on modification of properties of p-n-junctions during overgrowth

7/26/2019 ON MODIFICATION OF PROPERTIES OF P-N-JUNCTIONS DURING OVERGROWTH

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International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

DOI: 10.5121/ijitmc.2016.4203 35

ON MODIFICATION OF PROPERTIES OFP-N-JUNCTIONS DURING OVERGROWTH

E.L. Pankratov1, E.A. Bulaeva

1,2

1Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,

Russia2Nizhny Novgorod State University of Architecture and Civil Engineering,65 Il'insky

street, Nizhny Novgorod,603950, Russia

ABSTRACT

In this paper we consider influence of overgrowth of doped by diffusion and ion implantation areas of hete-

rostructures on distributions of concentrations of dopants. Several conditions to increase sharpness of p-n-

junctions (single and framework bipolar transistors), which were manufactured during considered technol-ogical process, have been determined. At the same time we analyzed influence of speed of overgrowth of

doped areas and mechanical stress in the considered heterostructure on distribution of concentrations of

dopants in the structure.

KEYWORDS

Diffusion-junction heterorectifier; implanted-junction heterorectifier; overgrowth of doped area; analytical

approach for modeling

1.INTRODUCTION

In the present time they are several approaches could be used to manufacturep-n-junctions diffu-sion of dopants in a homogenous sample or an epitaxial layer of heterostructure, implantation of

ions of dopants in the same situations or doping during epitaxial growth [1-7]. The same ap-

proaches could be used to manufacture systems of p-n-junctions: bipolar transistors and thyris-tors. The first and the second ways of doping are preferable in comparison with the third one be-

cause the approaches give us possibility to dope locally materials during manufacture integrated

circuits easily in comparison with epitaxial growth. Using diffusion and ion implantation in ho-mogenous sample to manufacturing p-n-junctions leads to production fluently varying and wide

distributions of dopants. One of actual problems is increasing sharpness of p-n- junctions [5,7].The increasing of sharpness gives us possibility to decrease switching time of p-n-junctions. In-

creasing of homogeneity of dopant distribution in enriched by the dopant area is also attracted an

interest [5]. The increasing of homogeneity gives us possibility to decrease local overheat of thedoped materials due to streaming of electrical current during operating of p-n-junction or to de-

crease depth ofp-n-junction for fixed value of local overheats. One way to increase sharpness ofp-n-junction based on using near-surficial (laser or microwave) types of annealing [8-15].

Framework the types of annealing one can obtain near-surficial heating of the doped materials. In

this situation due to the Arrhenius low one can obtain increasing of dopant diffusion coefficient ofnear-surficial area in comparison with volumetric dopant diffusion coefficient. The increasing ofdopant diffusion coefficient of near-surficial area leads to increasing of sharpness ofp-n-junction.

The second way to increase sharpness ofp-n-junction based on using high doping of materials. In

this case contribution of nonlinearity of diffusion process increases [4]. The third way to increase


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36

sharpness of p-n-junction is using of inhomogeneity of heterostructure [16,17]. Framework the

approach we consider simplest heterostructure, which consist of substrate and epitaxial layer. Onecan find increasing of sharpness of p-n-junction after annealing with appropriate annealing time

in the case, when dopant diffusion coefficient in the substrate is smaller, than in the epitaxial

layer. The fourth way to increase sharpness of p-n-junction is radiation processing of materials.The radiation processing leads to radiation-enhanced diffusion [18]. However using radiation

processing of materials leads to necessity of annealing of radiation defects. Density of elements ofintegrated circuits could be increases by using mismatch-induced stress [19]. However one could

obtain increased unsoundness of doped material (for example, to generation of dislocation of dis-

agreement) by using the approach [7].

Theconsideredapproaches gives a possibility to increase sharpness of p-n-junction with increas-ing of homogeneity of dopant distribution in enriched by the dopant area. Using combination of

the above approaches gives us possibility to increase sharpness of p-n-junction and increasing ofhomogeneity of dopant distribution in enriched area at one time.

z

D(z

),

P(z

)

D2

D3

0 Lza

fC impl(z)

fC diff(z)

P2

P3

Substrate

Epitaxial layer

Overlayer

D1

P1

Fig. 1. Substrate, epitaxial and overlayers framework heterostructure

Framework this paper we consider a substrate with known type of conductivity ( nor p) and anepitaxial layer, included into a heterostructure. The epitaxial layer have been doped by diffusion

or by ion implantation to manufacture another type of conductivity (por n). Farther we considerovergrowth of the epitaxial layer by an overlayer (see Fig. 1). The overlayer has type of conduc-

tivity, which coincide with type of conductivity of the substrate. In this paper we analyzed influ-ence of overgrowth of the doped epitaxial layer on distribution of dopants in the considered hete-

rostructure.

2.METHOD OF SOLUTION

In this section we calculate spatio-temporal distribution of concentration of dopant in the consi-dered heterostructure to solve our aim. To calculate the distribution we solved the following

boundary problem [1,20-22]

( ) ( ) ( ) ( )+

+

+

=

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxC ,,,,,,,,,,,, (1)


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37

( ) ( ) ( ) ( )

+

+

zz L

tvS

SL

tvS

S WdtWyxCtzyxTk

D

yWdtWyxCtzyx

Tk

D

x,,,,,,,,,,,, .

Boundary and initial conditions for our case could be written as

( )0

,,,

0

=

=xx

tzyxC,

( )0

,,,=

= xLxx

tzyxC,

( )0

,,,

0

=

=yy

tzyxC,

( )0

,,,=

= yLxy

tzyxC,

( )0

,,,=

= tvzz

tzyxC,

( )0

,,,=

= zLxz

tzyxC, C(x,y,z,0)=fC(x,y,z).

In the above relations we used the function C(x,y,z,t) as the spatio-temporal distribution of con-

centration of dopant; is the atomic volume; surface concentration of dopant on interface be-

tween layers of heterostructure could be determined as the following integral ( )

zL

tv

zdtzyxC ,,,

(we assume, that the interface between layers of heterostructure is perpendicular to the direction

Oz); surface gradient we denote as symbol S; (x,y,z,t) is the chemical potential (reason of ac-counting of the chemical potential is mismatch-induced stress); the parameters DandDSare the

coefficients of volumetric and surface diffusions. One can find the surface diffusions due to mis-match-induced stress. Diffusion coefficients depends on temperature and speed of heating andcooling of heterostructure, properties of materials of heterostructure, spatio-temporal distributions

of concentrations of dopant and radiation defects after ion implantation. Approximations of thesedependences could be approximated by the following functions [22,23]

( ) ( )

( )( ) ( )

( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD

SLSS

. (2)

FunctionsDL(x,y,z,T) andDLS(x,y,z,T) described dependences of diffusion coefficients on coordi-

nate (due to presents several layers in heterostructure, manufactured by using different materials)and temperature of annealing T(due to Arrhenius law); function P(x,y,z,T) describes dependence

of limit of solubility of dopant on coordinate and temperature; parameter could be integer anddepends on properties of materials of heterostructure [23]; V (x,y,z,t) is the spatio-temporal con-centration of radiation vacancies; V

*is the equilibrium concentration of vacancies. Concentration-

al dependence of dopant diffusion coefficients has been described in details in [23].

We determine distributions of concentrations of point defects in space and time as solutions of thefollowing system of equations [1,20-22]

( )( )

( )( )

( )( )

+

= Tzyxk

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIIIII

,,,,,,

,,,,,,

,,,,,,

,

( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz

tzyxITzyxD

ztzyxI

VII,,,,,,,,,

,,,,,,,,,

,

2

+

(3)


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38

( ) ( ) ( ) ( )

+

+

zz L

tvS

SIL

tvS

SIWdtWyxItzyx

Tk

D

yWdtWyxItzyx

Tk

D

x,,,,,,,,,,,,

( )( )

( )( )

( )( )

+

= Tzyxk

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVVVV ,,,

,,,,,,

,,,,,,

,,,,

( ) ( ) ( ) ( ) ( ) ( )+

+ tzyxVtzyxITzyxkz

tzyxVTzyxD

ztzyxV

VIV,,,,,,,,,

,,,,,,,,,

,

2

( ) ( ) ( ) ( )

+

+

zz L

tvS

VSL

tvS

VS WdtWyxVtzyxTk

D

yWdtWyxVtzyx

Tk

D

x,,,,,,,,,,,, .

Boundary and initial conditions for these equations could be written as

( )0

,,,

0

=

=xx

tzyxI

,

( )0

,,,=

= xLxx

tzyxI

,

( )0

,,,

0

=

=yy

tzyxI

,

( )0

,,,=

= yLyy

tzyxI

,

( )

0

,,,=

= tvzz

tzyxI

,

( )

0

,,,=

= zLzz

tzyxI

,

( )

0

,,,

0

=

=xx

tzyxV

,

( )

0

,,,=

= xLxx

tzyxV

,

( )0

,,,

0

=

=yy

tzyxV

,

( )0

,,,=

= yLyy

tzyxV

,

( )0

,,,=

= tvzz

tzyxV

,

( )0

,,,=

= zLzz

tzyxV

,

I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV(x,y,z). (4)

HereI(x,y,z,t) is the distribution of concentration of radiation interstitials in space and time;I*is

the equilibrium concentration interstitials;DI(x,y,z,T),DV(x,y,z,T),DIS(x,y,z,T),DVS(x,y,z,T) are thecoefficients of volumetric and surface diffusion; terms V

2(x,y,z,t) and I

2(x,y,z,t) corresponds to

generation divacancies and analogous complexes of interstitials (see, for example, [22] and ap-propriate references in this work); the functions kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) describeddependences of parameters of recombination of point defects and generation their complexes on

coordinate and temperature; kis the Boltzmann constant.

We calculate spatio-temporal distributions of concentrations of divacancies V(x,y,z,t) and diin-terstitials I(x,y,z,t) by solution of the equations [20-22]

( )( )

( )( )

( )( ) +

+

=

Tzyxky

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

III

II,,,

,,,,,,

,,,,,,

,,,

( ) ( ) ( )

( ) ( ) +

+

+

zI

I

L

tvIS

SI WdtWyxtzyxTk

D

xz

tzyxTzyxD

ztzyxI ,,,,,,

,,,,,,,,,

( ) ( ) ( ) ( )tzyxITzyxkWdtWyxtzyxTk

D

y II

L

tvIS

ISz

,,,,,,,,,,,, 2,

+

+

(5)

( )( )

( )( )

( )( ) +

+

=

Tzyxky

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

VVV

VV,,,

,,,,,,

,,,,,,

,,,

( ) ( ) ( )

( ) ( ) +

+

+

zV

V

L

tvVS

SV WdtWyxtzyxTk

D

xz

tzyxTzyxD

ztzyxV ,,,,,,

,,,,,,,,,


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39

( ) ( ) ( ) ( )tzyxVTzyxkWdtWyxtzyxTk

D

y VV

L

tvVS

S z

V ,,,,,,,,,,,, 2,

+

+

.

Boundary and initial conditions could be written as

( )0

,,,

0

=

=x

I

x

tzyx

,

( )0

,,,=

= xLx

I

x

tzyx

,

( )0

,,,

0

=

=y

I

y

tzyx

,

( )0

,,,=

= yLy

I

y

tzyx

,

( )0

,,,=

= tvz

I

z

tzyx

,

( )0

,,,=

= zLzz

tzyxI

,

( )0

,,,

0

=

=x

V

x

tzyx

,

( )0

,,,=

= xLxx

tzyxV

,

( )0

,,,

0

=

=y

V

y

tzyx

,

( )0

,,,=

= yLy

V

y

tzyx

,

( )0

,,,=

= tvz

V

z

tzyx

,

( )0

,,,=

= zLz

V

z

tzyx

,

I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV(x,y,z). (6)

Here DI(x,y,z,T), DV(x,y,z,T), DIS(x,y,z,T) and DVS(x,y,z,T) are the coefficients of volumetric

and surface diffusion; the functions kI(x,y,z,T) and kV(x,y,z,T) described dependences of parame-ters of decay of complexes of point defects on coordinate and temperature. One can determine

chemical potentialin the Eq.(1) by the following relation [20]

=E(z)ij[uij(x,y,z,t)+uji(x,y,z,t)]/2. (7)

HereEis the tension (Young) modulus;

+

=

i

j

j

i

ijx

u

x

uu

2

1is the deformation tensor; ijis the

stress tensor; ui, ujare the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement ten-

sor ( )tzyxu ,,,r

;xi,xjare the coordinates x,y,z. The relation (3) could be transformed to the fol-

lowing form

( ) ( ) ( ) ( ) ( ) ( )

+

+

+

=

ij

i

j

j

i

i

j

j

i

x

tzyxu

x

tzyxu

x

tzyxu

x

tzyxuzEtzyx 0

,,,,,,

2

1,,,,,,

2,,,

( )

( )

( )( ) ( ) ( )[ ]

+ ij

k

kij TtzyxTzzKx

tzyxu

z

z

00

,,,3,,,

21,

where is the Poisson coefficient; the parameter 0=(as-aEL)/aELdescribes the displacement pa-rameter with lattice distances of the substrate and the epitaxial layer as, aEL; K is the modulus of

uniform compression; the parameterbdescribes the thermal expansion; we assume, that the equi-librium temperature Tr coincides with the room temperature. Components of the displacement

vector could be described by solving the following system of equations [24]

( ) ( ) ( ) ( ) ( )

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz xz

xyxxx

+

+

=

,,,,,,,,,,,,2

2

( ) ( ) ( ) ( ) ( )

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz

yzyyyxy

+

+

=

,,,,,,,,,,,,2

2

( ) ( ) ( ) ( ) ( )

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz zz

zyzxz

+

+

=

,,,,,,,,,,,,2

2


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40

where( )

( )[ ]( ) ( ) ( )

( ) ( )

+

+

+=

k

k

ij

k

kij

i

j

j

i

ijx

tzyxuzK

x

tzyxu

x

tzyxu

x

tzyxu

z

zE ,,,,,,

3

,,,,,,

12

( ) ( ) ( )[ ]rTtzyxTzKz ,,, , (z) describes the density of materials of heterostructure. The ten-sor ijdescribes the Kronecker symbol. Accounting relation for ijin the previous system of equa-

tions last system of equation could be written as

( ) ( )

( ) ( )

( )[ ]( )

( ) ( )

( )[ ]

( )+

++

++=

yx

tzyxu

z

zEzK

x

tzyxu

z

zEzK

t

tzyxuz

yxx,,,

13

,,,

16

5,,,2

2

2

2

2

( )( )[ ]

( ) ( )( )

( )( )[ ]

( )

+++

+

++

zx

tzyxu

z

zEzK

z

tzyxu

y

tzyxu

z

zE zzy ,,,

13

,,,,,,

12

2

2

2

2

2

( ) ( ) ( )

( )[ ]

( ) ( )( ) ( )

( )+

+

+=

y

tzyxTzKz

yx

tzyxu

x

tzyxu

z

zE

t

tzyxuz x

yy ,,,,,,,,,

12

,,, 2

2

2

2

2

( )( )[ ]

( ) ( ) ( )( )[ ]

( ) ( )

+

++

+

+

+

+

2

2 ,,,

112

5,,,,,,

12 y

tzyxuzK

z

zE

y

tzyxu

z

tzyxu

z

zE

z

yzy

( ) ( )

( )[ ]

( )( )

( )

yx

tzyxuzK

zy

tzyxu

z

zEzK

yy

+

++

,,,,,,

16

22

(8)

( ) ( ) ( )

( )[ ]( ) ( ) ( ) ( )

+

+

+

+

+=

zy

tzyxu

zx

tzyxu

y

tzyxu

x

tzyxu

z

zE

t

tzyxuz

yxzzz,,,,,,,,,,,,

12

,,,22

2

2

2

2

2

2

( ) ( ) ( ) ( )

( ) ( ) ( )

+

+

+

+

z

tzyxTzzK

z

tzyxu

y

tzyxu

x

tzyxuzK

z

xyx ,,,,,,,,,,,,

( )( )

( ) ( ) ( ) ( )

+

+

z

tzyxu

y

tzyxu

x

tzyxu

z

tzyxu

z

zE

z

zyxz,,,,,,,,,,,,

616

1

.

Systems of conditions for these equations could be written as

( )0

,,,

0

=

=xx

tzyxur

;( )

0,,,

=

= xLxx

tzyxur

;( )

0,,,

0

=

=yy

tzyxur

;( )

0,,,

=

= yLyy

tzyxur

;

( )0

,,,=

= tvzz

tzyxur

;( )

0,,,

=

= zLzz

tzyxur

; ( )0

0,,, uzyxu rr

= ; ( )0

,,, uzyxu rr

= .

We calculate distribution of concentration of dopant in space in time by method of averaging of

function corrections [25-30]. To use the method we re-write Eqs. (1), (3) and (5) with accountappropriate initial distributions (see Appendix). In future we replace the required concentrations

in right sides of the obtained equations on their average values 1, which are not yet known. Theequations modified after the replacement and solution of these equations are presented in the Ap-

pendix.

We determined average values of the first-order approximations of the considered concentrations

by using the following standard relations [25-30]


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41

( )

=

0 0 011

,,,1 x y zL L L

tvzyx

tdxdydzdtzyxLLL

. (9)

Substitution of the solutions of the modified equations into the relation (9) gives a possibility to

obtain appropriate average values in the following form

( ) =

0 0 01

,,1 x y zL L L

tvC

zyx

C tdxdydzdzyxf

LLL ,

( )

4

3

4

1

2

3

2

4

2

3

14

44 a

Aa

a

aLLLBaB

a

Aa zyxI

+

++

+= ,

( )

=

zyxIII

L L L

tvI

IIV

V LLLStdxdydzdzyxf

S

x y z

0010 0 0100

1,,

1

,

( ) +

+

=

0 0 0

2011

,,1 x y z

II

L L L

tvzyxzyx

II

zyx

I tdxdydzdzyxfLLLLLL

S

LLL

R ,

( ) +

+

=

0 0 0

201

1,,

1 x y z

VV

L L L

tvzyxzyx

VV

zyx

V tdxdydzdzyxfLLLLLL

S

LLL

R .

Relations for calculations parameters Sij, ai,A,B, q,pare presented in the Appendix.

Weusedstandard iterative procedure of method of averaging of function corrections to calculatethe second- and higher-order approximations of the considered concentrations [25-30]. Frame-work the procedure to calculate approximations of the n-th order of the above concentrations we

replace the functions C(x,y,z,t),I(x,y,z,t), V(x,y,z,t), I(x,y,z,t) and V(x,y,z,t) in the Eqs. (1), (3),(5) with account initial distributions (see Eqs. (1a), (3a), (5a) in the Appendix) on the sums of thenot yet known average values of the considered approximations and approximations of the pre-

vious order, i.e. n+n-1(x,y,z,t). These obtained equations and their solutions, which calculatedare presented in the Appendix.

We calculate average values of the second-order approximations of required functions by using

the following standard relation [25-30]

( ) ( )[ ]

=

0 0 0 0122

,,,,,,1 x y zL L L

zyx

tdxdydzdtzyxtzyxLLL

. (10)

Substitution of the second-order approximations of concentrations of dopant into Eq.(10) leads to

the considered average values 2

2C=0, 2I =0, 2V =0,( )

4

3

4

1

2

3

2

4

2

3

24

44 b

Eb

b

bLLLFaF

b

Eb zyxV

+

++

+= ,

( )

00201

11021001200

2

22

2

IVVIV

VIVVzyxVIVVVVVVVI

SS

SSLLLSSSC

+

++

= .

Relations for parameters bi, C, F, r, sare presented in the Appendix.

Further we determine solutions of Eqs.(8). In this situation we determine approximation of dis-

placement vector. To determine the first-order approximations of the considered components

framework method of averaging of function corrections we replace the required values on their

not yet known average values 1i. The replacement leads to the following result


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42

( ) ( )

( ) ( ) ( )

x

tzyxTzzK

t

tzyxuz x

=

,,,,,,2

1

2

,

( ) ( )

( ) ( ) ( )

y

tzyxTzzK

t

tzyxuz

y

=

,,,,,,2

1

2

,

( ) ( ) ( ) ( ) ( )z

tzyxTzzKt

tzyxuz z

=

,,,,,,

2

1

2

.

Integration of the left and right sides of the previous relations on time tgives a possibility to ob-tain the required components to the following result

( ) ( ) ( )

( ) ( ) ( )

( )( )

( ) x

t

x uddzyxTxz

zzKddzyxT

xz

zzKtzyxu

00 00 0

1,,,,,,,,, +

=

,

( ) ( ) ( )

( ) ( ) ( )

( )( )

( ) y

t

y uddzyxTyz

zzKddzyxT

yz

zzKtzyxu

00 00 0

1,,,,,,,,, +

=

,

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( )

z

t

z uddzyxT

zz

zzKddzyxT

zz

zzKtzyxu

00 00 0

1,,,,,,,,, +

=

.

We calculate the second-order approximations of components of the displacement vector by stan-dard replacement of the required functions in the right sides of the Eqs.(8) on the standard sums

1i+ui(x,y,z,t) [19,26]. Equations for components of the displacement vector are presented in theAppendix. Solutions of these equations are also presented in the Appendix.

Frameworkthispaperallrequiredconcentrations(concentrationsofdopantandradiation defects)and components of displacement vector have been calculated as the appropriate second-order ap-

proximations by using the method of averaging of function corrections. The second-order approx-imation gives usually enough information on quantitative behavior of spatio- temporal distribu-

tions of concentrations of dopant and radiation defects and also several quantitative results. We

check all analytical results by using numerical approaches.

3.DISCUSSION

In this section we analyzed redistribution of dopant (for the ion doping of heterostructure) withaccount redistribution of radiation defects and their interaction with another defects. If growth

rate is small (v t


9/20


43

the overlayer increases (see Fig. 3). Qualitatively similar results could be obtained for diffusion

type of doping. Further we analyzed influence of mismatch-induced stress on distribution of con-centration

z

0.0

0.5

1.0

1.5

2.0

C(x,y,z,

)

0 Lz/4 Lz/2 3Lz/4 Lz

1

2

Fig. 2. Calculated spatial distributions of concentration of implanted dopant in homogenous sample (curve

1) and in heterostructure from Fig. 1 (curve 2) after annealing with the same continuance. Interfaces be-

tween layers of heterostructure are: a1=Lz/4 and a2=3Lz/4

z

0.00001

0.00010

0.00100

0.01000

0.10000

1.00000

C(x,y,z,

)

C(x,y,z,0)

Lz/40 Lz/2 3Lz/4 Lzx0

1

2

3

4

Fig. 3. Curves 1 and 2 are the calculated spatial distributions of concentration of implanted dopant in the

system of two layers: overlayer and epitaxial layer. Curves 3 and 4 are the calculated spatial distributions of

concentration of implanted dopant in the epitaxial layer only. Increasing of number of curves corresponds

to increasing of value of relationD1/D2. Coordinates of interfaces between layers of heterostructure are:

a1=Lz/4 (between overlayer and epitaxial layer) and a2=3Lz/4 (between epitaxial layer and substrate)


10/20


44

0.0

1.0

2.0

C(x,y,z

,)

0 Lz

12

3

Fig. 4. Spatial distributions of concentration of dopant in diffusion-junction rectifier after annealing with

equal continuance. Curve 1 corresponds to 00

0.0

1.0

2.0

C(x,y,z,

)

Lz-Lz 0

12

3

Fig. 5. Spatial distributions of concentration of dopant in implanted-junction rectifier after annealing with

equal continuance. Curve 1 corresponds to 00

z

0.0

0.2

0.4

0.6

0.8

1.0

Uz 1

2

0.0 a

Fig.6. Normalized dependences of component uzof displacement vector on coordinatezfor epitaxial layers

before radiation processing (curve 1) and after radiation processing (curve 2)


11/20


45

of dopant. We obtain during the analysis, thatp-n-junctions, manufactured near interface between

layers of heterostructures, have higher sharpness and higher homogeneity of concentration of do-pant in enriched area. Existing mismatch-induced stress leads to changing of distribution of con-

centration of dopant in directions, which are parallel to the considered interface. For example, for

00 one can obtain opposite effect (see Fig.5). It should be noted, that radiation processing of materials of heterostructure during ion doping

of materials gives a possibility to decrease mismatch-induced stress (see Fig. 6).

Further we analyzed influence of mismatch-induced stress on distribution of concentration of do-

pant. We obtain during the analysis, that p-n-junctions, manufactured near interface between lay-ers of heterostructures, have higher sharpness and higher homogeneity of concentration of dopantin enriched area. Existing mismatch-induced stress leads to changing of distribution of concentra-

tion of dopant in directions, which are parallel to the considered interface. For example, for 00 one canobtain opposite effect (see Fig. 5). It should be noted, that radiation processing of materials of

heterostructure during ion doping of materials gives a possibility to decrease mismatch-induced

stress (see Fig. 6). In this situation component of displacement vector perpendicular to interfacebetween materials of heterostructure became smaller after radiation processing in comparison

with analogous component of displacement vector in non processed heterostructure.

4.CONCLUSIONS

In this paper we analyzed influence of overgrowth of doped by diffusion or ion implantation areasof heterostructures on distributions of concentrations of dopants. We determine conditions to in-

crease sharpness if implanted-junction and diffusion- junction rectifiers (single rectifiers and rec-

tifiers framework bipolar transistors). At the same time we analyzed influence of overgrowth rateof doped areas and mismatch-induced stress in the considered heterostructure on distributions ofconcentrations of dopants.

ACKNOWLEDGEMENTS

This work is supported by the agreement of August 27, 2013 02..49.21.0003 between The

Ministry of education and science of the Russian Federation and Lobachevsky State University of

Nizhni Novgorod, educational fellowship for scientific research of Government of Russian, edu-cational fellowship for scientific research of Government of Nizhny Novgorod region of Russia

and educational fellowship for scientific research of Nizhny Novgorod State University of Archi-

tecture and Civil Engineering.

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[3] V.G. Gusev, Yu.M. Gusev. Electronics (Higher School, Moscow, 1991).

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[9] A.V. Dvurechensky, G.A. Kachurin, E.V. Nidaev, L.S. Smirnov. Impulse annealing of semiconductor

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APPENDIX

Eqs. (1), (3) and (5) with account initial distributions could be written as

( ) ( ) ( ) ( ) ( ) ( )++

+

+

=

tzyxf

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxCC ,,

,,,,,,,,,,,,

( ) ( ) ( ) ( )

+

+

zz L

tvS

SL

tvS

S WdtWyxCtzyxTk

D

yWdtWyxCtzyx

Tk

D

x,,,,,,,,,,,, (1a)


13/20


47

( )( )

( )( )

( )( ) ( )++

+

= tzyxf

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIIII

,,

,,,,,,

,,,,,,

,,,

( ) ( )

( ) ( ) ( ) ( ) ( )+

+ tzyxITzyxktzyxVtzyxITzyxk

z

tzyxITzyxD

z IIVII

,,,,,,,,,,,,,,,,,,

,,, 2,,

( ) ( ) ( ) ( )

+

+

zz L

tvS

ISL

tvS

IS WdtWyxItzyxTk

D

yWdtWyxItzyx

Tk

D

x,,,,,,,,,,,, (3a)

( )( )

( )( )

( )( ) ( )++

+

= tzyxf

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVVV

,,

,,,,,,

,,,,,,

,,,

( ) ( )

( ) ( ) ( ) ( ) ( )+

+ tzyxVTzyxktzyxVtzyxITzyxk

z

tzyxVTzyxD

z VVVIV

,,,,,,,,,,,,,,,,,,

,,, 2,,

( ) ( ) ( ) ( )

+

+

zz L

tvS

VSL

tvS

VS WdtWyxVtzyxTk

D

yWdtWyxVtzyx

Tk

D

x,,,,,,,,,,,,

( )( )

( )( )

( )( ) +

+

=

t

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxIII

II

,,,,,,

,,,,,,

,,,

( ) ( ) ( )

( ) ( ) +

+

+

zI

II

L

tvIS

SI WdtWyxtzyxTk

D

xz

tzyxTzyxD

zzyxf ,,,,,,

,,,,,,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkWdtWyxtzyxTk

D

y III

L

tvIS

S z

I ,,,,,,,,,,,,,,,,,, 2,

++

+

( )( )

( )( )

( )( )+

+

=

ty

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxVVV

VV

,,,,,,

,,,,,,

,,,(5a)

( ) ( ) ( )

( ) ( ) +

+

+

zV

VV

L

tvVS

SV WdtWyxtzyxTk

D

xz

tzyxTzyxD

zzyxf ,,,,,,

,,,,,,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkWdtWyxtzyxTk

D

y VVV

L

tvVS

S z

V ,,,,,,,,,,,,,,,,,, 2,

++

+

.

The first-order approximations of concentrations of dopant and radiation defects could by calcu-

lated by solution of the following equations

( )( ) ( ) +

+

=

tzyx

Tk

Dz

ytzyx

Tk

Dz

xt

tzyxCS

S

CS

S

C,,,,,,

,,,11

1

( ) ( )tzyxfC ,,+ (1b)

( ) ( ) ( ) ( ) ( )+

+

= tzyxftzyx

TkDz

ytzyx

TkD

xz

ttzyxI ISISISISI

,,,,,,,,,,, 111

( ) ( )TzyxkTzyxkVIVIIII

,,,,,,,11,

2

1 (3b)

( )( ) ( ) ( ) ( )+

+

= tzyxftzyx

Tk

Dz

ytzyx

Tk

D

xz

t

tzyxVVS

VS

VS

VS

V

,,,,,,,,

,,,11

1

( ) ( )TzyxkTzyxk VIVIVVV ,,,,,, ,11,2

1


14/20


48

( )( ) ( ) ( ) ( ) ( ) ( )+++=

tzyxftzyxITzyxktzyxITzyxkt

tzyxIIII

I

,,,,,,,,,,,,,,

,,, 2,

1

( ) ( )

+

+

tzyx

Tk

D

yztzyx

Tk

D

xz S

S

S

S I

I

I

I,,,,,, 11 (5b)

( )( ) ( ) ( ) ( ) ( ) ( )+++=

tzyxftzyxVTzyxktzyxVTzyxkt

tzyxVVVV

V

,,,,,,,,,,,,,,

,,, 2,

1

( ) ( )

+

+

tzyx

Tk

D

yztzyx

Tk

D

xz S

S

S

S V

V

V

V,,,,,, 11

The first-order approximations of the considered concentrations in the following form

( ) ( )( )

( ) ( )

( )

++

+

=

tCS

SC

V

zyxV

V

zyxV

TzyxPzyx

xtzyxC

02*

2

2*1

1

111

,,,,,,1

,,,1,,,,,,

( ) ( ) ( ) ( )( )

++

+

t

SCLS

VzyxV

VzyxVzyx

yd

TkzTzyxD

02*

2

2*111,,,,,,1,,,,,,

( )( )

( )zyxfdTzyxPTk

zTzyxD

C

CS

LS,,

,,,1,,, 1 +

+

(1c)

( ) ( ) ( ) ( )+

+

= zyxfdzyx

Tk

D

yzdzyx

Tk

D

xztzyxI

I

t

S

IS

I

t

S

IS

I,,,,,,,,,,,

01

011

( ) ( )t

VIVI

t

III dTzyxkdTzyxk0

,110

,

2

1,,,,,, (3c)

( ) ( ) ( ) ( )+

+

= zyxfdzyxTk

D

yzdzyxTk

D

xztzyxVV

t

SISV

t

SISV ,,,,,,,,,,,0

110

111

( ) ( )t

VIVI

t

VVV dTzyxkdTzyxk

0,11

0,

2

1,,,,,,

( ) ( ) ( ) +

+

=

t

S

St

S

S

I dzyx

Tk

D

xzdzyx

Tk

D

xztzyx I

I

I

I0

10

11,,,,,,,,,

( ) ( ) ( ) ( ) ( )+++ t

II

t

I dzyxITzyxkdzyxITzyxkzyxf

I0

2

,0

,,,,,,,,,,,,,, (5c)

( ) ( ) ( ) +

+

=

t

S

St

S

S

V dzyxTk

D

xzdzyx

Tk

D

xztzyx V

V

V

V0

10

11,,,,,,,,,

( ) ( ) ( ) ( ) ( )+++ t

VV

t

V dzyxVTzyxkdzyxVTzyxkzyxf

V0

2,

0

,,,,,,,,,,,,,, .

Relations for calculations parameters Sij, ai,A,B, q,pcould be written as

( ) ( ) ( ) ( ) =

0 0 011,

,,,,,,,,,x y zL L L

tv

ji

ij tdxdydzdtzyxVtzyxITzyxktS , ( )= 0000

2

004 VVIIIV SSSa


15/20


49

00IIS ,

0000

2

0000003 VVIIIVIIIV SSSSSa += , ( ) +=

0 0 0

222

002,,2

x y zL L L

tvIzyxIV tdxdydzdzyxfLLLSa

( ) ( ) +

2

0 0 0

2

000 0 0

2

00000000,,,, x

L L L

tvIIV

L L L

tvVIVIVIIVV

LtdxdydzdzyxfStdxdydzdzyxfSSSSx y zx y z

0022

VVzy SLL , ( ) =

0 0 0001 ,,

x y zL L L

tvIIV tdxdydzdzyxfSa , ++++= 3 323 32 qpqqpqB

4

2

6a

a+ , ( )

2

0 0 0000

,,

=

x y zL L L

tvIVV tdxdydzdzyxfSa ,

2

1

4

2

2

4

2

32 84

+= y

a

a

a

aA ,

=

2

4

2

3

24a

aq

2

4

2

1

4

222

3

4

3

2

3

4

2

32

22

4

02

4

31

0854

48

4a

aLLL

a

a

a

aa

a

a

a

aaLLLa

zyxzyx

, =

2

4

402

12

4

a

aap

( ) 244231

2 3623 aaaaaLLL zyx + , ( ) ( ) ( ) =

0 0 01

,,,,,,x y zL L L

tv

i

Ii tdxdydzdtzyxITzyxktR .

Equations for the second-order approximations of concentrations of dopant and radiation defects

could be written as

( )( )

( )[ ]( )

( ) ( )

( )

++

++

=

2*

2

2*1

122,,,,,,

1,,,

,,,1,,,

,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxD

xt

tzyxC CL

( ) ( ) ( )

( )( )[ ]

( )( )

++

++

+

y

tzyxC

TzyxP

tzyxC

V

tzyxV

V

tzyxV

yx

tzyxC C ,,,

,,,

,,,1

,,,,,,1

,,,112

2*

2

2*1

1

( )) ( ) ( )( )

( )[ ]( )

++

++

+

TzyxP

tzyxC

V

tzyxV

V

tzyxV

zTzyxD C

L,,,

,,,1

,,,,,,1,,, 12

2*

2

2*1

( ) ( )

( ) ( )[ ] +

+

+

zL

tv

CS

S

L WdtWyxCtzyx

Tk

D

xz

tzyxCTzyxD ,,,,,,

,,,,,,

22

1

( ) ( )[ ] ( ) ( )tzyxfWdtWyxCtzyxTk

D

y C

L

tvCS

Sz

,,,,,,,,22

+

+

+

(1d)

( )( )

( )( )

( )( )

+

= Tzyxk

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIVIII

,,,,,,

,,,,,,

,,,,,,

,

112

( )[ ] ( )[ ] ( ) ( )

( )[ ] +

+++

2

11

1

1111,,,

,,,,,,,,,,,, tzyxI

z

tzyxITzyxD

ztzyxVtzyxI

IIVI

( ) ( ) ( )[ ] +

+

+

zL

tvIS

IS

II WdtWyxItzyx

Tk

D

xTzyxk ,,,,,,,,,

12,

( ) ( )[ ]

+

+

zL

tvIS

IS WdtWyxItzyxTk

D

y,,,,,,

12 (3d)

( )( )

( )( )

( )( )

+

= Tzyxk

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVIVV

,,,,,,

,,,,,,

,,,,,,

,

112

( )[ ] ( )[ ] ( ) ( )

( )[ ] +

+++

2

11

1

1111,,,

,,,,,,,,,,,, tzyxV

z

tzyxVTzyxD

ztzyxVtzyxI

VVVI


16/20


50

( ) ( ) ( )[ ] +

+

+

zL

tvVS

VS

VV WdtWyxVtzyx

Tk

D

xTzyxk ,,,,,,,,,

12,

( ) ( )[ ]

+

+

zL

tvVS

VS WdtWyxVtzyxTk

D

y,,,,,,

12

( ) ( ) ( ) ( ) ( ) ( ) +

+

=

Tzyxky

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxII

III

II,,,

,,,,,,

,,,,,,

,,,,

112

( ) ( ) ( )[ ] ( ) ( )++

+

+

tzyxfWdtWyxtzyxTk

D

xtzyxI

I

z

I

I

L

tvIS

S ,,,,,,,,,,,

12

2

( ) ( )[ ] ( ) ( ) +

+

+

+

z

tzyxTzyxD

zWdtWyxtzyx

Tk

D

y

IL

tvIS

S

I

z

I

I

,,,,,,,,,,,, 112

( ) ( )tzyxITzyxkI

,,,,,,+ (5d)

( )( )

( )( )

( )( )+

+

=

Tzyxky

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxVV

VVV

VV,,,

,,,,,,

,,,,,,

,,,,

112

( ) ( ) ( )[ ] ( ) ( )++

+

+

tzyxfWdtWyxtzyxTk

D

xtzyxV V

z

V

V

L

tvVS

S ,,,,,,,,,,, 122

( ) ( )[ ] ( ) ( ) +

+

+

+

z

tzyxTzyxD

zWdtWyxtzyx

Tk

D

y

VL

tvVS

S

V

z

V

V

,,,,,,,,,,,, 112

( ) ( )tzyxVTzyxkV ,,,,,,+ .The second-order approximations of concentrations of dopant and radiation defects could be writ-ten as

( ) ( ) ( )[ ]

( )

( ) ( )

( )

++

++

=

tC

L

V

zyxV

V

zyxV

TzyxP

zyxCTzyxD

xtzyxC

02*

2

2*1

12

2

,,,,,,1

,,,

,,,1,,,,,,

( ) ( ) ( )

( )

( )[ ]

( )

++

++

+

tC

TzyxP

zyxC

V

zyxV

V

zyxV

y

d

x

zyxC

0

12

2*

2

2*1

1

,,,

,,,1

,,,,,,1

,,,

( ) ( )

( ) ( ) ( )

( )

++

+

t

LL

V

zyxV

V

zyxVTzyxD

zd

y

zyxCTzyxD

02*

2

2*1

1 ,,,,,,1,,,,,,

,,,

( ) ( )[ ]( )

( )[ ] +

+

++

t L

vC

SCz

WdWyxCTk

D

xd

TzyxP

zyxC

z

zyxC

012

121 ,,,,,,

,,,1

,,,

( ) ( ) ( )[ ] + +

+

t L

vCS

S

S ddWdWyxCzyx

Tk

D

yddzyx

z

012

,,,,,,,,,

( )zyxfC ,,+ (1e)

( ) ( ) ( )

( ) ( )

++=t

I

t

I d

y

zyxITzyxD

yd

x

zyxITzyxD

xtzyxI

0

1

0

12

,,,,,,

,,,,,,,,,

( ) ( )

( ) ( )[ ] ( )++ ++ zyxfdzyxITzyxkdz

zyxITzyxD

z I

t

III

t

I,,,,,,,,

,,,,,,

0

2

12,0

1

( ) ( )[ ] ( )

+ +

+

t

S

ISt L

vIS

IS zyxTk

D

yddWdWyxIzyx

Tk

D

x

z

0012

,,,,,,,,,

( )[ ] ( ) ( )[ ] ( )[ ] ++ +

t

VIVI

L

vI

dzyxVzyxITzyxkddWdWyxIz

01212,12

,,,,,,,,,,,,

(3e)


17/20


51

( ) ( ) ( )

( ) ( )

++=t

V

t

V dy

zyxVTzyxD

yd

x

zyxVTzyxD

xtzyxV

0

1

0

1

2

,,,,,,

,,,,,,,,,

( ) ( )

( ) ( )[ ] ( )++ ++ zyxfdzyxVTzyxkdz

zyxVTzyxD

z V

t

VVV

t

V ,,,,,,,,,,,

,,,0

2

12,0

1

( ) ( )[ ] ( )

+ +

+

t

SVS

t L

vVS

VS zyxTk

Dy

ddWdWyxVzyxTk

Dx

z

0012 ,,,,,,,,,

( )[ ] ( ) ( )[ ] ( )[ ] ++ +

t

VIVI

L

vI dzyxVzyxITzyxkddWdWyxV

z

01212,12

,,,,,,,,,,,,

( ) ( ) ( )

( ) ( )

+

+

=

tI

tI

I dy

zyxTzyxD

yd

x

zyxTzyxD

xtzyx

II0

1

0

1

2

,,,,,,

,,,,,,,,,

( ) ( )

( ) ( )[ ] +

+

+

t L

vIS

tI

z

IIWdWyxzyx

xd

z

zyxTzyxD

z 012

0

1 ,,,,,,,,,

,,,

( ) ( )[ ] ( )++ +

+

zyxfddWdWyxzyxTk

D

yd

Tk

D

I

z

I

IIt L

vIS

SS,,,,,,,,

012

( ) ( ) ( ) ( )++

t

I

t

II dzyxITzyxkdzyxITzyxk00

2

, ,,,,,,,,,,,, (5e)

( ) ( ) ( )

( ) ( )

+

+

=

tV

tV

V dy

zyxTzyxD

yd

x

zyxTzyxD

xtzyx

VV0

1

0

1

2

,,,,,,

,,,,,,,,,

( ) ( )

( ) ( )[ ] +

+

+

t L

vVS

tV

z

VVWdWyxzyx

xd

z

zyxTzyxD

z 012

0

1 ,,,,,,,,,

,,,

( ) ( )[ ] ( )++ +

+

zyxfddWdWyxzyxTk

D

yd

Tk

D

V

z

V

VVt L

vVS

SS,,,,,,,,

012

( ) ( ) ( ) ( )++t

V

t

VV dzyxVTzyxkdzyxVTzyxk

00

2

,,,,,,,,,,,,, .

Parameters bi, C, F, r, shave been determined by the following relations

00

2

0000

2

004

11IIVV

zyx

VVIV

zyx

SSLLL

SSLLL

b

= , ( ) +

++=

zyx

VVII

zyxIVVVLLL

SSLLLSSb 0000

100132

( ) ( )3333

10

2

00

1001

2

00

011001

0000 22zyx

IVIV

zyxIVVV

zyx

IV

zyxIVIIIV

zyx

VVIV

LLL

SSLLLSS

LLL

SLLLSSS

LLL

SS

++

++++

+ ,

( ) ( ) ( )++

++++

=0110

012

10011102

0000

2 22 IVIIzyxx

IV

IVVVzyxVIVVV

zyx

VVII SSLLLL

SSSLLLCSS

LLL

SSb

( ) ( )

++++++

+

zyx

IV

IVzyxVVIVIIIVzyx

zyx

IV

zy

VV

LLL

SSLLLSSSSLLL

LLL

S

LL

S 200

1001011001

0000 222

( )01

0010

2222

2

00

1102

2IV

zyx

IVIV

zyx

IVI

IVVVV SLLL

SS

LLL

SCSSC

+ , ( +

++=

xIV

zyx

VVVIV

II LS

LLL

CSSSb

10

0211

001

) ( )

++

++++

zyx

IVVVV

zyxIVVV

zyx

IVIIzyx

IVVVzyLLL

SSCLLLSS

LLL

SSLLLSSLL 1102

1001

0110

01012

22

( )zyx

IVIV

IVIVIzyxIIIVIVLLL

SSSSCLLLSSS

+++

2

0110

0100100100223 ,

( )

+= 01

2

0200

000 IV

zyx

VVIV

II S

LLL

SSSb


18/20


52

( ) ( )++

++

0110

1102

0110

1102 22IVIIzyx

zyx

IVVVV

IVIIzyx

zyx

IVVVV SSLLLLLL

SSCSSLLL

LLL

SSC

( )0110

1102

01

2

010122

IVIIzyx

zyx

IVVVV

IVIVIIV SSLLL

LLL

SSCSSCS ++

+ , +

=

zyx

VI

IVILLL

SC 1100

zyx

IV

zyx

IIII

zyx

III

LLLS

LLLSS

LLLS

+

11202000

2

1 ,110200

2

10011 IVVVVVVIVVIV SSSSC += ,4

2

2

4

2

32 48aa

aayE += ,

2

4

2

1

4

222

3

4

3

2

3

4

2

32

22

4

2

0

4

31

02

4

2

3

8544

84

24 b

bLLL

b

b

b

bb

bb

b

bbLLLb

b

br

zyxzyx

= , (

=

402

4

2

412

bbb

s

423118bbbbLLL

zyx ,

3 323 32

4

2

6rsrrsr

a

aF ++++

= .

Equations for components of the displacement vector could be presented in the form

( )

( )

( )

( )

( )[ ]

( )

( )

( )

( )[ ]

( )

+

++

++=

yx

tzyxu

z

zE

zKx

tzyxu

z

zE

zKt

tzyxu

z

yxx,,,

13

,,,

16

5,,, 12

2

1

2

2

2

2

( )( )[ ]

( ) ( )( )

( )( )[ ]

( )

+++

+

++

zx

tzyxu

z

zEzK

z

tzyxu

y

tzyxu

z

zE zzy ,,,

13

,,,,,,

12

1

2

2

1

2

2

1

2

( ) ( ) ( )

x

tzyxTzzK

,,,

( ) ( ) ( )

( )[ ]

( ) ( )( ) ( )

( )+

+

+=

y

tzyxTzzK

yx

tzyxu

x

tzyxu

z

zE

t

tzyxuz x

yy ,,,,,,,,,

12

,,,1

2

2

1

2

2

2

2

( )( )[ ]

( ) ( ) ( ) ( )( )[ ]

( ) +

++

+

+

+

+ zK

z

zE

y

tzyxu

y

tzyxu

z

tzyxu

z

zE

z

yzy

112

5,,,,,,,,,

12 21

2

11

( ) ( )( )[ ]

( ) ( ) ( )yx

tzyxuzK

zy

tzyxu

z

zEzK

yy

+

++

,,,,,,

16

12

12

( ) ( ) ( ) ( ) ( ) ( )

+

+

+

=

zy

tzyxu

zx

tzyxu

y

tzyxu

x

tzyxu

t

tzyxuz

yxzzz,,,,,,,,,,,,,,, 1

2

1

2

2

1

2

2

1

2

2

2

2

( )( )[ ]

( ) ( ) ( ) ( ) ( )

( )[ ]

+

+

+

+

+

+

z

zE

zz

tzyxu

y

tzyxu

x

tzyxuzK

zz

zE xyx

16

,,,,,,,,,

12

111

( ) ( ) ( ) ( )( ) ( )

( )

z

tzyxTzzK

z

tzyxu

y

tzyxu

x

tzyxu

z

tzyxuzyxz

,,,,,,,,,,,,,,,6 1

111 .

Integration of the left and the right sides of the above equations on time tleads to final relationsfor components of displacement vector

( )( )

( ) ( )

( )[ ] ( )

( ) ( )

( )( )[ ]

++

++=

z

zEzK

zddzyxu

xz

zEzK

ztzyxu

t

xx

13

1,,,

16

51,,,

0 012

2

2

( ) ( ) ( )

+

+

t

z

t

y

t

y ddzyxu

zddzyxu

yddzyxu

yx 0 012

2

0 012

2

0 01

2

,,,,,,,,,


19/20


53

( )( ) ( )[ ] ( )

( ) ( ) ( )

( )[ ] ( )

( )( )

++

+

+

z

zzK

z

zEzKddzyxu

zxzzz

zE tz

13,,,

1

12 0 01

2

( )( )

( ) ( )

( )[ ] ( )

( )

++

zddzyxu

xz

zEzK

zddzyxT

x x

t

1,,,

16

51,,,

0 012

2

0 0

( ) ( )( )[ ]

( )( )

( )

+

+

0 012

2

0 01

2

,,,2

1,,,

13

ddzyxuyz

ddzyxuyxz

zEzK

yy

( ) ( )

( ) ( )

( )( )[ ]

( )

++

+

+

0 01

2

0 012

2

,,,131

,,,

ddzyxuzxz

zEzK

z

zEddzyxu

z zz

( ) ( )

( )( )

( )

++

0 00

,,,1

ddzyxT

xz

zzKu

z x

( ) ( )

( ) ( )[ ] ( ) ( ) +

+

+=

t

x

t

xy ddzyxu

yxddzyxu

xzz

zEtzyxu

0 01

2

0 012

2

2,,,,,,

12,,,

( )( ) ( ) ( ) ( ) ( )

( )( )[ ] +

++

+

+ z

zE

zKddzyxuyzddzyxuyxz

zK t

x

t

y

112

5

,,,

1

,,,0 0

12

2

0 01

2

( )( )[ ] ( )

( )( )

( ) ( )

+

+

+

++

t

z

t

y ddzyxuy

ddzyxuzz

zE

zzz

zE

0 01

0 01 ,,,,,,

12

1

112

5

( ) ( )

( ) ( )

( ) ( ) ( )

( )( )[ ]

+

+

z

zEzKddzyxu

zyzddzyxT

z

zzK

t

y

t

16,,,

1,,,

0 01

2

0 0

( )( ) ( )[ ]

( ) ( ) ( ) ( )

( )

+

+

z

zzKddzyxu

yxddzyxu

xzz

zExx

0 01

2

0 012

2

,,,,,,12

( ) ( )

( ) ( )

( ) ( )

0 012

2

0 01

2

0 0

,,,1

,,,,,,

ddzyxuyz

ddzyxuyxz

zKddzyxT

xy

( ) ( )( )[ ]

( )( )

( ) ( )

+

+

++

0 01

0 01

,,,,,,1112

5

ddzyxu

yddzyxu

zz

zE

zz

zEzK

zy

( ) ( ) ( )

( )( )[ ]

( ) yy

uddzyxuzyz

zEzK

zz0

0 01

2

,,,16

1

2

1+

+

( ) ( )

( ) ( )[ ] ( ) ( )

+

+

+=

0 012

2

0 012

2

,,,,,,12

,,,

ddzyxuy

ddzyxuxzz

zEtzyxu

zzz

( ) ( ) ( )

+

+

+

+

0 01

0 01

2

0 01

2

,,,,,,,,,

ddzyxuxz

ddzyxuzy

ddzyxuzx

xyx

( ) ( ) ( )( ) ( )

( )( )

+

+

+

+

zzE

zzzzKddzyxu

zddzyxu

y xx

1611,,,,,,

0 01

0 01

( ) ( ) ( )

0 01

0 01

0 01

,,,,,,,,,6

ddzyxuy

ddzyxux

ddzyxuz

yxz

( ) ( ) ( )

( ) ( )

zz uddzyxTzz

zzKddzyxu

z0

0 00 01

,,,,,, +

.


20/20


54

AUTHORS

Pankratov Evgeny Leonidovichwas born at 1977. From 1985 to 1995 he was educated in a secondary

school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University:

from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Ra-

diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio-

physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro-structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo-

rod State University of Architecture and Civil Engineering. 2012-2015 Full Doctor course in Radiophysical

Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Professor of

Nizhny Novgorod State University. He has 155 published papers in area of his researches.

Bulaeva Elena Alexeevnawas born at 1991. From 1997 to 2007 she was educated in secondary school ofvillage Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school

Center for gifted children. From 2009 she is a student of Nizhny Novgorod State University of Architec-

ture and Civil Engineering (spatiality Assessment and management of real estate). At the same time she

is a student of courses Translator in the field of professional communication and Design (interior art) in

the University. Since 2014 E.A. Bulaeva is in a PhD program in Radiophysical Department of Nizhny

Novgorod State University. She has 105 published papers in area of her researches.

on modification of properties of p-n-junctions during overgrowth

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