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AN APPROACH FOR OPTIMIZATION OF MANUFACTURE MULTIEMITTER HETEROTRANSISTORS

E.L. Pankratov1 and E.A. Bulaeva2

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

2 Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky street, Nizhny Novgorod, 603950, Russia

ABSTRACT

In this paper we introduced an approach to model manufacture process of a multiemitter heterotransistor. The approach gives us also possibility to optimize manufacture process. p-n-junctions, which include into the multiemitter heterotransistor, have higher sharpness and higher homogeneity of dopants distributions in enriched by the dopant area.

KEYWORDS

Multiemitter heterotransistor, Approach to optimize technological processes.

1. INTRODUCTION

Logical elements often include into itself multiemitter transistors [1-4]. To manufacture bipolar transistors it could be used dopant diffusion, ion doping or epitaxial layer [1-8]. It is attracted an interest increasing of sharpness of p-n-junctions, which include into bipolar transistors and decreasing of dimensions of the transistors, which include into integrated circuits [1-4]. Dimensions of transistors will be decreased by using inhomogenous distribution of temperature during laser or microwave types of annealing [9,10], using defects of materials (for example, defects could be generated during radiation processing of materials) [11-14], native inhomogeneity (multilayer property) of heterostructure [12-14].

Framework this paper we consider a heterostructure, which consist of a substrate and three epitaxial layers (see Fig. 1). Some dopants are infused in the epitaxial layers to manufacture p and n types of conductivities during manufacture a multiemitter transistor. Let us consider two cases: infusion of dopant by diffusion or ion implantation. Farther in the first case we consider annealing of dopant so long, that after the annealing the dopants should achieve interfaces between epitaxial layers. In this case one can obtain increasing of sharpness of p-n-junctions [12-14]. The increasing of sharpness could be obtained under conditions, when values of dopant diffusion coefficients in last epitaxial layers is larger, than values of dopant diffusion coefficient in average epitaxial layer. At the same time homogeneity of dopant distributions in doped areas increases. It is attracted an interest higher doping of last epitaxial layers, than doping of average epitaxial layer, because the higher doping gives us possibility to increase sharpness of p-n-junctions. It is

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 1

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015

also practicably to choose materials of epitaxial layers and substrate so; those values of dopant diffusion coefficients in the substrate should be smaller, than in the epitaxial layer. In this situation one can obtain thinner transistor. In the case of ion doping of heterostructure it should be done annealing of radiation defects. It is practicably to choose so regimes of annealing of radiation defects, that dopant should achieves interface between epitaxial layers. At the same time the dopant could not diffuse into another epitaxial layer. If dopant did not achieves interface between epitaxial layers during annealing of radiation defects, additional annealing of dopant required.

Main aims of the present paper are analysis of dynamics of redistribution of dopants in considered heterostructure (Figs. 1) and optimization of annealing times of dopants. In some recent papers analogous analysis have been done [4,10,12-14]. However we can not find in literature any similar heterostructures as we consider in the paper.

n-type dopant

Epitaxial layer 1

p-type dopant

Epitaxial layer 2

n-type dopant

Epitaxial layer 3

Substrate

n-type dopant 1

n-type dopant 2

n-type dopant 3

n-type dopant 4

p-type dopant

n-type dopant

Epitaxial layer 1

Epitaxial layer 2

Epitaxial layer 3

Fig. 1a. Heterostructure, which consist of a substrate and three epitaxial layers. View from above Fig. 1b. Heterostructure, which consist of a substrate and three epitaxial layers. View from one side

2. Method of Analysis

Redistribution of dopant in the considered heterostructure has been described by the second Fick’s low [1-4]

Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; x, y, z and t are current coordinates and times; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient depends on properties of materials in layers of heterostructure, speed of heating and cooling of heterostructure (with account Arrhenius low), spatio-temporal distributions of concentrations of dopants and radiation defects. Two last dependences of dopant diffusion coefficient could be approximated by the following relation [15,16]

Here DL (x,y,z,T) is the spatial (due to inhomogeneity (многослойности) of heterostructure) and temperature (due to Arrhenius low) dependences of dopant diffusion coefficient; P (x,y,z,T) is the limit of solubility of dopant; parameter γ depends of properties of materials and could be integer in the interval γ ∈[1,3] [15]; V (x,y,z,t) is the spatio-temporal distribution of concentration of radiation vacancies; V * is the equilibrium distribution of vacancies. Concentrational dependence of dopant diffusion coefficient has been discussed in details in [15]. It should be noted, that radiation damage absents in the case of diffusion doping of haterostructure. In this situation ζ1= ζ2= 0. Spatio-temporal distributions of concentrations of point radiation defects could be determine by solution of the following system of equations [17-19]

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 3

( ) ( ) ( ) ( )      +       +       = z tzyx C D z y tzyx C D y x tzyx C D x t tzyx C C C C ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, (1) with boundary and initial conditions ( ) 0 ,,, 0 = ∂ ∂ = x x tzyx C , ( ) 0 ,,, = ∂ ∂ x Lx x tzyx C , ( ) 0 ,,, 0 = ∂ ∂ y y tzyx C , ( ) 0 ,,, = ∂ ∂ = y Ly y tzyx C , ( ) 0 ,,, 0 = ∂ ∂ = z z tzyx C , ( ) 0 ,,, = ∂ ∂ = z Lz z tzyx C , C (x,y,z,0)=fC (x,y,z). (2)

() ( ) () ( ) ( ) ()       + +       + = 2 * 2 2 * 1 ,,, ,,, 1 ,,, ,,, 1 ,,, V tzyx V V tzyx V Tzyx P tzyx C Tzyx D D L C ς ς ξ γ γ . (3)

( ) () ( ) () ( ) () ×       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ = ∂ ∂ tzyx I y tzyxI Tzyx D y x tzyxI Tzyx D x t tzyxI I I ,,, ,,, ,,, ,,, ,,, ,,, 2 () () ( ) ()()() tzyx TzyxVtzyxI k z tzyxI Tzyx D z Tzyx k VI I II ,,, ,,, ,,, ,,, ,,, ,,, , ,       ∂ ∂ ∂ ∂ + × (4) ( ) () ( ) () ( ) () ×       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ = ∂ ∂ tzyx V y tzyx V Tzyx D y x tzyx V Tzyx D x t tzyx V V V ,,, ,,, ,,, ,,, ,,, ,,, 2 () () ( ) ()()() tzyx VtzyxITzyx k z tzyx V Tzyx D z Tzyx k VI V VV ,,, ,,, ,,, ,,, ,,, ,,,       ∂ ∂ ∂ ∂ + ×

Here ρ=I,V; I (x,y,z,t) is the spatio-temporal distribution of concentration of interstitials; Dρ(x,y, z,T) are diffusion coefficients of interstitials and vacancies; terms V2(x,y, z,t) and I2(x,y,z,t) correspond to generation of divacancies and analogous complexes of interstitials; kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) parameters of recombination of point radiation defects and generation of complexes.

Spatio-temporal distributions of concentrations of divacancies ΦV(x,y,z,t) and analogous complexes of interstitials Φ I(x,y,z,t) will be determined by solving the following systems of equations [17-19]

Here DΦI(x,y,z,T) and DΦV(x,y,z,T) are diffusion coefficients of complexes of point radiation defects; kI(x,y,z,T) and kV(x,y,z,T) are parameters of decay of the complexes.

To determine spatio-temporal distributions of concentrations of complexes of radiation defects we used considered in Refs. [14,19] approach. Framework the approach we transform approximations of diffusion coefficients of complexes of point radiation defects to the following form:

the average values of appropriate parameters,

introduced the following dimensionless variables:

Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 4 with initial ρ(x,y,z,0)=fρ (x,y,z) (5a) and boundary conditions ( ) 0 ,,, = ∂ ∂ = y Ly y tzyx ρ , ( ) 0 ,,, 0 = ∂ ∂ = z z tzyx ρ , ( ) 0 ,,, = ∂ ∂ = z Lz z tzyx ρ . (5b)

Advanced

( ) () ( ) () ( ) +       Φ +       Φ = Φ Φ Φ y tzyx Tzyx D y x tzyx Tzyx D x t tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, () ( ) ()()()() tzyxITzyx k tzyx ITzyx k z tzyx Tzyx D z I II I I ,,, ,,, ,,, ,,, ,,, ,,, 2 +       Φ + Φ ∂ ∂ ∂ ∂ (6) ( ) () ( ) () ( ) +       Φ +       Φ = Φ Φ Φ y tzyx Tzyx D y x tzyx Tzyx D x t tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, () ( ) ()()()() tzyx V Tzyx k tzyx V Tzyx k z tzyx Tzyx D z V VV V V ,,, ,,, ,,, ,,, ,,, ,,, 2 +       Φ + Φ ∂ ∂ ∂ ∂ with boundary and initial conditions ( ) 0 ,,, 0 = ∂ Φ∂ x x tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ x Lx x tzyx ρ , ( ) 0 ,,, 0 = ∂ Φ∂ y y tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ y Ly y tzyx ρ , ( ) 0 ,,, 0 = ∂ Φ∂ z z tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ z Lz z tzyx ρ , Φ I(x,y,z,0)=fΦI(x,y,z), ΦV(x,y,z,0)=fΦV (x,y,z). (7)

Dρ (x,y,z,T) = D0ρ [1+ερ gρ

x,y,z,T)]

kI,V (x,y,z,T)=k0I,V[1+ζh(x,y,z,T)], where D0ρ and k0I,V are

0≤ερV <1, 0≤ζ<1, | gρ(x,y,z,T)|≤1, |h(x,y,z,T)|≤1. Let us

() ( ) * ,,, ,,, ~ ρ ρ ρ tzyx tzyx = , χ= x/Lx, η= y/Ly, ψ = z/Lz,         + + = 2 2 2 0 0 1 1 1 z y x VV L L L D D t ϑ , () VI z y xVI D D VI L L L k 00 ** 2 2 2 ,0 + + = ω , 2 2 2 2 1 z x y x x L L L L b + + = ,

(

and

We will determine solution of Eqs. (8) as the following power series

Substitution of the series into Eqs. (8) gives us possibility to obtain system of equations for initial-order approximations of concentrations of defects ( )ϑ χηψ ρ,,, ~ 000 and corrections for them

(i≥1, j≥1, k≥1). The equations are presented in Appendix.

Substitution of the series (9) into appropriate boundary and initial conditions gives us possibility to obtain boundary and initial conditions for the functions ( )ϑ χηψ ρ,,, ~ ijk . The conditions dard approaches [21,22], are presented in Appendix. and solutions for the functions ( )ϑ χηψ ρ,,, ~ ijk (i≥0, j≥0, k≥0), which have been obtained by stan

Farther we determine spatio-temporal distributions of concentrations of point radiation defects. To calculate the distributions we transform approximations of diffusion coefficients to the following form:

In this situation Eqs. (6) will be transform to the following form

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 5 2 2 2 2 1 z y x y y L L L L b + + = , 2 2 2 2 1 y z x z z L L L L b + + = . Using the above variables leads to transformation Eqs. (4) to the following form ( ) () [] ( ) × +       + = V I y II V I x D D b I T g D D b I 0 0 0 0 1 ,,, ~ ,,, 1 1 ,,, ~ χ∂ ϑ ηψ χ ∂ ηψ χ ε χ∂ ∂ ϑ∂ ϑ ηψ χ ∂ () [] ( ) () [] ( ) ×           + +           + × ψ∂ ϑ ηψ χ ∂ ηψ χ ε ψ∂ ∂ η∂ ϑ ψ ηχ ∂ ψ ηχ ε η∂ ∂ ,,, ~ ,,, 1 1 ,,, ~ ,,, 1 I T g b I T g II z II () [] ()() ϑ ψ ηχ ϑ ψ ηχ ψ ηχ ς ω ,,, ~ ,,, ~ ,,, 1 * * 0 0 V I T h I V D D V I + × (8) ( ) () [] ( ) × +       + = I V y VV I V x D D b V T g D D b V 0 0 0 0 1 ,,, ~ ,,, 1 1 ,,, ~ χ∂ ϑ ηψ χ ∂ ηψ χ ε χ∂ ∂ ϑ∂ ϑ ηψ χ ∂ () [] ( ) () [] ( ) ×       + +       + × ψ∂ ϑ ηψ χ ∂ ηψ χ ε ψ∂ ∂ η∂ ϑ ηψ χ ∂ ηψ χ ε η∂ ∂ ,,, ~ ,,, 1 ,,, ~ ,,, 1 V T g V T g VV VV () [] ()() ϑ ψ ηχ ϑ ψ ηχ ψ ηχ ς ω ,,, ~ ,,, ~ ,,, 1 1 * * 0 0 V I T h V I D D b I V z + × .

()() ∑∑∑ = 000 ,,, ,,, ijk ijk k j i ϑ ψ ηχ ρ ς ω ε ϑ ψ ρηχ ρ . (9)

( )ϑ χηψ ρ,,, ~

ijk

DΦI(x,y,z,T)=D0ΦI[1+εΦIgΦI(x,y,z,T)], DΦV(x,y,z,T)= D0ΦV[1+εΦVgΦV (x,y,z,T)].

( ) () [] ( ) () []        × + +       Φ + = Φ Φ Φ Φ Φ Tzyx g y x tzyx Tzyx g x t tzyx II I II I ,,, 1 ,,, ,,, 1 ,,, ε ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ( ) ()()()() tzyx ITzyx k tzyxI Tzyx k D y tzyx II I I I ,,, ,,, ,,, ,,, ,,, 2 , 0 +    Φ × Φ ∂ ∂

We determine solutions of these equations as the following power series

where ρ =I,V. Substitution of the series (10) into Eqs. (6) and appropriate boundary and initial conditions gives us possibility to obtain equations for initial-order approximations of concentrations of complexes of point radiation defects, corrections for them, boundary and initial conditions for all above functions. The equations, conditions and solutions, which have been obtained by standard approaches [21,22], are presented in the Appendix.

Analysis of spatio-temporal distributions of concentrations of radiation defects has been done by using the second-order approximations on parameters, which used in appropriate series, and have been defined more precisely by using numerical simulation.

Farther we determine solution of the Eq.(1). To calculate the solution we used the approach, which has been considered in [13,18]. Framework the approach we transform approximation of dopant diffusion coefficient DL(x,y,z,T) into following form:

We determine solution of the Eq.(1) as the following power series

Substitution of the series into Eq. (1) gives us possibility to obtain system of equations for initialorder approximation of dopant concentration

and corrections for them

(

j≥1). Substitution of the series (11) into appropriate boundary and initial conditions for the functions

Solutions of equations for the functions Cij

0), which has been calculated by standard approaches [21,22], are presented in the Appendix.

Analysis of spatio-temporal distribution of dopant concentration has been done analytically by using the second-order approximation on parameters, which has been used in appropriate series and have been defined more precisely by using numerical simulation.

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 6 ( ) () [] ( ) () []        × + +       Φ + = Φ Φ Φ Φ Φ Tzyx g y x tzyx Tzyx g x t tzyx V V V V V V ,,, 1 ,,, ,,, 1 ,,, ε ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ( ) ()()()() tzyx V Tzyx k tzyx V Tzyx k D y tzyx VV V V V ,,, ,,, ,,, ,,, ,,, 2 , 0 +    Φ × Φ ∂ ∂

()() ∑ Φ = Φ Φ 0 ,,, ,,, i i i tzyx tzyx ρρ ρ ε , (10)

DL(x,y,z,T) =D0L[1+ε L gL(x,y,z,T)].

()() ∑∑ = ∞ ∞ 00 , , ij ij j i L tx C tx C ξ ε . (11)

C00(x,y,z,t)

Cij

Cij(x,y,z,t) (i≥0, j≥0) ( ) 0 ,,, 0 = = x ij x tzyx C ∂ ∂ ; ( ) 0 ,,, = = x Lx ij x tzyx C ∂ ∂ ; ( ) 0 ,,, 0 = = y ij y tzyx C ∂ ∂ ; ( ) 0 ,,, = = y Ly ij y tzyx C ∂ ∂ ; ( ) 0 ,,, 0 = = z ij z tzyx C ∂ ∂ ; ( ) 0 ,,, = = z Lz ij z tzyx C ∂ ∂ ; C00(x,y,z,0)=fC (x,y,z); Cij(x,y,z,0)=0, i≥0, j≥0.

(

,y,

)

i≥1,

x,y,z,t) (i≥

j≥

(

3. Discussion

In this section we analyzed dynamics of redistribution of dopant with account relaxation of distributions of concentrations of radiation defects (in the case of ion doping of heterostructure).

The Figs. 2 show distributions of concentrations of infused and implanted dopants in one of last epitaxial layers in direction, which is perpendicular to interfaces between epitaxial layers, after annealing of dopant and/or radiation defects, respectively. In this case values of dopant diffusion coefficient in last epitaxial layers is larger, than value of dopant diffusion coefficient in average epitaxial layer. The figures show, that interfaces between epitaxial layers under the above conditions give us possibility to increase sharpness of p-n-junctions and at the same time to increase homogeneity of distributions of concentrations of dopants in enriched by the dopants areas.

The Figs. 3 show distributions of concentrations of infused and implanted dopants in one of last epitaxial layers in direction, which is perpendicular to interfaces between epitaxial layers, after annealing of dopant and/or radiation defects, respectively. In this case values of dopant diffusion coefficient in last epitaxial layers is smaller, than value of dopant diffusion coefficient in average epitaxial layer. The figures show, that presents of last epitaxial layers leads to increasing of homogeneity of dopant in the average epitaxial layers. However sharpness of p-n-junctions decreases in comparison with p-n-junctions for Figs. 2. In this case one can obtain accelerated diffusion in last epitaxial layers. The decreasing of sharpness of p-n-junctions is a reason to use higher level of doping of last epitaxial layer to use nonlinearity of dopant diffusion. In this case one obtain n + -p-n + and/or p + -n-p + structures.

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x 0.0 0.5 1.0 1.5 C ( x , Θ ) 1 2 3 4 5 6 0 L/4 L/2 3L/4 L Interface Epitaxial layer2 Epitaxial

layer1

Fig. 2a. Distributions of concentrations of infused dopant in heterostructure from Fig. 1. Increasing of number of curves corresponds to increasing of difference between values of dopant diffusion coefficient

Fig. 3a. Distributions of concentration of infused in average epitaxial layer dopant in the heterostructure in Fig. 1 without another epitaxial layers (curve 1) and with them for the case, when dopant diffusion coefficient in the average epitaxial layer is smaller in comparison with dopant diffusion coefficients in another layers (curves 2 and 3). Increasing of number of curves corresponds to increasing of difference between values of dopant diffusion coefficients epitaxial layers of heterostructure

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 8 x 0.0 0.5 1.0 1.5 2.0 C ( x , Θ ) 1 2 3 4 0 L/4 L/23L/4 L Interface Epitaxial layer2

Epitaxial layer1

x 0.0 0.5 1.0 1.5 C ( x , Θ ) 0 a=2L/3 L -a1 =-L/3 1 3 2

Fig. 2b. Distributions of concentrations of implanted dopant in heterostructure from Fig. 1. Increasing of number of curves corresponds to increasing of difference between values of dopant diffusion coefficient

x 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 C ( x , Θ ) fC(x) L/4 0 L/2 3L/4 L x0 1 2

Fig. 3b. Distributions of concentration of implanted in average epitaxial layer dopant in the heterostructure in Fig. 1 without another epitaxial layers (curve 1) and with them for the case, when dopant diffusion coefficient in the average epitaxial layer is smaller in comparison with dopant diffusion coefficients in another layers (curves 2 and 3). Increasing of number of curves corresponds to increasing of difference between values of dopant diffusion coefficients epitaxial layers of heterostructure

Optimal value of annealing time we determine framework recently introduced criterion [12-14]. Framework the criterion we approximate real distributions of dopant by step-wise function and minimize the following mean-squared error

at annealing time Θ. Dependences of the optimal value of annealing time from different parameters are presented on Figs. 4.

Fig. 4a. Dependences of dimensionless optimal value of annealing time of infused dopant, which has been calculated by minimization the mean-squared error (12), on different parameters of heterostructure. Curve 1 is the dependence of annealing time on relation a/L, ξ=γ=0 and equal to each other values of dopant diffusion coefficient in epitaxial layers. Curve 2 is the dependence of annealing time on parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of annealing time on parameter ξfor a/L=1/2 and ε=γ=0. Curve 4 is the dependence of annealing time on parameter γ for a/L=1/2 and ε = ξ= 0

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 9

()() []∫ Θ = L xd x x C L U 0 2 , 1 ψ (12)

0.00.10.20.30.40.5 a/L, ξ, ε, γ 0.0 0.1 0.2 0.3 0.4 0.5 Θ D 0 L -2 3 2 4 1

0.00.10.20.30.40.5 a/L, ξ, ε, γ 0.00 0.04 0.08 0.12 Θ D 0 L -2 3 2 4 1

Fig. 4b. Dependences of dimensionless optimal value time of additional annealing of implanted dopant, which has been calculated by minimization the mean-squared error (12), on different parameters of heterostructure. Curve 1 is the dependence of annealing time on relation a/L, ξ=γ=0 and equal to each other values of dopant diffusion coefficient in epitaxial layers. Curve 2 is the dependence of annealing time on parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of annealing time on parameter ξfor a/L=1/2 and ε=γ=0. Curve 4 is the dependence of annealing time on parameter γfor a/L=1/2 and ε=ξ=0

The figures show, that optimal values of annealing time after ion implantation are smaller, than optimal values of annealing time after infusion of dopant. Reason of the difference is necessity to use annealing of radiation defects after ion implantation. After that (if necessary) it could be used additional annealing of dopant. One can obtain spreading of distribution of concentration during annealing of radiation defects. Increasing of annealing time with increasing of thickness of epitaxial layer is the consequence of necessity of higher time for dopant to achieve interfaces between epitaxial layers of heterostructure. Decreasing of annealing time with increasing of values of parameters ε and ξ is the consequence of increasing of values of dopant diffusion coefficient in the area, where main part of dopant has been diffused. Increasing of annealing time with increasing of the parameter γ is the consequence of decreasing of the term [C (x,y,z,t)/ P (x,y,z,T)]γ in approximation of dopant diffusion coefficient (3) with increasing of the parameter γ, because dopant concentration did not usually achieved value of limit of solubility of dopant and parameter γis not smaller, than 1.

4. Conclusion

In this paper we introduce an approach of manufacturing multiemitter heterotransistors. Framework the approach the considered transistors became more compact and will include into itself p-n-junctions with higher sharpness.

ACKNOWLEDGEMENTS

This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, grant of Scientific School of Russia and educational fellowship for scientific research.

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 10

Equations and conditions for functions ( )ϑ ηψ χ ρ , ,, ~ ijk (i≥0, j≥0, k≥0) could be written as ( ) ( ) ( ) ( )         + + = 2 000 2 2 000 2 2 000 2 0 0 000 ,,, 1 ,,, 1 ,,, 1 ,,, ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ I b I b I b D D I z y x V I ; ( ) ( ) ( ) ( )         + + = 2 000 2 2 000 2 2 000 2 0 0 000 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ V b V b V b D D V z y x I V ; ( ) ( ) ( ) ( ) +         + + = 2 00 2 2 00 2 2 00 2 0 0 00 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ i z i y i x V I i I b I b I b D D I () ( ) () ( )      +         +         + η∂ ϑ ηψ χ ∂ ηψ χ η∂ ∂ χ∂ ϑ ηψ χ ∂ ηψ χ χ∂ ∂ ,,, ~ ,,, 1 ,,, ~ ,,, 1 100 100 0 0 i I y i I x V I I T g b I T g b D D

APPENDIX

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 11 () ( )              + ψ∂ ϑ ηψ χ ∂ ηψ χ ψ∂ ∂ ,,, ~ ,,, 1 100 i I z I T g b , i≥1; ( ) ( ) ( ) ( ) +         + + = 2 00 2 2 00 2 2 00 2 0 0 00 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ i z i y i x I V i V b V b V b D D V () ( ) () ( )      +         +         + η∂ ϑ ηψ χ ∂ ηψ χ η∂ ∂ χ∂ ϑ ηψ χ ∂ ηψ χ χ∂ ∂ ,,, ~ ,,, 1 ,,, ~ ,,, 1 100 100 0 0 i V y i V x I V V T g b V T g b D D () ()            + ψ∂ ϑ ψ ηχ ∂ ψ ηχ ψ∂ ∂ ,,, ~ ,,, 1 100 i V z V T g b , i≥1; ( ) ( ) ( ) ( )         + + = 2 010 2 2 010 2 2 010 2 0 0 010 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ I b I b I b D D I z y x V I ()() ϑ ηψ χ ϑ ηψ χ ,,, ~ ,,, ~ 000 000 * * V I I V ; ( ) ( ) ( ) ( )         + + = 2 010 2 2 010 2 2 010 2 0 0 010 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ψ ηχ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ψ ηχ ∂ ϑ∂ ϑ ψ ηχ ∂ V b V b V b D D V z y x I V ()() ϑ ηψ χ ϑ ηψ χ ,,, ~ ,,, ~ 000 000 * * V I I V ; ( ) ( ) ( ) ( )         + + = 2 020 2 2 020 2 2 020 2 0 0 020 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ I b I b I b D D I z y x V I ()() ()() ϑ ψ ηχ ϑ ψ ηχ ϑ ψ ηχ ϑ ψ ηχ ,,, ,,, ,,, ,,, 010 000 * * 000 010 * * V I I V V I I V ( ) ( ) ( ) ( )         + + = 2 020 2 2 020 2 2 020 2 0 0 020 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ψ ηχ ∂ ϑ∂ ϑ ψ ηχ ∂ V b V b V b D D V z y x I V ()() ()() ϑ ηψ χ ϑ ηψ χ ϑ ηψ χ ϑ ηψ χ ,,, ,,, ,,, ,,, 010 000 * * 000 010 * * V I V I V I V I ; ( ) ( ) ( ) ( ) +         + + = 2 110 2 2 110 2 2 110 2 0 0 110 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ I b I b I b D D I z y x V I () ( ) () ( )      +         +         + η∂ ϑ ψ ηχ ∂ ψ ηχ η∂ ∂ χ∂ ϑ ψ ηχ ∂ ψ ηχ χ∂ ∂ ,,, ~ ,,, 1 ,,, ~ ,,, 1 010 010 0 0 I T g b I T g b D D I y I x V I () ( ) ()() ×              + * * 000 100 * * 010 ,,, ~ ,,, ~ ,,, ~ ,,, 1 I V V I I V I T g b I z ϑ ηψ χ ϑ ηψ χ ψ∂ ϑ ηψ χ ∂ ηψ χ ψ∂ ∂ ( ) ( )ϑ ηψ χ ϑ ηψ χ ,,, ,,, 100 000 V I × ; ( ) ( ) ( ) ( ) +         + + = 2 110 2 2 110 2 2 110 2 0 0 110 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ψ ηχ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ψ ηχ ∂ V b V b V b D D V z y x I V () ( ) () ( )      +         +         + η∂ ϑ ηψ χ ∂ ηψ χ η∂ ∂ χ∂ ϑ ηψ χ ∂ ηψ χ χ∂ ∂ ,,, ~ ,,, 1 ,,, ~ ,,, 1 010 010 0 0 V T g b V T g b D D V y V x I V

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 12 () ( ) ()() ×              + * * 000 100 * * 010 ,,, ,,, ,,, ~ ,,, 1 I I V I I I V T g b V z ϑ ηψ χ ϑ ηψ χ ψ∂ ϑ ηψ χ ∂ ηψ χ ψ∂ ∂ ( ) ( )ϑ ηψ χ ϑ ηψ χ ,,, ~ ,,, ~ 100 000 V I × ; ( ) ( ) ( ) ( )         + + = 2 00 2 2 00 2 2 00 2 0 0 00 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ k z k y k x V I k I b I b I b D D I ( ) ( ) ( ) ( )         + + = 2 00 2 2 00 2 2 00 2 0 0 00 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ k z k y k x I V k I b I b I b D D V ; ( ) ( ) ( ) ( ) +         + + = 2 101 2 2 101 2 2 101 2 0 0 101 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ψ ηχ ∂ η∂ ϑ ψ ηχ ∂ χ∂ ϑ ψ ηχ ∂ ϑ∂ ϑ ψ ηχ ∂ I b I b I b D D I z y x V I () ( ) () ( )      +         +         + η∂ ϑ ηψ χ ∂ ηψ χ η∂ ∂ χ∂ ϑ ηψ χ ∂ ηψ χ χ∂ ∂ ,,, ~ ,,, 1 ,,, ~ ,,, 1 001 001 0 0 I T g b I T g b D D I y I x V I () ( )              + ψ∂ ϑ ηψ χ ∂ ηψ χ ψ∂ ∂ ,,, ~ ,,, 1 001 I T g b I z ; ( ) ( ) ( ) ( ) +         + + = 2 101 2 2 101 2 2 101 2 0 0 101 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 , ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ χ ∂ V b V b V b D D V z y x I V () ( ) () ( )      +         +         + η∂ ϑ ηψ χ ∂ ηψ χ η∂ ∂ χ∂ ϑ ηψ χ ∂ ηψ χ χ∂ ∂ ,,, ,,, 1 ,,, ,,, 1 001 001 0 0 I T g b I T g b D D V y V x I V () ( )              + ψ∂ ϑ ηψ χ ∂ ηψ χ ψ∂ ∂ ,,, ~ ,,, 1 001 I T g b V z ; ( ) ( ) ( ) ( ) +         + + = 2 011 2 2 011 2 2 011 2 0 0 011 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ηψ χ ∂ I b I b I b D D I z y x V I ()()()()() [ ϑ ηψ χ ϑ ηψ χ ηψ χ ϑ ηψ χ ϑ ηψ χ ,,, ,,, ,,, ,,, ,,, 000 000 000 001 * * V IT h V I I V ( ) ( )] ϑ ηψ χ ϑ ηψ χ ,,, ~ ,,, ~ 001 000 V I ( ) ( ) ( ) ( )         + + = 2 011 2 2 011 2 2 011 2 0 0 011 ,,, ~ 1 ,,, ~ 1 ,,, ~ 1 ,,, ~ ψ∂ ϑ ηψ χ ∂ η∂ ϑ ηψ χ ∂ χ∂ ϑ ηψ χ ∂ ϑ∂ ϑ ψ ηχ ∂ V b V b V b D D V z y x I V ()()()()() [ ϑ ηψ χ ϑ ηψ χ ηψ χ ϑ ηψ χ ϑ ηψ χ ,,, ,,, ,,, ,,, ,,, 000 000 000 001 * * V IT h V I I V ( ) ( )] ϑ ψ ηχ ϑ ψ ηχ ,,, ~ ,,, ~ 001 000 V I . ( ) 0 ,,, ~ 0 = χ χ∂ ϑ ψ ηχ ρ∂ ijk ; ( ) 0 ,,, ~ 1 = χ χ∂ ϑ ψ ηχ ρ∂ ijk ; ( ) 0 ,,, ~ 0 = η η∂ ϑ ψ ηχ ρ∂ ijk ; ( ) 0 ,,, ~ 1 = η η∂ ϑ ψ ηχ ρ∂ ijk ; ( ) 0 ,,, ~ 0 = =ψ ψ∂ ϑ ψηχ ρ∂ ijk ; ( ) 0 ,,, 1 = =ψ ψ∂ ϑ ψ ηχ ρ∂ ijk , i≥1, j≥1, k≥1; () ( ) * ,, ~0,,, 000 ρ χηψ ψ ηχ ρ ρ f = ; ( ) 0 ~0,,, = χηψ ρijk , i≥1, j≥1, k≥1. Solutions of equations for the functions ( )ϑ ηψ χ ρ,,, ijk (i≥0, j≥0, k≥0) could be written as

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 13 () ()()()() ∑ + = 1 000 2 1 ,,, ~ n n n e c c c F L L ϑ ψ η χ ϑ ψ ηχ ρ ρ ρ , where ()()()()∫∫∫ = 1 0 1 0 1 0 * ,, 1 udvdwdwvu fw cvc uc F n n ρ ρ ρ , cn(α) = cos (πn α), ( ) ( ) I V nI D D n e 0 0 22 exp ϑ π ϑ = , ( ) ( ) V I nV D D n e 0 0 22 exp ϑ π ϑ = ; () ()()()()()()()() × ∑ ∫∫∫∫ −= ∞ 1 0 1 0 1 0 1 0 2 0 00 ,,, * 2 ,,, ~ n V n n n n n n n zyx x i Twvu g vc us e e c c cn LLL b D ϑ ρ ρ ρ τ ϑ ψ η χ ρ π ϑ ψ ηχ ρ () ( ) ()()()()() × ∑ ∫ ∂ ∂ × 1 0 2 0 100 * 2 ,,, ~ n n n n n n zyx y i n e e c c LLL b D dudvdwd u wvu w c ϑ ρ ρ ρ τ ϑ ψ η χ ρ π τ τ ρ ()()()() ( ) () × ∑ ∫∫∫ ∂ ∂ × ∞ =1 2 0 1 0 1 0 1 0 100 * 2 ,,, ~ ,,, n n zyx z i V n n n cn LLL b D dudvdwd v wvu Twvu g w c v s u c χ ρ π τ τ ρ ρ ()()()()()()() ( ) () ∫∫∫∫ ∂ ∂ × ϑ ρ ρ τ τ ρ τ ϑ ψ η 0 1 0 1 0 1 0 100 ,,, ,,, Twvududvdwd g w wvu w s vc u c e e c c V i n n n n n n n , i≥1; () ()()()()()()()() × ∑ ∫∫∫∫ −= 1 0 1 0 1 0 1 0 * * 0 010 * 2 ,,, ~ n n n n In n n n zyx I w c vc u c e e c c c I V LLL I D I ϑ τ ϑ ψ η χ ϑ ψ ηχ ( ) ( ) τ τ τ Vwvududvdwd Iwvu ,,, ~ ,,, ~ 000 000 × ; () ()()()()()()()() × ∑ ∫∫∫∫ −= 1 0 1 0 1 0 1 0 * * 0 010 * 2 ,,, ~ n n n n n n n zyx I w c v c u c e e c c c V I VLLL D V ϑ τ ϑ ψ η χ ϑ ηψ χ ( ) ( ) τ τ τ Vwvududvdwd Iwvu ,,, ~ ,,, ~ 000 000 × ; () ()()()()()()()() × ∑ ∫∫∫∫ −= 1 0 1 0 1 0 1 0 * * 0 020 * 2 ,,, ~ n n n n In In n n n zyx I w c v c u c e e c c c I V LLL I D I ϑ τ ϑ ψ η χ ϑ ηψ χ ()() ()()()() × ∑ × ∞ 1 * * 0 000 010 * 2 ,,, ,,, n In n n n zyx I e c c c I V LLL I D Vwvududvdwd Iwvu ϑ ψ η χ τ τ τ ()()()()()()∫∫∫∫ × ϑ τ τ τ τ 0 1 0 1 0 1 0 010 000 ,,, ~ ,,, ~ Vwvududvdwd wvu Iw c vc u c e n n n ; () ()()()()()()()() × ∑ ∫∫∫∫ −= ∞ =1 0 1 0 1 0 1 0 * * 0 020 * 2 ,,, ~ n n n n n n n zyx I w c vc u c e e c c c V I VLLL D V ϑ τ ϑ ψ η χ ϑ ψ ηχ ()() ()()()()× ∑ × ∞ =1 * * 0 000 010 * 2 ,,, ~ ,,, ~ n n n n zyx I e c c c V I VLLL D Vwvududvdwd Iwvu ϑ ψ η χ τ τ τ ()()()()()()∫∫∫∫ × ϑ τ τ τ τ 0 1 0 1 0 1 0 010 000 ,,, ,,, Vwvududvdwd wvu Iw c v c u c e n n n ; ( ) ( ) ( ) ( ) 0 ,,, ~ ,,, ~ ,,, ~ ,,, ~ 002 001 002 001 = = = = ϑ ψ ηχ ϑ ψ ηχ ϑ ψ ηχ ϑ ψ ηχ V V I I ; () ()()()()()()()() × ∑ ∫∫∫∫ = 1 0 1 0 1 0 1 0 * * 2 0 110 * 2 ,,, ~ n n n n In n n n zyx x I w c vc us e e c c cn I V LLL Ib D I ϑ τ ϑ ψ η χ π ϑ ψ ηχ () ( ) ()()()() × ∑ + × ∞ 1 * * 2 0 010 * 2 ,,, ~ ,,, n In n n n zyx y I I e c c c I V LLL Ib D dudvdwd u Iwvu Twvu g ϑ ψ η χ π τ ∂ τ ∂ ()()()()() ( ) × + ∫∫∫∫ × 2 0 0 1 0 1 0 1 0 010 * 2 ,,, ~ ,,, zyx z I I n n n In LLL Ib D dudvdwd v Iwvu Twvu g w c vs u c en π τ ∂ τ ∂ τ ϑ

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 14 ()()()()()()() ( ) × ∑ ∫∫∫∫ × ∞ 1 0 1 0 1 0 1 0 010 * * ,,, ~ ,,, n I n n n n n dudvdwd w Iwvu Twvu g w s vc u c e c cn I V ϑ τ ∂ τ ∂ τ η χ ()() ()()()()()()()() × ∑ ∫∫∫∫ × ∞ =1 0 1 0 1 0 1 0 100 * * 0 ,,, ~ * 2 n n n In n n n zyx I In n Iwvu vc u c e e c c c I V LLL I D e c ϑ τ τ ϑ ψ η χ ϑ ψ ()() ()()()()()() × ∑ ∫∫∫∫ × 1 0 1 0 1 0 1 0 * * 0 000 * 2 ,,, ~ n n n n In n I n w c uvc c e e c I V L I D Vwvududvdwd w c ϑ τ ϑ ψ τ τ ( ) ( ) ( ) ( )η χ τ τ τ n n c c Vwvududvdwd Iwvu ,,, ~ ,,, ~ 100 000 × ; () ()()()()()()()() × ∑ ∫∫∫∫ = ∞ =1 0 1 0 1 0 1 0 * * 2 0 110 * 2 ,,, ~ n n n n Vn Vn n n n zyx x V w c v c u s e e c c cn V I LLL Ib D V ϑ τ ϑ ψ η χ π ϑ ηψ χ () ( ) ()()()() × ∑ + × ∞ 1 * * 2 0 010 * 2 ,,, ,,, n Vn n n n zyx y V V e c c cn V I LLL Vb D dudvdwd u Vwvu Twvu g ϑ ψ η χ π τ ∂ τ ∂ ()()()()() ( ) × + ∫∫∫∫ × * * 2 0 0 1 0 1 0 1 0 010 * 2 ,,, ,,, V I LLL Vb D dudvdwd v Vwvu Twvu g w c uvs c e zyx z V V n n n π τ ∂ τ ∂ τ ϑ ()()()()()()()() ( ) × ∑ ∫∫∫∫ × 1 0 1 0 1 0 1 0 010,,, ,,, n V n n n n n n dudvdwd w Vwvu Twvu g w s uvc c e c c cn ϑ τ ∂ τ ∂ τ ψ η χ () ()()()()()() × ∑ ∫∫ × ∞ 1 0 1 0 000 100 0 ,,, ~ ,,, ~ * 2 n n n zyx V Vwvududvdwd wvu Ivc e e c VLLL D e ϑ τ τ τ τ ϑ χ ϑ ()() ()()()()()()()() × ∑ ∫∫∫∫ × ∞ 1 0 1 0 1 0 1 0 000 * * 0 * * ,,, * 2 n n n n n n n V n n wvu Iw c vc u c e c c c V I L V D V I c c ϑ τ τ ψ η χ ψ η ( ) ( )ϑ τ τ Vne Vwvududvdwd ,,, ~ 100 × ; () ()()()()()()()() × ∑ ∫∫∫∫ = ∞ 1 0 1 0 1 0 1 0 2 101 ,,, * 2 ,,, n I n n In In n n n zyx x Twvu g v c u s e e c c cn LLL Ib I ϑ τ ϑ ψ η χ π ϑ ηψ χ () ( ) ()()()()()()× ∑ ∫∫ + × ∞ =1 0 1 0 2 001 * 2 ,,, ~ n n In n n n zyx y n u c e e c c c LLL Ib dudvdwd u Iwvu w c ϑ τ ϑ ψ η χ π τ ∂ τ ∂ ()()() ( ) ()()()()× ∑ + ∫∫ × 1 1 0 1 0 001 * 2 ,,, ~ ,,, n n n n z I n n e c c cn Ib dudvdwd v Iwvu Twvu g w c vsn ϑ ψ η χ π τ ∂ τ ∂ ()()()()() ( ) 2 0 1 0 1 0 1 0 001 1 ,,, ~ ,,, zyx I n n n In LLL dudvdwd w Iwvu Twvu g w s v c u c e ∫∫∫∫ × ϑ τ ∂ τ ∂ τ ; () ()()()()()()()() × ∑ ∫∫∫∫ = ∞ =1 0 1 0 1 0 1 0 2 101 ,,, * 2 ,,, ~ n V n n Vn n n n zyx x Twvu g vc us e e c c c LLL Vb V ϑ τ ϑ ψ η χ π ϑ ψ ηχ () ( ) ()()()()() × ∑ ∫ + × ∞ 1 0 2 001 2 ,,, ~ n n n n yzyx n e e c c cn bLLL dudvdwd u Vwvu w cn ϑ τ ϑ ψ η χ π τ ∂ τ ∂ ()()() ( ) ( ) () × ∑ + ∫∫∫ × ∞ =1 2 1 0 1 0 1 0 001 * 2 ,,, ~ * ,,, n n zyx z V n n n cn LLL Vb dudvdwd v Vwvu V Twvu g w c vs u c χ π τ ∂ τ ∂ ()()()()()()()() ( ) ∫∫∫∫ × ϑ τ ∂ τ ∂ τ ϑ ψ η 0 1 0 1 0 1 0 001 ,,, ~ ,,, dudvdwd w Vwvu Twvu g w s v c u c e e c c V n n n n n ; () ()()()()()()()() × ∑ ∫∫∫∫ −= =1 0 1 0 1 0 1 0 001 011 ,,, ~ * 2 ,,, ~ n n n n n n zyx Iwvu v c u c e e c c c LLL I I ϑ τ τ ϑ ψ η χ ϑ ηψ χ

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 15 ()() ()()()()()× ∑ ∫ × ∞ =1 0 * * * * 000 * 2 ,,, ~ n In n n n zyx n e e c c c I V LLL I I V Vwvududvdwd w c ϑ τ ϑ ψ η χ τ τ ()()()()() ()() × ∑ ∫∫∫ × ∞ =1 * * 1 0 1 0 1 0 001 000 * 2 ,,, ~ ,,, ~ n n n zyx n n n c c I V ILLL Vwvududvdwd Iwvu w c v c u c η χ τ τ τ ()()()()()()()()() ∫∫∫∫ × ϑ τ τ τ τ ϑ ψ 0 1 0 1 0 1 0 001 000 ,,, ~ ,,, ~ ,,, Vwvududvdwd Iwvu w cTwvuh vc u c e e c n n n n ; () ()()()()()()()() × ∑ ∫∫∫∫ −= 1 0 1 0 1 0 1 0 * * 011 * 2 ,,, ~ n n n n n n n zyx w c v c u c e e c c c V I VLLL V ϑ τ ϑ ψ η χ ϑ ηψ χ ()() ()()()()() × ∑ ∫ × =1 0 000 001 * 2 ,,, ~ ,,, ~ n n n n zyx e e c c c VLLL Vwvududvdwd Iwvu ϑ τ ϑ ψ η χ τ τ τ ()()()()() () × ∑ ∫∫∫ × ∞ 1 1 0 1 0 1 0 001 000 * * * 2 ,,, ,,, n n zyx n n n c VLLL Vwvududvdwd Iwvu w c uvc c V I χ τ τ τ ()()()()()()() × ∫∫∫∫ × ϑ τ τ τ τ 0 1 0 1 0 1 0 001 000 * * ,,, ~ ,,, ~ ,,, Vwvududvdwd wvu Iw cTwvuh vc u c e V I n n n ( ) ( ) ( )ϑ ψ η n n e c c × Equations for the functions Φρi(x,y,z,t) are ( ) ( ) ( ) ( ) +       Φ + Φ + Φ = Φ Φ 2 0 2 2 0 2 2 0 2 0 0 ,,, ,,, ,,, ,,, z tzyx y tzyx x tzyx D t tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) tzyxITzyx k tzyx I Tzyx k I II ,,, ,,, ,,, ,,, 2 , + ; ( ) ( ) ( ) ( ) +       Φ + Φ + Φ = Φ Φ 2 0 2 2 0 2 2 0 2 0 0 ,,, ,,, ,,, ,,, z tzyx y tzyx x tzyx D t tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) tzyx V Tzyx k tzyx V Tzyx k V VV ,,, ,,, ,,, ,,, 2 , + ; ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ Φ Φ 2 2 0 2 2 0 2 2 0 ,,, ,,, ,,, ,,, z tzyx D y tzyx D x tzyx D t tzyx iI I iI I iI I iI ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ () ( ) () ( ) +       Φ +       Φ + Φ Φ Φ Φ y tzyx Tzyx g y D x tzyx Tzyx g x D iI I I iI I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, 1 0 1 0 () ( )      Φ + Φ Φ z tzyx Tzyx g z D I I ∂ ∂ ∂ ∂ ,,, ,,, 1 0 , i≥1; ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ Φ Φ 2 2 0 2 2 0 2 2 0 ,,, ,,, ,,, ,,, z tzyx D y tzyx D x tzyx D t tzyx iV V iV V iV V iV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ () ( ) () ( ) +       Φ +       Φ + Φ Φ Φ Φ y tzyx Tzyx g y D x tzyx Tzyx g x D iV V V iV V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, 1 0 1 0 () ( )      Φ + Φ Φ z tzyx Tzyx g z D iV V V ∂ ∂ ∂ ∂ ,,, ,,, 1 0 , i≥1; ( ) 0 ,,, 0 = ∂ Φ∂ = x i x tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ = x Lx i x tzyx ρ , ( ) 0 ,,, 0 = ∂ Φ∂ = y i y tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ = y Ly i y tzyx ρ , ( ) 0 ,,, 0 = ∂ Φ∂ = z i z tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ = z Lz i z tzyx ρ , i≥0; Φ I0(x,y,z,0)=fΦI (x,y,z), ΦIi(x,y,z,0)=0,

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 16 ΦV0(x,y,z,0)=fΦV (x,y,z), ΦVi(x,y,z,0)=0, i≥1. Solutions of the equations could be written as () ()()()() ∑ + = Φ = Φ Φ 1 0 2 1 ,,, n n n zyx zyx t ezcycxc F LLL LLL tzyx ρ ρ ρ , where ()                 + + = Φ Φ 2 2 2 0 22 1 1 1 exp z y x n L L L t D n t e ρ ρ π , ()()()()∫∫∫ = Φ Φ Lx Ly Lz n n n n xdydzdzyx fzc y c x c F 000 ,, ρ ρ , cn(α)=cos(πn α/Lα); () ()()()()()()()()() × ∑ ∫∫∫∫ −= Φ ∞ = Φ Φ Φ 1 0000 2 .,, 2 ,,, n t Lx Ly Lz n n n n n n n n zyx i Twvu g w c vc us et ez cy cx cn LLL tzyx ρ ρ ρ ρ τ π ( ) ()()()()()()() × ∑ ∫∫∫ Φ × Φ Φ 1 000 2 1 2 ,,, n t Lx Ly n n n n n n n zyx iI vs u c et cxezcy cn LLL dudvdwd u wvu τ π τ ∂ τ ∂ ρ ρ ρ ()() ( ) ()()()()× ∑ ∫ Φ × ∞ Φ Φ 1 2 0 1 2 ,,, .,, n n n n n zyx Lz iI n t ez cy cx cn LLL dudvdwd v wvu Twvu g w c ρ ρ ρ π τ ∂ τ ∂ ()()()()() ( ) ∫∫∫∫ Φ × Φ Φ t Lx y L Lz iI n n n n dudvdwd w wvu Twvu g w s v c u c e 0000 1 ,,, .,, τ ∂ τ ∂ τ ρ ρ ρ , i≥1, where sn(α)=sin(πnα/Lα). System of equations for functions Cijk(x,y,z,t) (i≥0, j≥0, k≥0) could be written as ( ) ( ) ( ) ( )      ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 00 2 2 00 2 2 00 2 0 00 ,,, ,,, ,,, ,,, z tzyx C y tzyx C x tzyx C D t tzyx C L ; ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 0 2 0 2 0 2 0 2 0 2 0 0 ,,, ,,, ,,, ,,, z tzyx C D y tzyx C D x tzyx C D t tzyx C i L i L i L i () ( ) () ( ) +       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ + y tzyx C Tzyx g y D x tzyx C Tzyx g x D i L L i L L ,,, ,,, ,,, ,,, 10 0 10 0 () ( )      ∂ ∂ ∂ ∂ + z tzyx C Tzyx g z D i L L ,,, ,,, 10 0 , i≥1; ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 01 2 0 2 01 2 0 2 01 2 0 01 ,,, ,,, ,,, ,,, z tzyx C D y tzyx C D x tzyx C D t tzyx C L L L ( ) () ( ) ( ) () ( ) +       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ + y tzyx C Tzyx P tzyx C y D x tzyx C Tzyx P tzyx C x D L L ,,, ,,, ,,, ,,, ,,, ,,, 00 00 0 00 00 0 γ γ γ γ ( ) () ( )      ∂ ∂ ∂ ∂ + z tzyx C Tzyx P tzyx C z D L ,,, ,,, ,,, 00 00 0 γ γ ; ( ) ( ) ( ) ( ) × + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ L L L L D z tzyx C D y tzyx C D x tzyx C D t tzyx C 0 2 02 2 0 2 02 2 0 2 02 2 0 02 ,,, ,,, ,,, ,,, () ( ) () ( ) () ( ) () ( )      +       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ × y tzyx C Tzyx P tzyx C tzyx C y x tzyx C Tzyx P tzyx C tzyx C x ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, 00 1 00 01 00 1 00 01 γ γ γ γ

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 17 () ( ) () ( ) ( ) () ( )      +       ∂ ∂ ∂ ∂ +            ∂ ∂ ∂ ∂ + x tzyx C Tzyx P tzyx C x D z tzyx C Tzyx P tzyx C tzyx C z L ,,, ,,, ,,, ,,, ,,, ,,, ,,, 01 00 0 00 1 00 01 γ γ γ γ ( ) () ( ) ( ) () ( )          ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ + z tzyx C Tzyx P tzyx C z y tzyx C Tzyx P tzyx C y ,,, ,,, ,,, ,,, ,,, ,,, 01 00 01 00 γ γ γ γ ; ( ) ( ) ( ) ( ) × + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ L L L L D z tzyx C D y tzyx C D x tzyx C D t tzyx C 0 2 11 2 0 2 11 2 0 2 11 2 0 11 ,,, ,,, ,,, ,,, () ( ) () ( ) () ( ) () ( ) +            ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ × y tzyx C Tzyx P tzyx C tzyx C y x tzyx C Tzyx P tzyx C tzyx C x ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, 00 1 00 10 00 1 00 10 γ γ γ γ () ( ) () ( ) ( ) () ( )      +       ∂ ∂ ∂ ∂ +            ∂ ∂ ∂ ∂ + x tzyx C tzyx P tzyx C x D z tzyx C Tzyx P tzyx C tzyx C z L ,,, ,,, ,,, ,,, ,,, ,,, ,,, 10 00 0 00 1 00 10 γ γ γ γ () () () () () () ()           × ∂ ∂ +            ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ + Tzyx g x z tzyx C tzyx P tzyx C z y tzyx C tzyx P tzyx C y L ,,, ,,, ,,, ,,, ,,, ,,, ,,, 10 00 10 00 γ γ γ γ ( ) () ( ) () ( ) L L L D z tzyx C Tzyx g z y tzyx C Tzyx g y x tzyx C 0 01 01 01 ,,, ,,, ,,, ,,, ,,,            ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ +    ∂ ∂ × Solutions of equations for functions Cij(x,y,z,t) (i≥0, j≥0) with account appropriate conditions takes the form () ()()()() ∑ + = ∞ =1 00 2 1 ,,, n nC nC zyx zyx t ezcycxc F LLL LLL tzyx C , where ()                 + + = 2 2 2 0 22 1 1 1 exp z y x C nC L L L t D n t e π , ()()()()∫∫∫ = Lx y L Lz C n n n Cn xdydzdzyx fz c y c x c F 000 ,, ; () ()()()()()()()()() ∑ × ∫∫∫∫ −= ∞ 1 0000 2 0 ,,, 2 ,,, n t Lx y L Lz L n n n nC nC n n nnC zyx i Twvu g w c v c u s et ez cy cx c F LLL tzyx C τ π ( ) ()()()()()()() ∑ × ∫∫∫ ∂ ∂ × ∞ 1 000 2 10 2 ,,, n t Lx Ly n n nC nC n n nnC zyx i vs u c e t cxezcy c Fn LLL n dudvdwd u Cwvu τ π τ τ ()() ( ) ()()()()()× ∑ ∫ ∫ ∂ ∂ × ∞ 1 0 2 0 10 2 ,,, ,,, n t nC nC n n n zyx Lz i L n et cxezcy c LLL dudvdwd v Cwvu Twvu g w c τ π τ τ ()()()() ( ) ∫∫∫ ∂ ∂ × Lx y L Lz i L n n n nC dudvdwd w wvu C Twvu g w s v c u c Fn 000 10 ,,, ,,, τ τ , i≥1; () ()()()()()()() ( ) () ∑ × ∫∫∫∫ −= ∞ =1 0000 00 2 01 ,,, ,,, 2 ,,, n t Lx y L Lz n n nC nC n n nnC zyx Twvu P Cwvu v c u s et ez cy cx c Fn LLL tzyx C γ γ τ τ π () ( ) ()()()()()() ∑ × ∫∫ ∂ ∂ × ∞ =1 00 2 00 2 ,,, n t Lx n nC nC n n nnC zyx n u c e t cxezcy c Fn LLL dudvdwd u Cwvu w c τ π τ τ ()() ( ) () ( ) ()()() × ∑ ∫∫ ∂ ∂ × ∞ 1 2 00 00 00 2 ,,, ,,, ,,, n n n nnC zyx Ly Lz n n zcy cx c Fn LLL dudvdwd v Cwvu Twvu P Cwvu w c vs π τ τ τ γ γ ()()()()() ( ) () ( ) ∫∫∫∫ ∂ ∂ × t Lx y L Lz n n n nC nC dudvdwd w Cwvu Twvu P Cwvu w s v c u c et e 0000 00 00 ,,, ,,, ,,, τ τ τ τ γ γ ;

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 18 () ()()()()()()()() ( ) () ∑ × ∫∫∫∫ −= 1 000 1 00 0 2 02 ,,, ,,, 2 ,,, n t Lx y L Lz n n n nC nC n n nnC zyx Twvu P Cwvu w c vc us et ez cy cx c Fn LLL tzyx C γ γ τ τ π ( ) ()()()()()()() ∑ × ∫∫∫ ∂ ∂ × =1 000 2 00 2 ,,, n t Lx Ly n n nC nC n n nnC zyx vs u c et cxezcy c Fn LLL dudvdwd u Cwvu τ π τ τ () ( ) () ( ) ()()()() × ∑ ∫ ∂ ∂ × ∞ =1 2 0 00 1 00 2 ,,, ,,, ,,, n nC n n nnC zyx Lz n t ezcy cx c Fn LLL dudvdwd v Cwvu Twvu P Cwvu w c π τ τ τ γ γ ()()()() ( ) () ( ) () × ∑ ∫∫∫∫ ∂ ∂ × ∞ =1 2 0000 00 1 00 2 ,,, ,,, ,,, n n zyx t Lx Ly Lz n n n nC x cn LLL dudvdwd w Cwvu Twvu P Cwvu w s vc u c e π τ τ τ τ γ γ ()()()()()()() ( ) () ( ) ∫∫∫∫ ∂ ∂ × t Lx Ly Lz n n n nC nC n nnC dudvdwd u Cwvu Twvu P Cwvu w c vc us et ezcy c F 0000 01 00 ,,, ,,, ,,, τ τ τ τ γ γ ()()()()()()() ( ) () ( ) × ∑ ∫∫∫∫ ∂ ∂ ∞ =1 0000 01 00 2 ,,, ,,, ,,, 2 n t Lx Ly Lz n n nC nC n n nnC zyx v Cwvu Twvu P Cwvu vs u c et cxezcy c Fn LLL τ τ τ π γ γ () ()()()()()()()() × ∑ ∫∫∫∫ × ∞ 1 0000 2 2 n t Lx Ly Lz n n n nC nC n n nnC zyx n w s vc u c et cxezcy c Fn LLL dudvdwd w c τ π τ ( ) () ( ) τ τ τ γ γ dudvdwd w Cwvu Twvu P Cwvu ∂ ∂ × ,,, ,,, ,,, 01 00 ; () ()()()()()()()()() ∑ × ∫∫∫∫ −= 1 0000 2 11 ,,, 2 ,,, n t Lx y L Lz L n n n nC nC n n nnC zyx Twvu g w c v c u s et ez cy cx c Fn LLL tzyx C τ π ( ) ()()()()()()()() ∑ × ∫∫∫∫ ∂ ∂ × 1 0000 2 01 2 ,,, n t Lx y L Lz n n n nC nC n n n zyx w c vs u c et ez cy cx c LLL dudvdwd u Cwvu τ π τ τ () ( ) ()()()()() × ∑ ∫ ∂ ∂ × ∞ =1 0 2 01 2 ,,, ,,, n t nC nC n n n zyx L nC et ez cy cx c LLL dudvdwd v Cwvu Twvu g Fn τ π τ τ ()()()() ( ) () × ∑ ∫∫∫ ∂ ∂ × ∞ =1 2 000 01 2 ,,, ,,, n nnC zyx Lx Ly Lz L n n n nC x c Fn LLL dudvdwd w Cwvu Twvu g w s vc u c Fn π τ τ ()()()()()()() ( ) () ( ) ∫∫∫∫ ∂ ∂ × t Lx Ly Lz n n n nC nC n n dudvdwd u Cwvu Twvu P Cwvu w c vc us et ezcy c 0000 10 00 ,,, ,,, ,,, τ τ τ τ γ γ ()()()()() ( ) () ( ) × ∑ ∫∫∫∫ ∂ ∂ ∞ 1 0000 10 00 2 ,,, ,,, ,,, 2 n t Lx Ly Lz n n n nC n zyx dudvdwd v Cwvu Twvu P Cwvu w c vs u c e x LLL τ τ τ τ π γ γ ()()() ()()()()()()()() ∑ × ∫∫∫∫ × ∞ 1 0000 2 2 n t Lx Ly Lz n n n nC nC n n nnC zyx nC n nnC w s vc u c et ezcy cx c Fn LLL t ezcy c F τ π ( ) () ( ) ()()()()() ∑ × ∫ ∂ ∂ × ∞ 1 0 2 10 00 2 ,,, ,,, ,,, n t nC nC n n nnC zyx et ez cy cx c Fn LLL dudvdwd w Cwvu Twvu P Cwvu τ π τ τ τ γ γ ()()() ( ) () ( ) () × ∑ ∫∫∫ ∂ ∂ × ∞ =1 2 000 00 1 00 2 ,,, ,,, ,,, n nnC zyx Lx y L Lz n n n x c Fn LLL dudvdwd u Cwvu Twvu P Cwvu w c v c u s π τ τ τ γ γ ()()()()()()() ( ) () ( ) ∫∫∫∫ ∂ ∂ × t Lx y L Lz n n n nC nC n n dudvdwd v Cwvu Twvu P Cwvu w c v s u c et ez cy c 0000 00 1 00 ,,, ,,, ,,, τ τ τ τ γ γ ()()()() ( ) () ( ) ∑ × ∂ ∂ ∫∫∫∫ ∞ 1 00 0000 1 00 2 ,,, ,,, ,,, 2 n t Lx y L Lz n n n nC zyx dudvdwd w Cwvu Twvu P Cwvu w s v c u c e n LLL τ τ τ τ π γ γ ( ) ( ) ( ) ( )t ez cy cx c F nC n n nnC × .

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Author

Pankratov Evgeny Leonidovich was 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 Radiophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radiophysics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Microstructures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgorod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor course in Radiophysical Department of Nizhny Novgorod State University. He has 96 published papers in area of his researches.

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 19

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015

Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of village 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 Architecture 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. E.A. Bulaeva was a contributor of grant of President of Russia (grant № MK-548.2010.2). She has 29 published papers in area of her researches.