![International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
DOI : 10.5121/ijaceee.2015.3301 1
ON VERTICAL INTEGRATION FRAMEWORK
ELEMENT OF TRANSISTOR-TRANSISTOR LOGIC
E.L. Pankratov1
, 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
In this paper we introduce an approach to increase vertical integration of elements of transistor-transistor
logic with function AND-NOT. Framework the approach we consider a heterostructure with special confi-
guration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation.
Annealing of dopant and/or radiation defects should be optimized.
KEYWORDS
Transistor-transistor logic; optimization of manufacturing; decreasing of dimensions of transistor; analyti-
cal approach for modelling
1. INTRODUCTION
An actual and intensively solving problems of solid state electronics is increasing of integration
rate of elements of integrated circuits (p-n-junctions, their systems et al) [1-8]. Increasing of the
integration rate leads to necessity to decrease their dimensions. To decrease the dimensions are
using several approaches. They are widely using laser and microwave types of annealing of in-
fused dopants. These types of annealing are also widely using for annealing of radiation defects,
generated during ion implantation [9-17]. Using the approaches gives a possibility to increase
integration rate of elements of integrated circuits through inhomogeneity of technological para-
meters due to generating inhomogenous distribution of temperature. In this situation one can ob-
tain decreasing dimensions of elements of integrated circuits [18] with account Arrhenius law
[1,3]. Another approach to manufacture elements of integrated circuits with smaller dimensions is
doping of heterostructure by diffusion or ion implantation [1-3]. However in this case optimiza-
tion of dopant and/or radiation defects is required [18].
In this paper we consider a heterostructure presented in Figs. 1. The heterostructure consist of a
substrate and several epitaxial layers (see Figs. 1). Some sections have been manufactured in the
epitaxial layers so as it is shown on Figs. 1. Further we consider doping of these sections by dif-
fusion or ion implantation. The doping gives a possibility to manufacture transistors and p-n-
junction so as it is shown on Figs. 1. The manufacturing gives a possibility to prepare element of
transistor-transistor logic on Fig. 1a. After the considered doping dopant and/or radiation defects
should be annealed. Framework the paper we analyzed dynamics of redistribution of dopant
and/or radiation defects during their annealing. Similar logical element has been considered in
[19]. We introduce an approach to decrease dimensions of the element. However it is necessary to
complicate technological process.](https://image.slidesharecdn.com/onverticalintegrationframework-150910072543-lva1-app6891/85/On-vertical-integration-framework-1-320.jpg)
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Vertical integration of transistor-transistor logic
- 1. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 DOI : 10.5121/ijaceee.2015.3301 1 ON VERTICAL INTEGRATION FRAMEWORK ELEMENT OF TRANSISTOR-TRANSISTOR LOGIC E.L. Pankratov1 , 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 In this paper we introduce an approach to increase vertical integration of elements of transistor-transistor logic with function AND-NOT. Framework the approach we consider a heterostructure with special confi- guration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized. KEYWORDS Transistor-transistor logic; optimization of manufacturing; decreasing of dimensions of transistor; analyti- cal approach for modelling 1. INTRODUCTION An actual and intensively solving problems of solid state electronics is increasing of integration rate of elements of integrated circuits (p-n-junctions, their systems et al) [1-8]. Increasing of the integration rate leads to necessity to decrease their dimensions. To decrease the dimensions are using several approaches. They are widely using laser and microwave types of annealing of in- fused dopants. These types of annealing are also widely using for annealing of radiation defects, generated during ion implantation [9-17]. Using the approaches gives a possibility to increase integration rate of elements of integrated circuits through inhomogeneity of technological para- meters due to generating inhomogenous distribution of temperature. In this situation one can ob- tain decreasing dimensions of elements of integrated circuits [18] with account Arrhenius law [1,3]. Another approach to manufacture elements of integrated circuits with smaller dimensions is doping of heterostructure by diffusion or ion implantation [1-3]. However in this case optimiza- tion of dopant and/or radiation defects is required [18]. In this paper we consider a heterostructure presented in Figs. 1. The heterostructure consist of a substrate and several epitaxial layers (see Figs. 1). Some sections have been manufactured in the epitaxial layers so as it is shown on Figs. 1. Further we consider doping of these sections by dif- fusion or ion implantation. The doping gives a possibility to manufacture transistors and p-n- junction so as it is shown on Figs. 1. The manufacturing gives a possibility to prepare element of transistor-transistor logic on Fig. 1a. After the considered doping dopant and/or radiation defects should be annealed. Framework the paper we analyzed dynamics of redistribution of dopant and/or radiation defects during their annealing. Similar logical element has been considered in [19]. We introduce an approach to decrease dimensions of the element. However it is necessary to complicate technological process.
- 2. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 2 Fig. 1a. Composition element transistor-transistor logic. View from above. Black marked transistors and p- n-junction manufactured by using doping of appropriate sections of the epitaxial layer. Dimensions of these devices are decreased. Transistor 1 is a multiemitter transistor. Emitters have been marked by using letter E. The index indicates their number in the multiemitter transistor. D1 and D2 mean dopants of p and n types in p-n-junction. Red marked resistors (Ri) and wires have no decreasing of their dimensions Fig. 1b. Heterostructure, which consist of a substrate and epitaxial layer with sections, manufactured by using another materials. The figure shows integration of a multiemitter and homoemitter transistors. Dashed lines are illustrated wires
- 3. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 3 Substrate Base Collector Emitter Base Collector Emitter Fig. 1c. Heterostructure, which consist of a substrate and epitaxial layer with sections, manufactured by using another materials. The figure shows integration of two homoemitter transistors. Dashed lines are illu- strated wires 2. METHOD OF SOLUTION In this section we determine spatio-temporal distributions of concentrations of infused and im- planted dopants. To determine these distributions we calculate appropriate solutions of the second Fick's law [1,3,18] ( ) ( ) ( ) ( ) + + = 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) The function C(x,y,z,t) describes the spatio-temporal distribution of concentration of dopant; T is the temperature of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient could be changed with changing materials of heterostructure, with changing tempera- ture of materials (including annealing), with changing concentrations of dopant and radiation de- fects. We approximate dependences of dopant diffusion coefficient on parameters by the follow- ing relation with account results in Refs. [20-22] ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ . (3) Here the function DL (x,y,z,T) describes the spatial (in heterostructure) and temperature (due to Arrhenius law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) describes the limit of solubility of dopant. Parameter γ ∈[1,3] describes average quantity of charged defects interacted with atom of dopant [20]. The function V (x,y,z,t) describes the spatio-temporal distri- bution of concentration of radiation vacancies. Parameter V* describes the equilibrium distribution of concentration of vacancies. The considered concentrational dependence of dopant diffusion coefficient has been described in details in [20]. It should be noted, that using diffusion type of doping did not generation 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 [21,22]
- 4. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 4 ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ 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 spatio-temporal distribution of concentration of radiation interstitials; Dρ(x,y,z,T) are the diffusion coefficients of point radiation defects; terms V2 (x,y,z,t) and I2 (x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V(x,y,z,T) is the parameter of recombination of point radiation defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parame- ters of generation of simplest complexes of point radiation defects. Further we determine distributions in space and time of concentrations of divacancies ΦV(x,y,z,t) and diinterstitials ΦI(x,y,z,t) by solving the following system of equations [21,22] ( ) ( ) ( ) ( ) ( ) + 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)
- 5. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 5 Here DΦρ(x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects; kI(x,y,z,T) and kV (x,y,z,T) are the parameters of decay of these complexes. We calculate distributions of concentrations of point radiation defects in space and time by re- cently elaborated approach [18]. The approach based on transformation of approximations of dif- fusion coefficients in the following form: Dρ(x,y,z,T)=D0ρ[1+ερ gρ(x,y,z,T)], where D0ρ are the av- erage values of diffusion coefficients, 0≤ερ<1, |gρ(x,y,z,T)|≤1, ρ =I,V. We also used analogous transformation of approximations of parameters of recombination of point defects and parameters of generation of their complexes: 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, ( ) ( ) * ,,,,,, ~ 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) We determine solutions of Eqs.(8) with conditions (9) framework recently introduced approach [18], i.e. as the power series ( ) ( )∑ ∑ ∑Ω= ∞ = ∞ = ∞ =0 0 0 ,,,~,,,~ i j k ijk kji ϑφηχρωεϑφηχρ ρρ . (10) Substitution of the series (10) into Eqs.(8) and conditions (9) gives us possibility to obtain equa- tions for initial-order approximations of concentration of point defects ( )ϑφηχ ,,, ~ 000I and ( )ϑφηχ ,,, ~ 000V and corrections for them ( )ϑφηχ ,,, ~ ijkI and ( )ϑφηχ ,,, ~ ijkV , i ≥1, j ≥1, k ≥1. The equa- tions are presented in the Appendix. Solutions of the equations could be obtained by standard Fourier approach [24,25]. The solutions are presented in the Appendix.
- 6. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 6 Now we calculate distributions of concentrations of simplest complexes of point radiation defects in space and time. To determine the distributions we transform approximations of diffusion coef- ficients 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 ,,,,,,− . Farther we determine solutions of above equations as the following power series ( ) ( )∑ Φ=Φ ∞ = Φ 0 ,,,,,, i i i tzyxtzyx ρρρ ε . (11) Now we used the series (11) into Eqs.(6) and appropriate boundary and initial conditions. The using gives the possibility to obtain equations for initial-order approximations of concentrations of complexes of defects Φρ0(x,y,z,t), corrections for them Φρi(x,y,z,t) (for them i ≥1) and boundary and initial conditions for them. We remove equations and conditions to the Appendix. Solutions of the equations have been calculated by standard approaches [24,25] and presented in the Ap- pendix. Now we calculate distribution of concentration of dopant in space and time by using the ap- proach, which was used for analysis of radiation defects. To use the approach we consider follow- ing transformation of approximation of dopant diffusion coefficient: 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. Farther we consider solution of Eq.(1) as the following series: ( ) ( )∑ ∑= ∞ = ∞ =0 1 ,,,,,, i j ij ji L tzyxCtzyxC ξε . Using the relation into Eq.(1) and conditions (2) leads to obtaining equations for the functions Cij(x,y,z,t) (i ≥1, j ≥1), boundary and initial conditions for them. The equations are presented in the Appendix. Solutions of the equations have been calculated by standard approaches (see, for example, [24,25]). The solutions are 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. Usually the second-order approximations are enough good approximations to make qualitative analysis and to obtain quantitative results. All analytical results have been checked by numerical simulation.
- 7. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 7 3. DISCUSSION In this section we analyzed spatio-temporal distributions of concentrations of dopants. Figs. 2 shows typical spatial distributions of concentrations of dopants in neighborhood of interfaces of heterostructures. We calculate these distributions of concentrations of dopants under the follow- ing condition: value of dopant diffusion coefficient in doped area is larger, than value of dopant diffusion coefficient in nearest areas. In this situation one can find increasing of sharpness of p-n- junctions with increasing of homogeneity of distribution of concentration of dopant at one time. These both effects could be obtained in both situations, when p-n-junctions are single and frame- work their systems (transistors, thyristors). Changing relation between values of dopant diffusion coefficients leads to opposite result (see Figs. 3). Fig. 2a. Dependences of concentration of dopant, infused in heterostructure from Figs. 1, on coordinate in direction, which is perpendicular to interface between epitaxial layer substrate. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion coeffi- cient in the 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. 2b. Dependences of concentration of dopant, implanted in heterostructure from Figs. 1, on coordinate in direction, which is perpendicular to interface between epitaxial layer substrate. Difference between val- ues of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion coefficient in the substrate. Curve 1 corresponds to homogenous sample and annealing time Θ=0.0048 (Lx 2 +Ly 2 +Lz 2 )/D0. Curve 2 corresponds to homogenous sample and annealing time Θ=0.0057 (Lx 2 +Ly 2 + Lz 2 )/D0. Curves 3 and 4 correspond to heterostructure from Figs. 1; annealing times Θ=0.0048 (Lx 2 +Ly 2 + Lz 2 )/D0 and Θ=0.0057 (Lx 2 +Ly 2 + Lz 2 )/D0, respectively
- 8. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 8 Fig.3a. Distributions of concentration of dopant, infused in average section of epitaxial layer of heterostruc- ture from Figs. 1 in direction parallel to interface between epitaxial layer and substrate of heterostructure. Difference between values of dopant diffusion coefficients increases with increasing of number of curves. Value of dopant diffusion coefficient in this section is smaller, than value of dopant diffusion coefficient in nearest sections x 0.00000 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 C(x,Θ) fC(x) L/40 L/2 3L/4 Lx0 1 2 Substrate Epitaxial layer 1 Epitaxial layer 2 Fig.3b. Calculated distributions of implanted dopant in epitaxial layers of heterostructure. Solid lines are spatial distributions of implanted dopant in system of two epitaxial layers. Dushed lines are spatial distribu- tions of implanted dopant in one epitaxial layer. Annealing time increases with increasing of number of curves It should be noted, that framework the considered approach one shall optimize annealing of do- pant and/or radiation defects. To do the optimization we used recently introduced criterion [26- 34]. The optimization based on approximation real distribution by step-wise function ψ (x,y, z) (see Figs. 4). Farther the required values of optimal annealing time have been calculated by mi- nimization the following mean-squared error ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (12)
- 9. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 9 C(x,Θ) 0 Lx 2 1 3 4 Fig.4a. Distributions of concentration of infused dopant in depth of heterostructure from Fig. 1 for different values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of num- ber of curve corresponds to increasing of annealing time x C(x,Θ) 1 2 3 4 0 L Fig.4b. Distributions of concentration of implanted dopant in depth of heterostructure from Fig. 1 for dif- ferent values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of number of curve corresponds to increasing of annealing time We show optimal values of annealing time as functions of parameters on Figs. 5. It is known, that standard step of manufactured ion-doped structures is annealing of radiation defects. In the ideal case after finishing the annealing dopant achieves interface between layers of heterostructure. If the dopant has no enough time to achieve the interface, it is practicably to anneal the dopant addi- tionally. The Fig. 5b shows the described dependences of optimal values of additional annealing time for the same parameters as for Fig. 5a. Necessity to anneal radiation defects leads to smaller values of optimal annealing of implanted dopant in comparison with optimal annealing time of infused 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.5a. Dimensionless optimal annealing time of infused dopant as a function of several parameters. Curve 1 describes the dependence of the annealing time on the relation a/L and ξ =γ =0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 describes the dependence of the annealing time on value of parameter ε for a/L=1/2 and ξ =γ =0. Curve 3 describes the dependence of the annealing time on value of parameter ξ for a/L=1/2 and ε=γ =0. Curve 4 describes the dependence of the annealing time on value of parameter γ for a/L=1/2 and ε=ξ =0
- 10. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 10 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 ΘD0 L -2 3 2 4 1 Fig.5b. Dimensionless optimal annealing time of implanted dopant as a function of several parameters. Curve 1 describes the dependence of the annealing time on the relation a/L and ξ =γ =0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 describes the dependence of the annealing time on value of parameter ε for a/L=1/2 and ξ =γ =0. Curve 3 describes the dependence of the annealing time on value of parameter ξ for a/L=1/2 and ε=γ =0. Curve 4 describes the dependence of the annealing time on value of parameter γ for a/L=1/2 and ε=ξ =0 4. CONCLUSIONS In this paper we introduce an approach of vertical integration framework element of transistor- transistor logic. The approach gives us possibility to decrease area of the elements with smaller increasing of the element’s thickness. 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 and educational fellowship for scientific research of Government of Russian and of Nizhny Novgorod State University of Architecture and Civil Engineering. REFERENCES [1] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). [2] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990). [3] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). [4] D. Fathi, B. Forouzandeh. Accurate analysis of global interconnects in nano-fpgas. Nano. Vol. 4 (3). P. 171-176 (2009). [5] D. Fathi, B. Forouzandeh, N. Masoumi. Nano. New enhanced noise analysis in active mixers in nanoscale technologies. Vol. 4 (4). P. 233-238 (2009). [6] S.A. Chachuli, P.N.A. Fasyar, N. Soin, N.M. Kar, N. Yusop. Pareto ANOVA analysis for CMOS 0.18 µm two-stage Op-amp. Mat. Sci. Sem. Proc. Vol. 24. P. 9-14 (2014). [7] A.O. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudrik, P.M. Lit- vin, V.V. Milenin, A.V. Sachenko. Au-TiBx-n-6H-SiC Schottky barrier diodes: the features of cur- rent flow in rectifying and nonrectifying contacts. Semiconductors. Vol. 43 (7). P. 897-903 (2009). [8] Z. Li, J. Waldron, T. Detchprohm, C. Wetzel, R.F. Karlicek, Jr.T.P. Chow. Monolithic integration of light-emitting diodes and power metal-oxide-semiconductor channel high-electron-mobility transis- tors for light-emitting power integrated circuits in GaN on sapphire substrate. Appl. Phys. Lett. Vol. 102 (19). P. 192107-192109 (2013).
- 11. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 11 [9] Jung-Hui Tsai, Shao-Yen Chiu, Wen-Shiung Lour, Der-Feng Guo. High-performance InGaP/GaAs pnp δ-doped heterojunction bipolar transistor. Semiconductors. Vol. 43 (7). P. 971-974 (2009). [10] O.V. Alexandrov, A.O. Zakhar'in, N.A. Sobolev, E.I. Shek, M.M. Makoviychuk, E.O. Parshin. For- mation of donor centers after annealing of dysprosium and holmium implanted silicon. Semiconduc- tors. Vol. 32 (9). P. 1029-1032 (1998). [11] M.J. Kumar, T.V. Singh. Quantum confinement effects in strained silicon mosfets. Int. J. Nanoscience. Vol. 7 (2-3). P. 81-84 (2008). [12] P. Sinsermsuksakul, K. Hartman, S.B. Kim, J. Heo, L. Sun, H.H. Park, R. Chakraborty, T. Buonas- sisi, R.G. Gordon. Enhancing the efficiency of SnS solar cells via band-offset engineering with a zinc oxysulfide buffer layer. Appl. Phys. Lett. Vol. 102 (5). P. 053901-053905 (2013). [13] J.G. Reynolds, C.L. Reynolds, Jr.A. Mohanta, J.F. Muth, J.E. Rowe, H.O. Everitt, D.E. Aspnes. Shallow acceptor complexes in p-type ZnO. Appl. Phys. Lett. Vol. 102 (15). P. 152114-152118 (2013). [14] 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). [15] H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98 (9), 094901-094905 (2006). [16] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. Shallow p-n-junctions in Si prepared by pulse pho- ton annealing. Semiconductors. Vol.36 (5). P. 611-617 (2002). [17] 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 anneal- ing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003). [18] E.L. Pankratov, E.A. Bulaeva. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). [19] Yu.N. Erofeev. Pulse devices (Higher School, Moscow, 1989, in Russian). [20] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in Russian). [21] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [22] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [23] P.M. Fahey, P.B. Griffin, J.D. Plummer. Rev. Mod. Phys. 1989. Vol. 61. № 2. P. 289-388. [24] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in Russian). [25] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964). [26] E.L. Pankratov. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusion- coefficient nonuniformity. Russian Microelectronics. 2007. V.36 (1). P. 33-39. [27] E.L. Pankratov. Redistribution of dopant during annealing of radiative defects in a multilayer struc- ture by laser scans for production an implanted-junction rectifiers. Int. J. Nanoscience. Vol. 7 (4-5). P. 187–197 (2008). [28] E.L. Pankratov. Decreasing of depth of implanted-junction rectifier in semiconductor heterostruc- ture by optimized laser annealing. J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010). [29] E.L. Pankratov, E.A. Bulaeva. Application of native inhomogeneities to increase compactness of vertical field-effect transistors. J. Comp. Theor. Nanoscience. Vol. 10 (4). P. 888-893 (2013). [30] E.L. Pankratov, E.A. Bulaeva. Optimization of doping of heterostructure during manufacturing of p- i-n-diodes. Nanoscience and Nanoengineering. Vol. 1 (1). P. 7-14 (2013). [31] E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimensions of field-effect transistors. Uni- versal Journal of Materials Science. Vol. 1 (1). P.6-11 (2013). [32] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture a heterobipolar transistors in thin film structures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31 (2014). [33] E.L. Pankratov, E.A. Bulaeva. Application of native inhomogeneities to increase compactness of vertical field-effect transistors. J. Nanoengineering and Nanomanufacturing. Vol. 2 (3). P. 275-280 (2012). [34] E.L. Pankratov, E.A. Bulaeva. Influence of drain of dopant on distribution of dopant in diffusion- heterojunction rectifiers. J. Adv. Phys. Vol. 2 (2). P. 147-150 (2013).
- 12. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 12 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. 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,
- 13. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 13 ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 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+− ; ( ) ( ) ( ) ( ) ×+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 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+×
- 14. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 14 ( ) ( ) ( ) ( ) ×+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 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
- 15. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 15 ( )[ ] ( ) ( )[ ] ( ) ( )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 (π n χ), ( ) ( )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
- 16. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 16 ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∂ ∂ × ∞ = − 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 n χ); ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−= ∞ =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, +× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =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, +×
- 17. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 17 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =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 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =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
- 18. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 18 ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−× 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 functions Φρi(x,y,z,t), i ≥0 to describe concentrations of simplest complexes of radi- ation defects. ( ) ( ) ( ) ( ) + 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;
- 19. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 19 Boundary and initial conditions for the functions takes the form ( ) 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 ,,,,,,− , 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) (i ≥0, j ≥0), boundary and initial conditions could be writ- ten 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
- 20. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 20 ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + −− 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 ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + −− 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
- 21. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 21 ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + 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. Functions Cij(x,y,z,t) (i ≥0, j ≥0) could be approximated by the following series during solutions of the above equations ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 00 21 ,,, n nCnnnnC zyxzyx tezcycxcF LLLLLL tzyxC . Here ( ) ++−= 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
- 22. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 22 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× 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 ( ) ( ) ( ) τ ττ γ γ dudvdwd w wvuC TwvuP wvuC ∂ ∂ × ,,, ,,, ,,, 0100 ;
- 23. International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 23 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =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 .