modeling of manufacturing of a fieldeffect transistor to determine conditions to decrease length of...
TRANSCRIPT
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. Pankratov
1, E.A. Bulaeva
2
1Nizhny 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
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
a1
a2 -a
1L
x -a2 -a
1
c1 c1
c2
oxide
source drain
gate
c3
Fig. 2. Heterostructure with substrate and epitaxial layer with two or three sections. Sectional view
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
tzyxCD
zy
tzyxCD
yx
tzyxCD
xt
tzyxC ,,,,,,,,,,,, (1)
( ) ( ) ( ) ( )
∫∇
∂
∂Ω+
∫∇
∂
∂Ω+
zz L
S
SL
S
S WdtWyxCtzyxTk
D
yWdtWyxCtzyx
Tk
D
x 00
,,,,,,,,,,,, µµ
with boundary and initial conditions
( )0
,,,
0
=∂
∂
=xx
tzyxC,
( )0
,,,=
∂
∂
= xLxx
tzyxC,
( )0
,,,
0
=∂
∂
=yy
tzyxC,
( )0
,,,=
∂
∂
= yLxy
tzyxC,
( )0
,,,
0
=∂
∂
=zz
tzyxC,
( )0
,,,=
∂
∂
= zLxz
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
zdtzyxC0
,,, 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
tzyxCTzyxDD
L,,,
,,,1,,,
γ
γ
ξ , ( ) ( )( )
+=
TzyxP
tzyxCTzyxDD
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
ijx
u
x
uu
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
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
tzyxutzyx
,,,,,,
2
1,,,,,,,,,µ
( )( )
( )( ) ( ) ( ) ( )[ ]
−−
−
∂
∂
−+−
Ω×
ij
k
kij
ijTtzyxTzzK
x
tzyxu
z
zzE δβε
σ
δσδε 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
tzyxuz
z
tzyx
y
tzyx
x
tzyx
t
tzyxuz
z
tzyx
y
tzyx
x
tzyx
t
tzyxuz
zzzyzxz
yzyyyxy
xzxyxxx
,,,,,,,,,,,,
,,,,,,,,,,,,
,,,,,,,,,,,,
2
2
2
2
2
2
σσσρ
σσσρ
σσσρ
where ( )
( )[ ]( ) ( ) ( ) ( )
×∂
∂+
∂
∂−
∂
∂+
∂
∂
+=
k
k
ij
k
kij
i
j
j
i
ijx
tzyxu
x
tzyxu
x
tzyxu
x
tzyxu
z
zE ,,,,,,
3
,,,,,,
12δ
δ
σσ
( ) ( ) ( ) ( )[ ]r
TtzyxTzKzzK −−× ,,,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
zEzK
x
tzyxu
z
zEzK
t
tzyxuz
yxx,,,
13
,,,
16
5,,,2
2
2
2
2
σσρ
( )( )[ ]
( ) ( )( )
( )( )[ ]
( )( ) ×−
∂∂
∂
+++
∂
∂+
∂
∂
++ zK
zx
tzyxu
z
zEzK
z
tzyxu
y
tzyxu
z
zE zzy ,,,
13
,,,,,,
12
2
2
2
2
2
σσ
( ) ( )x
tzyxTz
∂
∂×
,,,β
( )( ) ( )
( )[ ]( ) ( )
( ) ( )( )
+∂
∂−
∂∂
∂+
∂
∂
+=
∂
∂
y
tzyxTzKz
yx
tzyxu
x
tzyxu
z
zE
t
tzyxuz xyy ,,,,,,,,,
12
,,, 2
2
2
2
2
βσ
ρ
( )( )[ ]
( ) ( ) ( )( )[ ]
( )( )
+∂
∂
++
+
∂
∂+
∂
∂
+∂
∂+
2
2 ,,,
112
5,,,,,,
12 y
tzyxuzK
z
zE
y
tzyxu
z
tzyxu
z
zE
z
yzy
σσ
( )( )
( )[ ]( )
( )( )
yx
tzyxuzK
zy
tzyxu
z
zEzK
yy
∂∂
∂+
∂∂
∂
+−+
,,,,,,
16
22
σ (4)
( )( ) ( ) ( ) ( ) ( )
×
∂∂
∂+
∂∂
∂+
∂
∂+
∂
∂=
∂
∂
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxu
t
tzyxuz
yxzzz,,,,,,,,,,,,,,,
22
2
2
2
2
2
2
ρ
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
35
( )( )[ ]
( )( ) ( ) ( ) ( )
×∂
∂−
∂
∂+
∂
∂+
∂
∂
∂
∂+
+×
z
tzyxT
z
tzyxu
y
tzyxu
x
tzyxuzK
zz
zE xyx ,,,,,,,,,,,,
12 σ
( ) ( ) ( )( )
( ) ( ) ( ) ( )
∂
∂−
∂
∂−
∂
∂−
∂
∂
+∂
∂+×
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxu
z
zE
zzzK zyxz
,,,,,,,,,,,,6
16
1
σβ .
Conditions for the displacement vector could be written as
( )0
,,,0=
∂
∂
x
tzyur
; ( )
0,,,
=∂
∂
x
tzyLux
r
; ( )
0,,0,
=∂
∂
y
tzxur
; ( )
0,,,
=∂
∂
y
tzLxuy
r
;
( )0
,0,,=
∂
∂
z
tyxur
; ( )
0,,,
=∂
∂
z
tLyxuz
r
; ( )00,,, uzyxurr
= ; ( )0,,, uzyxurr
=∞ .
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
tzyxTTc
,,,,,,,,,,,,λλλ
( )tzyxp ,,,+ (5)
with boundary and initial conditions
( )0
,,,
0
=∂
∂
=xx
tzyxT,
( )0
,,,=
∂
∂
= xLxx
tzyxT,
( )0
,,,
0
=∂
∂
=yy
tzyxT,
( )0
,,,=
∂
∂
= yLxy
tzyxT,
( )0
,,,
0
=∂
∂
=zz
tzyxT,
( )0
,,,=
∂
∂
= zLxz
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 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-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
TtzyxTtzyxT µε . (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
isistdxdydzdtzyxuM . The replacement leads to the following results
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
−∫ ∫
∂
∂−−∫ ∫
∂
∂=
∞ t zz
xdwd
x
wyxTwwKtdwd
x
wyxTwwKttzyxu
0 00 01
,,,,,,,,, τ
τβττ
τα
( )]101
,,,uxxux
twyx αφα +Φ− ,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
−∫ ∫
∂
∂−−∫ ∫
∂
∂=
∞ t zz
y dwdy
wyxTwwKtdwd
y
wyxTwwKttzyxu
0 00 01
,,,,,,,,, τ
τβττ
τα
( )]101
,,,uyyuy
twyx αφα +Φ− ,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
−∫ ∫
∂
∂−∫ ∫
∂
∂=
∞ t zz
zdwd
w
wyxTwwKtdwd
w
wyxTwwKttzyxu
0 00 01
,,,,,,,,, τ
τβττ
τα
( )]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−∞ΧΘ=
zL
z
xxz
ux
zdzzLL
L
0
3
20
1
8 ρα ,
( ) ( )[ ]( ) ( )∫ −
ΘΧ−∞ΧΘ=
zL
z
yyz
uy
zdzzLL
L
0
3
20
1
8 ρα ,
( ) ( )[ ]( ) ( )∫ −
ΘΧ−∞ΧΘ=
zL
z
zzz
uz
zdzzLL
L
0
3
20
1
8 ρα ,
where ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫
∂
∂−
Θ+=ΘΧ
Θ
− −0 0
,,,1
x
x
y
y
zL
L
L
L
L
z
i
isdxdydzd
s
zyxTzzKzL
tτ
τχ .
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)/
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
( ) ( )[ ] ( )( )
( ) ( )
×∂
∂
∂
∂+
∂
∂
++
∂
∂=
∂
∂
y
tzyxC
yx
tzyxC
TzP
tzyxCTzg
xD
t
tzyxCLLL
,,,,,,
,
,,,1,1
,,,0 γ
γ
ξε
( )[ ] ( )( )
( )[ ] ( )( )
×
++
∂
∂+
++×
TzyxP
tzyxCTzyxg
zDD
TzyxP
tzyxCTzg
LLLLLL,,,
,,,1,,,1
,,,
,,,1,1
00 γ
γ
γ
γ
ξεξε
( ) ( )[ ] ( )( )
( )
×∫
++
∂
∂Ω+
∂
∂×
zL
SLSLSLWdtWyxC
TzP
tzyxCTzg
xD
z
tzyxC
00 ,,,
,
,,,1,1
,,,γ
γ
ξεω (8)
( ) ( )( )
( )( )
∫∇
+
∂
∂Ω+
∇
×zL
S
SLL
S WdtWyxCTk
tzyx
TzP
tzyxC
yDD
Tk
tzyx
000 ,,,
,,,
,
,,,1
,,, µξω
µγ
γ
.
We determine solution of Eq.(1) as the following power series
( ) ( )∑ ∑ ∑=∞
=
∞
=
∞
=0 0 0
,,,,,,i j k
ijk
kji
LtzyxCtzyxC 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
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.
REFERENCES
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[2] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990).
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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
να
,,,,,,,,,,,,,,,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
tzyxTzg
zy
tzyxTxg
x
tzyxTzg i
T
i
T
i
Tass
,,,,,,,,,10
2
10
2
2
10
2
0α , i ≥1
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
40
( ) ( ) ( ) ( )( )
×+
∂
∂+
∂
∂+
∂
∂=
∂
∂
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxTdass
ass,,,
,,,,,,,,,,,,
00
0
2
01
2
2
01
2
2
01
2
0
01
ϕ
ϕαα
( ) ( ) ( )( )
( )
+
∂
∂−
∂
∂+
∂
∂+
∂
∂×
2
00
00
0
2
00
2
2
00
2
2
00
2 ,,,
,,,
,,,,,,,,,
x
tzyxT
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxTdass
ϕ
ϕα
( ) ( )
∂
∂+
∂
∂+
2
00
2
00,,,,,,
z
tzyxT
y
tzyxT
( ) ( ) ( ) ( )( )
×+
∂
∂+
∂
∂+
∂
∂=
∂
∂
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxTdass
ass,,,
,,,,,,,,,,,,
00
0
2
02
2
2
02
2
2
02
2
0
02
ϕ
ϕαα
( ) ( ) ( )( )
( )
×
∂
∂−
∂
∂+
∂
∂+
∂
∂×
+ x
tzyxT
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxTdass
,,,
,,,
,,,,,,,,,00
1
00
0
2
01
2
2
01
2
2
01
2
ϕ
ϕαϕ
( ) ( ) ( ) ( ) ( )
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂×
z
tzyxT
z
tzyxT
y
tzyxT
y
tzyxT
x
tzyxT ,,,,,,,,,,,,,,, 0100010001
( ) ( ) ( )( )
( )( )
( )
×+∂
∂+
∂
∂=
∂
∂Tzyxg
x
tzyxTTzyxg
tzyxT
tzyxT
x
tzyxT
t
tzyxTTTass
,,,,,,
,,,,,,
,,,,,,,,,2
00
2
00
01
2
11
2
0
11 α
( )( )
( )( )
( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂×
x
tzyxTTzyxg
xz
tzyxTTzyxg
zy
tzyxTTassT
,,,,,,
,,,,,,
,,, 01
0
00
2
00
2
α
( )( )[ ]
( )( )[ ]
( )+
∂
∂++
∂
∂++
∂
∂+
assTTz
tzyxTTzyxg
y
tzyxTTzyxg
x
tzyxT02
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
tzyxTϕϕϕ
( )( ) ( ) ( )
( )×+
∂
∂+
∂
∂+
∂
∂+×
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
tzyxT
TT dassdass
dass,,,
,,,,,,,,,
,,,00
0
2
10
2
2
10
2
2
10
2
00
0
0 ϕ
ϕ
ϕ
ϕϕ αα
α
( ) ( ) ( ) ( ) ( ) ( )−
∂
∂
∂
∂+
∂
∂+
∂
∂×
z
tzyxTTzyxg
zy
tzyxTTzyxg
x
tzyxTTzyxg
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
TTdass
ϕϕ
ϕαϕ
ϕϕα 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
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
41
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫+∫ ∫ ∫=∞
=1 00 0 000
2,,
1,,,
n
L
nnTnnn
zyx
L L L
T
zyx
xx y z
uctezcycxcLLL
udvdwdwvufLLL
tzyxT
( ) ( ) ( ) ( )×+∫ ∫ ∫ ∫+∫ ∫×
zyx
t L L L
asszyx
L L
TnnLLL
dudvdwdwvup
LLLudvdwdwvufwcvc
x y zy z 2,,,1,,
0 0 0 00 0
τν
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫−×∞
=1 0 0 0 0
,,,
n
t L L L
ass
nnnnTnTnnn
x y z
dudvdwdwvup
wcvcucetezcycxc τν
ττ ,
where cn(χ) = cos (π nχ/L), ( )
++−=
2220
22 111exp
zyx
assnTLLL
tnte απ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−=∞
=1 0 0 0 02
0
0,,,2,,,
n
t L L L
TnnnnTnTnnn
zyx
ass
i
x y z
TwvugwcvcusetezcycxcnLLL
tzyxT ταπ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−+∂
∂×
∞
=
−
1 0 0 02
010 2,,,
n
t L L
nnnTnTnnn
zyx
assix y
vsucetezcycxcnLLL
dudvdwdu
wvuTτ
απτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑+∫∂
∂×
∞
=
−
12
0
0
10 2,,,
,,,n
nTnnn
zyx
assL
i
nTtezcycxcn
LLLdudvdwd
v
wvuTwcTwvug
z απτ
τ
( ) ( ) ( ) ( ) ( )( )
∫ ∫ ∫ ∫∂
∂−× −
t L L Li
TnnnnT
x y z
dudvdwdw
wvuTTwvugwsvcuce
0 0 0 0
10 ,,,,,, τ
ττ , i ≥1,
where sn(χ) = sin (π n χ/L);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫∂
∂−=
∞
=1 0 0 0 02
00
2
2001
,,,2,,,
n
t L L L
nnnTnTnnn
zyx
d
ass
x y z
u
wvuTvcusetezcycxcn
LLL
TtzyxT
ττ
πα
ϕ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−+×∞
=1 0 0 0 020
00
2
,,, n
t L L L
nnnnTnTnnn
zyx
d
assn
x y z
wcvsucetezcycxcnLLL
T
wvuT
dudvdwdwc τ
πα
τ
τ ϕ
ϕ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−+∂
∂×
∞
=1 0 0 020
00
2
00
2 2
,,,
,,,
n
t L L
nnnTnTnnn
zyx
d
ass
x y
vcucetezcycxcnLLL
T
wvuT
dudvdwd
v
wvuTτ
πα
τ
ττ ϕ
ϕ
( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∫∂
∂×
∞
=1 0 0
0
0 00
2
00
2
2,,,
,,,
n
t L
nnTnTnnn
zyx
ass
d
L
n
xz
ucetezcycxcLLL
TwvuT
dudvdwd
u
wvuTws τ
αϕ
τ
ττ ϕ
ϕ
( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫ ∫
∂
∂×
∞
=+
1 0
0
0 01
00
2
00 2,,,
,,,
n
t
nTnTnnn
zyx
dass
L L
nnetezcycxc
LLL
T
wvuT
dudvdwd
u
wvuTwcvc
y z
τα
ϕτ
ττ ϕ
ϕ
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫
∂
∂×
∞
=+
1
0
0 0 01
00
2
00 2,,,
,,,
nnTnnn
zyx
ass
d
L L L
nnntezcycxc
LLLT
wvuT
dudvdwd
v
wvuTwcvcuc
x y z αϕ
τ
ττ ϕ
ϕ
( ) ( ) ( ) ( )( )
( )∫ ∫ ∫ ∫
∂
∂−×
+
t L L L
nnnnT
x y z
wvuT
dudvdwd
w
wvuTwcvcuce
0 0 0 01
00
2
00
,,,
,,,
τ
τττ
ϕ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫∂
∂−=
∞
=1 0 0 0 02
01
2
2002
,,,2,,,
n
t L L L
nnnTnTnnn
zyx
d
ass
x y z
u
wvuTvcusetezcycxcn
LLL
TtzyxT
ττ
πα
ϕ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−+×∞
=1 0 0 0 020
00
2
,,, n
t L L L
nnnnTnTnnn
zyx
d
assn
x y z
wcvsucetezcycxcnLLL
T
wvuT
dudvdwdwc τ
πα
τ
τ ϕ
ϕ
( )( )
( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫−+∂
∂×
∞
=1 0 020
00
2
01
2 2
,,,
,,,
n
t L
nnTnTnnn
zyx
d
ass
x
ucetezcycxcnLLL
T
wvuT
dudvdwd
v
wvuTτ
πα
τ
ττ ϕ
ϕ
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
42
( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫ ∫∂
∂×
∞
=1 02
0
0 0 00
2
01
2
2,,,
,,,
n
t
nTnTnnn
zyx
ass
L L
nnetezcycxc
LLLwvuT
dudvdwd
w
wvuTwsvc
y z
ταπ
τ
ττϕ
( ) ( ) ( ) ( ) ( )( )
( ) ×∑−∫ ∫ ∫∂
∂
∂
∂×
∞
=+
12
0
0 0 01
00
0100 2,,,
,,,,,,
nn
zyx
ass
d
L L L
nnndxc
LLLT
wvuT
dudvdwd
u
wvuT
u
wvuTwcvcucT
x y z απ
τ
τττ ϕ
ϕ
ϕ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
−∫ ∫ ∫ ∫∂
∂
∂
∂−×
+
t L L L
nnnnTnTnn
x y z
wvuT
dudvdwd
v
wvuT
v
wvuTwcvcucetezcyc
0 0 0 01
00
0100
,,,
,,,,,,
τ
ττττ
ϕ
( ) ( ) ( ) ( ) ( ) ( )( )
∑ ×∫ ∫ ∫ ∫∂
∂
∂
∂−−
∞
=+
10 0 0 01
00
0100
2
0
,,,
,,,,,,2
n
t L L L
nnnnT
zyx
assdx y z
wvuT
dudvdwd
w
wvuT
w
wvuTwcvcuce
LLL
T
τ
ττττ
απ
ϕ
ϕ
( ) ( ) ( ) ( )tezcycxc nTnnn× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
∑ ∫ ∫ ∫ ∫
×∂
∂−=
∞
=1 0 0 0 02
00
2
0
11
,,,2,,,
n
t L L L
nnnnTnTnnn
zyx
assx y z
w
wvuTwcvcucetezcycxc
LLLtzyxT
ττ
α
( ) ( ) ( ) ( ) ( ) ( )( )
+
∂
∂
∂
∂+
∂
∂+× τ
τ
τττdudvdwd
wvuT
wvuT
w
wvuTwg
wv
wvuTTwvugTwvug
TTT,,,
,,,,,,,,,,,,,,,
00
0100
2
00
2
( )( )[ ] ( )
( )( )
+
∂
∂
∂
∂+
∂
∂++
∂
∂× τ
τττdudvdwd
w
wvuTTwvug
wv
wvuTTwvug
u
wvuTTT
,,,,,,
,,,,,,1
,,,01
2
01
2
2
01
2
( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )
( )∑ ∫ ∫ ∫ ∫
+
∂
∂−+
∞
=+
1 0 0 0 02
00
2
1
00
100,,,
,,,
,,,2
n
t L L L
nnnTnTnnn
zyx
dassx y z
u
wvuT
wvuT
wvuTvcucetezcycxc
LLL
T τ
τ
ττ
αϕ
ϕ
( )( )
( ) ( )( )
( )( ) ×+
∂
∂+
∂
∂×
++
yx
ass
nLL
dudvdwdwcw
wvuT
wvuT
wvuT
v
wvuT
wvuT
wvuT0
2
00
2
1
00
10
2
00
2
1
00
10,,,
,,,
,,,,,,
,,,
,,, ατ
τ
τ
ττ
τ
τϕϕ
( ) ( ) ( ) ( )( ) ( ) ( )
∑ ×∫ ∫ ∫ ∫
∂
∂+
∂
∂+
∂
∂−×
∞
=1 0 0 0 02
10
2
2
10
2
2
10
2 ,,,,,,,,,2
n
t L L L
nnnnT
z
dx y z
w
wvuT
v
wvuT
u
wvuTwcvcuce
L
T ττττ
ϕ
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−+×
∞
=1 0 0 0
0
00
22,,, n
t L L
nnnTnTnnn
zyx
dass
nTnnn
x y
vcucetezcycxcLLL
Ttezcycxc
wvuT
dudvdwdτ
α
τ
τ ϕ
ϕ
( ) ( )( )
( )( )
( )∫
×+
∂
∂
∂
∂+
∂
∂×
zL
TTTnTwvug
w
wvuTTwvug
wu
wvuTTwvugwc
0
00
2
00
2
,,,,,,
,,,,,,
,,,ττ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−
∂
∂×
∞
=1 0 0 0
0
00
2
00
2
2,,,
,,,
n
t L L
nnnTnTnnn
zyx
dassx y
vcucetezcycxcLLL
T
wvuT
dudvdwd
v
wvuTτ
αϕ
τ
ττ ϕ
ϕ
( ) ( ) ( ) ( )( )∫
∂
∂+
∂
∂+
∂
∂×
+
zL
nwvuT
dudvdwd
w
wvuT
v
wvuT
u
wvuTwc
01
00
2
00
2
00
2
00
,,,
,,,,,,,,,
τ
ττττϕ
.
Integro-differential form of equations of systems (4) could be written as
( ) ( ) ( ) ( ) ( ) ( )
∫ ∫ ×
∂∂
∂−
∂∂
∂−
∂
∂−+=
t zzyx
xxwx
wyxu
yx
wyxu
x
wyxuttzyxutzyxu
0 0
22
2
2,,,,,,,,,
5,,,,,,τττ
τφ
( )( )[ ]
( ) ( ) ( ) ( )( )
−∫ ∫+
∂∂
∂−
∂∂
∂−
∂
∂−
+×
∞
0 0
22
2
2
1
,,,,,,,,,5
616
zzyx d
w
wdwE
wx
wyxu
yx
wyxu
x
wyxut
w
dwdwEτ
σ
τττ
σ
τ
( ) ( ) ( ) ( ) ( )∫ ∫
+
∂∂
∂+∫ ∫
∂∂
∂+
∂∂
∂+
∂
∂−
∞ t zx
zzyx
yx
wyxudwd
wx
wyxu
yx
wyxu
x
wyxuwKt
0 0
2
0 0
22
2
2,,,
2
1,,,,,,,,, ττ
τττ
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
43
( ) ( )( )
( ) ( )( ) ( )
×∫ ∫
∂
∂+
∂∂
∂−+−
+
∂
∂+
t zyxy
w
wyxu
wx
wyxutdt
w
wdwE
y
wyxu
0 02
22
2
2 ,,,,,,
2
1
1
,,, τττττ
σ
τ
( )( )
( ) ( ) ( )( )
( )∫ ∫
+
∂∂
∂−∫ ∫
+
∂
∂+
∂∂
∂−
+×
∞∞
0 0
2
0 02
22 ,,,
1
,,,,,,
21
zx
zyx
wx
wyxud
w
wdwE
y
wyxu
yx
wyxutd
w
wdwE ττ
σ
τττ
σ
( ) ( )( )
( ) ( ) ( )( )
( ) ×∫ ∫+∫ ∫∂
∂−−
+
∂
∂+
∞
0 00 02
2,,,
21
,,, zt zy
wKtdwdx
wyxTwKwt
td
w
wdwE
w
wyxuτ
τβττ
σ
τ
( ) ( ) ( ) ( ) ( ) ( )−Φ−∫ ∫∂
∂−+
∂
∂× tzyxdwd
wyxuwtdwd
x
wyxTw
x
t zx ,,,
,,,,,,1
0 02
2
ττ
τρττ
τβ
( )( )
∫ ∫∂
∂−
∞
0 02
2 ,,,zx dwd
wyxuwt τ
τ
τρ ,
( ) ( ) ( )( ) ( ) ( )
( )
×−∫ ∫+
∂∂
∂+
∂
∂−+=
21
,,,,,,
2
1,,,,,,
0 0
2
2
2
td
w
wdwE
yx
wyxu
x
wyxuttzyxutzyxu
t zxy
yyτ
σ
τττφ
( ) ( ) ( )( )
( ) ( )∫ ∫
−
∂∂
∂−
∂
∂+∫ ∫
+
∂∂
∂+
∂
∂×
∞ t zxy
zxy
yx
wyxu
y
wyxud
w
wdwE
yx
wyxu
x
wyxu
0 0
2
2
2
0 0
2
2
2,,,,,,
56
1
1
,,,,,, τττ
σ
ττ
( ) ( )( )
( )( ) ( ) ( )
×∫ ∫
∂∂
∂−
∂∂
∂−
∂
∂−−
+
∂∂
∂−
∞
0 0
22
2
22,,,,,,,,,
51
,,, zzxyz
wy
wyxu
yx
wyxu
y
wyxudt
w
wdwE
wy
wyxu τττττ
σ
τ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )+∫ ∫
∂
∂+∫ ∫
∂
∂−−
+×
∞
0 00 0
,,,,,,
61
zt z
dwdy
wyxTwKwtdwd
y
wyxTwwKt
td
w
wdwEτ
τβτ
τβττ
σ
( ) ( )( ) ( ) ( )
( ) ×∫ ∫−∫ ∫
∂∂
∂+
∂
∂+
∂∂
∂−+
∞
0 00 0
2
2
22 ,,,,,,,,, zt zyyy
wKtdwdwy
wyxu
y
wyxu
yx
wyxuwKt τ
ττττ
( ) ( ) ( )( )
( )∫
+
∂
∂−+
∂∂
∂+
∂
∂+
∂∂
∂×
tyyyy
z
zyxutdwd
wy
wyxu
y
wyxu
yx
wyxu
0
2
2
22 ,,,,,,,,,,,, τττ
τττ
( ) ( )( )[ ]
( )( )[ ]
( ) ( )+∫
∂
∂+
∂
∂
+−
+
∂
∂+
∞
0
,,,,,,
1212
,,,τ
ττ
σστ
τd
y
zyxu
z
zyxu
z
zEt
z
zEd
y
zyxuzyz
( )( )
( ) ( )( )
( )
Φ−∫ ∫∂
∂−−∫ ∫
∂
∂+
∞
tzyxdwdwyxu
wtdwdwyxu
wy
t zy
zy
,,,,,,,,,
10 0
2
2
0 02
2
ττ
τρττ
τ
τρ ,
( ) ( ) ( )( ) ( ) ( )
( )
×−∫ ∫+
∂∂
∂+
∂
∂−+=
21
,,,,,,
2
1,,,,,,
0 0
2
2
2t
dw
wdwE
wx
wyxu
x
wyxuttzyxutzyxu
t zxz
zzτ
σ
τττφ
( ) ( ) ( )( )
( ) ( )×∫ ∫
∂∂
∂+
∂
∂+∫ ∫
+
∂∂
∂+
∂
∂×
∞ t zyz
zxz
wy
wyxu
y
wyxud
w
wdwE
wx
wyxu
x
wyxu
0 0
2
2
2
0 0
2
2
2 ,,,,,,
1
,,,,,, τττ
σ
ττ
( )( )
( ) ( ) ( ) ( )( )
( ) ( ) ×∫ ∫−−∫ ∫+
∂∂
∂+
∂
∂−
−
+×
∞ t zzyz wtd
w
wdwE
wy
wyxu
y
wyxutd
t
w
wdwE
0 00 0
2
2
2
1
,,,,,,
221βττ
σ
τττ
τ
σ
( ) ( ) ( ) ( ) ( ) ( ) ( )∫
−
∂
∂−+∫ ∫
∂
∂+
∂
∂×
∞ tz
z
z
zyxutdwd
w
wyxTwwKtdwd
w
wyxTwK
00 0
,,,5
,,,,,, τττ
τβτ
τ
( ) ( ) ( )( )[ ]
( )( )
( ) ( )∫
−
∂
∂−
∂
∂
+−
+
∂
∂−
∂
∂−
∞
0
,,,,,,5
116
,,,,,,
x
zyxu
z
zyxu
z
zE
z
zEd
y
zyxu
x
zyxuxzyx
ττ
σστ
ττ
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
44
( )( ) ( ) ( ) ( ) ( )
−∫
∂
∂+
∂
∂+
∂
∂−+
∂
∂−
tzyxy
dz
zyxu
y
zyxu
x
zyxutzK
td
y
zyxu
0
,,,,,,,,,
6
,,,τ
τττττ
τ
( ) ( ) ( ) ( ) ( )
Φ−∫
∂
∂+
∂
∂+
∂
∂−
∞
tzyxdz
zyxu
y
zyxu
x
zyxuzKt
z
zyx ,,,,,,,,,,,,
10
ττττ
,
where ( ) ( ) ( )[ ]∫=Φz
i
iswdtwyxuwtzyx
s0
,,,,,, ρ , s =x,y,z, φ =L/Θ2E0, E0 is the average value of Young
modulus.
The second-order approximations of components of displacement vector could be written as
( ) ( ) ( ) ( ) ( )
∫ ∫
−
∂∂
∂−
∂
∂−++=
t zyx
xuxxyx
wyxu
x
wyxuttzyxutzyxu
0 0
1
2
2
1
2
1122
,,,,,,5
6
1,,,,,,
τττφα
( ) ( )( )
( ) ( ) ( )∫ ∫ ×
∂∂
∂−
∂∂
∂−
∂
∂−
+
∂∂
∂−
∞
0 0
1
21
2
2
1
2
1
2 ,,,,,,,,,5
1
,,, zzyxz
wx
wyxu
yx
wyxu
x
wyxud
w
wdwE
wx
wyxu ττττ
σ
τ
( )( )
( ) ( ) ( ) ( ) ( )−∫ ∫
∂∂
∂+
∂∂
∂+
∂
∂−+
+×
t zzyx dwd
wx
wyxu
yx
wyxu
x
wyxuwKt
td
w
wdwE
0 0
1
21
2
2
1
2,,,,,,,,,
61τ
τττττ
σ
( ) ( ) ( ) ( ) ( )∫ ∫
+
∂∂
∂+∫ ∫
∂∂
∂+
∂∂
∂+
∂
∂−
∞ t zx
zzyx
yx
wyxudwd
wx
wyxu
yx
wyxu
x
wyxuwKt
0 0
1
2
0 0
1
21
2
2
1
2,,,,,,,,,,,, τ
ττττ
( ) ( )( )
( ) ( ) ( ) ( )( )
×∫ ∫+
∂
∂+
∂∂
∂+
−
+
∂
∂+
t zyxy
w
wdwE
y
wyxu
yx
wyxud
t
w
wdwE
y
wyxu
0 02
1
2
1
2
2
1
2
1
,,,,,,
2
1
21
,,,
σ
τττ
τ
σ
τ
( ) ( ) ( ) ( ) ( )( )
( )( )
×∫ ∫+
−∫ ∫+
∂
∂+
∂∂
∂−+−×
∞
0 00 02
1
2
1
2
121
,,,,,,
2
1 zt zyx
w
wEtd
w
wdwE
w
wyxu
wx
wyxutdt
στ
σ
τττττ
( ) ( ) ( ) ( )×∫ ∫
∂
∂+
∂∂
∂−
∂
∂+
∂∂
∂×
∞
0 02
1
2
1
2
2
1
2
1
2 ,,,,,,
2
,,,,,, zyxyx
w
wyxu
wx
wyxutdwd
y
wyxu
yx
wyxu τττ
ττ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) (∫ −∫∂
∂−∫ ∫
∂
∂+
+×
∞ t zz
twdx
wyxTwwKdwd
x
wyxTwKwtd
w
wdwE
0 00 0
,,,,,,
1
τχτ
τχτ
σ
) ( ) ( ) ( ) ( ) ( )−∫ ∫
∂
∂+∫ ∫
∂
∂−−−
∞
0 02
1
2
0 02
1
2 ,,,,,, zx
t zx dwd
wyxuwtdwd
wyxuwtd τ
τ
τρτ
τ
τρτττ
( ) ( )tzyxtzyxxxux
,,,,,,102
Φ−Φ− α ,
( ) ( )( ) ( ) ( )
( )
−∫ ∫+
∂∂
∂+
∂
∂−++=
t zxy
yuyyd
w
wdwE
yx
wyxu
x
wyxuttzyxutzyxu
0 0
1
2
2
1
2
1221
,,,,,,
2,,,,,, τ
σ
τττα
( ) ( ) ( )( )
( ) ( )∫ ∫
−
∂∂
∂−
∂
∂+∫ ∫
+
∂∂
∂+
∂
∂−
∞ t zxy
zxy
yx
wyxu
y
wyxud
w
wdwE
yx
wyxu
x
wyxut
0 0
1
2
2
1
2
0 0
1
2
2
1
2,,,,,,
51
,,,,,,
2
τττ
σ
ττ
( ) ( )( )
( ) ( ) ( )×∫ ∫
∂∂
∂−
∂∂
∂−
∂
∂−
−
+
∂∂
∂−
∞
0 0
1
2
1
2
2
1
2
1
2 ,,,,,,,,,5
61
,,, zzxyz
wy
wyxu
yx
wyxu
y
wyxud
t
w
wdwE
wy
wyxu ττττ
τ
σ
τ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )−∫ ∫
∂
∂+∫ ∫
∂
∂−−
+×
∞
0 00 0
,,,,,,
61
zt z
dwdy
wyxTwwKtdwd
y
wyxTwKwt
td
w
wdwEτ
τχτ
τχττ
σ
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 zyyy
wKtdwdwy
wyxu
y
wyxu
yx
wyxuwKt τ
ττττ
( ) ( ) ( )( )
( )∫
+
∂
∂−+
∂∂
∂+
∂
∂+
∂∂
∂×
tyyyy
z
zyxutdwd
wy
wyxu
y
wyxu
yx
wyxu
0
11
2
2
1
2
1
2 ,,,
2
1,,,,,,,,, τττ
τττ
( ) ( )( )
( )( )[ ]
( ) ( )×−∫
∂
∂+
∂
∂
+−
+
∂
∂+
∞
20
111 ,,,,,,
121
,,,uy
zyz dy
zyxu
z
zyxu
z
zEt
z
zEd
y
zyxuατ
ττ
σστ
τ
( ) ( )110
,,,,,, φtzyxtzyxyy
Φ−Φ× ,
( ) ( )( ) ( ) ( )
( )
×∫ ∫+
∂∂
∂+
∂
∂++=
t zxz
zzzw
wdwE
wx
wyxu
x
wyxutzyxutzyxu
0 0
1
2
2
1
2
11221
,,,,,,
2
1,,,,,,
σ
ττφα
( ) ( ) ( ) ( )( )
( ) ( )∫ ∫
+
∂
∂−+∫ ∫
+
∂∂
∂+
∂
∂−−×
∞ t zz
zxz
y
wyxutd
w
wdwE
wx
wyxu
x
wyxutdt
0 02
1
2
0 0
1
2
2
1
2 ,,,
2
1
1
,,,,,,
2
τττ
σ
ττττ
( ) ( )( )
( ) ( ) ( ) ( ) ( )∫ ∫
+
∂
∂−∫ ∫
∂
∂−−
+
∂∂
∂+
∞
0 02
1
2
0 0
1
2,,,,,,
1
,,, zz
t zy
y
wyxudwd
w
wyxTwwKtd
w
wdwE
wy
wyxu ττ
τχττ
σ
τ
( ) ( )( )
( ) ( ) ( ) ( )( )[ ]
( )×∫ −+
+∫ ∫∂
∂+
+
∂∂
∂+
∞ tzy
tz
zEdwd
w
wyxTwwKt
td
w
wdwE
wy
wyxu
00 0
1
2
16
,,,
21
,,,τ
στ
τχτ
σ
τ
( ) ( ) ( ) ( ) ( )∫
−
∂
∂−
∂
∂−
∂
∂−
∂
∂−
∂
∂×
∞
0
11111,,,,,,
56
,,,,,,,,,5
x
zyxu
z
zyxutd
y
zyxu
x
zyxu
z
zyxuxzyxz
τττ
τττ
( ) ( )( )[ ]
( ) ( ) ( ) ( )×∫
∂
∂+
∂
∂+
∂
∂−+
+
∂
∂−
tzyxy
dz
zyxu
y
zyxu
x
zyxut
z
zEtd
y
zyxu
0
1111 ,,,,,,,,,
16
,,,τ
ττττ
στ
τ
( ) ( )( ) ( ) ( )
( ) −Φ−∫
∂
∂+
∂
∂+
∂
∂−×
∞
tzyxdz
zyxu
y
zyxu
x
zyxuzKtzK
zz
zyx ,,,,,,,,,,,,
020
111 αττττ
( )tzyxz
,,,1
Φ− .
Calculation of the average values αuβ2 leads to the following results
( ) ( ) ( ) ( ) ( )( )
(
∫ ∫ ∫ ∫ −+
∂∂
∂−
∂∂
∂−
∂
∂−Θ=
Θ
0 0 0 0
1
21
2
2
1
22
21
,,,,,,,,,5
12
1 x y zL L L
z
zyx
zuxL
z
zdzE
zx
tzyxu
yx
tzyxu
x
tzyxutL
σα
) ( ) ( ) ( ) ( ) ( )( )
×∫ ∫ ∫ ∫+
−
∂∂
∂−
∂∂
∂−
∂
∂Θ−−
∞
0 0 0 0
1
21
2
2
1
22
1
,,,,,,,,,5
12
x y zL L Lzzyx
z
zL
zx
tzyxu
yx
tzyxu
x
tzyxuzEtdxdydz
σ
( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∫
∂∂
∂+
∂∂
∂+
∂
∂−−Θ+×
Θ
0 0 0 0
1
21
2
2
1
22 ,,,,,,,,,
2
1 x y zL L Lzyx
zzx
tzyxu
yx
tzyxu
x
tzyxuzLttdxdydzd
( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∫
∂∂
∂+
∂∂
∂+
∂
∂−
Θ−×
∞
0 0 0 0
1
21
2
2
1
22 ,,,,,,,,,
2
x y zL L Lzyx
zzx
tzyxu
yx
tzyxu
x
tzyxuzLtdxdydzdzK
( ) ( ) ( ) ( ) ( )( )
×∫ ∫ ∫ ∫+
−
∂
∂+
∂∂
∂−Θ+×
Θ
0 0 0 02
1
2
1
22
1
,,,,,,
2
1 x y zL L Lzyx ydzd
z
zL
y
tzyxu
yx
tzyxuzEttdxdydzdzK
σ
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 02
1
2
1
22
x y zL L Lyx
ztdxdyd
z
zdzE
z
tzyxu
zx
tzyxuzLttdxd
σ
( ) ( )( )
( ) ( )( )
( )∫ ∫ ∫ ∫ ×+
−−Θ+∫ ∫ ∫ ∫
∂
∂+
∂∂
∂
+
−×
Θ∞
0 0 0 0
2
0 0 0 02
1
2
1
2
14
1,,,,,,
1
x y zx y z L L Lz
L L Lyxz
z
zLttdxdydzd
y
tzyxu
yx
tzyxu
z
zLzE
σσ
( ) ( )( ) ( ) ( ) ( )
×∫ ∫ ∫ ∫∂
∂−+
∂
∂+
∂∂
∂×
∞
0 0 0 02
1
2
2
1
2
1
2 ,,,
2
,,,,,, x y zL L Lx
z
yx
t
tzyxuzzLtdxdydzdzE
z
tzyxu
zx
tzyxu ρ
( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ −Θ
+∫ ∫ ∫ ∫ −−Θ−Θ×∞Θ
0 0 0 0
2
0 0 0 01
2
2,,,
x y zx y z L L L
z
L L L
xzzLtdxdydzdtzyxuzzLttdxdydzd ρ
( )( ) ( ) ( ) ( )
∫ −Θ
ΘΧ−∞ΧΘ
+∂
∂×
zL
zxx
x zdzzLLtdxdydzdt
tzyxu
020
2
2
1
2
42
1
2
,,,ρ ,
( ) ( )( )
( )( ) ( )
×Θ
−∫ ∫ ∫ ∫
∂∂
∂+
∂
∂
+
−−Θ=
Θ
2
,,,,,,
12
12
0 0 0 0
1
2
2
1
2
2
2
x y zL L Lxyz
zuytdxdydzd
yx
tzyxu
x
tzyxu
z
zLzEL
στα
( )( )
( )( ) ( )
( )( )
×∫ ∫ ∫ ∫+
−+∫ ∫ ∫ ∫
∂∂
∂+
∂
∂
+
−×
Θ∞
0 0 0 00 0 0 0
1
2
2
1
2
112
1,,,,,,
1
x y zx y z L L Lz
L L Lxyz
z
zLzEtdxdydzd
yx
tzyxu
x
tzyxu
z
zLzE
σσ
( ) ( ) ( )( ) ( ) ×∫ ∫ ∫ ∫
Θ−−Θ
∂∂
∂−
∂∂
∂−
∂
∂×
∞
0 0 0 0
221
2
1
2
2
1
2
12
,,,,,,,,,5
x y zL L Lzxy
zEtdxdydzdzy
tzyxu
yx
tzyxu
y
tzyxuτ
( )( ) ( ) ( ) ( ) ( )
×∫ ∫ ∫ ∫−Θ+
∂∂
∂−
∂∂
∂−
∂
∂
+
−×
Θ
0 0 0 0
21
2
1
2
2
1
2
2
,,,,,,,,,5
1
x y zL L Lzxyz
zKttdxdydzd
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
zzLzKtdxdydzd
zy
tzyxu
y
tzyxu
yx
tzyxuzL
( ) ( ) ( ) ( )∫ ∫ ∫ ∫
+
∂
∂+
∂∂
∂+
∂
∂+
∂∂
∂Θ×
Θ
0 0 0 0
11
2
2
1
2
1
22 ,,,
2
1,,,,,,,,,
2
x y zL L Lyyyy
z
tzyxutdxdydzd
zy
tzyxu
y
tzyxu
yx
tzyxu
( ) ( )( )
( )( ) ( ) ( )
( )×∫ ∫ ∫ ∫
+
∂
∂+
∂
∂Θ−−Θ
+
∂
∂+
∞
0 0 0 0
112
21
1
,,,,,,
21
,,, x y zL L Lzyz
z
zdzE
y
tzyxu
z
tzyxutdxdyd
z
zdzE
y
tzyxu
στ
σ
( ) ( )( ) ( )
×∫ ∫ ∫ ∫∂
∂Θ+∫ ∫ ∫ ∫
∂
∂−Θ−×
∞Θ
0 0 0 02
1
22
0 0 0 02
1
2
2,,,
2
,,,
2
1 x y zx y z L L Ly
L L Ly
t
tzyxutdxdydzd
t
tzyxuzttdxdyd ρ
( ) ( ) ( ) ( )( ) ( )
×∞Χ
Θ+ΘΧ
−∫ ∫ ∫ ∫ −−×Θ
22,,,
022
0 0 0 01
yyL L L
yz
x y z
tdxdydzdtzyxuzzLtdxdydzdz ρρ
( ) ( )1
0
4
−
∫ −Θ×zL
zzdzzLL ρ ,
( ) ( )( )
( )( ) ( )
×Θ
−∫ ∫ ∫ ∫
∂∂
∂+
∂
∂
+
−−Θ=
Θ
2
,,,,,,
12
12
0 0 0 0
1
2
2
1
22
2
x y zL L Lxzz
zztdxdydzd
zx
tzyxu
x
tzyxu
z
zLzEtL
σα
( ) ( )( )
( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫−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 Lxzz zEttdxdydzd
zx
tzyxu
x
tzyxu
z
zLzE
σ
( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ×
∂∂
∂+
∂
∂−
∂∂
∂+
∂
∂×
∞
0 0 0 0
1
2
2
1
21
2
2
1
2 ,,,,,,,,,,,, x y zL L Lyzyz
zy
tzyxu
y
tzyxutdxdydzd
zy
tzyxu
y
tzyxu
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
47
( ) ( )( )[ ]
( ) ( ) ( )∫ ∫ ∫ ∫ ×
∂
∂−
∂
∂−
∂
∂+
+
−Θ×
Θ
0 0 0 0
1112,,,,,,,,,
512
1
12
x y zL L Lyxzz
y
tzyxu
x
tzyxu
z
tzyxutdxdydzd
z
zLzE
σ
( )( )
( )( ) ( ) ( )
∫ ∫ ∫ ∫ ×
∂
∂−
∂
∂−
∂
∂Θ−−Θ
+×
∞
0 0 0 0
111
22
,,,,,,,,,5
121
x y zL L Lyxz
y
tzyxu
x
tzyxu
z
tzyxutdtxdyd
z
zdzE
σ
( )( )
( )( ) ( ) ( ) ( )
×∫ ∫ ∫ ∫
∂
∂+
∂
∂+
∂
∂−Θ+
+×
Θ
0 0 0 0
1112 ,,,,,,,,,
21
x y zL L Lzyx
z
tzyxu
y
tzyxu
x
tzyxuzKttdxdyd
z
zdzE
σ
( )( )
( ) ( ) ( )×∫ ∫ ∫ ∫
∂
∂+
∂
∂+
∂
∂Θ−
ΘΧ−×
∞
0 0 0 0
111
22 ,,,,,,,,,
22
x y zL L Lzyxz
z
tzyxu
y
tzyxu
x
tzyxuzKtdxdydzd
( ) ( ) ( )( )
( ) ×
∫ −∞Χ
Θ+∫ ∫ ∫ ∫ −−×Θ zx y z L
z
zL L L
zzzLtdxdydzdztzyxuzLtdxdydzd
0
02
0 0 0 01
42
,,, 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
tzyxCD
t
tzyxCL
;
( )( )
( )( )
( )
+
∂
∂
∂
∂+
∂
∂
∂
∂=
∂
∂ −−
y
tzyxCTzg
yx
tzyxCTzg
xD
t
tzyxCi
L
i
LL
i,,,
,,,,
,,,,
100100
0
00
( )( ) ( ) ( ) ( )
∂
∂+
∂
∂+
∂
∂+
∂
∂
∂
∂+ −
2
00
2
2
00
2
2
00
2
100,,,,,,,,,,,,
,z
tzyxC
y
tzyxC
x
tzyxC
z
tzyxCTzg
z
iiii
L, i ≥1;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
010
2
2
010
2
2
010
2
0
010,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxCD
t
tzyxCL
( )( )
( ) ( )( )
( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
xD
L
,,,
,,,
,,,,,,
,,,
,,,000000000000
0 γ
γ
γ
γ
( )( )
( )
∂
∂
∂
∂+
z
tzyxC
TzyxP
tzyxC
z
,,,
,,,
,,, 000000
γ
γ
;
( ) ( ) ( ) ( )×+
∂
∂+
∂
∂+
∂
∂=
∂
∂LL
Dz
tzyxC
y
tzyxC
x
tzyxCD
t
tzyxC02
020
2
2
020
2
2
020
2
0
020,,,,,,,,,,,,
( )( )
( )( )
( )( )
×
∂
∂
∂
∂+
∂
∂
∂
∂×
−
y
tzyxCtzyxC
yy
tzyxC
TzyxP
tzyxCtzyxC
x
,,,,,,
,,,
,,,
,,,,,, 000
010
000
1
000
010 γ
γ
( )( )
( )( )
( )( ) ( )
( )
×∂
∂+
∂
∂
∂
∂+
×
−−
TzyxP
tzyxC
xD
z
tzyxC
TzyxP
tzyxCtzyxC
zTzyxP
tzyxCL
,,,
,,,,,,
,,,
,,,,,,
,,,
,,,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
tzyxCD
t
tzyxCL
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
48
( )[ ] ( )( )
( )( )
+
∇
∫
++
∂
∂Ω+
Tk
tzyxWdtWyxC
TzyxP
tzyxCTzyxg
xD S
L
SLSLSSL
z ,,,,,,
,,,
,,,1,,,1
0000
000
0
µξε
γ
γ
( )[ ] ( )( )
( )( )
∇
∫
++
∂
∂Ω+
Tk
tzyxWdtWyxC
TzyxP
tzyxCTzyxg
yD S
L
SLSLSSL
z ,,,,,,
,,,
,,,1,,,1
0000
000
0
µξε
γ
γ
;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
002
2
2
002
2
2
002
2
0
002,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxCD
t
tzyxCL
( )[ ] ( )( )
( )( )
+
∇
∫
++
∂
∂Ω+
Tk
tzyxWdtWyxC
TzyxP
tzyxCTzyxg
xD S
L
SLSLSSL
z ,,,,,,
,,,
,,,1,,,1
0001
000
0
µξε
γ
γ
( )[ ] ( )( )
( )( )
+
∇
∫
++
∂
∂Ω+
Tk
tzyxWdtWyxC
TzyxP
tzyxCTzyxg
yD S
L
SLSLSSL
z ,,,,,,
,,,
,,,1,,,1
0001
000
0
µξε
γ
γ
( )[ ] ( )( )
( )( )
×∫
++
∂
∂Ω+
−zL
SLSLSSLWdtWyxC
TzyxP
tzyxCtzyxCTzyxg
xD
0000
1
000
0010 ,,,,,,
,,,,,,1,,,1
γ
γγξε
( ) ( ) ( )( )
( ) ( ) ×
∫∇
+
∂
∂+∇
×−
zLS
S
S WdtWyxCTk
tzyx
TzyxP
tzyxCtzyxC
yTk
tzyx
0000
1
000
001,,,
,,,
,,,
,,,,,,1
,,, µξ
µγ
γγ
( )[ ]SLLSLS
DTzyxg0
,,,1 Ω+× ε ;
( ) ( ) ( ) ( ) ( )
×
∂
∂
∂
∂+
∂
∂+
∂
∂+
∂
∂=
∂
∂
x
tzyxC
xz
tzyxC
y
tzyxC
x
tzyxCD
t
tzyxCL
,,,,,,,,,,,,,,,010
2
110
2
2
110
2
2
110
2
0
110
( ) ( )( )
( )( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
×
LLLLD
z
tzyxCTzyxg
zy
tzyxCTzyxg
yTzyxg
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
tzyxCtzyxC
xD
z
tzyxCL
,,,
,,,,,,
,,,
,,,,,,
,,,1
000000
1
000
1000
100
γ
γ
γ
γ
( )( )
( )( )
( )( )
LD
z
tzyxC
TzyxP
tzyxCtzyxC
zy
tzyxCtzyxC 0
000
1
000
100
000
100
,,,
,,,
,,,,,,
,,,,,,
∂
∂
∂
∂+
∂
∂×
−
γ
γ
;
( ) ( ) ( ) ( ) ( )
×
∂
∂
∂
∂+
∂
∂+
∂
∂+
∂
∂=
∂
∂
y
tzyxC
xz
tzyxC
y
tzyxC
x
tzyxCD
t
tzyxCL
,,,,,,,,,,,,,,,000
2
100
2
2
100
2
2
100
2
0
101
( ) ( ) ( ) ( ) ( )+
∂
∂
∂
∂+
∂
∂
∂
∂+
×
LLLLD
z
tzyxCTzyxg
zy
tzyxCTzyxg
yTzyxg
0
000000,,,
,,,,,,
,,,,,,
( )[ ] ( ) ( )( )
( )
×∫
++
∂
∂Ω+
−zL
SLSLSSLWdtWyxC
TzP
tzyxCtzyxCTzyxg
xD
0100
1
000
1000,,,
,
,,,,,,1,,,1
γ
γ
ξε
( )( )[ ] ( )
( )( )
( )( )
( )[ ] ( )
×∫+∂
∂+
∇
∫×
×
++
∂
∂Ω+
∇
×−
zz L
LSLS
SL
SLSLSSL
S
WdtWyxCTzyxgxTk
tzyxWdtWyxC
TzP
tzyxCtzyxCTzyxg
yD
Tk
tzyx
0100
0100
1
000
1000
,,,,,,1,,,
,,,
,
,,,,,,1,,,1
,,,
εµ
ξεµ
γ
γ
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
49
( ) ( ) ( )[ ] ( )
×∇
+∂
∂+Ω
∫∇
×Tk
tzyxTzyxg
yDWdtWyxC
Tk
tzyxS
LSLSSL
LS
z ,,,,,,1,,,
,,,0
0100
µε
µ
( ) ( )SL
LL
DWdtWyxCWdtWyxCzz
00
1000
100,,,,,, Ω
∫∫× ;
( ) ( ) ( ) ( ) ( )( )
×
∂
∂+
∂
∂+
∂
∂+
∂
∂=
∂
∂
TzyxP
tzyxC
xz
tzyxC
y
tzyxC
x
tzyxCD
t
tzyxCL
,,,
,,,,,,,,,,,,,,,000
2
011
2
2
011
2
2
011
2
0
011
γ
γ
( ) ( )( )
( ) ( )( )
( )+
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂×
LD
z
tzyxC
TzyxP
tzyxC
zy
tzyxC
TzyxP
tzyxC
yx
tzyxC0
001000001000001 ,,,
,,,
,,,,,,
,,,
,,,,,,γ
γ
γ
γ
( ) ( )( )
( ) ( ) ( )
×
∂
∂
∂
∂+
∂
∂
∂
∂+
−
y
tzyxCtzyxC
yx
tzyxC
TzyxP
tzyxCtzyxC
xD
L
,,,,,,
,,,
,,,
,,,,,, 000
001
000
1
000
0010 γ
γ
( )( )
( )( )
( )( ) ( )
×∇
∂
∂+
∂
∂
∂
∂+
×
−−
Tk
tzyx
xz
tzyxC
TzyxP
tzyxCtzyxC
zTzyxP
tzyxCS
,,,,,,
,,,
,,,,,,
,,,
,,, 000
1
000
001
1
000 µγ
γ
γ
γ
( )[ ] ( )( )
( ) ( )×
∇
∂
∂+Ω
∫
++×
Tk
tzyx
xDWdtWyxC
TzyxP
tzyxCTzyxg S
SL
L
SLSLS
z ,,,,,,
,,,
,,,1,,,1 0
0010
000µ
ξεγ
γ
( )[ ] ( ) ( )( )
( ) ×Ω+Ω
∫
++×
−
SLSL
L
SLSLSDDWdtWyxC
TzyxP
tzyxCtzyxCTzyxg
z
000
000
1
000
010,,,
,,,
,,,,,,1,,,1
γ
γ
ξε
( )[ ] ( )( )
( )( )
( )
∫∇
++
∂
∂×
−zL
S
SLSLSWdtWyxC
Tk
tzyx
TzyxP
tzyxCtzyxCTzyxg
y 0000
1
000
010,,,
,,,
,,,
,,,,,,1,,,1
µξε
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,,,
nnCnnnnC
zyxzyx
C tezcycxcFLLLLLL
FtzyxC ,
where ( )
++−=
2220
22 111exp
zyx
LnCLLL
tDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=x y zL L L
CnnnnCudvdwdwvufwcvcucF
0 0 0
,, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 02
0
00 ,,,2
,,,n
t L L L
LnnnCnCnnnnC
zyx
L
i
x y z
TwvugvcusetezcycxcFnLLL
DtzyxC τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=
−
1 0 02
0100 2,,,
n
t L
nnCnCnnnnC
zyx
Li
n
x
ucetezcycxcFnLLL
Ddudvdwd
u
wvuCwc τ
πτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫∂
∂×
∞
=
−
12
0
0 0
1002,,,
,,,n
nCnnn
zyx
L
L Li
Lnntezcycxc
LLL
Ddudvdwd
v
wvuCTwvugwcvs
y z πτ
τ
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
50
( ) ( ) ( ) ( ) ( )( )
∫ ∫ ∫ ∫∂
∂−× −
t L L Li
LnnnnCnC
x y z
dudvdwdw
wvuCTwvugwsvcuceFn
0 0 0 0
100 ,,,,,, τ
ττ , i ≥1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 0
000
2
0
010,,,
,,,2,,,
n
t L L L
nnnCnCnnnnC
zyx
Lx y z
TwvuP
wvuCvcusetezcycxcFn
LLL
DtzyxC
γ
γ ττ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=1 0 02
00002,,,
n
t L
nnCnCnnnnC
zyx
L
n
x
ucetezcycxcFnLLL
Ddudvdwd
u
wvuCwc τ
πτ
τ
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ×∑−∫ ∫ ∫∂
∂×
∞
=12
0
0 0 0
0000002,,,
,,,
,,,
nnn
zyx
L
L L L
nnnycxcn
LLL
Ddudvdwd
v
wvuC
TwvuP
wvuCwcwcvs
y z z πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( )( )
( )( )
∫ ∫ ∫ ∫∂
∂−×
t L L L
nnnnCnCnnC
x y z
dudvdwdw
wvuC
TwvuP
wvuCwsvcucetezcF
0 0 0 0
000000 ,,,
,,,
,,,τ
τττ
γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 00102
0
020 ,,,2
,,,n
t L L L
nnnCnCnnnnC
zyx
Lx y z
wvuCvcusetezcycxcFnLLL
DtzyxC ττ
π
( ) ( )( )
( ) ( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=
−
12
0000
1
0002,,,
,,,
,,,
nnCnnnnC
zyx
L
ntezcycxcFn
LLL
Ddudvdwd
u
wvuC
TwvuP
wvuCwc
πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( )( )
( )−∫ ∫ ∫ ∫
∂
∂−×
−t L L L
nnnnC
x y z
dudvdwdv
wvuC
TwvuP
wvuCwvuCwcvcuse
0 0 0 0
000
1
000
010
,,,
,,,
,,,,,, τ
ττττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫−−∞
=
−
1 0 0 0 0
1
000
010,,,
,,,,,,
n
t L L L
nnnnCnCnnnnC
x y z
TwvuP
wvuCwvuCwcvcusetezcycxcFn
γ
γ τττ
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∂
∂×
∞
=1 02
0000
2
0 2,,,2
n
t
nCnCnnnnC
zyx
L
zyx
L etezcycxcFnLLL
Ddudvdwd
w
wvuC
LLL
Dτ
πτ
τπ
( ) ( ) ( ) ( )( )
( ) ( ) ×∑−∫ ∫ ∫∂
∂×
∞
=12
0
0 0 0
0100002,,,
,,,
,,,
nnnC
zyx
LL L L
nnnxcFn
LLL
Ddudvdwd
u
wvuC
TwvuP
wvuCwcvcus
x y z πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( )−∫ ∫ ∫ ∫
∂
∂−×
t L L L
nnnnCnCnn
x y z
dudvdwdv
wvuC
TwvuP
wvuCwcvsucetezcyc
0 0 0 0
010000,,,
,,,
,,,τ
τττ
γ
γ
( ) ( ) ( ) ( ) ( )( )
( )×∑ ∫ ∫ ∫ ∫
∂
∂−−
∞
=1 0 0 0 0
010000
2
0,,,
,,,
,,,2
n
t L L L
nnnnCnC
zyx
Lx y z
dudvdwdw
wvuC
TwvuP
wvuCwsvcuceFn
LLL
Dτ
τττ
πγ
γ
( ) ( ) ( ) ( )tezcycxcnCnnn
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∇−
Ω−=
∞
=1 0 0 0 02
0
001
,,,2,,,
n
t L L LS
nnCnCnnnnC
zyx
Lx y z
Tk
wvuusetezcycxcFn
LLL
DtzyxC
τµτ
π
( )[ ] ( )( )
( ) ( ) ( ) −∫
++× ττ
τξε
γ
γ
dudvdvcwdwcWdWvuCTwvuP
wvuCTwvug
nn
L
SLSLS
z
0000
000 ,,,,,,
,,,1,,,1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∫−Ω
−∞
=1 0 0 0 0 00002
0 ,,,2
n
t L L L L
nnnnCnCnnnnC
zyx
Lx y z z
WdWvuCwcvsucetezcycxcFnLLL
Dττ
π
( )[ ] ( )( )
( )τ
τµτξε
γ
γ
dudvdwdTk
wvu
TwP
wvuCTwvug S
SLSLS
,,,
,
,,,1,,,1 000
∇
++× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∇−
Ω−=
∞
=1 0 0 0 02
0
002
,,,2,,,
n
t L L LS
nnnCnCnnnnC
zyx
Lx y z
Tk
wvuvcusetezcycxcFn
LLL
DtzyxC
τµτ
π
( )[ ] ( )( )
( ) ( ) ×Ω
−∫
++×
zyx
L
n
L
SLSLSLLL
DdudvdwdwcWdWvuC
TwvuP
wvuCTwvug
z
2
0
0001
000 ,,,,,,
,,,1,,,1
πττ
τξε
γ
γ
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
51
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∫∇
−×∞
=1 0 0 0 0 0001
,,,,,,
n
t L L L LS
nnnnCnCnnnnC
x y z z
WdWvuCTk
wvuwcvsucetezcycxcFn τ
τµτ
( )[ ] ( )( )
( ) ( ) ( ) ×∑Ω
−
++×
∞
=12
00002
,,,
,,,1,,,12
nnnnnC
zyx
L
SLSLSzcycxcF
LLL
Ddudvdwd
TwvuP
wvuCTwvug
πτ
τξε
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( ) ×∫ ∫ ∫ ∫ ∫
+−×
−t L L L L
SnnnnCnC
x y z z
WdWvuCTwvuP
wvuCtzyxCwcvcuseten
0 0 0 0 0000
1
000
001,,,
,,,
,,,,,,1 τ
τξτ
γ
γγ
( )[ ] ( ) ( ) ( ) ( ) ( ) ×∑Ω
−∇
+×∞
=12
02,,,
,,,1n
nCnnnnC
zyx
LS
LSLStezcycxcF
LLL
Ddudvdwd
Tk
wvuTwvug
πτ
τµε
( ) ( ) ( ) ( ) ( ) ( )( )
( )∫ ∫ ∫ ∫ ×∫
+
∇×
−t L L L L
S
S
nnn
x y z z
WdWvuCTwP
wvuCtzyxC
Tk
wvuwcvcusn
0 0 0 0 0000
1
000
001,,,
,
,,,,,,1
,,,τ
τξ
τµγ
γγ
( )[ ] ( ) ττε deudvdwdTwvugnCLSLS
−+× ,,,1 ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 02
0
110 ,,,2
,,,n
t L L L
LnnnCnCnnnnC
zyx
Lx y z
TwvugvcusetezcycxcFnLLL
DtzyxC τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=1 0 02
00102,,,
n
t L
nnCnCnnnnC
zyx
L
n
x
ucetezcycxcFnLLL
Ddudvdwd
u
wvuCwc τ
πτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∫∂
∂×
∞
=12
0
0 0
0102,,,
,,,n
nnnnC
zyx
L
L L
LnnzcycxcFn
LLL
Ddudvdwd
v
wvuCTwvugwcvs
y z πτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫
∂
∂−×
zyx
Lt L L L
LnnnnCnCLLL
Ddudvdwd
w
wvuCTwvugwsvcucete
x y z
2
0
0 0 0 0
0102,,,
,,,π
ττ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−×∞
=1 0 0 0 0100 ,,,,,,
n
t L L L
LnnnnCnCnnnnC
x y z
wvuCTwvugwcvcusetezcycxcFn ττ
( )( )
( ) ( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=
−
12
0000
1
0002,,,
,,,
,,,
nnCnnnnC
zyx
L tezcycxcFnLLL
Ddudvdwd
u
wvuC
TwvuP
wvuC πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( )( )
( )×∫ ∫ ∫ ∫
∂
∂−×
−t L L L
LnnnC
x y z
v
wvuC
TwvuP
wvuCwvuCTwvugvsuce
0 0 0 0
000
1
000
100
,,,
,,,
,,,,,,,,,
ττττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞
=1 0 0 02
02
n
t L L
nnnCnCnnnnC
zyx
L
n
x y
vcucetezcycxcFnLLL
Ddudvdwdwc τ
πτ
( ) ( ) ( ) ( )( )
( )−∫
∂
∂×
−zL
Lndudvdwd
w
wvuC
TwvuP
wvuCwvuCTwvugws
0
000
1
000
100
,,,
,,,
,,,,,,,,, τ
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−∞
=1 0 0 0 02
0 ,,,2
n
t L L L
LnnnnCnCnnnnC
zyx
Lx y z
TwvugwcvcusetezcycxcFnLLL
Dτ
π
( )( )
( ) ( ) ( ) ( ) ( )×∑−∂
∂×
∞
=12
01000002,,,
,,,
,,,
nnCnnn
zyx
L tezcycxcnLLL
Ddudvdwd
u
wvuC
TwvuP
wvuC πτ
ττγ
γ
( ) ( ) ( ) ( ) ( )( )
( )×∫ ∫ ∫ ∫
∂
∂−×
t L L L
LnnnCnC
x y z
vdwdu
wvuC
TwvuP
wvuCTwvugwcvseF
0 0 0 0
100000,,,
,,,
,,,,,,
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×∞
=1 0 0 0 02
02
n
t L L L
nnnnCnCnnnnC
zyx
L
n
x y z
wsvcucetezcycxcFnLLL
Dduduc τ
πτ
( )( )
( )( )
τττ
γ
γ
dudvdwdu
wvuC
TwP
wvuCTwvug
L∂
∂×
,,,
,
,,,,,, 100000 ;
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
52
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 02
0
101 ,,,2
,,,n
t L L L
LnnnCnCnnnnC
zyx
Lx y z
TwvugvcusetezcycxcFnLLL
DtzyxC τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=1 0 02
0000 2,,,
n
t L
nnCnCnnnnC
zyx
L
n
x
ucetezcycxcFLLL
Ddudvdwd
u
wvuCwc τ
πτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫∂
∂×
∞
=12
0
0 0 0
0002,,,
,,,n
nn
zyx
L
L L L
Lnnnycxcn
LLL
Ddudvdwd
v
wvuCTwvugwcwcvsn
y z z πτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )−∫ ∫ ∫ ∫
∂
∂−×
t L L L
LnnnnCnCnnC
x y z
dudvdwdw
wvuCTwvugwsvcucetezcF
0 0 0 0
000,,,
,,, ττ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]×∑ ∫ ∫ ∫ ∫ +−Ω−∞
=1 0 0 0 02
0 ,,,12
n
t L L L
LSLSnnnCnCnnnnC
zyx
Lx y z
TwvugvcusetezcycxcFnLLL
Dετ
π
( ) ( )( )
( ) ( )−
∇∫
+
∇× τ
τµτ
τξ
τµγ
γ
dudvdwdTk
wvuWdWvuC
TwvuP
wvuC
Tk
wvuS
L
S
Sz ,,,
,,,,,,
,,,1
,,,
0100
000
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−Ω−∞
=1 0 0 0 02
0 ,,,12
n
t L L L
LSLSnnnCnCnnnnC
zyx
Lx y z
TwvugvcusetezcycxcFnLLL
Dετ
π
( ) ( ) ( )( )
( ) ( ) −∫∇
+×
−
τττµτ
τξγ
γ
dudvdwdWdWvuCTk
wvu
TwvuP
wvuCwvuCwc
zLS
Sn0
000
1
000
100 ,,,,,,
,,,
,,,,,,1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ ∫ ∫ ∫ ∫ +−Ω−∞
=1 0 0 0 02
0 ,,,12
n
t L L L
LSLSnnnCnCnnnnC
zyx
Lx y z
TwvugvsucetezcycxcFnLLL
Dετ
π
( )( )
( )( )
( )( ) ττ
ττξ
τµγ
γ
dudvdwdWdWvuCTwvuP
wvuCwvuC
Tk
wvuwc
zL
S
S
n ∫
+
∇×
−
0000
1
000
100 ,,,,,,
,,,,,,1
,,, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 0
000
2
0
011,,,
,,,2,,,
n
t L L L
nnnCnCnnnnC
zyx
Lx y z
TwvuP
wvuCvcusetezcycxcFn
LLL
DtzyxC
γ
γ ττ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=1 0 02
00012,,,
n
t L
nnCnCnnnnC
zyx
L
n
x
ucetezcycxcFnLLL
Ddudvdwd
u
wvuCwc τ
πτ
τ
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ×∑−∫ ∫ ∫∂
∂×
∞
=12
0
0 0 0
0010002,,,
,,,
,,,
nnnnC
zyx
L
L L L
nnnycxcFn
LLL
Ddudvdwd
v
wvuC
TwvuP
wvuCwcwcvs
y z z πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( )×−∫ ∫ ∫ ∫
∂
∂−×
zyx
Lt L L L
nnnnCnCnLLL
Ddudvdwd
w
wvuC
TwvuP
wvuCwsvcucetezc
x y z
2
0
0 0 0 0
0010002,,,
,,,
,,, πτ
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )( )
( )×∑ ∫ ∫ ∫ ∫
∂
∂−×
∞
=
−
1 0 0 0 0
000
1
000
001
,,,
,,,
,,,,,,
n
t L L L
nnnnCnC
x y z
dudvdwdu
wvuC
TwvuP
wvuCwvuCwcvcuseF τ
ττττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×∞
=1 0 0 0 00012
0 ,,,2
n
t L L L
nnnCnCnnn
zyx
L
nCnnn
x y z
wvuCvsucetezcycxcLLL
Dtezcycxcn ττ
π
( ) ( )( )
( ) ( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=
−
12
0000
1
0002,,,
,,,
,,,
nnCnnn
zyx
L
nnCtezcycxcn
LLL
Ddudvdwd
v
wvuC
TwvuP
wvuCwcFn
πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( )( )
( )−∫ ∫ ∫ ∫
∂
∂−×
−t L L L
nnnnCnC
x y z
dudvdwdw
wvuC
TwvuP
wvuCwvuCwsvcuceF
0 0 0 0
000
1
000
001
,,,
,,,
,,,,,, τ
ττττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ ∫ ∫ ∫ ∫ +−Ω−∞
=1 0 0 0 02
0 ,,,12
n
t L L L
LSLSnnnnCnCnnnnC
zyx
Lx y z
TwvugwcvcusetezcycxcFnLLL
Dετ
π
Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
53
( )( )
( ) ( ) ×∑Ω−∫∇
+×
∞
=12
0
0010
0002
,,,,,,
,,,
,,,1
nnC
zyx
LL
S
SF
LLL
DdudvdwdWdWvuC
Tk
wvu
TwvuP
wvuC z πττ
τµτξ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )( )
×∫ ∫ ∫ ∫
++−×
t L L L
SLSLSnnnCnCnnn
x y z
TwvuP
wvuCTwvugvsucetezcycxcn
0 0 0 0
000
,,,
,,,1,,,1
γ
γ τξετ
( ) ( ) ( ) ( ) ( ) ( )∑ ×Ω−∫∇
×∞
=12
0
0010
2,,,
,,,
nnnnnC
zyx
LL
S
nzcycxcF
LLL
DdudvdwdWdWvuC
Tk
wvuwc
z πττ
τµ
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )( )
×∫ ∫ ∫ ∫
++−×
−t L L L
SLSLSnnnnCnC
x y z
TwvuP
wvuCwvuCTwgwcvcuseten
0 0 0 0
1
000
010,,,
,,,,,,1,1
γ
γ ττξετ
( ) ( ) ( ) ( ) ( ) ( ) ×∑Ω−∫∇
×∞
=12
0
0000
2,,,
,,,
nnCnnn
zyx
LL
S tezcycxcnLLL
DdudvdwdWdWvuC
Tk
wvu z πττ
τµ
( ) ( ) ( ) ( ) ( )[ ] ( ) ×∫ ∫ ∫ ∫ ∫+−×t L L L L
LSLSnnnnCnC
x y z z
WdWvuCTwvugwcvsuceF0 0 0 0 0
000 ,,,,,,1 τετ
( )( )
( )( )
ττµτ
τξγ
γ
dudvdwdTk
wvu
TwP
wvuCwvuC 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.