![International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3
23
ON OPTIMIZATION OF MANUFACTURING OF ELE-
MENTS OF AN BINARY-ROM CIRCUIT TO INCREASE
THEIR INTEGRATION RATE
E.L. Pankratov
1
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, Russia
1
Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia
ABSTRACT
In this paper we introduce an approach to increase integration rate of elements of an binary-ROM circuit.
Framework the approach we consider a heterostructure with special configuration. Several specific areas
of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radia-
tion defects should be optimized.
KEYWORDS
Binary-ROM circuit; increasing of integration rate of elements; prognosis of technological process; opti-
mization of manufacturing.
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 pa-
rameters due to generating inhomogenous distribution of temperature. In this situation one can
obtain 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. The heterostructure consist of a substrate and several
epitaxial layers. Some sections have been manufactured in the epitaxial layers. Further we con-
sider doping of these sections by diffusion or ion implantation. The doping gives a possibility to
manufacture field-effect transistors framework an binary-ROM circuit so as it is shown on Figs.
1. The manufacturing gives a possibility to increase density of elements of the binary-ROM cir-
cuit [4]. After the considered doping dopant and/or radiation defects should be annealed. Frame-
work the paper we analyzed dynamics of redistribution of dopant and/or radiation defects during
their annealing. We introduce an approach to decrease dimensions of the element. However it is
necessary to complicate technological process.](https://image.slidesharecdn.com/1318jares02-200205061624/85/on-optimization-of-manufacturing-of-elements-of-an-binaryrom-circuit-to-increase-their-integration-rate-1-320.jpg)
![International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3
24
Fig. 1. The considered binary-ROM circuit [4]
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)
( ) 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)
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. Dopant diffusion coefficient will
have different values in different materials. The value will be changed during heating and cooling
of heterostructure (with account Arrhenius law). Approximation of dopant diffusion coefficient
by the following function [20-22]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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ON OPTIMIZATION OF MANUFACTURING OF ELEMENTS OF AN BINARY-ROM CIRCUIT TO INCREASE THEIR INTEGRATION RATE
- 1. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 23 ON OPTIMIZATION OF MANUFACTURING OF ELE- MENTS OF AN BINARY-ROM CIRCUIT TO INCREASE THEIR INTEGRATION RATE E.L. Pankratov 1 Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, Russia 1 Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia ABSTRACT In this paper we introduce an approach to increase integration rate of elements of an binary-ROM circuit. Framework the approach we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radia- tion defects should be optimized. KEYWORDS Binary-ROM circuit; increasing of integration rate of elements; prognosis of technological process; opti- mization of manufacturing. 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 pa- rameters due to generating inhomogenous distribution of temperature. In this situation one can obtain 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. The heterostructure consist of a substrate and several epitaxial layers. Some sections have been manufactured in the epitaxial layers. Further we con- sider doping of these sections by diffusion or ion implantation. The doping gives a possibility to manufacture field-effect transistors framework an binary-ROM circuit so as it is shown on Figs. 1. The manufacturing gives a possibility to increase density of elements of the binary-ROM cir- cuit [4]. After the considered doping dopant and/or radiation defects should be annealed. Frame- work the paper we analyzed dynamics of redistribution of dopant and/or radiation defects during their annealing. We introduce an approach to decrease dimensions of the element. However it is necessary to complicate technological process.
- 2. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 24 Fig. 1. The considered binary-ROM circuit [4] 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) ( ) 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) 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. Dopant diffusion coefficient will have different values in different materials. The value will be changed during heating and cooling of heterostructure (with account Arrhenius law). Approximation of dopant diffusion coefficient by the following function [20-22]
- 3. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 25 ( ) ( ) ( ) ( ) ( ) ( ) ++ += 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] ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( )−− ∂ ∂ ∂ ∂ + tzyxVtzyxITzyxk z tzyxI TzyxD z VII ,,,,,,,,, ,,, ,,, , ( ) ( )tzyxITzyxk II ,,,,,, 2 ,− (4) ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( )+− ∂ ∂ ∂ ∂ + tzyxVtzyxITzyxk z tzyxV TzyxD z VIV ,,,,,,,,, ,,, ,,, , ( ) ( )tzyxVTzyxk VV ,,,,,, 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]
- 4. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 26 ( ) ( ) ( ) ( ) ( ) + 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) 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 = , ( )=tzyxV ,,, ~ ( ) * ,,, VtzyxV= , VIVI DDkL 00,0 2 =ω , VIDDkL 00,0 2 ρρρ =Ω , 2 00 LtDD VI=ϑ , χ=x/Lx, η=y/Ly, 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 ( ) ( )[ ] ( ) ( ) ×− ∂ ∂ + ∂ ∂ + ∂ ∂ × ϑφηχω φ ϑφηχ φηχε φη ϑφηχ ,,, ~,,, ~ ,,,1 ,,, ~ 00 0 00 0 I I Tg DD D DD DI II VI I VI I ( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχε ,,, ~ ,,,1,,, ~ ,,,1 2 ,,,, ITgVTg IIIIIVIVI +Ω−+× (8)
- 5. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 27 ( ) ( )[ ] ( ) ( )[ ]{ ×+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ Tg V Tg DD DV VVVV VI V ,,,1 ,,, ~ ,,,1 ,,, ~ 00 0 φηχε ηχ ϑφηχ φηχε χϑ ϑφηχ ( ) ( )[ ] ( ) ( ) ×− ∂ ∂ + ∂ ∂ + ∂ ∂ × ϑφηχω φ ϑφηχ φηχε φη ϑφηχ ,,, ~,,, ~ ,,,1 ,,, ~ 00 0 00 0 I V Tg DD D DD DV VV VI V VI V ( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχε ,,, ~ ,,,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. 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
- 6. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 28 ( ) ( )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+ εL gL(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. 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 compactness of field-effect transistors with increasing of homogeneity of distribution of concentration of dopant at one time. Changing relation between values of dopant diffusion coefficients leads to opposite result (see Figs. 3). 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 min- imization the following mean-squared error
- 7. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 29 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 ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (12) 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.
- 8. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 30 Fig.3a. Distributions of concentration of dopant, infused in average section of epitaxial layer of heterostructure 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 dif- fusion 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 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
- 9. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 31 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 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 4. CONCLUSIONS In this paper we introduce an approach to increase integration rate of element of an binary-ROM circuit. The approach gives us possibility to decrease area of the elements with smaller increasing of the element’s thickness. 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).
- 10. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 32 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.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] F. Jiang, D. Wu, L. Zhou, Y. Huang, J. Wu, Zh. Jin, X. Liu. A wideband calibration-free 1.5-GS/s 8- bit analog-to-digital converter with low latency. Analog. Integr. Circ. Sig. Process. Vol. 81. P. 341- 348 (2014). [5] D. Fathi, B. Forouzandeh, N. Masoumi. New enhanced noise analysis in active mixers in nanoscale technologies. Nano. 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 0.18µm 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-TiB x -n-6H-SiC Schottky barrier diodes: Specific features of charge transport in rectifying and nonrectifying contacts. Semiconductors. Vol. 43 (7). P. 865-871 (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). [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. 939-942 (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 upon annealing of dysprosium-and holmium-implanted silicon. Semiconduc- tors. Vol. 32 (9). P. 921-923 (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. Buonassisi, R.G. Gordon. Enhancing the efficiency of SnS solar cells via band-offset engineering with a zinc oxy- sulfide 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. Shal- low acceptor complexes in p-type ZnO. Appl. Phys. Lett. Vol. 102 (15). P. 152114-152118 (2013).
- 11. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 33 [14] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Chong. Dopant distribution in the recrystallization 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 formed in silicon using pulsed photon 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 annealing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003). [18] 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 dis- tributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). [19] Yu.N. Erofeev. Pulse devices (Higher School, Moscow, 1989). [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] .M. Fahey, P.B. Griffin, J.D. Plummer. Point defects and dopant diffusion in silicon. Rev. Mod. Phys. Vol. 61 (2). P. 289-388 (1989). [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. Vol. 36 (1). P. 33-39 (2007). [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. On approach to optimize manufacturing of bipolar heterotransistors framework circuit of an operational amplifier to increase their integration rate. Influence mismatch-induced Stress. J. Comp. Theor. Nanoscience. Vol. 14 (10). P. 4885–4899 (2017). [29] E.L. Pankratov. On optimization of manufacturing of two-phase logic circuit based on heterostruc- tures to increase density of their elements. Influence of miss-match induced stress. Advanced science, engineering and medicine. Vol. 9 (9). P. 787–801 (2017). [30] E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimensions of field-effect transistors. Univer- sal Journal of Materials Science. Vol. 1 (1). P.6-11 (2013). [31] E.L. Pankratov, E.A. Bulaeva. An approach to manufac-ture of bipolar transistors in thin film struc- tures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31 (2014). [32] E.L. Pankratov, E.A. Bulaeva. Application of native inhomogeneities to increase compactness of ver- tical field-effect transistors. J. Nanoengineering and Nanomanufacturing. Vol. 2 (3). P. 275-280 (2012).
- 12. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 34 [33] E.L. Pankratov, E.A. Bulaeva. An approach to increase the integration rate of planar drift heterobipo- lar transistors. Materials science in semiconductor processing. Vol. 34. P. 260-268 (2015). APPENDIX Equations for the functions ( )ϑφηχ ,,, ~ ijkI and ( )ϑφηχ ,,, ~ ijkV , i ≥0, j ≥0, k ≥0 and conditions for them ( ) ( ) ( ) ( ) ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 000 2 2 000 2 2 000 2 0 0000 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( ) ( ) ( ) ( ) ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 000 2 2 000 2 2 000 2 0 0000 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ; ( ) ( ) ( ) ( ) ×+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ V Iiii V Ii D DIII D DI 0 0 2 00 2 2 00 2 2 00 2 0 000 ,,, ~ ,,, ~ ,,, ~ , ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑχ ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ × −− η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 100100 i I i I I Tg I Tg ( ) ( ) ∂ ∂ ∂ ∂ + − φ ϑφηχ φηχ φ ,,, ~ ,,, 100i I I Tg , i ≥1, ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ Tg VVV D DV V iii I Vi ,,, ,,, ~ ,,, ~ ,,, ~ , ~ 2 00 2 2 00 2 2 00 2 0 000 φηχ χφ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑχ ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ × −− Tg V Tg D D D DV V i V I V I Vi ,,, ,,, ~ ,,, ,,, ~ 100 0 0 0 0100 φηχ φη ϑφηχ φηχ ηχ ϑφηχ ( ) I Vi D DV 0 0100 ,,, ~ ∂ ∂ × − φ ϑφηχ , i ≥1, ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 010 2 2 010 2 2 010 2 0 0010 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000000,, VITg VIVI+− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 010 2 2 010 2 2 010 2 0 0010 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000000,, VITg VIVI+− ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 020 2 2 020 2 2 020 2 0 0020 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,,1 010000000010,, VIVITg VIVI ++− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 020 2 2 020 2 2 020 2 0 0020 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV V I
- 13. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 35 ( )[ ] ( ) ( ) ( ) ( )[ ]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+× ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 110 2 2 110 2 2 110 2 0 0110 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 010010 0 0 V Tg V Tg D D VV I V ( ) ( ) ( )[ ]×+− ∂ ∂ ∂ ∂ + Tg V Tg VVVVV ,,,1 ,,, ~ ,,, ,, 010 φηχε φ ϑφηχ φηχ φ ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχ ,,, ~ ,,, ~ ,,, ~ ,,, ~ 100000000100 IVIV +× ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 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 ( )[ ] ( ) ( )ϑφηχϑφηχϑφηχε ,,, ~ ,,, ~ ,,,1 000001,, VVg 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
- 14. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 36 ( ) ( ) ( )[ ] ( ) ( )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 ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 000 21 ,,,~ n nn ecccF LL ϑφηχϑφηχρ ρρ , where ( ) ( ) ( ) ( )∫ ∫ ∫= 1 0 1 0 1 0 * ,,coscoscos 1 udvdwdwvufwnvnunF nn ρρ πππ ρ , cn(χ) = cos (π n χ), ( ) ( )IVnI DDne 00 22 exp ϑπϑ −= , ( ) ( )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,
- 15. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 37 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−= ∞ =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 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, +× ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 110 2,,, ~ n nnnnVnVnnn I V ucvcuseecccn D D V ϑ τϑφηχπϑφηχ
- 16. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 38 ( ) ( ) ( ) ( ) ( ) ( )×∑− ∂ ∂ × ∞ = − 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 0001 2 ,,, ~ n nInnn V I n ecccn D D dudvdwd u wvuI wc ϑφηχπτ 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 nnnnI V I Inn cccen D D dudvdwd v wvuI 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 VIVIIIII ,,, ~ ,,, ~ ,,,1,,, ~ ,,,1 000001,,010,, +++× .
- 17. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 39 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; 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 , ( ) ( )[ ]222 0 22 exp −−− ΦΦ ++−= zyxn LLLtDnte ρρ π , cn(x) = cos (π n x/Lx);
- 18. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 40 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=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 ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + −− 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
- 19. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 41 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ × 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. Functions Cij(x,y,z,t) (i ≥0, j ≥0) could be approximated by the following series during solutions of the above equations ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 0 00 2 ,,, n nCnnnnC zyxzyx C tezcycxcF LLLLLL F 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;
- 20. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 42 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 201 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑− ∂ ∂ × ∞ =1 2 0000 2,,, ,,, ,,, n nCnnnnC zyx tezcycxcFn LLL dudvdwd u wvuC TwvuP wvuC π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∫ ∫ ∫ ∂ ∂ −× ∞ =1 2 0 0 0 0 0000 2,,, ,,, ,,, n nC zyx t L L L nnnnC ten LLL dudvdwd v wvuC TwvuP wvuC wcvsuce x y z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnnnnC x y z dudvdwd w wvuC TwvuP wvuC wsvcucezcycxcF 0 0 0 0 0000 ,,, ,,, ,,, τ ττ τ γ γ ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 202 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( )×∑− ∂ ∂ × ∞ = − 1 2 00 1 00 01 2,,, ,,, ,,, ,,, n nnnC zyx ycxcF LLL dudvdwd u wvuC TwvuP wvuC wvuC π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× −t L L L nnnCnCn x y z v wvuC TwvuP wvuC wvuCvsucetezcn 0 0 0 0 00 1 00 01 ,,, ,,, ,,, ,,, ττ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−× ∞ =1 0 0 0 2 2 n t L L nnnCnCnnnnC zyx n x y vcucetezcycxcFn LLL dudvdwdwc τ π τ ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∂ ∂ × ∞ = − 1 2 0 00 1 00 01 2,,, ,,, ,,, ,,, n n zyx L n xcn LLL dudvdwd w wvuC TwvuP wvuC wvuCws z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnCnnnC x y z u wvuC wvuCwcvcusetezcycF 0 0 0 0 00 01 ,,, ,,, τ ττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−× ∞ = − 1 0 0 2 1 00 2 ,,, ,,, n t L nnCnCnnnnC zyx x ucetezcycxcFn LLL dudvdwd TwvuP wvuC τ π τ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∂ ∂ × ∞ = − 1 2 0 0 00 1 00 01 2,,, ,,, ,,, ,,, n zyx L L nn n LLL dudvdwd v wvuC TwvuP wvuC wvuCwcvs y z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫−× −t L L L nnnnCnCnnnnC x y z TwvuP wvuC wvuCwsvcucetezcycxcF 0 0 0 0 1 00 01 ,,, ,,, ,,, γ γ τ ττ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 0 2 00 2,,, n t L nnCnCnnnnC zyx x usetezcycxcF LLL dudvdwd w wvuC τ π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∫ ∂ ∂ × ∞ =1 2 0 0 0100 2,,, ,,, ,,, n nCn zyx L L nn texc LLL dudvdwd u wvuC TwvuP wvuC wcvcn y z π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnnC x y z dudvdwd v wvuC TwvuP wvuC wcvsuceycF 0 0 0 0 0100 ,,, ,,, ,,, τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−× ∞ =1 0 0 0 0 2 2 n t L L L nnnnCnCnnnnC zyx n x y z wsvcucetezcycxcFn LLL zcn τ π
- 21. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.3 43 ( ) ( ) ( ) τ ττ γ γ dudvdwd w wvuC TwvuP wvuC ∂ ∂ × ,,, ,,, ,,, 0100 ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 211 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( ) ×∑− ∂ ∂ × ∞ =1 2 01 2,,, ,,, n nCnnnnC zyx L tezcycxcFn LLL dudvdwd u wvuC Twvug π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫ ∂ ∂ −× 2 0 0 0 0 01 2,,, ,,, zyx t L L L LnnnnC LLL dudvdwd v wvuC Twvugwcvsuce x y z π τ τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∂ ∂ −× ∞ =1 0 0 0 0 01 ,,, ,,, n t L L L LnnnnCnC x y z dudvdwd w wvuC Twvugwsvcuceten τ τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−× ∞ =1 0 0 0 2 2 n t L L nnnCnCnnnnC zyx nnnnC x y vcusetezcycxcF LLL zcycxcF τ π ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∂ ∂ × ∞ =1 2 0 1000 2,,, ,,, ,,, n nnnC zyx L n ycxcFn LLL dudvdwd u wvuC TwvuP wvuC wcn z π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnCn x y z dudvdwd v wvuC TwvuP wvuC wcvsucetezc 0 0 0 0 1000 ,,, ,,, ,,, τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−− ∞ =1 0 0 0 0 00 2 ,,, ,,,2 n t L L L nnnnCnCnnnnC zyx x y z TwvuP wvuC wsvcucetezcycxcFn LLL γ γ τ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 0 2 10 2,,, n t L nnCnCnnnnC zyx x usetezcycxcFn LLL dudvdwd w wvuC τ π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∂ ∂ × ∞ = − 1 2 0 0 00 1 00 10 2,,, ,,, ,,, ,,, n zyx L L nn n LLL dudvdwd u wvuC TwvuP wvuC wvuCwcvc y z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× −t L L L nnnnCnCnnnnC x y z v wvuC TwvuP wvuC wcvsucetezcycxcF 0 0 0 0 00 1 00 ,,, ,,, ,,, ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−× ∞ =1 0 0 210 2 ,,, n t L nnCnCnnnnC zyx x ucetezcycxcFn LLL dudvdwdwvuC τ π ττ ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∂ ∂ × −y z L L nn dudvdwd w wvuC TwvuP wvuC wvuCwsvc 0 0 00 1 00 10 ,,, ,,, ,,, ,,, τ ττ τ γ γ .