ON APPROACH OF OPTIMIZATION OF FORMATION OF INHOMOGENOUS DISTRIBUTIONS OF DOPANT IN A FIELD-EFFECT HETEROTRANSISTORS

1. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 DOI:10.5121/ijcsa.2015.5602 17 ON APPROACH OF OPTIMIZATION OF FORMATION OF INHOMOGENOUS DISTRIBUTIONS OF DOPANT IN A FIELD-EFFECT HETEROTRANSISTORS E.L. Pankratov1 and E.A. Bulaeva1,2 1 Nizhny Novgorod State University,23 Gagarin avenue,Nizhny Novgorod,603950,Russia 2 Nizhny Novgorod State University of Architecture and Civil Engineering,65 Il'insky street,Nizhny Novgorod, 603950, Russia ABSTRACT We introduce an approach of manufacturing of a field-effect heterotransistor with inhomogenous doping of channel. The inhomogenous distribution of concentration of dopant gives a possibility to change speed of transport of charge carriers and to decrease length of channel. KEYWORDS Field-effect heterotransistor; inhomogenous doping of channel; decreasing of dimensions of transistor 1.INTRODUCTION Development of solid state electronic devices leads to necessity to decrease dimensions of elements of integrated circuits. In the present time are have been elaborated several approaches to decrease dimensions of these elements. One of them is growth thin film devices [1-4]. The second one is diffusion or ion doping of required areas of homogenous sample or heterostructures and laser or ion doping of dopant and /or radiation defects [5-7]. Using one of these types of annealing gives a possibility to generate inhomogenous distribution of temperature. In this situation one can obtain inhomogeneity of structure, which should be doped. The inhomogeneity leads to decreasing of dimensions of the considered devices. Another way to modify properties of materials is radiation processing [8,9]. In this paper we consider manufacturing a field-effect heterotransistor with inhomogenous doping of channel. Inhomogenous doping of channel during technological process gives us possibility to change speed of transport of charge carriers [10] and to prevent the effect of closing the source to the drain after decreasing of length of channel [11]. To illustrate the introduced approach of optimization of doping of channel we consider heterostructure, which presented in Fig. 1. One of ends of the channel has been doped by diffusion or ion implantation. After that the dopant and/or radiation defects have been annealed by using electro-magnetic (microwave) irradiation. In this situation one can find inhomogenous distribution of concentration of dopant. This type of annealing gives a possibility to heat near-surficial area to decrease dopant diffusion deep heterostructure. After finishing of the annealing channel will be overgrown. After the overgrowth it should be manufactured gate. Main aim of the present paper is optimization of annealing time to dope of required part of channel.

2. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 18 Fig. 1. Structure of the field-effect heterotransistor 2.METHOD OF SOLUTION We solved our aims by solving the following second Fick’s law [10,12-14] to determine spatio- temporal distribution of concentration of dopant ( ) ( ) ( ) ( )       +      +      = 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 concentration of dopant; the parameter DС is the dopant diffusion coefficient. Dopant diffusion coefficient 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). Depen- dences of dopant diffusion coefficient on coordinate in heterostructure, temperature of annealing and concentrations of dopant and radiation defects could be written as [15-17] ( ) ( ) ( ) ( ) ( ) ( )         ++      += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ . (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 average) with each atom of dopant. More detailed information about concentrational dependence of dopant diffusion coefficient is presented in [15]. The function V (x,y,z,t) describes distribution of concentration of radiation vacancies in space and time. The parameter V* describes the equilibrium 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 [16,17]

3. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 19 ( ) ( ) ( ) ( ) ( ) ( ) ×−      ∂ ∂ ∂ ∂ +      ∂ ∂ ∂ ∂ = ∂ ∂ 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 intersti- tials in space and time. The functions Dρ(x,y,z,T) describe dependences of the diffusion coefficients 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 parameters of generation of simplest complexes of point radiation defects on coordinate and temperature. 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 [16,17] ( ) ( ) ( ) ( ) ( ) +      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) ( ) ( ) ( ) ( ) ( ) +      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)

4. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 20 The functions DΦρ(x,y,z,T) describe dependences of the diffusion coefficients of the above complexes 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 [18-20]. To use the approach let us transform dependences of diffusion 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 va- riables: ( ) ( ) * ,,,,,, ~ ItzyxItzyxI = , χ = x/Lx, η = y /Ly, ( ) ( ) * ,,,,,, ~ VtzyxVtzyxV = , 2 00 LtDD VI=ϑ , VIVI DDkL 00,0 2 =ω , VI DDkL 00,0 2 ρρρ =Ω , φ= z/Lz. The introduction leads to transformation 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 , ( ) 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 [18-20] ( ) ( )∑ ∑ ∑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 initial- order approximations of concentration of point defects ( )ϑφηχ ,,, ~ 000I and ( )ϑφηχ ,,, ~ 000V , corrections ( )ϑφηχ ,,, ~ ijkI and ( )ϑφηχ ,,, ~ ijkV and conditions for them for all i≥1, j≥1, k≥1. The equations are

5. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 21 presented in the Appendix. We calculate solutions of the equations by standard Fourier approach [21,22]. 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 average 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 [21,22] 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 ( ) ( )∑ ∑= ∞ = ∞ =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 substitution 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 [21,22]. The solutions have been presented in the Appendix.

6. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 22 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 qualita- tive 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 compact and more homogenous distribution of concentration of dopant in direction, which is perpendicular 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 [18-20]. 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 Fig.2. The infused dopant concentration distributions. The considered direction is perpendicular to interface 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 larger, than value of dopant diffusion coefficient in substrate

7. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 23 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 ( ) ( )[ ]∫ ∫ ∫ −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 annealing 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 annealing of dopant. 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 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

8. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 24 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 4.CONCLUSIONS In this paper we introduce an approach to manufacture of a field-effect heterotransistor with inhomogenous doping of channel. Some recommendations to optimize technological process to manufacture more compact distribution of concentration of dopant have been formulated. 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 Univer- sity of Nizhni Novgorod, educational fellowship for scientific research of Government of Russia, educational fellowship for scientific research of Government of Nizhny Novgorod Region and of Nizhny Novgorod State University of Architecture and Civil Engineering. REFERENCES [1] G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Issue 2. P. 10-17 (2006). [2] A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Electron- ics. 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. Lyt- vyn, 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] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the re- crystallization transient at the maximum melt depth induced by laser annealing. Appl. Phys. Lett. 89 (17), 172111-172114 (2006).

9. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 25 [6] H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98 (9), 094901-094905 (2006). [7] 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). [8] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in Russian). [9] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [10] I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980). [11] M.V. Dunga, L. Chung-Hsun, X. Xuemei, D.D. Lu, A.M. Niknejad, H. Chenming. Modeling Ad- vanced FET Technology in a Compact Model. IEEE Transactions on Electron Devices. Vol. 53 (9). P. 157-162 (2006). [12] V.G. Gusev, Yu.M. Gusev. Electronics (Moscow: Vysshaya shkola, 1991, in Russian). [13] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). [14] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). [15] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [16] 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. [17] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [18] E.L. Pankratov. Influence of mechanical stress in semiconductor heterostructure on density of p-n- junctions. Applied Nanoscience. Vol. 2 (1). P. 71-89 (2012). [19 ]E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar transistors. Analytical approaches to model technological approaches and ways of optimization of distributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). [20] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012). [21] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in Russian). [22] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964). Authors: 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 110 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 74 published papers in area of her researches.

10. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 26 APPENDIX Equations for the functions ( )ϑφηχ ,,, ~ ijkI and ( )ϑφηχ ,,, ~ ijkV , i ≥0, j ≥0, k ≥0 and conditions for them ( ) ( ) ( ) ( ) 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 ++−

11. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 27 ( ) ( ) ( ) ( ) −      ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 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+− ; ( ) ( ) ( ) ( ) ×+         ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 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+−

12. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 28 ( ) ( ) ( ) ( ) −      ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 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 ( )[ ] ( ) ( )[ ] ( ) ( )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

13. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 29 ( ) ( ) ( ) ( ) ( )∑+= ∞ =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, +× ;

14. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 30 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =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 +× ;

15. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 31 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =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 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =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 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

16. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 32 ( ) ( ) ( ) ( )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 , ( ) 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 ,,,,,,− ,

17. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 33 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 ;

18. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 34 ( ) ( ) ( ) ( )+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 02 2 02 02 2 02 02 2 0 02 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL ( ) ( ) ( ) ( ) ( ) ( ) ( )       × ∂ ∂ +      ∂ ∂ ∂ ∂ + −− 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.

19. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 35 Solutions of the above equations with account boundary and initial have been calculated by Fourier 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 ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∫ ∫ ∂ ∂ × ∞ = − 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

20. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 36 ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∂ ∂ × ∞ = − 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 ( ) ( ) ( ) 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

21. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015 37 ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∂ ∂ × ∞ =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 .

ON APPROACH OF OPTIMIZATION OF FORMATION OF INHOMOGENOUS DISTRIBUTIONS OF DOPANT IN A FIELD-EFFECT HETEROTRANSISTORS

ON APPROACH OF OPTIMIZATION OF FORMATION OF INHOMOGENOUS DISTRIBUTIONS OF DOPANT IN A FIELD-EFFECT HETEROTRANSISTORS

ON APPROACH OF OPTIMIZATION OF FORMATION OF INHOMOGENOUS DISTRIBUTIONS OF DOPANT IN A FIELD-EFFECT HETEROTRANSISTORS

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