NANO-MICRO LETTERS

NANO-MICRO LETTERS
Vol. 2, No. 4
256-267 (2010)
Optimization of Pulse Laser Annealing to Increase
Sharpness of Implanted-junction Rectifier in Semiconductor Heterostructure
E. L. Pankratov*
It has been recently shown that inhomogeneity of a semiconductor heterostructure leads to
increasing of sharpness of diffusion-junction and implanted-junction rectifiers, which are formed in
the semiconductor heterostructure. It has been also shown that together with increasing of the
sharpness, homogeneity of impurity distribution in doped area increases. The both effect could be
increased by formation of an inhomogeneous distribution of temperature (for example, by laser
annealing). Some conditions on correlation between inhomogeneities of the semiconductor
heterostructure and temperature distribution have been considered. Annealing time has been
optimized for pulse laser annealing.
Keywords: Pulse laser; Implanted-juction rectifier; Heterostructure
Citation: E. L. Pankratov, “Optimization of Pulse Laser Annealing to Increase Sharpness of Implanted-junction Rectifier
in Semiconductor Heterostructure”, Nano-Micro Lett. 2, 256-267 (2010). doi:10.3786/nml.v2i4.p256-267
Increasing performance and reliability of microelectronic
thickness L-a and known type of conductivity (n or p). The
devices and integrated circuits has attracted great interest
second layer of the SH is an EL (0d x d a) with diffusion
recently. One way to increase performance of semiconductor
coefficient D1 and thickness a. Let us consider a dopant, which is
devices is decreasing capacitance of p-n-junctions [1,2]. The
implanted across the boundary x=0 into the EL to produce the
increase of homogeneity of dopant distribution in doped areas of
opposite type of conductivity (p or n). At the time t=0 annealing
a semiconductor structure allows to operate with higher current
of radiation defects is started with continuance 4. The annealing
densities and to decrease local overheats or to decrease depth of
of radiation defects after production of the implanted-junction
p-n-junction [1-3]. Another actual problem is the increase of
rectifier leads to a decrease of quantity of the defects and an
exactness of theoretical description of dynamics of technological
increase of depth of the p-n-junction. The increasing is unwanted,
process. The increase leads to higher predictability of dopant
because the process leads to deviation of characteristics
dynamics and, as following, higher reproducibility of parameters
implanted-junction rectifier from scheduled values. It has been
of solid state electronic devices.
recently shown, that inhomogeneity of a SH leads to increasing
Different types of technological processes could be used for
of sharpness of diffusion-junction (see, for example, [6,7]) and
production p-n-junctions (see, for example, [1-5]). One of them
implanted-junction (see, for example, [8]) rectifiers, which are
is dopant diffusion into a semiconductor sample or in an epitaxial
formed in the SH. It has been also shown, that together with
layer (EL). Another one is ion implantation in the same cases. In
increasing of the sharpness homogeneity of dopant distribution
this paper we consider a semiconductor heterostructure (SH),
in doped area increases. To increase the both effects heating of
which is presented in Fig. 1. The SH consists of two layers. First
surface region (the thickness of the heated surface region is
of them is a substrate (ad x d L) with diffusion coefficient D2,
approximately equal to the thickness EL) of the SH attracting an
Nizhny Novgorod State University of Architecture and Civil Engineering,65 Il'insky street, Nizhny Novgorod, 603950, Russia
*Corresponding author. E-mail: [email protected]
E. L. Pankratov
Nano-Micro Lett. 2, 256-267 (2010)
257
interest. One way to produce the inhomogeneous distribution of
Here V(x,t) and V* are spatiotemporal and equilibrium
temperature is pulse laser annealing. Another advantage of this
distributions of concentrations of vacancies. P(x,T) is the limit of
type of annealing is local heating of the surface of the SH. The
solubility of dopant in SH. The fitting parameters ], [ and J
advantage is useful for production of elements of integrated
depend on properties of layers of SH. Parameter ] characterizes
circuits with decreasing spread of dopant across the interface of
degree of radiation damage of SH. Parameter [ characterizes
the SH. Some theoretical analysis of spatiotemporal distribution
doping degree of SH. Parameter J usually is equal to an integer
of temperature during laser annealing has been done in previous
value in the interval J [1,3] (see [2]). In the following let us
works. However, the analysis has been done in simplified
consider the limiting case, when the number of different
limiting cases.
complexes (for example, complexes of defects) is negligible in
The main aim of the present paper is to determine the
comparison with the number of point defects. Spatiotemporal
conditions, which correspond to increasing of recently detected
distribution of vacancies concentration is described by the
effect, i.e. to increase of the sharpness of the p-n-junction and the
following system of equation [3]
homogeneity of impurity concentration in doped areas at the
same time. The accompanying aim is to develop mathematical
approaches for analysis of dopant redistribution during annealing
by laser pulses.
Method of Solution
Spatiotemporal distribution of dopant concentration in the
considered SH (see Fig. 1) has been described by the second
Fick's law [1-3,5]
w C x, t w ª
w C x, t º
w t
w x¬
w x
« D x, T , V , C x, t w J C C x, t »
¼
,
(1)
w t
where C(x,t) is the spatiotemporal distribution of dopant
concentration. JC(x,t) is the spatiotemporal distribution of dopant
flow. D(x,T,V,C(x,t)) is the diffusion coefficient of dopant in the
SH. The diffusion coefficient depends on dynamical properties
of dopant in materials of the SH, on temperature T of annealing
and on concentrations of radiation defects and dopant. It has
been shown in Ref. [2] that in high-doped materials interaction
between dopant atoms and point defects increases. If the point
defects have nonzero charge J e with e an elementary charge,
then the interaction leads to concentrational dependence of the
diffusion coefficient. The concentrational dependence of the
diffusion coefficient could be approximated by the following
I x, t w t
w ª
DI x, T w x «¬
D x, T , V , C x, t (2)
w I x, t º
wx
» k I ,V x, T ¼
ª¬ I x, t V x, t I *V * º¼
w J I x, t w t
k I ,V x, T * *
¬ª I x, t V x, t I V ¼º
V x, t w V x, t º
w ª
k I ,V x, T DV x, T «
w t
w x¬
w x »¼
(3)
ª¬ I x, t V x, t I *V * º¼
w J V x, t w t
k I ,V x, T ª¬ I x, t V x, t I *V * º¼
where I(x,t) and I* are the spatiotemporal and the equilibrium
distributions of interstitials, respectively. JI(x,t) and JV(x,t) are
spatiotemporal distributions of flows of interstitials and
vacancies, respectively. DV(x,T) and DI(x,T) are diffusion
coefficients of vacancies and interstitials, respectively. kI,V(x,T) is
the parameter of recombination of point defects. Spatiotemporal
distribution of temperature could be estimated by using the
second Fourier's law
c T w T x, t wt
w T x, t º
ª
O x, T p x, t «
wx¬
w x »¼
w
p x, t function (see, for example, [9,10] and [2])
J
V x, t º ª
C x, t º .
ª
DL x, T 1 9
*
«¬
»¼ «¬1 [ P J x, T »¼
V
w
°
°
°
°
°
°
°
°°
®
°w
°
°
°
°
°
°
°
°¯
w J T x, t ,
(4)
wx
where c(T) is heat capacitance. For the most interesting (in our
case) interval of values of temperature one can consider
approximately constant value of heat capacitance (c(T)|cass).
O(x,T) is the heat conduction coefficient. Temperature
dependence of the heat conduction coefficient can be
DOI:10.3786/nml.v2i4.p256-267
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Nano-Micro Lett. 2, 256-267 (2010)
258
E. L. Pankratov
approximated by the following power law: O(x,T)=Oass(x)[1+P
Substitution of the average value of the functions U (x,t) (U = F,T;
(Td/T(x,t))M] (see appropriate figures in [12]), Td is Debye
F =I,V,L) and their partial derivatives in the right side of the Eqs.
P
M
parameters.
(6) instead of the considered functions gives us possibility to
D(x,T)=O(x,T)/c(T) is thermal diffusivity. p(x,t) is the bulk
obtained the first-order approximations U1 (x,t) of the functions U
density of heat power, which is allocated in MS. The power
(x,t). To decrease steps of the iterative process, let us consider
could be approximated by the function: p(x,t)=P0G(x/ L)sin(S
more accurate initial-order approximation (see, for example, [8]).
t/4), t[0,4/2], G(x/L) is Dirac G-function, 4 is the continuance
As such approximation we consider the solutions of the
of the laser pulse, S is the lateral area of SH, and P0 is the power
equations of the system (6), which correspond to average values
of the laser pulse. JT(x,t) is spatiotemporal distribution of heat
of diffusion coefficients D0L, D0I and D0V, thermal diffusivity
temperature
[12],
and
are
fitting
flow. The similar time dependence of power has been considered
D0ass and zero parameter of recombination. The solutions can be
in [13]. However, the approximation considered in our work
written in the form
leads to simplification of analysis of mass and heat transport.
The Eqs. (1), (3) and (4) are complemented by the following
2
I x, t L
boundary and initial conditions in the form
(5)
where Tr is the equilibrium distribution of temperature, which
coincides with room temperature.
nI
nI
n 0
f
¦ F c x e t ,
L
nV
JI(0,t)=0, I(L,t)=0, I(x,0)=fI(x); JV(0,t)=0, V(L,t)=0, V(x,0)=fV(x),
C(x,0)=fC(x); JT(0,t)=0, T(L,t)=Tr, T(x,0)=fT(x),
¦ F c x e t ,
2
V x, t JC(0,t)=0, C(L,t)=0,
f
nV
n 0
2
C x, t L
(7)
f
¦ F c x e t ,
nC
nC
n 0
2 f
T x, t Tr ¦ FnT t cnT x enT t ,
Ln1
First of all let us estimate spatiotemporal distribution of
temperature. The parabolic equation has been transformed as
where Dass(x)=O ass(x)/cass(x).
v, t w T u , t T x, t T ³
³ wt d ud v PT
D v
T v, t 1
³
³ p u, t d u d v PT c
D v
T x, t T
P M 1 T
x
1
T
M
v
d
L
x
M
³ f v c v d v,
F
FnT t 0
ass
L
FnF
n
0
M
r
where
v
t
L
0
0
³ enT W ³ cn v p v, W Cass
d vdW,
M
d
ass
L
cn(x)=cos[S (n+0.5)x/L], enT(t) =exp[-S 2(n+0.5)2D0asst/L2],
0
ass
M 1
M 1
r
enF(t)=exp[-S 2(n+0.5)2D0Ft/L2].
M
d
I x, t x
1
v
³ D v, T ³ k I ,V u , T I u , t V u , t d u d v x
v
L
I
1
0
³ D v, T ³
L
us possibility to obtain the first-order approximations (in the
dudv
modified method of averaging of function corrections) of the
appropriate functions. The algorithm is presented in details in [8]
³ D v, T ³ k u , T I u , t V u , t d u d v I ,V
L
0
V
1
v
³ D v, T ³
L
w V u, t wt
0
V
x
V
³ D v, T ªV
¬
L
*
P³
L
C
P
J
J
v
V v , t º¼
v, t w C v, t dv
v, T w v
³
0
w C u, t wt
substitute the sums D2U+U1 (x,t) instead of the functions U (x,t) in
dudv
the right side of the equations of the system (6). The substitution
gives us possibility to obtained the second-order approximation
(6)
functional corrections (see, for example, [14].
DOI:10.3786/nml.v2i4.p256-267
approximations of the functions U (x,t), by using the method of
standard procedure (see, for example, [8]), i.e. one shall
Let us determine the solution of the system Eqs. (6) by
averaging
and will not be considered in this paper. The second-order
averaging of function corrections can be determined by using the
dudv
*
L
x
equations of the system (6) instead of the functions U (x,t) gives
v
1
x
C x, t wt
0
I
x
V x, t w I u, t Substitution of the Eqs. (7) into the right side of the
of the functions U2 (x,t). The algorithm is presented in details in
[8,14] and will not be considered in this paper.
The parameter D2U is determined by the following relation
[8,14],
http://www.nmletters.org
E. L. Pankratov
259
M ij U M i 1 j U
D2U
L4
,
Nano-Micro Lett. 2, 256-267 (2010)
(8)
where
4 L
M ij U
³ ³ U x, t d x d t .
j
i
0 0
Furthemore let us analyze the dynamics of redistribution of
FIG. 2. Calculated distribution of dopant after annealing with continuance 4 =
dopant in the considered MS (see Fig. 1). The obtained analytical
0.0025D0L/L2 (curves 1 and 3) and 4 = 0.005D0L/L2 (curves 2 and 4). Curves 1
relations give us possibility to analyze the redistribution during
and 2 are corresponding to spatial of distribution of dopant in homogeneous
annealing
numerical
material. Curves 3 and 4 are correspond to spatial distribution of dopant in SH
approaches of the Eqs. (1), (3) and (4) leads to increase the
for D1L/D2L=4. Solid lines are analytical results. Dashed lines are numerical
exactness of the spatiotemporal distribution of dopant
results. Coordinate of interface is equal to a=L/2.
of
dopant
demonstratively.
Using
concentration.
in EL. The effects increase at the same time with the increase of
differences between properties of layers of SH. The increase of
Discussion
sharpness gives us possibility to decrease transit time of charge
Let us analyze the dynamics of redistribution of dopant in
carriers through the p-n-junction. The increase of homogeneity
the SH (Fig. 1) for step-wise approximations of spatial
of dopant distribution leads to a decrease of local overheats in
distribution of diffusion coefficients of radiations defects and
doped area. Dopant distributions on the figure are qualitatively
dopant and thermal diffusivity. In the case the approximations
similar with analogous distributions in Fig. 3 in reference [8].
can be written as Dass(x)=Dass1[1(x)-1(x-a)]+Dass2 1(x-a) and
But spreading of the distributions in the present paper is higher,
DF,L(x)=DF,L1[1(x)-1(x-a)]+DF,L21(x-a), where 1(x) is the unit
because diffusion coefficient of dopant in EL is larger in
function; DF,L1, DF,L2, Dass1 and Dass2 are diffusion coefficient and
comparison with the same diffusion coefficient in [8] due to in
thermal diffusivity of the EL and substrate, respectively. Spatial
homogeneity of distribution of temperature. Figure 2 in the
distributions of dopant concentration for some values of
present paper shows that inhomogeneity of the SH leads to
annealing time and the difference between diffusion coefficients
increasing the sharpness of p-n-junction (if the junction was
of EL and the substrate are presented in Fig. 2. For simplification
formed near the interface) and homogeneity of dopant
of analysis we consider the following normalization,
distribution in doped area. The calculated spatial distribution of
L
³
dopant and experimental one has been compared in Fig. 3. In this
C x, t d x
1.
0
situation we obtain satisfactory agreement between calculated
and measured results. The satisfactory agreement suggests our
Figure 2 shows that semi-insulating property of interface of
SH gives us possibility to increase sharpness of p-n-junction and
at the same time to increase homogeneity of dopant distribution
FIG. 3. Calculated distribution of dopant (solid line). Squares are experimental
distribution of boron concentration in silicon (see [15]) for dose F=2u1015 cm-2.
FIG. 1. Semiconductor heterostructure, which consist of an epitaxial layer
The boron distribution has been annealed by 20 laser pulses with continuance 23
(x[0,a]) and a substrate (x[a,L]).
ns, repetition rate 1 hertz and density of power 0.5 J/cm2.
DOI:10.3786/nml.v2i4.p256-267
http://www.nmletters.org
Nano-Micro Lett. 2, 256-267 (2010)
260
E. L. Pankratov
FIG. 4. Distributions of implanted dopant distributions (curves 1-3) for different
FIG. 5. Dependences compromise continuance of laser pulses 4 on some
values of annealing times. Increasing of number of curves corresponds to
parameters of SH. Curve 1 is the dependence of 4 on the ratio D1L/D2L for ]
increasing of value of annealing time. Curve 4 is idealised approximation of
=0 and a=L/2. Curve 2 is the dependence of 4 on the parameter ] for
curves 1-3.
D1L/D2L=1 and a=L/2. Curve 3 is the dependency of the 4 on the ration a/L for
D1L/D2L=1 and ] =0. Curve 4 is the dependence of 4 on the ratio a/L for
accurate choice of model for the present paper.
D1L/D2L=1 and ] =0.
The increase of annealing time leads to the increase of the
homogeneity of dopant distribution and to decrease of the
deceleration of dopant diffusion in the substrate is the reason of
sharpness of the p-n-junction. Optimization of continuance of
the increase of sharpness of the p-n-junction and the increase of
laser pulse leads to an increase of the effects at the same time. It
homogeneity of dopant distribution in doped area. The obtained
should be noted, that two limiting cases of annealing of radiation
dependences of optimal annealing times are quantitatively
defects could be considered. The first of them is the limiting case
differing from analogous dependences
of large continuance of laser pulse (spreading of distribution of
inhomogeneity of temperature distribution. It should be noted,
dopant is larger, than thickness of EL). The second of them is the
that annealing by laser pulse with optimal continuance could be
limiting case of small continuance of laser pulse (spreading of
substituted by some laser pulses with smaller continuance, but
distribution of dopant is smaller, than thickness of EL).
with high frequency.
in [8]
due
to
Optimization of annealing time in the second limiting case is
Dependences of optimal annealing time on several
necessary, because the increasing of the continuance of laser
parameters are presented in Fig. 5. The figure shows that
pulse leads to shifting the p-n-junction to the interface of the MS.
increasing of the thickness of the EL leads to increasing of the
Let us to use the earlier introduced criterion (see, for example,
compromise annealing time. Increasing of the ratio D1L/D2L and
[6-8] for optimization of annealing time. To use the criterion to
the parameter ] leads to decreasing of the annealing time. It
optimize the continuance of laser pulse we approximate
should be noted, that annealing by laser pulse with optimal
spatiotemporal distribution of dopant concentration by step-wise
continuance could be substituted by some laser pulses with
function (see Fig. 4 and appropriate parts of Refs. [6-8]). To
smaller continuance, but with high frequency. It should be noted,
estimate the optimal continuance of laser pulse the mean squared
that optimal annealing time for the laser annealing case is smaller
error between the real spatiotemporal distribution of dopant
than optimal annealing time for the volumetric annealing case
concentration and step-wise approximation function should be
(see Fig. 5). It has been obtained that annealing times for Fig. 2
minimized. Dependences of optimal continuance of laser pulse
almost equal to optimal values of annealing times in Fig. 5. The
on several parameters are presented in Fig. 5. The figure shows
difference could be explained by two reasons. The first of them is
that the increase of the thickness of the EL leads to an increase of
insufficient continuance of annealing in Fig. 2. The second one is
the compromise continuance of laser pulse. The increase of
finite exactness of mathematical approach.
continuance of laser pulse is obtained due to the increase of
continuance of dopant diffusion to interface between layers of
SH. The increasing of the ratio D1L/D2L and the parameter ] leads
to a decrease of the annealing time. The decrease could be
obtained due to the increase of diffusion coefficient of EL, i.e.
acceleration of dopant diffusion in the layer. At the same time
dopant diffusion coefficient in the substrate decreases. The
DOI:10.3786/nml.v2i4.p256-267
Conclusion
In this paper we consider pulse laser annealing of radiation
defects to increase sharpness of implanted-junction rectifier in
semiconductor heterostructure. With the increase of the
sharpness, homogeneity of dopant distribution in doped area
increases. We optimized continuance of laser pulse to obtain
http://www.nmletters.org
E. L. Pankratov
maximal compromise between the increase of sharpness of the
Nano-Micro Lett. 2, 256-267 (2010)
261
6.
p-n-junction and the increase of homogeneity of dopant
distribution in doped area. The optimal continuance has been
1103/PhysRevB.72.075201
7.
analyzed as a function of several parameters of heterostructure.
E. L. Pankratov and B. Spagnolo, Eur. Phys. J. B 46, 15
(2005). doi:10.1140/epjb/e2005-00233-1
8.
This work has been supported by grant of President of Russia
(project № MK-548.2010.2).
E. L. Pankratov, Phys. Rev. B 72, 075201 (2005). doi:10.
E. L. Pankratov, Phys. Lett. A 372, 1897 (2008).
doi:10.1016/j.physleta.2007.10.058
9.
E. I. Zorin, P. V. Pavlov and D. I. Tetelbaum, “Ion doping
of semiconductors”,
Received 11 October 2010; accepted 6 December 2010; published
online 12 December 2010.
(Energiya, Moscow, 1975, in
Russian).
10. H. Ryssel and I. Ruge, Ion implantation, (Teubner,
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DOI:10.3786/nml.v2i4.p256-267
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Nano-Micro Lett. 2, 256-267 (2010)
262
E. L. Pankratov
Appendix
Final analytical formulae describing curves at Fig. (2):
Relations for dopant concentration
J 1
2 S
C2 x , t uP J
u
J 2
L
§ S 2 D0 L ·
¨ 2 ¸
© L ¹
d u d w sn v d v
Dc v, T2 P
J
f
¦ n 0.5 F
e t nC nC
n 0
D0 L
v, T 2
w V2 u , t d u d w d v
2S
V
V
¦ n 0.5 F ³ ªV
¬
nC
2
n 0
*
L
x
2 J 1 S
L
¯
J 2
L
f
L
J
v
uFnC ³
L
u
*
sn w J 1
ª f F c w e t º d w D 2S
¦ nC n
0L
nC
J
2
»¼
P w, T2 «¬ n 0
L
dw
dv
DC w, T2 P
2
J
v, T V sn 0.5 u *
v w
e t ³
nC nC
V sn 0.5 u d u
v
f
w
*
v
¦ n 0.5 F
n 0
*
V2 v, t ¼º
½
D0 L 2 ¦ n 0.5 FnC ³
D 2C ¾
*
L n0
DC u , T2 ¬ªV V2 u , t ¼º
L
¿
2S
J 1
ª f n 2 F c u e t º u
nC n
nC
³ ³ ªV * V w, t º ³ «¬¦
»¼
L L ¬
2
¼0 n0
x v
x
f
L
P ³ ®P
Dc v, T2 wu
J 1
³³ D
0 L
c
u, T ª¬V
V2 u , t º¼
*
2
sn u 2
u
J
ª f F c u e t º d u ¦ nC n nC »¼
J
P v , T2 «¬ n 0
3
J 1
2 S f

D
P
J
®
¦ n 0.5 enC t u
0L
J 4
L
n 0
¯
v
f
¦ n 0.5 F ³
w V2 w, t V sn 0.5 w wu
ª¬V * V2 w, t º¼
nC
n 0
L
*
2
u
,
2
where D2C is determined by solving the following equation for concrete values of parameter J
J 1

2 S
D
P
®
³ ³ 2C
J 2
L4 0 0 ¯
L
P
D 2C
4 L
sn v J
f
¦ n 0.5 FnC enC t ³ PJ v, T ª«¬¦ FmC cm v emC t º»¼ d v n 0
m 0
L
2
x
f
J
½ 2S
D0 C 2 ¦ n 0.5 FnC ³
¾ ® D0 L 2
*
L n0
Dc v, T2 ª¬V V2 v, t º¼ ¿ ¯
L
L
2S
u
u
w V2 v, t J
v, T P x, T J
2
w
u FnC ³
L
P D0 L
2 S
J 4
J 1
FnC P D0 L
3
L
2 S
x
f
sn 0.5 v ¦ n 0.5 F ³ D v, T u
nC
n 0
L
c
2
J
f
¦ n n 0.5 enC t ³ ª«¬¦ FmC cm x emC t º»¼ u
n 0
m 0
L
x
f
2
3 4 L
J 5
4L
sn v xV
³ ³ D x, T ªV
0 0
c
2
¬
*
*
x
f
³ ¦ n 0.5 e t u
V x, t º
¼
2
nC
0 n 0
J
ª f F c x e t º d v d w d x d t 2S D0 L V * 2 f n 0.5 F u
¦
¦ nC n nC »¼
nC
3
J
P v, T2 «¬ n 0
L4
n 0
x v
x
³ ³ D x, T ªV
0 0
2
2
4 L
u
J 1
*
sn v d v x d x d t
P
*
V dv
*
¬ªV V2 v, t ¼º
wt
V sn 0.5 v d v
x
f
c
2
¬
*
V2 x, t º¼
DOI:10.3786/nml.v2i4.p256-267
sn 0.5 u d u d v d x d t
³ ³ D u, T ªV
0 L
c
2
¬
*
V2 u , t º¼
J 1
P
2 S
J 3
4L

¦ n 0.5 F ®³ e t u
¯
4
f
nC
n 0
nC
0
http://www.nmletters.org
E. L. Pankratov
L
u³ L x 0
Nano-Micro Lett. 2, 256-267 (2010)
263
J
J
4
L
ª f F c x e t º sn x d x d t L e t ª f F c x e t º sn v d v d t ½ ;
nC n
nC
nC n
nC
³ nC ³ «¬¦
«¬ ¦
»¼ P J x, T »¼ P J v, T ¾¿
n 0
n 0
0
0
2
2
Relations for temperature
4
L
x
p u, - sn v 2 D ass 0 f

D
T
S
n
e
c
u
d
u
d
0.5
>Tr ®
¦
r
nT
n
2T
³
³
³
M ³
M
P Td L ¯
P Td L n 0
cass
D ass v L
0
0
4
f
2
x
1
T2 x, t L
¦ c v e t ³ e - ³ c u n
L
nT
nT
n 0
n
0
4
u ³ enT - ³ cn u 0
cass
0
L
u cn u ³
p u, - 0
cass
cass
0
p u, - L
p u, - º
d u d -»
¼
º
d u d - Tr »
¼
M 1
M
v
³
p u, t cass
0
M
x
1
1
º
ªT 2 f c v e t u
d u d- » d v ¦ n nT
r
M ³
P Td L D ass v «¬
L n0
¼
dudv
M
½
°
M ¾
P M 1 Td °
¿
Tr
M 1
4
ª2 f
c
x
e
t
¦ n nT ³ enT - u
M
P M 1 Td «¬ L n 0
0
1
2 8M D ass2 0
³ ®¯S P T M L4
0
d
v
4
f
L
¦ n 0.5 ³ e - ³
n 0
0
s v ª
p u, - 2 f
º
ucn u d u d - ³ n
T
cn v enT t ³ enT - ³ cn u d u d- »
¦
r
«
D ass v ¬
L n0
cass
¼
L
0
0
4
w
4
um
L
³ emT - ³ cm u 2
0
p u, - cass
0
u cn u d u d - Tr @
v
M
³
p u, t cass
0
u
S
2
L
¦ m cm w emT t ³ emT - ³ cm u m 0
0
f
4
L
n 0
0
0
0
uS ¦ n 0.5 ³ enT - ³ cn u M
u³
0
p u, t cass
M 1
u
1 ½
M
¾
P Td ¿
f
¦ c v e t u
m
w
1
M
p u, - cass
p u, - cass
½ d wd v
L
2 D ass 0

®D 2T M u
P Td L
¿ D ass v P M 1 Td ¯
d u d- ¾
1
M
4 L
p u, - 2 f
ª
T
c
v
e
t
cn u du u
¦
r
n
nT
³
³
«
D ass v ¬
L n0
cass
L
0 0
x
d u d -³
sn v x
1
4
L
p u,- 2 f
ª
º
dudv
T ¦ cn x enT t ³ enT - ³ cn u d u d- »
M « r
L n0
cass
P M 1 Td ¬
¼
0
0
1
ª
Tr «1 ¬
mT
m 0
4
L
p u, - 2 f
ª
º
T
c
v
e
t
e
cn u d u d -» u
¦
r
n
nT
nT
³
³
M ³
«
P Td L D ass v ¬
L n0
cass
¼
0
0
1
uenT - d- @ d v Tr v
M 1
u
4
L
p u, - ª2 f
u
c
v
e
t
e
¦
n
nT
nT
³
³
M ³
«
P Td L D ass v ¬ L n 0
cass
0
0
1
4
2
L Td
L
cass
0
p u,- 2ª
2 f
º
d u d v «Tr ¦ cn w enT t ³ enT - ³ cn u d u d- » u
P¬
L n0
cass
¼
0
0
4
f
M
3
d u d- d v p u, - nT
M 1
Tr
M 1
M 1
u
4
L
p u, - 2 D ass 0 f
º x
D
S
n
e
cn u d ud-u
0.5
® 2T
¦
nT
³
³
³
M »
M
P M 1 Td ¼ L ¯
P Td L n 0
cass
0
0
Tr
DOI:10.3786/nml.v2i4.p256-267
M
http://www.nmletters.org
Nano-Micro Lett. 2, 256-267 (2010)
264
E. L. Pankratov
M
sn v ª
p u, - 2 f
1
º
Tr ¦ cn v enT t ³ enT - ³ cn u d u d- » d v Tr M
«
D ass v ¬
L n0
cass
P Td
¼
L
0
0
4
x
u³
2
L
4
f
L
L
¦ cn v enT t ³ enT - ³ cn u n 0
0
4
uenT t enT - ³
0
³
cass
0
p u, - L
p u, - º
cn u d u d - »
¼
cass
0
M 1
º
d u d -»
¼
M
v
³
p u, t cass
0
M
½
°
M ¾
P M 1 Td °
¿
Tr
M 1
v
³
0
1
³ D v >T
r
L
ass
ªT 2 f c x u
¦n
r
M
P M 1 Td «¬
L n0
1
dudv
p u, t w
du
cass
dw
1
,
D ass w P Td
M
where D2T is determined by solving the following equation for concrete values of parameter M
ª
D 2T
Tr «1 ¬
4 L
4
L
p u, - 2 D ass 0 f
1
º

D
S
n
e
cn u d ud-u
0.5
® 2T
¦
nT
³
³
³
³
M »
M
M
P M 1 Td ¼ P Td L4 0 0 ¯
P Td L n 0
cass
0
0
Tr
M
M
s v ª
p u, - 2 f
1
º
u³ n
T
cn v enT t ³ enT - ³ cn u d u d- » d v Tr ¦
r
M
«
D ass v ¬
L n0
cass
P Td
¼
L
0
0
4
x
4
L
0
0
p u, - u enT t ³ enT - ³ cn u L
u ³ cn u p u, - cass
0
º
d u d -»
¼
cass
M
³
M
M 1
dudv
4
L
p u, - 2 f
ª
u
T
c
x
e
t
e
¦
n
nT
nT
³
³
M « r
P M 1 Td ¬
L n0
cass
0
0
1
p u, - s v
¦ ³ e - ³ c u c d u d - ³ D v u
3 M D ass2 0
³ ®4S P T M L4
0 ¯
d
x
f
4
L
x
n
nT
n
n 0 0
0
L
p u, - 0
u
u
2º
M 1
L »¼
2
L
³
L
0
4 L
cass
4
f
m
L
mT
p u, - mT
cass
m
0
º
cn u d u d- »
¼
¦ m c v e t ³ e - u
2
m
mT
2 D ass 0
¯
P Td
f
0
ass
4
f
M
d
L
¦ m cm v emT t ³ emT - ³ cm u 2
m 0
0
0
4
f
ª
T
c
x
e
t
¦
nT
³ enT - u
« r n0 n
¬
0
p u, - cass
½
°
1
°
¿
P L 4Td
d u d- ¾ d x d t 4
L
x
n
nT
n 0
DOI:10.3786/nml.v2i4.p256-267
mT
m 0
p u, - s v ª
d u d -³
T
n 0.5 ³ e - ³ c u ¦
L
c
D v «¬

M
0
M
4
f
1
2
m 0
M 1
ass
4
L
f
p u, - ª
T
c
v
e
t
e
cn u dud-u
¦
r
n
nT
nT
³
³
M ³
«
P Td L D ass v ¬
cass
n 0
0
0
x
1
p u, - 1
¦ m c v e t ³ e - ³ c u c d u d- P T
u³ ³ ®D 2 T Tr S
0 0
d u d- d v L
ass
4
L
p u, - 2 f
ª
º
u n 0.5 «Tr ¦ cn v enT t ³ enT - ³ cn u d u d- »
L n0
cass
¬
¼
0
0
u ³ cm u 1
M
cass
0
ª 2 f c v u
n
³ D v «¬ L ¦
n 0
L
ass
x
x
4
L
1
1
º
ª2 f
d u d - Tr » d v ¦ cn v enT t ³ enT - ³ cn u u
M ³
P Td L D ass v «¬ L n 0
¼
0
0
p u, t v
½
ucn u d u d- @ M ¾
P M 1 Td ¿
Tr
M 1
L
0
n
0
r
ass
L
ass
2
M
u
f
¦ c v e t u
L
n
nT
n 0
http://www.nmletters.org
E. L. Pankratov
4
p u, - L
u ³ enT - ³ cn u 0
u cn v enT t @
M
p u, t v
³
cass
0
M
½
u M M ¾
P M 1 Td ¿
Td
Tr
u
p u, - cass
M 1
º
³
4
L
p u,- 2 f
ª
T
e
dud-u
³ cn u ¦
r
nT
³
M ³
«
P Td L D ass v ¬
L n0 0
cass
0
x
1
1
M 1
4
L
p u,- 2 f
ª
º
dudv
T
c
x
e
t
e
cn u d u d- » u
¦
r
n
nT
nT
³
³
«
P M 1 ¬
L n0
cass
¼
0
0
1
p v, t x
M
d u d- » d v ¼
cass
0
1
dv
cass
0
xd xdt
D ass x 4
L
2 D ass 0 f

D
S
T
n
0.5
e
®
¦
³ ³ 2T r P T M L n 0
³ nT
³ cn u u
L4 0 0 ¯
0
0
d
1
4 L
4
4
L
u enT t enT - cn u ³
³
0
u
p u, - cass
0
2 D ass 0
M
P Td L
p u, - cass
M
sn v ª
p u, - 2 f
1
º
Tr ¦ cn v enT t ³ enT - ³ cn u d u d- » d v M
«
D ass v ¬
L n0
cass
P Td
¼
L
0
0
x
d u d -³
L
4
L
p u, - 2 f
ª
º
u «Tr ¦ cn v enT t ³ enT - ³ cn u d u d -»
L n0
cass
¬
¼
0
0
S
Nano-Micro Lett. 2, 256-267 (2010)
265
f
4
n 0
0
º
d u d- »
¼
L
¦ n 0.5 ³ e - ³ c u nT
M 1
n
v
³
p u, t dudv
cass
0
cass
ass
4 L
³ ³ ^D
Tr 2T
0 0
4
L
sn v ª
2 f
T
c
v
e
t
e
¦ n nT ³ nT
r
³ cn u u
D ass v «¬
Ln0
0
0
L
x
d u d -³
M
x
4
L
p u, - 1 ª
2 f
º
º
d u d- » d v ³ «Tr ¦ cn v enT t ³ enT - ³ cn u d u d -»
P L¬
L n0
cass
¼
¼
0
0
L
4
p u, - 2 f
ª
º
u M T
c
x
e
t
e
cn u d u d- »
¦
n
nT
nT
³
³
M « r
Td
P M 1 Td ¬
Ln0
cass
¼
0
0
1
L
ªT 2 f c x u
¦
M 1 TdM «¬ r L n 0 n
M 1
M 1
1
³ D v u
1
½
1
°
d xdt M ¾
M
P P M 1 Td °
P L4 M 1 Td
¿
Tr
1
p u, - 0
M
x
1
M 1
M
p u, t v
³
dv
du
D ass v cass
0
u
M 1
½°
d xdt ;
M ¾
P M 1 Td °
¿
Tr
M 1
Relations for temperature
I 2 x, t D0 I
4S
D0V
2
4S
4
L
2
x v w
n 0
L 0 0
x v w
f
³ ³ ³ k y, T ¦ F
nI
2
n 0
L 0 0
cn y enI t ¦ m FmV cm y emV t d y
u
4
2
L
w
L 0 0
f
f
dudv
DI y , T2 x
³
L
1
DI v, T2 ³ k I ,V y, T ¦ FnI cn y enI t ¦ FmV cm y emV t d y d w 2D0 I
0
n 0
DOI:10.3786/nml.v2i4.p256-267
m 0
dudv
DI w, T2 DI v , T2 dudv
DI w, T2 DI v, T2 m 0
3
n 0
wd w
wd w
2
u D0 I ¦ n FnI enI t ³ ³ ³ k I ,V y , T2 sn y d y
2
m 0
f
x v w
f
f
2
I ,V
4
L
f
³ ³ ³ k I ,V y, T ¦ n FnI cn y enI t ¦ FmV cm y emV t d y

v
u
³ k I ,V u , T2 ®D 2 I ³
¯
0
S
2
L
f
¦ nF
nI
n 0
L
u
enI t ³
L
4S
5
L
3
u
1
DI w, T u
sn w d w ½
¾u
DI w, T2 ¿
http://www.nmletters.org
Nano-Micro Lett. 2, 256-267 (2010)

4
¯
L
u ®D 2V 1
³ D w, T 2
L
V
w
f
f
S
0
n 0
m 0
L
³ k I ,V y, T ¦ FnI cn y enI t ¦ FmV cm y emV t d y d w 2D0 I
4S
D0 I
2
2
x v w
f
³ ³ ³ k y, T ¦ n
I ,V
4
L
2
n 0
L 0 0
x v w
f
2
FnI cn y enI t ¦ FmV cm y emV t d y
m 0
f
f
2
4
L
n 0
L 0 0
3
n 0
2
L
L 0 0
w
f
x
dudv
DV y , T2 ³
L
DI v , T2 f
n 0
0

4
¯
L
u
u
L
1
³ D w, T 2
u enI t ³
m 0
L
V
dudv
¯
S
2
L
4S
u
u
n 0
L
¦ nFnI enI t ³
u
5
L
DI w, T L
f
3
1
³ k I ,V u , T2 ®D 2 I ³
0
DI w, T2 DI v , T2
v
1
³ k I ,V y, T ¦ FnI cn y enI t ¦ FmV cm y emV t d y d w 2D0 I
u ®D 2V u
dudv
DI w, T2 DI v, T2 m 0
x v w
f
4
nI
n 0
wd w
wd w
³ ³ ³ k I ,V y, T2 ¦ FnI cn y enI t ¦ m FmV cm y emV t d y
u D0V ¦ n FnV enV t ³ ³ ³ k I ,V y , T2 sn y d y
u
¦ nF
¾d u d v ,
V2 x, t 2
2
f
DI w, T2 ¿
L
4S
E. L. Pankratov
sn w d w ½
u
u enI t ³
D0V
u
266
u
sn w d w ½
¾u
DI w, T2 ¿
w
f
f
S
0
n 0
m 0
L
³ k I ,V y, T ¦ FnI cn y enI t ¦ FmV cm y emV t d y d w 2D0 I
2
f
¦ nF
nI
u
n 0
sn w d w ½
¾d u d v ,
DI w, T2 ¿
where D2I and D2V are determined by the following relations
D 2V
1ª
§
« L4 S ¨ 1 2¬
u S I 11 WI M I 11 u 4 ©
I 01
S I 10 ·
S
S
S I 10 SV 01 º ª
§ S ·
§ S ·
S I 00 ¨ 1 V 10 ¸ S I 10 SV 11 WV M I 11 ¨ 1 I 10 ¸ S I 01 L u
»
«
L4 ¼ ¬
© L4 ¹
© L4 ¹
2
SV 00
L4
S
S
I 00
WV SV 11 M V 11 I 00 SV 11 WV M V 111 V 00 u
¸
L4 ¹ L4
L4
L4
WI M I 11 SV 01
I 11
S I 10 º ª§
¨1 L4 »¼ «¬©
SV 10 ·
¸ S I 00 S I 10 SV 00
L4 ¹
2
º 4 S M > V 11 V 11
¼»
1
§
º ªS §1 º ½2 ,
WV ¨ 1 W
S
M
S
S
¸
¸ I 10 V 00 » ¾
I
I 11
I 11
I 00 ¨
© L4 ¹ L4
¼» ¬« © L4 ¹
¼ ¿
S I 01 ·
D2I
SV 10 ·
SV 01
QI S I 11 M I 111 D 2V S I 10
L4 D 2V S I 00 S I 01
DOI:10.3786/nml.v2i4.p256-267
1
,
http://www.nmletters.org
E. L. Pankratov
Nano-Micro Lett. 2, 256-267 (2010)
267
where
4 L
x
x
³ ³ D x, T ³
S U ij
U
0 0
2 D0 I
S
2
f
v
n 0
L
¦ n FnI enI t ³
2
L
4 D0 I
4 D0V
2 D0 I
WV
4 D0V
2 D0V
S
S
S
2
L
³³
» «
v
2
nV
n 0
enV t ³
L
2 4 L x u
1
V
¦F
mV
u
f
f
0
n 0
m 0
L
f
n 0
m 0
f
³ ³ ³ ³ k I ,V w, T2 ¦ FnI cn w enI t ¦ m FmV cm w emV t d w
n 0
4 L
f
¦ n FnI enI t ³ ³
2
4
L
n 0
4 D0 I
S
0 0
m 0
x
x
³
D x, T S 2 4L x u
4
L
v sn v d v
DI v, T2 0
I
2 4 L x u
L
3
f
f
n 0
m 0
f
f
³ ³ ³ ³ k I ,V w, T2 ¦ FnI cn w enI t ¦ m FmV cm w emV t d w
n 0
0 0 0 0
4 L
f
¦ n FnV enV t ³ ³
n 0
DOI:10.3786/nml.v2i4.p256-267
0 0
m 0
x
2
4
DI v, T2 DI x, T vdud v xd xdt
DI v, T2 DI x, T DV
x
x, T ³
0
d xdt ,
2
4
vdud v xd xdt
2
³ ³ ³ ³ k I ,V w, T2 ¦ n FnI cn wenI t ¦ FmV cm wemV t d w
0000
» d wd u d vd xd t ,
f
f
DI u , T2 DV u , T2 ¼
³ ³ ³ ³ k I ,V w, T2 ¦ n FnI cn w enI t ¦ FmV cm w emV t d w
0 0 0 0
d wd u
j
2
4
cm w emV t ³ k I ,V w, T2 ¦ FnI cn w enI t ¦ FmV cm w u
2
sn u d u º
f
m 0
³ D u, T L
v
¦n F
¦
FnI cn w enI t n 0
DI u , T ¼ ¬ L
f
f
L 0
i
0 0 0 0
3
L
2
k I ,V w, T2 2
L
S
v u
sn u d u º ª 4
2 4 L x u
4
S
¬L
0
uemV t d w d u 2 D0V
WI
ª4
k I ,V v, T2 «
v sn v d v
DV v, T2 vdud v
vd ud v xd xd t
DV v, T2 DV x, T xd xdt
DV v, T2 DV x, T d xdt.
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