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.

1. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
31
MODELING OF MANUFACTURING OF A FIELD-
EFFECT 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 in-
tegrated circuits and their discrete analogs [1-7]. To solve this problem attracted an interest wide-
ly used laser and microwave types of annealing [8-14]. These types of annealing leads to genera-
tion 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 rectifi-
ers [12-18]. It is known, that radiation processing of materials is also leads to modification of dis-
tribution 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 heterostruc-
ture [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

2. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
32
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
a1a2-a1Lx-a2-a1
c1 c1
c2
oxide
source drain
gate
c3
Fig. 2. Heterostructure with substrate and epitaxial layer with two or three sections. Sectional view

3. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
33
2. METHOD OF SOLUTION
To solve our aim we determine spatio-temporal distribution of concentration of dopant in the con-
sidered heterostructure and make the analysis. We determine spatio-temporal distribution of con-
centration of dopant with account mechanical stress by solving the second Fick’s law [1-4,22,23]
( ) ( ) ( ) ( ) +





∂
∂
∂
∂
+





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
(1)
( ) ( ) ( ) ( ) 





∫∇
∂
∂
Ω+





∫∇
∂
∂
Ω+
zz L
S
S
L
S
S
WdtWyxCtzyx
Tk
D
y
WdtWyxCtzyx
Tk
D
x 00
,,,,,,,,,,,, µµ
with boundary and initial conditions
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C(x,y,z,0)=fC(x,y,z).
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; Ω is the atomic vo-
lume; ∇S is the operators of surficial gradient; ( )∫
zL
zdtzyxC
0
,,, is the surficial concentration of
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. Concen-
traional dependence of dopant diffusion coefficients have been approximated by the following
relations [24]
( ) ( )
( )





+=
TzyxP
tzyxC
TzyxDD L
,,,
,,,
1,,, γ
γ
ξ , ( ) ( )
( )





+=
TzyxP
tzyxC
TzyxDD SLSS
,,,
,,,
1,,, γ
γ
ξ . (2)
Here DL (x,y,z,T) and DLS (x,y,z,T) are spatial (due to inhomogeneity of heterostructure) and tem-
perature (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 depen-
dence 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)
where E is the Young modulus; σij is the stress tensor;








∂
∂
+
∂
∂
=
i
j
j
i
ij
x
u
x
u
u
2
1
is the deformation
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
( )tzyxu ,,,
r
; xi, xj are the coordinates x, y, z. Relation (3) could be transform to the following form

4. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
34
( ) ( ) ( ) ( ) ( )




×








∂
∂
+
∂
∂








∂
∂
+
∂
∂
=
i
j
j
i
i
j
j
i
x
tzyxu
x
tzyxu
x
tzyxu
x
tzyxu
tzyx
,,,,,,
2
1,,,,,,
,,,µ
( )
( )
( )
( ) ( ) ( ) ( )[ ]




−−





−
∂
∂
−
+−
Ω
× ij
k
kij
ij TtzyxTzzK
x
tzyxu
z
z
zE δβε
σ
δσ
δε 000 ,,,3
,,,
212
,
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 solv-
ing of the following equations [23]
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )










∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
zzzyzxz
yzyyyxy
xzxyxxx
,,,,,,,,,,,,
,,,,,,,,,,,,
,,,,,,,,,,,,
2
2
2
2
2
2
σσσ
ρ
σσσ
ρ
σσσ
ρ
where
( )
( )[ ]
( ) ( ) ( ) ( )
×
∂
∂
+








∂
∂
−
∂
∂
+
∂
∂
+
=
k
k
ij
k
kij
i
j
j
i
ij
x
tzyxu
x
tzyxu
x
tzyxu
x
tzyxu
z
zE ,,,,,,
3
,,,,,,
12
δ
δ
σ
σ
( ) ( ) ( ) ( )[ ]rTtzyxTzKzzK −−× ,,,β , ρ (z) is the density of materials, δij is the Kronecker sym-
bol. With account of the relation the last system of equation takes the form
( )
( )
( ) ( )
( )[ ]
( )
( ) ( )
( )[ ]
( )
+
∂∂
∂






+
−+
∂
∂






+
+=
∂
∂
yx
tzyxu
z
zE
zK
x
tzyxu
z
zE
zK
t
tzyxu
z
yxx
,,,
13
,,,
16
5,,,
2
2
2
2
2
σσ
ρ
( )
( )[ ]
( ) ( )
( ) ( )
( )[ ]
( )
( ) ×−
∂∂
∂






+
++








∂
∂
+
∂
∂
+
+ zK
zx
tzyxu
z
zE
zK
z
tzyxu
y
tzyxu
z
zE zzy ,,,
13
,,,,,,
12
2
2
2
2
2
σσ
( ) ( )
x
tzyxT
z
∂
∂
×
,,,
β
( )
( ) ( )
( )[ ]
( ) ( )
( ) ( ) ( ) +
∂
∂
−








∂∂
∂
+
∂
∂
+
=
∂
∂
y
tzyxT
zKz
yx
tzyxu
x
tzyxu
z
zE
t
tzyxu
z xyy ,,,,,,,,,
12
,,, 2
2
2
2
2
β
σ
ρ
( )
( )[ ]
( ) ( ) ( )
( )[ ]
( )
( )
+
∂
∂






+
+
+














∂
∂
+
∂
∂
+∂
∂
+ 2
2
,,,
112
5,,,,,,
12 y
tzyxu
zK
z
zE
y
tzyxu
z
tzyxu
z
zE
z
yzy
σσ
( ) ( )
( )[ ]
( )
( )
( )
yx
tzyxu
zK
zy
tzyxu
z
zE
zK
yy
∂∂
∂
+
∂∂
∂






+
−+
,,,,,,
16
22
σ
(4)
( )
( ) ( ) ( ) ( ) ( )
×








∂∂
∂
+
∂∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxu
t
tzyxu
z
yxzzz
,,,,,,,,,,,,,,,
22
2
2
2
2
2
2
ρ

5. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
35
( )
( )[ ]
( )
( ) ( ) ( ) ( )
×
∂
∂
−














∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
+
×
z
tzyxT
z
tzyxu
y
tzyxu
x
tzyxu
zK
zz
zE xyx ,,,,,,,,,,,,
12 σ
( ) ( ) ( )
( )
( ) ( ) ( ) ( )














∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
+∂
∂
+×
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxu
z
zE
z
zzK zyxz ,,,,,,,,,,,,
6
16
1
σ
β .
Conditions for the displacement vector could be written as
( ) 0
,,,0
=
∂
∂
x
tzyu
r
;
( ) 0
,,,
=
∂
∂
x
tzyLu x
r
;
( ) 0
,,0,
=
∂
∂
y
tzxu
r
;
( ) 0
,,,
=
∂
∂
y
tzLxu y
r
;
( ) 0
,0,,
=
∂
∂
z
tyxu
r
;
( ) 0
,,,
=
∂
∂
z
tLyxu z
r
; ( ) 00,,, uzyxu
rr
= ; ( ) 0,,, uzyxu
rr
=∞ .
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]
( ) ( ) ( ) ( ) ( ) +





∂
∂
∂
∂
+





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
z
tzyxT
zy
tzyxT
yx
tzyxT
xt
tzyxT
Tc
,,,,,,,,,,,,
λλλ
( )tzyxp ,,,+ (5)
with boundary and initial conditions
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxT
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxT
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxT
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxT
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxT
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxT
, T (x,y,z,0)=fT (x,y,z).
Here λ is the heat conduction coefficient. Value of the coefficient depends on materials of hetero-
structure and temperature. Temperature dependence of heat conduction coefficient in most inter-
est area could be approximated by the following function: λ(x,y,z,T)=λass(x,y,z)[1+µ Td
ϕ
/
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-n-
junctions 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 ap-
proach 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
( ) ( )∑ ∑=
∞
=
∞
=0 0
,,,,,,
i j
ij
ji
T tzyxTtzyxT µε . (6)

6. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
36
Substitution of the series into Eq.(6) gives us possibility to obtain system of equations for the ini-
tial-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 depen-
dence. 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 correc-
tions [12-14,16,27]. It is practicably to transform differential equations of system (4) to integro-
differential form. The integral equations are presented in the Appendix. We determine the first-
order 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)
where ( )∫ ∫ ∫ ∫=
Θ
− −0 0
,,,
x
x
y
y
zL
L
L
L
L
isis tdxdydzdtzyxuM . The replacement leads to the following results
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )



−∫ ∫
∂
∂
−−∫ ∫
∂
∂
=
∞ t zz
x dwd
x
wyxT
wwKtdwd
x
wyxT
wwKttzyxu
0 00 0
1
,,,,,,
,,, τ
τ
βττ
τ
α
( )] 101 ,,, uxxux twyx αφα +Φ− ,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )



−∫ ∫
∂
∂
−−∫ ∫
∂
∂
=
∞ t zz
y dwd
y
wyxT
wwKtdwd
y
wyxT
wwKttzyxu
0 00 0
1
,,,,,,
,,, τ
τ
βττ
τ
α
( )] 101 ,,, uyyuy twyx αφα +Φ− ,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )



−∫ ∫
∂
∂
−∫ ∫
∂
∂
=
∞ t zz
z dwd
w
wyxT
wwKtdwd
w
wyxT
wwKttzyxu
0 00 0
1
,,,,,,
,,, τ
τ
βττ
τ
α
( )] 101 ,,, zzuz twyx αφα +Φ− .
Substitution of the above relations into relations (7) gives us possibility to obtain values of the
parameters αuβ1. Appropriate relations could be written as
( ) ( )[ ]
( ) ( )∫ −
ΘΧ−∞Χ
Θ= zL
z
xxz
ux
zdzzLL
L
0
3
20
1
8 ρ
α ,
( ) ( )[ ]
( ) ( )∫ −
ΘΧ−∞Χ
Θ= zL
z
yyz
uy
zdzzLL
L
0
3
20
1
8 ρ
α ,
( ) ( )[ ]
( ) ( )∫ −
ΘΧ−∞ΧΘ
= zL
z
zzz
uz
zdzzLL
L
0
3
20
1
8 ρ
α ,
where ( ) ( ) ( ) ( ) ( )
∫ ∫ ∫ ∫
∂
∂
−





Θ
+=ΘΧ
Θ
− −0 0
,,,
1
x
x
y
y
zL
L
L
L
L
z
i
is dxdydzd
s
zyxT
zzKzL
t
τ
τ
χ .

7. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
37
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)/
4L3
Θ. 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
( ) ( )[ ] ( )
( )
( ) ( )



×
∂
∂
∂
∂
+






∂
∂






++
∂
∂
=
∂
∂
y
tzyxC
yx
tzyxC
TzP
tzyxC
Tzg
x
D
t
tzyxC
LLL
,,,,,,
,
,,,
1,1
,,,
0 γ
γ
ξε
( )[ ] ( )
( )
( )[ ] ( )
( )


×





++
∂
∂
+









++×
TzyxP
tzyxC
Tzyxg
z
DD
TzyxP
tzyxC
Tzg LLLLLL
,,,
,,,
1,,,1
,,,
,,,
1,1 00 γ
γ
γ
γ
ξεξε
( ) ( )[ ] ( )
( )
( )



×∫





++
∂
∂
Ω+



∂
∂
×
zL
SLSLSL WdtWyxC
TzP
tzyxC
Tzg
x
D
z
tzyxC
0
0 ,,,
,
,,,
1,1
,,,
γ
γ
ξεω (8)
( ) ( )
( )
( )
( )






∫
∇






+
∂
∂
Ω+


∇
×
zL
S
SLL
S
WdtWyxC
Tk
tzyx
TzP
tzyxC
y
DD
Tk
tzyx
0
00 ,,,
,,,
,
,,,
1
,,, µ
ξω
µ
γ
γ
.
We determine solution of Eq.(1) as the following power series
( ) ( )∑ ∑ ∑=
∞
=
∞
=
∞
=0 0 0
,,,,,,
i j k
ijk
kji
L tzyxCtzyxC ωξε . (9)
Substitution of the series into Eq.(8) gives us possibility to obtain systems of equations for initial-
order 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 ap-
propriate boundary and initial conditions gives us possibility to obtain boundary and initial condi-
tions 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 compari-
son 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 manufactur-
ing 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

8. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
38
the increasing of compactness one can obtain decreasing of length of channel of field- effect tran-
sistor 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].
x, y
C(x,Θ),C(y,Θ)
2
3
1
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 us-
ing 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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[4] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
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[7] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. "Shallow p−n junctions in Si prepared by pulse pho-
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[10] J. A. Sharp, N. E. B. Cowern, R. P. Webb, K. J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M. A.
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APPENDIX
Equations for functions Tij(x,y,z,t) (i≥0, j≥0)
( ) ( ) ( ) ( ) ( )
ass
ass
tzyxp
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxT
ν
α
,,,,,,,,,,,,,,,
2
00
2
2
00
2
2
00
2
0
00
+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
( ) ( ) ( ) ( ) +





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
0
2
2
0
2
2
0
2
0
0 ,,,,,,,,,,,,
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxT iii
ass
i
α
( ) ( ) ( ) ( ) ( ) ( )












∂
∂
∂
∂
+
∂
∂
+
∂
∂
+ −−−
z
tzyxT
zg
zy
tzyxT
xg
x
tzyxT
zg i
T
i
T
i
Tass
,,,,,,,,, 10
2
10
2
2
10
2
0α , i≥1

10. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
40
( ) ( ) ( ) ( )
( )
×+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxT dass
ass
,,,
,,,,,,,,,,,,
00
0
2
01
2
2
01
2
2
01
2
0
01
ϕ
ϕ
α
α
( ) ( ) ( )
( )
( )




+





∂
∂
−





∂
∂
+
∂
∂
+
∂
∂
×
2
00
00
0
2
00
2
2
00
2
2
00
2
,,,
,,,
,,,,,,,,,
x
tzyxT
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT dass
ϕ
ϕ
α
( ) ( )










∂
∂
+





∂
∂
+
2
00
2
00 ,,,,,,
z
tzyxT
y
tzyxT
( ) ( ) ( ) ( )
( )
×+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxT dass
ass
,,,
,,,,,,,,,,,,
00
0
2
02
2
2
02
2
2
02
2
0
02
ϕ
ϕ
α
α
( ) ( ) ( )
( )
( )



×
∂
∂
−





∂
∂
+
∂
∂
+
∂
∂
× +
x
tzyxT
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT dass ,,,
,,,
,,,,,,,,, 00
1
00
0
2
01
2
2
01
2
2
01
2
ϕ
ϕ
αϕ
( ) ( ) ( ) ( ) ( )



∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
z
tzyxT
z
tzyxT
y
tzyxT
y
tzyxT
x
tzyxT ,,,,,,,,,,,,,,, 0100010001
( ) ( ) ( )
( )
( )
( )
( )



×+
∂
∂
+
∂
∂
=
∂
∂
Tzyxg
x
tzyxT
Tzyxg
tzyxT
tzyxT
x
tzyxT
t
tzyxT
TTass ,,,
,,,
,,,
,,,
,,,,,,,,,
2
00
2
00
01
2
11
2
0
11
α
( )
( )
( )
( )
( )




+





∂
∂
∂
∂
+










∂
∂
∂
∂
+
∂
∂
×
x
tzyxT
Tzyxg
xz
tzyxT
Tzyxg
zy
tzyxT
TassT
,,,
,,,
,,,
,,,
,,, 01
0
00
2
00
2
α
( )
( )[ ] ( )
( )[ ] ( )
+



∂
∂
++
∂
∂
++
∂
∂
+ assTT
z
tzyxT
Tzyxg
y
tzyxT
Tzyxg
x
tzyxT
02
01
2
2
01
2
2
01
2
,,,
,,,1
,,,
,,,1
,,,
α
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ×





∂
∂
+
∂
∂
+
∂
∂
+ +++ 2
00
2
1
00
10
2
00
2
1
00
10
2
00
2
1
00
10 ,,,
,,,
,,,,,,
,,,
,,,,,,
,,,
,,,
z
tzyxT
tzyxT
tzyxT
y
tzyxT
tzyxT
tzyxT
x
tzyxT
tzyxT
tzyxT
ϕϕϕ
( )
( ) ( ) ( )
( )
×+





∂
∂
+
∂
∂
+
∂
∂
+×
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
tzyxT
T
T dassdass
dass
,,,
,,,,,,,,,
,,, 00
0
2
10
2
2
10
2
2
10
2
00
0
0 ϕ
ϕ
ϕ
ϕ
ϕ αα
α
( ) ( ) ( ) ( ) ( ) ( ) −














∂
∂
∂
∂
+
∂
∂
+
∂
∂
×
z
tzyxT
Tzyxg
zy
tzyxT
Tzyxg
x
tzyxT
Tzyxg TTT
,,,
,,,
,,,
,,,
,,,
,,, 00
2
10
2
2
10
2
( ) ( ) ( ) ( ) ( ) ( ) ×





∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
−
z
tzyxT
z
tzyxT
y
tzyxT
y
tzyxT
x
tzyxT
x
tzyxT ,,,,,,,,,,,,,,,,,, 001000100010
( )
( )
( )
( ) ( ) ( )
×














∂
∂
+





∂
∂
+





∂
∂
−× ++
2
00
2
00
2
00
1
00
1
00
0 ,,,,,,,,,
,,,
,,,
,,, z
tzyxT
y
tzyxT
x
tzyxT
tzyxT
Tzyxg
tzyxT
T Tdass
ϕϕ
ϕ
αϕ
ϕ
ϕα dass T0× .
Conditions for the functions Tij(x,y,z,t) (i≥0, j≥0)
( )
0
,,,
0
=
∂
∂
=x
ij
x
tzyxT
,
( )
0
,,,
=
∂
∂
= xLx
ij
x
tzyxT
,
( )
0
,,,
0
=
∂
∂
=y
ij
y
tzyxT
,
( )
0
,,,
=
∂
∂
= yLx
ij
y
tzyxT
,
( )
0
,,,
0
=
∂
∂
=z
ij
z
tzyxT
,
( )
0
,,,
=
∂
∂
= zLx
ij
z
tzyxT
, T00(x,y,z,0)=fT(x,y,z), Tij(x,y,z,0)=0, i≥1, j≥1.
Solutions of the equations for the functions Tij(x,y,z,t) (i≥0, j≥0) with account appropriate boun-
dary and initial conditions could be written as

11. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
41
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫+∫ ∫ ∫=
∞
=1 00 0 0
00
2
,,
1
,,,
n
L
nnTnnn
zyx
L L L
T
zyx
xx y z
uctezcycxc
LLL
udvdwdwvuf
LLL
tzyxT
( ) ( ) ( ) ( ) ×+∫ ∫ ∫ ∫+∫ ∫×
zyx
t L L L
asszyx
L L
Tnn
LLL
dudvdwd
wvup
LLL
udvdwdwvufwcvc
x y zy z 2,,,1
,,
0 0 0 00 0
τ
ν
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ∫ ∫ ∫ ∫−×
∞
=1 0 0 0 0
,,,
n
t L L L
ass
nnnnTnTnnn
x y z
dudvdwd
wvup
wcvcucetezcycxc τ
ν
τ
τ ,
where cn(χ)=cos(πnχ/L), ( )
















++−= 2220
22 111
exp
zyx
assnT
LLL
tnte απ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−=
∞
=1 0 0 0 0
2
0
0 ,,,2,,,
n
t L L L
TnnnnTnTnnn
zyx
ass
i
x y z
Twvugwcvcusetezcycxcn
LLL
tzyxT τ
απ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−+
∂
∂
×
∞
=
−
1 0 0 0
2
010
2
,,,
n
t L L
nnnTnTnnn
zyx
assi
x y
vsucetezcycxcn
LLL
dudvdwd
u
wvuT
τ
απ
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑+∫
∂
∂
×
∞
=
−
1
2
0
0
10
2
,,,
,,,
n
nTnnn
zyx
ass
L
i
nT tezcycxcn
LLL
dudvdwd
v
wvuT
wcTwvug
z απ
τ
τ
( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫
∂
∂
−× −
t L L L
i
TnnnnT
x y z
dudvdwd
w
wvuT
Twvugwsvcuce
0 0 0 0
10 ,,,
,,, τ
τ
τ , i≥1,
where sn(χ)=sin(πnχ/L);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−=
∞
=1 0 0 0 0
2
00
2
2001
,,,2
,,,
n
t L L L
nnnTnTnnn
zyx
d
ass
x y z
u
wvuT
vcusetezcycxcn
LLL
T
tzyxT
τ
τ
π
α
ϕ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−+×
∞
=1 0 0 0 0
20
00
2
,,, n
t L L L
nnnnTnTnnn
zyx
d
assn
x y z
wcvsucetezcycxcn
LLL
T
wvuT
dudvdwd
wc τ
π
α
τ
τ ϕ
ϕ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−+
∂
∂
×
∞
=1 0 0 0
20
00
2
00
2
2
,,,
,,,
n
t L L
nnnTnTnnn
zyx
d
ass
x y
vcucetezcycxcn
LLL
T
wvuT
dudvdwd
v
wvuT
τ
π
α
τ
ττ ϕ
ϕ
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−∫
∂
∂
×
∞
=1 0 0
0
0 00
2
00
2
2
,,,
,,,
n
t L
nnTnTnnn
zyx
ass
d
L
n
xz
ucetezcycxc
LLL
T
wvuT
dudvdwd
u
wvuT
ws τ
αϕ
τ
ττ ϕ
ϕ
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )×∑ ∫ −−∫ ∫ 





∂
∂
×
∞
=
+
1 0
0
0 0
1
00
2
00
2
,,,
,,,
n
t
nTnTnnn
zyx
dass
L L
nn etezcycxc
LLL
T
wvuT
dudvdwd
u
wvuT
wcvc
y z
τ
α
ϕ
τ
ττ ϕ
ϕ
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )×∑−∫ ∫ ∫ 





∂
∂
×
∞
=
+
1
0
0 0 0
1
00
2
00
2
,,,
,,,
n
nTnnn
zyx
ass
d
L L L
nnn tezcycxc
LLL
T
wvuT
dudvdwd
v
wvuT
wcvcuc
x y z αϕ
τ
ττ ϕ
ϕ
( ) ( ) ( ) ( )
( )
( )∫ ∫ ∫ ∫ 





∂
∂
−× +
t L L L
nnnnT
x y z
wvuT
dudvdwd
w
wvuT
wcvcuce
0 0 0 0
1
00
2
00
,,,
,,,
τ
ττ
τ ϕ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−=
∞
=1 0 0 0 0
2
01
2
2002
,,,2
,,,
n
t L L L
nnnTnTnnn
zyx
d
ass
x y z
u
wvuT
vcusetezcycxcn
LLL
T
tzyxT
τ
τ
π
α
ϕ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−+×
∞
=1 0 0 0 0
20
00
2
,,, n
t L L L
nnnnTnTnnn
zyx
d
assn
x y z
wcvsucetezcycxcn
LLL
T
wvuT
dudvdwd
wc τ
π
α
τ
τ ϕ
ϕ
( )
( )
( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫−+
∂
∂
×
∞
=1 0 0
20
00
2
01
2
2
,,,
,,,
n
t L
nnTnTnnn
zyx
d
ass
x
ucetezcycxcn
LLL
T
wvuT
dudvdwd
v
wvuT
τ
π
α
τ
ττ ϕ
ϕ

12. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
42
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )×∑ ∫ −−∫ ∫
∂
∂
×
∞
=1 0
2
0
0 0 00
2
01
2
2
,,,
,,,
n
t
nTnTnnn
zyx
ass
L L
nn etezcycxc
LLLwvuT
dudvdwd
w
wvuT
wsvc
y z
τ
απ
τ
ττ
ϕ
( ) ( ) ( ) ( ) ( )
( )
( ) ×∑−∫ ∫ ∫
∂
∂
∂
∂
×
∞
=
+
1
2
0
0 0 0
1
00
0100
2
,,,
,,,,,,
n
n
zyx
ass
d
L L L
nnnd xc
LLL
T
wvuT
dudvdwd
u
wvuT
u
wvuT
wcvcucT
x y z απ
τ
τττ ϕ
ϕ
ϕ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
−∫ ∫ ∫ ∫
∂
∂
∂
∂
−× +
t L L L
nnnnTnTnn
x y z
wvuT
dudvdwd
v
wvuT
v
wvuT
wcvcucetezcyc
0 0 0 0
1
00
0100
,,,
,,,,,,
τ
τττ
τ ϕ
( ) ( ) ( ) ( ) ( ) ( )
( )
∑ ×∫ ∫ ∫ ∫
∂
∂
∂
∂
−−
∞
=
+
10 0 0 0
1
00
0100
2
0
,,,
,,,,,,
2
n
t L L L
nnnnT
zyx
assd
x y z
wvuT
dudvdwd
w
wvuT
w
wvuT
wcvcuce
LLL
T
τ
τττ
τ
α
π ϕ
ϕ
( ) ( ) ( ) ( )tezcycxc nTnnn× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
∑ ∫ ∫ ∫ ∫



×
∂
∂
−=
∞
=1 0 0 0 0
2
00
2
0
11
,,,2
,,,
n
t L L L
nnnnTnTnnn
zyx
ass
x y z
w
wvuT
wcvcucetezcycxc
LLL
tzyxT
τ
τ
α
( ) ( ) ( ) ( ) ( ) ( )
( )
+










∂
∂
∂
∂
+
∂
∂
+× τ
τ
τττ
dudvdwd
wvuT
wvuT
w
wvuT
wg
wv
wvuT
TwvugTwvug TTT
,,,
,,,,,,,,,
,,,,,,
00
0100
2
00
2
( )
( )[ ] ( )
( )
( )
+










∂
∂
∂
∂
+
∂
∂
++
∂
∂
× τ
τττ
dudvdwd
w
wvuT
Twvug
wv
wvuT
Twvug
u
wvuT
TT
,,,
,,,
,,,
,,,1
,,, 01
2
01
2
2
01
2
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
∑ ∫ ∫ ∫ ∫ 


+
∂
∂
−+
∞
=
+
1 0 0 0 0
2
00
2
1
00
100 ,,,
,,,
,,,
2
n
t L L L
nnnTnTnnn
zyx
dass
x y z
u
wvuT
wvuT
wvuT
vcucetezcycxc
LLL
T τ
τ
τ
τ
α
ϕ
ϕ
( )
( )
( ) ( )
( )
( )
( ) ×+


∂
∂
+
∂
∂
× ++
yx
ass
n
LL
dudvdwdwc
w
wvuT
wvuT
wvuT
v
wvuT
wvuT
wvuT 0
2
00
2
1
00
10
2
00
2
1
00
10 ,,,
,,,
,,,,,,
,,,
,,, α
τ
τ
τ
ττ
τ
τ
ϕϕ
( ) ( ) ( ) ( )
( ) ( ) ( )
∑ ×∫ ∫ ∫ ∫ 





∂
∂
+
∂
∂
+
∂
∂
−×
∞
=10 0 0 0
2
10
2
2
10
2
2
10
2
,,,,,,,,,
2
n
t L L L
nnnnT
z
d
x y z
w
wvuT
v
wvuT
u
wvuT
wcvcuce
L
T τττ
τ
ϕ
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−+×
∞
=1 0 0 0
0
00
22
,,, n
t L L
nnnTnTnnn
zyx
dass
nTnnn
x y
vcucetezcycxc
LLL
T
tezcycxc
wvuT
dudvdwd
τ
α
τ
τ ϕ
ϕ
( ) ( )
( )
( )
( )
( )∫




×+





∂
∂
∂
∂
+
∂
∂
×
zL
TTTn Twvug
w
wvuT
Twvug
wu
wvuT
Twvugwc
0
00
2
00
2
,,,
,,,
,,,
,,,
,,,
ττ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−



∂
∂
×
∞
=1 0 0 0
0
00
2
00
2
2
,,,
,,,
n
t L L
nnnTnTnnn
zyx
dass
x y
vcucetezcycxc
LLL
T
wvuT
dudvdwd
v
wvuT
τ
α
ϕ
τ
ττ ϕ
ϕ
( ) ( ) ( ) ( )
( )∫














∂
∂
+





∂
∂
+





∂
∂
× +
zL
n
wvuT
dudvdwd
w
wvuT
v
wvuT
u
wvuT
wc
0
1
00
2
00
2
00
2
00
,,,
,,,,,,,,,
τ
ττττ
ϕ
.
Integro-differential form of equations of systems (4) could be written as
( ) ( ) ( ) ( ) ( ) ( )




∫ ∫ ×








∂∂
∂
−
∂∂
∂
−
∂
∂
−+=
t z
zyx
xx
wx
wyxu
yx
wyxu
x
wyxu
ttzyxutzyxu
0 0
22
2
2
,,,,,,,,,
5,,,,,,
τττ
τφ
( )
( )[ ]
( ) ( ) ( ) ( )
( )
−∫ ∫
+







∂∂
∂
−
∂∂
∂
−
∂
∂
−
+
×
∞
0 0
22
2
2
1
,,,,,,,,,
5
616
z
zyx
d
w
wdwE
wx
wyxu
yx
wyxu
x
wyxut
w
dwdwE
τ
σ
τττ
σ
τ
( ) ( ) ( ) ( ) ( )
∫ ∫ 


+
∂∂
∂
+∫ ∫








∂∂
∂
+
∂∂
∂
+
∂
∂
−
∞ t z
x
z
zyx
yx
wyxu
dwd
wx
wyxu
yx
wyxu
x
wyxu
wKt
0 0
2
0 0
22
2
2
,,,
2
1,,,,,,,,, τ
τ
τττ

13. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
43
( ) ( )
( )
( ) ( )
( ) ( )
×∫ ∫








∂
∂
+
∂∂
∂
−+−
+



∂
∂
+
t z
yxy
w
wyxu
wx
wyxu
tdt
w
wdwE
y
wyxu
0 0
2
22
2
2
,,,,,,
2
1
1
,,, ττ
τττ
σ
τ
( )
( )
( ) ( ) ( )
( )
( )
∫ ∫ 


+
∂∂
∂
−∫ ∫
+







∂
∂
+
∂∂
∂
−
+
×
∞∞
0 0
2
0 0
2
22
,,,
1
,,,,,,
21
z
x
z
yx
wx
wyxu
d
w
wdwE
y
wyxu
yx
wyxut
d
w
wdwE τ
τ
σ
ττ
τ
σ
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ×∫ ∫+∫ ∫
∂
∂
−−
+



∂
∂
+
∞
0 00 0
2
2
,,,
21
,,, zt z
y
wKtdwd
x
wyxT
wKwt
t
d
w
wdwE
w
wyxu
τ
τ
βττ
σ
τ
( ) ( ) ( ) ( ) ( ) ( )−Φ−∫ ∫
∂
∂
−+
∂
∂
× tzyxdwd
wyxu
wtdwd
x
wyxT
w x
t z
x
,,,
,,,,,,
1
0 0
2
2
τ
τ
τ
ρττ
τ
β
( )
( )



∫ ∫
∂
∂
−
∞
0 0
2
2
,,,z
x
dwd
wyxu
wt τ
τ
τ
ρ ,
( ) ( ) ( )
( ) ( ) ( )
( )



×−∫ ∫
+







∂∂
∂
+
∂
∂
−+=
21
,,,,,,
2
1
,,,,,,
0 0
2
2
2
t
d
w
wdwE
yx
wyxu
x
wyxu
ttzyxutzyxu
t z
xy
yy τ
σ
ττ
τφ
( ) ( ) ( )
( )
( ) ( )
∫ ∫




−
∂∂
∂
−
∂
∂
+∫ ∫
+







∂∂
∂
+
∂
∂
×
∞ t z
xy
z
xy
yx
wyxu
y
wyxu
d
w
wdwE
yx
wyxu
x
wyxu
0 0
2
2
2
0 0
2
2
2
,,,,,,
5
6
1
1
,,,,,, ττ
τ
σ
ττ
( ) ( )
( )
( )
( ) ( ) ( )
×∫ ∫








∂∂
∂
−
∂∂
∂
−
∂
∂
−−
+



∂∂
∂
−
∞
0 0
22
2
22
,,,,,,,,,
5
1
,,, z
zxyz
wy
wyxu
yx
wyxu
y
wyxu
dt
w
wdwE
wy
wyxu τττ
ττ
σ
τ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) +∫ ∫
∂
∂
+∫ ∫
∂
∂
−−
+
×
∞
0 00 0
,,,,,,
61
zt z
dwd
y
wyxT
wKwtdwd
y
wyxT
wwKt
t
d
w
wdwE
τ
τ
βτ
τ
βττ
σ
( ) ( )
( ) ( ) ( )
( ) ×∫ ∫−∫ ∫








∂∂
∂
+
∂
∂
+
∂∂
∂
−+
∞
0 00 0
2
2
22
,,,,,,,,, zt z
yyy
wKtdwd
wy
wyxu
y
wyxu
yx
wyxu
wKt τ
τττ
τ
( ) ( ) ( )
( )
( )
∫ 


+
∂
∂
−+








∂∂
∂
+
∂
∂
+
∂∂
∂
×
t
yyyy
z
zyxu
tdwd
wy
wyxu
y
wyxu
yx
wyxu
0
2
2
22
,,,,,,,,,,,, τ
ττ
τττ
( ) ( )
( )[ ]
( )
( )[ ]
( ) ( ) +∫ 





∂
∂
+
∂
∂
+
−
+



∂
∂
+
∞
0
,,,,,,
1212
,,,
τ
ττ
σσ
τ
τ
d
y
zyxu
z
zyxu
z
zEt
z
zE
d
y
zyxu zyz
( )
( )
( ) ( )
( )
( )




Φ−∫ ∫
∂
∂
−−∫ ∫
∂
∂
+
∞
tzyxdwd
wyxu
wtdwd
wyxu
w y
t z
y
z
y
,,,
,,,,,,
1
0 0
2
2
0 0
2
2
τ
τ
τ
ρττ
τ
τ
ρ ,
( ) ( ) ( )
( ) ( ) ( )
( )



×−∫ ∫
+






∂∂
∂
+
∂
∂
−+=
21
,,,,,,
2
1
,,,,,,
0 0
2
2
2
t
d
w
wdwE
wx
wyxu
x
wyxu
ttzyxutzyxu
t z
xz
zz τ
σ
ττ
τφ
( ) ( ) ( )
( )
( ) ( )
×∫ ∫








∂∂
∂
+
∂
∂
+∫ ∫
+






∂∂
∂
+
∂
∂
×
∞ t z
yz
z
xz
wy
wyxu
y
wyxu
d
w
wdwE
wx
wyxu
x
wyxu
0 0
2
2
2
0 0
2
2
2
,,,,,,
1
,,,,,, ττ
τ
σ
ττ
( )
( )
( ) ( ) ( ) ( )
( )
( ) ( )×∫ ∫−−∫ ∫
+







∂∂
∂
+
∂
∂
−
−
+
×
∞ t zz
yz
wtd
w
wdwE
wy
wyxu
y
wyxut
d
t
w
wdwE
0 00 0
2
2
2
1
,,,,,,
221
βττ
σ
ττ
τ
τ
σ
( ) ( ) ( ) ( ) ( ) ( ) ( )
∫ 


−
∂
∂
−+∫ ∫
∂
∂
+
∂
∂
×
∞ t
z
z
z
zyxu
tdwd
w
wyxT
wwKtdwd
w
wyxT
wK
00 0
,,,
5
,,,,,, τ
ττ
τ
βτ
τ
( ) ( ) ( )
( )[ ]
( )
( )
( ) ( )
∫ 


−
∂
∂
−
∂
∂
+
−
+



∂
∂
−
∂
∂
−
∞
0
,,,,,,
5
116
,,,,,,
x
zyxu
z
zyxu
z
zE
z
zE
d
y
zyxu
x
zyxu xzyx ττ
σσ
τ
ττ

14. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
44
( )
( ) ( ) ( ) ( ) ( ) −∫ 





∂
∂
+
∂
∂
+
∂
∂
−+


∂
∂
−
t
zyxy
d
z
zyxu
y
zyxu
x
zyxu
tzK
t
d
y
zyxu
0
,,,,,,,,,
6
,,,
τ
τττ
ττ
τ
( ) ( ) ( ) ( ) ( )




Φ−∫ 





∂
∂
+
∂
∂
+
∂
∂
−
∞
tzyxd
z
zyxu
y
zyxu
x
zyxu
zKt z
zyx
,,,
,,,,,,,,,
1
0
τ
τττ
,
where ( ) ( ) ( )[ ]∫=Φ
z
i
is wdtwyxuwtzyx s
0
,,,,,, ρ , s=x,y,z, φ=L/Θ2
E0, E0 is the average value of Young
modulus.
The second-order approximations of components of displacement vector could be written as
( ) ( ) ( ) ( ) ( )




∫ ∫




−
∂∂
∂
−
∂
∂
−++=
t z
yx
xuxx
yx
wyxu
x
wyxu
ttzyxutzyxu
0 0
1
2
2
1
2
1122
,,,,,,
5
6
1
,,,,,,
ττ
τφα
( ) ( )
( )
( ) ( ) ( )
∫ ∫ ×








∂∂
∂
−
∂∂
∂
−
∂
∂
−
+



∂∂
∂
−
∞
0 0
1
2
1
2
2
1
2
1
2
,,,,,,,,,
5
1
,,, z
zyxz
wx
wyxu
yx
wyxu
x
wyxu
d
w
wdwE
wx
wyxu τττ
τ
σ
τ
( )
( )
( ) ( ) ( ) ( ) ( ) −∫ ∫








∂∂
∂
+
∂∂
∂
+
∂
∂
−+
+
×
t z
zyx
dwd
wx
wyxu
yx
wyxu
x
wyxu
wKt
t
d
w
wdwE
0 0
1
2
1
2
2
1
2
,,,,,,,,,
61
τ
τττ
ττ
σ
( ) ( ) ( ) ( ) ( )
∫ ∫ 


+
∂∂
∂
+∫ ∫








∂∂
∂
+
∂∂
∂
+
∂
∂
−
∞ t z
x
z
zyx
yx
wyxu
dwd
wx
wyxu
yx
wyxu
x
wyxu
wKt
0 0
1
2
0 0
1
2
1
2
2
1
2
,,,,,,,,,,,, τ
τ
τττ
( ) ( )
( )
( ) ( ) ( ) ( )
( )
×∫ ∫
+







∂
∂
+
∂∂
∂
+
−
+



∂
∂
+
t z
yxy
w
wdwE
y
wyxu
yx
wyxu
d
t
w
wdwE
y
wyxu
0 0
2
1
2
1
2
2
1
2
1
,,,,,,
2
1
21
,,,
σ
ττ
τ
τ
σ
τ
( ) ( ) ( ) ( ) ( )
( )
( )
( )
×∫ ∫
+
−∫ ∫
+







∂
∂
+
∂∂
∂
−+−×
∞
0 00 0
2
1
2
1
2
121
,,,,,,
2
1 zt z
yx
w
wEt
d
w
wdwE
w
wyxu
wx
wyxu
tdt
σ
τ
σ
ττ
τττ
( ) ( ) ( ) ( )
×∫ ∫








∂
∂
+
∂∂
∂
−








∂
∂
+
∂∂
∂
×
∞
0 0
2
1
2
1
2
2
1
2
1
2
,,,,,,
2
,,,,,, z
yxyx
w
wyxu
wx
wyxut
dwd
y
wyxu
yx
wyxu ττ
τ
ττ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) (∫ −∫
∂
∂
−∫ ∫
∂
∂
+
+
×
∞ t zz
twd
x
wyxT
wwKdwd
x
wyxT
wKwtd
w
wdwE
0 00 0
,,,,,,
1
τ
χτ
τ
χτ
σ
) ( ) ( ) ( ) ( ) ( ) −∫ ∫
∂
∂
+∫ ∫
∂
∂
−−−
∞
0 0
2
1
2
0 0
2
1
2
,,,,,, z
x
t z
x
dwd
wyxu
wtdwd
wyxu
wtd τ
τ
τ
ρτ
τ
τ
ρτττ
( ) ( )}tzyxtzyx xxux ,,,,,, 102 Φ−Φ−α ,
( ) ( )
( ) ( ) ( )
( )



−∫ ∫
+







∂∂
∂
+
∂
∂−
++=
t z
xy
yuyy d
w
wdwE
yx
wyxu
x
wyxut
tzyxutzyxu
0 0
1
2
2
1
2
122
1
,,,,,,
2
,,,,,, τ
σ
τττ
α
( ) ( ) ( )
( )
( ) ( )
∫ ∫




−
∂∂
∂
−
∂
∂
+∫ ∫
+







∂∂
∂
+
∂
∂
−
∞ t z
xy
z
xy
yx
wyxu
y
wyxu
d
w
wdwE
yx
wyxu
x
wyxut
0 0
1
2
2
1
2
0 0
1
2
2
1
2
,,,,,,
5
1
,,,,,,
2
ττ
τ
σ
ττ
( ) ( )
( )
( ) ( ) ( ) ×∫ ∫








∂∂
∂
−
∂∂
∂
−
∂
∂
−
−
+



∂∂
∂
−
∞
0 0
1
2
1
2
2
1
2
1
2
,,,,,,,,,
5
61
,,, z
zxyz
wy
wyxu
yx
wyxu
y
wyxu
d
t
w
wdwE
wy
wyxu τττ
τ
τ
σ
τ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) −∫ ∫
∂
∂
+∫ ∫
∂
∂
−−
+
×
∞
0 00 0
,,,,,,
61
zt z
dwd
y
wyxT
wwKtdwd
y
wyxT
wKwt
t
d
w
wdwE
τ
τ
χτ
τ
χττ
σ

15. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
45
( ) ( )
( ) ( ) ( )
( )∫ ∫ ×−∫ ∫








∂∂
∂
+
∂
∂
+
∂∂
∂
−−
∞
0 00 0
1
2
2
1
2
1
2
,,,,,,,,, zt z
yyy
wKtdwd
wy
wyxu
y
wyxu
yx
wyxu
wKt τ
τττ
τ
( ) ( ) ( )
( )
( )
∫ 


+
∂
∂
−+








∂∂
∂
+
∂
∂
+
∂∂
∂
×
t
yyyy
z
zyxu
tdwd
wy
wyxu
y
wyxu
yx
wyxu
0
11
2
2
1
2
1
2
,,,
2
1,,,,,,,,, τ
ττ
τττ
( ) ( )
( )
( )
( )[ ]
( ) ( )
×−∫ 





∂
∂
+
∂
∂
+
−
+



∂
∂
+
∞
2
0
111 ,,,,,,
121
,,,
uy
zyz
d
y
zyxu
z
zyxu
z
zEt
z
zE
d
y
zyxu
ατ
ττ
σσ
τ
τ
( ) ( )} 110 ,,,,,, φtzyxtzyx yy Φ−Φ× ,
( ) ( )
( ) ( ) ( )
( )



×∫ ∫
+






∂∂
∂
+
∂
∂
++=
t z
xz
zzz
w
wdwE
wx
wyxu
x
wyxu
tzyxutzyxu
0 0
1
2
2
1
2
1122
1
,,,,,,
2
1
,,,,,,
σ
ττ
φα
( ) ( ) ( ) ( )
( )
( ) ( )
∫ ∫ 


+
∂
∂
−+∫ ∫
+






∂∂
∂
+
∂
∂
−−×
∞ t z
z
z
xz
y
wyxu
td
w
wdwE
wx
wyxu
x
wyxut
dt
0 0
2
1
2
0 0
1
2
2
1
2
,,,
2
1
1
,,,,,,
2
τ
ττ
σ
ττ
ττ
( ) ( )
( )
( ) ( ) ( ) ( ) ( )
∫ ∫ 


+
∂
∂
−∫ ∫
∂
∂
−−
+



∂∂
∂
+
∞
0 0
2
1
2
0 0
1
2
,,,,,,
1
,,, z
z
t z
y
y
wyxu
dwd
w
wyxT
wwKtd
w
wdwE
wy
wyxu τ
τ
τ
χττ
σ
τ
( ) ( )
( )
( ) ( ) ( ) ( )
( )[ ]
( )×∫ −
+
+∫ ∫
∂
∂
+
+



∂∂
∂
+
∞ tz
y
t
z
zE
dwd
w
wyxT
wwKt
t
d
w
wdwE
wy
wyxu
00 0
1
2
16
,,,
21
,,,
τ
σ
τ
τ
χτ
σ
τ
( ) ( ) ( ) ( ) ( )
∫ 


−
∂
∂
−
∂
∂
−





∂
∂
−
∂
∂
−
∂
∂
×
∞
0
11111 ,,,,,,
5
6
,,,,,,,,,
5
x
zyxu
z
zyxut
d
y
zyxu
x
zyxu
z
zyxu xzyxz ττ
τ
τττ
( ) ( )
( )[ ]
( ) ( ) ( ) ( ) ×∫ 





∂
∂
+
∂
∂
+
∂
∂
−+
+



∂
∂
−
t
zyxy
d
z
zyxu
y
zyxu
x
zyxu
t
z
zEt
d
y
zyxu
0
1111 ,,,,,,,,,
16
,,,
τ
τττ
τ
σ
τ
τ
( ) ( )
( ) ( ) ( )
( )−Φ−∫ 





∂
∂
+
∂
∂
+
∂
∂
−×
∞
tzyxd
z
zyxu
y
zyxu
x
zyxu
zKtzK zz
zyx
,,,
,,,,,,,,,
02
0
111
ατ
τττ
( )}tzyxz ,,,1Φ− .
Calculation of the average values αuβ2 leads to the following results
( ) ( ) ( ) ( ) ( )
( )
(




∫ ∫ ∫ ∫ −
+







∂∂
∂
−
∂∂
∂
−
∂
∂
−Θ=
Θ
0 0 0 0
1
2
1
2
2
1
2
2
2
1
,,,,,,,,,
5
12
1 x y zL L L
z
zyx
zux L
z
zdzE
zx
tzyxu
yx
tzyxu
x
tzyxu
tL
σ
α
) ( ) ( ) ( ) ( ) ( )
( )
×∫ ∫ ∫ ∫
+
−








∂∂
∂
−
∂∂
∂
−
∂
∂Θ
−−
∞
0 0 0 0
1
2
1
2
2
1
22
1
,,,,,,,,,
5
12
x y zL L L
zzyx
z
zL
zx
tzyxu
yx
tzyxu
x
tzyxu
zEtdxdydz
σ
( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫








∂∂
∂
+
∂∂
∂
+
∂
∂
−−Θ+×
Θ
0 0 0 0
1
2
1
2
2
1
2
2 ,,,,,,,,,
2
1 x y zL L L
zyx
z
zx
tzyxu
yx
tzyxu
x
tzyxu
zLttdxdydzd
( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫








∂∂
∂
+
∂∂
∂
+
∂
∂
−
Θ
−×
∞
0 0 0 0
1
2
1
2
2
1
22
,,,,,,,,,
2
x y zL L L
zyx
z
zx
tzyxu
yx
tzyxu
x
tzyxu
zLtdxdydzdzK
( ) ( ) ( ) ( ) ( )
( )
×∫ ∫ ∫ ∫
+
−








∂
∂
+
∂∂
∂
−Θ+×
Θ
0 0 0 0
2
1
2
1
2
2
1
,,,,,,
2
1 x y zL L L
zyx
ydzd
z
zL
y
tzyxu
yx
tzyxu
zEttdxdydzdzK
σ

16. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
46
( ) ( ) ( ) ( ) ( )
( )
×
Θ
−∫ ∫ ∫ ∫
+







∂
∂
+
∂∂
∂
−−Θ+×
Θ
41
,,,,,,
2
1 2
0 0 0 0
2
1
2
1
2
2
x y zL L L
yx
z tdxdyd
z
zdzE
z
tzyxu
zx
tzyxu
zLttdxd
σ
( )( )
( )
( ) ( )
( )
( )∫ ∫ ∫ ∫ ×
+
−
−Θ+∫ ∫ ∫ ∫








∂
∂
+
∂∂
∂
+
−
×
Θ∞
0 0 0 0
2
0 0 0 0
2
1
2
1
2
14
1,,,,,,
1
x y zx y z L L L
z
L L L
yxz
z
zL
ttdxdydzd
y
tzyxu
yx
tzyxu
z
zL
zE
σσ
( ) ( )
( ) ( ) ( ) ( )×∫ ∫ ∫ ∫
∂
∂
−+








∂
∂
+
∂∂
∂
×
∞
0 0 0 0
2
1
2
2
1
2
1
2
,,,
2
,,,,,, x y zL L L
x
z
yx
t
tzyxuz
zLtdxdydzdzE
z
tzyxu
zx
tzyxu ρ
( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ −
Θ
+∫ ∫ ∫ ∫ −−Θ−Θ×
∞Θ
0 0 0 0
2
0 0 0 0
1
2
2
,,,
x y zx y z L L L
z
L L L
xz zLtdxdydzdtzyxuzzLttdxdydzd ρ
( )
( ) ( ) ( ) ( ) 



∫ −Θ



ΘΧ−∞Χ
Θ
+
∂
∂
×
zL
zxx
x
zdzzLLtdxdydzd
t
tzyxu
0
20
2
2
1
2
4
2
1
2
,,,
ρ ,
( ) ( )
( )
( )
( ) ( )




×
Θ
−∫ ∫ ∫ ∫








∂∂
∂
+
∂
∂
+
−
−Θ=
Θ
2
,,,,,,
12
1 2
0 0 0 0
1
2
2
1
2
2
2
x y zL L L
xyz
zuy tdxdydzd
yx
tzyxu
x
tzyxu
z
zL
zEL
σ
τα
( )
( )
( )
( ) ( )
( )
( )
×∫ ∫ ∫ ∫
+
−
+∫ ∫ ∫ ∫








∂∂
∂
+
∂
∂
+
−
×
Θ∞
0 0 0 00 0 0 0
1
2
2
1
2
112
1,,,,,,
1
x y zx y z L L L
z
L L L
xyz
z
zL
zEtdxdydzd
yx
tzyxu
x
tzyxu
z
zL
zE
σσ
( ) ( ) ( )
( ) ( ) ×∫ ∫ ∫ ∫
Θ
−−Θ








∂∂
∂
−
∂∂
∂
−
∂
∂
×
∞
0 0 0 0
2
21
2
1
2
2
1
2
12
,,,,,,,,,
5
x y zL L L
zxy
zEtdxdydzd
zy
tzyxu
yx
tzyxu
y
tzyxu
τ
( )
( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫−Θ+








∂∂
∂
−
∂∂
∂
−
∂
∂
+
−
×
Θ
0 0 0 0
21
2
1
2
2
1
2
2
,,,,,,,,,
5
1
x y zL L L
zxyz zK
ttdxdydzd
zy
tzyxu
yx
tzyxu
y
tzyxu
z
zL
σ
( )
( ) ( ) ( )
( )( )×∫ ∫ ∫ ∫ −−








∂∂
∂
+
∂
∂
+
∂∂
∂
−×
∞
0 0 0 0
1
2
2
1
2
1
2
,,,,,,,,, x y zL L L
z
yyy
z zLzKtdxdydzd
zy
tzyxu
y
tzyxu
yx
tzyxu
zL
( ) ( ) ( ) ( )
∫ ∫ ∫ ∫ 


+
∂
∂
+








∂∂
∂
+
∂
∂
+
∂∂
∂Θ
×
Θ
0 0 0 0
11
2
2
1
2
1
22
,,,
2
1,,,,,,,,,
2
x y zL L L
yyyy
z
tzyxu
tdxdydzd
zy
tzyxu
y
tzyxu
yx
tzyxu
( ) ( )
( )
( )
( ) ( ) ( )
( )
×∫ ∫ ∫ ∫
+






∂
∂
+
∂
∂Θ
−−Θ
+



∂
∂
+
∞
0 0 0 0
11
2
21
1
,,,,,,
21
,,, x y zL L L
zyz
z
zdzE
y
tzyxu
z
tzyxu
tdxdyd
z
zdzE
y
tzyxu
σ
τ
σ
( ) ( )
( ) ( )
×∫ ∫ ∫ ∫
∂
∂Θ
+∫ ∫ ∫ ∫
∂
∂
−Θ−×
∞Θ
0 0 0 0
2
1
22
0 0 0 0
2
1
2
2 ,,,
2
,,,
2
1 x y zx y z L L L
y
L L L
y
t
tzyxu
tdxdydzd
t
tzyxu
zttdxdyd ρ
( ) ( ) ( ) ( )
( ) ( )
×


∞Χ
Θ+
ΘΧ
−∫ ∫ ∫ ∫ −−×
Θ
22
,,,
022
0 0 0 0
1
yy
L L L
yz
x y z
tdxdydzdtzyxuzzLtdxdydzdz ρρ
( ) ( )
1
0
4
−




∫ −Θ×
zL
z zdzzLL ρ ,
( ) ( )
( )
( )
( ) ( )




×
Θ
−∫ ∫ ∫ ∫ 





∂∂
∂
+
∂
∂
+
−
−Θ=
Θ
2
,,,,,,
12
1 2
0 0 0 0
1
2
2
1
2
2
2
x y zL L L
xzz
zz tdxdydzd
zx
tzyxu
x
tzyxu
z
zL
zEtL
σ
α
( )( )
( )
( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫−Θ+∫ ∫ ∫ ∫ 





∂∂
∂
+
∂
∂
+
−
×
Θ∞
0 0 0 0
2
0 0 0 0
1
2
2
1
2
2
1,,,,,,
1
x y zx y z L L LL L L
xzz
zEttdxdydzd
zx
tzyxu
x
tzyxu
z
zL
zE
σ
( ) ( ) ( ) ( )
∫ ∫ ∫ ∫ ×








∂∂
∂
+
∂
∂
−








∂∂
∂
+
∂
∂
×
∞
0 0 0 0
1
2
2
1
2
1
2
2
1
2
,,,,,,,,,,,, x y zL L L
yzyz
zy
tzyxu
y
tzyxu
tdxdydzd
zy
tzyxu
y
tzyxu

17. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
47
( ) ( )
( )[ ]
( ) ( ) ( )
∫ ∫ ∫ ∫ ×





∂
∂
−
∂
∂
−
∂
∂
+
+
−
Θ×
Θ
0 0 0 0
1112
,,,,,,,,,
5
12
1
12
x y zL L L
yxzz
y
tzyxu
x
tzyxu
z
tzyxu
tdxdydzd
z
zLzE
σ
( )
( )
( )
( ) ( ) ( )
∫ ∫ ∫ ∫ ×





∂
∂
−
∂
∂
−
∂
∂Θ
−−Θ
+
×
∞
0 0 0 0
111
2
2 ,,,,,,,,,
5
121
x y zL L L
yxz
y
tzyxu
x
tzyxu
z
tzyxu
tdtxdyd
z
zdzE
σ
( )
( )
( ) ( ) ( ) ( ) ( )
×∫ ∫ ∫ ∫ 





∂
∂
+
∂
∂
+
∂
∂
−Θ+
+
×
Θ
0 0 0 0
1112 ,,,,,,,,,
21
x y zL L L
zyx
z
tzyxu
y
tzyxu
x
tzyxuzK
ttdxdyd
z
zdzE
σ
( )
( )
( ) ( ) ( )
×∫ ∫ ∫ ∫ 





∂
∂
+
∂
∂
+
∂
∂Θ
−
ΘΧ
−×
∞
0 0 0 0
111
2
2 ,,,,,,,,,
22
x y zL L L
zyxz
z
tzyxu
y
tzyxu
x
tzyxu
zKtdxdydzd
( ) ( ) ( )
( )
( ) ×



∫ −


∞Χ
Θ+∫ ∫ ∫ ∫ −−×
Θ zx y z L
z
z
L L L
zz zLtdxdydzdztzyxuzLtdxdydzd
0
02
0 0 0 0
1 4
2
,,, ρ
( ) } 1−
Θ× Lzdzρ .
System of equations for the functions Cijk(x,y,z,t) (i≥0, j≥0, k≥0) could be written as
( ) ( ) ( ) ( )






∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
000
2
2
000
2
2
000
2
0
000 ,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
L ;
( )
( )
( )
( )
( )








+





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂ −−
y
tzyxC
Tzg
yx
tzyxC
Tzg
x
D
t
tzyxC i
L
i
LL
i ,,,
,
,,,
,
,,, 100100
0
00
( )
( ) ( ) ( ) ( )










∂
∂
+
∂
∂
+
∂
∂
+










∂
∂
∂
∂
+ −
2
00
2
2
00
2
2
00
2
100 ,,,,,,,,,,,,
,
z
tzyxC
y
tzyxC
x
tzyxC
z
tzyxC
Tzg
z
iiii
L , i≥1;
( ) ( ) ( ) ( )
+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
010 ,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
L
( )
( )
( ) ( )
( )
( )




+





∂
∂
∂
∂
+





∂
∂
∂
∂
+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
x
D L
,,,
,,,
,,,,,,
,,,
,,, 000000000000
0 γ
γ
γ
γ
( )
( )
( )









∂
∂
∂
∂
+
z
tzyxC
TzyxP
tzyxC
z
,,,
,,,
,,, 000000
γ
γ
;
( ) ( ) ( ) ( )
×+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
LL D
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
02
020
2
2
020
2
2
020
2
0
020 ,,,,,,,,,,,,
( )
( )
( )
( )
( )
( )







×
∂
∂
∂
∂
+





∂
∂
∂
∂
×
−
y
tzyxC
tzyxC
yy
tzyxC
TzyxP
tzyxC
tzyxC
x
,,,
,,,
,,,
,,,
,,,
,,, 000
010
000
1
000
010 γ
γ
( )
( )
( )
( )
( )
( ) ( )
( )


×
∂
∂
+










∂
∂
∂
∂
+


×
−−
TzyxP
tzyxC
x
D
z
tzyxC
TzyxP
tzyxC
tzyxC
zTzyxP
tzyxC
L
,,,
,,,,,,
,,,
,,,
,,,
,,,
,,, 000
0
000
1
000
010
1
000
γ
γ
γ
γ
γ
γ
( ) ( )
( )
( ) ( )
( )
( )










∂
∂
∂
∂
+





∂
∂
∂
∂
+


∂
∂
×
z
tzyxC
TzyxP
tzyxC
zy
tzyxC
TzyxP
tzyxC
yx
tzyxC ,,,
,,,
,,,,,,
,,,
,,,,,, 010000010000010
γ
γ
γ
γ
;
( ) ( ) ( ) ( )
+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
001 ,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
L

18. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
48
( )[ ] ( )
( )
( )
( )
+







 ∇
∫





++
∂
∂
Ω+
Tk
tzyx
WdtWyxC
TzyxP
tzyxC
Tzyxg
x
D S
L
SLSLSSL
z ,,,
,,,
,,,
,,,
1,,,1
0
000
000
0
µ
ξε γ
γ
( )[ ] ( )
( )
( )
( )







 ∇
∫





++
∂
∂
Ω+
Tk
tzyx
WdtWyxC
TzyxP
tzyxC
Tzyxg
y
D S
L
SLSLSSL
z ,,,
,,,
,,,
,,,
1,,,1
0
000
000
0
µ
ξε γ
γ
;
( ) ( ) ( ) ( ) +





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
002 ,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
L
( )[ ] ( )
( )
( )
( )
+







 ∇
∫





++
∂
∂
Ω+
Tk
tzyx
WdtWyxC
TzyxP
tzyxC
Tzyxg
x
D S
L
SLSLSSL
z ,,,
,,,
,,,
,,,
1,,,1
0
001
000
0
µ
ξε γ
γ
( )[ ] ( )
( )
( )
( )
+







 ∇
∫





++
∂
∂
Ω+
Tk
tzyx
WdtWyxC
TzyxP
tzyxC
Tzyxg
y
D S
L
SLSLSSL
z ,,,
,,,
,,,
,,,
1,,,1
0
001
000
0
µ
ξε γ
γ
( )[ ] ( )
( )
( )
( )




×∫





++
∂
∂
Ω+
−
zL
SLSLSSL WdtWyxC
TzyxP
tzyxC
tzyxCTzyxg
x
D
0
000
1
000
0010 ,,,
,,,
,,,
,,,1,,,1 γ
γ
γ
ξε
( ) ( ) ( )
( )
( ) ( ) ×



∫
∇






+
∂
∂
+


∇
×
−
zL
S
S
S
WdtWyxC
Tk
tzyx
TzyxP
tzyxC
tzyxC
yTk
tzyx
0
000
1
000
001 ,,,
,,,
,,,
,,,
,,,1
,,, µ
ξ
µ
γ
γ
γ
( )[ ]} SLLSLS DTzyxg 0,,,1 Ω+× ε ;
( ) ( ) ( ) ( ) ( )







×
∂
∂
∂
∂
+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
x
tzyxC
xz
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
L
,,,,,,,,,,,,,,, 010
2
110
2
2
110
2
2
110
2
0
110
( ) ( )
( )
( )
( )
+










∂
∂
∂
∂
+





∂
∂
∂
∂
+




× LLLL D
z
tzyxC
Tzyxg
zy
tzyxC
Tzyxg
y
Tzyxg 0
010010 ,,,
,,,
,,,
,,,,,,
( )
( )
( ) ( )
( )
( ) ( )
( )






×
∂
∂
+





∂
∂
∂
∂
+





∂
∂
∂
∂
+
TzyxP
tzyxC
zy
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
x ,,,
,,,,,,
,,,
,,,,,,
,,,
,,, 000100000100000
γ
γ
γ
γ
γ
γ
( ) ( ) ( )
( )
( ) ( )
( )






×
∂
∂
+





∂
∂
∂
∂
+







∂
∂
×
−−
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x
D
z
tzyxC
L
,,,
,,,,,,
,,,
,,,
,,,
,,, 1
000000
1
000
1000
100
γ
γ
γ
γ
( )
( )
( )
( )
( )
( )
LD
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC
tzyxC 0
000
1
000
100
000
100
,,,
,,,
,,,
,,,
,,,
,,,










∂
∂
∂
∂
+


∂
∂
×
−
γ
γ
;
( ) ( ) ( ) ( ) ( )







×
∂
∂
∂
∂
+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
y
tzyxC
xz
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
L
,,,,,,,,,,,,,,, 000
2
100
2
2
100
2
2
100
2
0
101
( ) ( ) ( ) ( ) ( ) +










∂
∂
∂
∂
+





∂
∂
∂
∂
+




× LLLL D
z
tzyxC
Tzyxg
zy
tzyxC
Tzyxg
y
Tzyxg 0
000000 ,,,
,,,
,,,
,,,,,,
( )[ ] ( ) ( )
( )
( )



×∫





++
∂
∂
Ω+
−
zL
SLSLSSL WdtWyxC
TzP
tzyxC
tzyxCTzyxg
x
D
0
100
1
000
1000 ,,,
,
,,,
,,,1,,,1 γ
γ
ξε
( )
( )[ ] ( )
( )
( )
( )
( )
( )[ ] ( )



×∫+
∂
∂
+


∇
∫×
×










++
∂
∂
Ω+


∇
×
−
zz L
LSLS
S
L
SLSLSSL
S
WdtWyxCTzyxg
xTk
tzyx
WdtWyxC
TzP
tzyxC
tzyxCTzyxg
y
D
Tk
tzyx
0
100
0
100
1
000
1000
,,,,,,1
,,,
,,,
,
,,,
,,,1,,,1
,,,
ε
µ
ξε
µ
γ
γ

19. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
49
( ) ( ) ( )[ ] ( )



×
∇
+
∂
∂
+Ω



∫
∇
×
Tk
tzyx
Tzyxg
y
DWdtWyxC
Tk
tzyx S
LSLSSL
L
S
z ,,,
,,,1,,,
,,,
0
0
100
µ
ε
µ
( ) ( ) SL
LL
DWdtWyxCWdtWyxC
zz
0
0
100
0
100 ,,,,,, Ω



∫∫× ;
( ) ( ) ( ) ( ) ( )
( )





×
∂
∂
+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
TzyxP
tzyxC
xz
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
L
,,,
,,,,,,,,,,,,,,, 000
2
011
2
2
011
2
2
011
2
0
011
γ
γ
( ) ( )
( )
( ) ( )
( )
( ) +










∂
∂
∂
∂
+





∂
∂
∂
∂
+


∂
∂
× LD
z
tzyxC
TzyxP
tzyxC
zy
tzyxC
TzyxP
tzyxC
yx
tzyxC
0
001000001000001 ,,,
,,,
,,,,,,
,,,
,,,,,,
γ
γ
γ
γ
( ) ( )
( )
( ) ( ) ( )







×
∂
∂
∂
∂
+





∂
∂
∂
∂
+
−
y
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x
D L
,,,
,,,
,,,
,,,
,,,
,,, 000
001
000
1
000
0010 γ
γ
( )
( )
( )
( )
( )
( ) ( )



×
∇
∂
∂
+










∂
∂
∂
∂
+


×
−−
Tk
tzyx
xz
tzyxC
TzyxP
tzyxC
tzyxC
zTzyxP
tzyxC S ,,,,,,
,,,
,,,
,,,
,,,
,,, 000
1
000
001
1
000 µ
γ
γ
γ
γ
( )[ ] ( )
( )
( ) ( )×


∇
∂
∂
+Ω



∫





++×
Tk
tzyx
x
DWdtWyxC
TzyxP
tzyxC
Tzyxg S
SL
L
SLSLS
z ,,,
,,,
,,,
,,,
1,,,1 0
0
010
000 µ
ξε γ
γ
( )[ ] ( ) ( )
( )
( ) ×Ω+Ω



∫





++×
−
SLSL
L
SLSLS DDWdtWyxC
TzyxP
tzyxC
tzyxCTzyxg
z
00
0
000
1
000
010 ,,,
,,,
,,,
,,,1,,,1 γ
γ
ξε
( )[ ] ( )
( )
( )
( )
( )








∫
∇






++
∂
∂
×
−
zL
S
SLSLS WdtWyxC
Tk
tzyx
TzyxP
tzyxC
tzyxCTzyxg
y 0
000
1
000
010 ,,,
,,,
,,,
,,,
,,,1,,,1
µ
ξε γ
γ
.
Boundary and initial conditions for functions Cijk(x,t) (i≥0, j≥0, k≥0) could be written as
( )
0
,,,
0
=
=x
ijk
x
tzyxC
∂
∂
;
( )
0
,,,
=
= xLx
ijk
x
tzyxC
∂
∂
;
( )
0
,,,
0
=
=y
ijk
y
tzyxC
∂
∂
;
( )
0
,,,
=
= yLy
ijk
y
tzyxC
∂
∂
;
( )
0
,,,
0
=
=z
ijk
z
tzyxC
∂
∂
;
( )
0
,,,
=
= zLz
ijk
z
tzyxC
∂
∂
; C000(x,y,z,0)=fC (x,y,z); Cijk(x,y,z,0)=0.
Solutions of equations for functions Cijk(x,t) (i≥0, j≥0, k≥0) could be written as
( ) ( ) ( ) ( ) ( )∑+=
∞
=1
0
000
2
,,,
n
nCnnnnC
zyxzyx
C
tezcycxcF
LLLLLL
F
tzyxC ,
where ( )
















++−= 2220
22 111
exp
zyx
LnC
LLL
tDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=
x y zL L L
CnnnnC udvdwdwvufwcvcucF
0 0 0
,, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
2
0
00 ,,,
2
,,,
n
t L L L
LnnnCnCnnnnC
zyx
L
i
x y z
TwvugvcusetezcycxcFn
LLL
D
tzyxC τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂
×
∞
=
−
1 0 0
2
0100 2,,,
n
t L
nnCnCnnnnC
zyx
Li
n
x
ucetezcycxcFn
LLL
D
dudvdwd
u
wvuC
wc τ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∫
∂
∂
×
∞
=
−
1
2
0
0 0
100 2,,,
,,,
n
nCnnn
zyx
L
L L
i
Lnn tezcycxc
LLL
D
dudvdwd
v
wvuC
Twvugwcvs
y z π
τ
τ

20. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
50
( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫
∂
∂
−× −
t L L L
i
LnnnnCnC
x y z
dudvdwd
w
wvuC
TwvugwsvcuceFn
0 0 0 0
100 ,,,
,,, τ
τ
τ , i≥1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
000
2
0
010
,,,
,,,2
,,,
n
t L L L
nnnCnCnnnnC
zyx
L
x y z
TwvuP
wvuC
vcusetezcycxcFn
LLL
D
tzyxC γ
γ
τ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂
×
∞
=1 0 0
2
0000 2,,,
n
t L
nnCnCnnnnC
zyx
L
n
x
ucetezcycxcFn
LLL
D
dudvdwd
u
wvuC
wc τ
π
τ
τ
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ×∑−∫ ∫ ∫
∂
∂
×
∞
=1
2
0
0 0 0
000000 2,,,
,,,
,,,
n
nn
zyx
L
L L L
nnn ycxcn
LLL
D
dudvdwd
v
wvuC
TwvuP
wvuC
wcwcvs
y z z π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
∫ ∫ ∫ ∫
∂
∂
−×
t L L L
nnnnCnCnnC
x y z
dudvdwd
w
wvuC
TwvuP
wvuC
wsvcucetezcF
0 0 0 0
000000 ,,,
,,,
,,,
τ
ττ
τ γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
0102
0
020 ,,,
2
,,,
n
t L L L
nnnCnCnnnnC
zyx
L
x y z
wvuCvcusetezcycxcFn
LLL
D
tzyxC ττ
π
( ) ( )
( )
( ) ( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
2
0000
1
000 2,,,
,,,
,,,
n
nCnnnnC
zyx
L
n tezcycxcFn
LLL
D
dudvdwd
u
wvuC
TwvuP
wvuC
wc
π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) −∫ ∫ ∫ ∫
∂
∂
−×
−t L L L
nnnnC
x y z
dudvdwd
v
wvuC
TwvuP
wvuC
wvuCwcvcuse
0 0 0 0
000
1
000
010
,,,
,,,
,,,
,,, τ
ττ
ττ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫−−
∞
=
−
1 0 0 0 0
1
000
010
,,,
,,,
,,,
n
t L L L
nnnnCnCnnnnC
x y z
TwvuP
wvuC
wvuCwcvcusetezcycxcFn γ
γ
τ
ττ
( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ −−
∂
∂
×
∞
=1 0
2
0000
2
0 2,,,2
n
t
nCnCnnnnC
zyx
L
zyx
L
etezcycxcFn
LLL
D
dudvdwd
w
wvuC
LLL
D
τ
π
τ
τπ
( ) ( ) ( ) ( )
( )
( ) ( )×∑−∫ ∫ ∫
∂
∂
×
∞
=1
2
0
0 0 0
010000 2,,,
,,,
,,,
n
nnC
zyx
L
L L L
nnn xcFn
LLL
D
dudvdwd
u
wvuC
TwvuP
wvuC
wcvcus
x y z π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) −∫ ∫ ∫ ∫
∂
∂
−×
t L L L
nnnnCnCnn
x y z
dudvdwd
v
wvuC
TwvuP
wvuC
wcvsucetezcyc
0 0 0 0
010000 ,,,
,,,
,,,
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( )
( )
( ) ×∑ ∫ ∫ ∫ ∫
∂
∂
−−
∞
=1 0 0 0 0
010000
2
0 ,,,
,,,
,,,2
n
t L L L
nnnnCnC
zyx
L
x y z
dudvdwd
w
wvuC
TwvuP
wvuC
wsvcuceFn
LLL
D
τ
ττ
τ
π
γ
γ
( ) ( ) ( ) ( )tezcycxc nCnnn× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∇
−
Ω
−=
∞
=1 0 0 0 0
2
0
001
,,,2
,,,
n
t L L L
S
nnCnCnnnnC
zyx
L
x y z
Tk
wvu
usetezcycxcFn
LLL
D
tzyxC
τµ
τ
π
( )[ ] ( )
( )
( ) ( ) ( ) −∫





++× ττ
τ
ξε γ
γ
dudvdvcwdwcWdWvuC
TwvuP
wvuC
Twvug nn
L
SLSLS
z
0
000
000
,,,
,,,
,,,
1,,,1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∫−
Ω
−
∞
=1 0 0 0 0 0
0002
0
,,,
2
n
t L L L L
nnnnCnCnnnnC
zyx
L
x y z z
WdWvuCwcvsucetezcycxcFn
LLL
D
ττ
π
( )[ ] ( )
( )
( )
τ
τµτ
ξε γ
γ
dudvdwd
Tk
wvu
TwP
wvuC
Twvug S
SLSLS
,,,
,
,,,
1,,,1 000 ∇






++× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∇
−
Ω
−=
∞
=1 0 0 0 0
2
0
002
,,,2
,,,
n
t L L L
S
nnnCnCnnnnC
zyx
L
x y z
Tk
wvu
vcusetezcycxcFn
LLL
D
tzyxC
τµ
τ
π
( )[ ] ( )
( )
( ) ( ) ×
Ω
−∫





++×
zyx
L
n
L
SLSLS
LLL
D
dudvdwdwcWdWvuC
TwvuP
wvuC
Twvug
z
2
0
0
001
000
,,,
,,,
,,,
1,,,1
π
ττ
τ
ξε γ
γ

21. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
51
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∫
∇
−×
∞
=1 0 0 0 0 0
001 ,,,
,,,
n
t L L L L
S
nnnnCnCnnnnC
x y z z
WdWvuC
Tk
wvu
wcvsucetezcycxcFn τ
τµ
τ
( )[ ] ( )
( )
( ) ( ) ( )×∑
Ω
−





++×
∞
=1
2
0000 2
,,,
,,,
1,,,12
n
nnnnC
zyx
L
SLSLS zcycxcF
LLL
D
dudvdwd
TwvuP
wvuC
Twvug
π
τ
τ
ξε γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ×∫ ∫ ∫ ∫ ∫





+−×
−t L L L L
SnnnnCnC
x y z z
WdWvuC
TwvuP
wvuC
tzyxCwcvcuseten
0 0 0 0 0
000
1
000
001 ,,,
,,,
,,,
,,,1 τ
τ
ξτ γ
γ
γ
( )[ ] ( ) ( ) ( ) ( ) ( ) ×∑
Ω
−
∇
+×
∞
=1
2
02,,,
,,,1
n
nCnnnnC
zyx
LS
LSLS tezcycxcF
LLL
D
dudvdwd
Tk
wvu
Twvug
π
τ
τµ
ε
( ) ( ) ( ) ( ) ( ) ( )
( )
( )∫ ∫ ∫ ∫ ×∫





+
∇
×
−t L L L L
S
S
nnn
x y z z
WdWvuC
TwP
wvuC
tzyxC
Tk
wvu
wcvcusn
0 0 0 0 0
000
1
000
001 ,,,
,
,,,
,,,1
,,,
τ
τ
ξ
τµ
γ
γ
γ
( )[ ] ( ) ττε deudvdwdTwvug nCLSLS −+× ,,,1 ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
2
0
110 ,,,
2
,,,
n
t L L L
LnnnCnCnnnnC
zyx
L
x y z
TwvugvcusetezcycxcFn
LLL
D
tzyxC τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂
×
∞
=1 0 0
2
0010 2,,,
n
t L
nnCnCnnnnC
zyx
L
n
x
ucetezcycxcFn
LLL
D
dudvdwd
u
wvuC
wc τ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∫
∂
∂
×
∞
=1
2
0
0 0
010 2,,,
,,,
n
nnnnC
zyx
L
L L
Lnn zcycxcFn
LLL
D
dudvdwd
v
wvuC
Twvugwcvs
y z π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫
∂
∂
−×
zyx
L
t L L L
LnnnnCnC
LLL
D
dudvdwd
w
wvuC
Twvugwsvcucete
x y z
2
0
0 0 0 0
010 2,,,
,,,
π
τ
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−×
∞
=1 0 0 0 0
100 ,,,,,,
n
t L L L
LnnnnCnCnnnnC
x y z
wvuCTwvugwcvcusetezcycxcFn ττ
( )
( )
( ) ( ) ( ) ( ) ( ) ×∑−
∂
∂
×
∞
=
−
1
2
0000
1
000 2,,,
,,,
,,,
n
nCnnnnC
zyx
L
tezcycxcFn
LLL
D
dudvdwd
u
wvuC
TwvuP
wvuC π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( )×∫ ∫ ∫ ∫
∂
∂
−×
−t L L L
LnnnC
x y z
v
wvuC
TwvuP
wvuC
wvuCTwvugvsuce
0 0 0 0
000
1
000
100
,,,
,,,
,,,
,,,,,,
ττ
ττ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−−×
∞
=1 0 0 0
2
02
n
t L L
nnnCnCnnnnC
zyx
L
n
x y
vcucetezcycxcFn
LLL
D
dudvdwdwc τ
π
τ
( ) ( ) ( ) ( )
( )
( ) −∫
∂
∂
×
−
zL
Ln dudvdwd
w
wvuC
TwvuP
wvuC
wvuCTwvugws
0
000
1
000
100
,,,
,,,
,,,
,,,,,, τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−
∞
=1 0 0 0 0
2
0
,,,
2
n
t L L L
LnnnnCnCnnnnC
zyx
L
x y z
TwvugwcvcusetezcycxcFn
LLL
D
τ
π
( )
( )
( ) ( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=1
2
0100000 2,,,
,,,
,,,
n
nCnnn
zyx
L
tezcycxcn
LLL
D
dudvdwd
u
wvuC
TwvuP
wvuC π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( ) ×∫ ∫ ∫ ∫
∂
∂
−×
t L L L
LnnnCnC
x y z
vdwd
u
wvuC
TwvuP
wvuC
TwvugwcvseF
0 0 0 0
100000 ,,,
,,,
,,,
,,,
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−×
∞
=1 0 0 0 0
2
02
n
t L L L
nnnnCnCnnnnC
zyx
L
n
x y z
wsvcucetezcycxcFn
LLL
D
duduc τ
π
τ
( )
( )
( )
( )
τ
ττ
γ
γ
dudvdwd
u
wvuC
TwP
wvuC
TwvugL
∂
∂
×
,,,
,
,,,
,,, 100000
;

22. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
52
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
2
0
101 ,,,
2
,,,
n
t L L L
LnnnCnCnnnnC
zyx
L
x y z
TwvugvcusetezcycxcFn
LLL
D
tzyxC τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂
×
∞
=1 0 0
2
0000 2,,,
n
t L
nnCnCnnnnC
zyx
L
n
x
ucetezcycxcF
LLL
D
dudvdwd
u
wvuC
wc τ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫
∂
∂
×
∞
=1
2
0
0 0 0
000 2,,,
,,,
n
nn
zyx
L
L L L
Lnnn ycxcn
LLL
D
dudvdwd
v
wvuC
Twvugwcwcvsn
y z z π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∫ ∫ ∫ ∫
∂
∂
−×
t L L L
LnnnnCnCnnC
x y z
dudvdwd
w
wvuC
TwvugwsvcucetezcF
0 0 0 0
000 ,,,
,,, τ
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]×∑ ∫ ∫ ∫ ∫ +−Ω−
∞
=1 0 0 0 0
2
0
,,,1
2
n
t L L L
LSLSnnnCnCnnnnC
zyx
L
x y z
TwvugvcusetezcycxcFn
LLL
D
ετ
π
( ) ( )
( )
( ) ( ) −
∇
∫





+
∇
× τ
τµ
τ
τ
ξ
τµ
γ
γ
dudvdwd
Tk
wvu
WdWvuC
TwvuP
wvuC
Tk
wvu S
L
S
S
z ,,,
,,,
,,,
,,,
1
,,,
0
100
000
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−Ω−
∞
=1 0 0 0 0
2
0
,,,1
2
n
t L L L
LSLSnnnCnCnnnnC
zyx
L
x y z
TwvugvcusetezcycxcFn
LLL
D
ετ
π
( ) ( ) ( )
( )
( ) ( ) −∫
∇






+×
−
ττ
τµτ
τξ γ
γ
dudvdwdWdWvuC
Tk
wvu
TwvuP
wvuC
wvuCwc
zL
S
Sn
0
000
1
000
100 ,,,
,,,
,,,
,,,
,,,1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]×∑ ∫ ∫ ∫ ∫ +−Ω−
∞
=1 0 0 0 0
2
0
,,,1
2
n
t L L L
LSLSnnnCnCnnnnC
zyx
L
x y z
TwvugvsucetezcycxcFn
LLL
D
ετ
π
( )
( )
( )
( )
( )
( ) ττ
τ
τξ
τµ
γ
γ
dudvdwdWdWvuC
TwvuP
wvuC
wvuC
Tk
wvu
wc
zL
S
S
n ∫





+
∇
×
−
0
000
1
000
100 ,,,
,,,
,,,
,,,1
,,,
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
000
2
0
011
,,,
,,,2
,,,
n
t L L L
nnnCnCnnnnC
zyx
L
x y z
TwvuP
wvuC
vcusetezcycxcFn
LLL
D
tzyxC γ
γ
τ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂
×
∞
=1 0 0
2
0001 2,,,
n
t L
nnCnCnnnnC
zyx
L
n
x
ucetezcycxcFn
LLL
D
dudvdwd
u
wvuC
wc τ
π
τ
τ
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ×∑−∫ ∫ ∫
∂
∂
×
∞
=1
2
0
0 0 0
001000 2,,,
,,,
,,,
n
nnnC
zyx
L
L L L
nnn ycxcFn
LLL
D
dudvdwd
v
wvuC
TwvuP
wvuC
wcwcvs
y z z π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ×−∫ ∫ ∫ ∫
∂
∂
−×
zyx
L
t L L L
nnnnCnCn
LLL
D
dudvdwd
w
wvuC
TwvuP
wvuC
wsvcucetezc
x y z
2
0
0 0 0 0
001000 2,,,
,,,
,,, π
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0 0 0 0
000
1
000
001
,,,
,,,
,,,
,,,
n
t L L L
nnnnCnC
x y z
dudvdwd
u
wvuC
TwvuP
wvuC
wvuCwcvcuseF τ
ττ
ττ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−×
∞
=1 0 0 0 0
0012
0
,,,
2
n
t L L L
nnnCnCnnn
zyx
L
nCnnn
x y z
wvuCvsucetezcycxc
LLL
D
tezcycxcn ττ
π
( ) ( )
( )
( ) ( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
2
0000
1
000 2,,,
,,,
,,,
n
nCnnn
zyx
L
nnC tezcycxcn
LLL
D
dudvdwd
v
wvuC
TwvuP
wvuC
wcFn
π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) −∫ ∫ ∫ ∫
∂
∂
−×
−t L L L
nnnnCnC
x y z
dudvdwd
w
wvuC
TwvuP
wvuC
wvuCwsvcuceF
0 0 0 0
000
1
000
001
,,,
,,,
,,,
,,, τ
ττ
ττ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]×∑ ∫ ∫ ∫ ∫ +−Ω−
∞
=1 0 0 0 0
2
0
,,,1
2
n
t L L L
LSLSnnnnCnCnnnnC
zyx
L
x y z
TwvugwcvcusetezcycxcFn
LLL
D
ετ
π

23. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
53
( )
( )
( ) ( ) ×∑Ω−∫
∇






+×
∞
=1
2
0
0
010
000 2
,,,
,,,
,,,
,,,
1
n
nC
zyx
L
L
S
S F
LLL
D
dudvdwdWdWvuC
Tk
wvu
TwvuP
wvuC z π
ττ
τµτ
ξ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )
( )
×∫ ∫ ∫ ∫ 





++−×
t L L L
SLSLSnnnCnCnnn
x y z
TwvuP
wvuC
Twvugvsucetezcycxcn
0 0 0 0
000
,,,
,,,
1,,,1 γ
γ
τ
ξετ
( ) ( ) ( ) ( ) ( ) ( )∑ ×Ω−∫
∇
×
∞
=1
2
0
0
010
2
,,,
,,,
n
nnnnC
zyx
L
L
S
n zcycxcF
LLL
D
dudvdwdWdWvuC
Tk
wvu
wc
z π
ττ
τµ
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )
( )
×∫ ∫ ∫ ∫ 





++−×
−t L L L
SLSLSnnnnCnC
x y z
TwvuP
wvuC
wvuCTwgwcvcuseten
0 0 0 0
1
000
010
,,,
,,,
,,,1,1 γ
γ
τ
τξετ
( ) ( ) ( ) ( ) ( ) ( ) ×∑Ω−∫
∇
×
∞
=1
2
0
0
000
2
,,,
,,,
n
nCnnn
zyx
L
L
S
tezcycxcn
LLL
D
dudvdwdWdWvuC
Tk
wvu z π
ττ
τµ
( ) ( ) ( ) ( ) ( )[ ] ( ) ×∫ ∫ ∫ ∫ ∫+−×
t L L L L
LSLSnnnnCnC
x y z z
WdWvuCTwvugwcvsuceF
0 0 0 0 0
000 ,,,,,,1 τετ
( )
( )
( )
( )
τ
τµτ
τξ γ
γ
dudvdwd
Tk
wvu
TwP
wvuC
wvuC S
S
,,,
,
,,,
,,,1
1
000
010
∇






+×
−
.
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 Ra-
diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio-
physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro-
structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo-
rod State University of Architecture and Civil Engineering. 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 Architec-
ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she
is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in
the University. 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.