research article nonlinear electrical conductivity properties of au...

Research ArticleNonlinear Electrical Conductivity Properties of Au FilmsPrepared by Sputtering

Qingyun Meng1 Yixin Kang1 Xiaoyu Zhai1 Ziwen Yin1 and Dongpeng Yan12

1 State Key Laboratory of Chemical Resource Engineering Beijing University of Chemical Technology Beijing 100029 China2 Key Laboratory of Theoretical and Computational Photochemistry Ministry of Education College of ChemistryBeijing Normal University Beijing 100875 China

Correspondence should be addressed to Qingyun Meng mqybuct163com Ziwen Yin yinziwen1989163comand Dongpeng Yan yandpmailbucteducn

Received 10 March 2014 Accepted 7 July 2014 Published 13 August 2014

Academic Editor Ugur Serincan

Copyright copy 2014 Qingyun Meng et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Metal-based films with tunable electrical conductivity have played an important role in developing new types of electric devices forfuture application In this work a sputtering method was used to obtain Au films on silicon substrate in a hypobaric atmosphereScanning electron microscope (SEM) shows that the interspaces between the Au nanoparticles were highly uniform and orderlydistributed with the width of several nanometers at the surface By measuring the I-V curves of the films with thickness lessthan 20 nm the nonlinear behaviors of electrical resistivity became gradually obvious as the decrease of the film thickness Forexample upon the thickness reducing to 10 nm remarkable discontinuous step phenomenon appeared Moreover a computationalsimulation was carried on the electrical conductivity of films under normal temperature based on the Coulomb blockade theoryand scattering theory in which the electric current was in the range from 0 to 15times 10minus5 AThe computational results were consistentwell with the experimental observations which confirm that the nonlinear and step phenomenon can be assigned to the Coulombblockade effect when electrons transfer occurs in the interspaces between the nanoparticles

1 Introduction

In recent years due to the rapid development of electric andelectronic devices extensive study has been focused on theinterface behaviors ofmetal-metal andmetal-semiconductorUnder low-temperature condition (such as 77K) it wasfound that there was Coulomb blockade phenomenon innanocrystalline metal with island structures [1 2] Whenthe thickness of metal film was at the nanometer scale thestructures became discontinuous [3] which were composedof isolated nanoparticles while the thickness of film wasclose to De Broglie wavelength the energy level became indiscrete state and the motion of electrons was perpendicularto the surface of films which presented quantum size effect[4 5] In the latter case the resistivity of film was very largeand the temperature coefficient of resistance (TCR) usuallybecame a minus value which presented Ohms feature at lowelectric field but presented non-Ohms feature [6 7] at highelectric field A large number of experimental reports show

the jumping transmission of current carriers in interspaceThat is if the particle size in the nanoscaled film is smallenough electron transferwill be restrained and the resistivityof films will show a jump with the increasing of voltageHowever up to date there was no report about the Coulombblockade phenomenon at room temperature which hasrestricted such effect into the practical applications Coulombblockade phenomenon and macroscopic quantum tunnelingphenomenon will play a significant role in the research andproduction of single electrondevice [8 9] whichwill improvethe degree of integration of integrated circuit In this workthe nonlinear electrical conductivity of the nanostructuredAu films was studied at room temperature We furtherperformed theoretical calculation which agreed well withexperimental results By the combination of experimentaland computational study on the discontinuous electricalconductivity of films this work may pave an alternative wayof metal films for further device application

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2014 Article ID 437082 6 pageshttpdxdoiorg1011552014437082

2 Journal of Nanomaterials

2 Experiments

21 Experimental Materials and Instrument Single-side-polished silicon wafer was purchased from Tianjin ZhaoyiJinke Technology Co Ltd Au sputtering target (purity9999) was purchased from Beijing Zhongke Jinyu Tech-nology Co Ltd HF solution (40 or higher) was purchasedfrom Beijing Chemical Works anhydrous ethanol solution(mass fraction 997 or higher) was purchased from BeijingChemical Works graphite electrode was purchased fromBeijing Jixing Shengrsquoan Industry amp Trade Co Ltd

The Au nanoparticles were prepared on a DH2010 typevacuum evaporation coating machine (Hangzhou DahuaInstrument Co Ltd) and a JZ-BJJY-type 2 DC sputteringapparatus (Beijing Zhongke Jinyu Technology Co Ltd)Surface morphology was observed on an S-4700 type iceemission scanning electron microscope (SEM HITASCHI)The electric measurement was performed on a TR-KDY-type 1 four-point probe resistivity meter (Beijing Tong DeTechnology Co Ltd) The I-V curve was measured on anHP4145 type semiconductor parameter analyzer and 122 typenanoampere table (Agilent companies in the United States)The thickness of film was estimated on an SE200BA-M300type ellipsometer (American Stress Technologies Inc) Theelectrical conductivity of Au films with different thicknesseswas measured at room temperature by the nanoampere tablein the range from plusmn2 times 10minus3 A to plusmn1 times 10minus15 A and theelementary accuracy was 05 to 10

22 Preparation for Au Nanometer Films The electrolytic cellanode corrosion device was used to corrode the silicon wafer(4 cm times 08 cm) The total volume of solution was 200mlcontaining HF solution and anhydrous ethanol solution andthe mixing proportion (bulk factor) was 18 1 The corrosioncurrent was 45mA and the corrosion time was 45 minutesCorroded silicon wafer was put into LPCVD to achieve thegrowth of SiO

2

films with pressure of 760 torr temperature of650∘C and the oxidation time of 30 minutes in 9999 pureoxygen environment Finally low vacuum DC sputteringmethod was employed to grow Au nanometer films under 5times 10minus2 PaMultiplemeasurements were performed to confirmthe results

3 Results and Discussion

31 Volt-Ampere Characteristics of Au Films with DifferentThickness Figure 1(a) shows the typical I-V curve of Au filmwith the thickness of 20 nm in which the slope of the I-Vcurve shows gradual change characteristic and the thresholdvoltage is located at 18 V which presents the I-V behaviorsof typical semiconductors Figure 1(b) shows the I-V curveof Au film with the thickness of 13 nm and there are severalpeaks of the current in the range from6 to 91 VTherefore theI-V curve of Au nanometer films presents discrete electricalconductivity Figure 1(c) is the I-V curve of Au film with thethickness of 9 nm and it is found that the curve possessessteps structure and the threshold voltage is 75 V There isno step until the voltage increases to 75 V and there is only

one peak at the voltage of 4V Figure 1(d) is the I-V curveof Au thin film with the thickness of 4 nm There is a jumpwhen the voltage is 05 V and the steps appear in successionwith the increasing of voltageThese results also show that thecurrent values (I) of the films are in the range of the order ofmagnitude of microampere

32 Surface Morphology of Au Nanometer Films Figure 2shows the FE-SEM micrographs of Au nanometer films withthe thickness of 13 9 and 4 nm fromwhich it can be observedthat the as-prepared two-dimensional metal thin film ishighly uniform which is composed of discontinuous metalparticle clusters There are interspaces whose separationdistance is in the range from 1 to 5 nm between the particleson the surface of the samples and such an interspacemay giverise to the discontinuous current change upon the increasingvoltage

33 Theoretical Calculation of the Electrical Conductivityof Films According to FE-SEM micrographs of films inFigure 2 it is supposed that there are metal island capacitorsat the surface of film (shown in Figure 3) In a single modelS I and D stand for the source quantum island and drainelectrode respectivelyThe applied voltage between S andD is119881119863119878

and electronswill transfer inmetal island capacitor If thequantum island can accommodate 119873 electronics and in theexisting119873minus1 when the119873 electrons can be completedmetal-quantum island-metal tunnel 119881

119863119878

must meet the followingconditions according to the theory of Coulomb blockade [10ndash13]

119881119863119878

ge min((119873 minus 12) 119890119862119878

(119873 minus 12) 119890

119862119863

) (1)

In Formula (1) 119890 represents the electric quantity of asingle electron and 119862

119878

and 119862119863

are the capacitances of sourceelectrode and drain electrode respectively

The radius of metal island is 119903 the width between metalisland and electrode is l the thickness of films is 119863 theapplied voltage on the both ends of each unit is 119881

119863119878

and thetransverse and longitudinal number of two-dimensional filmsare 119898 and 119899 respectively Based on the scattering theory theelectric current between any two metal islands on the surfaceof films is 119868 [14 15] which can be estimated as follows

119868 = 119873 sdot 119890 = 119890119863Vint120590 (Ω) 119889Ω (2)

In Formula (2) 119873 is the number of scattered particlesduring unit time and in unit scattering angle 119889Ω in a certaindirectionΩ and 119890 is the electric quantity of a single electronand V is the speed of electronic transmission 120590(Ω) is thescattering cross-section inΩ direction

According to 119879matrix theory

119889120590119878119863

119889Ω

=

4120587

ℏV1003816100381610038161003816119879119878119863

1003816100381610038161003816

2

120588119863

(119864119865

) (3)

in which 120590119878119863

is the scattering cross-section from sourceelectrode to drain electrode and ℏ is Planck constant

Journal of Nanomaterials 3

0 1 2 3 4 5 6

15 120583

12 120583

9000n

6000n

3000n

00

I(A

)

U (V)

I experiment

(a)

0 2 4 6 8 10

I(A

)

U (V)

I experiment

40 120583

30 120583

20 120583

10 120583

00

(b)

0 5 10 15 20

I(A

)

U (V)

I experiment

150 120583

120 120583

90 120583

60 120583

30 120583

00

(c)

0 1 2 3 4 5 6

I(A

)

U (V)

I experiment

40 120583

30 120583

20 120583

10 120583

00

(d)

Figure 1 Volt-ampere characteristics curve of samples with different thickness (20 nm (a) 13 nm (b) 9 nm (c) and 4 nm (d))

120588119863

(119864119865

) is the density of electronic state of Fermi level Afterdragging Formula (3) in Formula (2) we can get the followingformulas

119868 =

4120587119890119863

ℏ

1003816100381610038161003816119879119878119863

1003816100381610038161003816

2

int119899 (119864) 119889119864 (4)

119899 (119864) = 119891 (119864) [1 minus 119891 (119864 minus 119864119862

+ Δ119864119878minus119868

)]

minus 119891 (119864 + Δ119864119868minus119863

) [1 minus 119891 (119864 minus 119864119862

)]

(5)

where 119891(119909) = 1(1 + 119890(119909minus119864119865)119870119861119879) and in the formulas 119870119861

is Boltzman constant and the first part on the right of theequal sign of Formula (5) indicates that electrons are scatteredfrom source electrode to quantum island and the second partsuggests that electrons are scattered from quantum island to

drain electrode Using WKB method we can further get thefollowing formula

1003816100381610038161003816119879119878119863

1003816100381610038161003816

2

sim 119890minus120573119897

120573 =

2radic21205831198640

ℏ

(6)

where 120583 is the effective mass of transmission electron and 1198640

is effective barrier height among source electrode-quantum-drain electrode In an ideal conditionCS is equal to119862119863 so wecan obtain that the energy level separation between sourceelectrode and quantum island is equal to the energy levelseparation between quantum island and drain electrode

Δ119864119878rarr119868

= 119864119862

minus 119890 (

119862119863

119862

)119881119863119878

= Δ119864119868rarr119863

(7)

where 119864119862

= 1198902

2119862 and in the formula 119864119862

is the energy ofan electron which transfers from source electrode to drainelectrode and goes through quantum island and119862 is the total


(a) (b)

(c)

Figure 2 FE-SEMmicrographs of Au film with different thickness ((a) 13 nm (b) 9 nm and (c) 4 nm)

Metal

Quantum island

Metal

Figure 3 Metal island capacitor model

capacitance of metallic capacitor So we can further get thefollowing formula

119868 =

4120587119890119863

ℏ

sdot 119890minus(2radic21205831198640ℏ)119897

times [

119864119862

+ Δ119864

1 minus 119890(119864119862+Δ119864)119870119861119879

minus

119864119862

minus Δ119864

1 minus 119890(119864119862minusΔ119864)119870119861119879

]

119862119878

= 119862119863

=

1205760

119878

119889

=

21205760

119903119863

119889

(8)

If we supposed the applied total voltage at the surface of filmwas 119881

119881 = 119898 sdot 119881119878119863

sdot

1198970

119897

(9)

in which 1198970

is the average width of interspace It can beknown that 119897 is less than 5 nm from FE-SEM micrographsSo we would obtain the relationship between electric current

and voltage under a bias voltage We dragged the specificparameters in the formula to carry on a calculation andthe calculation results were compared with the experimentalresults as shown in Figures 4(a) and 4(b)

34 Discussion and Analysis about the Electrical Conductivityof Films By comparing Figures 1(a)ndash1(d) it is clear thatat room temperature electrical conductivity transfers intononlinearity from linearity and became step phenomenonwith the gradual decrease of filmsrsquo thickness with the range ofcurrent of 0 to 15 times 10minus5 A In the experiment nonlinearityphenomenon can be observed only when the thickness offilms dropped to below 20 nm and step phenomenon canbe observed below 10 nm Under the temperature of 09 Kthe systematic capacitance 119862 would be less than 10minus15 F orderof magnitude and the charging energy (119864

119862

= 1198902

2119862) of asingle electron would exceed the energy (119870

119861

119879 = 77625 120583eV)of electronrsquos thermal motion which showed that thermalmotion was covered up in low atmospheric pressure In thisway the phenomenon of electron tunneling can be observedAt room temperature (119879 = 300K) 119870

119861

119879 = 25875MeV onlyif the charging energy of a single electron had exceeded thisenergy we could observe electron tunneling phenomenonIn this case the capacitance needed to meet the condition(119862 le 31 times 10minus18 F) and the total capacitance reduced to 1300of the one in low temperature So we would observe thephenomenon at 09 K only when the craft size of capacitorwas less than 1 micron And we did the experiment at room


0 5 10 15 20

I(A

)

150 120583

120 120583

90 120583

60 120583

30 120583

00

U (V)

I experimentI theory

(a) Au film with the thickness of 9 nm

0 1 2 3 4 5 6

I(A

)

U (V)


40 120583

30 120583

20 120583

10 120583

00

(b) Au film with the thickness of 4 nm

Figure 4 Comparing simulation I-V curves with experimental results

temperature which meant that we could only reduce thesize of the capacitor in order to observe electron tunnelingphenomenon So we needed to reduce the thickness ofconductive films and only when the thickness of films wascontrolled less than 20 nanometers we could observe theelectron tunneling phenomenon

From FE-SEM micrographs it could be known thatthe films were discontinuous and the particle size wasuniform distribution When electrons transferred at thesurface of films electrons would be scattered in interspaceand there would be Coulomb blockade effect Figure 4(a)was in comparison with Figure 4(b) and we could find thatwhen the thickness was below 10 nm the experimental resultswere close to the calculation ones Coulomb blockade effectcaused by the contact-potential barrier in interspace can bethe origin of the nonlinearity and step phenomena of theelectrical conductivity of film conductor

4 Conclusion

At room temperature when the thickness of Au films withdiscontinuous surface structure was below 20 nanometersnonlinearity of resistivity can be observed When the thick-ness was below 10 nanometers step phenomena can be obvi-ously observed The working current was in the range from 0to 15 times 10minus5 A The computational electrical conductivity offilms based on the Coulomb blockade theory and scatteringtheory was consistent well with experimental result Theresults showed that the nonlinear electrical conductivity offilms conductor was caused by Coulomb blockade effectwhen electrons transferred in interspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the 973 Program (Grant no2014CB932103) the 863 Program (Grant no 2013AA032501)the National Natural Science Foundation of China (NSFC)the Scientific Fund from Beijing Municipal Commission ofEducation (20111001002) the Fundamental Research Fundsfor the Central Universities the 111 Project (Grant B07004)and Program for Changjiang Scholars and the InnovativeResearch Team in University (PCSIRT IRT1205)

References

[1] W Chen H Ahmed and K Nakazoto ldquoCoulomb blockade at77 K in nanoscale metallic islands in a lateral nanostructurerdquoApplied Physics Letters vol 66 no 24 pp 3383ndash3384 1995

[2] M Aslam I S Mulla and K Vijayamohanan ldquoInsulator-metal transition in Coulomb blockade nanostructuresrdquoAppliedPhysics Letters vol 79 no 5 pp 689ndash691 2001

[3] X K Zhao and J H Fendler ldquoSize quantization in semiconduc-tor particulate filmsrdquo Journal of Physical Chemistry vol 95 no9 pp 3716ndash3723 1991

[4] G R Wang L Wang Q Rendeng J Wang J Luo and CZhong ldquoCorrelation between nanostructural parameters andconductivity properties for molecularly-mediated thin filmassemblies of gold nanoparticlesrdquo Journal of Materials Chemis-try vol 17 no 5 pp 457ndash462 2007

[5] D V Averin and Y V Nazarov ldquoVirtual electron diffusionduring quantum tunneling of the electric chargerdquo PhysicalReview Letters vol 65 no 19 pp 2446ndash2449 1990

[6] L J Geerligs D V Averin and J E Mooij ldquoObservation ofmacroscopic quantum tunneling through the Coulomb energybarrierrdquo Physical Review Letters vol 65 no 24 pp 3037ndash30401990

[7] A T Tilke F C Simmel R H Blick H Lorenz and J P Kott-haus ldquoCoulomb blockade in silicon nanostructuresrdquo Progress inQuantum Electronics vol 25 no 3 pp 97ndash138 2001


[8] D V Talapin J Lee M V Kovalenko and E V ShevchenkoldquoProspects of colloidal nanocrystals for electronic and optoelec-tronic applicationsrdquo Chemical Reviews vol 110 no 1 pp 389ndash458 2010

[9] C J B Ford T J Thornton R Newbury et al ldquoTransport inGaAs heterojunction ring structuresrdquo Superlattices and Micro-structures vol 4 no 4-5 pp 541ndash544 1988

[10] A Zabet-Khosousi P Trudeau Y Suganuma A Dhirani andB Statt ldquoMetal to insulator transition in films of molecularlylinked gold nanoparticlesrdquo Physical Review Letters vol 96 no15 Article ID 156403 2006

[11] T B Tran I S Beloborodov J Hu X M Lin T F Rosenbaumand H M Jaeger ldquoSequential tunneling and inelastic cotun-neling in nanoparticle arraysrdquo Physical Review B vol 78 no 7Article ID 075437 2008

[12] X M Lin C M Sorensen and K J Klabunde ldquoDigestiveripening nanophase segregation and superlattice formation ingold nanocrystal colloidsrdquo Journal of Nanoparticle Research vol2 no 2 pp 157ndash164 2000

[13] C P Collier R J Saykally J J Shiang S E Henrichs and JR Heath ldquoReversible tuning of silver quantum dot monolayersthrough the metal-insulator transitionrdquo Science vol 277 no5334 pp 1978ndash1981 1997

[14] A Messiah Quantum Mechanics vol 2 North-Holland Ams-terdam The Netherlands 1970

[15] M B Isichenko ldquoPercolation statistical topography and trans-port in random mediardquo Reviews of Modern Physics vol 64 no4 pp 961ndash1043 1992

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CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


2 Experiments

21 Experimental Materials and Instrument Single-side-polished silicon wafer was purchased from Tianjin ZhaoyiJinke Technology Co Ltd Au sputtering target (purity9999) was purchased from Beijing Zhongke Jinyu Tech-nology Co Ltd HF solution (40 or higher) was purchasedfrom Beijing Chemical Works anhydrous ethanol solution(mass fraction 997 or higher) was purchased from BeijingChemical Works graphite electrode was purchased fromBeijing Jixing Shengrsquoan Industry amp Trade Co Ltd

The Au nanoparticles were prepared on a DH2010 typevacuum evaporation coating machine (Hangzhou DahuaInstrument Co Ltd) and a JZ-BJJY-type 2 DC sputteringapparatus (Beijing Zhongke Jinyu Technology Co Ltd)Surface morphology was observed on an S-4700 type iceemission scanning electron microscope (SEM HITASCHI)The electric measurement was performed on a TR-KDY-type 1 four-point probe resistivity meter (Beijing Tong DeTechnology Co Ltd) The I-V curve was measured on anHP4145 type semiconductor parameter analyzer and 122 typenanoampere table (Agilent companies in the United States)The thickness of film was estimated on an SE200BA-M300type ellipsometer (American Stress Technologies Inc) Theelectrical conductivity of Au films with different thicknesseswas measured at room temperature by the nanoampere tablein the range from plusmn2 times 10minus3 A to plusmn1 times 10minus15 A and theelementary accuracy was 05 to 10

22 Preparation for Au Nanometer Films The electrolytic cellanode corrosion device was used to corrode the silicon wafer(4 cm times 08 cm) The total volume of solution was 200mlcontaining HF solution and anhydrous ethanol solution andthe mixing proportion (bulk factor) was 18 1 The corrosioncurrent was 45mA and the corrosion time was 45 minutesCorroded silicon wafer was put into LPCVD to achieve thegrowth of SiO

2

films with pressure of 760 torr temperature of650∘C and the oxidation time of 30 minutes in 9999 pureoxygen environment Finally low vacuum DC sputteringmethod was employed to grow Au nanometer films under 5times 10minus2 PaMultiplemeasurements were performed to confirmthe results

3 Results and Discussion

31 Volt-Ampere Characteristics of Au Films with DifferentThickness Figure 1(a) shows the typical I-V curve of Au filmwith the thickness of 20 nm in which the slope of the I-Vcurve shows gradual change characteristic and the thresholdvoltage is located at 18 V which presents the I-V behaviorsof typical semiconductors Figure 1(b) shows the I-V curveof Au film with the thickness of 13 nm and there are severalpeaks of the current in the range from6 to 91 VTherefore theI-V curve of Au nanometer films presents discrete electricalconductivity Figure 1(c) is the I-V curve of Au film with thethickness of 9 nm and it is found that the curve possessessteps structure and the threshold voltage is 75 V There isno step until the voltage increases to 75 V and there is only

one peak at the voltage of 4V Figure 1(d) is the I-V curveof Au thin film with the thickness of 4 nm There is a jumpwhen the voltage is 05 V and the steps appear in successionwith the increasing of voltageThese results also show that thecurrent values (I) of the films are in the range of the order ofmagnitude of microampere

32 Surface Morphology of Au Nanometer Films Figure 2shows the FE-SEM micrographs of Au nanometer films withthe thickness of 13 9 and 4 nm fromwhich it can be observedthat the as-prepared two-dimensional metal thin film ishighly uniform which is composed of discontinuous metalparticle clusters There are interspaces whose separationdistance is in the range from 1 to 5 nm between the particleson the surface of the samples and such an interspacemay giverise to the discontinuous current change upon the increasingvoltage

33 Theoretical Calculation of the Electrical Conductivityof Films According to FE-SEM micrographs of films inFigure 2 it is supposed that there are metal island capacitorsat the surface of film (shown in Figure 3) In a single modelS I and D stand for the source quantum island and drainelectrode respectivelyThe applied voltage between S andD is119881119863119878

and electronswill transfer inmetal island capacitor If thequantum island can accommodate 119873 electronics and in theexisting119873minus1 when the119873 electrons can be completedmetal-quantum island-metal tunnel 119881

119863119878

must meet the followingconditions according to the theory of Coulomb blockade [10ndash13]

119881119863119878

ge min((119873 minus 12) 119890119862119878

(119873 minus 12) 119890

119862119863

) (1)

In Formula (1) 119890 represents the electric quantity of asingle electron and 119862

119878

and 119862119863

are the capacitances of sourceelectrode and drain electrode respectively

The radius of metal island is 119903 the width between metalisland and electrode is l the thickness of films is 119863 theapplied voltage on the both ends of each unit is 119881

119863119878

and thetransverse and longitudinal number of two-dimensional filmsare 119898 and 119899 respectively Based on the scattering theory theelectric current between any two metal islands on the surfaceof films is 119868 [14 15] which can be estimated as follows

119868 = 119873 sdot 119890 = 119890119863Vint120590 (Ω) 119889Ω (2)

In Formula (2) 119873 is the number of scattered particlesduring unit time and in unit scattering angle 119889Ω in a certaindirectionΩ and 119890 is the electric quantity of a single electronand V is the speed of electronic transmission 120590(Ω) is thescattering cross-section inΩ direction

According to 119879matrix theory

119889120590119878119863

119889Ω

=

4120587

ℏV1003816100381610038161003816119879119878119863

1003816100381610038161003816

2

120588119863

(119864119865

) (3)

in which 120590119878119863

is the scattering cross-section from sourceelectrode to drain electrode and ℏ is Planck constant


0 1 2 3 4 5 6

15 120583

12 120583

9000n

6000n

3000n

00

I(A

)

U (V)

I experiment

(a)

0 2 4 6 8 10

I(A

)

U (V)

I experiment

40 120583

30 120583

20 120583

10 120583

00

(b)

0 5 10 15 20

I(A

)

U (V)

I experiment

150 120583

120 120583

90 120583

60 120583

30 120583

00

(c)

0 1 2 3 4 5 6

I(A

)

U (V)

I experiment

40 120583

30 120583

20 120583

10 120583

00

(d)


120588119863

(119864119865


119868 =

4120587119890119863

ℏ

1003816100381610038161003816119879119878119863

1003816100381610038161003816

2

int119899 (119864) 119889119864 (4)

119899 (119864) = 119891 (119864) [1 minus 119891 (119864 minus 119864119862

+ Δ119864119878minus119868

)]

minus 119891 (119864 + Δ119864119868minus119863

) [1 minus 119891 (119864 minus 119864119862

)]

(5)




1003816100381610038161003816119879119878119863

1003816100381610038161003816

2

sim 119890minus120573119897

120573 =

2radic21205831198640

ℏ

(6)



Δ119864119878rarr119868

= 119864119862

minus 119890 (

119862119863

119862

)119881119863119878

= Δ119864119868rarr119863

(7)

where 119864119862

= 1198902




(a) (b)

(c)


Metal

Quantum island

Metal



119868 =

4120587119890119863

ℏ


times [

119864119862

+ Δ119864

1 minus 119890(119864119862+Δ119864)119870119861119879

minus

119864119862

minus Δ119864

1 minus 119890(119864119862minusΔ119864)119870119861119879

]

119862119878

= 119862119863

=

1205760

119878

119889

=

21205760

119903119863

119889

(8)


119881 = 119898 sdot 119881119878119863

sdot

1198970

119897

(9)

in which 1198970




119862

= 1198902


119861


119861



0 5 10 15 20

I(A

)

150 120583

120 120583

90 120583

60 120583

30 120583

00

U (V)



0 1 2 3 4 5 6

I(A

)

U (V)


40 120583

30 120583

20 120583

10 120583

00





4 Conclusion




Acknowledgments


References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0 1 2 3 4 5 6

15 120583

12 120583

9000n

6000n

3000n

00

I(A

)

U (V)

I experiment

(a)

0 2 4 6 8 10

I(A

)

U (V)

I experiment

40 120583

30 120583

20 120583

10 120583

00

(b)

0 5 10 15 20

I(A

)

U (V)

I experiment

150 120583

120 120583

90 120583

60 120583

30 120583

00

(c)

0 1 2 3 4 5 6

I(A

)

U (V)

I experiment

40 120583

30 120583

20 120583

10 120583

00

(d)


120588119863

(119864119865


119868 =

4120587119890119863

ℏ

1003816100381610038161003816119879119878119863

1003816100381610038161003816

2

int119899 (119864) 119889119864 (4)

119899 (119864) = 119891 (119864) [1 minus 119891 (119864 minus 119864119862

+ Δ119864119878minus119868

)]

minus 119891 (119864 + Δ119864119868minus119863

) [1 minus 119891 (119864 minus 119864119862

)]

(5)




1003816100381610038161003816119879119878119863

1003816100381610038161003816

2

sim 119890minus120573119897

120573 =

2radic21205831198640

ℏ

(6)



Δ119864119878rarr119868

= 119864119862

minus 119890 (

119862119863

119862

)119881119863119878

= Δ119864119868rarr119863

(7)

where 119864119862

= 1198902




(a) (b)

(c)


Metal

Quantum island

Metal



119868 =

4120587119890119863

ℏ


times [

119864119862

+ Δ119864

1 minus 119890(119864119862+Δ119864)119870119861119879

minus

119864119862

minus Δ119864

1 minus 119890(119864119862minusΔ119864)119870119861119879

]

119862119878

= 119862119863

=

1205760

119878

119889

=

21205760

119903119863

119889

(8)


119881 = 119898 sdot 119881119878119863

sdot

1198970

119897

(9)

in which 1198970




119862

= 1198902


119861


119861



0 5 10 15 20

I(A

)

150 120583

120 120583

90 120583

60 120583

30 120583

00

U (V)



0 1 2 3 4 5 6

I(A

)

U (V)


40 120583

30 120583

20 120583

10 120583

00





4 Conclusion




Acknowledgments


References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(a) (b)

(c)


Metal

Quantum island

Metal



119868 =

4120587119890119863

ℏ


times [

119864119862

+ Δ119864

1 minus 119890(119864119862+Δ119864)119870119861119879

minus

119864119862

minus Δ119864

1 minus 119890(119864119862minusΔ119864)119870119861119879

]

119862119878

= 119862119863

=

1205760

119878

119889

=

21205760

119903119863

119889

(8)


119881 = 119898 sdot 119881119878119863

sdot

1198970

119897

(9)

in which 1198970




119862

= 1198902


119861


119861



0 5 10 15 20

I(A

)

150 120583

120 120583

90 120583

60 120583

30 120583

00

U (V)



0 1 2 3 4 5 6

I(A

)

U (V)


40 120583

30 120583

20 120583

10 120583

00





4 Conclusion




Acknowledgments


References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0 5 10 15 20

I(A

)

150 120583

120 120583

90 120583

60 120583

30 120583

00

U (V)



0 1 2 3 4 5 6

I(A

)

U (V)


40 120583

30 120583

20 120583

10 120583

00





4 Conclusion




Acknowledgments


References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



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