Optimization of XOR Circuits
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The paper describes an approach of increasing of integration rate of elements of integrated circuits. The approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one should manufacture a heterostructure with specific configuration. After that several special areas of the heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical approaches of modeling of technological process.
![International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016
DOI:10.5121/ijfcst.2016.6101 1
OPTIMIZATION OF TECHNOLOGICAL PROCESS TO
DECREASE DIMENSIONS OF CIRCUITS XOR, MANU-
FECTURED BASED ON FIELD-EFFECT HETEROTRAN-
SISTORS
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
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of do-
pant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
KEYWORDS
Circuits XOR; increasing of density of elements; optimization of technological process.
1. INTRODUCTION
One of intensively solving problems of solid state electronics is improvement of frequency cha-
racteristics of electronic devices and their reliability. Another intensively solving problem is in-
creasing of integration rate of integrated circuits with decreasing of their dimensions [1-9]. To
solve these problems they were used searching materials with higher values of charge carriers
motilities, development new and elaboration existing technological approaches [1-14]. In the
present paper we consider circuit XOR from [15]. Based on recently considered approaches [16-
23] we consider an approach to decrease dimensions of the circuit. The approach based on manu-
facturing a heterostructure, which consist of a substrate and an epitaxial layer. The epitaxial layer
manufactured with several sections. To manufacture these sections another materials have been
used. These sections have been doped by diffusion or ion implantation. The doping gives a possi-
bility to generate another type of conductivity (p or n). After finishing of manufacturing of the
circuit XOR these sections will be used by sources, drains and gates (see Fig. 1). Dopant and rad-
iation defects should be annealed after finishing the dopant diffusion and the ion implantation.
Our aim framework the paper is analysis of redistribution of dopant and redistribution of radiation
defects to prognosis technological process. The accompanying aim of the present paper is devel-
opment of analytical approach for prognosis technological processes with account all required
influenced factors.](https://image.slidesharecdn.com/optimization-of-technological-process-to-decrease-dimensions-of-circuits-xor-manufectured-based-on-f-160212060446/85/optimization-of-technological-process-to-decrease-dimensions-of-circuits-xor-manufectured-based-on-field-effect-heterotransistors-1-320.jpg)
![International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016
2
2. METHOD OF SOLUTION
We solve our aim by calculation and analysis distribution of concentrations of dopants in space
and time. The required distribution has been determined by solving the second Fick's law in the
following form [24,25]
( ) ( ) ( ) ( )
+
+
=
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC
CCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,,
. (1)
Boundary and initial conditions for the equations are
Fig. 1. Structure of epitaxial layer. View from top](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Optimization of XOR Circuits
- 1. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 DOI:10.5121/ijfcst.2016.6101 1 OPTIMIZATION OF TECHNOLOGICAL PROCESS TO DECREASE DIMENSIONS OF CIRCUITS XOR, MANU- FECTURED BASED ON FIELD-EFFECT HETEROTRAN- SISTORS 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 The paper describes an approach of increasing of integration rate of elements of integrated circuits. The approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one should manufacture a heterostructure with specific configuration. After that several special areas of the heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of do- pant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical approaches of modeling of technological process. KEYWORDS Circuits XOR; increasing of density of elements; optimization of technological process. 1. INTRODUCTION One of intensively solving problems of solid state electronics is improvement of frequency cha- racteristics of electronic devices and their reliability. Another intensively solving problem is in- creasing of integration rate of integrated circuits with decreasing of their dimensions [1-9]. To solve these problems they were used searching materials with higher values of charge carriers motilities, development new and elaboration existing technological approaches [1-14]. In the present paper we consider circuit XOR from [15]. Based on recently considered approaches [16- 23] we consider an approach to decrease dimensions of the circuit. The approach based on manu- facturing a heterostructure, which consist of a substrate and an epitaxial layer. The epitaxial layer manufactured with several sections. To manufacture these sections another materials have been used. These sections have been doped by diffusion or ion implantation. The doping gives a possi- bility to generate another type of conductivity (p or n). After finishing of manufacturing of the circuit XOR these sections will be used by sources, drains and gates (see Fig. 1). Dopant and rad- iation defects should be annealed after finishing the dopant diffusion and the ion implantation. Our aim framework the paper is analysis of redistribution of dopant and redistribution of radiation defects to prognosis technological process. The accompanying aim of the present paper is devel- opment of analytical approach for prognosis technological processes with account all required influenced factors.
- 2. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 2 2. METHOD OF SOLUTION We solve our aim by calculation and analysis distribution of concentrations of dopants in space and time. The required distribution has been determined by solving the second Fick's law in the following form [24,25] ( ) ( ) ( ) ( ) + + = z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC CCC ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,,,,,,,,,,, . (1) Boundary and initial conditions for the equations are Fig. 1. Structure of epitaxial layer. View from top
- 3. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 3 ( ) 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) Here the function C(x,y,z,t) describes the distribution of concentration of dopant in space and time. DС describes distribution the dopant diffusion coefficient in space and as a function of tem- perature of annealing. Dopant diffusion coefficient will be changed with changing of materials of heterostructure, heating and cooling of heterostructure during annealing of dopant or radiation defects (with account Arrhenius law). Dependences of dopant diffusion coefficient on coordinate in heterostructure, temperature of annealing and concentrations of dopant and radiation defects could be written as [26-28] ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ . (3) Here function DL (x,y,z,T) describes dependences of dopant diffusion coefficient on coordinate and temperature of annealing T. Function P (x,y,z,T) describes the same dependences of the limit of solubility of dopant. The parameter γ is integer and usually could be varying in the following interval γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in aver- age) with each atom of dopant. Ref.[26] describes more detailed information about dependence of dopant diffusion coefficient on concentration of dopant. Spatio-temporal distribution of concen- tration of radiation vacancies described by the function V (x,y,z,t). The equilibrium distribution of concentration of vacancies has been denoted as V* . It is known, that doping of materials by diffu- sion did not leads to radiation damage of materials. In this situation ζ1=ζ2=0. We determine spa- tio-temporal distributions of concentrations of radiation defects by solving the following system of equations [27,28] ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ 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 ,
- 4. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 4 ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxρ , ( ) 0 ,,, = ∂ ∂ = zLz z tzyxρ , ρ (x,y,z,0)=fρ (x,y,z). (5) Here ρ =I,V. We denote spatio-temporal distribution of concentration of radiation interstitials as I (x,y,z,t). Dependences of the diffusion coefficients of point radiation defects on coordinate and temperature have been denoted as Dρ(x,y,z,T). The quadric on concentrations terms of Eqs. (4) describes generation divacancies and diinterstitials. Parameter of recombination of point radiation defects and parameters of generation of simplest complexes of point radiation defects have been denoted as the following functions kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T), respectively. Now let us calculate distributions of concentrations of divacancies ΦV(x,y,z,t) and diinterstitials ΦI(x,y,z,t) in space and time by solving the following system of equations [27,28] ( ) ( ) ( ) ( ) ( ) + 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) The functions DΦρ(x,y,z,T) describe dependences of the diffusion coefficients of the above com- plexes of radiation defects on coordinate and temperature. The functions kI(x,y,z,T) and kV(x,y,z, T) describe the parameters of decay of these complexes on coordinate and temperature. To describe physical processes they are usually solving nonlinear equations with space and time varying coefficients. In this situation only several limiting cases have been analyzed [29-32]. One way to solve the problem is solving the Eqs. (1), (4), (6) by the Bubnov-Galerkin approach [33] after appropriate transformation of these transformation. To determine the spatio-temporal distri- bution of concentration of dopant we transform the Eq.(1) to the following integro- differential form ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ++=∫ ∫ ∫ t y L z L L x L y L z L zyx y zx y z V wvxV V wvxV TwvxDudvdwdtwvuC LLL zyx 0 2* 2 2*1 ,,,,,, 1,,,,,, τ ς τ ς
- 5. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 5 ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ++ +× t x L z L L zy x z TzyxP wyuC TwyuD LL zy d x wvxC TwvxP wvxC 0 ,,, ,,, 1,,, ,,, ,,, ,,, 1 γ γ γ γ τ ξτ ∂ τ∂τ ξ ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫+ ++× t x L y L L zx x y TzvuD LL zx d y wyuC V wyuV V wyuV 0 2* 2 2*1 ,,, ,,,,,,,,, 1 τ ∂ τ∂τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) + + ++× yx LL yx d z zvuC TzyxP zvuC V zvuV V zvuV τ ∂ τ∂τ ξ τ ς τ ς γ γ ,,, ,,, ,,, 1 ,,,,,, 1 2* 2 2*1 ( )∫ ∫ ∫+ x L y L z L zyx x y z udvdwdwvuf LLL zyx ,, . (1a) Now let us determine solution of Eq.(1a) by Bubnov-Galerkin approach [33]. To use the ap- proach we consider solution of the Eq.(1a) as the following series ( ) ( ) ( ) ( ) ( )∑= = N n nCnnnnC tezcycxcatzyxC 0 0 ,,, . Here ( ) ( )[ ]222 0 22 exp −−− ++−= zyxCnC LLLtDnte π , cn(χ) =cos(π nχ/Lχ). Number of terms N in the series is finite. The above series is almost the same with solution of linear Eq.(1) (i.e. for ξ=0) and averaged dopant diffusion coefficient D0. Substitution of the series into Eq.(1a) leads to the following result ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ × ∑+−=∑ == t y L z L N n nCnnnnC zy N n nCnnn C y z ewcvcxca LL zy tezsysxs n azyx 0 11 32 1 γ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ++ × = N n nnnCL vcxsaTwvxD V wvxV V wvxV TwvxP 1 2* 2 2*1 ,,, ,,,,,, 1 ,,, τ ς τ ς ξ γ ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ × ∑+−× = t x L z L N m mCmmmmC zx nCn x z TwyuP ewcycuca LL zx dewcn 0 1 ,,, 1 γ γ ξ τττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ++× = ττ τ ς τ ς dewcysucn V wyuV V wyuV TwyuD N n nCnnnL 1 2* 2 2*1 ,,,,,, 1,,, ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑+−× = t x L y L N n nCnnnnCL yx nC x y ezcvcuca TzvuP TzvuD LL yx a 0 1,,, 1,,, γ γ τ ξ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×+∑ ++× = zyx N n nCnnnnC LLL zyx dezsvcucan V zvuV V zvuV ττ τ ς τ ς 1 2* 2 2*1 ,,,,,, 1 ( )∫ ∫ ∫× x L y L z Lx y z udvdwdwvuf ,, ,
- 6. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 6 where sn(χ)=sin(πnχ/Lχ). We used condition of orthogonality to determine coefficients an in the considered series. The coefficients an could be calculated for any quantity of terms N. In the common case the relations could be written as ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ × ∑+−=∑− == t L L L N n nCnnnnCL zy N n nC nCzyx x y z ezcycxcaTzyxD LL te n aLLL 0 0 0 0 1 2 1 65 222 1,,, 2 γ τ ππ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ++ × = N n nCnnn nC ezcycxs n a V zyxV V zyxV TzyxP 1 2* 2 2*1 2 ,,,,,, 1 ,,, τ τ ς τ ς ξ γ ( ) ( )[ ] ( ) ( )[ ] ( )×∫ ∫ ∫ ∫− −+ −+× t L L L Ln z nn y n x y z TzyxDdxdydzdzc n L zszyc n L ysy 0 0 0 0 ,,,11 τ ππ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ++ ∑+× = *1 1 ,,, 1 ,,, 1,,, V zyxV TzyxP ezcycxcaTzyxD N n nCnnnnCL τ ς ξ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ × −+ ++ + = N n nC n x n n a xc n L xsx V zyxV V zyxV V zyxV 1 2* 2 2*12* 2 2 1 ,,,,,, 1 ,,, π τ ς τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( )[ ] ×− −+× 22 2 12 2 π τ π τ π yx n z nnCnnn zx LL dxdydzdzc n L zszezcysxc LL ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ++ ∑+× = t L L L N n nCnnnnC x y z V zyxV TzyxP ezcycxca 0 0 0 0 2* 2 2 1 ,,, 1 ,,, 1 τ ς ξ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ −+ + = N n n x nnnn nC L xc n L xsxzsycxc n a TzyxD V zyxV 1 *1 1,,, ,,, π τ ς ( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ × −++ −+× = N n L n x nnCn y n x xc n L xsxdxdydzdeyc n L ysy 1 0 11 π ττ π ( ) ( )[ ] ( ) ( )[ ] ( )∫ ∫ −+ −+× y z L L n z nn y n xdydzdzyxfzc n L zszyc n L ysy 0 0 ,,11 ππ . As an example for γ=0 we obtain ( ) ( )[ ] ( ) ( )[ ] ( ) ( ){∫ ∫ ∫ + −+ −+= x y zL L L nn y nn y nnC xsxydzdzyxfzc n L zszyc n L ysya 0 0 0 ,,11 ππ ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ∫ ∫ ∫ ∫ × −+ −× t L L L Ln y nnn x n x y z TzyxDyc n L ysyycxs n xd n L xc 0 0 0 0 ,,,12 2 1 ππ
- 7. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 7 ( ) ( )[ ] ( ) ( ) ( ) ( ) × + ++ −+× TzyxPV zyxV V zyxV zc n L zsz n y n ,,, 1 ,,,,,, 11 2* 2 2*1 γ ξτ ς τ ς π ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )×∫ ∫ ∫ ∫ −++× t L L L nnn y nnnCnCn x y z zcysxc n L xsxxcedexdydzdzc 0 0 0 0 21 π τττ ( ) ( )[ ] ( ) ( ) ( ) ( ) × ++ + −+× 2* 2 2*1 ,,,,,, 1 ,,, 11 V zyxV V zyxV TzyxP zc n L zsz n y n τ ς τ ς ξ π γ ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ){∫ ∫ ∫ × −++× t L L nnn x nnnCL x y ysycxc n L xsxxcedxdydzdTzyxD 0 0 0 1,,, π ττ ( )[ ] ( ) ( ) ( ) ( ) ( )∫ ++ + −+× zL Lnn y V zyxV TzyxP TzyxDzsyc n L y 0 2* 2 2 ,,, 1 ,,, 1,,,21 τ ς ξ π γ ( ) ( ) 1 65 222 *1 ,,, − − + te n LLL dxdydzd V zyxV nC zzz π τ τ ς . For γ = 1 one can obtain the following relation to determine required parameters ( ) ( ) ( ) ( )∫ ∫ ∫+±−= x y zL L L nnnnn n n nC xdydzdzyxfzcycxca 0 0 0 2 ,,4 2 αβ α β , where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ × ++= t L L L nnnnC zy n x y z V zyxV V zyxV zcycxse n LL 0 0 0 0 2* 2 2*12 ,,,,,, 12 2 τ ς τ ςτ π ξ α ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×+ −+ −+× n LL dxdydzdzc n L zszyc n L ysy TzyxP TzyxD zx n z nn y n L 2 2 11 ,,, ,,, π ξ τ ππ ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ×∫ ∫ ∫ ∫ −− −+× t L L L n z n L nn x nnnC x y z zc n L zsz TzyxP TzyxD zcxc n L xsxxce 0 0 0 0 1 ,,, ,,, 1 ππ τ ( ) ( ) ( ) ( ) ( ) ( ) ×+ ++× n LL dxdydyszd V zyxV V zyxV TzyxP TzyxD yx n L 22* 2 2*1 2 2 ,,,,,, 1 ,,, ,,, π ξ τ τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ++× t L L L L nnnnC x y z V zyxV V zyxV TzyxP TzyxD zsycxce 0 0 0 0 2* 2 2*1 ,,,,,, 1 ,,, ,,, 2 τ ς τ ςτ ( ) ( )[ ] ( ) ( )[ ] τ ππ dxdydzdyc n L ysyxc n L xsx n y nn x n −+ −+× 11 , ( )×∫= t nC zy n e n LL 0 2 2 τ π β ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ∫ ∫ ∫ × ++ −+× x y zL L L nn y nnn V zyxV V zyxV zcyc n L ysyycxs 0 0 0 2* 2 2*1 ,,,,,, 112 τ ς τ ς π
- 8. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 8 ( ) ( ) ( )[ ] ( ) ( ) ( )×∫ ∫ ∫+ −+× t L L nnnC zx n z nL x y ysxce n LL dxdydzdzc n L zszTzyxD 0 0 0 2 2 2 1,,, τ π τ π ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ×∫ ++ −+× zL nLn x n V zyxV V zyxV zcTzyxDxc n L xsx 0 2* 2 2*1 ,,,,,, 1,,,1 τ ς τ ς π ( ) ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫ −++ −+× t L n x nnC yx n z n x xc n L xsxe n LL dxdydzdzc n L zsz 0 0 2 1 2 1 π τ π τ π ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ∫ ∫ × ++ −+× y z L L Ln y nn V zyxV V zyxV TzyxDyc n L ysyxc 0 0 2* 2 2*1 ,,,,,, 1,,,1 τ ς τ ς π ( ) ( ) ( ) 65222 2 nteLLLdxdydyczdzs nCzyxnn πτ −× . The same approach could be used for calculation parameters an for different values of parameterγ. However the relations are bulky and will not be presented in the paper. Advantage of the ap- proach is absent of necessity to join dopant concentration on interfaces of heterostructure. The same Bubnov-Galerkin approach has been used for solution the Eqs.(4). Previously we trans- form the differential equations to the following integro- differential form ( ) ( ) ( ) +∫ ∫ ∫ ∂ ∂ =∫ ∫ ∫ t y L z L I zy x L y L z L zyx y zx y z dvdwd x wvxI TwvxD LL zy udvdwdtwvuI LLL zyx 0 ,,, ,,,,,, τ τ ( ) ( ) ( ) ( )×∫ ∫ ∫−∫ ∫ ∫ ∂ ∂ + x L y L z L VI zyx t x L z L I zx x y zx z twvuITwvuk LLL zyx dudwd x wyuI TwyuD LL zx ,,,,,, ,,, ,,, , 0 τ τ ( ) ( ) ( ) ×−∫ ∫ ∫ ∂ ∂ +× zyx t x L y L I yx LLL zyx dudvdTzvuD z zvuI LL yx udvdwdtwvuV x y0 ,,, ,,, ,,, τ τ ( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫× x L y L z L I zyx x L y L z L II x y zx y z udvdwdwvuf LLL zyx udvdwdtwvuITwvuk ,,,,,,,, 2 , (4a) ( ) ( ) ( ) +∫ ∫ ∫ ∂ ∂ =∫ ∫ ∫ t y L z L V zy x L y L z L zyx y zx y z dvdwd x wvxV TwvxD LL zy udvdwdtwvuV LLL zyx 0 ,,, ,,,,,, τ τ ( ) ( ) ( ) ×∫ ∫ ∫ ∂ ∂ +∫ ∫ ∫ ∂ ∂ + t x L y L yx t x L z L V zx x yx z z zvuV LL yx dudwd x wyuV TwyuD LL zx 00 ,,,,,, ,,, τ τ τ ( ) ( ) ( ) ( ) −∫ ∫ ∫−× x L y L z L VI zyx V x y z udvdwdtwvuVtwvuITwvuk LLL zyx dudvdTzvuD ,,,,,,,,,,,, , τ ( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− x L y L z L V zyx x L y L z L VV zyx x y zx y z udvdwdwvuf LLL zyx udvdwdtwvuVTwvuk LLL zyx ,,,,,,,, 2 , .
- 9. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 9 We determine spatio-temporal distributions of concentrations of point defects as the same series ( ) ( ) ( ) ( ) ( )∑= = N n nnnnn tezcycxcatzyx 1 0 ,,, ρρρ . Parameters anρ should be determined in future. Substitution of the series into Eqs.(4a) leads to the following results ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑ == N n t y L z L InnnI zyx N n nInnn nI y z vdwdTwvxDzcyca LLL zy tezsysxs n azyx 1 01 33 ,,, π π ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−× = N n t x L z L InnnInnI zyx nnI x z dudwdTwyuDzcxceysa LLL zx xsde 1 0 ,,, ττ π ττ ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫−∑ ∫ ∫ ∫− = x L y L z L II N n t x L y L InnnInnI zyx x y zx y TvvukdudvdTzvuDycxcezsa LLL yx ,,,,,, , 1 0 ττ π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∑− ∑× == x L y L z L N n nnnnI zyxzyx N n nInnnnI x y z wcvcuca LLL zyx LLL zyx udvdwdtewcvcuca 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫+∑× = x L y L z L I N n VInVnnnnVnI x y z udvdwdwvufudvdwdTvvuktewcvcucate ,,,,, 1 , zyx LLLzyx× ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑ == N n t y L z L VnnnV zyx N n nVnnn nV y z vdwdTwvxDzcyca LLL zy tezsysxs n azyx 1 01 33 ,,, π π ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−× = N n t x L z L VnnnVnnV zyx nnV x z dudwdTwyuDzcxceysa LLL zx xsde 1 0 ,,, ττ π ττ ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫−∑ ∫ ∫ ∫− = x L y L z L VV N n t x L y L VnnnVnnV zyx x y zx y TvvukdudvdTzvuDycxcezsa LLL yx ,,,,,, , 1 0 ττ π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∑− ∑× == x L y L z L N n nnnnI zyxzyx N n nInnnnV x y z wcvcuca LLL zyx LLL zyx udvdwdtewcvcuca 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫+∑× = x L y L z L V N n VInVnnnnVnI x y z udvdwdwvufudvdwdTvvuktewcvcucate ,,,,, 1 , zyx LLLzyx× . We used orthogonality condition of functions of the considered series framework the heterostruc- ture to calculate coefficients anρ. The coefficients an could be calculated for any quantity of terms N. In the common case equations for the required coefficients could be written as ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− == N n t L L n y nyn nI x N n nI nIzyx x y yc n L ysyLxc n a L te n aLLL 1 0 0 0 2 1 65 222 12 2 221 2 1 πππ
- 10. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 10 ( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫ −+× = N n t L n nI y L nIn z nI xz xsx n a L dexdydzdzc n L zszTzyxD 1 0 0 2 0 2 2 1 1 2 ,,, π ττ π ( )[ ] ( ) ( ) ( )[ ] ( )[ ]×∫ ∫ − −++ −++ y z L L nn z nzIn x x yczdzc n L zszLTzyxDxc n L L 0 0 2112 2 2,,,12 ππ ( ) ( ) ( ) ( )[ ] ( ) −∫ −++× zL nIn z nzInI dexdydzdzc n L zszLTzyxDdexdyd 0 12 2 2,,, ττ π ττ ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = N n t L L n y nyn x nx nI z x y yc n L ysyLxc n L xsxL n a L 1 0 0 0 2 12 2 212 2 2 2 1 πππ ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ +−+−∫ −× = N n L n x xnInI L nIIn xz xc n L LteadexdydzdTzyxDzc 1 0 2 0 12 2 2,,,21 π ττ ( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫ +−+ −+++ y z L L n z zIIn y nyn zc n L LTzyxkyc n L ysyLxsx 0 0 , 12 2 ,,,12 2 22 ππ ( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ + −++−+ = N n L L yn x nxnVnInVnIn x y Lxc n L xsxLteteaaxdydzdzsz 1 0 0 12 2 22 π ( ) ( )[ ] ( ) ( ) ( )[ ]∫ × −++ −++ zL n z nzVIn y n zdzc n L zszLTzyxkyc n L ysy 0 , 12 2 2,,,12 2 2 ππ ( ) ( )[ ] ( ) ( )[ ] ( )×∑ ∫ ∫ ∫ −+ −++× = N n L L L In y nn x n x y z Tzyxfyc n L ysyxc n L xsxxdyd 1 0 0 0 ,,,11 ππ ( ) ( )[ ] xdydzdzc n L zszL n z nz −++× 12 2 2 π ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− == N n t L L n y nyn nV x N n nV nVzyx x y yc n L ysyLxc n a L te n aLLL 1 0 0 0 2 1 65 222 12 2 221 2 1 πππ ( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫ −+× = N n t L n nV y L nVn z nV xz xsx n a L dexdydzdzc n L zszTzyxD 1 0 0 2 0 2 2 1 1 2 ,,, π ττ π ( )[ ] ( ) ( ) ( )[ ] ( )[ ]×∫ ∫ − −++ −++ y z L L nn z nzVn x x yczdzc n L zszLTzyxDxc n L L 0 0 2112 2 2,,,12 ππ ( ) ( ) ( ) ( )[ ] ( ) −∫ −++× zL nVn z nzVnV dexdydzdzc n L zszLTzyxDdexdyd 0 12 2 2,,, ττ π ττ
- 11. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 11 ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = N n t L L n y nyn x nx nV z x y yc n L ysyLxc n L xsxL n a L 1 0 0 0 2 12 2 212 2 2 2 1 πππ ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ +−+−∫ −× = N n L n x xnVnV L nVVn xz xc n L LteadexdydzdTzyxDzc 1 0 2 0 12 2 2,,,21 π ττ ( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫ +−+ −+++ y z L L n z zVVn y nyn zc n L LTzyxkyc n L ysyLxsx 0 0 , 12 2 ,,,12 2 22 ππ ( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ + −++−+ = N n L L yn x nxnVnInVnIn x y Lxc n L xsxLteteaaxdydzdzsz 1 0 0 12 2 22 π ( ) ( )[ ] ( ) ( ) ( )[ ]∫ × −++ −++ zL n z nzVIn y n zdzc n L zszLTzyxkyc n L ysy 0 , 12 2 2,,,12 2 2 ππ ( ) ( )[ ] ( ) ( )[ ] ( )×∑ ∫ ∫ ∫ −+ −++× = N n L L L Vn y nn x n x y z Tzyxfyc n L ysyxc n L xsxxdyd 1 0 0 0 ,,,11 ππ ( ) ( )[ ] xdydzdzc n L zszL n z nz −++× 12 2 2 π . In the final form relations for required parameters could be written as ( ) − +− + ± + −= A yb yb Ab b Ab a nInV nI 2 3 4 2 3 4 3 4 44 λγ , nInI nInInInInI nV a aa a χ λδγ ++ −= 2 , where ( ) ( ) ( ) ( )[ ] ( ){∫ ∫ ∫ ++ −++= x y zL L L ynn x nxnn Lysyxc n L xsxLTzyxkte 0 0 0 , 212 2 2,,,2 π γ ρρρρ ( )[ ] ( ) ( )[ ] xdydzdzc n L zszLyc n L n z nzn y −++ −+ 12 2 212 2 ππ , ( )×∫= t n x n e nL 0 2 2 1 τ π δ ρρ ( ) ( )[ ] ( ) ( )[ ] ( ) [∫ ∫ ∫ − −+ −+× x y zL L L n z nn y n ydzdTzyxDzc n L zszyc n L ysy 0 0 0 1,,,1 2 1 2 ρ ππ ( )] ( ) ( ) ( )[ ] ( )[ ] {∫ ∫ ∫ ∫ +− −+++− t L L L znn x nxn y n x y z Lycxc n L xsxLe nL dxdxc 0 0 0 0 2 21122 2 1 2 π τ π τ ρ ( ) ( )[ ] ( ) ( ) ( ){∫ ∫ ++ −++ t L nn z n z n x xsxe nL dxdydzdTzyxDzc n L zsz 0 0 2 2 2 1 ,,,12 2 2 τ π τ π ρρ
- 12. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 12 ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ×∫ ∫ − −++ −++ y z L L nn y nyn x x zdTzyxDzcyc n L ysyLxc n L L 0 0 ,,,211 2 12 ρ ππ ( )te n LLL dxdyd n zyx ρ π τ 65 222 −× , ( ) ( )[ ] ( )[ ]∫ ∫ +−+ −+= x yL L n y yn x nnIV yc n L Lxc n L xsx 0 0 12 2 1 ππ χ ( )} ( ) ( ) ( )[ ] ( ) ( )tetexdydzdzc n L zszLTzyxkysy nVnI L n z nzVIn z ∫ −+++ 0 , 12 2 2,,,2 π , ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]∫ ∫ ∫ × −+ −+ −+= x y zL L L n z nn y nn x nn zc n L zszyc n L ysyxc n L xsx 0 0 0 111 πππ λ ρ ( ) xdydzdTzyxf ,,,ρ× , 22 4 nInInInV b χγγγ −= , nInInVnInInInInV b γχδχδδγγ −−= 2 3 2 , 2 2 3 48 bbyA −+= , ( ) 22 2 2 nInInVnInInVnInVnInInV b χλλδχδγγλδγ −+−+= , ×= nI b λ21 nInInVnInV λχδδγ −× , 4 33 323 32 3b b qpqqpqy −++−−+= , 2 4 2 342 9 3 b bbb p − = , ( ) 3 4 2 4132 3 3 542792 bbbbbbq +−= . We determine distributions of concentrations of simplest complexes of radiation defects in space and time as the following functional series ( ) ( ) ( ) ( ) ( )∑=Φ = Φ N n nnnnn tezcycxcatzyx 1 0 ,,, ρρρ . Here anΦρ are the coefficients, which should be determined. Let us previously transform the Eqs. (6) to the following integro-differential form ( ) ( ) ( ) ×∫ ∫ ∫ Φ =∫ ∫ ∫Φ Φ t y L z L I I x L y L z L I zyx y zx y z dvdwd x wvx TwvxDudvdwdtwvu LLL zyx 0 ,,, ,,,,,, τ ∂ τ∂ ( ) ( ) ( )∫ ∫ ∫ ×+∫ ∫ ∫ Φ +× ΦΦ t x L y L I yx t x L z L I I zxzy x yx z TzvuD LL yx dudwd y wyu TwyuD LL zx LL zy 00 ,,, ,,, ,,, τ ∂ τ∂ ( ) ( ) ( ) −∫ ∫ ∫+ Φ × x L y L z L II zyx I x y z udvdwdwvuITwvuk LLL zyx dudvd z zvu ττ ∂ τ∂ ,,,,,, ,,, 2 , (6a) ( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− Φ x L y L z L I zyx x L y L z L I zyx x y zx y z udvdwdwvuf LLL zyx udvdwdwvuITwvuk LLL zyx ,,,,,,,, τ ( ) ( ) ( ) ×∫ ∫ ∫ Φ =∫ ∫ ∫Φ Φ t y L z L V V x L y L z L V zyx y zx y z dvdwd x wvx TwvxDudvdwdtwvu LLL zyx 0 ,,, ,,,,,, τ ∂ τ∂ ( ) ( ) ( )∫ ∫ ∫ ×+∫ ∫ ∫ Φ +× ΦΦ t x L y L V yx t x L z L V V zxzy x yx z TzvuD LL yx dudwd y wyu TwyuD LL zx LL zy 00 ,,, ,,, ,,, τ ∂ τ∂
- 13. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 13 ( ) ( ) ( ) −∫ ∫ ∫+ Φ × x L y L z L VV zyx V x y z udvdwdwvuVTwvuk LLL zyx dudvd z zvu ττ ∂ τ∂ ,,,,,, ,,, 2 , ( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− Φ x L y L z L V zyx x L y L z L V zyx x y zx y z udvdwdwvuf LLL zyx udvdwdwvuVTwvuk LLL zyx ,,,,,,,, τ . Substitution of the previously considered series in the Eqs.(6a) leads to the following form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−=∑− = Φ = Φ N n t y L z L nnnInIn zyx N n nInnn In y z wcvctexsan LLL zy tezsysxs n a zyx 1 01 33 π π ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−× = ΦΦΦ N n t x L z L InnIn zyx I x z dudwdTwvuDwcuca LLL zx dvdwdTwvxD 1 0 ,,,,,, τ π τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) +∑ ∫ ∫ ∫−× = ΦΦΦΦ N n t x L y L InnInnIn zyx Inn x y dudvdTzvuDvcuctezsan LLL yx teysn 1 0 ,,, τ π ( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+ Φ x L y L z L I x L y L z L II zyx x y zx y z udvdwdwvufudvdwdwvuITwvuk LLL zyx ,,,,,,,, 2 , τ ( ) ( )∫ ∫ ∫−× x L y L z L I zyxzyx x y z udvdwdwvuITwvuk LLL zyx LLL zyx τ,,,,,, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−=∑− = Φ = Φ N n t y L z L nnnVnVn zyx N n nVnnn Vn y z wcvctexsan LLL zy tezsysxs n a zyx 1 01 33 π π ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−× = ΦΦ N n t x L z L Vnn zyx V x z dudwdTwvuDwcucn LLL zx dvdwdTwvxD 1 0 ,,,,,, τ π τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−× = ΦΦΦΦ N n t x L y L VnnVnn zyx VnnVn x y dudvdTzvuDvcuctezsn LLL yx teysa 1 0 ,,, τ π ( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+× ΦΦ x L y L z L V x L y L z L VV zyx Vn x y zx y z udvdwdwvufudvdwdwvuVTwvuk LLL zyx a ,,,,,,,, 2 , τ ( ) ( )∫ ∫ ∫−× x L y L z L V zyxzyx x y z udvdwdwvuVTwvuk LLL zyx LLL zyx τ,,,,,, . We used orthogonality condition of functions of the considered series framework the heterostruc- ture to calculate coefficients anΦρ. The coefficients anΦρ could be calculated for any quantity of terms N. In the common case equations for the required coefficients could be written as ( ) ( )[ ] ( ) ( )[ ] ×∑∫ ∫ ∫ −++−−=∑− == Φ Φ N n t L L n y nyn x N n In Inzyx x y yc n L ysyLxc L te n aLLL 1 0 0 01 65 222 12 2 221 2 1 πππ
- 14. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 14 ( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫ −+× = ΦΦ Φ N n t L n L Inn z nI In xz xsxdexdydzdzc n L zszTzyxD n a 1 0 00 2 2 2 1 1 2 ,,, π ττ π ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫ −+− −++ Φ y z L L n z nInn x x xdydzdzc n L zszTzyxDycxc n L L 0 0 1 2 ,,,2112 2 ππ ( ) ( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫ +−+ −+−× = ΦΦ Φ N n t L L n y nn x n In xy In In x y yc n L ysyxc n L xsx n a L d Ln e a 1 0 0 0 22 12 2 21 22 1 πππ τ τ } ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ +−+∫ −+ = Φ Φ ΦΦ N n t L n x In In L InIny xz xc n L e n a dexdydzdTzyxDycL 1 0 0 33 0 1 2 1 ,,,21 π τ π ττ ( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫ +−∫ −++ zy L n z II L n y nn zc n L TzyxktzyxIyc n L ysyxsx 0 , 2 0 1 2 ,,,,,,1 2 ππ ( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫ +− −+−+ = Φ Φ N n t L L n y n x nIn In n x y yc n L xc n L xsxe n a xdydzdzsz 1 0 0 0 33 1 2 1 2 1 ππ τ π ( )} ( ) ( )[ ] ( ) ( ) ×∑+∫ −++ = Φ N n In L In z nn n a xdydzdtzyxITzyxkzc n L zszysy z 1 33 0 1 ,,,,,,1 2 ππ ( ) ( ) ( )[ ] ( ) ( )[ ] ( )[ ]∫ ∫ ∫ ∫ +− −+ −+× Φ t L L L n z n y nn x nIn x y z zc n L yc n L ysyxc n L xsxe 0 0 0 0 1 2 1 2 1 2 πππ τ ( )} ( ) xdydzdzyxfzsz In ,,Φ + ( ) ( )[ ] ( ) ( )[ ] ×∑∫ ∫ ∫ −++−−=∑− == Φ Φ N n t L L n y nyn x N n Vn Vnzyx x y yc n L ysyLxc L te n aLLL 1 0 0 01 65 222 12 2 221 2 1 πππ ( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫ −+× = ΦΦ Φ N n t L n L Vnn z nV Vn xz xsxdexdydzdzc n L zszTzyxD n a 1 0 00 2 2 2 1 1 2 ,,, π ττ π ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫ −+− −++ Φ y z L L n z nVnn x x xdydzdzc n L zszTzyxDycxc n L L 0 0 1 2 ,,,2112 2 ππ ( ) ( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫ +−+ −+−× = ΦΦ Φ N n t L L n y nn x n Vn xy Vn Vn x y yc n L ysyxc n L xsx n a L d Ln e a 1 0 0 0 22 12 2 21 22 1 πππ τ τ } ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ +−+∫ −+ = Φ Φ ΦΦ N n t L n x Vn Vn L VnVny xz xc n L e n a dexdydzdTzyxDycL 1 0 0 33 0 1 2 1 ,,,21 π τ π ττ
- 15. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 15 ( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫ +−∫ −++ zy L n z VV L n y nn zc n L TzyxktzyxVyc n L ysyxsx 0 , 2 0 1 2 ,,,,,,1 2 ππ ( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫ +− −+−+ = Φ Φ N n t L L n y n x nVn Vn n x y yc n L xc n L xsxe n a xdydzdzsz 1 0 0 0 33 1 2 1 2 1 ππ τ π ( )} ( ) ( )[ ] ( ) ( ) ×∑+∫ −++ = Φ N n Vn L Vn z nn n a xdydzdtzyxVTzyxkzc n L zszysy z 1 33 0 1 ,,,,,,1 2 ππ ( ) ( ) ( )[ ] ( ) ( )[ ] ( )[ ]∫ ∫ ∫ ∫ +− −+ −+× Φ t L L L n z n y nn x nVn x y z zc n L yc n L ysyxc n L xsxe 0 0 0 0 1 2 1 2 1 2 πππ τ ( )} ( ) xdydzdzyxfzsz Vn ,,Φ + . 3. DISCUSSION In this section we analyzed redistribution of dopant with account redistribution of radiation de- fects. The analysis shown, that presents of interface between materials of heterostructure gives a possibility to increase absolute value of gradient of concentration of dopant outside of enriched are by the dopant (see Figs. 2 and 3). At the same time homogeneity of concentration of dopant in enriched area increases (see Figs. 2 and 3). The effects could be find, when dopant diffusion coef- ficient in the doped area is larger, than in the nearest areas. Otherwise absolute value of gradient of concentration of dopant decreases outside enriched by the dopant area (see Fig. 4). However the decreasing could be partially or fully compensated by using high doping of materials. The high doping leads to significant nonlinearity of diffusion of dopant. To increase compactness of the considered circuits XOR it is attracted an interest the first relation between values of dopant diffusion coefficient. Fig.2a. Spatial distributions of infused dopant concentration in the considered heterostructure. The consi- dered direction perpendicular to the interface between epitaxial layer substrate. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves
- 16. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 16 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. Spatial distributions of infused dopant concentration in the considered heterostructure. Curves 1 and 3 corresponds to annealing time Θ=0.0048(Lx 2 +Ly 2 +Lz 2 )/D0. Curves 2 and 4 corresponds to annealing time Θ=0.0057(Lx 2 +Ly 2 +Lz 2 )/D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corres- ponds to the considered heterostructure. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves 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.3. Implanted dopant distributions in heterostructure in heterostructure with two epitaxial layers (solid lines) and with one epitaxial layer (dushed lines) for different values of annealing time. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves Increasing of annealing time leads to acceleration of diffusion. In this situation one can find in- creasing quantity of dopant in materials near doped sections. If annealing time is small, the do- pant can not achieves nearest interface between layers of heterostructure. These effects are shown by Figs. 5 and 6. We used recently introduced criterion [16-23] to estimate compromise value of annealing time. Framework the criterion we approximate real distribution of concentration of do- pant by idealized step-wise distribution ψ (x,y,z), which would be better to use for minimization dimensions of elements of the considered circuit XOR [19-26]. Farther the required compromise annealing time has been calculated by minimization the following mean-squared error
- 17. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 17 ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (8) C(x,Θ) 0 Lx 2 1 3 4 Fig. 4. Distributions of concentrations of infused dopant in the considered heterostructure. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of concentrations of dopant for differ- ent values of annealing time for increasing of annealing time with increasing of number of curve x C(x,Θ) 1 2 3 4 0 L Fig. 5. Distributions of concentrations of implanted dopant in the considered heterostructure. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of concentrations of dopant for differ- ent values of annealing time for increasing of annealing time with increasing of number of curve We analyzed optimal value of annealing time. The analysis shows, that optimal value of anneal- ing time for ion type of doping is smaller, than optimal value of annealing time for diffusion type of doping. It is known, that ion doping of materials leads to radiation damage of doped materials. In this situation radiation defects should be annealed. The annealing leads to the above difference between optimal values of annealing time of dopant. The annealing of implanted dopant is neces- sary in the case, when the dopant did not achieved nearest interface between layers of heterostruc- ture during annealing of radiation defects.
- 18. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 18 It should be noted, that using diffusion type of doping did not leads to radiation damage of mate- rials. However radiation damage of materials during ion doping gives a possibility to decrease mismatch-induced stress in heterostructure [34]. 4. CONCLUSIONS In this paper we introduced an approach to decrease dimensions of a circuit XOR. The approach based on optimization of manufacturing field-effect heterotransistors, which includes into itself the considered circuit. 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, educational fellowship for scientific research of Government of Russian, edu- cational fellowship for scientific research of Government of Nizhny Novgorod region of Russia and of Nizhny Novgorod State University of Architecture and Civil Engineering. REFERENCES [1] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. Shallow p-n-junctions in Si prepared by pulse photon annealing. Semiconductors. Vol. 36 (5). P. 611-617 (2002). [2] M.J. Kumar, T.V. Singh. Quantum confinement effects in strained silicon MOSFETs MOSFETs. Int. J. Nanoscience. Vol. 7 (2-3). P. 81-84 (2008). [3] D. Fathi, B. Forouzandeh. Accurate analysis of global interconnects in nano-FPGAs. Nano. Vol. 4 (3). P. 171-176 (2009). [4] D. Fathi, B. Forouzandeh, N. Masoumi. New enhanced noise analysis in active mixers in nanoscale technologies. Nano. Vol. 4 (4). P. 233-238 (2009). [5] 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). [6] 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). С. 971-974 (2009). [7] 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). [8] 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). [9] S.A. Chachuli, P.N.A. Fasyar, N. Soin, N.M. Karim, N. Yusop. Pareto ANOVA analysis for CMOS 0.18 µm two-stage Op-amp. Mat. Sci. Sem. Proc. Vol. 24. P. 9-14 (2014). [10] N.I. Volokobinskaya, I.N. Komarov, T.V. Matioukhina, V.I. Rechetniko, A.A. Rush, I.V. Falina, A.S. Yastrebov. Investigation of technological processes of manufacturing of the bipolar power high- voltage transistors with a grid of inclusions in the collector region. Semiconductors. Vol. 35 (8). P. 1013-1017 (2001). [11] G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Issue 2. P. 10-17 (2006). [12] M.V. Dunga, L. Chung-Hsun, X. Xuemei, D.D. Lu, A.M. Niknejad, H. Chenming. Modeling ad- vanced FET technology in a compact model. IEEE Transactions on Electron Devices. Vol. 53 (9). P. 157-162 (2006). [13] A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Electron- ics. Issue 1. P. 34-38 (2008). [14] C. Senthilpari, K. Diwakar, A.K. Singh. Low Energy, Low Latency and High Speed Array Divider Circuit Using a Shannon Theorem Based Adder Cell. Recent Patents on Nanotechnology. Vol. 3 (1). P. 61-72 (2009).
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- 20. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 20 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. Since 2014 E.A. Bulaeva is in a PhD program in Radiophysical Department of Nizhny Novgorod State University. She has 90 published papers in area of her researches.