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

MODELING OF MANUFACTURING OF A FIELDEFFECT TRANSISTOR TO DETERMINE CONDITIONS TO DECREASE LENGTH OF CHANNEL E.L. Pankratov1, 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 introduce an approach to model technological process of manufacture of a field-effect heterotransistor. The modeling gives us possibility to optimize the technological process to decrease length of channel by using mechanical stress. As accompanying results of the decreasing one can find decreasing of thickness of the heterotransistors and increasing of their density, which were comprised in integrated circuits.

KEYWORDS Modelling of manufacture of a field-effect heterotransistor, Accounting mechanical stress; Optimization the technological process, Decreasing length of channel

1. INTRODUCTION One of intensively solved problems of the solid-state electronics is decreasing of elements of integrated circuits and their discrete analogs [1-7]. To solve this problem attracted an interest widely used laser and microwave types of annealing [8-14]. These types of annealing leads to generation of inhomogeneous distribution of temperature. In this situation one can obtain increasing of sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area [8-14]. The first effects give us possibility to decrease switching time of p-n-junction and value of local overheats during operating of the device or to manufacture more shallow p-n-junction with fixed maximal value of local overheat. An alternative approach to laser and microwave types of annealing one can use native inhomogeneity of heterostructure and optimization of annealing of dopant and/or radiation defects to manufacture diffusive –junction and implanted-junction rectifiers [12-18]. It is known, that radiation processing of materials is also leads to modification of distribution of dopant concentration [19]. The radiation processing could be also used to increase of sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area [20, 21]. Distribution of dopant concentration is also depends on mechanical stress in heterostructure [15]. In this paper we consider a field-effect heterotransistor. Manufacturing of the transistor based on combination of main ideas of Refs. [5] and [12-18]. Framework the combination we consider a 31


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

heterostructure with a substrate and an epitaxial layer with several sections. Some dopants have been infused or implanted into the section to produce required types of conductivity. Farther we consider optimal annealing of dopant and/or radiation defects. Manufacturing of the transistor has been considered in details in Ref. [17]. Main aim of the present paper is analysis of influence of mechanical stress on length of channel of the field-effect transistor.

Fig. 1. Heterostructure with substrate and epitaxial layer with two or three sections

c2 c1

oxide source

drain c1 a1 a2-a1

c3 gate

Lx-a2-a1 Fig. 2. Heterostructure with substrate and epitaxial layer with two or three sections. Sectional view

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

2. METHOD OF SOLUTION To solve our aim we determine spatio-temporal distribution of concentration of dopant in the considered heterostructure and make the analysis. We determine spatio-temporal distribution of concentration of dopant with account mechanical stress by solving the second Fick’s law [1-4,22,23] ∂ C ( x, y , z , t ) ∂  ∂ C ( x, y , z , t )  ∂  ∂ C ( x, y , z , t )  ∂  ∂ C ( x , y , z , t )  = D + D + D  + (1) ∂t ∂x ∂x ∂y ∂z   ∂y  ∂z  Lz Lz  D   D  ∂ ∂ S S +Ω ∇ S µ ( x , y , z , t ) ∫ C ( x, y , W , t ) d W  + Ω ∇ S µ ( x, y , z , t ) ∫ C ( x, y , W , t ) d W    ∂ x k T ∂ y k T 0 0   with boundary and initial conditions ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) =0, =0, =0, = 0, ∂y ∂x ∂ x ∂ y x=Ly x =0 x= L y =0 x

∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) =0, = 0 , C (x,y,z,0)=fC(x,y,z). ∂z ∂z z =0 x = Lz

Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; Ω is the atomic voLz

lume; ∇S is the operators of surficial gradient; ∫ C ( x, y, z , t ) d z is the surficial concentration of 0

dopant on interface between layers of heterostructure; µ (x,y,z,t) is the chemical potential; D and DS are coefficients of volumetric and surficial (due to mechanical stress) of diffusion. Values of the diffusion coefficients depend on properties of materials of heterostructure, speed of heating and cooling of heterostructure, spatio-temporal distribution of concentration of dopant. Concentraional dependence of dopant diffusion coefficients have been approximated by the following relations [24]   C γ ( x, y , z , t )  C γ ( x, y , z , t )  , . ( ) D = DL (x, y , z , T ) 1 + ξ γ D = D x , y , z , T 1 + ξ S SL S  P ( x, y, z , T )  P γ (x, y , z , T )   

(2)

Here DL (x,y,z,T) and DLS (x,y,z,T) are spatial (due to inhomogeneity of heterostructure) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing; P (x,y,z,T) is the limit of solubility of dopant; parameter γ could be integer in the following interval γ ∈[1,3] [24]; V (x,y,z,t) is the spatio-temporal distribution of concentration of radiation of vacancies; V* is the equilibrium distribution of vacancies. Concentrational dependence of dopant diffusion coefficients has been discussed in details in [24]. Chemical potential could be determine by the following relation [22]

µ =E(z)Ωσij[uij(x,y,z,t)+uji(x,y,z,t)]/2,

(3)

∂u j  1  ∂u  is the deformation where E is the Young modulus; σij is the stress tensor; uij =  i + 2  ∂ x j ∂ xi  tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector r u ( x, y, z , t ) ; xi, xj are the coordinates x, y, z. Relation (3) could be transform to the following form

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

 ∂ u i ( x, y , z , t )

µ ( x, y , z , t ) =  

× E (z )

∂ xj

+

∂ u j (x, y , z, t )   1  ∂ u i ( x, y, z , t ) ∂ u j (x, y , z , t )  +   × ∂ xi ∂ xj ∂ xi   2  

 σ (z )δ ij  ∂ u k ( x, y, z , t )  Ω − ε 0δ ij + − 3 ε 0  − K (z ) β ( z )[T ( x, y, z , t ) − T0 ] δ ij  ,  2 1 − 2σ ( z )  ∂ xk  

where σ is the Poisson coefficient; ε0=(as-aEL)/aEL is the mismatch strain; as, aEL are the lattice spacings for substrate and epitaxial layer, respectively; K is the modulus of uniform compression; β is the coefficient of thermal expansion; Tr is the equilibrium temperature, which coincide (for our case) with room temperature. Components of displacement vector could be obtained by solving of the following equations [23]  ∂ 2 u x ( x, y, z , t ) ∂ σ xx ( x, y , z , t ) ∂ σ xy ( x, y , z, t ) ∂ σ xz ( x, y, z , t ) ρ ( z ) = + +  ∂t2 ∂x ∂y ∂z  2  ∂ u y ( x, y, z , t ) ∂ σ yx (x, y , z , t ) ∂ σ yy ( x, y, z , t ) ∂ σ yz ( x, y , z, t ) = + +  ρ (z ) ∂t2 ∂x ∂y ∂z  2  ∂ u z (x, y , z, t ) ∂ σ zx (x, y , z, t ) ∂ σ zy ( x, y, z , t ) ∂ σ zz ( x, y, z , t )  ρ (z ) = + + ∂t2 ∂x ∂y ∂z 

where σ ij =

∂ u k ( x, y , z , t ) E (z )  ∂ u i ( x, y, z, t ) ∂ u j (x, y , z , t ) δ ij ∂ u k ( x, y, z, t )  + − ×   + δ ij 2 [1 + σ ( z )]  ∂ xj ∂ xi 3 ∂ xk ∂ xk 

× K ( z ) − β ( z ) K (z ) [T ( x, y, z , t ) − Tr ] , ρ (z) is the density of materials, δij is the Kronecker symbol. With account of the relation the last system of equation takes the form 2 ∂ 2 u x (x, y, z, t )  E ( z )  ∂ u y ( x, y, z, t ) 5E ( z )  ∂ 2 u x (x, y, z, t )  = K ( z ) + + K ( z ) − + ρ (z )   ∂t2 ∂ x2 ∂ x∂ y 6 [1 + σ ( z )] 3 [1 + σ ( z )]  

+

2 E ( z )  ∂ u y ( x, y , z , t ) ∂ 2 u z ( x, y , z , t )   E ( z )  ∂ 2 u z ( x, y , z , t ) ( ) + + K z + − K (z ) ×     2 [1 + σ ( z )]  ∂ y2 ∂ z2 3[1 + σ ( z )] ∂ x∂ z  

× β (z )

ρ (z )

∂ 2 u y ( x, y , z , t ) ∂t2

∂ T ( x, y , z , t ) ∂x

2 E ( z )  ∂ u y ( x, y , z , t ) ∂ 2 u x ( x, y , z , t )  ∂ T ( x, y , z , t ) = + +   − β (z ) K (z ) 2 ∂x ∂ x∂ y ∂y 2[1 + σ ( z )]  

2  ∂ u y (x, y , z , t ) ∂  E (z )  ∂ u y (x, y, z , t ) ∂ u z ( x, y, z , t )    5 E ( z ) + + + K ( z ) +    +  ∂ z  2[1 + σ (z )]  ∂z ∂y ∂ y2    12[1 + σ ( z )] 2 ∂ 2 u y (x, y , z , t )  E (z )  ∂ u y (x, y , z , t ) ( ) + K (z ) − + K z  6 [1 + σ ( z )] ∂ y∂ z ∂ x∂ y 

ρ (z )

(4)

2 ∂ 2 u z ( x, y , z , t )  ∂ 2 u z ( x , y , z , t ) ∂ 2 u z ( x, y , z , t ) ∂ 2 u x ( x, y , z , t ) ∂ u y ( x, y , z , t )  = + + +  × ∂t2 ∂ x2 ∂ y2 ∂ x∂ z ∂ y∂ z  

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

 ∂ u x ( x, y, z , t ) ∂ u y (x, y , z , t ) ∂ u x ( x, y , z, t )   ∂ T ( x, y, z, t ) E (z ) ∂  + + + × K (z )   − 2 [1 + σ (z )] ∂ z  ∂x ∂y ∂z ∂z    1 ∂  E ( z )  ∂ u z ( x, y, z, t ) ∂ u x ( x, y, z, t ) ∂ u y ( x, y, z, t ) ∂ u z (x, y, z, t )   × K (z ) β (z ) + − − −  6  . 6 ∂ z 1 + σ (z )  ∂z ∂x ∂y ∂z   Conditions for the displacement vector could be written as r r r r ∂ u (x, L y , z , t ) ∂ u (L x , y , z , t ) ∂ u (0, y, z , t ) ∂ u (x,0, z , t ) = 0; =0; =0; =0; ∂x ∂x ∂y ∂y r r ∂ u (x, y , Lz , t ) r r r r ∂ u (x, y ,0, t ) =0; = 0 ; u (x, y, z ,0 ) = u 0 ; u ( x, y, z , ∞ ) = u 0 . ∂z ∂z

×

Farther let us analyze spatio-temporal distribution of temperature during annealing of dopant. We determine spatio-temporal distribution of temperature as solution of the second law of Fourier [25] c (T )

∂ T ( x, y , z , t ) ∂  ∂ T ( x, y , z , t )  ∂  ∂ T ( x, y , z , t )  ∂  ∂ T ( x, y , z , t )  = λ + λ + λ + ∂t ∂x ∂x ∂y ∂z  ∂y  ∂z   (5) + p (x, y, z , t )

with boundary and initial conditions ∂ T ( x, y , z , t ) ∂ T ( x, y , z , t ) ∂ T ( x, y , z , t ) ∂ T ( x, y , z , t ) = 0, =0, =0, =0, ∂x ∂y ∂x ∂y y =0 x =0 x = Lx x = Ly ∂ T ( x, y , z , t ) ∂ T ( x, y , z , t ) = 0, = 0 , T (x,y,z,0)=fT (x,y,z). ∂z ∂z z =0 x = Lz Here λ is the heat conduction coefficient. Value of the coefficient depends on materials of heterostructure and temperature. Temperature dependence of heat conduction coefficient in most interest area could be approximated by the following function: λ(x,y,z,T)=λass(x,y,z)[1+µ Tdϕ/ Tϕ(x,y,z,t)] (see, for example, [25]). c(T)=cass[1-ϑ exp(-T(x,y,z,t)/Td)] is the heat capacitance; Td is the Debye temperature [25]. The temperature T(x,y,z,t) is approximately equal or larger, than Debye temperature Td for most interesting for us temperature interval. In this situation one can write the following approximate relation: c(T)≈cass. p(x,y,z,t) is the volumetric density of heat, which escapes in the heterostructure. Framework the paper it is attracted an interest microwave annealing of dopant. This approach leads to generation inhomogenous distribution of temperature [12-14,26]. Such distribution of temperature gives us possibility to increase sharpness of p-njunctions with increasing of homogeneity of dopant distribution in enriched area. In this situation it is practicably to choose frequency of electro-magnetic field. After this choosing thickness of scin-layer should be approximately equal to thickness of epitaxial layer. First of all we calculate spatio-temporal distribution of temperature. We calculate spatio-temporal distribution of temperature by using recently introduce approach [15, 17,18]. Framework the approach we transform approximation of thermal diffusivity α ass(x,y,z) =λass(x,y,z)/cass =α0ass[1+εT gT(x,y,z)]. Farther we determine solution of Eqs. (5) as the following power series ∞

i =0

j =0

T ( x, y, z , t ) = ∑ ε Ti ∑ µ j Tij (x, y , z, t ) .

(6) 35


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

Substitution of the series into Eq.(6) gives us possibility to obtain system of equations for the initial-order approximation of temperature T00(x,y,z,t) and corrections for them Tij(x,y,z,t) (i≥1, j≥1). The equations are presented in the Appendix. Substitution of the series (6) in boundary and initial conditions for spatio-temporal distribution of temperature gives us possibility to obtain boundary and initial conditions for the functions Tij(x,y,z,t) (i≥0, j≥0). The conditions are presented in the Appendix. Solutions of the equations for the functions Tij(x,y,z,t) (i≥0, j≥0) have been obtain as the second-order approximation on the parameters ε and µ by standard approaches [28,29] and presented in the Appendix. Recently we obtain, that the second-order approximation is enough good approximation to make qualitative analysis and to obtain some quantitative results (see, for example, [15,17,18]). Analytical results have allowed to identify and to illustrate the main dependence. To check our results obtained we used numerical approaches. Farther let us estimate components of the displacement vector. The components could be obtained by using the same approach as for calculation distribution of temperature. However, relations for components could by calculated in shorter form by using method of averaging of function corrections [12-14,16,27]. It is practicably to transform differential equations of system (4) to integrodifferential form. The integral equations are presented in the Appendix. We determine the firstorder approximations of components of the displacement vector by replacement the required functions uβ(x,y,z,t) on their average values αuβ1. The average values αuβ1 have been calculated by the following relations

αus1=Ms1/4LΘ,

(7)

Θ Lx L y Lz

where M s i = ∫ ∫ ∫ ∫ u s i (x, y, z , t ) d z d y d x d t . The replacement leads to the following results 0 − Lx − L y 0

t z  ∞z ∂ T (x, y, w,τ ) ∂ T ( x, y, w,τ ) u x1 (x, y, z, t ) = t ∫ ∫ K (w)α (w) d w d τ − ∫ (t − τ )∫ K (w) β (w) d wdτ − ∂x ∂x 0 0  00 − α ux1 Φ x 0 (x, y, w, t ) φ + α ux1 ,

]

t z  ∞z ∂ T (x, y, w,τ ) ∂ T (x, y, w,τ ) u y1 (x, y, z, t ) = t ∫ ∫ K (w)α (w) d w d τ − ∫ (t − τ )∫ K (w) β (w) d wdτ − ∂ ∂y y 0 0 0 0  − α uy1 Φ y 0 ( x, y, w, t ) φ + α uy1 ,

]

t z  ∞z ∂ T (x, y, w,τ ) ∂ T ( x, y, w, τ ) u z 1 (x, y, z , t ) = t ∫ ∫ K (w)α (w) d w d τ ∫ (t − τ ) ∫ K (w) β (w) d wdτ − ∂w ∂w 0 0  00 − α uz1 Φ z 0 (x, y, w, t ) φ + α z 1 .

]

Substitution of the above relations into relations (7) gives us possibility to obtain values of the parameters αuβ1. Appropriate relations could be written as

α ux1 = Θ

[

], α

Lz Χ x 0 (∞ ) − Χ x 2 (Θ ) Lz

8 L3 ∫ (Lz − z ) ρ (z ) d z 0

uy1

[

], α

Lz Χ y 0 (∞ ) − Χ y 2 (Θ ) Lz

8 L3 ∫ (Lz − z ) ρ ( z ) d z 0

uz1

=

[

],

Θ Lz Χ z 0 (∞ ) − Χ z 2 (Θ ) Lz

8 L3 ∫ (Lz − z ) ρ ( z ) d z 0

t  ∂ T ( x, y , z,τ )  where Χ s i (Θ ) = ∫ 1 +  ∫ ∫ ∫ (Lz − z ) K ( z ) χ ( z ) d z d y d x dτ . Θ  − Lx − L y 0 ∂s 0 Θ

i

Lx L y Lz

36


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

The second-order approximations of components of the displacement vector by replacement of the required functions uβ(x,y,z,t) on the following sums αus2+us1(x,y,z,t), where αus2=(Mus2-Mus1)/ 4L3Θ. Results of calculations of the second-order approximations us2(x,y,z,t) and their average values αus2 are presented in the Appendix, because the relations are bulky. Spatio-temporal distribution of concentration of dopant we obtain by solving the Eq.(1). To solve the equations we used recently introduce approach [15,17,18] and transform approximations of dopant diffusion coefficients DL (x,y,z,T) and DSL (x,y,z, T) to the following form: DL(x,y,z,T)=D0L[1+ε L gL(x,y,z,T)] and DSL(x,y,z,T)=D0SL[1+ εSLgSL(x,y,z,T)]. We also introduce the following dimensionless parameter: ω = D0SL/ D0L. In this situation the Eq.(1) takes the form  ∂ C ( x, y , z , t ) ∂  C γ ( x, y , z , t )  ∂ C ( x, y , z , t )  ∂  ∂ C ( x, y , z , t ) = D0 L × [1 + ε L g L (z , T )]1 + ξ +  ∂t ∂x P γ ( z, T )  ∂x ∂y   ∂y   C γ ( x, y , z , t )   ∂  C γ ( x, y , z , t )  ( ) × [1 + ε L g L ( z , T )]1 + ξ γ D + D [ 1 + ε g x , y , z , T ] 1 + ξ   0 L 0 L L L   × P (x, y , z , T )   ∂z P γ ( x, y , z , T )     C γ (x, y , z , t )  Lz ∂ C ( x, y , z , t )  ∂  ∫ C (x, y ,W , t ) d W × (8)  + ω Ω D0 L  1 + ε S L g S L ( z, T ) 1 + ξ S ∂z ∂x P γ ( z , T )  0    ∇ µ (x, y, z, t )  C γ (x, y, z, t )  ∇ S µ (x, y, z, t ) Lz ∂  × S ∫ C ( x, y , W , t ) d W  . D0 L + ω Ω D0 L 1 + ξ S  γ ∂ y  kT kT 0 P ( z, T )    ×

[

]

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

i =0

j =0

k =0

C ( x, y, z , t ) = ∑ ε Li ∑ ξ j ∑ ω k Cijk ( x, y, z , t ) .

(9)

Substitution of the series into Eq.(8) gives us possibility to obtain systems of equations for initialorder approximation of concentration of dopant C000(x,y,z,t) and corrections for them Cijk(x, y,z,t) (i ≥1, j ≥1, k ≥1).The equations are presented in the Appendix. Substitution of the series into appropriate boundary and initial conditions gives us possibility to obtain boundary and initial conditions for all functions Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0). Solutions of equations for the functions Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0) have been calculated by standard approaches [28,29] and presented in the Appendix. Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects has been done analytically by using the second-order approximations framework recently introduced power series. The approximation is enough good approximation to make qualitative analysis and to obtain some quantitative results. All obtained analytical results have been checked by comparison with results, calculated by numerical simulation.

3. DISCUSSION In this section we analyzed redistribution of dopant under influence of mechanical stress based on relations calculated in previous section. Fig. 3 show spatio-temporal distribution of concentration of dopant in a homogenous sample (curve 1) and in heterostructure with negative and positive value of the mismatch strain ε0 (curve 2 and 3, respectively). The figure shows, that manufacturing field-effect heterotransistors gives us possibility to make more compact field-effect transistors in direction, which is parallel to interface between layers of heterostructure. As a consequence of 37


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

the increasing of compactness one can obtain decreasing of length of channel of field- effect transistor and increasing of density of the transistors in integrated circuits. It should be noted, that presents of interface between layers of heterostructure give us possibility to manufacture more thin transistors [17].

C(x,Θ), C(y,Θ)

2 1

3

x, y Fig. 3. Distributions of concentration of dopant in directions, which is perpendicular to interface between layers of heterostructure. Curve 1 is a dopant distribution in a homogenous sample. Curve 2 corresponds to negative value of the mismatch strain. Curve 3 corresponds to positive value of the mismatch strain

4. CONCLUSION In this paper we consider possibility to decrease length of channel of field-effect transistor by using mechanical stress in heterostructure. At the same time with decreasing of the length it could be increased density of transistors in integrated circuits.

ACKNOWLEDGEMENTS This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research.

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Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 [10] J. A. Sharp, N. E. B. Cowern, R. P. Webb, K. J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M. A. Foad, F. Cristiano, P. F. Fazzini. "Deactivation of ultrashallow boron implants in preamorphized silicon after nonmelt laser annealing with multiple scans" Appl.Phys. Lett. Vol. 89, 192105 (2006). [11] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N. Drozdov, V.D. Skupov. "Diffusion processes in semiconductor structures during microwave annealing" Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003). [12] E.L. Pankratov. "Redistribution of dopant during microwave annealing of a multilayer structure for production p-n-junction" J. Appl. Phys. Vol. 103 (6). P. 064320-064330 (2008). [13] E.L. Pankratov. "Optimization of near-surficial annealing for decreasing of depth of p-n-junction in semiconductor heterostructure" Proc. of SPIE. Vol. 7521, 75211D (2010). [14] E.L. Pankratov. "Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure by optimized laser annealing" J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010). [15] E.L. Pankratov. "Influence of mechanical stress in semiconductor heterostructure on density of p-njunctions" Applied Nanoscience. Vol. 2 (1). P. 71-89 (2012). [16] E.L. Pankratov, E.A. Bulaeva. "Application of native inhomogeneities to increase compactness of vertical field-effect transistors" J. Comp. Theor. Nanoscience. Vol. 10 (4). P. 888-893 (2013). [17] E.L. Pankratov, E.A. Bulaeva. "An approach to decrease dimensions of field-effect transistors " Universal Journal of Materials Science. Vol. 1 (1). P.6-11 (2013). [18] E.L. Pankratov, E.A. Bulaeva. "Doping of materials during manufacture p-n-junctions and bipolar transistors. Analytical approaches to model technological approaches and ways of optimization of distributions of dopants" Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). [19] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in Russian). [20] E.L. Pankratov. "Decreasing of depth of p-n-junction in a semiconductor heterostructure by serial radiation processing and microwave annealing" J. Comp. Theor. Nanoscience. Vol. 9 (1). P. 41-49 (2012). [21] E.L. Pankratov, E.A. Bulaeva. "Increasing of sharpness of diffusion-junction heterorectifier by using radiation processing" Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012). [22] Y.W. Zhang, A.F. Bower. "Numerical simulations of islands formation in a coherent strained epitaxial thin film system" Journal of the Mechanics and Physics of Solids. Vol. 47 (11). P. 2273-2297 (1999). [23] L.D. Landau, E.M. Lefshits. Theoretical physics. 7 (Theory of elasticity) (Physmatlit, Moscow, 2001, in Russian). [24] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [25] K.V. Shalimova. Physics of semiconductors (Moscow: Energoatomizdat, 1985, in Rus-sian). [26] V.E. Kuz’michev. Laws and formulas of physics. Kiev: Naukova Dumka, 1989 (in Russian). [27] Yu.D. Sokolov. About the definition of dynamic forces in the mine lifting Applied Mechanics. Vol.1 (1). P. 23-35 (1955). [28] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in Russian). [29] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).

APPENDIX Equations for functions Tij(x,y,z,t) (i≥0, j≥0)  ∂ 2T (x, y , z , t ) ∂ 2T00 ( x, y , z, t ) ∂ 2T00 (x, y , z , t )  p ( x, y, z , t ) ∂ T00 (x, y , z, t ) = α 0 ass  00 2 + + + ∂t ∂x ∂ y2 ∂ z2 ν ass  

 ∂ 2T ( x, y , z, t ) ∂ 2Ti 0 (x, y , z , t ) ∂ 2Ti 0 ( x, y, z , t )  ∂ Ti 0 ( x, y , z, t ) = α 0 ass  i 0 2 + + + ∂t ∂x ∂ y2 ∂ z2    ∂ 2T (x, y, z, t ) ∂ 2 Ti −10 (x, y, z , t ) ∂  ∂Ti −10 (x, y, z , t )   + α 0 ass  g T (z ) i −10 2 + gT (x ) +  gT (z )   , i ≥1 2 ∂x ∂y ∂z  ∂z   39


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

 ∂ 2T01 ( x, y , z, t ) ∂ 2T01 ( x, y, z, t ) ∂ 2T01 ( x, y, z , t )  ∂ T01 ( x, y, z , t ) α 0 ass Tdϕ = α 0 ass  + + × + ϕ 2 2 2 ∂t ∂x ∂y ∂z   T00 (x, y , z , t ) 2  ∂ 2 T00 ( x, y, z, t ) ∂ 2 T00 ( x, y, z, t ) ∂ 2T00 ( x, y , z, t )  α 0 ass Tdϕ  ∂ T00 ( x, y, z , t )  × + + −    + ϕ ∂ x2 ∂ y2 ∂ z2 ∂x    T00 ( x, y , z, t ) 

2

 ∂ T (x, y , z , t )   ∂ T00 (x, y , z, t )  +  00  +  ∂y ∂z    

2

  

 ∂ 2T (x, y, z, t ) ∂ 2T02 (x, y, z, t ) ∂ 2T02 (x, y, z, t )  ∂ T02 (x, y, z, t ) α 0 assTdϕ = α 0ass  02 2 + + + ×  ϕ ∂t ∂x ∂ y2 ∂ z2   T00 (x, y, z, t )  ∂ 2T ( x, y, z, t ) ∂ 2T01 ( x, y , z, t ) ∂ 2T01 (x, y , z, t )  ϕ α 0 assTdϕ  ∂ T00 ( x, y, z, t ) ×  01 2 + + − ×   ϕ +1 ∂x ∂ y2 ∂ z2 ∂x   T00 (x, y , z, t )  ∂ T ( x, y, z, t ) ∂ T00 ( x, y , z, t ) ∂ T01 ( x, y, z , t ) ∂ T00 ( x, y, z, t ) ∂ T01 ( x, y, z , t )  × 01 + +  ∂x ∂y ∂y ∂z ∂z 

∂ 2 T00 (x, y, z, t ) ∂ T11 (x, y, z, t ) ∂ 2T11 (x, y, z, t ) T01 (x, y, z, t )  = α 0 ass + g ( x , y , z , T ) + g T (x, y, z, T ) ×  T ∂t ∂ x2 T00 (x, y, z, t )  ∂ x2 ×

 ∂  ∂ 2 T00 (x, y, z , t ) ∂  ∂ T00 (x, y, z , t )   ∂ T01 (x, y, z, t )  +  g T ( x, y , z , T )  α 0 ass +   g T (x, y, z , T ) + 2 ∂y ∂z  ∂z ∂x  ∂ x    

+

∂ 2 T01 (x, y, z , t ) ∂ 2 T01 (x, y, z , t ) ∂ 2 T01 (x, y, z , t )  [ ( ) ] [ ( ) ] 1 g x , y , z , T 1 g x , y , z , T + + + +  α 0 ass + T T ∂ x2 ∂ y2 ∂ z2 

 T ( x, y, z, t ) ∂ 2T00 ( x, y, z, t ) T10 ( x, y, z , t ) ∂ 2T00 ( x, y, z , t ) T10 (x, y , z , t ) ∂ 2T00 (x, y , z, t )  +  ϕ10+1 + ϕ +1 + ϕ +1 × T00 (x, y , z, t ) T00 ( x, y, z, t ) ∂ x2 ∂ y2 ∂ z2  T00 (x, y , z , t )  2 2 2 ϕ ϕ   ( ) ( ) ( ) α T ∂ T x , y , z , t ∂ T x , y , z , t ∂ T x , y , z , t α T 10 10 0 ass d × α 0 assTdϕ + ϕ 0 ass d  10 2 + + × + ϕ ( T00 ( x, y , z, t )  ∂x ∂ y2 ∂ z2 T x , y, z, t )  00  ∂ 2T ( x , y , z , t ) ∂ 2T (x, y , z , t ) ∂  ∂ T00 ( x, y, z , t )   ×  g T ( x, y, z, T ) 10 2 + g T ( x, y, z , T ) 10 2 +  g T ( x, y , z , T )  − ∂x ∂y ∂z  ∂z     ∂ T ( x, y , z, t ) ∂ T00 ( x, y , z, t ) ∂ T10 (x, y , z , t ) ∂ T00 (x, y , z , t ) ∂ T10 ( x, y, z, t ) ∂ T00 ( x, y, z, t )  −  10 + + × ∂x ∂x ∂y ∂y ∂z ∂z   2 2 2 g T ( x, y, z , T )  ∂ T00 ( x, y, z , t )   ∂ T00 (x, y, z , t )   ∂ T00 ( x, y, z , t )   × ϕ +1 −   +  +  × ∂x ∂y ∂z T00 (x, y, z , t ) T00ϕ +1 ( x, y, z , t )        

ϕ α 0 ass Tdϕ

× α 0 assϕ Tdϕ . Conditions for the functions Tij(x,y,z,t) (i≥0, j≥0) ∂ Tij (x, y, z , t ) ∂ Tij (x, y, z, t ) ∂ Tij ( x, y, z, t ) ∂ Tij (x, y , z , t ) = 0, =0, =0, = 0, ∂x ∂x ∂y ∂y x =0 x=L y =0 x= L x

∂ Tij (x, y , z, t ) ∂z

= 0, z =0

∂ Tij ( x, y, z , t ) ∂z

y

= 0 , T00(x,y,z,0)=fT(x,y,z), Tij(x,y,z,0)=0, i ≥1, j ≥1. x = Lz

Solutions of the equations for the functions Tij(x,y,z,t) (i≥0, j≥0) with account appropriate boundary and initial conditions could be written as 40


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

T00 (x, y , z, t ) = Ly

Lz

0

0

Lx ∞ 1 Lx L y Lz 2 ∑ cn ( x )cn ( y )cn (z )enT (t ) ∫ cn (u ) × ∫ ∫ ∫ f T (u , v, w) d w d v d u + Lx L y Lz 0 0 0 Lx L y Lz n =1 0

× ∫ c n (v ) ∫ c n (w) f T (u , v, w) d w d v d u +

t L x L y L z p (u , v , w, τ ) 1 2 × d wd vd u dτ + ∫∫∫∫ ν ass Lx L y Lz 0 0 0 0 Lx L y Lz

t

Lx

Ly

Lz

n =1

0

0

0

0

× ∑ c n (x ) c n ( y ) c n (z ) enT (t )∫ enT (− τ ) ∫ c n (u ) ∫ c n (v ) ∫ c n (w)

p (u, v, w,τ )

ν ass

d wd v d u dτ ,

  1 1 1  where cn(χ) = cos (π nχ/L), enT (t ) = exp − π 2 n 2α 0 ass t  2 + 2 + 2  ;    Lx L y Lz  

Ti 0 ( x, y, z, t ) = 2 ×

π α 0 ass 2 x

L L y Lz

t

Lx

Ly

0

0

0

Lz

∑ n c n ( x ) c n ( y ) c n (z ) e nT (t )∫ e nT (− τ ) ∫ s n (u ) ∫ c n (v ) ∫ c n (w) g T (u, v, w, T ) ×

n =1

0

Ly Lx π α 0 ass ∂ Ti −10 (u , v, w,τ ) d wd vd u dτ + 2 ∑ n cn ( x )cn ( y ) cn (z )enT (t )∫ enT (− τ ) ∫ cn (u ) ∫ sn (v ) × 2 ∂u Lx L y Lz n =1 0 0 0 t

Lz

× ∫ g T (u , v, w, T )cn (w ) 0

∂ Ti −10 (u , v, w,τ ) π α 0 ass ∞ d wd v d u dτ + 2 ∑ n cn (x )cn ( y )cn ( z ) enT (t ) × ∂v Lx L y L2z n =1

t

Lx

Ly

Lz

0

0

0

0

× ∫ e nT (− τ ) ∫ c n (u ) ∫ c n (v ) ∫ s n (w) g T (u , v, w, T )

∂ Ti −10 (u , v, w,τ ) d w d v d u d τ , i ≥1, ∂w

where sn(χ) = sin (π n χ/L); Ly Lx L z ∂ 2 T (u , v, w, τ ) t 2π T ϕ ∞ 00 T01 ( x, y, z, t ) = α 0 ass 2 d ∑ n c n ( x ) c n ( y ) c n ( z ) e nT (t )∫ e nT (− τ ) ∫ s n (u ) ∫ c n (v ) ∫ × L x L y L z n =1 ∂u2 0 0 0 0 × cn (w) ×

Ly Lx Lz t 2π Tdϕ ∞ d wd vd u dτ ( ) ( ) ( ) ( ) ( ) ( ) + α n c x c y c z e t e − τ c u ∑ ∫ ∫ ∫ sn (v ) ∫ cn (w) × n n n nT nT n 0 ass ϕ 2 T00 (u , v, w,τ ) Lx L y Lz n =1 0 0 0 0

Ly Lx t ∂ 2T00 (u , v, w,τ ) d w d v d u d τ 2π Tdϕ ∞ ( ) ( ) ( ) ( ) ( ) ( ) + α n c x c y c z e t e − τ c u ∑ ∫ ∫ ∫ cn (v ) × 0 ass n n n nT nT n ∂ v2 T00ϕ (u , v, w,τ ) Lx L y L2z n =1 0 0 0 Lz

× ∫ sn (w) 0

Lx t ϕ α 0 ass ∞ ∂ 2T00 (u , v, w,τ ) d w d v d u d τ − 2 Tdϕ ∑ cn (x )cn ( y )cn ( z )enT (t )∫ enT (− τ ) ∫ cn (u ) × ϕ 2 ∂u T00 (u , v, w,τ ) Lx L y Lz n =1 0 0

2

t  ∂ T (u , v, w,τ )  d w d v d u d τ α Tϕ ∞ × ∫ cn (v ) ∫ cn (w)  00 − 2 ϕ 0 ass d ∑ cn (x ) cn ( y )cn ( z )enT (t ) ∫ enT (− τ ) ×  ϕ +1 ∂u Lx L y Lz n =1 0 0 0   T00 (u , v, w,τ ) Ly

Lz

2

Ly Lx Lz  ∂ T (u , v, w,τ )  d w d v d u d τ ϕ α 0 ass ∞ × ∫ cn (u ) ∫ cn (v ) ∫ cn (w)  00 − 2 Tdϕ ∑ cn ( x )cn ( y )cn (z )enT (t ) ×  ϕ +1 ∂v Lx L y Lz n =1 0 0 0   T00 (u , v, w,τ ) 2

Ly Lx Lz t  ∂ T (u , v, w, τ )  d w d v d u d τ × ∫ e nT (− τ ) ∫ c n (u ) ∫ c n (v ) ∫ c n (w)  00 ;  ϕ +1 ∂w 0 0 0 0   T00 (u , v, w, τ ) Ly Lx L z ∂ 2 T (u , v, w, τ ) t 2π T ϕ ∞ 01 T02 (x, y , z, t ) = α 0 ass 2 d ∑ n c n ( x ) c n ( y ) c n ( z ) e nT (t )∫ e nT (− τ ) ∫ s n (u ) ∫ c n (v ) ∫ × ∂u2 L x L y L z n =1 0 0 0 0

× cn (w) ×

Ly Lx Lz t 2π Tdϕ ∞ d wd v d u dτ ( ) ( ) ( ) ( ) ( ) ( ) + α n c x c y c z e t e − τ c u ∑ ∫ nT ∫ n ∫ sn (v ) ∫ cn (w) × 0 ass n n n nT ϕ 2 T00 (u , v, w,τ ) Lx L y Lz n =1 0 0 0 0

Lx t ∂ 2T01 (u , v, w,τ ) d w d v d u d τ 2π Tdϕ ∞ + α n cn (x ) cn ( y )cn ( z ) enT (t )∫ enT (− τ ) ∫ cn (u ) × 0 ass ϕ 2 2 ∑ ∂v T00 (u , v, w,τ ) Lx L y Lz n =1 0 0

41


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

Lz

0

0

× ∫ cn (v ) ∫ sn (w)

t ∞ πα ∂ 2T01 (u , v, w,τ ) d w d v d u d τ − 2 2 0 ass ∑ cn ( x )cn ( y ) cn (z )enT (t )∫ enT (− τ ) × 2 ϕ ∂w T00 (u , v, w,τ ) Lx L y Lz n =1 0

Lx

Ly

Lz

0

0

0

× Tdϕ ∫ cn (u ) ∫ cn (v ) ∫ cn (w )

∂ T00 (u , v, w,τ ) ∂ T01 (u , v, w,τ ) d w d v d u d τ π α 0 ass ∞ − 2Tdϕ ∑ cn ( x ) × ϕ +1 ∂u ∂u T00 (u , v, w,τ ) Lx L2y Lz n =1

t

Lx

Ly

Lz

0

0

0

0

× cn ( y )cn ( z )enT (t )∫ enT (− τ ) ∫ cn (u ) ∫ cn (v ) ∫ cn (w) − 2π

∂ T00 (u , v, w,τ ) ∂ T01 (u , v, w,τ ) d w d v d u d τ − ∂v ∂v T00ϕ +1 (u , v, w,τ )

Ly Lx Lz Tdϕ α 0 ass ∞ t ∂ T00 (u , v, w,τ ) ∂ T01 (u , v, w,τ ) d w d v d u d τ ( ) ( ) τ e − c u × ∑ ∫ ∫ ∫ cn (v ) ∫ cn (w) nT n 2 Lx L y Lz n =1 0 T00ϕ +1 (u , v, w,τ ) ∂w ∂w 0 0 0

× c n ( x ) c n ( y ) c n (z ) e nT (t ) ; T11 (x, y , z , t ) =

Ly Lx Lz t  ∂ 2 T00 (u , v, w,τ ) 2 α 0 ass ∞ × ∑ c n (x ) c n ( y ) c n (z ) e nT (t )∫ e nT (− τ ) ∫ c n (u ) ∫ c n (v ) ∫ c n (w) L x L y L z n =1 ∂ w2 0 0 0 0 

× g T (u , v, w, T ) + g T (u , v, w, T )

×

∂ 2 T00 (u , v, w,τ ) ∂  ∂ T00 (u , v, w,τ )   T01 (u , v, w,τ ) + d wd vd u dτ +  g T (w)  2 ∂v ∂w ∂w   T00 (u , v, w,τ )

∂ 2T01 (u, v, w,τ ) ∂ 2T (u, v, w,τ ) ∂  ∂ T01 (u, v, w,τ )  + [1 + gT (u, v, w, T )] 01 2 + gT (u, v, w, T )  d w d v d u d τ + 2 ∂u ∂v ∂w ∂w 

Ly Lx Lz  t ∞ T (u , v, w, τ ) ∂ 2 T00 (u , v, w,τ ) + ∑ c n ( x ) c n ( y ) c n ( z ) e nT (t )∫ e nT (− τ ) ∫ c n (u ) ∫ c n (v ) ∫  ϕ10+1 L x L y L z n =1 ∂u2 0 0 0 0  T00 (u , v, w, τ ) T (u , v, w,τ ) ∂ 2T00 (u , v, w,τ ) T10 (u , v, w,τ ) ∂ 2 T00 (u , v, w, τ )  α 0 ass × ϕ10+1 + ϕ +1 ×  c n (w) d w d v d u d τ + 2 2 T00 (u , v, w,τ ) ∂v T00 (u , v, w,τ ) ∂w Lx L y 

+2

×2 ×

α 0 ass Tdϕ

Ly Lx Lz  ∂ 2 T10 (u, v, w,τ ) ∂ 2T10 (u , v, w,τ ) ∂ 2T10 (u , v, w,τ )  Tdϕ ∞ t ( ) ( ) ( ) ( ) e c u c v c w + + − τ ∑ ∫ nT ∫ n ∫ n ∫ n  × L z n =1 0 ∂u2 ∂ v2 ∂ w2 0 0 0  

Ly Lx t α Tϕ ∞ d wd v d u dτ cn ( x )cn ( y )cn (z )enT (t ) + 2 0 ass d 2 ∑ cn ( x )cn ( y )cn (z )enT (t )∫ enT (− τ ) ∫ cn (u ) ∫ cn (v ) × ϕ T00 (u, v, w,τ ) Lx L y Lz n =1 0 0 0

Lz  ∂ 2 T00 (u , v, w, τ ) ∂ T00 (u , v, w, τ )  ∂  × ∫ c n (w)  g T (u , v, w, T ) +  g T (u , v, w, T )  + g T (u , v, w, T ) × 2 ∂u ∂w ∂w 0   2 ϕ Ly Lx t ∂ T00 (u , v, w,τ )  d w d v d u d τ α T ∞ × − 2ϕ 0 ass d ∑ cn ( x )cn ( y )cn ( z ) enT (t )∫ enT (− τ ) ∫ cn (u ) ∫ cn (v ) ×  ϕ 2 ∂v Lx L y Lz n =1 0 0 0  T00 (u , v, w,τ ) Lz  ∂ T (u , v, w,τ )  2  ∂ T (u , v, w,τ )  2  ∂ T (u , v, w,τ )  2  d w d v d u d τ 00 00 × ∫ cn (w)  00 .  +  +   ϕ +1 u v ∂ ∂ ∂w 0        T00 (u , v, w,τ ) Integro-differential form of equations of systems (4) could be written as z  ∂ 2 u ( x, y , w,τ )  t ∂ 2u y ( x, y, w,τ ) ∂ 2u z (x, y , w,τ )  x u x (x, y , z, t ) = u x (x, y , z, t ) + φ ∫ (t − τ )∫ 5 − − × ∂ x2 ∂ x∂ y ∂ x∂ w 0 0  

×

2 E (w) d w d τ t ∞ z  ∂ 2 u x (x, y , w,τ ) ∂ u y (x, y , w,τ ) ∂ 2 u z (x, y , w,τ )  E (w) d w − ∫ ∫ 5 − − dτ −  6 [1 + σ (w)] 6 0 0  ∂ x2 ∂ x∂ y ∂ x∂w  1 + σ (w)

∞z  ∂ 2 u x ( x, y, w,τ ) ∂ 2 u y ( x, y, w,τ ) ∂ 2 u z ( x, y, w,τ )  1 t z  ∂ 2 u x (x, y , w,τ ) − t ∫ ∫ K (w)  + + +  d wdτ + ∫ ∫  2 ∂x ∂ x∂ y ∂ x∂w 2 00 ∂ x∂ y 00  

42


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

+ ×

z  ∂ 2 u ( x , y , w, τ ) ∂ 2 u y ( x, y, w,τ )  E (w) d w ∂ 2 u y ( x, y, w,τ )  1t x ( ) ( ) t − d + t − + τ τ τ   × ∫ ∫ ∂ y2 ∂ x∂ w ∂ w2 20 0  1 + σ (w)  

2 ∞ z  ∂ 2 u ( x, y , w,τ ) E (w) d w t ∞ z  ∂ 2u x ( x, y, w,τ ) ∂ u y (x, y , w,τ )  E (w) d w x τ dτ − ∫ ∫  + d − + ∫∫  1 + σ (w) 2 0 0  ∂ x∂ y ∂ y2 ∂ x∂ w 00  1 + σ (w)

z ∞z ∂ 2 u y (x, y , w,τ )  E (w) d w ∂ T ( x, y , w,τ ) t t + d τ − ∫ (t − τ )∫ β (w) K (w) d w d τ + t ∫ ∫ K (w) ×  2 ∂w ∂x 2 0 0 00  1 + σ (w) 2 t z ∂ u x ( x, y, w,τ ) ∂ T (x, y , w,τ ) × β (w) d w d τ + ∫ (t − τ )∫ ρ (w ) d w d τ − Φ x 1 ( x, y , z , t ) − ∂x ∂τ 2 0 0

∞z

− t ∫ ∫ ρ (w) 00

 ∂ 2 u x (x, y , w,τ ) d wdτ , 2 ∂τ 

2  1 t z  ∂ u ( x, y, w, τ ) ∂ 2 u x (x, y, w, τ )  E (w) d w t y u y (x, y, z, t ) = u y ( x, y, z, t ) + φ  ∫ (t − τ )∫  + dτ − ×  2 ∂ x∂ y 2 ∂x 0  2 0  1 + σ (w)  2 2 ∞ z  ∂ u ( x , y , w,τ ) ∂ 2 u x ( x, y , w,τ )  E (w) d w 1 t z  ∂ u y (x, y , w,τ ) ∂ 2 u x (x, y , w,τ ) y × ∫∫ + d τ + ∫ ∫ 5 − −  2 ∂x ∂ x∂ y 6 0 0  ∂ y2 ∂ x∂ y 00  1 + σ (w )  2 ∞ z  ∂ u ( x, y, w, τ ) ∂ 2 u x (x, y, w, τ ) ∂ 2 u z (x, y, w, τ )  ∂ 2 u z (x, y, w, τ )  E (w) d w y t d 5 − ( − ) − − − τ τ × ∫∫  2 ∂ y∂w ∂ x∂ y ∂ y∂ w 00 ∂y  1 + σ (w)   t z ∞ z E (w) d w t ∂ T ( x, y, w,τ ) ∂ T (x, y, w,τ ) × d τ − ∫ (t − τ )∫ K (w) β (w) d w d τ + t ∫ ∫ β (w)K (w) d wdτ + 1 + σ (w) 6 0 ∂y ∂y 0 00 t

z

 ∂ 2 u y (x, y , w,τ )

0

0



+ ∫ (t − τ )∫ K (w) 

∂ x∂ y

∂ 2 u y ( x, y, w,τ )

+

∂ y2

∞z ∂ 2 u y ( x, y, w,τ )  +  d w d τ − t ∫ ∫ K (w) × ∂ y∂ w 00 

t  ∂ 2 u y (x, y , w,τ ) ∂ 2u y (x, y , w,τ ) ∂ 2u y (x, y , w,τ )   ∂ u y (x, y , z,τ ) × + + +  d w d τ + ∫ (t − τ )  2 ∂ x∂ y ∂y ∂ y∂w ∂z 0   

+ ∞z

∂ u z ( x , y , z ,τ )  E (z ) t E (z ) ∞  ∂ u y ( x, y , z ,τ ) ∂ u z (x, y , z ,τ )  − + ∫  dτ dτ + ∂y 2 [1 + σ ( z )] 2 [1 + σ (z )] 0  ∂z ∂y  

+ ∫ ∫ ρ (w) 00

∂ 2 u y (x, y, w, τ ) ∂τ

2

t

z

d w d τ − ∫ (t − τ )∫ ρ (w) 0

0

∂ 2 u y (x, y, w, τ ) ∂τ

2

 d w d τ − Φ y 1 (x, y, z , t ) , 

z  ∂ 2 u ( x, y, w, τ )  1 t ∂ 2 u x (x, y, w, τ )  E (w) d w t z u z (x, y, z, t ) = u z (x, y, z, t ) + φ  ∫ (t − τ )∫  + dτ − ×  2 ( ) x w w 0 ∂ ∂ 1 + 2 σ ∂ x  2 0   2 2 2 ∞ z  ∂ u ( x, y , w,τ ) t z  ∂ u ( x, y , w,τ ) ∂ 2 u y (x, y , w,τ )  ∂ u x (x, y , w,τ )  E (w) d w z z τ × ∫∫ + d + + ∫ ∫  ×  ∂ x2 ∂ x∂ w ∂ y2 ∂ y∂ w 00 00   1 + σ (w)  2 t z E (w) d w (t − τ ) t ∞ z  ∂ 2 u z ( x, y , w,τ ) ∂ u y ( x, y , w,τ )  E (w) d w × dτ − ∫ ∫  + d τ − ∫ (t − τ )∫ β (w) ×  2 1 + σ (w) 2 2 0 0  ∂y ∂ y∂w 0 0  1 + σ (w)

× K (w) −

∞z t  ∂ u ( x, y , z ,τ ) ∂ T (x, y , w,τ ) ∂ T (x, y , w,τ ) d w d τ + t ∫ ∫ K (w) β (w) d w d τ + ∫ (t − τ ) 5 z − ∂w ∂w ∂z 00 0 

∂ u x (x, y , z,τ ) ∂ u y (x, y , z,τ )  E (z ) E ( z ) ∞  ∂ u z ( x, y, z,τ ) ∂ u x ( x, y, z ,τ ) − − − − ∫ 5  dτ ∂x ∂y 6 [1 + σ (z )] 1 + σ ( z ) 0  ∂z ∂x  43


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

t ∂ u y (x, y , z,τ )   ∂ u x ( x, y, z ,τ ) ∂ u y ( x, y , z,τ ) ∂ u z ( x, y, z ,τ )  t + +  d τ + K (z )∫ (t − τ )   dτ − ∂y 6 ∂x ∂y ∂z 0    ∞  ∂ u ( x, y , z ,τ )  ∂ u y ( x, y, z ,τ ) ∂ u z ( x, y, z ,τ )  − t K ( z )∫  x + +  d τ − Φ z 1 ( x, y , z , t ) , ∂x ∂y ∂z 0  

[

z

]

where Φ s i (x, y, z, t ) = ∫ ρ (w) u s (x, y, w, t ) d w , s =x,y,z, φ =L/Θ2E0, E0 is the average value of Young i

0

modulus. The second-order approximations of components of displacement vector could be written as z  ∂ 2 u ( x, y , w,τ )  1 t ∂ 2u y1 ( x, y, w,τ ) x1 u x 2 ( x, y, z, t ) = α ux 2 + u x1 ( x, y , z , t ) + φ 1  ∫ (t − τ ) ∫ 5 − − ∂ x2 ∂ x∂ y 0  6 0  ∞ z  ∂ 2 u ( x, y , w, τ ) ∂ 2 u y1 (x, y , w, τ ) ∂ 2 u z1 ( x, y, w, τ )  ∂ 2 u z1 ( x, y, w, τ )  E (w) d w x1 τ − d − 5 − − × ∫∫  ∂ x∂ w ∂ x2 ∂ x∂ y ∂ x∂ w 00  1 + σ (w)  

×

z  ∂ 2u x1 (x, y , w,τ ) ∂ 2u y1 ( x, y, w,τ ) ∂ 2 u z1 (x, y , w,τ )  E (w) d w t t d τ + ∫ (t − τ )∫ K (w)  + +  d wd τ − 1 + σ (w) 6 0 ∂ x2 ∂ x∂ y ∂ x∂ w 0  

∞z t z  ∂ 2 u ( x, y , w,τ )  ∂ 2 u x1 ( x, y , w,τ ) ∂ 2 u y1 (x, y , w,τ ) ∂ 2 u z1 ( x, y, w,τ )  x1 τ − t ∫ ∫ K (w)  + + d w d + + ∫ ∫  ∂ x2 ∂ x∂ y ∂ x∂ w ∂ x∂ y 00 00   2 ∂ 2u y1 ( x, y, w,τ )  E (w) d w (t − τ ) 1 t z  ∂ 2 u x1 ( x, y, w,τ ) ∂ u y1 ( x, y, w,τ )  E (w) d w + dτ + ∫ ∫  + ×   ∂ y2 2 0 0  ∂ x∂ y ∂ y2  1 + σ (w) 2  1 + σ (w)

z  ∂ 2 u ( x, y , w,τ ) ∂ 2 u y1 ( x, y , w,τ )  E (w) d w 1t t ∞ z E (w) x1 ( ) t − + dτ − ∫ ∫ × τ ∫ ∫  2 20 ∂ x∂w ∂w 2 0 0 1 + σ (w) 0   1 + σ (w)

× (t − τ ) d τ +

2  ∂ 2u x1 ( x, y, w,τ ) ∂ 2 u y1 ( x, y, w,τ )  t ∞ z  ∂ 2u x1 (x, y , w,τ ) ∂ u y1 (x, y , w,τ )  τ d w d × + − + ∫ ∫    × 2 0 0  ∂ x∂ y ∂ y2 ∂ x∂ w ∂ w2    ∞z t z E (w) d w ∂ T ( x, y , w,τ ) ∂ T (x, y , w,τ ) × d τ + t ∫ ∫ χ (w) K (w) d w d τ − ∫ ∫ K (w) χ (w) d w (t − 1 + σ (w) ∂x ∂x 00 00 t

z

0

0

− τ ) d τ − ∫ (t − τ )∫ ρ (w)

∞z ∂ 2 u x1 (x, y , w,τ ) ∂ 2 u x1 ( x, y, w,τ ) ( ) τ ρ d w d + t w d w dτ − ∫ ∫ ∂τ 2 ∂τ 2 00 − α ux 2 Φ x 0 ( x, y , z , t ) − Φ x 1 (x, y , z , t )},

 t t − τ z  ∂ 2 u y1 (x, y, w, τ ) ∂ 2 u x1 (x, y, w, τ )  E (w) d w u y 2 ( x, y, z , t ) = α uy 2 + u y1 ( x, y , z , t ) + ∫ + dτ −  ∫ ∂ x2 ∂ x∂ y 0 2 0   1 + σ (w) 2 2 t z  ∂ u ( x, y , w,τ ) ∂ 2u x1 (x, y , w,τ ) t ∞ z  ∂ u y1 ( x, y , w,τ ) ∂ 2u x1 ( x, y, w,τ )  E (w) d w y1 τ − ∫∫ + d + 5 − − ∫∫  2 2 2 0 0  ∂x ∂ x∂ y ∂y ∂ x∂ y 00  1 + σ (w)  2 ∞ z  ∂ u ( x, y , w,τ ) ∂ 2 u z1 (x, y , w,τ )  E (w ) d w t − τ ∂ 2 u x1 ( x, y , w,τ ) ∂ 2 u z1 (x, y , w,τ )  y1 d − 5 − − τ ∫∫ ×  ∂ y∂ w ∂ y2 ∂ x∂ y ∂ y∂w 00  1 + σ (w) 6   z ∞z E (w ) d w t t ∂ T (x, y, w,τ ) ∂ T (x, y, w,τ ) × d τ − ∫ (t − τ )∫ χ (w) K (w) d w d τ + t ∫ ∫ K (w ) χ (w ) d wdτ − 1 + σ (w ) 6 0 ∂y ∂y 0 00

44


Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, 2015 t z ∞z  ∂ 2 u y1 ( x, y , w, τ ) ∂ 2 u y1 ( x, y, w, τ ) ∂ 2 u y1 ( x, y, w,τ )  − ∫ (t − τ )∫ K (w)  + +  d w d τ − t ∫ ∫ K (w) × 2 ∂ x∂ y ∂y ∂ y∂w 0 0 00  

 ∂ 2 u y1 (x, y , w,τ ) ∂ 2 u y1 (x, y , w,τ ) ∂ 2 u y1 ( x, y , w, τ )   ∂ u y1 ( x , y , z , τ ) 1t × + + +  d w d τ + ∫ (t − τ )  2 ∂ x∂ y ∂y ∂ y∂w 20 ∂z    ∂ u z1 ( x, y , z , τ )  E (z ) t E ( z ) ∞  ∂ u y1 ( x , y , z , τ ) ∂ u z 1 ( x , y , z , τ )  + − + ∫  d τ − α uy 2 ×  dτ ∂y ∂z ∂y 1 + σ ( z ) 2 [1 + σ ( z )] 0    × Φ y 0 (x, y , z , t ) − Φ y 1 ( x, y, z, t )} φ 1 ,  1 t z  ∂ 2 u z1 ( x, y, w,τ ) ∂ 2 u x1 (x, y , w,τ )  E (w) d w u z 2 ( x, y , z , t ) = α z 2 + u z1 ( x, y , z , t ) + φ 1  ∫ ∫  + ×  ∂ x2 ∂ x∂w  2 0 0   1 + σ (w) z  ∂ 2 u ( x, y , w,τ ) 1t t ∞ z  ∂ 2 u z1 ( x, y , w,τ ) ∂ 2 u x1 (x, y , w,τ )  E (w) d w z1 ( ) × (t − τ ) d τ − ∫ ∫  + d + t − + τ τ ∫ ∫   2 00 20 ∂ x2 ∂ x∂ w ∂ y2 0  1 + σ (w) +

t z ∞ z  ∂ 2 u ( x, y , w,τ ) ∂ 2 u y1 (x, y, w,τ )  E (w) d w ∂ T ( x, y, w,τ ) z1 τ τ χ τ d − t − K w w d w d − + ( ) ( ) ( )  ∫ ∫ ∫∫ ( ) σ ∂ y∂w 1 + w ∂ w ∂ y2 0 0 0 0  

+

∞z ∂ 2 u y1 ( x, y , w,τ )  E (w) d w ∂ T (x, y, w,τ ) E (z ) t t d τ + t ∫ ∫ K (w) χ (w) d wdτ +  ∫ (t − τ ) × ∂ y∂w 2 ∂w 6 [1 + σ (z )] 0 00  1 + σ (w)

 ∂ u (x, y , z,τ ) ∂ u x1 (x, y , z ,τ ) ∂ u y1 ( x, y, z,τ )  t ∞  ∂ u z1 ( x, y, z ,τ ) ∂ u x1 ( x, y , z,τ ) × 5 z1 − − − −  d τ − ∫ 5 ∂z ∂x ∂y 6 0 ∂z ∂x   t ∂ u y1 ( x, y, z ,τ )   ∂ u ( x, y, z,τ ) ∂ u y1 ( x, y, z ,τ ) ∂ u z1 ( x, y , z,τ )  t E (z ) − + ∫ (t − τ )  x1 + +  dτ  dτ × ∂y ∂x 6 [1 + σ ( z )] 0 ∂y ∂z    ∞  ∂ u ( x, y , z , τ ) ∂ u y 1 ( x , y , z , τ ) ∂ u z 1 ( x, y , z , τ )  × K ( z ) − t K ( z ) ∫  x1 + +  d τ − α z 2 Φ z 0 ( x, y , z , t ) − ∂x ∂y ∂z 0   − Φ z 1 (x, y , z, t )} .

Calculation of the average values αuβ2 leads to the following results 2 Lx L y Lz   1 Θ ∂ 2 u x1 ( x , y , z , t ) ∂ u y1 ( x , y , z , t ) ∂ 2 u z 1 ( x , y , z , t )  E ( z ) d z 2 (L z − α ux 2 = Lz  ∫ (Θ − t ) ∫ ∫ ∫ 5 − −  ∂ x2 ∂ x∂ y ∂ x∂ z 0 0 0  12 0  1 + σ (z )   ∂ 2u x1 (x, y , z, t ) ∂ 2u y1 ( x, y, z , t ) ∂ 2u z1 ( x, y , z , t )  (Lz − z ) Θ 2 ∞ Lx L y Lz ) ( ) − z d yd xd t − − − × ∫ ∫ ∫ ∫ E z 5  12 0 0 0 0 ∂ x2 ∂ x∂ y ∂ x∂ z   1 + σ ( z )

×d z d y d x d t +

Lx L y Lz  ∂ 2 u x1 ( x , y , z , t ) ∂ 2 u y 1 ( x , y , z , t ) ∂ 2 u z 1 ( x , y , z , t )  1Θ 2 ( ) ( ) Θ − t L − z + + ∫ ∫∫ ∫ z  × 20 ∂ x2 ∂ x∂ y ∂ x∂ z 0 0 0  

× K (z ) d z d y d x d t −

 ∂ 2u x1 (x, y , z, t ) ∂ 2u y1 (x, y , z, t ) ∂ 2u z1 ( x, y, z , t )  Θ 2 ∞ Lx L y Lz + + ∫ ∫ ∫ ∫ ( Lz − z )  × 2 000 0 ∂ x2 ∂ x∂ y ∂ x∂ z  

Lx L y Lz  ∂ 2 u x1 (x, y , z , t ) ∂ 2 u y1 (x, y , z , t )  Lz − z 1Θ 2 × K ( z ) d z d y d x d t + ∫ (Θ − t ) ∫ ∫ ∫ E ( z )  + d zd y ×  20 ∂ x∂ y ∂ y2 0 0 0   1 + σ (z )

45


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

×d x d t +

Lx L y Lz  ∂ 2 u x1 ( x , y , z , t ) ∂ 2 u y 1 ( x , y , z , t )  E ( z ) d z Θ2 1Θ 2 + d y d x d t − × ∫ (Θ − t ) ∫ ∫ ∫ (Lz − z )   ∂ x∂ z ∂ z2 20 4 0 0 0   1 + σ (z )

(Lz − z )  ∂ 2u x1 (x, y, z , t ) + ∂ 2u y1 (x, y, z , t )  d z d y d x d t + 1 Θ (Θ − t )2 L L L Lz − z × ∫ ∫∫ ∫   1 + σ (z )  ∂ x∂ y ∂ y2 40 0 0 0 0 0 0 0 1 + σ (z )  ∞L L L  ∂ 2 u x1 ( x , y , z , t ) ∂ 2 u y1 ( x , y , z , t )  ρ ( z ) ∂ 2 u x1 ( x , y , z , t ) ( ) ( ) × + + − × E z d z d y d x d t L z ∫ ∫ ∫ ∫  z 2 2

∞ Lx L y Lz

x y

× ∫ ∫ ∫ ∫ E (z )

x y

∂ x∂ z



∂z



Θ

Lx L y Lz

0

0 0 0

z

z

∂t

2

0 0 0 0

× Θ 2 d z d y d x d t − ∫ (Θ − t ) ∫ ∫ ∫ (Lz − z ) ρ (z ) u x1 (x, y , z, t ) d z d y d x d t +

Θ 2 ∞ Lx L y Lz ∫ ∫ ∫ ∫ ( Lz − z ) × 2 000 0

Lz   ∂ u x1 (x, y, z , t ) 1 Θ2 ( ) ( ) d z d y d x d t + Χ ∞ − Χ Θ 4 Θ L ∫ (L z − z ) ρ ( z ) d  x 0 x 2  2 2 ∂t2 0   2

×

 z , 

Lx L y Lz  1 Θ (Lz − z )  ∂ 2 u y1 (x, y, z, t ) ∂ 2 u x1 (x, y, z, t )  Θ2 2 α uy 2 = L z  ∫ (Θ − τ ) ∫ ∫ ∫ E (z ) + d z d y d x d t − ×   1 + σ (z )  ∂ x∂ y 2 ∂ x2 0 0 0  2 0  ∞ Lx L y Lz (L − z )  ∂ 2 u y1 (x, y, z, t ) ∂ 2 u x1 (x, y, z, t ) Lz − z 1 Θ Lx L y Lz d z d y d x d t × ∫ ∫ ∫ ∫ E (z ) z + + ×   ∫ ∫ ∫ ∫ E (z ) 2 1 + σ (z )  12 0 0 0 0 1 + σ (z ) ∂x ∂ x∂ y 0 0 0 0   ∂ 2 u y1 (x, y, z , t ) ∂ 2 u x1 (x, y, z , t ) ∂ 2 u z1 (x, y, z , t )  Θ 2 ∞ Lx L y Lz 2 × 5 − − d z d y d x ( Θ − τ ) d t −  ∫ ∫ ∫ ∫ E (z ) × ∂ y2 ∂ x∂ y ∂ y∂ z 12 0 0 0 0   2 Lx L y Lz Θ L − z  ∂ u y1 ( x, y , z , t ) ∂ 2 u x1 ( x, y , z , t ) ∂ 2 u z 1 ( x, y , z , t )  K (z ) 2 × z 5 − − ×   d z d y d x d t + ∫ (Θ − t ) ∫ ∫ ∫ 2 1 + σ (z )  ∂y ∂ x∂ y ∂ y∂ z 2 0 0 0 0  ∞ Lx L y Lz  ∂ 2 u y 1 ( x , y , z , t ) ∂ 2 u y1 ( x , y , z , t ) ∂ 2 u y 1 ( x , y , z , t )  × ( Lz − z )  + + d z d y d x d t − ∫ ∫ ∫ ∫ K ( z ) ( Lz − z ) ×  ∂ x∂ y ∂ y2 ∂ y∂ z 0 0 0 0   2 2 2 Θ 2  ∂ u y1 ( x , y , z , t ) ∂ u y1 ( x , y , z , t ) ∂ u y1 ( x , y , z , t )  1 Θ L x L y L z  ∂ u y1 ( x , y , z , t ) × + + d z d y d x d t + + ∫ ∫ ∫ ∫   2  ∂ x∂ y ∂ y2 ∂ y∂ z 2 0 0 0 0 ∂z  ∂ u ( x, y , z , t )  E ( z ) d z Θ 2 ∞ L x L y L z  ∂ u y1 ( x , y , z , t ) ∂ u z 1 ( x , y , z , t )  E ( z ) d z 2 τ d y d x d t + z1 ( Θ − ) − + × ∫ ∫ ∫ ∫   2 0 0 0 0 ∂y ∂z ∂y  1 + σ (z )  1 + σ (z ) 2 Lx L y Lz ∂ 2 u y1 ( x , y , z , t ) 1Θ Θ 2 ∞ L x L y L z ∂ u y 1 ( x, y , z , t ) 2 d z d y d x d t × d y d x d t − ∫ (Θ − t ) ∫ ∫ ∫ ρ ( z ) + × ∫∫∫∫ 20 ∂t2 2 0000 ∂t2 0 0 0 Θ Lx L y Lz

Χ y 2 (Θ )

0 0 0 0

2

× ρ ( z ) d z d y d x d t − ∫ ∫ ∫ ∫ (L z − z ) ρ ( z ) u y 1 ( x, y , z , t ) d z d y d x d t −

+ Θ2

Χ y 0 (∞ )  × 2 

−1

Lz   ×  4 Θ L ∫ (L z − z ) ρ ( z ) d z  , 0  

 1 Θ  2 0

Lx L y Lz

α z 2 = L z  ∫ (Θ − t )2 ∫ ∫ ∫ E (z ) 0 0 0

− z )  ∂ 2 u z1 ( x, y, z , t ) ∂ 2 u x1 (x, y, z , t )  Θ2 + d z d y d x d t − ×   1 + σ (z )  ∂ x2 ∂ x∂ z 2 

(L z

(Lz − z )  ∂ 2u z1 (x, y, z , t ) + ∂ 2u x1 (x, y, z , t )  d z d y d x d t + 1 Θ (Θ − t )2 L L L E (z ) × ∫ ∫∫ ∫   1 + σ (z )  ∂ x2 ∂ x∂ z 20 0 0 0 0 0 0 0  ∞ L L L  ∂ 2 u ( x, y , z , t )  ∂ 2 u z1 ( x, y , z , t ) ∂ 2 u y 1 ( x, y , z , t )  ∂ 2 u y1 ( x , y , z , t )  z1 ∞ Lx L y Lz

x y

× ∫ ∫ ∫ ∫ E (z ) × 

∂y

2

+

x y z

∂ y∂ z

d zd yd xdt − ∫ ∫ ∫ ∫ 0 0 0 0  

∂ y2

+

z

∂ y∂ z

×  46


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

× Θ2

E ( z ) (L z − z ) 1 Θ Lx L y Lz  ∂ u (x, y, z, t ) ∂ u x1 (x, y, z, t ) ∂ u y1 (x, y, z, t )  − − d z d y d x d t + ∫ ∫ ∫ ∫ 5 z1 × 2 [1 + σ (z )] 12 0 0 0 0  ∂z ∂x ∂y 

×

E (z ) d z Θ 2 ∞ L x L y L z  ∂ u z 1 ( x , y , z , t ) ∂ u x1 ( x , y , z , t ) ∂ u y 1 ( x , y , z , t )  2 − − d y d x (Θ − t ) d t − ∫ ∫ ∫ ∫ 5 × 1 + σ (z ) 12 0 0 0 0  ∂z ∂x ∂y 

×

Lx L y L z Θ E (z ) d z K (z )  ∂ u x1 (x, y , z , t ) ∂ u y1 (x, y , z , t ) ∂ u z1 ( x, y, z, t )  2 d y d x d t + ∫ (Θ − t ) ∫ ∫ ∫ + +  × 1 + σ (z ) 2  ∂x ∂y ∂z 0 0 0 0 

× d zd yd xdt −

Χ z 2 (Θ ) 2

 ∂ u x1 ( x , y , z , t ) ∂ u y1 ( x , y , z , t ) ∂ u z 1 ( x , y , z , t )  Θ 2 ∞ Lx L y Lz + + ∫ ∫ ∫ ∫ K (z )  × ∂x ∂y ∂z 2 0000  

Θ Lx L y Lz

× d z d y d x d t − ∫ ∫ ∫ ∫ ( Lz − z ) u z 1 ( x , y , z , t ) ρ ( z ) d z d y d x d t + Θ 2 0 0 0 0

Χ z 0 (∞ )   L z  4 ∫ ( L z − z ) × 2  0

× ρ (z ) d z Θ L} . −1

System of equations for the functions Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0) could be written as  ∂ 2 C 000 (x, y , z , t ) ∂ 2 C 000 ( x, y, z , t ) ∂ 2 C000 (x, y, z , t )  ∂ C 000 (x, y , z , t ) = D0 L  + + ; ∂t ∂ x2 ∂ y2 ∂ z2  

  ∂  ∂ C i 00 ( x, y , z, t ) ∂ C i −100 ( x, y, z , t )  ∂  ∂ C i −100 ( x, y, z , t )  = D0 L    g L (z, T ) +  g L (z , T ) +  ∂t ∂x ∂y  ∂y    ∂ x  +

∂ Ci −100 (x, y, z, t )    ∂ 2 Ci 00 (x, y, z, t ) ∂ 2 Ci 00 (x, y, z, t ) ∂ 2 Ci 00 (x, y, z, t )   ∂  ( ) g z , T + +   , i ≥1;  L  +  ∂z  ∂z ∂ x2 ∂ y2 ∂ z2    

 ∂ 2 C 010 ( x, y , z, t ) ∂ 2 C 010 (x, y , z, t ) ∂ 2 C 010 (x, y , z , t )  ∂ C 010 ( x, y, z, t ) = D0 L  + + + ∂t ∂ x2 ∂ y2 ∂ z2   γ γ  ∂  C 000 (x, y, z , t ) ∂ C 000 (x, y, z, t ) ∂  C 000 (x, y, z, t ) ∂ C 000 (x, y, z , t )  + D0 L  +  γ   γ + ∂x ∂y  ∂ x  P ( x, y, z , T )  ∂ y  P ( x, y , z , T )  γ ∂  C 000 (x, y , z , t ) ∂ C000 ( x, y , z, t )   +   ; ∂ z  P γ ( x, y , z , T ) ∂z  2 2  ∂ C 020 ( x, y, z , t ) ∂ C 020 ( x, y , z , t ) ∂ 2 C 020 ( x, y , z , t )  ∂ C 020 ( x, y, z , t ) = D0 L  + +  + D0 L × ∂t ∂ x2 ∂ y2 ∂ z2  

γ −1  ∂  (x, y, z, t ) ∂ C 000 (x, y, z, t ) ∂  C 000 ∂ C 000 (x, y , z, t ) × ×  C 010 (x, y , z , t ) γ + C 010 ( x, y , z, t ) P (x, y , z , T ) ∂y ∂y  ∂ x   ∂y

γ −1 γ −1 γ  ∂ C 000 (x, y, z, t ) ∂  (x, y, z, t ) ∂ C000 (x, y, z, t )  ( x, y, z , t ) C 000 C 000 ( ) + C x , y , z , t + D ×      010 0 L γ γ γ ∂z P (x, y, z, T )  ∂ z  P ( x, y, z , T )    ∂ x P (x, y, z, T ) γ γ (x, y, z, t ) ∂ C 010 (x, y, z, t )  ∂  C 000 (x, y, z, t ) ∂ C 010 (x, y, z, t )   ∂ C 010 (x, y , z , t )  ∂  C 000 ×  γ +  γ  ; + ∂x ∂y ∂z  ∂ y  P ( x, y , z , T )  ∂ z  P ( x, y , z , T )   2 2 2  ∂ C 001 ( x, y, z , t ) ∂ C 001 (x, y , z , t ) ∂ C 001 ( x, y, z , t )  ∂ C 001 (x, y , z , t ) = D0 L  + + + ∂t ∂ x2 ∂ y2 ∂ z2  

×

47


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

+ Ω D0 SL

γ  C 000 ( x, y , z , t )  L z ∇ S µ (x, y, z , t )  ∂   1 + ε S L g S L (x, y, z , T ) 1 + ξ S γ +  ∫ C 000 (x, y, W , t ) d W ∂ x  P (x, y, z , T )  0 kT  

+ Ω D0 SL

∂   1 + ε S L g S L ( x, y , z , T ) ∂ y 

[

]

]1 + ξ

[

S

γ C 000 ( x, y , z , t )  L z ∇ S µ (x, y, z , t )  ;  ∫ C 000 ( x, y , W , t ) d W γ P ( x, y , z , T )  0 kT 

 ∂ 2 C 002 ( x, y , z , t ) ∂ 2 C 002 (x, y , z , t ) ∂ 2 C 002 ( x, y, z , t )  ∂ C 002 ( x, y , z , t ) = D0 L  + + + ∂t ∂ x2 ∂ y2 ∂ z2   + Ω D0 SL

γ  C 000 ( x, y , z , t )  L z ∇ S µ (x, y , z , t )  ∂  1 + g ( x , y , z , T ) 1 + ε ξ  +   ∫ C 001 (x, y, W , t ) d W SL SL S γ ∂ x  P ( x, y, z , T )  0 kT  

+ Ω D0 SL

γ  ( x, y , z , t )  L z C 000 ∇ S µ (x, y, z , t )  ∂   1 + ε S L g S L (x, y, z , T ) 1 + ξ S γ +  ∫ C 001 ( x, y, W , t ) d W ∂ y  P (x, y, z , T )  0 kT  

[

+ Ω D0 SL ×

]

[

]

∂   1 + ε S L g S L (x, y , z , T ) ∂ x 

]1 + ξ

[

S

γ ( x, y , z , t ) C 001

γ −1 ( x, y , z , t )  L z C 000  ∫ C 000 (x, y , W , t ) d W × γ P ( x, y , z , T )  0

γ −1 ( x, y , z , t )  ∇ S µ ( x, y , z , t ) L z C ( x , y , W , t ) d W × ∇ S µ ( x, y , z , t )  ∂   C000 γ ( ) ξ + 1 + C x , y , z , t ∫ 000   S 001  γ kT P ( x, y , z , T )  kT 0  ∂ y 

[

]

× 1 + ε S L g S L (x, y , z , T ) } Ω D0 SL ;  ∂ 2 C110 (x, y , z , t ) ∂ 2 C110 ( x, y , z , t ) ∂ 2 C110 ( x, y , z, t )   ∂  ∂ C010 ( x, y, z , t ) ∂ C110 ( x, y, z, t ) = D0 L  + + × +  ∂t ∂ x2 ∂ y2 ∂ z2 ∂x    ∂ x 

 ∂  ∂ C010 (x, y, z, t )  ∂  ∂ C010 (x, y, z, t )   × g L (x, y, z, T ) +  g L (x, y, z, T ) +  g L (x, y, z, T )   D0 L + ∂y ∂z  ∂ y   ∂z    γ γ  ∂  C γ (x, y , z, t ) ∂ C100 (x, y , z , t )  ∂  C000 (x, y, z, t ) ∂ C100 (x, y, z, t ) + ∂  C000 ( x, y , z , t ) × +   000 +   γ   γ γ ∂x ∂y  ∂ x  P ( x, y , z , T )  ∂ y  P ( x, y , z , T )  ∂ z  P (x, y , z , T ) γ −1 γ −1  ∂  (x, y, z, t ) ∂ C000 (x, y, z, t ) + ∂  C000 ( x, y , z , t ) × ∂ C (x, y , z , t )   C 000 × 100   D0 L +  C100 ( x, y , z , t ) γ   γ P ( x, y , z , T ) ∂z ∂x  ∂ x     ∂ y  P ( x, y , z , T ) γ −1 (x, y, z, t ) ∂ C 000 (x, y, z, t )  ∂ C 000 (x, y , z , t )  ∂  C 000 ( ) + , , , C x y z t  100   D0 L ;  γ ∂y ∂z P ( x, y , z , T )  ∂z     ∂ 2 C100 (x, y, z , t ) ∂ 2 C100 (x, y , z , t ) ∂ 2 C100 (x, y , z, t )   ∂  ∂ C000 (x, y , z, t ) ∂ C101 (x, y , z, t ) = D0 L  + + × +  ∂t ∂ x2 ∂ y2 ∂ z2 ∂y    ∂ x   ∂  ∂ C000 (x, y , z , t )  ∂  ∂ C000 ( x, y, z , t )   × g L (x, y, z , T ) +  g L ( x, y , z , T ) +  g L ( x, y , z , T )   D0 L + ∂y ∂z  ∂ y   ∂z    γ −1  ( x , y , z , t )  L z C ( x, y , W , t ) d W × C000 ∂  + Ω D0 SL  1 + ε S L g S L ( x, y, z , T ) 1 + ξ S C100 (x, y , z , t )  ∫ 100 ∂x P γ (z, T )  0  γ −1  ( x, y , z , t )  ∇ µ ( x, y , z , t )  C 000 ∂  × S  + Ω D0 SL  1 + ε S L g S L ( x, y, z, T ) 1 + ξ S C100 ( x, y , z, t ) × γ kT ∂ y  P (z , T )    Lz Lz ∇ µ ( x, y , z , t )  ∂  × ∫ C100 ( x, y , W , t ) d W S +  1 + ε S L g S L ( x, y , z, T ) ∫ C100 ( x, y , W , t ) d W × kT 0 0  ∂x

× C100 (x, y , z , t )

[

]

[

]

[

]

48


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

×

 ∇ S µ ( x, y , z , t ) L z ∇ S µ ( x, y , z , t ) ∂  × ∫ C100 (x, y ,W , t ) d W  Ω D0 SL +  1 + ε S L g S L ( x, y , z , T ) kT ∂y kT 0 

[

]

Lz Lz  × ∫ C100 ( x, y, W , t ) d W ∫ C100 ( x, y , W , t ) d W  Ω D0 SL ; 0 0 

γ  ∂ 2 C011 (x, y, z , t ) ∂ 2C011 ( x, y , z , t ) ∂ 2 C011 (x, y, z , t )   ∂  C000 (x, y, z, t ) × ∂ C011 (x, y, z , t ) = D0 L  + + +  γ 2 2 2 ( ∂t ∂ x ∂ y ∂ z ∂ x P x , y, z , T )    

×

γ γ (x, y, z, t ) ∂ C001 (x, y, z, t ) + ∂  C000 (x, y, z, t ) ∂ C001 (x, y, z, t ) D + ∂ C001 (x, y, z , t )  ∂  C000 +  γ  0L   γ  ∂x ∂y ∂z    ∂ y  P ( x, y , z , T )  ∂ z  P ( x, y , z , T )

 ∂  C γ −1 ( x, y , z , t ) ∂ C 000 (x, y , z, t )  ∂  ∂ C000 ( x, y , z, t ) + D0 L  C001 ( x, y , z , t ) 000 × + C001 (x, y , z , t ) γ P ( x, y , z , T ) ∂x ∂y  ∂ x   ∂y γ −1 (x, y, z , t ) ∂ C 000 (x, y, z , t )   ∂  ∇ S µ (x, y, z, t ) C γ −1 (x, y , z , t )  ∂  C 000 ( ) × 000 + C x , y , z , t ×    001  + γ γ P ( x, y , z , T )  ∂ z  P ( x, y , z , T ) ∂z kT   ∂ x    C γ (x, y , z , t )  Lz ∂ ∇ S µ ( x, y, z, t ) × 1 + ε S L g S L (x, y , z, T ) 1 + ξ S 000 × ∫ C010 (x, y ,W , t ) d W  Ω D0 SL +   γ P ( x, y , z , T )  0 ∂x kT  

[

]

  C γ −1 ( x, y, z , t )  Lz × 1 + ε S L g S L (x, y , z , T ) 1 + ξ S C010 ( x, y, z , t ) 000  ∫ C000 ( x, y ,W , t ) d W  Ω D0 SL + D0 SL Ω × γ P ( x, y , z , T )  0  

[

×

]

∂   1 + ε S L g S L (x, y, z, T ) ∂ y 

] 1 + ξ C (x, y, z, t ) CP (x(,x,yy, z, ,zT, t)) ∇ µ (kxT, y, z, t ) ∫ C (x, y,W , t ) d W  .

[

γ −1 000

S

010

γ

S

Lz

000

0    Boundary and initial conditions for functions Cijk(x,t) (i ≥0, j ≥0, k ≥0) could be written as ∂ Cijk ( x, y, z , t ) ∂ Cijk ( x, y, z , t ) ∂ Cijk ( x, y, z , t ) ∂ C ijk ( x, y, z, t ) =0; =0; = 0; = 0; ∂x ∂x ∂y ∂y x= L y =0 y=L x =0 x

y

∂ Cijk ( x, y , z , t ) ∂ Cijk ( x, y , z, t ) =0; = 0 ; C000(x,y,z,0)=fC (x,y,z); Cijk(x,y,z,0)=0. ∂z ∂z z =0 z = Lz Solutions of equations for functions Cijk(x,t) (i ≥0, j ≥0, k ≥0) could be written as C 000 ( x, y, z , t ) =

∞ F0 C 2 + ∑ FnC cn (x ) cn ( y )cn (z )enC (t ) , Lx L y Lz Lx L y Lz n =1

Ly Lx Lz   1 1 1  where enC (t ) = exp − π 2 n 2 D0 L t  2 + 2 + 2  , FnC = ∫ c n (u ) ∫ cn (v ) ∫ c n (w) f C (u, v, w) d w d v d u ;  L L L  0 0 0  y z   x Ly Lx Lz t 2π D ∞ C i 00 (x, y , z , t ) = − 2 0 L ∑ n FnC c n (x ) c n ( y ) c n (z ) e nC (t )∫ e nC (− τ ) ∫ s n (u ) ∫ c n (v ) ∫ g L (u , v, w, T ) × L x L y L z n =1 0 0 0 0

× cn (w)

Lx t ∂ Ci −100 (u , v, w,τ ) 2π D0 L ∞ d wd v d u dτ − ∑ n FnC cn (x )cn ( y )cn ( z )enC (t )∫ enC (− τ ) ∫ cn (u ) × 2 ∂u Lx L y Lz n =1 0 0

Ly

Lz

0

0

× ∫ sn (v ) ∫ cn (w) g L (u , v, w, T )

∂ Ci −100 (u , v, w,τ ) 2π D0 L ∞ d wd vd u dτ − ∑ cn ( x )cn ( y )cn (z )enC (t ) × ∂v Lx L y L2z n =1

49


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

Lx

Ly

Lz

0

0

0

0

× n FnC ∫ e nC (− τ ) ∫ c n (u ) ∫ c n (v ) ∫ s n (w) g L (u , v, w, T ) C010 ( x, y , z , t ) = − × cn (w)

∂ C i −100 (u , v, w,τ ) d w d v d u d τ , i ≥1; ∂w

γ Ly Lx Lz t (u , v, w,τ ) × 2π D0 L ∞ C000 ( ) ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − s u c v τ ∑ ∫ ∫ ∫ ∫ nC n n n nC nC n n 2 γ Lx L y Lz n =1 0 0 0 0 P (u , v, w, T )

Lx t ∂ C 000 (u , v, w,τ ) 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) d wd vd u dτ − n F c x c y c z e t e − τ ∑ ∫ ∫ cn (u ) × nC n n n nC nC ∂u Lx L2y Lz n =1 0 0

Ly

Lz

Lz

0

0

0

× ∫ sn (v ) ∫ cn (w) ∫ cn (w )

γ (u , v, w,τ ) ∂ C000 (u, v, w,τ ) d w d v d u d τ − 2π D0 L ∞ n c (x )c ( y ) × C000 ∑ n n γ P (u , v, w, T ) ∂v Lx Ly L2z n =1

t

Lx

Ly

Lz

0

0

0

× FnC c n (z ) e nC (t )∫ e nC (− τ ) ∫ c n (u ) ∫ c n (v ) ∫ s n (w) 0

C020 ( x, y , z , t ) = − × cn (w )

γ (u, v, w,τ ) ∂ C 000 (u, v, w,τ ) C 000 d wd vd u dτ ; γ P (u , v, w, T ) ∂w

Ly Lx Lz t 2π D0 L ∑ n FnC cn ( x )cn ( y )cn ( z )enC (t )∫ enC (− τ ) ∫ s n (u ) ∫ cn (v ) ∫ C 010 (u , v, w,τ ) × 2 Lx L y Lz n =1 0 0 0 0 ∞

γ −1 (u, v, w,τ ) ∂ C000 (u, v, w,τ ) d w d v d u d τ − 2π D0 L ∞ n F c (x )c ( y )c (z )e (t ) × C 000 ∑ nC n n n nC γ P (u , v, w, T ) ∂u Lx L2y Lz n =1

t

Lx

Ly

Lz

0

0

0

× ∫ enC (− τ ) ∫ sn (u ) ∫ cn (v ) ∫ cn (w) C 010 (u , v, w,τ ) 0

γ −1 (u, v, w,τ ) ∂ C000 (u, v, w,τ ) d w d v d u d τ − C 000 P γ (u , v, w, T ) ∂v

t

Lx

Ly

n =1

0

0

0

Lz

− ∑ n FnC cn ( x )cn ( y )cn ( z )enC (t ) ∫ enC (− τ ) ∫ sn (u ) ∫ cn (v ) ∫ cn (w)C010 (u , v, w,τ )

×

t 2π D0 L ∂ C 000 (u , v, w,τ ) 2π D d w d v d u d τ − 2 0 L ∑ n FnC cn (x )cn ( y )cn ( z ) enC (t ) ∫ enC (− τ ) × 2 Lx L y Lz ∂w Lx L y Lz n =1 0 ∞

Lx

Ly

Lz

0

0

0

× ∫ s n (u ) ∫ cn (v ) ∫ cn (w)

γ (u, v, w,τ ) ∂ C010 (u, v, w,τ ) d w d v d u d τ − 2π D0 L ∞ n F c (x ) × C000 ∑ nC n γ P (u , v, w, T ) ∂u Lx L2y Lz n =1 Ly

Lx

t

Lz

× cn ( y )cn (z )enC (t )∫ enC (− τ ) ∫ cn (u ) ∫ s n (v ) ∫ cn (w ) 0

0

γ −1 (u, v, w,τ ) × C000 P γ (u , v, w, T )

0

0

0

γ (u, v, w,τ ) ∂ C010 (u, v, w,τ ) d w d v d u d τ − C 000 P γ (u , v, w, T ) ∂v

γ (u, v, w,τ ) ∂ C010 (u, v, w,τ ) d w d v d u d τ × C 000 2π D0 L ( ) ( ) ( ) ( ) n F e − τ c u c v s w ∑ ∫ n ∫ n ∫ n nC ∫ nC Lx L y L2z n =1 P γ (u , v, w, T ) ∂w 0 0 0 0 ∞

Ly

Lx

t

Lz

× cn ( x )cn ( y )cn ( z )enC (t ) ;

C001 ( x, y, z , t ) = −

L y Lz Lx t 2π D0 L Ω ∞ ∇ S µ (u , v, w,τ ) ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − s u × τ ∑ ∫ ∫ ∫ ∫ nC n n n nC nC n 2 Lx L y Lz n =1 kT 0 0 0 0

 C γ (u , v, w,τ )  Lz × 1 + ε S L g S L (u , v, w, T ) 1 + ξ S 000  ∫ C000 (u , v,W ,τ ) d W cn (w) d w cn (v ) d v d u d τ − P γ (u , v, w, T )  0  Ly Lx Lz Lz t 2π D0 L Ω ∞ ( ) ( ) ( ) ( ) ( ) ( ) − n F c x c y c z e t e − τ c u ∑ ∫ nC ∫ n ∫ sn (v ) ∫ cn (w) ∫ C 000 (u , v,W ,τ ) d W × nC n n n nC 2 Lx L y Lz n =1 0 0 0 0 0

[

]

 C γ (u , v, w, τ )  ∇ S µ (u , v, w,τ ) × 1 + ε S L g S L (u , v, w, T ) 1 + ξ S 000 γ d wd vd u dτ ;  P (w, T )  kT  Ly Lx Lz t 2π D Ω ∞ ∇ µ (u , v, w,τ ) C002 ( x, y , z , t ) = − 2 0 L ∑ n FnC cn (x ) cn ( y )cn ( z )enC (t )∫ enC (− τ ) ∫ sn (u ) ∫ cn (v ) ∫ S × Lx L y Lz n =1 kT 0 0 0 0

[

]

 C γ (u , v, w,τ )  Lz π D0 L Ω × 1 + ε S L g S L (u , v, w, T ) 1 + ξ S 000 × ∫ C001 (u , v,W ,τ ) d W cn (w ) d w d v d u d τ −  γ P (u , v, w, T )  0 Lx L2y Lz 

[

]

50


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

t

n =1

0

Lx

Ly

Lz

0

0

0

× ∑ n FnC cn ( x )cn ( y )cn ( z ) enC (t )∫ enC (− τ ) ∫ cn (u ) ∫ sn (v ) ∫ cn (w)

∇ S µ (u , v, w,τ ) Lz ∫ C001 (u , v,W ,τ ) d W × kT 0

 C (u , v, w,τ )  2π D Ω ∞ × 2 1 + ε S L g S L (u , v, w, T ) 1 + ξ S 000 d w d v d u d τ − 2 0 L ∑ FnC cn ( x )cn ( y )cn ( z ) ×  γ P (u , v, w, T )  Lx L y Lz n =1  γ −1 Ly Lx Lz t   Lz γ (x, y, z , t ) C000γ (u, v, w,τ ) ∫ C000 (u, v,W ,τ )d W × × n enC (t )∫ enC (− τ ) ∫ sn (u ) ∫ cn (v ) ∫ cn (w)1 + ξ S C001 P (u , v, w, T )  0 0 0 0 0  γ

[

]

[

]∇ µ (kuT, v, w,τ ) d w d v d u d τ − 2Lπ LD LΩ ∑ F c (x )c ( y )c (z )e (t ) ×

× 1 + ε S L g S L (u , v, w, T )

S

x

t Lx

Ly

Lz

0 0

0

0

× n ∫ ∫ sn (u ) ∫ cn (v ) ∫ cn (w)

0L 2 y z

n =1

nC n

n

n

nC

γ −1 (u, v, w,τ )  LzC (u, v,W ,τ )d W × C000 ∇ S µ (u, v, w,τ )  γ 1 + ξ S C001 (x, y, z , t )  ∫ 000 kT P γ (w, T )  0 

[

]

× 1 + ε S L g S L (u , v, w, T ) d w d v d u e nC (− τ ) d τ ; Ly Lx Lz t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − τ s u c v ∑ ∫ ∫ ∫ ∫ g L (u , v, w, T ) × nC n n n nC nC n n L2x L y L z n =1 0 0 0 0

C110 (x, y , z , t ) = − × cn (w)

Lx t ∂ C 010 (u , v, w,τ ) 2π D0 L ∞ d wd vd u dτ − ∑ n FnC cn ( x )cn ( y )cn (z )enC (t )∫ enC (− τ ) ∫ cn (u ) × 2 ∂u Lx L y Lz n =1 0 0

Ly

Lz

× ∫ sn (v ) ∫ cn (w) g L (u , v, w, T ) 0

0

∂ C010 (u , v, w,τ ) 2π D0 L ∞ d wd v d u d τ − ∑ n FnC cn ( x )cn ( y )cn ( z )× ∂v Lx L y L2z n =1

t

Lx

Ly

Lz

0

0

0

0

× enC (t )∫ enC (− τ ) ∫ cn (u ) ∫ cn (v ) ∫ s n (w) g L (u , v, w, T ) ∞

∂ C010 (u , v, w,τ ) 2π D d w d v d u d τ − 2 0L × ∂w Lx L y Lz

t

Lx

Ly

Lz

0

0

0

0

× ∑ n FnC cn ( x )cn ( y )cn ( z ) enC (t )∫ enC (− τ ) ∫ sn (u ) ∫ cn (v ) ∫ cn (w) g L (u , v, w, T )C100 (u , v, w,τ ) × n =1

C000 (u , v, w,τ ) ∂ C000 (u , v, w,τ ) 2 π D0 L ∞ d wd v d u dτ − ∑ n FnC cn ( x )cn ( y )cn ( z ) enC (t ) × γ P (u , v, w, T ) ∂u Lx L2y Lz n =1 γ −1

×

t

Lx

Ly

Lz

0

0

0

0

× ∫ enC (− τ ) ∫ cn (u ) ∫ s n (v ) ∫ g L (u , v, w, T )C100 (u , v, w,τ )

× cn (w) d w d v d u d τ −

Ly Lx t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − τ c u ∑ ∫ nC ∫ n ∫ cn (v ) × nC n n n nC Lx L y L2z n =1 0 0 0

Lz

× ∫ sn (w) g L (u , v, w, T ) C100 (u , v, w,τ ) 0

− ×

γ −1 (u, v, w,τ ) ∂ C000 (u, v, w,τ ) × C000 P γ (u , v, w, T ) ∂v

γ −1 C000 (u, v, w,τ ) ∂ C000 (u, v, w,τ ) d w d v d u d τ − P γ (u , v, w, T ) ∂w

Ly Lx Lz t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − τ s u ∑ ∫ ∫ ∫ cn (v ) ∫ cn (w) g L (u , v, w, T ) × nC n n n nC nC n 2 Lx L y Lz n =1 0 0 0 0

γ (u, v, w,τ ) ∂ C100 (u, v, w,τ ) d w d v d u d τ − 2π D0 L ∞ n c (x )c ( y )c (z )e (t )× C 000 ∑ n n n nC γ P (u , v, w, T ) ∂u Lx L2y Lz n =1 t

Lx L y

Lz

0

0 0

0

× FnC ∫ enC (− τ ) ∫ ∫ s n (v ) ∫ cn (w) g L (u , v, w, T )

× cn (u ) d u d τ −

γ (u, v, w,τ ) ∂ C100 (u, v, w,τ ) d w d v × C000 γ P (u , v, w, T ) ∂u

Ly Lx Lz t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − τ c u c v ∑ ∫ ∫ ∫ ∫ s n (w ) × nC n n n nC nC n n Lx L y L2z n =1 0 0 0 0

× g L (u , v, w, T )

γ (u , v, w,τ ) ∂ C100 (u, v, w,τ ) C 000 d wd vd u dτ ; P γ (w, T ) ∂u

51


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

C101 ( x, y , z , t ) = −

× cn (w )

Ly Lx Lz t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − τ s u ∑ ∫ nC ∫ n ∫ cn (v ) ∫ g L (u , v, w, T ) × nC n n n nC 2 Lx L y Lz n =1 0 0 0 0

Lx t ∂ C000 (u , v, w,τ ) 2π D0 L ∞ d wd v d u dτ − ∑ FnC cn (x ) cn ( y )cn ( z ) enC (t ) ∫ enC (− τ ) ∫ cn (u ) × 2 ∂u Lx L y Lz n =1 0 0

Ly

Lz

Lz

0

0

0

× n ∫ s n (v ) ∫ cn (w ) ∫ cn (w)g L (u , v, w, T )

∂ C000 (u , v, w,τ ) 2π D0 L ∞ d wd vd u dτ − ∑ n cn (x )cn ( y ) × ∂v Lx L y L2z n =1

t

Lx

Ly

Lz

0

0

0

0

× FnC cn ( z )enC (t )∫ enC (− τ ) ∫ cn (u ) ∫ cn (v ) ∫ s n (w ) g L (u , v, w, T )

−Ω

×

∂ C000 (u , v, w,τ ) d wd vd u dτ − ∂w

Ly Lx Lz t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − τ s u c v ∑ ∫ ∫ ∫ ∫ 1 + ε S L g S L (u , v, w, T ) × nC n n n nC nC n n L2x L y Lz n =1 0 0 0 0

[

]

γ ∇ S µ (u , v, w,τ )  C000 (u, v, w,τ ) LzC (u, v,W ,τ ) d W ∇ S µ (u, v, w,τ ) d w d v d u d τ − + ξ 1 S   ∫ 100 kT P γ (u , v, w, T )  0 kT 

−Ω

Ly Lx Lz t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) ( ) n F c x c y c z e t e − τ s u c v ∑ ∫ ∫ ∫ ∫ 1 + ε S L g S L (u , v, w, T ) × nC n n n nC nC n n L2x L y Lz n =1 0 0 0 0

[

]

 C γ −1 (u , v, w,τ )  ∇ S µ (u , v, w,τ ) L z × cn (w) 1 + ξ S C100 (u , v, w,τ ) 000 ∫ C000 (u , v,W ,τ ) d W d w d v d u d τ −  P γ (u , v, w, T )  kT 0  Ly Lx Lz t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) −Ω n F c x c y c z e t e − τ c u ∑ ∫ nC ∫ n ∫ s n (v ) ∫ 1 + ε S L g S L (u , v, w, T ) × nC n n n nC 2 L x L y L z n =1 0 0 0 0

[

× c n (w )

γ −1 (u , v , w , τ )  L z ∇ S µ (u , v , w , τ )  C 000 1 + ξ S C 100 (u , v , w , τ ) γ  ∫ C 000 (u , v , W , τ ) d W d w d v d u d τ ; kT P (u , v , w , T )  0 

C011 ( x, y, z , t ) = − × cn (w )

Ly Lx Lz t 2π D0 L ∞ C γ (u , v, w,τ ) × ∑ n FnC cn ( x )cn ( y )cn (z ) enC (t )∫ enC (− τ ) ∫ s n (u ) ∫ cn (v ) ∫ 000 γ 2 Lx L y Lz n =1 0 0 0 0 P (u , v, w, T )

Lx t ∂ C001 (u , v, w,τ ) 2π D0 L ∞ d wd v d u dτ − ∑ n FnC cn (x ) cn ( y )cn ( z )enC (t )∫ enC (− τ ) ∫ cn (u ) × 2 ∂u Lx L y Lz n =1 0 0

Ly

Lz

Lz

0

0

0

× ∫ sn (v ) ∫ cn (w) ∫ cn (w ) t

Lx

0

0

γ (u, v, w,τ ) ∂ C001 (u, v, w,τ ) d w d v d u d τ − 2π D0 L ∞ n F c (x )c ( y ) × C 000 ∑ nC n n γ P (u , v, w, T ) ∂v Lx L y L2z n =1 Ly

Lz

0

0

× cn (z )enC (t )∫ enC (− τ ) ∫ cn (u ) ∫ cn (v ) ∫ sn (w) ∞

t

Lx

Ly

n =1

0

0

0

Lz 0

γ (u, v, w,τ ) ∂ C001 (u, v, w,τ ) d w d v d u d τ − 2π D0 L × C000 P γ (u , v, w, T ) ∂w L2x Ly Lz

× ∑ FnC ∫ enC (− τ ) ∫ sn (u ) ∫ cn (v ) ∫ cn (w)C001 (u , v, w,τ ) × n cn (x )cn ( y )cn (z )enC (t ) −

× n FnC cn (w)

γ −1 (u , v, w,τ ) ∂ C000 (u, v, w,τ ) d w d v d u d τ × C000 γ P (u , v, w, T ) ∂u

Ly Lx Lz t 2π D0 L ∞ ( ) ( ) ( ) ( ) ( ) ( ) ( ) c x c y c z e t e − τ c u s v ∑ ∫ ∫ ∫ ∫ C001 (u , v, w,τ ) × n n n nC nC n n Lx L2y Lz n =1 0 0 0 0

γ −1 (u, v, w,τ ) ∂ C000 (u, v, w,τ ) d w d v d u d τ − 2π D0 L ∞ n c (x )c ( y )c (z )e (t ) × C000 ∑ n n n nC γ P (u , v, w, T ) ∂v Lx L y L2z n =1

t

Lx

Ly

Lz

0

0

0

0

× FnC ∫ enC (− τ ) ∫ cn (u ) ∫ cn (v ) ∫ sn (w)C001 (u , v, w,τ ) −Ω

]

γ −1 (u, v, w,τ ) ∂ C000 (u, v, w,τ ) d w d v d u d τ − C000 γ P (u , v, w, T ) ∂w

Ly Lx Lz t 2π D0 L ∞ ∑ n FnC c n ( x ) c n ( y ) c n (z ) e nC (t )∫ e nC (− τ ) ∫ s n (u ) ∫ c n (v ) ∫ c n (w) 1 + ε S L g S L (u, v, w, T ) × 2 L x L y L z n =1 0 0 0 0

[

]

52


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

 C γ (u , v, w,τ )  ∇ S µ (u , v, w,τ ) Lz 2π D0 L ∞ × 1 + ξ S 000 ∑ FnC × ∫ C 010 (u , v,W ,τ ) d W d w d v d u d τ − Ω  γ P (u , v, w, T )  kT Lx L2y Lz n =1 0  Ly Lx Lz t  C γ (u , v, w,τ )  × n cn (x )cn ( y )cn (z )enC (t )∫ enC (− τ ) ∫ cn (u ) ∫ s n (v ) ∫ 1 + ε S L g S L (u , v, w, T ) 1 + ξ S 000 × P γ (u , v, w, T )  0 0 0 0  ∇ µ (u , v, w,τ ) Lz 2π D0 L ∞ × cn (w) S ∑ FnC cn ( x )cn ( y ) cn (z ) × ∫ C010 (u , v,W ,τ ) d W d w d v d u d τ − Ω 2 kT Lx Ly Lz n =1 0

[

]

Ly Lx Lz t  C γ −1 (u , v, w,τ )  × n enC (t )∫ enC (− τ ) ∫ sn (u ) ∫ cn (v ) ∫ cn (w) 1 + ε S L g S L (w, T ) 1 + ξ S C 010 (u , v, w,τ ) 000 × P γ (u , v, w, T )  0 0 0 0  ∇ µ (u , v, w,τ ) Lz 2π D0 L ∞ × S ∑ n cn (x )cn ( y ) cn (z ) enC (t ) × ∫ C000 (u , v,W ,τ ) d W d w d v d u d τ − Ω kT Lx L2y Lz n =1 0

[

t

Lx

Ly

Lz

0

0

0

0

]

[

]

Lz

× FnC ∫ e nC (− τ ) ∫ c n (u ) ∫ s n (v ) ∫ c n (w ) 1 + ε S L g S L (u , v, w, T ) ∫ C 000 (u , v, W ,τ ) d W × 0

 C (u , v, w,τ )  ∇ S µ (u , v, w,τ ) × 1 + ξ S C 010 (u , v, w,τ ) 000 γ d wd vd u dτ .  P (w, T )  kT  γ −1

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. 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.

53

MODELING OF MANUFACTURING OF A FIELDEFFECT TRANSISTOR TO DETERMINE CONDITIONS TO DECREASE LENGTH OF  

In this paper we introduce an approach to model technological process of manufacture of a field-effect heterotransistor. The modeling gives...

MODELING OF MANUFACTURING OF A FIELDEFFECT TRANSISTOR TO DETERMINE CONDITIONS TO DECREASE LENGTH OF  

In this paper we introduce an approach to model technological process of manufacture of a field-effect heterotransistor. The modeling gives...

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