![International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
DOI: 10.5121/ijoe.2017.6201 1
ON OPTIMIZATION OF MANUFACTURING OF AN
AMPLIFIER TO INCREASE DENSITY OF BIPOLAR
TRANSISTOR FRAMEWORK THE AMPLIFIER
E.L. Pankratov and E.A. Bulaeva
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
ABSTRACT
In this paper we consider a possibility to increase density of bipolar heterotransistor framework an ampli-
fier due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping finished by opti-
mized annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account re-
distribution of radiation defects (after implantation of ions of dopant) for optimization of the above an-
nealing have been done by using recently introduced analytical approach. The approach gives a possibil-
ity to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
KEYWORDS
Bipolar heterotransistor; increasing of density of transistor; an amplifier circuit
1. INTRODUCTION
In the present time an actual question is decreasing of dimensions of solid state electronic devic-
es. To decrease the dimensions are could be increased density of elements of integrated circuits
and decreased dimensions of these elements. To date, several methods to decrease dimensions of
elements of integrated circuits have been developed. One of them is growth of thin films struc-
tures [1-5]. The second approach is diffusion or ion doping of required areas of samples or
heterostructures and father laser or microwave annealing of dopant and/or radiation defects [6-8].
Using of the above approaches of annealing leads to generation of inhomogenous distribution of
temperature and consequently to decreasing of dimensions of elements of integrated circuits.
Another approach to change properties of doped materials is radiation processing [9,10].
In the present paper we consider circuits of an amplifier [11] (see Fig. 1). Based on recently for-
mulated recommendations to decrease dimensions of single transistors (both bipolar and field-
effect) [12-17] we formulate recommendations to increase density of bipolar framework the con-
sidered circuits. We assume, that the considered circuits were manufactured framework the con-
sidered in Fig.1 heterostructure. The heterostructure consist of a substrate and an epitaxial layer.
The epitaxial layer includes into itself several sections, manufactured by using other materials.
These sections were doped by diffusion or by ion implantation to generate required types of con-
ductivity (n or p) in the required materials. Framework this paper we analyzed redistribution of
dopants during annealing of these dopants and/or radiation defects to formulate conditions,
which correspond to decreasing of the considered circuits.](https://image.slidesharecdn.com/6217ijoe01-170508052427/85/on-optimization-of-manufacturing-of-an-amplifier-to-increase-density-of-bipolar-transistor-framework-the-amplifier-1-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
2
Fig. 1. Structure of inverter [11]. View from top on epitaxial layer
2. METHOD OF SOLUTION
To solve our aims let us determine spatio-temporal distributions of concentrations of dopants.
The required distributions we determined by solving the second Fick's law [9,10,18,19]
( ) ( ) ( ) ( )
+
+
=
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC
CCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,,
. (1)
Boundary and initial conditions for the equations are
( ) 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)=f (x,y,z). (2)
In the Eqs. (1) and (2) the function C(x,y,z,t) describes the spatio-temporal distribution of con-
centration of dopant; the parameter DС is the dopant diffusion coefficient. Dopant diffusion coef-
ficient will be changed with changing of materials of heterostructure, heating and cooling of
heterostructure during annealing of dopant or radiation defects (with account Arrhenius law).
Dependences of dopant diffusion coefficient on coordinate in heterostructure, temperature of an-
nealing and concentrations of dopant and radiation defects could be written as [10,18]
( ) ( )
( )
( ) ( )
( )
++
+= 2*
2
2*1
,,,,,,
1
,,,
,,,
1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxC
TzyxDD LC ςςξ γ
γ
. (3)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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ON OPTIMIZATION OF MANUFACTURING OF AN AMPLIFIER TO INCREASE DENSITY OF BIPOLAR TRANSISTOR FRAMEWORK THE AMPLIFIER
- 1. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 DOI: 10.5121/ijoe.2017.6201 1 ON OPTIMIZATION OF MANUFACTURING OF AN AMPLIFIER TO INCREASE DENSITY OF BIPOLAR TRANSISTOR FRAMEWORK THE AMPLIFIER E.L. Pankratov and E.A. Bulaeva Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia ABSTRACT In this paper we consider a possibility to increase density of bipolar heterotransistor framework an ampli- fier due to decreasing of their dimensions. The considered approach based on doping of required areas of heterostructure with specific configuration by diffusion or ion implantation. The doping finished by opti- mized annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account re- distribution of radiation defects (after implantation of ions of dopant) for optimization of the above an- nealing have been done by using recently introduced analytical approach. The approach gives a possibil- ity to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces between layers of the heterostructure with account nonlinearity of these transports and variation in time of their parameters. KEYWORDS Bipolar heterotransistor; increasing of density of transistor; an amplifier circuit 1. INTRODUCTION In the present time an actual question is decreasing of dimensions of solid state electronic devic- es. To decrease the dimensions are could be increased density of elements of integrated circuits and decreased dimensions of these elements. To date, several methods to decrease dimensions of elements of integrated circuits have been developed. One of them is growth of thin films struc- tures [1-5]. The second approach is diffusion or ion doping of required areas of samples or heterostructures and father laser or microwave annealing of dopant and/or radiation defects [6-8]. Using of the above approaches of annealing leads to generation of inhomogenous distribution of temperature and consequently to decreasing of dimensions of elements of integrated circuits. Another approach to change properties of doped materials is radiation processing [9,10]. In the present paper we consider circuits of an amplifier [11] (see Fig. 1). Based on recently for- mulated recommendations to decrease dimensions of single transistors (both bipolar and field- effect) [12-17] we formulate recommendations to increase density of bipolar framework the con- sidered circuits. We assume, that the considered circuits were manufactured framework the con- sidered in Fig.1 heterostructure. The heterostructure consist of a substrate and an epitaxial layer. The epitaxial layer includes into itself several sections, manufactured by using other materials. These sections were doped by diffusion or by ion implantation to generate required types of con- ductivity (n or p) in the required materials. Framework this paper we analyzed redistribution of dopants during annealing of these dopants and/or radiation defects to formulate conditions, which correspond to decreasing of the considered circuits.
- 2. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 2 Fig. 1. Structure of inverter [11]. View from top on epitaxial layer 2. METHOD OF SOLUTION To solve our aims let us determine spatio-temporal distributions of concentrations of dopants. The required distributions we determined by solving the second Fick's law [9,10,18,19] ( ) ( ) ( ) ( ) + + = z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC CCC ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,,,,,,,,,,, . (1) Boundary and initial conditions for the equations are ( ) 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)=f (x,y,z). (2) In the Eqs. (1) and (2) the function C(x,y,z,t) describes the spatio-temporal distribution of con- centration of dopant; the parameter DС is the dopant diffusion coefficient. Dopant diffusion coef- ficient will be changed with changing of materials of heterostructure, heating and cooling of heterostructure during annealing of dopant or radiation defects (with account Arrhenius law). Dependences of dopant diffusion coefficient on coordinate in heterostructure, temperature of an- nealing and concentrations of dopant and radiation defects could be written as [10,18] ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ . (3)
- 3. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 3 Here function DL (x,y,z,T) describes dependences of dopant diffusion coefficient on coordinate and temperature of annealing T. Function P (x,y,z,T) describes the same dependences of the limit of solubility of dopant. The parameter γ is integer and usually could be varying in the following interval γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in av- erage) with each atom of dopant. More detailed information about concentrational dependence of dopant diffusion coefficient is presented in [18]. The function V (x,y,z,t) describes distribution of concentration of radiation vacancies in space and time. The parameter V* describes the equilibri- um distribution of concentration of vacancies. It should be noted, that diffusion type of doping gives a possibility to obtain doped materials without radiation defects. In this situation ζ1=ζ2=0. We determine spatio-temporal distributions of concentrations of radiation defects by solving the following system of equations [10,13] ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ Tzyxk y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI IIII ,,, ,,, ,,, ,,, ,,, ,,, , ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxI TzyxD z tzyxI VII ,,,,,,,,, ,,, ,,,,,, , 2 − ∂ ∂ ∂ ∂ +× (4) ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ Tzyxk y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VVVV ,,, ,,, ,,, ,,, ,,, ,,, , ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxV TzyxD z tzyxV VIV ,,,,,,,,, ,,, ,,,,,, , 2 − ∂ ∂ ∂ ∂ +× . Boundary and initial conditions for these equations are ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxρ , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxρ , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxρ , ( ) 0 ,,, = ∂ ∂ = yLy y tzyxρ , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxρ , ( ) 0 ,,, = ∂ ∂ = zLz z tzyxρ , ρ (x,y,z,0)=fρ (x,y,z). (5) Here ρ =I,V. The function I(x,y,z,t) describes the distribution of concentration of radiation inter- stitials in space and time. The functions Dρ(x,y,z,T) describe dependences of the diffusion coeffi- cients of point radiation defects on coordinate and temperature. The quadric on concentrations terms of Eqs. (4) describes generation divacancies and diinterstitials. The function kI,V(x,y,z,T) describes dependence of the parameter of recombination of point radiation defects on coordinate and temperature. The function kI,I(x,y,z,T) and kV,V(x,y,z,T) describes dependences of the parame- ters of generation of simplest complexes of point radiation defects on coordinate and tempera- ture. Now let us calculate distributions of concentrations of divacancies ΦV(x,y,z,t) and diinterstitials ΦI(x,y,z,t) in space and time by solving the following system of equations [10,19] ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk z tzyx TzyxD z III I I ,,,,,,,,,,,, ,,, ,,, 2 , −+ Φ + Φ ∂ ∂ ∂ ∂ (6)
- 4. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 4 ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk z tzyx TzyxD z VVV V V ,,,,,,,,,,,, ,,, ,,, 2 , −+ Φ + Φ ∂ ∂ ∂ ∂ . Boundary and initial conditions for these equations are ( ) 0 ,,, 0 = ∂ Φ∂ =x x tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = xLx x tzyxρ , ( ) 0 ,,, 0 = ∂ Φ∂ =y y tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = yLy y tzyxρ , ( ) 0 ,,, 0 = ∂ Φ∂ =z z tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = zLz z tzyxρ , ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (7) The functions DΦρ(x,y,z,T) describe dependences of the diffusion coefficients of the above com- plexes of radiation defects on coordinate and temperature. The functions kI(x,y,z,T) and kV(x,y,z, T) describe the parameters of decay of these complexes on coordinate and temperature. We calculate distributions of concentrations of point radiation defects in space and time by using recently elaborated approach [12,13]. To use the approach let us transform dependences of diffu- sion coefficients of point defects in space and time to the following form: Dρ(x,y,z,T)=D0ρ [1+ερ gρ(x,y,z,T)], where D0ρ are the average values of diffusion coefficients, 0≤ερ<1, |gρ(x,y,z, T)|≤1,ρ =I,V. Let us also transform dependences of another parameters to the similar form: kI,V(x, y,z,T)= k0I,V[1+εI,V gI,V(x,y,z,T)], kI,I(x,y,z,T)=k0I,I [1+εI,I gI,I(x,y,z,T)] and kV,V (x,y,z,T) = k0V,V [1+εV,V gV,V(x,y,z,T)], where k0ρ1,ρ2 are the their average values, 0≤εI,V <1, 0≤εI,I <1, 0≤εV,V<1, | gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1, |gV,V(x,y,z,T)|≤1. Let us introduce the following dimensionless variables: ( ) ( ) * ,,,,,, ~ ItzyxItzyxI = , χ = x/Lx, η = y /Ly, φ = z/Lz, ( ) ( ) * ,,,,,, ~ VtzyxVtzyxV = , 2 00 LtDD VI=ϑ , VIVI DDkL 00,0 2 =ω , VI DDkL 00,0 2 ρρρ =Ω . The introduction leads to trans- formation of Eqs.(4) and conditions (5) to the following form ( ) ( )[ ] ( ) ( )[ ]{ ×+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ Tg I Tg DD DI IIII VI I ,,,1 ,,, ~ ,,,1 ,,, ~ 00 0 φηχε ηχ ϑφηχ φηχε χϑ ϑφηχ ( ) ( )[ ] ( ) ( ) ×− ∂ ∂ + ∂ ∂ + ∂ ∂ × ϑφηχ φ ϑφηχ φηχε φη ϑφηχ ,,, ~,,, ~ ,,,1 ,,, ~ 00 0 00 0 I I Tg DD D DD DI II VI I VI I ( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,, ~ ,,,1,,, ~ ,,,1 2 ,,,, ITgVTg IIIIIVIVI +Ω−+× (8) ( ) ( )[ ] ( ) ( )[ ]{ ×+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ Tg V Tg DD DV VVVV VI V ,,,1 ,,, ~ ,,,1 ,,, ~ 00 0 φηχε ηχ ϑφηχ φηχε χϑ ϑφηχ ( ) ( )[ ] ( ) ( ) ×− ∂ ∂ + ∂ ∂ + ∂ ∂ × ϑφηχ φ ϑφηχ φηχε φη ϑφηχ ,,, ~,,, ~ ,,,1 ,,, ~ 00 0 00 0 I V Tg DD D DD DV VV VI V VI V ( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,, ~ ,,,1,,, ~ ,,,1 2 ,,,, VTgVTg VVVVVVIVI +Ω−+× ( ) 0 ,,,~ 0 = ∂ ∂ =χ χ ϑφηχρ , ( ) 0 ,,,~ 1 = ∂ ∂ =χ χ ϑφηχρ , ( ) 0 ,,,~ 0 = ∂ ∂ =η η ϑφηχρ , ( ) 0 ,,,~ 1 = ∂ ∂ =η η ϑφηχρ ,
- 5. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 5 ( ) 0 ,,,~ 0 = ∂ ∂ =φ φ ϑφηχρ , ( ) 0 ,,,~ 1 = ∂ ∂ =φ φ ϑφηχρ , ( ) ( ) * ,,, ,,,~ ρ ϑφηχ ϑφηχρ ρf = . (9) Let us determine solutions of Eqs.(8) as the following power series [12,13] ( ) ( )∑ ∑ ∑Ω= ∞ = ∞ = ∞ =0 0 0 ,,,~,,,~ i j k ijk kji ϑφηχρωεϑφηχρ ρρ . (10) After substitution of the series (10) into Eqs.(8) and conditions (9) we obtain equations for ini- tial-order approximations of concentration of point defects ( )ϑφηχ ,,, ~ 000I and ( )ϑφηχ ,,, ~ 000V , correc- tions ( )ϑφηχ ,,, ~ ijkI and ( )ϑφηχ ,,, ~ ijkV and conditions for them for all i≥1, j≥1, k≥1. The equations are presented in the Appendix. We calculate solutions of the equations by standard Fourier ap- proach [20,21]. The solutions are presented in the Appendix. Farther we determine spatio-temporal distributions of concentrations of simplest complexes of point radiation defects. To determine the distributions we transform approximations of diffusion coefficients in the following form: DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)], where D0Φρ are the av- erage values of diffusion coefficients. In this situation the Eqs.(6) could be written as ( ) ( )[ ] ( ) ( ) ( )++ Φ += Φ ΦΦΦ tzyxITzyxk x tzyx Tzyxg x D t tzyx II I III I ,,,,,, ,,, ,,,1 ,,, 2 ,0 ∂ ∂ ε ∂ ∂ ∂ ∂ ( )[ ] ( ) ( )[ ] ( ) − Φ ++ Φ ++ ΦΦΦΦΦΦ z tzyx Tzyxg z D y tzyx Tzyxg y D I III I III ∂ ∂ ε ∂ ∂ ∂ ∂ ε ∂ ∂ ,,, ,,,1 ,,, ,,,1 00 ( ) ( )tzyxITzyxkI ,,,,,,− ( ) ( )[ ] ( ) ( ) ( )++ Φ += Φ ΦΦΦ tzyxITzyxk x tzyx Tzyxg x D t tzyx II V VVV V ,,,,,, ,,, ,,,1 ,,, 2 ,0 ∂ ∂ ε ∂ ∂ ∂ ∂ ( )[ ] ( ) ( )[ ] ( ) − Φ ++ Φ ++ ΦΦΦΦΦΦ z tzyx Tzyxg z D y tzyx Tzyxg y D V VVV V VVV ∂ ∂ ε ∂ ∂ ∂ ∂ ε ∂ ∂ ,,, ,,,1 ,,, ,,,1 00 ( ) ( )tzyxITzyxkI ,,,,,,− . We determine spatio-temporal distributions of concentrations of complexes of radiation defects as the following power series ( ) ( )∑ Φ=Φ ∞ = Φ 0 ,,,,,, i i i tzyxtzyx ρρρ ε . (11) Equations for the functions Φρi(x,y,z,t) and conditions for them could be obtained by substitution of the series (11) into Eqs.(6) and appropriate boundary and initial conditions. We present the equations and conditions in the Appendix. Solutions of the equations have been calculated by standard approaches [20,21] and presented in the Appendix. Now we calculate distribution of concentration of dopant in space and time by using the same approach, which was used for calculation the same distributions of another concentrations. To use the approach we transform spatio-temperature approximation of dopant diffusion coefficient to the form: DL(x,y,z,T)=D0L[1+εLgL(x,y,z,T)], where D0L is the average value of dopant diffusion coefficient, 0≤εL<1, |gL(x,y,z,T)|≤1. Now we solve the Eq.(1) as the following power series
- 6. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 6 ( ) ( )∑ ∑= ∞ = ∞ =0 1 ,,,,,, i j ij ji L tzyxCtzyxC ξε . The equations for the functions Cij(x,y,z,t) and conditions for them have been obtained by substi- tution of the series into Eq.(1) and conditions (2). We presented the equations and conditions for them in the Appendix. We solve the equations by standard Fourier approach [20,21]. The solu- tions have been presented in the Appendix. We analyzed distributions of concentrations of dopant and radiation defects in space and time analytically by using the second-order approximations on all parameters, which have been used in appropriate series. The approximations are usually enough good approximations to make qual- itative analysis and to obtain quantitative results. We check all results of analytical modeling by comparison with results of numerical simulation. 3. DISCUSSION In this section based on recently calculated relations we analyzed redistribution of dopant with account redistribution of radiation defects. These relations give us possibility to obtain spatial distributions of concentration of dopant. Typical spatial distributions of concentration of dopant in directions, which is perpendicular to channel, are presented in Fig. 2. Curve 1 of this paper is a typical distribution of concentration of dopant in directions of channel. The figure shows, that presents of interface between layers of heterostructure gives us possibility to obtain more com- pact and more homogenous distribution of concentration of dopant in direction, which is perpen- dicular to the interface. However in this situation one shall to optimize annealing time. Reason of the optimization is following. If annealing time is small, dopant can not achieves the interface. If annealing time is large, dopant will diffuse into another layers of heterostructure too deep. We calculate optimal value of annealing time by using recently introduced criterion [12-17]. Frame- work the criterion we approximate real distribution of concentration of dopant by ideal step-wise function ψ (x,y,z). Farther the required optimal value of dopant concentration by minimization of the following mean-squared error ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (12) Dependences of optimal value of annealing time are presented on Figs. 3. Optimal values of an- nealing time of implanted dopant should be smaller in comparison with the same annealing time of infused dopant. Reason of the difference is necessity to anneal radiation defects before anneal- ing of dopant.
- 7. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 7 Fig.2. The infused dopant concentration distributions. The considered direction is perpendicular to inter- face between epitaxial layer and substrate. Increasing of number of distributions corresponds to increasing of difference between values of dopant diffusion coefficient in layers of heterostructure. The distributions have been calculated under condition, when value of dopant diffusion coefficient in epitaxial layer is larg- er, than value of dopant diffusion coefficient in substrate x 0.0 0.5 1.0 1.5 2.0 C(x,Θ) 2 3 4 1 0 L/4 L/2 3L/4 L Epitaxial layer Substrate Fig.3. Fig.2. The implanted dopant concentration distributions. The considered direction is perpendicular to interface between epitaxial layer and substrate. Curves 1 and 3 corresponds to annealing time Θ = 0.0048 (Lx 2 +Ly 2 +Lz 2 )/D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057(Lx 2 +Ly 2 +Lz 2 )/D0. Curves 1 and 2 have been calculated for homogenous sample. Curves 3 and 4 have been calculated for heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate 0.0 0.1 0.2 0.3 0.4 0.5 a/L, ξ, ε, γ 0.0 0.1 0.2 0.3 0.4 0.5 ΘD0L -2 3 2 4 1
- 8. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 8 Fig.3a. Optimal annealing time of infused dopant as dependences of several parameters. Curve 1 is the dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and ξ=γ=0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of the considered annealing time on the parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of the considered annealing time on the parameter ξ for a/L=1/2 and ε=γ=0. Curve 4 is the dependence of the considered annealing time on parameter γ for a/L=1/2 and ε=ξ=0 4. CONCLUSIONS In this paper we consider a possibility to increase density of elements in circuit of an amplifier based on bipolar heterotransistors. Several conditions to increase the density have been formulat- ed. Analysis of redistribution of dopant with account redistribution of radiation defects (after im- plantation of ions of dopant) for optimization of the above annealing have been done by using recently introduced analytical approach. The approach gives a possibility to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces between layers of the heterostructure with account nonlinearity of these transports and variation in time of their parameters. 0.0 0.1 0.2 0.3 0.4 0.5 a/L, ξ, ε, γ 0.00 0.04 0.08 0.12 ΘD0L -2 3 2 4 1 Fig.3a. Optimal annealing time of implanted dopant as dependences of several parameters. Curve 1 is the dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and ξ=γ=0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of the considered annealing time on the parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of the considered annealing time on the parameter ξ for a/L=1/2 and ε=γ=0. Curve 4 is the dependence of the considered annealing time on parameter γ for a/L=1/2 and ε=ξ=0 ACKNOWLEDGEMENTS This work is supported by the agreement of August 27, 2013 № 02.В.49.21.0003 between The Ministry of education and science of the Russian Federation and Lobachevsky State University of Nizhni Novgorod. REFERENCES 1. G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Is- sue 2. P. 10-17 (2006). 2. A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Electronics. Issue 1. P. 34-38 (2008). 3. O.A. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudryk, P.M.
- 9. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 9 Lytvyn, V.V. Milenin, A.V. Sachenko. Influence of displacement of the electron-hole equilibrium on the process of transition metals diffusion in GaAs. Semiconductors. Vol. 43 (7). P. 897-903 (2009). 4. N.I. Volokobinskaya, I.N. Komarov, T.V. Matioukhina, V.I. Rechetniko, A.A. Rush, I.V. Falina, A.S. Yastrebov. Investigation of technological processes of manufacturing of the bipolar power high-voltage transistors with a grid of inclusions in the collector region. Semiconductors. Vol. 35 (8). P. 1013-1017 (2001). 5. A. Subramaniam, K. D. Cantley, E.M. Vogel. Active and Passive Electronic Components. Vol. 2013, ID 525017 (2013). 6. K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the recrystallization transient at the maximum melt depth induced by laser annealing. Appl. Phys. Lett. 89 (17), 172111-172114 (2006). 7. H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98 (9), 094901-094905 (2006). 8. Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N. Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave annealing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003). 9. V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in Russian). 10. V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). 11. S. Kargarrazi, L. Lanni, C.-M. Zetterling. Solid-State Electronics. Vol. 116. P. 33-37 (2016). 12. E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by us- ing radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012). 13. E.L. Pankratov, E.A. Bulaeva. About some ways to decrease quantity of defects in materials for solid state electronic devices and diagnostics of their realization. Reviews in Theoretical Science. Vol. 3 (2). P. 177-215 (2015). 14. E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimentions of logical elements based on bipolar transistor. Int. J. Comp. Sci. Appl. Vol. 5 (4). P. 1-18 (2015). 15. E.L. Pankratov, E.A. Bulaeva. On approach to decrease dimensions of element OR manufactured by using field-effect heterotransistor. Nano Science and Nano Technology: An Indian Journal. Vol. 9 (4). P. 43-60 (2015). 16. E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to de- crease surface of chip. Int. J. Mod. Phys. B. Vol. 29 (5). P. 1550023-1-1550023-12 (2015). 17. E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor heterostructure by radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101 (2014). 18. Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). 19. P.M. Fahey, P.B. Griffin, J.D. Plummer. Diffusion and point defects in silicon. Rev. Mod. Phys. 1989. V. 61. № 2. P. 289-388. 20. A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in Russian). 21. H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964). Authors: Pankratov Evgeny Leonidovich is a Full Doctor of Science, an Associate Professor of Nizhny Novgorod State University. He has 186 published papers in area of his researches. Bulaeva Elena Alexeevna is a PhD student of Nizhny Novgorod State University. She has 134 published papers in area of her researches. APPENDIX Equations for the functions ( )ϑφηχ ,,, ~ ijkI and ( )ϑφηχ ,,, ~ ijkV , i ≥0, j ≥0, k≥0 and conditions for them
- 10. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 10 ( ) ( ) ( ) ( ) 2 000 2 0 0 2 000 2 0 0 2 000 2 0 0000 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ I D DI D DI D DI V I V I V I ( ) ( ) ( ) ( ) 2 000 2 0 0 2 000 2 0 0 2 000 2 0 0000 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ V D DV D DV D DV I V I V I V ; ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ Tg III D DI I iii V Ii ,,, ,,, ~ ,,, ~ ,,, ~ , ~ 2 00 2 2 00 2 2 00 2 0 000 φηχ χφ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑχ ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ × −− Tg I Tg D D D DI I i I V I V Ii ,,, ,,, ~ ,,, ,,, ~ 100 0 0 0 0100 φηχ φη ϑφηχ φηχ ηχ ϑφηχ ( ) V Ii D DI 0 0100 ,,, ~ ∂ ∂ × − φ ϑφηχ , i≥1, ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ Tg VVV D DV V iii I Vi ,,, ,,, ~ ,,, ~ ,,, ~ , ~ 2 00 2 2 00 2 2 00 2 0 000 φηχ χφ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑχ ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ × −− Tg V Tg D D D DV V i V I V I Vi ,,, ,,, ~ ,,, ,,, ~ 100 0 0 0 0100 φηχ φη ϑφηχ φηχ ηχ ϑφηχ ( ) I Vi D DV 0 0100 ,,, ~ ∂ ∂ × − φ ϑφηχ , i≥1, ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 010 2 2 010 2 2 010 2 0 0010 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000000,, VITg VIVI+− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 010 2 2 010 2 2 010 2 0 0010 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000000,, VITg VIVI+− ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 020 2 2 020 2 2 020 2 0 0020 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,,1 010000000010,, VIVITg VIVI ++− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 020 2 2 020 2 2 020 2 0 0020 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV V I ( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,,1 010000000010,, VIVITg VIVI ++− ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 001 2 2 001 2 2 001 2 0 0001 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( )ϑφηχφηχε ,,, ~ ,,,1 2 000,, ITg IIII+− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 001 2 2 001 2 2 001 2 0 0001 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( )[ ] ( )ϑφηχφηχε ,,, ~ ,,,1 2 000,, VTg IIII+− ;
- 11. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 11 ( ) ( ) ( ) ( ) ×+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ V I V I D DIII D DI 0 0 2 110 2 2 110 2 2 110 2 0 0110 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ ( ) ( ) ( ) ( ) ( )[ × ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ × Tg I Tg I Tg III ,,, ,,, ~ ,,, ,,, ~ ,,, 010010 φηχ φη ϑφηχ φηχ ηχ ϑφηχ φηχ χ ( ) ( ) ( ) ( ) ( )[ ]×+− ∂ ∂ × ϑφηχϑφηχϑφηχϑφηχ φ ϑφηχ ,,, ~ ,,, ~ ,,, ~ ,,, ~,,, ~ 100000000100 010 VIVI I ( )[ ]Tg IIII ,,,1 ,, φηχε+× ( ) ( ) ( ) ( ) ×+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ I V I V D DVVV D DV 0 0 2 110 2 2 110 2 2 110 2 0 0110 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ ( ) ( ) ( ) ( ) ( )[ × ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ × Tg V Tg V Tg IVV ,,, ,,, ~ ,,, ,,, ~ ,,, 010010 φηχ φη ϑφηχ φηχ ηχ ϑφηχ φηχ χ ( ) ( ) ( ) ( ) ( )[ ]×+− ∂ ∂ × ϑφηχϑφηχϑφηχϑφηχ φ ϑφηχ ,,, ~ ,,, ~ ,,, ~ ,,, ~,,, ~ 100000000100 010 IVIV V ( )[ ]Tg VVVV ,,,1 ,, φηχε+× ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 002 2 2 002 2 2 002 2 0 0002 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000001,, IITg IIII+− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 002 2 2 002 2 2 002 2 0 0002 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000001,, VVЕg VVVV+− ; ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 101 2 2 101 2 2 101 2 0 0101 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 001001 0 0 I Tg I Tg D D II V I ( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχε φ ϑφηχ φηχ φ ,,, ~ ,,, ~ ,,,1 ,,, ~ ,,, 000100 001 VITg I Tg III +− ∂ ∂ ∂ ∂ + ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 101 2 2 101 2 2 101 2 0 0101 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 001001 0 0 V Tg V Tg D D VV I V ( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχε φ ϑφηχ φηχ φ ,,, ~ ,,, ~ ,,,1 ,,, ~ ,,, 100000 001 VITg V Tg VVV +− ∂ ∂ ∂ ∂ + ; ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ϑφηχ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ ,,, ~,,, ~ ,,, ~ ,,, ~ ,,, ~ 0102 011 2 2 011 2 2 011 2 0 0011 I III D DI V I
- 12. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 12 ( )[ ] ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεϑφηχφηχε ,,, ~ ,,, ~ ,,,1,,, ~ ,,,1 000001,,000,, VITgITg VIVIIIII +−+× ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ϑφηχ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ ,,, ~,,, ~ ,,, ~ ,,, ~ ,,, ~ 0102 011 2 2 011 2 2 011 2 0 0011 V VVV D DV I V ( )[ ] ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεϑφηχφηχε ,,, ~ ,,, ~ ,,,1,,, ~ ,,,1 001000,,000,, VItgVTg VIVIVVVV +−+× ; ( ) 0 ,,,~ 0 = ∂ ∂ =x ijk χ ϑφηχρ , ( ) 0 ,,,~ 1 = ∂ ∂ =x ijk χ ϑφηχρ , ( ) 0 ,,,~ 0 = ∂ ∂ =η η ϑφηχρijk , ( ) 0 ,,,~ 1 = ∂ ∂ =η η ϑφηχρijk , ( ) 0 ,,,~ 0 = ∂ ∂ =φ φ ϑφηχρijk , ( ) 0 ,,,~ 1 = ∂ ∂ =φ φ ϑφηχρijk (i≥0, j≥0, k≥0); ( ) ( ) * 000 ,,0,,,~ ρφηχφηχρ ρf= , ( ) 00,,,~ =φηχρijk (i≥1, j≥1, k≥1). Solutions of the above equations could be written as ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 000 21 ,,,~ n nn ecccF LL ϑφηχϑφηχρ ρρ , where ( ) ( ) ( ) ( )∫ ∫ ∫= 1 0 1 0 1 0 * ,,coscoscos 1 udvdwdwvufwnvnunF nn ρρ πππ ρ , ( ) ( )IVnI DDne 00 22 exp ϑπϑ −= , cn(χ)=cos(πnχ), ( ) ( )VInV DDne 00 22 exp ϑπϑ −= ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∂ ∂ −−= ∞ = − 1 0 1 0 1 0 1 0 100 0 0 00 ,,, ~ 2,,, ~ n i nnnInIn V I i u wvuI vcuseecccn D D I ϑ τ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−× ∞ =1 0 1 0 1 00 0 2,,, n nnnInIn V I In vsuceecccn D D dudvdwdTwvugwc ϑ τϑφηχπτ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫ ∂ ∂ × ∞ = − 1 00 0 1 0 100 2 ,,, ~ ,,, n nInIn V Ii In eecccn D D dudvdwd v wvuI Twvugwc ϑ τϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∂ ∂ × − 1 0 1 0 1 0 100 ,,, ~ ,,, τ τ dudvdwd w wvuI Twvugwsvcuc i Innn , i ≥1, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 00 ,,,2,,, ~ n VnnnInVn I V i Twvugvcuseecccn D D V ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−− ∂ ∂ × ∞ = − 1 0 1 0 1 00 0100 , ~ n nnnInVn I Vi n vsuceecccn D D dudvdwd u uV wc ϑ τϑφηχτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∂ ∂ × ∞ = − 1 0 0 1 0 100 2 , ~ ,,,2 n nVn I Vi Vn ecccn D D dudvdwd v uV Twvugwc ϑφηχπτ τ π ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× − ϑ τ τ τ 0 1 0 1 0 1 0 100 , ~ ,,, dudvdwd w uV Twvugwsvcuce i VnnnnI , i≥1, where sn(χ)=sin(π nχ); ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−= ∞ =1 0 1 0 1 0 1 0 010 2,,,~ n nnnnnnnn wcvcuceeccc ϑ ρρ τϑφηχϑφηχρ ( )[ ] ( ) ( ) τττε dudvdwdwvuVwvuITwvug VIVI ,,, ~ ,,, ~ ,,,1 000000,,+× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ×∑ ∫ ∫ ∫ ∫ +−−= ∞ =1 0 1 0 1 0 1 0 , 0 0 020 12,,,~ n VInnnnnnnn V I wcvcuceeccc D D ϑ ρρ ετϑφηχϑφηχρ ( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,, 010000000010, +× ;
- 13. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 13 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 0 001 2,,,~ n nnnnnnnn wcvcuceeccc ϑ ρρ τϑφηχϑφηχρ ( )[ ] ( ) ττρε ρρρρ dudvdwdwvuTwvug ,,,~,,,1 2 000,,+× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−= ∞ =1 0 1 0 1 0 1 0 002 2,,,~ n nnnnnnnn wcvcuceeccc ϑ ρρ τϑφηχϑφηχρ ( )[ ] ( ) ( ) ττρτρε ρρρρ dudvdwdwvuwvuTwvug ,,,~,,,~,,,1 000001,,+× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 110 2,,, ~ n nnnnInInnn V I ucvcuseecccn D D I ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ×∑− ∂ ∂ × ∞ = − 1 0 0100 2 ,,, ~ ,,, n nInnn V Ii I ecccn D D dudvdwd u wvuI Twvug ϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫ ∂ ∂ −× − V Ii InnnnI D D dudvdwd v wvuI Twvugucvsuce 0 0 0 1 0 1 0 1 0 100 2 ,,, ~ ,,, πτ τ τ ϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∂ ∂ −× ∞ = − 1 0 1 0 1 0 1 0 100 ,,, ~ ,,, n i InnnnInI dudvdwd w wvuI Twvugusvcuceen ϑ τ τ τϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−× ∞ =1 0 1 0 1 0 1 0 ,12 n VInnnnInnnInnnn vcvcuceccecccc ϑ ετφηϑχφηχ ( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,, 100000000100, +× ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 110 2,,, ~ n nnnnVnVnnn I V ucvcuseecccn D D V ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( )×∑− ∂ ∂ × ∞ = − 1 0 0100 2 ,,, ~ ,,, n nVnnn I Vi V ecccn D D dudvdwd u wvuV Twvug ϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫ ∂ ∂ −× − I Vi VnnnnV D D dudvdwd v wvuV Twvugucvsuce 0 0 0 1 0 1 0 1 0 100 2 ,,, ~ ,,, πτ τ τ ϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∂ ∂ −× ∞ = − 1 0 1 0 1 0 1 0 100 ,,, ~ ,,, n i VnnnnVnV dudvdwd w wvuV Twvugusvcuceen ϑ τ τ τϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−−× ∞ =1 0 1 0 1 0 1 0 ,, ,,,12 n VIVInnnVnnnInnnn Twvugvcuceccecccc ϑ ετφηϑχφηχ ( ) ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuIwcn ,,, ~ ,,, ~ ,,, ~ ,,, ~ 100000000100 +× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 101 ,,,2,,, ~ n InnnInInnn V I Twvugvcuseecccn D D I ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 1 00 0001 2 ,,, ~ n nnInInnn V I n uceecccn D D dudvdwd u wvuI wc ϑ τϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∂ ∂ × ∞ =1 0 0 1 0 1 0 001 2 ,,, ~ ,,, n nnnnI V I Inn cccen D D dudvdwd v wvuI Twvugwcvs φηχϑπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫ ∂ ∂ −× ∞ =10 1 0 1 0 1 0 001 2 ,,, ~ ,,, n nnnInnnnI cccdudvdwd w wvuI Twvugwsvcuce φηχτ τ τ ϑ ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−× ϑ τττετϑ 0 1 0 1 0 1 0 000100,, ,,, ~ ,,, ~ ,,,1 dudvdwdwvuVwvuITwvugwcvcucee VIVInnnnInI
- 14. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 14 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 101 ,,,2,,, ~ n VnnnVnVnnn I V Twvugvcuseecccn D D V ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 1 00 0001 2 ,,, ~ n nnVnInnn I V n uceecccn D D dudvdwd u wvuV wc ϑ τϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∂ ∂ × ∞ =1 0 0 1 0 1 0 001 2 ,,, ~ ,,, n nnnnV I V Vnn cccen D D dudvdwd v wvuV Twvugwcvs φηχϑπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫ ∂ ∂ −× ∞ =10 1 0 1 0 1 0 001 2 ,,, ~ ,,, n nnnVnnnnV cccdudvdwd w wvuV Twvugwsvcuce φηχτ τ τ ϑ ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−× ϑ τττετϑ 0 1 0 1 0 1 0 000100,, ,,, ~ ,,, ~ ,,,1 dudvdwdwvuVwvuITwvugwcvcucee VIVInnnnVnV ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){∑ ∫ ∫ ∫ ∫ ×−−= ∞ =1 0 1 0 1 0 1 0 000011 ,,, ~ 2,,, ~ n nnnnInInnn wvuIwcvcuceecccI ϑ ττϑφηχϑφηχ ( )[ ] ( ) ( )[ ] ( ) ( )} τττετε dudvdwdwvuVwvuITwvugwvuITwvug VIVIIIII ,,, ~ ,,, ~ ,,,1,,, ~ ,,,1 000001,,010,, +++× ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){∑ ∫ ∫ ∫ ∫ ×−−= ∞ =1 0 1 0 1 0 1 0 000011 ,,, ~ 2,,, ~ n nnnnVnVnnn wvuIwcvcuceecccV ϑ ττϑφηχϑφηχ ( )[ ] ( ) ( )[ ] ( ) ( )} τττετε dudvdwdwvuVwvuITwvugwvuITwvug VIVIVVVV ,,, ~ ,,, ~ ,,,1,,, ~ ,,,1 000001,,010,, +++× . Equations for the functions Φρi(x,y,z,t), boundary and initial conditions for them could be written as ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ 2 0 2 2 0 2 2 0 2 0 0 ,,,,,,,,,,,, z tzyx y tzyx x tzyx D t tzyx III I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk III ,,,,,,,,,,,, 2 , −+ ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ 2 0 2 2 0 2 2 0 2 0 0 ,,,,,,,,,,,, z tzyx y tzyx x tzyx D t tzyx VVV V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk VVV ,,,,,,,,,,,, 2 , −+ ; ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ 2 2 2 2 2 2 0 ,,,,,,,,,,,, z tzyx y tzyx x tzyx D t tzyx iIiIiI I iI ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) + Φ + Φ + − Φ − ΦΦ y tzyx Tzyxg yx tzyx Tzyxg x D iI I iI II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, 11 0 ( ) ( ) Φ + − Φ z tzyx Tzyxg z iI I ∂ ∂ ∂ ∂ ,,, ,,, 1 , i≥1, ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ 2 2 2 2 2 2 0 ,,,,,,,,,,,, z tzyx y tzyx x tzyx D t tzyx iViViV V iV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) + Φ + Φ + − Φ − ΦΦ y tzyx Tzyxg yx tzyx Tzyxg x D iV V iV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, 11 0 ( ) ( ) Φ + − Φ z tzyx Tzyxg z iV V ∂ ∂ ∂ ∂ ,,, ,,, 1 , i≥1; ( ) 0 ,,, 0 = ∂ Φ∂ =x i x tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = xLx i x tzyxρ , ( ) 0 ,,, 0 = ∂ Φ∂ =y i y tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = yLy i y tzyxρ ,
- 15. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 15 ( ) 0 ,,, 0 = ∂ Φ∂ =z i z tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = zLz i z tzyxρ , i≥0; Φρ0(x,y,z,0)=fΦρ (x,y,z), Φρi(x,y,z,0)=0, i≥1. Solutions of the above equations could be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑+∑+=Φ ∞ = ∞ = ΦΦ 11 0 221 ,,, n nnn n nnnnn zyxzyx zcycxcn L tezcycxcF LLLLLL tzyx ρρρ ( ) ( ) ( ) ( ) ( ) ( ) ( )[∫ ∫ ∫ ∫ −−× ΦΦ t L L L IInnnnn x y z wvuITwvukwcvcucete 0 0 0 0 2 , ,,,,,, ττρρ ( ) ( )] ττ dudvdwdwvuITwvukI ,,,,,,− , where ( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ x y zL L L nnnn udvdwdwvufwcvcucF 0 0 0 ,,ρρ , ( ) ++−= ΦΦ 2220 22 111 exp zyx n LLL tDnte ρρ π , cn(x)=cos(π n x/Lx); ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ ∞ = ΦΦΦ 1 0 0 0 0 2 ,,, 2 ,,, n t L L L nnnnnnn zyx i x y z Twvugvcusetezcycxcn LLL tzyx ρρρ τ π ρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −− Φ × ∞ = ΦΦ − 1 0 2 1 2,,, n t nnnnn zyx iI n etezcycxcn LLL dudvdwd u wvu wc τ π τ ∂ τ∂ ρρ ρ ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫ Φ −× ∞ = − ΦΦ 1 2 0 0 0 0 1 2,,, ,,, n zyx t L L L iI nnnn n LLL dudvdwd v wvu Twvugwcvsuce x y z π τ ∂ τ∂ τ ρ ρρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ Φ −× Φ − ΦΦ t L L L iI nnnnn x y z dudvdwdTwvug w wvu wsvcucete 0 0 0 0 1 ,,, ,,, τ ∂ τ∂ τ ρ ρ ρρ ( ) ( ) ( )zcycxc nnn× , i≥1, where sn(x)=sin(π n x/Lx). Equations for the functions Cij(x,y,z,t), boundary and initial conditions for them could be written as ( ) ( ) ( ) ( ) 2 00 2 02 00 2 02 00 2 0 00 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ; ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 0 2 2 0 2 2 0 2 0 0 ,,,,,,,,,,,, z tzyxC y tzyxC x tzyxC D t tzyxC iii L i ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + −− y tzyxC Tzyxg y D x tzyxC Tzyxg x D i LL i LL ,,, ,,, ,,, ,,, 10 0 10 0 ( ) ( ) ∂ ∂ ∂ ∂ + − z tzyxC Tzyxg z D i LL ,,, ,,, 10 0 , i≥1; ( ) ( ) ( ) ( )+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 01 2 02 01 2 02 01 2 0 01 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL ( ) ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + y tzyxC TzyxP tzyxC y D x tzyxC TzyxP tzyxC x D LL ,,, ,,, ,,,,,, ,,, ,,, 0000 0 0000 0 γ γ γ γ ( ) ( ) ( ) ∂ ∂ ∂ ∂ + z tzyxC TzyxP tzyxC z D L ,,, ,,, ,,, 0000 0 γ γ ; ( ) ( ) ( ) ( )+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 02 2 02 02 2 02 02 2 0 02 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL
- 16. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 16 ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + −− TzyxP tzyxC tzyxC yx tzyxC TzyxP tzyxC tzyxC x D L ,,, ,,, ,,, ,,, ,,, ,,, ,,, 1 00 01 00 1 00 010 γ γ γ γ ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ × − z tzyxC TzyxP tzyxC tzyxC zy tzyxC ,,, ,,, ,,, ,,, ,,, 00 1 00 01 00 γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ × − TzyxP tzyxC x D z tzyxC TzyxP tzyxC tzyxC zy tzyxC L ,,, ,,,,,, ,,, ,,, ,,, ,,, 00 0 00 1 00 01 00 γ γ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ × z tzyxC TzyxP tzyxC zy tzyxC TzyxP tzyxC yx tzyxC ,,, ,,, ,,,,,, ,,, ,,,,,, 0100010001 γ γ γ γ ; ( ) ( ) ( ) ( )+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 11 2 02 11 2 02 11 2 0 11 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + −− TzyxP tzyxC tzyxC yx tzyxC TzyxP tzyxC tzyxC x ,,, ,,, ,,, ,,, ,,, ,,, ,,, 1 00 10 00 1 00 10 γ γ γ γ ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ × − LD z tzyxC TzyxP tzyxC tzyxC zy tzyxC 0 00 1 00 10 00 ,,, ,,, ,,, ,,, ,,, γ γ ( ) ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + y tzyxC TzyxP tzyxC yx tzyxC TzyxP tzyxC x D L ,,, ,,, ,,,,,, ,,, ,,, 10001000 0 γ γ γ γ ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + x tzyxC Tzyxg x D z tzyxC TzyxP tzyxC z LL ,,, ,,, ,,, ,,, ,,, 01 0 1000 γ γ ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + z tzyxC Tzyxg zy tzyxC Tzyxg y LL ,,, ,,, ,,, ,,, 0101 ; ( ) 0 ,,, 0 = =x ij x tzyxC ∂ ∂ , ( ) 0 ,,, = = xLx ij x tzyxC ∂ ∂ , ( ) 0 ,,, 0 = =y ij y tzyxC ∂ ∂ , ( ) 0 ,,, = = yLy ij y tzyxC ∂ ∂ , ( ) 0 ,,, 0 = =z ij z tzyxC ∂ ∂ , ( ) 0 ,,, = = zLz ij z tzyxC ∂ ∂ , i ≥0, j ≥0; C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i ≥1, j ≥1. Solutions of the above equations with account boundary and initial have been calculated by Fou- rier approach. The result of calculation could be written as ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 00 21 ,,, n nCnnnnC zyxzyx tezcycxcF LLLLLL tzyxC , where ( ) ++−= 2220 22 111 exp zyx CnC LLL tDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫= x y zL L L nCnnnC udvdwdwcwvufvcucF 0 0 0 ,, ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 20 ,,, 2 ,,, n t L L L LnnnCnCnnnnC zyx i x y z TwvugvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −− ∂ ∂ × ∞ = − 1 0 2 10 2,,, n t nCnCnnnnC zyx i n etezcycxcFn LLL dudvdwd u wvuC wc τ π τ τ
- 17. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 17 ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∫ ∫ ∂ ∂ × ∞ = − 1 2 0 0 0 10 2,,, ,,, n nCnC zyx L L L i Lnnn teFn LLL dudvdwd v wvuC Twvugvcvsuc x y z π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× − t L L L i LnnnnCnnn x y z dudvdwd w wvuC Twvugvsvcucezcycxc 0 0 0 0 10 ,,, ,,, τ τ τ , i ≥1; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 201 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑− ∂ ∂ × ∞ =1 2 0000 2,,, ,,, ,,, n nCnnnnC zyx tezcycxcFn LLL dudvdwd u wvuC TwvuP wvuC π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∫ ∫ ∫ ∂ ∂ −× ∞ =1 2 0 0 0 0 0000 2,,, ,,, ,,, n nC zyx t L L L nnnnC ten LLL dudvdwd v wvuC TwvuP wvuC wcvsuce x y z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnnnnC x y z dudvdwd w wvuC TwvuP wvuC wsvcucezcycxcF 0 0 0 0 0000 ,,, ,,, ,,, τ ττ τ γ γ ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 202 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( )×∑− ∂ ∂ × ∞ = − 1 2 00 1 00 01 2,,, ,,, ,,, ,,, n nnnC zyx ycxcF LLL dudvdwd u wvuC TwvuP wvuC wvuC π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× −t L L L nnnCnCn x y z v wvuC TwvuP wvuC wvuCvsucetezcn 0 0 0 0 00 1 00 01 ,,, ,,, ,,, ,,, ττ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−−× ∞ =1 0 0 0 2 2 n t L L nnnCnCnnnnC zyx n x y vcucetezcycxcFn LLL dudvdwdwc τ π τ ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∂ ∂ × ∞ = − 1 2 0 00 1 00 01 2,,, ,,, ,,, ,,, n n zyx L n xcn LLL dudvdwd w wvuC TwvuP wvuC wvuCws z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnCnnnC x y z u wvuC wvuCwcvcusetezcycF 0 0 0 0 00 01 ,,, ,,, τ ττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−× ∞ = − 1 0 0 2 1 00 2 ,,, ,,, n t L nnCnCnnnnC zyx x ucetezcycxcFn LLL dudvdwd TwvuP wvuC τ π τ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∂ ∂ × ∞ = − 1 2 0 0 00 1 00 01 2,,, ,,, ,,, ,,, n zyx L L nn n LLL dudvdwd v wvuC TwvuP wvuC wvuCwcvs y z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫−× −t L L L nnnnCnCnnnnC x y z TwvuP wvuC wvuCwsvcucetezcycxcF 0 0 0 0 1 00 01 ,,, ,,, ,,, γ γ τ ττ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 0 2 00 2,,, n t L nnCnCnnnnC zyx x usetezcycxcF LLL dudvdwd w wvuC τ π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∫ ∂ ∂ × ∞ =1 2 0 0 0100 2,,, ,,, ,,, n nCn zyx L L nn texc LLL dudvdwd u wvuC TwvuP wvuC wcvcn y z π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnnC x y z dudvdwd v wvuC TwvuP wvuC wcvsuceycF 0 0 0 0 0100 ,,, ,,, ,,, τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−× ∞ =1 0 0 0 0 2 2 n t L L L nnnnCnCnnnnC zyx n x y z wsvcucetezcycxcFn LLL zcn τ π
- 18. International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017 18 ( ) ( ) ( ) τ ττ γ γ dudvdwd w wvuC TwvuP wvuC ∂ ∂ × ,,, ,,, ,,, 0100 ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 211 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( ) ×∑− ∂ ∂ × ∞ =1 2 01 2,,, ,,, n nCnnnnC zyx L tezcycxcFn LLL dudvdwd u wvuC Twvug π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫ ∂ ∂ −× 2 0 0 0 0 01 2,,, ,,, zyx t L L L LnnnnC LLL dudvdwd v wvuC Twvugwcvsuce x y z π τ τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∂ ∂ −× ∞ =1 0 0 0 0 01 ,,, ,,, n t L L L LnnnnCnC x y z dudvdwd w wvuC Twvugwsvcuceten τ τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−× ∞ =1 0 0 0 2 2 n t L L nnnCnCnnnnC zyx nnnnC x y vcusetezcycxcF LLL zcycxcF τ π ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∂ ∂ × ∞ =1 2 0 1000 2,,, ,,, ,,, n nnnC zyx L n ycxcFn LLL dudvdwd u wvuC TwvuP wvuC wcn z π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnCn x y z dudvdwd v wvuC TwvuP wvuC wcvsucetezc 0 0 0 0 1000 ,,, ,,, ,,, τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−− ∞ =1 0 0 0 0 00 2 ,,, ,,,2 n t L L L nnnnCnCnnnnC zyx x y z TwvuP wvuC wsvcucetezcycxcFn LLL γ γ τ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 0 2 10 2,,, n t L nnCnCnnnnC zyx x usetezcycxcFn LLL dudvdwd w wvuC τ π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∂ ∂ × ∞ = − 1 2 0 0 00 1 00 10 2,,, ,,, ,,, ,,, n zyx L L nn n LLL dudvdwd u wvuC TwvuP wvuC wvuCwcvc y z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× −t L L L nnnnCnCnnnnC x y z v wvuC TwvuP wvuC wcvsucetezcycxcF 0 0 0 0 00 1 00 ,,, ,,, ,,, ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−× ∞ =1 0 0 210 2 ,,, n t L nnCnCnnnnC zyx x ucetezcycxcFn LLL dudvdwdwvuC τ π ττ ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∂ ∂ × −y z L L nn dudvdwd w wvuC TwvuP wvuC wvuCwsvc 0 0 00 1 00 10 ,,, ,,, ,,, ,,, τ ττ τ γ γ .