![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016
DOI:10.5121/msej.2016.3101 1
MODIFICATION OF DOPANT CONCENTRA-
TION PROFILE IN A FIELD-EFFECT HETERO-
TRANSISTOR FOR MODIFICATION ENERGY
BAND DIAGRAM
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 consider an approach of manufacturing more compact field-effect heterotransistors. The
approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer
with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or
ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we intro-
duce an approach of modification of energy band diagram by additional doping of channel of the transis-
tors. We also consider an analytical approach to model and optimize technological process.
KEYWORDS
Modification of profile of dopant; decreasing of dimension of field-effect transistor; modification of energy
band diagram
1. INTRODUCTION
Development of solid state electronic leads to increasing performance of the appropriate electron-
ic devices [1-11]. At the same time one can find increasing integration rate of integrated circuits
[1-3,5,7]. In this situation dimensions of elements of integrated circuits decreases. To increase
performance of solid-state electronics devices are now elaborating new technological processes of
manufacturing of solid state electronic devices. Another ways to increase the performance are
optimization of existing technological processes and determination new materials with higher
values of charge carriers mobilities. To decrease dimensions of elements of integrated circuits
they are elaborating new and optimizing existing technological processes. Framework this paper
we introduce an approach to decrease dimensions of field-effect heterotransistors. At the same
time we introduce an approach of modification of energy band diagram for regulation of transport
of charge carriers. The approach based on manufacturing a field-effect transistor in the hetero-
structure from Fig. 1. The heterostructure consists of a substrate and an epitaxial layer. They are
have been considered four sections in the epitaxial layer. The sections have been doped by diffu-
sion or ion implantation. Left and right sections will be considered in future as source and drain,
respectively. Both average sections became as channel of transistor. Using one section instead
two sections leads to simplification of structure of transistor. However using additional doped
section framework the channel of the considered transistor gives us possibility to modify energy
band diagram in the structure. After finishing of the considered doping annealing of dopant and/or
radiation defects should be done. Several conditions for achievement of decreasing of dimensions
of the field-effect transistor have been formulated.](https://image.slidesharecdn.com/msej1-160407094049/85/modification-of-dopant-concentration-profile-in-a-fieldeffect-heterotransistor-for-modification-energy-band-diagram-1-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016
2
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 [12,13]
( ) ( ) ( ) ( )
+
+
=
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC
CCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,,
. (1)
Boundary and initial conditions for the equations are
Fig. 1. Heterostructure, which consist of a substrate and an epitaxial layer.
View from side
( ) 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 [14-16]
( ) ( )
( )
( ) ( )
( )
++
+= 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-](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSISTOR FOR MODIFICATION ENERGY BAND DIAGRAM
- 1. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 DOI:10.5121/msej.2016.3101 1 MODIFICATION OF DOPANT CONCENTRA- TION PROFILE IN A FIELD-EFFECT HETERO- TRANSISTOR FOR MODIFICATION ENERGY BAND DIAGRAM 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 consider an approach of manufacturing more compact field-effect heterotransistors. The approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we intro- duce an approach of modification of energy band diagram by additional doping of channel of the transis- tors. We also consider an analytical approach to model and optimize technological process. KEYWORDS Modification of profile of dopant; decreasing of dimension of field-effect transistor; modification of energy band diagram 1. INTRODUCTION Development of solid state electronic leads to increasing performance of the appropriate electron- ic devices [1-11]. At the same time one can find increasing integration rate of integrated circuits [1-3,5,7]. In this situation dimensions of elements of integrated circuits decreases. To increase performance of solid-state electronics devices are now elaborating new technological processes of manufacturing of solid state electronic devices. Another ways to increase the performance are optimization of existing technological processes and determination new materials with higher values of charge carriers mobilities. To decrease dimensions of elements of integrated circuits they are elaborating new and optimizing existing technological processes. Framework this paper we introduce an approach to decrease dimensions of field-effect heterotransistors. At the same time we introduce an approach of modification of energy band diagram for regulation of transport of charge carriers. The approach based on manufacturing a field-effect transistor in the hetero- structure from Fig. 1. The heterostructure consists of a substrate and an epitaxial layer. They are have been considered four sections in the epitaxial layer. The sections have been doped by diffu- sion or ion implantation. Left and right sections will be considered in future as source and drain, respectively. Both average sections became as channel of transistor. Using one section instead two sections leads to simplification of structure of transistor. However using additional doped section framework the channel of the considered transistor gives us possibility to modify energy band diagram in the structure. After finishing of the considered doping annealing of dopant and/or radiation defects should be done. Several conditions for achievement of decreasing of dimensions of the field-effect transistor have been formulated.
- 2. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 2 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 [12,13] ( ) ( ) ( ) ( ) + + = z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC CCC ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,,,,,,,,,,, . (1) Boundary and initial conditions for the equations are Fig. 1. Heterostructure, which consist of a substrate and an epitaxial layer. View from side ( ) 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 [14-16] ( ) ( ) ( ) ( ) ( ) ( ) ++ += 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-
- 3. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 3 age) with each atom of dopant. Ref.[14] 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 [15,16] ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ 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 [15,16] ( ) ( ) ( ) ( ) ( ) + 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 ∂ ∂ ∂ ∂ .
- 4. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 4 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 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 [17]. 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
- 5. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 5 ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ × ∑+−× = 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 ,, , 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
- 6. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 6 ( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ × −++ −+× = 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 ( ) ( )[ ] ( ) ( ) ( ) ( ) × + ++ −+× 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
- 7. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 7 ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×+ −+ −+× 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 ( ) ( ) ( )[ ] ( ) ( ) ( )×∫ ∫ ∫+ −+× 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
- 8. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 8 ( ) ( ) ( ) ×−∫ ∫ ∫ ∂ ∂ +× 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 ( ) ( ) ( ) ( ) ( )∑= = 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
- 9. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 9 ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−× = 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 ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = 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
- 10. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 10 ( ) ( )[ ] ( ) ( ) ( )[ ]∫ × −++ −++ 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 ( ) ( )[ ] xdydzdzc n L zszL n z nz −++× 12 2 2 π .
- 11. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 11 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 . Here anΦρ are the coefficients, which should be determined. Let us previously transform the Eqs. (6) to the following integro-differential form
- 12. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 12 ( ) ( ) ( ) ×∫ ∫ ∫ 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
- 13. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 13 ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−× = 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 ( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫ −+× = 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
- 14. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 14 ( )} ( ) 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 ( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫ +−∫ −++ 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 the present paper we analyzed redistribution of infused and implanted dopants in heterostruc- ture, which have been presented on Fig. 1. The analysis has been done by using relations, calcu- lated in the previous section. First of all we consider situation, when dopant diffusion coefficient in doped area is larger, than in nearest areas. It has been shown, that in this case distribution of concentration of dopant became more compact in comparison with wise versa situation (see Figs. 2 and 3 for diffusion and ion types of doping). In the wise versa situation (when dopant diffusion coefficient in doped area is smaller, than in nearest areas) we obtained spreading of distribution of concentration of dopant. In this case outside of doping material one can find higher spreading in comparison with wise versa situation (see Fig. 4). It should be noted, that properties of layers of multilayer structure varying in space: varying lay- ers with larger and smaller values of the diffusion coefficient. In this situation with account re- sults, which shown on Figs. 2-4, layers of the considered heterostructure with smaller value of
- 15. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 15 dopant should has smaller level of doping in comparison with nearest layers to exclude changing of type of doping in the nearest layers. For a more complete doping of each section and at the same to decrease diffusion of dopant into nearest sections it is attracted an interest optimization of annealing of dopant and /or radiation defects. Let us optimize annealing of dopant and/or radia- tion defects by using recently introduce criterion [18-23]. Framework the criterion we approx- imate real distribution of concentration of dopant by idealized step-wise distribution of concentra- tion ψ (x,y,z) (see Figs. 5 and 6 for diffusive or ion types of doping). Farther we determine optim- al annealing time by minimization of mean-squared error Fig.2. Spatial distributions of infused dopant concentration in the considered heterostructure. The considered direction perpendicular to the interface between epitaxial layer substrate. Differ- ence between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves x 0.0 0.5 1.0 1.5 2.0 C(x,Θ) 2 3 4 1 0 L/4 L/2 3L/4 L Epitaxial layer Substrate Fig.3. 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 corres- ponds 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 corresponds to the considered heterostructure. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves
- 16. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 16 Fig.4. 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 ( ) ( )[ ]∫ ∫ ∫ −Θ= xL yL zL zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (8) Dependences of optimal annealing time are shown on Figs. 7 and 8 for diffusion and ion types of doping. It should be noted, that after finishing of ion doping of material it is necessary to anneal radiation defects. In the ideal case after finishing of the annealing dopant achieves nearest inter- face between materials of heterostructure. If the dopant has no time to achieve the interface, it is practicably to additionally anneal the dopant. The Fig. 8 shows dependences of the exactly addi- tional annealing time of implanted dopant. Necessity of annealing of radiation defects leads to smaller values of optimal annealing time for ion doping in comparison with values of optimal annealing time for diffusion type of doping. Using diffusion type of doping did not leads to so large damage in comparison with damage during ion type of doping. C(x,Θ) 0 Lx 2 1 3 4 Fig. 5. 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 concentra- tions of dopant for different values of annealing time for increasing of annealing time with in- creasing of number of curve
- 17. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 17 x C(x,Θ) 1 2 3 4 0 L Fig. 6. 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 concentra- tions of dopant for different values of annealing time for increasing of annealing time with in- creasing of number of curve Dependences of optimal annealing time are shown on Figs. 7 and 8 for diffusion and ion types of doping. It should be noted, that after finishing of ion doping of material it is necessary to anneal radiation defects. In the ideal case after finishing of the annealing dopant achieves nearest inter- face between materials of heterostructure. If the dopant has no time to achieve the interface, it is practicably to additionally anneal the dopant. The Fig. 8 shows dependences of the exactly addi- tional annealing time of implanted dopant. Necessity of annealing of radiation defects leads to smaller values of optimal annealing time for ion doping in comparison with values of optimal annealing time for diffusion type of doping. Using diffusion type of doping did not leads to so large damage in comparison with damage during ion type of doping. 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.7. Optimal annealing time of infused dopant as dependences of several parameters. Curve 1 is the dependence of the considered annealing time on dimensionless thickness of epitax- ial layer a/L and ξ=γ=0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of the considered annealing time on the parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of the considered annealing time on the para- meter ξ for a/L=1/2 and ε=γ=0. Curve 4 is the dependence of the considered annealing time on parameter γ for a/L=1/2 and ε=ξ=0
- 18. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 18 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.8. Optimal annealing time of implanted dopant as dependences of several parameters. Curve 1 is the dependence of the considered annealing time on dimensionless thickness of epi- taxial layer a/L and ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of the considered annealing time on the para- meter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of the considered annealing time on the parameter ξ for a/L=1/2 and ε= γ= 0. Curve 4 is the dependence of the considered annealing time on parameter γ for a/L=1/2 and ε=ξ=0 4. CONCLUSIONS In this paper we introduced an approach to manufacture of field-effect of transistors which gives a possibility to decrease their dimensions. The decreasing based on manufacturing the transistors in a heterostructure with specific configuration, doping of required areas of the heterostructure by diffusion or ion implantation and optimization of annealing of dopant and/or radiation defects. Framework the approach we introduce an approach of additional doping of channel. The addi- tional doping gives us possibility to modify energy band diagram. We also consider an analytical approach to model and optimize technological process. 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. J.Y. Fang, G.Y. Lee, J.I. Chyi, C.P. Hsu, Y.W. Kang, K.C. Fang, W.L. Kao, D.J. Yao, C.H. Hsu, Y.F. Huang, C.C. Chen, S.S. Li, J.A. Yeh, F. Ren, Y.L. Wang. Viscosity-dependent drain current noise of AlGaN/GaN high electron mobility transistor in polar liquids. J. Appl. Phys. Vol. 114 (20). P. 204503-204507 (2013). 2. G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Is- sue 2. P. 10-17 (2006). 3. A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Elec- tronics. Issue 1. P. 34-38 (2008). 4. A.O. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudrik, P.M. Litvin, V.V. Milenin, A.V. Sachenko. Au–TiBx−n-6H-SiC Schottky barrier diodes: the features of current flow in rectifying and nonrectifying contacts. Semiconductors. Vol. 43 (7). P. 897-903 (2009).
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