(119909 119910 119911 119905)
= 119891Φ119881
(119909 119910 119911) + int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
(3c)
Average values of the first-order approximations of therequired approximations could be calculated by the followingrelation [14ndash16]
1205721120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
1205881(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
(6)
Substitution of relations (1c) and (3c) into relation (6)gives a possibility to calculate the required average values inthe following form
1205721119868= radic
(1198863+ 119860)2
411988624
minus 4(119861 +Θ1198863119861 + Θ
21198711199091198711199101198711199111198861
1198864
)
minus1198863+ 119860
41198864
1205721119881
=1
11987811986811988100
[Θ
1205721119868
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 120572111986811987811986811986800
minus Θ119871119909119871119910119871119911]
1205721Φ119868
=1198771198681
Θ119871119909119871119910119871119911
+11987811986811986820
Θ119871119909119871119910119871119911
+1
119871119909119871119910119871119911
sdot int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119868
(119909 119910 119911) 119889119911 119889119910 119889119909
1205721Φ119881
=1
119871119909119871119910119871119911
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119881
(119909 119910 119911) 119889119911 119889119910 119889119909
+(1198771198811+ 11987811988111988120
)
Θ119871119909119871119910119871119911
(7)
Here
119878120588120588119894119895
= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896120588120588(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119881
119895
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
1198864= 11987811986811986800
(1198782
11986811988100minus 11987811986811986800
11987811988111988100
)
1198863= 11987811986811988100
11987811986811986800
+ 1198782
11986811988100minus 11987811986811986800
11987811988111988100
1198862= int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119881(119909 119910 119911) 119889119911 119889119910 119889119909119878
119868119881001198782
11986811988100+ 211987811988111988100
11987811986811986800
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 1198782
11986811988100int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909 + 119878
11986811988100Θ1198712
1199091198712
1199101198712
119911minus Θ1198712
1199091198712
1199101198712
11991111987811988111988100
Journal of Nanoscience 7
1198861= 11987811986811988100
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
119860 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
1198860= 11987811988111988100
[int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909]
2
119861 =Θ1198862
61198864
+3
radicradic1199022 + 1199013 minus 119902 minus3
radicradic1199022 + 1199013 + 119902
119902 =Θ31198862
2411988624
(41198860minus Θ119871119909119871119910119871119911
11988611198863
1198864
) minus Θ2 1198860
811988624
(4Θ1198862minus Θ21198862
3
1198864
) minusΘ31198863
2
5411988634
minus 1198712
1199091198712
1199101198712
119911
Θ41198862
1
811988624
119877120588119894= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896119868(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
119901 = Θ2 411988601198864
1211988624
minusΘ1198862
181198864
minusΘ11988611198863
1211988624
119871119909119871119910119871119911
(8)
We determine approximations with the second andhigher orders of concentrations of radiations defects frame-work standard iterative procedure of method of averagingof function corrections [14ndash16] For the framework of theprocedure we determine the approximation of 119899th orderby replacement of the concentrations of radiation defects119868(119909 119910 119911 119905) 119881(119909 119910 119911 119905) Φ
119868(119909 119910 119911 119905) and Φ
119881(119909 119910 119911 119905) in
right sides of (1b) and (3b) on the following sums 120572119899120588
+
120588119899minus1
(119909 119910 119911 119905) The replacement gives a possibility to obtainthe second-order approximations of concentrations of radia-tion defects
1205971198682(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119868119868(119909 119910 119911 119879)
sdot [1205721119868+ 1198681(119909 119910 119911 119905)]
2
1205971198812(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119881119881(119909 119910 119911 119879)
sdot [1205721119868+ 1198811(119909 119910 119911 119905)]
2
(1d)
120597Φ2119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119909]
8 Journal of Nanoscience
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905) + 119896
119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 119905) +120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ2119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905) + 119896
119881(119909 119910 119911 119879)
sdot 119881 (119909 119910 119911 119905) +120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119881
(119909 119910 119911) 120575 (119905)
(3d)
Integration of left and right sides of (1d) and (3d) gives apossibility to obtain relations for the second-order approxi-mations of the required concentrations of radiation defectsin the following form
1198682(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868(119909 119910 119911
119879)1205971198681(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119868119868(119909 119910 119911 119879) [120572
2119868+ 1198681(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [1205722119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119868119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
(1e)
1198812(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119881119881
(119909 119910 119911 119879) [1205722119881+ 1198811(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [120572119881119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119881119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
Φ2119868(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119868
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 120591)
120597119911119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 + int
119905
0
119896119868119868(119909 119910 119911 119879)
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 3
120597119881 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597119881 (119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
119868 (119909 119910 119911 0) = 119891119868(119909 119910 119911)
119881 (119909 119910 119911 0) = 119891119881(119909 119910 119911)
119881 (1199091+ 119881119899119905 1199101+ 119881119899119905 1199111+ 119881119899119905 119905)
= 119881lowast(1 +
2ℓ120596
119896119879radic11990921+ 11991021+ 11991121
)
(2)
Here 119868 and 119881lowast are the equilibrium distributions of concen-trations of interstitials and vacancies respectively 120596 = 119886
3119886 is the atomic spacing ℓ is the specific surface energy Theabove boundary conditions correspond to absence of flow ofpoint defects through external boundary of heterostructureand absorption of these defects by pores (last condition)The above initial conditions correspond to distributions ofconcentration of the above defects after finishing radiationprocessing To take into account porosity we assume thatporous are approximately cylindrical with average dimen-sions 119903 = radic1199092
1+ 11991021and 119911
1[13] With time small pores
decompose into vacancies and the vacancies are absorbedby large pores [10] The large pores take spherical formduring the absorption [10] Distribution of concentrationof vacancies which was formed due to porosity could bedetermined by summing all pores that is
119881 (119909 119910 119911 119905)
=
119897
sum
119894=0
119898
sum
119895=0
119899
sum
119896=0
119881119901(119909 + 119894120572 119910 + 119895120573 119911 + 119896120594 119905)
119877 = radic1199092 + 1199102 + 1199112
(3)
Here 120572 120573 and 120594 are averaged distances between centers ofpores in 119909 119910 and 119911 directions respectively 119897 119898 and 119899 arequantities of pores in the same directions
We determine distributions of concentrations of divacan-ciesΦ
119881(119909 119910 119911 119905) and diinterstitialsΦ
119868(119909 119910 119911 119905) in space and
time by solving the following system of equations [9 11 1214]
120597Φ119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
minus 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]120597Φ119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
minus 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
(4)
The first the second and the third terms of both equa-tions describe diffusion of point defects with the diffusioncoefficients 119863
Φ119868
(119909 119910 119911 119879) for diinterstitials and 119863Φ119881
(119909 119910
119911 119879) for divacancies The fourth terms of both equationscorrespond to generation of new diinterstitials and divacan-cies The fifth terms of the above equations correspond todecay of existing diinterstitials and divacanciesThe functions119896119868(119909 119910 119911 119879) and 119896
119881(119909 119910 119911 119879) describe the parameters of
decay of the above complexes on coordinate and tempera-ture The last terms of both equations describe correctionto diffusion due to porosity of material The functions119863Φ119868119878(119909 119910 119911 119879) and 119863
Φ119881119878(119909 119910 119911 119879) describe dependencies
of diffusion coefficients of defects due to porositymaterials oncoordinate and temperature Boundary and initial conditionsfor (4) could be written as
120597Φ119868(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
4 Journal of Nanoscience
120597Φ119868(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597Φ119868(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
120597Φ119881(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
120597Φ119881(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597Φ119881(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
Φ119868(119909 119910 119911 0) = 119891
Φ119868
(119909 119910 119911)
Φ119881(119909 119910 119911 0) = 119891
Φ119881
(119909 119910 119911)
(5)
The above boundary conditions correspond to absence offlow of point defects through external boundary of het-erostructure The above initial conditions correspond todistributions of concentration of the above defects afterfinishing radiation processing
Wedetermine distributions of concentrations of radiationdefects in space and time by method of averaging of functioncorrections [14ndash16] To use the approach we write (1) and (4)on account of initial distributions of defects that is
120597119868 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+ 119891119868(119909 119910 119911) 120575 (119905) +
120597
120597119909[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597119881 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881 (119909 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+ 119891119881(119909 119910 119911) 120575 (119905) +
120597
120597119909[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(1a)
120597Φ119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
minus 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
Journal of Nanoscience 5
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
minus 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
(3a)
Farther we replace the required concentrations in rightsides of (1a) and (3a) on their not yet known average values1205721120588The replacement gives us possibility to obtain the follow-
ing equations for determining the first-order approximationsof concentrations of radiation defects in the following form
1205971198681(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119868(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119868119896119868119868(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119881(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119881119896119881119881
(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
(1b)
120597Φ1119868(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597Φ1119881(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119881
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(3b)
Integration of the left and right sides of (1b) and (3b) ontime gives a possibility to obtain the first-order approxima-tions of concentrations of radiation defects in the final form
1198681(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119911int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119868int
119905
0
119896119868119868(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
6 Journal of Nanoscience
1198811(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119911int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119881int
119905
0
119896119881119881
(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
(1c)
Φ1119868(119909 119910 119911 119905)
= 119891Φ119868
(119909 119910 119911) + int
119905
0
119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
Φ1119881(119909 119910 119911 119905)
= 119891Φ119881
(119909 119910 119911) + int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
(3c)
Average values of the first-order approximations of therequired approximations could be calculated by the followingrelation [14ndash16]
1205721120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
1205881(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
(6)
Substitution of relations (1c) and (3c) into relation (6)gives a possibility to calculate the required average values inthe following form
1205721119868= radic
(1198863+ 119860)2
411988624
minus 4(119861 +Θ1198863119861 + Θ
21198711199091198711199101198711199111198861
1198864
)
minus1198863+ 119860
41198864
1205721119881
=1
11987811986811988100
[Θ
1205721119868
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 120572111986811987811986811986800
minus Θ119871119909119871119910119871119911]
1205721Φ119868
=1198771198681
Θ119871119909119871119910119871119911
+11987811986811986820
Θ119871119909119871119910119871119911
+1
119871119909119871119910119871119911
sdot int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119868
(119909 119910 119911) 119889119911 119889119910 119889119909
1205721Φ119881
=1
119871119909119871119910119871119911
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119881
(119909 119910 119911) 119889119911 119889119910 119889119909
+(1198771198811+ 11987811988111988120
)
Θ119871119909119871119910119871119911
(7)
Here
119878120588120588119894119895
= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896120588120588(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119881
119895
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
1198864= 11987811986811986800
(1198782
11986811988100minus 11987811986811986800
11987811988111988100
)
1198863= 11987811986811988100
11987811986811986800
+ 1198782
11986811988100minus 11987811986811986800
11987811988111988100
1198862= int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119881(119909 119910 119911) 119889119911 119889119910 119889119909119878
119868119881001198782
11986811988100+ 211987811988111988100
11987811986811986800
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 1198782
11986811988100int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909 + 119878
11986811988100Θ1198712
1199091198712
1199101198712
119911minus Θ1198712
1199091198712
1199101198712
11991111987811988111988100
Journal of Nanoscience 7
1198861= 11987811986811988100
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
119860 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
1198860= 11987811988111988100
[int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909]
2
119861 =Θ1198862
61198864
+3
radicradic1199022 + 1199013 minus 119902 minus3
radicradic1199022 + 1199013 + 119902
119902 =Θ31198862
2411988624
(41198860minus Θ119871119909119871119910119871119911
11988611198863
1198864
) minus Θ2 1198860
811988624
(4Θ1198862minus Θ21198862
3
1198864
) minusΘ31198863
2
5411988634
minus 1198712
1199091198712
1199101198712
119911
Θ41198862
1
811988624
119877120588119894= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896119868(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
119901 = Θ2 411988601198864
1211988624
minusΘ1198862
181198864
minusΘ11988611198863
1211988624
119871119909119871119910119871119911
(8)
We determine approximations with the second andhigher orders of concentrations of radiations defects frame-work standard iterative procedure of method of averagingof function corrections [14ndash16] For the framework of theprocedure we determine the approximation of 119899th orderby replacement of the concentrations of radiation defects119868(119909 119910 119911 119905) 119881(119909 119910 119911 119905) Φ
119868(119909 119910 119911 119905) and Φ
119881(119909 119910 119911 119905) in
right sides of (1b) and (3b) on the following sums 120572119899120588
+
120588119899minus1
(119909 119910 119911 119905) The replacement gives a possibility to obtainthe second-order approximations of concentrations of radia-tion defects
1205971198682(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119868119868(119909 119910 119911 119879)
sdot [1205721119868+ 1198681(119909 119910 119911 119905)]
2
1205971198812(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119881119881(119909 119910 119911 119879)
sdot [1205721119868+ 1198811(119909 119910 119911 119905)]
2
(1d)
120597Φ2119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119909]
8 Journal of Nanoscience
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905) + 119896
119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 119905) +120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ2119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905) + 119896
119881(119909 119910 119911 119879)
sdot 119881 (119909 119910 119911 119905) +120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119881
(119909 119910 119911) 120575 (119905)
(3d)
Integration of left and right sides of (1d) and (3d) gives apossibility to obtain relations for the second-order approxi-mations of the required concentrations of radiation defectsin the following form
1198682(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868(119909 119910 119911
119879)1205971198681(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119868119868(119909 119910 119911 119879) [120572
2119868+ 1198681(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [1205722119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119868119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
(1e)
1198812(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119881119881
(119909 119910 119911 119879) [1205722119881+ 1198811(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [120572119881119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119881119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
Φ2119868(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119868
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 120591)
120597119911119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 + int
119905
0
119896119868119868(119909 119910 119911 119879)
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
4 Journal of Nanoscience
120597Φ119868(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597Φ119868(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597Φ119868(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
120597Φ119881(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119909
100381610038161003816100381610038161003816100381610038161003816119909=119871119909
= 0
120597Φ119881(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119910
100381610038161003816100381610038161003816100381610038161003816119910=119871119910
= 0
120597Φ119881(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=0
= 0
120597Φ119881(119909 119910 119911 119905)
120597119911
100381610038161003816100381610038161003816100381610038161003816119911=119871119911
= 0
Φ119868(119909 119910 119911 0) = 119891
Φ119868
(119909 119910 119911)
Φ119881(119909 119910 119911 0) = 119891
Φ119881
(119909 119910 119911)
(5)
The above boundary conditions correspond to absence offlow of point defects through external boundary of het-erostructure The above initial conditions correspond todistributions of concentration of the above defects afterfinishing radiation processing
Wedetermine distributions of concentrations of radiationdefects in space and time by method of averaging of functioncorrections [14ndash16] To use the approach we write (1) and (4)on account of initial distributions of defects that is
120597119868 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
120597119868 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+ 119891119868(119909 119910 119911) 120575 (119905) +
120597
120597119909[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597119881 (119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
120597119881 (119909 119910 119911 119905)
120597119911]
minus 119896119868119881 (119909 119879) 119868 (119909 119910 119911 119905) 119881 (119909 119910 119911 119905)
+ 119891119881(119909 119910 119911) 120575 (119905) +
120597
120597119909[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
minus 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(1a)
120597Φ119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
minus 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
Journal of Nanoscience 5
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
minus 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
(3a)
Farther we replace the required concentrations in rightsides of (1a) and (3a) on their not yet known average values1205721120588The replacement gives us possibility to obtain the follow-
ing equations for determining the first-order approximationsof concentrations of radiation defects in the following form
1205971198681(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119868(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119868119896119868119868(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119881(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119881119896119881119881
(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
(1b)
120597Φ1119868(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597Φ1119881(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119881
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(3b)
Integration of the left and right sides of (1b) and (3b) ontime gives a possibility to obtain the first-order approxima-tions of concentrations of radiation defects in the final form
1198681(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119911int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119868int
119905
0
119896119868119868(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
6 Journal of Nanoscience
1198811(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119911int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119881int
119905
0
119896119881119881
(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
(1c)
Φ1119868(119909 119910 119911 119905)
= 119891Φ119868
(119909 119910 119911) + int
119905
0
119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
Φ1119881(119909 119910 119911 119905)
= 119891Φ119881
(119909 119910 119911) + int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
(3c)
Average values of the first-order approximations of therequired approximations could be calculated by the followingrelation [14ndash16]
1205721120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
1205881(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
(6)
Substitution of relations (1c) and (3c) into relation (6)gives a possibility to calculate the required average values inthe following form
1205721119868= radic
(1198863+ 119860)2
411988624
minus 4(119861 +Θ1198863119861 + Θ
21198711199091198711199101198711199111198861
1198864
)
minus1198863+ 119860
41198864
1205721119881
=1
11987811986811988100
[Θ
1205721119868
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 120572111986811987811986811986800
minus Θ119871119909119871119910119871119911]
1205721Φ119868
=1198771198681
Θ119871119909119871119910119871119911
+11987811986811986820
Θ119871119909119871119910119871119911
+1
119871119909119871119910119871119911
sdot int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119868
(119909 119910 119911) 119889119911 119889119910 119889119909
1205721Φ119881
=1
119871119909119871119910119871119911
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119881
(119909 119910 119911) 119889119911 119889119910 119889119909
+(1198771198811+ 11987811988111988120
)
Θ119871119909119871119910119871119911
(7)
Here
119878120588120588119894119895
= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896120588120588(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119881
119895
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
1198864= 11987811986811986800
(1198782
11986811988100minus 11987811986811986800
11987811988111988100
)
1198863= 11987811986811988100
11987811986811986800
+ 1198782
11986811988100minus 11987811986811986800
11987811988111988100
1198862= int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119881(119909 119910 119911) 119889119911 119889119910 119889119909119878
119868119881001198782
11986811988100+ 211987811988111988100
11987811986811986800
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 1198782
11986811988100int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909 + 119878
11986811988100Θ1198712
1199091198712
1199101198712
119911minus Θ1198712
1199091198712
1199101198712
11991111987811988111988100
Journal of Nanoscience 7
1198861= 11987811986811988100
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
119860 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
1198860= 11987811988111988100
[int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909]
2
119861 =Θ1198862
61198864
+3
radicradic1199022 + 1199013 minus 119902 minus3
radicradic1199022 + 1199013 + 119902
119902 =Θ31198862
2411988624
(41198860minus Θ119871119909119871119910119871119911
11988611198863
1198864
) minus Θ2 1198860
811988624
(4Θ1198862minus Θ21198862
3
1198864
) minusΘ31198863
2
5411988634
minus 1198712
1199091198712
1199101198712
119911
Θ41198862
1
811988624
119877120588119894= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896119868(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
119901 = Θ2 411988601198864
1211988624
minusΘ1198862
181198864
minusΘ11988611198863
1211988624
119871119909119871119910119871119911
(8)
We determine approximations with the second andhigher orders of concentrations of radiations defects frame-work standard iterative procedure of method of averagingof function corrections [14ndash16] For the framework of theprocedure we determine the approximation of 119899th orderby replacement of the concentrations of radiation defects119868(119909 119910 119911 119905) 119881(119909 119910 119911 119905) Φ
119868(119909 119910 119911 119905) and Φ
119881(119909 119910 119911 119905) in
right sides of (1b) and (3b) on the following sums 120572119899120588
+
120588119899minus1
(119909 119910 119911 119905) The replacement gives a possibility to obtainthe second-order approximations of concentrations of radia-tion defects
1205971198682(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119868119868(119909 119910 119911 119879)
sdot [1205721119868+ 1198681(119909 119910 119911 119905)]
2
1205971198812(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119881119881(119909 119910 119911 119879)
sdot [1205721119868+ 1198811(119909 119910 119911 119905)]
2
(1d)
120597Φ2119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119909]
8 Journal of Nanoscience
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905) + 119896
119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 119905) +120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ2119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905) + 119896
119881(119909 119910 119911 119879)
sdot 119881 (119909 119910 119911 119905) +120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119881
(119909 119910 119911) 120575 (119905)
(3d)
Integration of left and right sides of (1d) and (3d) gives apossibility to obtain relations for the second-order approxi-mations of the required concentrations of radiation defectsin the following form
1198682(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868(119909 119910 119911
119879)1205971198681(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119868119868(119909 119910 119911 119879) [120572
2119868+ 1198681(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [1205722119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119868119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
(1e)
1198812(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119881119881
(119909 119910 119911 119879) [1205722119881+ 1198811(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [120572119881119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119881119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
Φ2119868(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119868
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 120591)
120597119911119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 + int
119905
0
119896119868119868(119909 119910 119911 119879)
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 5
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ119881(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
minus 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+120597
120597119909[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
120597120583 (119909 119910 119911 119905)
120597119911]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
(3a)
Farther we replace the required concentrations in rightsides of (1a) and (3a) on their not yet known average values1205721120588The replacement gives us possibility to obtain the follow-
ing equations for determining the first-order approximationsof concentrations of radiation defects in the following form
1205971198681(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119868(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119868119896119868119868(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891119881(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
minus 1205722
1119881119896119881119881
(119909 119910 119911 119879)
minus 12057211198681205721119881119896119868119881
(119909 119910 119911 119879)
(1b)
120597Φ1119868(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119868
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 119905)
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905)
120597Φ1119881(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+ 119891Φ119881
(119909 119910 119911) 120575 (119905)
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
+ 119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 119905)
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905)
(3b)
Integration of the left and right sides of (1b) and (3b) ontime gives a possibility to obtain the first-order approxima-tions of concentrations of radiation defects in the final form
1198681(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119911int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119868int
119905
0
119896119868119868(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
6 Journal of Nanoscience
1198811(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119911int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119881int
119905
0
119896119881119881
(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
(1c)
Φ1119868(119909 119910 119911 119905)
= 119891Φ119868
(119909 119910 119911) + int
119905
0
119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
Φ1119881(119909 119910 119911 119905)
= 119891Φ119881
(119909 119910 119911) + int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
(3c)
Average values of the first-order approximations of therequired approximations could be calculated by the followingrelation [14ndash16]
1205721120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
1205881(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
(6)
Substitution of relations (1c) and (3c) into relation (6)gives a possibility to calculate the required average values inthe following form
1205721119868= radic
(1198863+ 119860)2
411988624
minus 4(119861 +Θ1198863119861 + Θ
21198711199091198711199101198711199111198861
1198864
)
minus1198863+ 119860
41198864
1205721119881
=1
11987811986811988100
[Θ
1205721119868
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 120572111986811987811986811986800
minus Θ119871119909119871119910119871119911]
1205721Φ119868
=1198771198681
Θ119871119909119871119910119871119911
+11987811986811986820
Θ119871119909119871119910119871119911
+1
119871119909119871119910119871119911
sdot int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119868
(119909 119910 119911) 119889119911 119889119910 119889119909
1205721Φ119881
=1
119871119909119871119910119871119911
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119881
(119909 119910 119911) 119889119911 119889119910 119889119909
+(1198771198811+ 11987811988111988120
)
Θ119871119909119871119910119871119911
(7)
Here
119878120588120588119894119895
= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896120588120588(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119881
119895
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
1198864= 11987811986811986800
(1198782
11986811988100minus 11987811986811986800
11987811988111988100
)
1198863= 11987811986811988100
11987811986811986800
+ 1198782
11986811988100minus 11987811986811986800
11987811988111988100
1198862= int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119881(119909 119910 119911) 119889119911 119889119910 119889119909119878
119868119881001198782
11986811988100+ 211987811988111988100
11987811986811986800
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 1198782
11986811988100int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909 + 119878
11986811988100Θ1198712
1199091198712
1199101198712
119911minus Θ1198712
1199091198712
1199101198712
11991111987811988111988100
Journal of Nanoscience 7
1198861= 11987811986811988100
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
119860 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
1198860= 11987811988111988100
[int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909]
2
119861 =Θ1198862
61198864
+3
radicradic1199022 + 1199013 minus 119902 minus3
radicradic1199022 + 1199013 + 119902
119902 =Θ31198862
2411988624
(41198860minus Θ119871119909119871119910119871119911
11988611198863
1198864
) minus Θ2 1198860
811988624
(4Θ1198862minus Θ21198862
3
1198864
) minusΘ31198863
2
5411988634
minus 1198712
1199091198712
1199101198712
119911
Θ41198862
1
811988624
119877120588119894= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896119868(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
119901 = Θ2 411988601198864
1211988624
minusΘ1198862
181198864
minusΘ11988611198863
1211988624
119871119909119871119910119871119911
(8)
We determine approximations with the second andhigher orders of concentrations of radiations defects frame-work standard iterative procedure of method of averagingof function corrections [14ndash16] For the framework of theprocedure we determine the approximation of 119899th orderby replacement of the concentrations of radiation defects119868(119909 119910 119911 119905) 119881(119909 119910 119911 119905) Φ
119868(119909 119910 119911 119905) and Φ
119881(119909 119910 119911 119905) in
right sides of (1b) and (3b) on the following sums 120572119899120588
+
120588119899minus1
(119909 119910 119911 119905) The replacement gives a possibility to obtainthe second-order approximations of concentrations of radia-tion defects
1205971198682(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119868119868(119909 119910 119911 119879)
sdot [1205721119868+ 1198681(119909 119910 119911 119905)]
2
1205971198812(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119881119881(119909 119910 119911 119879)
sdot [1205721119868+ 1198811(119909 119910 119911 119905)]
2
(1d)
120597Φ2119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119909]
8 Journal of Nanoscience
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905) + 119896
119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 119905) +120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ2119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905) + 119896
119881(119909 119910 119911 119879)
sdot 119881 (119909 119910 119911 119905) +120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119881
(119909 119910 119911) 120575 (119905)
(3d)
Integration of left and right sides of (1d) and (3d) gives apossibility to obtain relations for the second-order approxi-mations of the required concentrations of radiation defectsin the following form
1198682(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868(119909 119910 119911
119879)1205971198681(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119868119868(119909 119910 119911 119879) [120572
2119868+ 1198681(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [1205722119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119868119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
(1e)
1198812(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119881119881
(119909 119910 119911 119879) [1205722119881+ 1198811(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [120572119881119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119881119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
Φ2119868(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119868
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 120591)
120597119911119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 + int
119905
0
119896119868119868(119909 119910 119911 119879)
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
6 Journal of Nanoscience
1198811(119909 119910 119911 119905)
=120597
120597119909int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119911int
119905
0
119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591
minus 1205722
1119881int
119905
0
119896119881119881
(119909 119910 119911 119879) 119889120591
minus 12057211198681205721119881int
119905
0
119896119868119881
(119909 119910 119911 119879) 119889120591
(1c)
Φ1119868(119909 119910 119911 119905)
= 119891Φ119868
(119909 119910 119911) + int
119905
0
119896119868(119909 119910 119911 119879) 119868 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
Φ1119881(119909 119910 119911 119905)
= 119891Φ119881
(119909 119910 119911) + int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591
+ int
119905
0
119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 120591) 119889120591
+120597
120597119909int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591
(3c)
Average values of the first-order approximations of therequired approximations could be calculated by the followingrelation [14ndash16]
1205721120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
1205881(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
(6)
Substitution of relations (1c) and (3c) into relation (6)gives a possibility to calculate the required average values inthe following form
1205721119868= radic
(1198863+ 119860)2
411988624
minus 4(119861 +Θ1198863119861 + Θ
21198711199091198711199101198711199111198861
1198864
)
minus1198863+ 119860
41198864
1205721119881
=1
11987811986811988100
[Θ
1205721119868
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 120572111986811987811986811986800
minus Θ119871119909119871119910119871119911]
1205721Φ119868
=1198771198681
Θ119871119909119871119910119871119911
+11987811986811986820
Θ119871119909119871119910119871119911
+1
119871119909119871119910119871119911
sdot int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119868
(119909 119910 119911) 119889119911 119889119910 119889119909
1205721Φ119881
=1
119871119909119871119910119871119911
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891Φ119881
(119909 119910 119911) 119889119911 119889119910 119889119909
+(1198771198811+ 11987811988111988120
)
Θ119871119909119871119910119871119911
(7)
Here
119878120588120588119894119895
= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896120588120588(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119881
119895
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
1198864= 11987811986811986800
(1198782
11986811988100minus 11987811986811986800
11987811988111988100
)
1198863= 11987811986811988100
11987811986811986800
+ 1198782
11986811988100minus 11987811986811986800
11987811988111988100
1198862= int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119881(119909 119910 119911) 119889119911 119889119910 119889119909119878
119868119881001198782
11986811988100+ 211987811988111988100
11987811986811986800
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
minus 1198782
11986811988100int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909 + 119878
11986811988100Θ1198712
1199091198712
1199101198712
119911minus Θ1198712
1199091198712
1199101198712
11991111987811988111988100
Journal of Nanoscience 7
1198861= 11987811986811988100
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
119860 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
1198860= 11987811988111988100
[int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909]
2
119861 =Θ1198862
61198864
+3
radicradic1199022 + 1199013 minus 119902 minus3
radicradic1199022 + 1199013 + 119902
119902 =Θ31198862
2411988624
(41198860minus Θ119871119909119871119910119871119911
11988611198863
1198864
) minus Θ2 1198860
811988624
(4Θ1198862minus Θ21198862
3
1198864
) minusΘ31198863
2
5411988634
minus 1198712
1199091198712
1199101198712
119911
Θ41198862
1
811988624
119877120588119894= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896119868(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
119901 = Θ2 411988601198864
1211988624
minusΘ1198862
181198864
minusΘ11988611198863
1211988624
119871119909119871119910119871119911
(8)
We determine approximations with the second andhigher orders of concentrations of radiations defects frame-work standard iterative procedure of method of averagingof function corrections [14ndash16] For the framework of theprocedure we determine the approximation of 119899th orderby replacement of the concentrations of radiation defects119868(119909 119910 119911 119905) 119881(119909 119910 119911 119905) Φ
119868(119909 119910 119911 119905) and Φ
119881(119909 119910 119911 119905) in
right sides of (1b) and (3b) on the following sums 120572119899120588
+
120588119899minus1
(119909 119910 119911 119905) The replacement gives a possibility to obtainthe second-order approximations of concentrations of radia-tion defects
1205971198682(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119868119868(119909 119910 119911 119879)
sdot [1205721119868+ 1198681(119909 119910 119911 119905)]
2
1205971198812(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119881119881(119909 119910 119911 119879)
sdot [1205721119868+ 1198811(119909 119910 119911 119905)]
2
(1d)
120597Φ2119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119909]
8 Journal of Nanoscience
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905) + 119896
119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 119905) +120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ2119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905) + 119896
119881(119909 119910 119911 119879)
sdot 119881 (119909 119910 119911 119905) +120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119881
(119909 119910 119911) 120575 (119905)
(3d)
Integration of left and right sides of (1d) and (3d) gives apossibility to obtain relations for the second-order approxi-mations of the required concentrations of radiation defectsin the following form
1198682(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868(119909 119910 119911
119879)1205971198681(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119868119868(119909 119910 119911 119879) [120572
2119868+ 1198681(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [1205722119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119868119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
(1e)
1198812(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119881119881
(119909 119910 119911 119879) [1205722119881+ 1198811(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [120572119881119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119881119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
Φ2119868(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119868
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 120591)
120597119911119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 + int
119905
0
119896119868119868(119909 119910 119911 119879)
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 7
1198861= 11987811986811988100
int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909
119860 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
1198860= 11987811988111988100
[int
119871119909
0
int
119871119910
0
int
119871119911
0
119891119868(119909 119910 119911) 119889119911 119889119910 119889119909]
2
119861 =Θ1198862
61198864
+3
radicradic1199022 + 1199013 minus 119902 minus3
radicradic1199022 + 1199013 + 119902
119902 =Θ31198862
2411988624
(41198860minus Θ119871119909119871119910119871119911
11988611198863
1198864
) minus Θ2 1198860
811988624
(4Θ1198862minus Θ21198862
3
1198864
) minusΘ31198863
2
5411988634
minus 1198712
1199091198712
1199101198712
119911
Θ41198862
1
811988624
119877120588119894= int
Θ
0
(Θ minus 119905) int
119871119909
0
int
119871119910
0
int
119871119911
0
119896119868(119909 119910 119911 119879) 119868
119894
1(119909 119910 119911 119905) 119889119911 119889119910 119889119909 119889119905
119901 = Θ2 411988601198864
1211988624
minusΘ1198862
181198864
minusΘ11988611198863
1211988624
119871119909119871119910119871119911
(8)
We determine approximations with the second andhigher orders of concentrations of radiations defects frame-work standard iterative procedure of method of averagingof function corrections [14ndash16] For the framework of theprocedure we determine the approximation of 119899th orderby replacement of the concentrations of radiation defects119868(119909 119910 119911 119905) 119881(119909 119910 119911 119905) Φ
119868(119909 119910 119911 119905) and Φ
119881(119909 119910 119911 119905) in
right sides of (1b) and (3b) on the following sums 120572119899120588
+
120588119899minus1
(119909 119910 119911 119905) The replacement gives a possibility to obtainthe second-order approximations of concentrations of radia-tion defects
1205971198682(119909 119910 119911 119905)
120597119905=
120597
120597119909[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119868119868(119909 119910 119911 119879)
sdot [1205721119868+ 1198681(119909 119910 119911 119905)]
2
1205971198812(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 119905)
120597119911]
minus 119896119868119881
(119909 119910 119911 119879) [1205721119868+ 1198681(119909 119910 119911 119905)]
sdot [1205721119881+ 1198811(119909 119910 119911 119905)] +
120597
120597119909
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911119889120591 minus 119896
119881119881(119909 119910 119911 119879)
sdot [1205721119868+ 1198811(119909 119910 119911 119905)]
2
(1d)
120597Φ2119868(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119909]
8 Journal of Nanoscience
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905) + 119896
119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 119905) +120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ2119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905) + 119896
119881(119909 119910 119911 119879)
sdot 119881 (119909 119910 119911 119905) +120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119881
(119909 119910 119911) 120575 (119905)
(3d)
Integration of left and right sides of (1d) and (3d) gives apossibility to obtain relations for the second-order approxi-mations of the required concentrations of radiation defectsin the following form
1198682(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868(119909 119910 119911
119879)1205971198681(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119868119868(119909 119910 119911 119879) [120572
2119868+ 1198681(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [1205722119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119868119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
(1e)
1198812(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119881119881
(119909 119910 119911 119879) [1205722119881+ 1198811(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [120572119881119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119881119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
Φ2119868(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119868
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 120591)
120597119911119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 + int
119905
0
119896119868119868(119909 119910 119911 119879)
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
8 Journal of Nanoscience
+120597
120597119910[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119868119868(119909 119910 119911 119879) 119868
2(119909 119910 119911 119905) + 119896
119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 119905) +120597
120597119909[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119868
(119909 119910 119911) 120575 (119905)
120597Φ2119881(119909 119910 119911 119905)
120597119905
=120597
120597119909[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881
(119909 119910 119911 119879)120597Φ1119881(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 119905)
120597119911]
+ 119896119881119881
(119909 119910 119911 119879)1198812(119909 119910 119911 119905) + 119896
119881(119909 119910 119911 119879)
sdot 119881 (119909 119910 119911 119905) +120597
120597119909[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909]
+120597
120597119910[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119910]
+120597
120597119911[119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911] + 119891Φ119881
(119909 119910 119911) 120575 (119905)
(3d)
Integration of left and right sides of (1d) and (3d) gives apossibility to obtain relations for the second-order approxi-mations of the required concentrations of radiation defectsin the following form
1198682(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119868(119909 119910 119911 119879)
1205971198681(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119868(119909 119910 119911
119879)1205971198681(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119868119868(119909 119910 119911 119879) [120572
2119868+ 1198681(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [1205722119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119868(119909 119910 119911)
+120597
120597119909[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119868119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119868119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
(1e)
1198812(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119909119889120591
+120597
120597119910int
119905
0
119863119881(119909 119910 119911 119879)
1205971198811(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863119881(119909 119910 119911
119879)1205971198811(119909 119910 119911 120591)
120597119911119889120591
minus int
119905
0
119896119881119881
(119909 119910 119911 119879) [1205722119881+ 1198811(119909 119910 119911 120591)]
2119889120591
minus int
119905
0
119896119868119881
(119909 119910 119911 119879) [120572119881119868+ 1198681(119909 119910 119911 120591)] [120572
2119881
+ 1198811(119909 119910 119911 120591)] 119889120591 + 119891
119881(119909 119910 119911)
+120597
120597119909[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119909] +
120597
120597119910[119863119881119878
119881119896119879
sdot1205971205832(119909 119910 119911 119905)
120597119910] +
120597
120597119911[119863119881119878
119881119896119879
1205971205832(119909 119910 119911 119905)
120597119911]
Φ2119868(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119868
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119868
(119909 119910 119911
119879)120597Φ1119868(119909 119910 119911 120591)
120597119910119889120591
+120597
120597119911int
119905
0
119863Φ119868
(119909 119910 119911 119879)120597Φ1119868(119909 119910 119911 120591)
120597119911119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 + int
119905
0
119896119868119868(119909 119910 119911 119879)
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 9
times 1198682(119909 119910 119911 120591) 119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119868119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119868(119909 119910 119911 119879)
sdot 119868 (119909 119910 119911 120591) 119889120591
Φ2119881(119909 119910 119911 119905) =
120597
120597119909int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119909119889120591
+ 119891Φ119881
(119909 119910 119911) +120597
120597119910int
119905
0
119863Φ119881
(119909 119910 119911 119879)
times120597Φ1119881(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911int
119905
0
119863Φ119881
(119909 119910 119911
119879)120597Φ1119881(119909 119910 119911 120591)
120597119911119889120591
+ int
119905
0
119896119881(119909 119910 119911 119879)119881 (119909 119910 119911 120591) 119889120591 +
120597
120597119909
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119909119889120591 +
120597
120597119910
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119910119889120591 +
120597
120597119911
sdot int
119905
0
119863Φ119881119878
119881119896119879
1205971205832(119909 119910 119911 120591)
120597119911119889120591 + int
119905
0
119896119881119881
(119909 119910 119911 119879)
sdot 1198812(119909 119910 119911 120591) 119889120591
(3e)
Wedetermine average values of the second-order approx-imations by the standard relation [14ndash16]
1205722120588=
1
Θ119871119909119871119910119871119911
sdot int
Θ
0
int
119871119909
0
int
119871119910
0
int
119871119911
0
[1205882(119909 119910 119911 119905) minus 120588
1(119909 119910 119911 119905)] 119889119911 119889119910 119889119909 119889119905
(9)
Substitution of relations (1e) and (3e) in relation (9) givesa possibility to obtain relations for the required values 120572
2120588
1205722119862= 0
1205722Φ119868
= 0
1205722Φ119881
= 0
1205722119881
= radic(1198873+ 119864)2
411988724
minus 4(119865 +Θ1198863119865 + Θ
21198711199091198711199101198711199111198871
1198874
) minus1198873+ 119864
41198874
1205722119868=119862119881minus 1205722
211988111987811988111988100
minus 1205722119881(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) minus 11987811988111988102
minus 11987811986811988111
11987811986811988101
+ 120572211988111987811986811988100
(10)
where
1198874=
1
Θ119871119909119871119910119871119911
1198782
1198681198810011987811988111988100
minus1
Θ119871119909119871119910119871119911
1198782
1198811198810011987811986811986800
1198873= minus (2119878
11988111988101+ 11987811986811988110
+ Θ119871119909119871119910119871119911)11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
+11987811986811988100
11987811988111988100
Θ119871119909119871119910119871119911
(11987811986811988101
+ 211987811986811986810
+ 11987811986811988101
+ Θ119871119909119871119910119871119911)
+ (211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911)1198782
11986811988100
Θ119871119909
1
119871119910119871119911
minus1198782
1198681198810011987811986811988110
Θ3119871311990911987131199101198713119911
1198872=11987811986811986800
11987811988111988100
Θ119871119909119871119910119871119911
(11987811988111988102
+ 11987811986811988111
+ 119862119881) minus (Θ119871
119909119871119910119871119911
minus 211987811988111988101
+ 11987811986811988110
)2
+11987811986811988101
11987811988111988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911
+ 211987811986811986810
+ 11987811986811988101
) +11987811986811988100
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 11987811986811988101
+ 211987811986811986810
+ 211987811986811988101
) (Θ119871119909119871119910119871119911+ 211987811988111988101
+ 11987811986811988110
)
minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
1198782
11986811988100+
1198621198681198782
11986811988100
Θ2119871211990911987121199101198712119911
minus 211987811986811988110
sdot11987811986811988100
11987811986811988101
Θ119871119909119871119910119871119911
1198871= 11987811986811986800
11987811986811988111
+ 11987811988111988102
+ 119862119881
Θ119871119909119871119910119871119911
(211987811988111988101
+ 11987811986811988110
+ Θ119871119909119871119910119871119911) +
11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
10 Journal of Nanoscience
+ 11987811986811988101
) (211987811988111988101
+11987811986811988110
+ Θ119871119909119871119910119871119911)
minus11987811986811988100
Θ119871119909119871119910119871119911
(311987811986811988101
+ 211987811986811986810
+ Θ119871119909119871119910119871119911) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 211986211986811987811986811988100
11987811986811988101
minus11987811986811988110
1198782
11986811988101
Θ119871119909119871119910119871119911
1198870=
11987811986811986800
Θ119871119909119871119910119871119911
(11987811986811988100
+ 11987811988111988102
)2
minus11987811986811988101
Θ119871119909119871119910119871119911
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
) (119862119881
minus 11987811988111988102
minus 11987811986811988111
) + 21198621198681198782
11986811988101minus119862119881minus 11987811988111988102
minus 11987811986811988111
Θ119871119909119871119910119871119911
sdot 11987811986811988101
(Θ119871119909119871119910119871119911+ 211987811986811986810
+ 11987811986811988101
)
119862119868=
12057211198681205721119881
Θ119871119909119871119910119871119911
11987811986811988100
+1205722
111986811987811986811986800
Θ119871119909119871119910119871119911
minus11987811986811986820
Θ119871119909119871119910119871119911
minus11987811986811988111
Θ119871119909119871119910119871119911
119862119881= 1205721119868120572111988111987811986811988100
+ 1205722
111988111987811988111988100
minus 11987811988111988102
minus 11987811986811988111
119864 = radic8119910 + Θ21198862
3
11988624
minus 4Θ1198862
1198864
119865 =Θ1198862
61198864
+3
radicradic1199032 + 1199043 minus 119903 minus3
radicradic1199032 + 1199043 + 119903
119903 =Θ31198872
2411988724
(41198870minus Θ119871119909119871119910119871119911
11988711198873
1198874
) minus 1198870
Θ2
811988724
(4Θ1198872
minus Θ21198872
3
1198874
) minusΘ31198873
2
5411988734
minus 1198712
1199091198712
1199101198712
119911
Θ41198872
1
811988724
119904 = minusΘ1198872
181198874
+Θ2(411988701198874minus Θ11987111990911987111991011987111991111988711198873)
1211988724
(11)
For the framework of this paper the required spatiotem-poral distributions of concentrations of radiations defectshave been determined by using the second-order approxi-mations by using method of averaging of function correc-tions The approximations are usually good enough to makequalitative analysis and to obtain some quantitative resultsAll obtained results have been checked by comparison withresults of numerical simulations The results of numericalsimulations have been obtained by solving (1) and (4) by usingstandard explicit difference scheme
3 Discussion
In this section we analyzed distributions of concentrationsof point defects and their simplest complexes In the pre-vious section we analytically take into account porosity of
000
025
050
075
100
Distribution of point defectsin nonporous material
0
Distribution of pointdefects in porous material
C(xΘ
)
Lx4 Lx2 3Lx4 Lx
Figure 2 Distributions of concentrations of point radiation defectsfor fixed value of annealing time Curve 1 corresponds to implan-tation of ions of dopant through nonporous epitaxial layer Curve2 corresponds to implantation of ions of dopant through porousepitaxial layer Solid lines are the analytical results Dashed lines arethe numerical results
0000
0025
0050
0075
0100Distribution of complexes of point
defects in nonporous material
Distribution of complexes of point defects in porous material
C(xΘ
)
0 Lx4 Lx2 3Lx4 Lx
Figure 3 Distributions of concentrations of simplest complexes ofpoint radiation defects for fixed value of annealing time Curve 1corresponds to implantation of ions of dopant through nonporousepitaxial layer Curve 2 corresponds to implantation of ions ofdopant through porous epitaxial layer Solid lines are the analyticalresults Dashed lines are the numerical results
materials in comparison with cited similar works In thissituation we obtain decreasing quantity of radiation defects(one can find decreasing both types of accounted defectspoint defects and their simplest complexes) in comparisonwith nonporous materials Probably this effect could beobtained due to draining of these defects to pores Typicaldistributions of concentrations of point radiation defectsand their simplest complexes are presented in Figures 2and 3 respectively In this situation using overlayer overdevice area gives a possibility to increase radiation resistanceof the devices during radiation processing Using porousoverlayer gives a possibility to obtain larger increase ofradiation resistance Figures 2 and 3 also show that quantity
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 11
of point defects is larger than quantity of simplest complexesof point defects This effect could be found because onlypart of point defects could generate their complexes Itshould be also noted that we have also analytically takeninto account larger quantity of effects which could befound during relaxation of distributions of concentrations ofradiation defects diffusion of radiation defects (on accountof porosity stimulated diffusion) recombination of defectsand generation and decay of complexes of defects Analysisof nonlinearity of the relaxation shows that the nonlinearitygives a possibility to decrease quantity of radiation defectsaftermultistage radiation processing in comparisonwith one-stage radiation processing Distributions of concentrationsof radiation defects during comparison of multistage andone-stage radiation processing will be qualitatively similar todistributions in Figures 2 and 3
4 Conclusion
In the present paper we analyzed redistributions of radiationsdefects in material with porous and nonporous overlayerafter radiation processing It has been shown that presenceof porous overlayer gives a possibility to decrease quantity ofradiation defects
Competing Interests
The authors declare that the grant scholarship andorfunding mentioned in Acknowledgments section do notlead to any competing interests Additionally the authorsdeclare that there are no competing interests regarding thepublication of this paper
Acknowledgments
This work is supported by the Agreement of August 272013 no 02B49210003 betweenTheMinistry of Educationand Science of the Russian Federation and LobachevskyState University of Nizhni Novgorod educational fellowshipfor scientific research of Government of Russia educationalfellowship for scientific research of Government of NizhnyNovgorod region of Russia and educational fellowship forscientific research of Nizhny Novgorod State University ofArchitecture and Civil Engineering
References
[1] A A Lebedev A M Ivanov and N B Strokan ldquoRadiationresistance of SiC and nuclear-radiation detectors based on SiCfilmsrdquo Semiconductors vol 38 no 2 pp 129ndash150 2004
[2] E V Kalinina V G Kossov R R Yafaev A M Strelrsquochukand G N Violina ldquoA high-temperature radiation-resistantrectifier based on p+-n junctions in 4H-SiC ion-implanted withaluminumrdquo Semiconductors vol 44 no 6 pp 778ndash788 2010
[3] A E Belyaev N S Boltovets A V Bobylrsquo et al ldquoRadiationeffects and interphase interactions in ohmic and barrier con-tacts to indium phosphide as induced by rapid thermal anneal-ing and irradiation with 120574-ray 60Co photonsrdquo Semiconductorsvol 44 no 12 pp 1559ndash1566 2010
[4] G P Gaydar ldquoOn the kinetics of electron processes in 60Co 120574-irradiated n-Ge single crystalsrdquo Semiconductors vol 48 no 9pp 1171ndash1175 2014
[5] P A Aleksandrov N E Belova K D Demakov and S GShemardov ldquoOn the generation of charge-carrier recombina-tion centers in the sapphire substrates of silicon-on-sapphirestructuresrdquo Semiconductors vol 49 no 8 pp 1099ndash1103 2015
[6] Y W Zhang and A F Bower ldquoNumerical simulations of islandformation in a coherent strained epitaxial thin film systemrdquoJournal of the Mechanics and Physics of Solids vol 47 no 11 pp2273ndash2297 1999
[7] M Kitayama T Narushima W C Carter R M Cannon andAM Glaeser ldquoWulff shape of alumina I modeling the kineticsof morphological evolutionrdquo Journal of the American CeramicSociety vol 83 no 10 pp 2561ndash2571 2000
[8] M Kitayama T Narushima and A M Glaeser ldquoWulff shapeof alumina II experimental measurements of pore shapeevolution ratesrdquo Journal of the American Ceramic Society vol83 no 10 pp 2572ndash2583 2000
[9] E L Pankratov ldquoApplication of porous layers and optimizationof annealing of dopant and radiation defects to increase sharp-ness of p-n-junctions in a bipolar heterotransistorsrdquo Journal ofNanoelectronics and Optoelectronics vol 6 no 2 pp 188ndash2062011
[10] P G Cheremskoy V V Slesov and V I Betekhtin Pore in SolidBodies Energoatomizdat Moscow Russia 1990 (Russian)
[11] V L Vinetskiy and G A Kholodarrsquo Radiative Physics of Semi-conductors Naukova Dumka Kiev Ukraine 1979 (Russian)
[12] P M Fahey P B Griffin and J D Plummer ldquoPoint defects anddopant diffusion in siliconrdquo Reviews of Modern Physics vol 61no 2 pp 289ndash384 1989
[13] M G Mynbaeva E N Mokhov A A Lavrentrsquoev and KD Mynbaev ldquoHigh-temperature diffusion doping of poroussilicon carbiderdquo Technical Physics Letters vol 34 no 9 pp 731ndash733 2008
[14] E L Pankratov and E A Bulaeva ldquoAbout influence of bufferporous layers between epitaxial layers of heterostructure ondistributions of concentrations of dopants in heterobipolartransistorsrdquo International Journal of Multiphysics vol 9 no 2pp 109ndash136 2015
[15] Y D Sokolov ldquoAbout determination of dynamical forces inminersquos hoisting ropesrdquoAppliedMechanics vol 1 no 1 pp 23ndash351955
[16] E L Pankratov and E A Bulaeva ldquoOptimization of manu-facturing of emitter-coupled logic to decrease surface of chiprdquoInternational Journal of Modern Physics B vol 29 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials