INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERS
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In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the considered p-n-junction have been formulated.
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016
DOI : 10.5121/ijitmc.2016.4101 1
INFLUENCE OF OVERLAYERS ON DEPTH OF IM-
PLANTED-HETEROJUNCTION RECTIFIERS
E.L. Pankratov1
, E.A. Bulaeva1,2
1
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
2
Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky
street, Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
KEYWORDS
Heterostructures; implanted-junction rectifiers: overlayers.
1. INTRODUCTION
In the present time integration rate of elements of integrated circuits intensively increasing. At the
same time dimensions of the above elements decreases. To increase integration rate of elements
of integrated circuits and to increase dimensions of the same elements are have been elaborated
different approaches [1-10]. In the present paper we consider a heterostructure with two layers: a
substrate and an epitaxial layer. The heterostructure could include into itself a third layer: an
overlayer (see Figs. 1). We assume, that type of conductivity of the substrate (n or p) is known.
The epitaxial layer has been doped by ion implantation to manufacture required type of conduc-
tivity (p or n). Farther we consider annealing of radiation defects. Main aim of the present paper
is comparison two ways of ion doping of the epitaxial layer: ion doping of epitaxial layer and ion
doping of epitaxial layer through the overlayer.
Fig. 1a. Heterostructure, which consist of a substrate and an epitaxial layer with initial distribution of
implanted dopant](https://image.slidesharecdn.com/ijitmc1-160309083510/85/INFLUENCE-OF-OVERLAYERS-ON-DEPTH-OF-IMPLANTED-HETEROJUNCTION-RECTIFIERS-1-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016
2
x
D(x)
D1
D2
0 La
fC(x)
D0
Substrate
Epitaxial layer
b
Overlayer
Fig. 1b. Heterostructure, which consist of a substrate, an epitaxial layer and an overlayer with initial distri-
bution of implanted dopant
2. METHOD OF SOLUTION
To solve our aim we determine spatio-temporal distributions of concentrations of dopants. We
calculate the required distributions by solving the second Fick's law in the following form [1]
( ) ( ) ( ) ( )
+
+
=
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC
CCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,,
. (1)
Boundary and initial conditions for the equations are
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C(x,y,z,0)=f (x,y,z). (2)
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 [11-13]
( ) ( )
( )
( ) ( )
( )
++
+= 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](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERS
- 1. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 DOI : 10.5121/ijitmc.2016.4101 1 INFLUENCE OF OVERLAYERS ON DEPTH OF IM- PLANTED-HETEROJUNCTION RECTIFIERS E.L. Pankratov1 , E.A. Bulaeva1,2 1 Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia 2 Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky street, Nizhny Novgorod, 603950, Russia ABSTRACT In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the considered p-n-junction have been formulated. KEYWORDS Heterostructures; implanted-junction rectifiers: overlayers. 1. INTRODUCTION In the present time integration rate of elements of integrated circuits intensively increasing. At the same time dimensions of the above elements decreases. To increase integration rate of elements of integrated circuits and to increase dimensions of the same elements are have been elaborated different approaches [1-10]. In the present paper we consider a heterostructure with two layers: a substrate and an epitaxial layer. The heterostructure could include into itself a third layer: an overlayer (see Figs. 1). We assume, that type of conductivity of the substrate (n or p) is known. The epitaxial layer has been doped by ion implantation to manufacture required type of conduc- tivity (p or n). Farther we consider annealing of radiation defects. Main aim of the present paper is comparison two ways of ion doping of the epitaxial layer: ion doping of epitaxial layer and ion doping of epitaxial layer through the overlayer. Fig. 1a. Heterostructure, which consist of a substrate and an epitaxial layer with initial distribution of implanted dopant
- 2. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 2 x D(x) D1 D2 0 La fC(x) D0 Substrate Epitaxial layer b Overlayer Fig. 1b. Heterostructure, which consist of a substrate, an epitaxial layer and an overlayer with initial distri- bution of implanted dopant 2. METHOD OF SOLUTION To solve our aim we determine spatio-temporal distributions of concentrations of dopants. We calculate the required distributions by solving the second Fick's law in the following form [1] ( ) ( ) ( ) ( ) + + = z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC CCC ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,,,,,,,,,,, . (1) Boundary and initial conditions for the equations are ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = yLx y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLx z tzyxC , C(x,y,z,0)=f (x,y,z). (2) 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 [11-13] ( ) ( ) ( ) ( ) ( ) ( ) ++ += 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
- 3. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 3 interval γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in aver- age) with each atom of dopant. Ref.[11] 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 [12,13] ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ Tzyxk y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI IIII ,,, ,,, ,,, ,,, ,,, ,,, , ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxI TzyxD z tzyxI VII ,,,,,,,,, ,,, ,,,,,, , 2 − ∂ ∂ ∂ ∂ +× (4) ( ) ( ) ( ) ( ) ( ) ( )×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ Tzyxk y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VVVV ,,, ,,, ,,, ,,, ,,, ,,, , ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxV TzyxD z tzyxV VIV ,,,,,,,,, ,,, ,,,,,, , 2 − ∂ ∂ ∂ ∂ +× . Boundary and initial conditions for these equations are ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxρ , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxρ , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxρ , ( ) 0 ,,, = ∂ ∂ = yLy y tzyxρ , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxρ , ( ) 0 ,,, = ∂ ∂ = zLz z tzyxρ , ρ (x,y,z,0)=fρ (x,y,z). (5) Here ρ =I,V. 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 [12,13] ( ) ( ) ( ) ( ) ( ) + 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
- 4. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 4 ( ) 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 determine spatio-temporal distribution 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 ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ++ +× 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 [14]. 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
- 5. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ++× = 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 ,, , 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
- 6. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 6 ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ∫ ∫ ∫ ∫ × −+ −× 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 ( ) ( )[ ] ( ) ( ) ( ) ( ) × + ++ −+× 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
- 7. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 7 ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ∫ ∫ ∫ × ++ −+× 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 ( ) ( ) ( )[ ] ( ) ( ) ( )×∫ ∫ ∫+ −+× 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 parameter γ. 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 , . We determine spatio-temporal distributions of concentrations of point defects as the same series
- 8. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 8 ( ) ( ) ( ) ( ) ( )∑= = 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 ( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫ −+× = 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
- 9. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 9 ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = 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 ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = 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
- 10. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 10 ( ) ( )[ ] 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 ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ×∫ ∫ − −++ −++ 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 .
- 11. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 11 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∂ ( ) ( ) ( ) −∫ ∫ ∫+ 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
- 12. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 12 ( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+× 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 ( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫ −+× = 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
- 13. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 13 } ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ +−+∫ −+ = 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 ( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫ +−∫ −++ 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 compare spatial distributions of concentrations of dopant in a heterostructure with overlayer and without the overlayer. Fig. 2 shows result of the above comparison. One can find from the figure, that using the overlayer gives a possibility to manufacture more shallow p-n- junction in comparison with p-n-junction in the heterostructure without the overlayer. At the same time analysis of redistribution of radiation defects shows, that implantation of ions of dopant through the overlayer gives a possibility to decrease quantity of radiation defects in the epitaxial layer. Fig.2. Distributions of concentration of implanted dopant in the considered heterostructure. Curves 1 and 3 describe distributions of concentration of implanted dopant in the heterostructure with overlayer. Curve 1 describes distribution of concentration of implanted dopant for the case, when dopant diffusion coefficient in the overlayer is larger, than in the epitaxial layer. Curve 2 describes distribution of concentration of implanted dopant for the case, when dopant diffusion coefficient in the overlayer is smaller, than in the epitaxial layer. Curve 2 describes distribution of concentration of implanted dopant in the heterostructure without overlayer.
- 14. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 14 It should be noted, that manufacturing the considered implanted-junction rectifier near interface between layers of heterostructure gives a possibility to increase sharpness of the p-n-junction and at the same time to increase homogeneity of concentration of the dopant in the enriched area [15- 19]. In this situation it is practicably to choose appropriate thickness of epitaxial layer. 4. CONCLUSIONS In this paper we analyzed distributions of concentration of implanted dopant in heterostructure with overlayer and without overlayer during manufacture an implanted -heterojunction rectifier. We determine conditions, which correspond to decrease depth of the p-n-junction. 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 Russia, educa- tional fellowship for scientific research of Government of Nizhny Novgorod region of Russia and educational fellowship for scientific research of Nizhny Novgorod State University of Architec- ture and Civil Engineering. REFERENCES [1] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). [2] D. Fathi, B. Forouzandeh. Accurate analysis of global interconnects in nano-FPGAs. Nano. Vol. 4 (3). P. 171-176 (2009). [3] D. Fathi, B. Forouzandeh, N. Masoumi. New enhanced noise analysis in active mixers in nanoscale technologies. Nano. Vol. 4 (4). P. 233-238 (2009). [4] 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). [5] Jung-Hui Tsai, Shao-Yen Chiu, Wen-Shiung Lour, Der-Feng Guo. High-performance InGaP/GaAs pnp δ-doped heterojunction bipolar transistor Semiconductors. Vol. 43 (7). P. 971-974 (2009). [6] J.Y. Choi, Ch. Park. A Thyristor-Only Input ESD Protection Scheme for CMOS RF ICs. Circuits and Systems. Vol. 2 (3). P. 170-182 (2011). [7] 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). [8] 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). [9] R. Laajimi, M. Masmoudi. High-Performance CMOS Current Mirrors: Application to Linear Volt- age-to-Current Converter Used for Two-Stage Operational Amplifier. Circuits and Systems. Vol. 3 (4). P. 311-316 (2012). [10] A. Ghorbani, A. Ghanaatian. A Novel High CMRR, Low Power and Low Voltage COS with QFG. Circuits and Systems. Vol. 3 (3). P. 263-268 (2012). [11] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [12] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [13] P.M. Fahey, P.B. Griffin, J.D. Plummer. Point defects and dopant diffusion in silicin. Rev. Mod. Phys. Vol. 61 (2). P. 289-388 (1989). [14] M.L. Krasnov, A.I. Kiselev, G.I. Makarenko. Integral equations ("Science", Moscow, 1976). [15] E.L. Pankratov. Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure by optimized laser annealing. J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010). [16] 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).
- 15. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016 15 [17] 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. J. Comp. Theor. Nanoscience. Vol. 12 (6). P. 976-990 (2014). [18] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture 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). [19] E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimensions of field-effect transistors without p-n-junctions. Int. J. Mod. Phys. B. Vol. 28 (27). P. 1450190-1-1450190-17 (2014). AUTHORS Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University: from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Ra- diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio- physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro- structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo- rod State University of Architecture and Civil Engineering. 2012-2015 Full Doctor course in Radiophysical Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Professor of Nizhny Novgorod State University. He has 135 published papers in area of his researches. Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school “Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architec- ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in the University. 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.