![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
DOI:10.5121/msej.2016.3301 1
ON DECREASING OF DIMENSIONS OF FIELD-
EFFECT TRANSISTORS WITH SEVERAL
SOURCES
E.L. Pankratov, E.A. Bulaeva
Nizhny Novgorod State University, 2 Gagarin avenue, Nizhny Novgorod, 603950, Russia
ABSTRACT
We analyzed mass and heat transport during manufacturing field-effect heterotransistors with several
sources to decrease their dimensions. Framework the result of manufacturing it is necessary to manufacture
heterostructure with specific configuration. After that it is necessary to dope required areas of the hetero-
structure by diffusion or ion implantation to manufacture the required type of conductivity (p or n). After
the doping it is necessary to do optimize annealing. We introduce an analytical approach to prognosis mass
and heat transport during technological processes. Using the approach leads to take into account nonlineari-
ty of mass and heat transport and variation in space and time (at one time) physical parameters of these
processes
KEYWORDS
Field-effect transistor, transistor with several channels, increasing of compactness of transistors
1. INTRODUCTION
Now several problems of solid state electronic intensively solving. The problems are increasing
of density of elements of integrated circuits and at the same time decreasing of dimensions of
these elements [1-4], increasing performance [5-7] and increasing reliability [8,9]. Now one can
find intensive development of both power electronic devices and logical elements. In this paper
we consider an approach to manufacture more compact field-effect heterotransistor with several
sources. Framework the approach it is necessary to manufacture a heterostructure. The hetero-
structure consist of a substrate and an epitaxial layer (see Fig. 1). Several sections have been
manufactured into the epitaxial layer. These sections manufactured by using other materials (see
Fig. 1). The sections have been doped by diffusion or ion implantation to obtain required type of
conductivity (n or p). After the doping one can manufacture a field-effect transistor framework
the considered heterostructure so as it is shown on the Fig. 1. The doping should be finished by
annealing of dopant and/or radiation defects. The annealing should be optimized. The optimiza-
tion attracted an interest to manufacture more compact distributions of concentrations of dopant.
Framework the paper we formulate conditions to increase compactness and at the same time to
increase homogeneity of distribution of concentration of dopant in enriched by the dopant area.
Fig. 1. Structure of a field-effect heterotransistor. Top side of the structure](https://image.slidesharecdn.com/3316msej01-200916123219/85/on-decreasing-of-dimensions-of-fieldeffect-transistors-with-several-sources-1-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
2
2. METHOD OF SOLUTION
To solve our aim we determine distribution of concentration of dopant C(x,y,z,t) in space and
time. To determine the distribution we solve the following boundary problem
( ) ( ) ( ) ( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
(1)
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( )
0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C(x,y,z,0)=fC(x,y,z). (2)
Here T is the temperature of annealing; DС is the dopant diffusion coefficient. Dopant diffusion
coefficient takes another value in other materials. Heating and cooling of heterostructure (see
Arrhenius law) also leads to changing of value of diffusion coefficient. We consider following
approximation of concentrational dependences of dopant diffusion coefficient [10-12]
( ) ( )
( )
( ) ( )
( )
++
+= 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; function P (x,y,z,T) describes dependences of limit of solubility on coordinate
and temperature; parameter γ ∈[1,3] is integer and depends on properties of materials; function V
(x,y,z,t) describes distribution of concentration of radiation vacancies on space and time with
equilibrium distribution V*
. Dependence of dopant diffusion coefficient on concentration of do-
pant has been investigated and described in details in [10]. Diffusion of dopant gives a possibility
to dope materials without generation radiation defects. In this situation ζ1= ζ2= 0. Ion doping of
dopant leads to generation radiation defects. Distributions of concentrations of point radiation
defects have been determined by solving the following boundary problem [11,12]
( ) ( ) ( ) ( ) ( ) ( ) ×−
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
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
−
∂
∂
∂
∂
+×
( ) 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)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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ON DECREASING OF DIMENSIONS OF FIELDEFFECT TRANSISTORS WITH SEVERAL SOURCES
- 1. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 DOI:10.5121/msej.2016.3301 1 ON DECREASING OF DIMENSIONS OF FIELD- EFFECT TRANSISTORS WITH SEVERAL SOURCES E.L. Pankratov, E.A. Bulaeva Nizhny Novgorod State University, 2 Gagarin avenue, Nizhny Novgorod, 603950, Russia ABSTRACT We analyzed mass and heat transport during manufacturing field-effect heterotransistors with several sources to decrease their dimensions. Framework the result of manufacturing it is necessary to manufacture heterostructure with specific configuration. After that it is necessary to dope required areas of the hetero- structure by diffusion or ion implantation to manufacture the required type of conductivity (p or n). After the doping it is necessary to do optimize annealing. We introduce an analytical approach to prognosis mass and heat transport during technological processes. Using the approach leads to take into account nonlineari- ty of mass and heat transport and variation in space and time (at one time) physical parameters of these processes KEYWORDS Field-effect transistor, transistor with several channels, increasing of compactness of transistors 1. INTRODUCTION Now several problems of solid state electronic intensively solving. The problems are increasing of density of elements of integrated circuits and at the same time decreasing of dimensions of these elements [1-4], increasing performance [5-7] and increasing reliability [8,9]. Now one can find intensive development of both power electronic devices and logical elements. In this paper we consider an approach to manufacture more compact field-effect heterotransistor with several sources. Framework the approach it is necessary to manufacture a heterostructure. The hetero- structure consist of a substrate and an epitaxial layer (see Fig. 1). Several sections have been manufactured into the epitaxial layer. These sections manufactured by using other materials (see Fig. 1). The sections have been doped by diffusion or ion implantation to obtain required type of conductivity (n or p). After the doping one can manufacture a field-effect transistor framework the considered heterostructure so as it is shown on the Fig. 1. The doping should be finished by annealing of dopant and/or radiation defects. The annealing should be optimized. The optimiza- tion attracted an interest to manufacture more compact distributions of concentrations of dopant. Framework the paper we formulate conditions to increase compactness and at the same time to increase homogeneity of distribution of concentration of dopant in enriched by the dopant area. Fig. 1. Structure of a field-effect heterotransistor. Top side of the structure
- 2. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 2 2. METHOD OF SOLUTION To solve our aim we determine distribution of concentration of dopant C(x,y,z,t) in space and time. To determine the distribution we solve the following boundary problem ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, (1) ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = yLx y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLx z tzyxC , C(x,y,z,0)=fC(x,y,z). (2) Here T is the temperature of annealing; DС is the dopant diffusion coefficient. Dopant diffusion coefficient takes another value in other materials. Heating and cooling of heterostructure (see Arrhenius law) also leads to changing of value of diffusion coefficient. We consider following approximation of concentrational dependences of dopant diffusion coefficient [10-12] ( ) ( ) ( ) ( ) ( ) ( ) ++ += 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; function P (x,y,z,T) describes dependences of limit of solubility on coordinate and temperature; parameter γ ∈[1,3] is integer and depends on properties of materials; function V (x,y,z,t) describes distribution of concentration of radiation vacancies on space and time with equilibrium distribution V* . Dependence of dopant diffusion coefficient on concentration of do- pant has been investigated and described in details in [10]. Diffusion of dopant gives a possibility to dope materials without generation radiation defects. In this situation ζ1= ζ2= 0. Ion doping of dopant leads to generation radiation defects. Distributions of concentrations of point radiation defects have been determined by solving the following boundary problem [11,12] ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ 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 − ∂ ∂ ∂ ∂ +× ( ) 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)
- 3. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 3 Here ρ=I,V; distribution of concentration of radiation interstitials in space and time describes by the function I (x,y,z,t); terms of Eqs.(4) with quadric concentrations of point radiation defects (V2 (x,y,z,t) and I2 (x,y,z,t)) correspond to generation of simplest complexes of radiation defects (divacancies and diinterstitials); temperature and spatial dependences of diffusion coefficients of point radiation defects describe by functions Dρ(x,y,z,T); temperature and spatial dependences of parameter of recombination of point radiation defects describe by function kI,V(x,y,z,T); functions kI,I(x,y,z,T) and kV,V(x,y,z,T) describe spatial and temperature dependences of parameters of gener- ation of simplest complexes of point radiation defects. Concentrations of divacancies ΦV (x,y,z,t) and dinterstitials ΦI (x,y,z,t) have been calculated by solution of the following boundary problem [11,12] ( ) ( ) ( ) ( ) ( ) + 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 ∂ ∂ ∂ ∂ ( ) 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) Functions DΦρ(x,y,z,T) describe spatial and temperature dependences of diffusion coefficients of simplest complexes of point radiation defects; functions kI(x,y,z,T) and kV (x,y,z,T) describe spatial and temperature dependences of parameters of decay of the above complexes. Now let us consider equivalent integro-differential form of Eq.(1) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ++=∫ ∫ ∫ t y L z L L x L y L z Lzyx 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 zxzy x z y wyuC TwyuD LL zx LL zy d x wvxC TwvxP wvxC 0 ,,, ,,, ,,, ,,, ,,, 1 ∂ τ∂ τ ∂ τ∂τ ξ γ γ ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ×+ + ++× t x L y L L yx x y TzvuD LL yx d TzyxP wyuC V wyuV V wyuV 0 2* 2 2*1 ,,, ,,, ,,, 1 ,,,,,, 1 τ τ ξ τ ς τ ς γ γ ( ) ( ) ( ) ( ) ( ) ( ) ×+ + ++× zyx LLL zyx 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 Lx y z udvdwdwvuf ,, . (1a)
- 4. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 4 We used the Bubnov-Galerkin approach [13] to calculate solution of the above equation. To use the approach we consider 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 n χ/Lχ). Number of terms N of the above series is finite. The series is almost coincides with solution of Eq.(1) in the linear case (i.e. with ξ = 0) and average value of dopant diffusion coefficient D0. Framework the approach we substitute the above series into Eq.(1a). After the substitution we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ × ∑+−=∑ == t y L z L N n nCnnnnC N n nCnnn C y z TwvxP ewcvcxcatezsysxs n azyx 0 11 32 ,,, 1 γ γ ξ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ++× = zy N n nCnnnnCL LL zy dewcvcxsanTwvxD V wvxV V wvxV 1 2* 2 2*1 ,,, ,,,,,, 1 ττ τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ × ∑+ ++− = t x L z L N m mCmmmmC zx x z ewcycuca V wyuV V wyuV LL zx 0 1 2* 2 2*1 1 ,,,,,, 1 γ τ τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∑ × = t x L y L L yx N n nCnnnnCL x y TzvuD LL yx dewcysucanTwyuD TwyuP 01 ,,,,,, ,,, ττ ξ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ++ ∑+× = 2* 2 2*1 1 ,,,,,, 1 ,,, 1 V zvuV V zvuV ezcvcuca TzvuP N n nCnnnnC τ ς τ ςτ ξ γ γ ( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑× = x L y L z Lzyx N n nCnnnnC x y z udvdwdwvuf LLL zyx dezsvcucan ,, 1 ττ . (8) Here sn(χ)=sin(πnχ/Lχ). To determine coefficients an it is necessary to use orthogonality condi- tion of terms of the above series framework scale of heterostructure. Using the condition leads to the following equations to calculate of coefficients an for any quantity of terms N ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ × ∑+−=∑− == t L L L N n nCnnnnCL N n nC nCzyx x y z TzyxP ezcycxcaTzyxDte n aLLL 0 0 0 0 11 65 222 ,,, 1,,, γ γ ξ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ −+ ++× = N n n y nnn nCzy yc n L ysyycxs n a V zyxV V zyxVLL 1 2* 2 2*12 12 ,,,,,, 1 2 π τ ς τ ς π ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∑− −+× = t L L L N n nCnnnnCn z nnCn x y z ezcycxcadxdydzdzc n L zszezc 0 0 0 0 1 1 γ ττ π τ ( ) ( ) ( ) ( ) ( ) ( ) ++ ++ +× *12* 2 2*1 ,,, 1 ,,,,,, 1,,,1 ,,, V zyxV V zyxV V zyxV TzyxD TzyxP L τ ς τ ς τ ς ξ γ ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )∑ × −+ −+ + = N n nnnn z nn x n znC zcysxczc n L zszxc n L xsx n La V zyxV 1 22* 2 2 211 2 ,,, πππ τ ς ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ × ∑+−× = t L L L N n nCnnnnC yx nC x y z TzyxP ezcycxca LL dxdydzde 0 0 0 0 1 2 ,,, 1 2 γ γ ξ τ π ττ
- 5. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 5 ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ −+ ++× = N n n x nn nC L xc n L xsxxc n a V zyxV V zyxV TzyxD 1 *12* 2 2 1 ,,,,,, 1,,, π τ ς τ ς ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ × −++ −+× = N n L n x nnCn y nnn x xc n L xsxdxdydzdeyc n L ysyzsyc 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 . (9) Now we consider several examples. For γ=0 we obtain ( ) ( )[ ] ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ×+ −+ −+= x y zL L L x nn y nn y nnC n L xsxydzdzyxfzc n L zszyc n L ysya 0 0 0 ,,11 πππ ( )[ ]} ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){ ∫ ∫ ∫ ∫ + −+−× t L L L nn y nLnnnn x y z ysyzc n L zszTzyxDzcycxs n xdxc 0 0 0 0 1,,,2 2 1 π ( )[ ] ( ) ( ) ( ) ( ) ( ) × + ++ −+ xdydzdzc TzyxPV zyxV V zyxV yc n L nn y ,,, 1 ,,,,,, 11 2* 2 2*1 γ ξτ ς τ ς π ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ×∫ ∫ ∫ ∫ −+ −++× t L L L n y nnnn y nnnCnC x y z zc n L zszzcysxc n L xsxxcede 0 0 0 0 121 ππ τττ ( ) ( ) ( ) ( ) ( ) + ++ +× τ τ ς τ ς ξ γ dxdydzd V zyxV V zyxV TzyxP TzyxDL 2* 2 2*1 ,,,,,, 1 ,,, 1,,, ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×∫ ∫ ∫ ∫ −+ −++ t L L L Ln y nnn x nnnC x y z TzyxDyc n L ysyycxc n L xsxxce 0 0 0 0 ,,,11 ππ τ ( ) ( ) ( ) ( ) ( ) 1 65 222 *12* 2 2 ,,,,,, 1 ,,, 12 − − ++ +× n LLL dxdydzd V zyxV V zyxV TzyxP zs zzz n π τ τ ς τ ς ξ γ .(10) For γ=1 calculation of parameters an leads to the following results ( ) ( ) ( ) ( )∫ ∫ ∫+±−= x y zL L L nnnnn n n nC xdydzdzyxfzcycxca 0 0 0 2 ,,4 2 αβ α β . (11) Here ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ × ++= t L L L L nn zy n x y z V zyxV V zyxV TzyxP TzyxD ycxs n LL 0 0 0 0 2* 2 2*12 ,,,,,, 1 ,,, ,,, 2 2 τ ς τ ς π ξ α ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×+ −+ −+× n LL dexdydzdzc n L zszyc n L ysyzc zx nCn z nn y nn 2 2 11 π ξ ττ ππ ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ×∫ ∫ ∫ ∫ −− −+× t L L L n z n L nn x nnnC x y z zc n L zsz TzyxP TzyxD ysxc n L xsxxce 0 0 0 0 1 ,,, ,,, 21 ππ τ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫+ ++× t L L nnnC yx n x y ycxce n LL dxdydzd V zyxV V zyxV zc 0 0 0 22* 2 2*1 2 ,,,,,, 1 τ π ξ τ τ ς τ ς
- 6. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){∫ × −+ ++× zL nn x n L n ysxc n L xsx V zyxV V zyxV TzyxP TzyxD zs 0 2* 2 2*1 1 ,,,,,, 1 ,,, ,,, 2 π τ ς τ ς ( )[ ] τ π dxdydzdyc n L y n y −+× 1 , ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫= t L L L LnnnnC zy n x y z TzyxDzcycxse n LL 0 0 0 0 2 ,,,2 2 τ π β ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) × +− −+ ++× ysyyc n L zdzc n L zsz V zyxV V zyxV nn y n z n 11 ,,,,,, 1 2* 2 2*1 ππ τ ς τ ς ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ∫ ∫ ∫ ∫ ++ −++× t L L L nnn x nnnC zx x y z V zyxV zcysxc n L xsxxce n LL dxdyd 0 0 0 0 *12 ,,, 121 2 τ ς π τ π τ ( ) ( ) ( ) ( ) ( )[ ] ( )∫ ×+ −+ + t nC yx n z nL e n LL dxdydzdzc n L zszTzyxD V zyxV 0 22* 2 2 2 1,,, ,,, τ π τ π τ ς ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ∫ ∫ ∫ × ++ −+ −+× x y zL L L n y nn x n V zyxV V zyxV yc n L ysyxc n L xsx 0 0 0 2* 2 2*1 ,,,,,, 111 τ ς τ ς ππ ( ) ( ) ( ) ( ) ( ) 65222 ,,,2 nteLLLdxdxcydyczdTzyxDzs nCzyxnnLn πτ −× . It could be used the same approach to calculate values of parameters an for larger values of the parameterγ. However the relations became more bulky and will not be present in the paper. The considered approach gives a possibility to calculate distributions of concentrations of dopant and radiation defects without joining of the above concentration on interfaces of the considered hete- rostructure. We solved equations of the system (4) by using Bubnov-Galerkin approach. To use the approach we previously transform the differential equations to the following integro-differential form ( ) ( ) ( ) ×+∫ ∫ ∫ ∂ ∂ =∫ ∫ ∫ zx t y L z L I zy x L y L z Lzyx LL zx dvdwd x wvxI TwvxD LL zy udvdwdtwvuI LLL zyx y zx y z 0 ,,, ,,,,,, τ τ ( ) ( ) ( ) ( ) −∫ ∫ ∫ ∂ ∂ +∫ ∫ ∫ ∂ ∂ × t x L y L I yx t x L z L I x yx z dudvd z zvuI TzvuD LL yx dudwd x wyuI TwyuD 00 ,,, ,,, ,,, ,,, τ τ τ τ ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫− x L y L z L II zyx x L y L z L VI zyx x y zx y z Twvuk LLL zyx udvdwdtwvuVtwvuITwvuk LLL zyx ,,,,,,,,,,,, ,, ( ) ( )∫ ∫ ∫+× x L y L z L I zyx x y z udvdwdwvuf LLL zyx udvdwdtwvuI ,,,,,2 (4a) ( ) ( ) ( ) ×+∫ ∫ ∫ ∂ ∂ =∫ ∫ ∫ zx t y L z L V zy x L y L z Lzyx LL zx dvdwd x wvxV TwvxD LL zy udvdwdtwvuV LLL zyx y zx y z 0 ,,, ,,,,,, τ τ ( ) ( ) ( ) ( ) −∫ ∫ ∫ ∂ ∂ +∫ ∫ ∫ ∂ ∂ × t x L y L V yx t x L z L V x yx z dudvd z zvuV TzvuD LL yx dudwd x wyuV TwyuD 00 ,,, ,,, ,,, ,,, τ τ τ τ ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫− x L y L z L VV zyx x L y L z L VI zyx x y zx y z Twvuk LLL zyx udvdwdtwvuVtwvuITwvuk LLL zyx ,,,,,,,,,,,, ,, ( ) ( )∫ ∫ ∫+× x L y L z L V zyx x y z udvdwdwvuf LLL zyx udvdwdtwvuV ,,,,,2 . Now we consider solutions of the above integro-differential equations as the following series
- 7. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 7 ( ) ( ) ( ) ( ) ( )∑= = N n nnnnn tezcycxcatzyx 1 0 ,,, ρρρ . Here anρ are coefficients, which should be determined. After substitution of the series into Eqs. (4a) one can obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑ == N n t y L z L InnnInI zyx N n nInnn nI y z dvdwdTwvxDzcycea LLL zy tezsysxs n azyx 1 01 33 ,,, ττ π π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑−∑ ∫ ∫ ∫−× == N n nnI zyx N n t x L z L InnnInnI zyx n zsa LLL yx dudwdTwyuDzcxceysa LLL zx xs x z 11 0 ,,, π ττ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑−∫ ∫ ∫× = x L y L z L N n nInnnnI zyx t x L y L InnnI x y zx y tewcvcuca LLL zyx dudvdTzvuDycxce 2 10 ,,, ττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑ ∑−× = = x L y L z L N n N n nnnnVnInnnnI zyx II x y z wcvcucatewcvcuca LLL zyx udvdwdTvvuk 1 1 , ,,, ( ) ( ) ( )∫ ∫ ∫+× x L y L z L I zyx VInV x y z udvdwdwvuf LLL zyx udvdwdTvvukte ,,,,,, (12) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑ == N n t y L z L VnnnVnV zyx N n nVnnn nV y z dvdwdTwvxDzcycea LLL zy tezsysxs n azyx 1 01 33 ,,, ττ π π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∑ ∫ ∫ ∫−× == N n nnV zyx N n t x L z L VnnnVnnV zyx n zsa LLL yx dudwdTwyuDzcxceysa LLL zx xs x z 11 0 ,,, π ττ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑−∫ ∫ ∫× = x L y L z L N n nVnnnnV zyx t x L y L VnnnV x y zx y tewcvcuca LLL zyx dudvdTzvuDycxce 2 10 ,,, ττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑ ∑−× = = x L y L z L N n N n nnnnVnInnnnI zyx VV x y z wcvcucatewcvcuca LLL zyx udvdwdTvvuk 1 1 , ,,, ( ) ( ) ( )∫ ∫ ∫+× x L y L z L V zyx VInV x y z udvdwdwvuf LLL zyx udvdwdTvvukte ,,,,,, . We use orthogonality condition of functions in the above series on scale of the heterostructure to calculate coefficients anρ. Using the condition gives a possibility to obtain equations for calcula- tion the above coefficients for any quantity N of terms of considered series ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− == N n t L L n y nynnI nI x N n nI nIzyx x y yc n L ysyLxce n a L te n aLLL 1 0 0 0 2 1 65 222 12 2 221 2 1 π τ ππ ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ++−−∫ −+× = N n t L xn xnI y L n z nI xz Lxc n L n a L dxdydzdzc n L zszTzyxD 1 0 0 2 0 12 2 1 1 2 ,,, ππ τ π ( )} ( ) ( ) ( )[ ] ( )[ ] ( ) −∫ ∫ − −+++ y z L L nInn z nzIn dexdydyczdzc n L zszLTzyxDxsx 0 0 2112 2 2,,,2 ττ π ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = N n t L L n y nyn x nxnI nI z x y yc n L ysyLxc n L xsxLe n a L 1 0 0 0 2 12 2 212 2 2 2 1 ππ τ π
- 8. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 8 ( )[ ] ( ) ( ) ( )[ ] ( ) ×∑ ∫ +−+−∫ −× = N n L nn x xnInI L In xz xsxxc n L LteadxdydzdTzyxDzc 1 0 2 0 212 2 2,,,21 π τ ( ) ( )[ ] ( ) ( )[ ] ( ) −∫ ∫ +−+ −++× y z L L nn z zIIn y ny xdydzdzszzc n L LTzyxkyc n L ysyL 0 0 , 212 2 ,,,12 2 2 ππ ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ −++ −++− = N n L L n y nyn x nxnVnInVnI x y yc n L ysyLxc n L xsxLteteaa 1 0 0 12 2 212 2 2 ππ ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ −++∫ −++× = N n L n x n L n z nzVI xz xc n L xsxxdydzdzc n L zszLTzyxk 1 00 , 112 2 2,,, ππ ( ) ( )[ ] ( ) ( ) ( )[ ]∫ ∫ −++ −+× y z L L n z nzIn y n xdydzdzc n L zszLTzyxfyc n L ysy 0 0 12 2 2,,,1 ππ (13) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− == N n t L L n y nynnV nV x N n nV nVzyx x y yc n L ysyLxce n a L te n aLLL 1 0 0 0 2 1 65 222 12 2 221 2 1 π τ ππ ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ++−−∫ −+× = N n t L xn xnV y L n z nV xz Lxc n L n a L dxdydzdzc n L zszTzyxD 1 0 0 2 0 12 2 1 1 2 ,,, ππ τ π ( )} ( ) ( ) ( )[ ] ( )[ ] ( ) −∫ ∫ − −+++ y z L L nVnn z nzVn dexdydyczdzc n L zszLTzyxDxsx 0 0 2112 2 2,,,2 ττ π ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = N n t L L n y nyn x nxnV nV z x y yc n L ysyLxc n L xsxLe n a L 1 0 0 0 2 12 2 212 2 2 2 1 ππ τ π ( )[ ] ( ) ( ) ( )[ ] ( ) ×∑ ∫ +−+−∫ −× = N n L nn x xnVnV L Vn xz xsxxc n L LteadxdydzdTzyxDzc 1 0 2 0 212 2 2,,,21 π τ ( ) ( )[ ] ( ) ( )[ ] ( ) −∫ ∫ +−+ −++× y z L L nn z zVVn y ny xdydzdzszzc n L LTzyxkyc n L ysyL 0 0 , 212 2 ,,,12 2 2 ππ ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ −++ −++− = N n L L n y nyn x nxnVnInVnI x y yc n L ysyLxc n L xsxLteteaa 1 0 0 12 2 212 2 2 ππ ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ −++∫ −++× = N n L n x n L n z nzVI xz xc n L xsxxdydzdzc n L zszLTzyxk 1 00 , 112 2 2,,, ππ ( ) ( )[ ] ( ) ( ) ( )[ ]∫ ∫ −++ −+× y z L L n z nzVn y n xdydzdzc n L zszLTzyxfyc n L ysy 0 0 12 2 2,,,1 ππ . Final relations for the above parameters takes the form ( ) − +− + ± + −= 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 . (14) Here ( ) ( ) ( ) ( )[ ] ( )∫ ∫ ∫ ×++ −++= x y zL L L y ynn x nxnn n L Lysyxc n L xsxLTzyxkte 0 0 0 , 2 212 2 2,,,2 ππ γ ρρρρ
- 9. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 9 ( )[ ]} ( ) ( )[ ] xdydzdzc n L zszLyc n z nzn −++−× 12 2 212 π , ( ) ( ){∫ ∫ ∫ += t L L nn x n x y ysye nL 0 0 0 2 2 1 τ π δ ρρ ( )[ ] ( ) ( )[ ] ( ) ( )[ ] ×+∫ − −+ −+ y L nn z nn y L dxdxcydzdTzyxDzc n L zszyc n L z π τ ππ ρ 2 1 21,,,1 2 1 2 0 ( ) ( ) ( )[ ] ( )[ ] ( ) ( )[ ]∫ ∫ ∫ ∫ +−− −++× t L L L n z nn x nxn x y z zc n L TzyxDycxc n L xsxLe n 0 0 0 0 2 12 2 ,,,21122 2 1 ππ τ ρρ ( ) } ( ) ( ) ( )[ ] ( ){∫ ∫ ∫ + −+++++ t L L nn x xnn z zn x y ysyxc n L Lxsxe nL dxdydzdLzsz 0 0 0 2 122 2 1 2 π τ π τ ρ ( )[ ] ( )[ ] ( ) ( )te n LLL dxdydzdTzyxDzcyc n L L n zyx L nn y y z ρρ π τ π 65 222 0 ,,,211 2 −∫ − −++ , ( ) ×= tenInIVχ ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( ){∫ ∫ ∫ ++ +−+ −+× x y zL L L nzVInn y yn x nnV zszLTzyxkysyyc n L Lxc n L xsxte 0 0 0 , 2,,,212 2 1 ππ ( )[ ] xdydzdzc n L n z −+ 12 2π , ( ) ( )[ ] ( ) ( )[ ] {∫ ∫ ∫ × −+ −+= x y zL L L n y nn x nn zyc n L ysyxc n L xsx 0 0 0 11 ππ λ ρ ( ) ( )[ ] ( ) xdydzdTzyxfzc n L zs n z n ,,,1 ρ π −+× , 22 4 nInInInVb χγγγ −= , −−= 2 3 2 nInInInInVb χδδγγ nInInV γχδ− , ( ) 22 2 2 nInInVnInInVnInVnInInVb χλλδχδγγλδγ −+−+= , nInInVnInVnIb λχδδγλ −= 21 , 2 2 3 48 bbyA −+= , 2 4 2 342 9 3 b bbb p − = , 3 4 2 4132 3 3 54 2792 b bbbbb q +− = , −++−−+= 3 323 32 qpqqpqy 43 3bb− . Now we will calculate distributions of concentrations of simplest complexes of radiation defects as the following functional series ( ) ( ) ( ) ( ) ( )∑=Φ = Φ N n nnnnn tezcycxcatzyx 1 0 ,,, ρρρ (15) with not yet known coefficients anΦρ. To calculate these coefficient we transform the Eqs.(6) to the following integro-differential form ( ) ( ) ( ) +∫ ∫ ∫ Φ =∫ ∫ ∫ Φ Φ t y L z L I I zy x L y L z L I zyx y zx y z dvdwd x wvx TwvxD LL zy udvdwdtwvu LLL zyx 0 ,,, ,,,,,, τ ∂ τ∂ ( ) ( ) ( ) ( ) ∫ ∫ ∫ × Φ +∫ ∫ ∫ Φ + ΦΦ t x L y L I I t x L z L I I zx x yx z dudvd z zvu TzvuDdudwd y wyu TwyuD LL zx 00 ,,, ,,, ,,, ,,, τ ∂ τ∂ τ ∂ τ∂ ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫+× x L y L z L I zyx x L y L z L II zyxyx x y zx y z Twvuk LLL zyx udvdwdwvuITwvuk LLL zyx LL yx ,,,,,,,,, 2 , τ ( ) ( )∫ ∫ ∫+× Φ x L y L z L I zyx x y z udvdwdwvuf LLL zyx udvdwdwvuI ,,,,, τ (6a) ( ) ( ) ( ) +∫ ∫ ∫ Φ =∫ ∫ ∫ Φ Φ t y L z L V V zy x L y L z L V zyx y zx y z dvdwd x wvx TwvxD LL zy udvdwdtwvu LLL zyx 0 ,,, ,,,,,, τ ∂ τ∂ ( ) ( ) ( ) ( ) ∫ ∫ ∫ × Φ +∫ ∫ ∫ Φ + ΦΦ t x L y L V V t x L z L V V zx x yx z dudvd z zvu TzvuDdudwd y wyu TwyuD LL zx 00 ,,, ,,, ,,, ,,, τ ∂ τ∂ τ ∂ τ∂
- 10. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 10 ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫+× x L y L z L V zyx x L y L z L VV zyxyx x y zx y z Twvuk LLL zyx udvdwdwvuVTwvuk LLL zyx LL yx ,,,,,,,,, 2 , τ ( ) ( )∫ ∫ ∫+× Φ x L y L z L V zyx x y z udvdwdwvuf LLL zyx udvdwdwvuV ,,,,, τ . Further we substitute the previously considered series in the Eqs.(6a). In this situation we obtain the following equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−=∑− = ΦΦ = Φ N n t y L z L InnInIn zyx N n nInnn In y z TwvxDvctexsan LLL zy tezsysxs n a zyx 1 01 33 ,,, π π ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−× = ΦΦΦ N n t x L z L InnInnIn zyx n x z dudwdTwvuDwcucteysna LLL zx dvdwdwc 1 0 ,,, τ π τ ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫+∑ ∫ ∫ ∫− = ΦΦΦ x L y L z L II N n t x L y L InnInnIn zyx x y zx y TwvukdudvdTzvuDvcuctezsan LLL yx ,,,,,, , 1 0 τ π ( ) ( ) ( ) ×+∫ ∫ ∫−× zyx x L y L z L I zyxzyx LLL zyx udvdwdwvuITwvuk LLL zyx LLL zyx udvdwdwvuI x y z ττ ,,,,,,,,,2 ( )∫ ∫ ∫× Φ x L y L z L I x y z udvdwdwvuf ,, (16) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑− = ΦΦ = Φ N n t y L z L VnnVnVn zyx N n nVnnn Vn y z TwvxDvctexsan LLL zy tezsysxs n a zyx 1 01 33 ,,, π π ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−× = ΦΦΦ N n t x L z L VnnVnnVn zyx n x z dudwdTwvuDwcucteysna LLL zx dvdwdwc 1 0 ,,, τ π τ ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫+∑ ∫ ∫ ∫− = ΦΦΦ x L y L z L VV N n t x L y L VnnVnnVn zyx x y zx y TwvukdudvdTzvuDvcuctezsan LLL yx ,,,,,, , 1 0 τ π ( ) ( ) ( ) ×+∫ ∫ ∫−× zyx x L y L z L V zyxzyx LLL zyx udvdwdwvuVTwvuk LLL zyx LLL zyx udvdwdwvuV x y z ττ ,,,,,,,,,2 ( )∫ ∫ ∫× Φ x L y L z L V x y z udvdwdwvuf ,, . We use orthogonality condition of functions in the above series on scale of the heterostructure to calculate coefficients anΦρ. Using the condition gives a possibility to obtain equations for calcula- tion the above coefficients for any quantity N of terms of considered series ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− = Φ = Φ Φ N n t L L n y nyn In x N n In Inzyx 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 ( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ ++−∫ −+× = Φ ΦΦ N n t L xn y In L Inn z nI xz Lxsx Ln a dexdydzdzc n L zszTzyxD 1 0 0 2 0 2 2 1 1 2 ,,, π ττ π ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×−∫ ∫ −+− − + ΦΦ x L L Inn z nIn n x L dexdydzdzc n L zszTzyxDyc n xc L y z π ττ ππ 1 1 2 ,,,21 2 12 0 0
- 11. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 11 ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∑ ∫ ∫ ∫ ∫ +−+ −+× = ΦΦ Φ N n t L L L Iyn y nn x nIn In x y z TzyxDLyc n L ysyxc n L xsxe n a 1 0 0 0 0 2 ,,,12 2 21 22 ππ τ ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ×∑∫ ∫ ∫ −+ +−+−× = Φ N n t L L n y nnn x Inn x y yc n L ysyxsxxc n L edxdydzdyc 10 0 0 1 2 1 2 21 ππ ττ ( ) ( ) ( )[ ] ( ) ( ) ( ){∑∫ ∫ +−∫ +−× = Φ Φ N n t L nIn L nn z II In xz xsxexdydzdzszzc n L TzyxktzyxI n a 10 00 , 2 33 1 2 ,,,,,, τ ππ ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫ +− +− −+ Φ y z L L n z Inn yIn n x zc n L tzyxITzyxkysyyc n L n a xc n L 0 0 33 1 2 ,,,,,,1 2 1 2 ππππ ( )} ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −+ −+++ = Φ Φ N n t L L n y nn x nIn In n x y yc n L ysyxc n L xsxe n a xdydzdzsz 1 0 0 0 33 1 2 1 2 ππ τ π ( )[ ] ( ) ( )∫ +−× Φ zL Inn z xdydzdzyxfzszzc n L 0 ,,1 2π (17) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− = Φ = Φ Φ N n t L L n y nyn Vn x N n Vn Vnzyx 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 ( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ ++−∫ −+× = Φ ΦΦ N n t L xn y Vn L Vnn z nV xz Lxsx Ln a dexdydzdzc n L zszTzyxD 1 0 0 2 0 2 2 1 1 2 ,,, π ττ π ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×−∫ ∫ −+− − + ΦΦ x L L Vnn z nVn n x L dexdydzdzc n L zszTzyxDyc n xc L y z π ττ ππ 1 1 2 ,,,21 2 12 0 0 ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∑ ∫ ∫ ∫ ∫ +−+ −+× = ΦΦ Φ N n t L L L Vyn y nn x nVn Vn x y z TzyxDLyc n L ysyxc n L xsxe n a 1 0 0 0 0 2 ,,,12 2 21 22 ππ τ ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ×∑∫ ∫ ∫ −+ +−+−× = Φ N n t L L n y nnn x Vnn x y yc n L ysyxsxxc n L edxdydzdyc 10 0 0 1 2 1 2 21 ππ ττ ( ) ( ) ( )[ ] ( ) ( ) ( ){∑∫ ∫ +−∫ +−× = Φ Φ N n t L nVn L nn z VV Vn xz xsxexdydzdzszzc n L TzyxktzyxV n a 10 00 , 2 33 1 2 ,,,,,, τ ππ ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫ +− +− −+ Φ y z L L n z Vnn yVn n x zc n L tzyxVTzyxkysyyc n L n a xc n L 0 0 33 1 2 ,,,,,,1 2 1 2 ππππ ( )} ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −+ −+++ = Φ Φ N n t L L n y nn x nVn Vn n x y yc n L ysyxc n L xsxe n a xdydzdzsz 1 0 0 0 33 1 2 1 2 ππ τ π ( )[ ] ( ) ( )∫ +−× Φ zL Vnn z xdydzdzyxfzszzc n L 0 ,,1 2π .
- 12. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 12 Fig.2a. Spatial distributions of concentration of dopant after infusion and annealing with the same anneal- ing time before and after interface between layers of heterostructure. Curve 1 is the distribution of concen- tration of dopant in homogenous sample with averaged dopant diffusion coefficient D0. Curves 2-6 are the distribution of concentration of dopant in heterostructure with increasing difference between values of dif- fusion coefficient. Value of dopant diffusion coefficient in the substrate is smaller, than in epitaxial the layer 3. DISCUSSION In this section analysis of distributions of concentration of dopant, infused (see Fig. 2a) or im- planted (see Fig. 2b) into epitaxial layer have been done. Annealing time is the same for all curves of these figures. One can find from these figures, that absolute value of gradient of con- centration of dopant increases due to presents an interface between layers of heterostructure. In this situation dimensions of the considered transistors decreases. At the same time homogeneity of concentration of dopant in enriched area increases. 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 concentration of dopant after implantation and annealing with the same an- nealing time before and after interface between layers of heterostructure. Annealing time for curves 1 and 3 is equal to Θ = 0.0048(Lx 2 +Ly 2 +Lz 2 )/D0. Annealing time for curves 2 and 4 is equal to Θ = 0.0057(Lx 2 +Ly 2 +Lz 2 )/D0. Curves 1 and 2 are the distributions of concentration of dopant in homogenous sample. Curves 3 and 4 are distributions of concentration of dopant in heterostructure under condition, when value of dopant diffusion coefficient in the substrate is smaller, than in epitaxial the layer
- 13. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 13 To estimate optimal annealing time we estimate decreasing of absolute value of gradient of con- centration of dopant near interface between layers of the heterostructure with increasing of an- nealing time. Decreasing of annealing time leads to increasing inhomogeneity of distribution of concentration of dopant. Estimation of the compromise value of annealing time has been done by using recently introduced criterion [14,15]. To use the criterion we approximate real spatial dis- tribution of concentration of dopant by idealized step-wise function ψ(x,y,z). After that we esti- mate the required optimal annealing time by minimization of mean-square error [16-20] ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (18) Minimization of the above mean-squared error leads to dependences of optimal annealing time on parameters, which are presented on Figs. 3. It should be noted, that radiation defects, generated during ion implantation, should be annealed. After ideal optimization of annealing time the im- planted dopant should achieve the interface between layers of heterostructure. If annealing time is smaller, it is attracted an interest to make additional annealing to achieve the interface. The Fig. 3b shows dependences of additional annealing time. The figures shows, that optimal annealing time of implanted dopant is smaller in comparison with optimal annealing time of infused dopant. If the considered heterostructure have been doped by diffusion, any radiation damage of materials of layers is absent. On the other hand radiation processing of materials of heterostructure (including of ion implantation) leads to decreasing of mismatch-induced stress in the processed heterostructure [20]. 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. 3a. Dependences of optimal annealing time of infused dopant on several parameters. Dependence of the optimal annealing time on normalized thickness of the epitaxial layer a/L describes by curve 1 for ξ=γ= 0 for equal to each other values of dopant diffusion coefficient in both parts of heterostructure. Curve 2 describes dependence of the optimal annealing time on the parameter ε for a/L=1/2 and ξ= γ = 0. Curve 3 describes dependence of the optimal annealing time on the parameter ξ for a/L=1/2 and ε= γ = 0. Curve 4 describes dependence of the optimal annealing time on parameter γ for a/L=1/2 and ε=ξ=0
- 14. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 14 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.3b. Dependences of optimal annealing time of implanted dopant on several parameters. Dependence of the optimal annealing time on normalized thickness of the epitaxial layer a/L describes by curve 1 for ξ=γ= 0 for equal to each other values of dopant diffusion coefficient in both parts of heterostructure. Curve 2 describes dependence of the optimal annealing time on the parameter ε for a/L=1/2 and ξ= γ = 0. Curve 3 describes dependence of the optimal annealing time on the parameter ξ for a/L=1/2 and ε= γ = 0. Curve 4 describes dependence of the optimal annealing time on parameter γ for a/L=1/2 and ε=ξ=0 4. CONCLUSIONS In this paper we formulate several recommendations to optimize manufacture heterotransistor with several source based on prognosis of time varying of spatial distributions of concentrations of infused and implanted dopants in specific heterostructure. We also introduce analytical ap- proach to prognosis diffusion and ion types of doping with account variation in space and time parameters of technological parameters and nonlinearity of mass and heat transport. 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 and educational fellowship for scientific research of Government of Nizhny Novgorod region of Rus- sia. REFERENCES [1] Z. Ramezani, A.A. Orouji. A silicon-on-insulator metal semiconductor field-effect transistor with an L- shaped buried oxide for high output-power density. Mat. Sci. Sem. Proc. Vol. 19. P. 124-129 (2014). [2] Ch. Dong, J. Shi, J. Wu, Y. Chen, D. Zhou, Z. Hu, H. Xie, R. Zhan, Zh. Zou. Improvements in passiva- tion effect of amorphous InGaZnO thin film transistors. Mat. Sci. Sem. Proc. Vol. 20. P. 7-11 (2014). [3] V. Papageorgiou, A. Khalid, C. Li, M.J. Steer, D.R.S. Cumming. Integration techniques of pHEMTs and planar Gunn diodes on GaAs substrates. Solid-State Electronics. Vol. 102. P. 87-92 (2014). [4] R. Grassi, A. Gnudi, V. Di Lecce, E. Gnani, S. Reggiani, G. Baccarani. Boosting the voltage gain of graphene FETs through a differential amplifier scheme with positive feedback. Solid-State Electronics. Vol. 100. P. 54-60 (2014). [5] 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: Specific features of charge transport in rectifying and nonrectifying contacts. Semiconductors. Vol. 43 (7). P. 897-903 (2009). [6] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990).
- 15. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016 15 [7] T.T.A. Tuan, D.-H. Kuo, C.-C. Li, G.-Z. Li. Effect of temperature dependence on electrical characteri- zation of p-n GaN diode fabricated by RF magnetron sputtering. Mat. Sci. Appl. Vol. 6. P. 809-817 (2015). [8] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). [9] R. Pal, R. Pandey, N. Pandey, R.Ch. Tiwari. Single CDBA Based Voltage Mode Bistable Multivibrator and Its Applications. Circuits and Systems. Vol. 6 (11). P. 237-251 (2015). [10] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [11] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [12] P.M. Fahey, P.B. Griffin, J.D. Plummer. Point defects and dopant diffusion in silicon. Rev. Mod. Phys. Vol. 61 (2). P. 289-388 (1989). [13] M.L. Krasnov, A.I. Kiselev, G.I. Makarenko. Integral equations ("Science", Moscow, 1976). [14] E.L. Pankratov. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusion- coefficient nonuniformity. Russian Microelectronics. Vol. 36 (1). P. 33-39 (2007). [15] E.L. Pankratov. Redistribution of dopant during annealing of radiative defects in a multilayer structure by laser scans for production an implanted-junction rectifiers. Int. J. Nanoscience. Vol. 7 (4-5). P. 187– 197 (2008). [16] 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). [17] E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to decrease surface of chip. International Journal of Modern Physics B. Vol. 29 (5). P. 1550023-1-1550023-12 (2015). [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. Increasing of sharpness of diffusion-junction heterorectifier by using radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1250035 (2012). [20] E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor heterostructure by radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101 (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 155 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 103 published papers in area of her researches.