![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016
DOI:10.5121/msej.2016.3203 23
DEPENDENCE OF CHARGE CARRIERS MOBILITY IN THE
P-N-HETEROJUNCTIONS ON COMPOSITION OF MULTI-
LAYER STRUCTURE
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 manufacturing a p-n-junctions by dopant diffusion or ion implantation into a
multilayer structure. We introduce an approach to increase sharpness of these p-n-junctions and at the
same time to increase homogeneity of distributions of dopants in enriched by these dopants areas. We con-
sider influence of the above changing of distribution of dopant on charge carrier mobility. We also consid-
er an approach to decrease value of mismatch-induced stress in the considered multilayer structure by us-
ing a buffer layer. The decreasing gives a possibility to increase value of charge carrier mobility.
KEYWORDS:
Multilayer structure; charge carriers mobility; p-n-heterojunctions
1. INTRODUCTION
In the present time one of the actual questions is increasing of performance of solid state electron-
ic devices (diodes, field-effect and bipolar transistors, ...) [1-6]. To increase the performance it is
attracted an interest searching of materials with higher values of the charge carrier mobility [7-
10]. An alternative approach to increase the performance is development of new technological
processes and optimization of existing one. Framework this paper we introduce an approach of
manufacturing of heterodiodes. The approach gives a possibility to decrease of switching time. At
the same using the approach gives a possibility to decrease dimensions of these diodes. With de-
creasing of switching time and dimensions of these diodes value of mismatch- induced stress also
decreases.
In this paper we consider a heterostructure on Fig. 1. The heterostructure includes into itself a
substrate and an epitaxial layer. A buffer layer has been grown between these substrate and epi-
taxial layer. We consider doping of the epitaxial layer by diffusion or ion implantation after man-
ufacturing the heterostructure. The doping gives a possibility to produce required type of conduc-
tivity (p or n). We assume, that type of conductivity of substrate is known: p or n. After doping of
the epitaxial layer optimized annealing of dopant and/or radiation defects has been considered. In
this paper we analyzed possibility to increase sharpness of p-n- junction with increasing of homo-
geneity of concentration of dopant in enriched by the dopant area. Framework the paper we ana-
lyzed influence of changing of distribution of concentration of dopant in space and time on charge
carrier mobility with account mismatch-induced stress in the considered heterostructure.](https://image.slidesharecdn.com/msej3-160708040235/85/Dependence-of-Charge-Carriers-Mobility-in-the-P-N-Heterojunctions-on-Composition-of-Multilayer-Structure-1-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016
24
2. METHOD OF SOLUTION
To analyze influence of changing of distribution of concentration of dopant in space and time on
charge carrier mobility we solve the second Fick's law in the following form [1,11-15]
( ) ( ) ( ) ( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
(1)
( ) ( ) ( ) ( )
∫∇
∂
∂
Ω+
∫∇
∂
∂
Ω+
zz L
S
S
L
S
S
WdtWyxCtzyx
Tk
D
y
WdtWyxCtzyx
Tk
D
x 0
1
0
1 ,,,,,,,,,,,, µµ .
Fig. 1. Heterostructure, which consist of a substrate, epitaxial layers and buffer layer
Boundary and initial conditions for our case could be written as
( ) 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).
The function C(x,y,z,t) describes distribution of concentration of dopant in space and time. Ω is
the atomic volume of dopant; ∇s is the symbol of surficial gradient. The function ( )∫
zL
zdtzyxC
0
,,,
describes distribution of surficial concentration of dopant in space and time on interface between
layers of heterostructure (in this situation we assume, that Z-axis is perpendicular to interface be-
tween layers of heterostructure). The function µ1(x,y,z,t) describes the chemical potential due to
the presence of mismatch-induced stress in space and time. The functions D and DS describe dis-
tributions of coefficients of volumetric and surficial diffusions on coordinate and temperature.
Values of dopant diffusions coefficients depends on properties of materials of heterostructure,
speed of heating and cooling of materials during annealing and spatio-temporal distribution of
concentration of dopant. Dependences of dopant diffusions coefficients on parameters could be
approximated by the following relations [13-15]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Dependence of Charge Carriers Mobility in the P-N-Heterojunctions on Composition of Multilayer Structure
- 1. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 DOI:10.5121/msej.2016.3203 23 DEPENDENCE OF CHARGE CARRIERS MOBILITY IN THE P-N-HETEROJUNCTIONS ON COMPOSITION OF MULTI- LAYER STRUCTURE 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 manufacturing a p-n-junctions by dopant diffusion or ion implantation into a multilayer structure. We introduce an approach to increase sharpness of these p-n-junctions and at the same time to increase homogeneity of distributions of dopants in enriched by these dopants areas. We con- sider influence of the above changing of distribution of dopant on charge carrier mobility. We also consid- er an approach to decrease value of mismatch-induced stress in the considered multilayer structure by us- ing a buffer layer. The decreasing gives a possibility to increase value of charge carrier mobility. KEYWORDS: Multilayer structure; charge carriers mobility; p-n-heterojunctions 1. INTRODUCTION In the present time one of the actual questions is increasing of performance of solid state electron- ic devices (diodes, field-effect and bipolar transistors, ...) [1-6]. To increase the performance it is attracted an interest searching of materials with higher values of the charge carrier mobility [7- 10]. An alternative approach to increase the performance is development of new technological processes and optimization of existing one. Framework this paper we introduce an approach of manufacturing of heterodiodes. The approach gives a possibility to decrease of switching time. At the same using the approach gives a possibility to decrease dimensions of these diodes. With de- creasing of switching time and dimensions of these diodes value of mismatch- induced stress also decreases. In this paper we consider a heterostructure on Fig. 1. The heterostructure includes into itself a substrate and an epitaxial layer. A buffer layer has been grown between these substrate and epi- taxial layer. We consider doping of the epitaxial layer by diffusion or ion implantation after man- ufacturing the heterostructure. The doping gives a possibility to produce required type of conduc- tivity (p or n). We assume, that type of conductivity of substrate is known: p or n. After doping of the epitaxial layer optimized annealing of dopant and/or radiation defects has been considered. In this paper we analyzed possibility to increase sharpness of p-n- junction with increasing of homo- geneity of concentration of dopant in enriched by the dopant area. Framework the paper we ana- lyzed influence of changing of distribution of concentration of dopant in space and time on charge carrier mobility with account mismatch-induced stress in the considered heterostructure.
- 2. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 24 2. METHOD OF SOLUTION To analyze influence of changing of distribution of concentration of dopant in space and time on charge carrier mobility we solve the second Fick's law in the following form [1,11-15] ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, (1) ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ zz L S S L S S WdtWyxCtzyx Tk D y WdtWyxCtzyx Tk D x 0 1 0 1 ,,,,,,,,,,,, µµ . Fig. 1. Heterostructure, which consist of a substrate, epitaxial layers and buffer layer Boundary and initial conditions for our case could be written as ( ) 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). The function C(x,y,z,t) describes distribution of concentration of dopant in space and time. Ω is the atomic volume of dopant; ∇s is the symbol of surficial gradient. The function ( )∫ zL zdtzyxC 0 ,,, describes distribution of surficial concentration of dopant in space and time on interface between layers of heterostructure (in this situation we assume, that Z-axis is perpendicular to interface be- tween layers of heterostructure). The function µ1(x,y,z,t) describes the chemical potential due to the presence of mismatch-induced stress in space and time. The functions D and DS describe dis- tributions of coefficients of volumetric and surficial diffusions on coordinate and temperature. Values of dopant diffusions coefficients depends on properties of materials of heterostructure, speed of heating and cooling of materials during annealing and spatio-temporal distribution of concentration of dopant. Dependences of dopant diffusions coefficients on parameters could be approximated by the following relations [13-15]
- 3. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 25 ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ , ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD SLSS ςςξ γ γ . (2) Here DL (x,y,z,T) and DLS (x,y,z,T) are the spatial (due to accounting all layers of heterostruicture) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing. Function P (x,y,z,T) describes spatial and temperature dependence of the limit of solubility of dopant. Parameter γ ∈[1,3] with integer values depends on properties of materials [13]. The function V (x,y,z,t) describes distribution of concentration of radiation vacan- cies in space and time; V* is the equilibrium distribution of vacancies. One can find description of dependence of dopant diffusion coefficient on concentration of dopant in [13]. To determine dis- tributions of concentration of point radiation defects in space and time we solve the following boundary problem [11,12,14,15] ( ) ( ) ( ) ( ) ( ) ( ) ×− + = Tzyxk y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI IIII ,,, ,,, ,,, ,,, ,,, ,,, , ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( ) +− +× tzyxVtzyxITzyxk z tzyxI TzyxD z tzyxI VII ,,,,,,,,, ,,, ,,,,,, , 2 ∂ ∂ ∂ ∂ (3) ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ zL S IS zL S IS WdtWyxItzyx Tk D y WdtWyxItzyx Tk D x 00 ,,,,,,,,,,,, µµ ( ) ( ) ( ) ( ) ( ) ( ) ×− + = Tzyxk y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VVVV ,,, ,,, ,,, ,,, ,,, ,,, , ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( ) +− +× tzyxVtzyxITzyxk z tzyxV TzyxD z tzyxV VIV ,,,,,,,,, ,,, ,,,,,, , 2 ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ zL S VS zL S VS WdtWyxVtzyx Tk D y WdtWyxVtzyx Tk D x 00 ,,,,,,,,,,,, µµ ( ) 0 ,,, 0 = =x x tzyxI ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxI ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =z z tzyxI ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =x x tzyxV ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxV ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =z z tzyxV ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxV ∂ ∂ I(x,y,z,0)=fI (x,y,z), V(x,y,z,0)=fV(x,y,z). (4) The function I(x,y,z,t) describes distribution of concentration of radiation interstitials in space and time with the equilibrium distribution I* . The functions DI(x,y,z,T), DV(x,y,z,T), DIS(x,y,z,T), DVS(x,y,z,T) describe dependences of coefficients of volumetric and surficial diffusions of intersti- tials and vacancies on coordinate and temperature. Terms V2 (x,y,z,t) and I2 (x,y,z,t) gives a possi- bility to take into account generation of divacancies and diinterstitials [15]. The functions
- 4. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 26 kI,V(x,y,z,T), kI,I(x,y, z,T) and kV,V(x,y,z,T) describe dependences of parameters of recombination of point radiation defects and generation of their complexes on coordinate and temperature. We determine spatio-temporal distributions of divacancies ΦV(x,y,z,t) and diinterstitials ΦI(x,y, z,t) by solving the following boundary problem [11,12,14,15] ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ×+ ∫Φ∇ ∂ ∂ Ω+ Φ + Φ Φ TzyxkWdtWyxtzyx Tk D xz tzyx TzyxD z I zL IS SII I ,,,,,,,,, ,,, ,,, 0 1µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( )tzyxITzyxkWdtWyxtzyx Tk D y tzyxI II zL IS SI ,,,,,,,,,,,,,,, 2 , 0 1 + ∫ Φ∇ ∂ ∂ Ω+× Φ µ (5) ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ×+ ∫Φ∇ ∂ ∂ Ω+ Φ + Φ Φ TzyxkWdtWyxtzyx Tk D xz tzyx TzyxD z V zL VS SVV V ,,,,,,,,, ,,, ,,, 0 1µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( )tzyxVTzyxkWdtWyxtzyx Tk D y tzyxV VV zL VS SV ,,,,,,,,,,,,,,, 2 , 0 1 + ∫ Φ∇ ∂ ∂ Ω+× Φ µ ( ) 0 ,,, 0 = Φ =x I x tzyx ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxI ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxI ∂ ∂ , ( ) 0 ,,, 0 = Φ =z I z tzyx ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxI ∂ ∂ , ( ) 0 ,,, 0 = Φ =x V x tzyx ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxV ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =z z tzyxV ∂ ∂ , ( ) 0 ,,, = Φ = zLz V z tzyx ∂ ∂ , ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (6) The functions DΦI(x,y,z,T), DΦV(x,y,z,T), DΦIS (x,y,z,T) and DΦVS(x,y,z,T) describe dependences of coefficients of volumetric and surficial diffusions of complexes of radiation defects on coordinate and temperature. The functions kI(x,y,z,T) and kV(x,y,z,T) describe dependences of parameters of decay of complexes of radiation defects on coordinate and temperature. Chemical potential µ1 in Eq.(1) could be determine by the following relation [11] µ1=E(z)Ωσij [uij(x,y,z,t)+uji(x,y,z,t)]/2, (7) where E(z) is the Young modulus, σij is the stress tensor; ∂ ∂ + ∂ ∂ = i j j i ij x u x u u 2 1 is the deformation tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector ( )tzyxu ,,, r ; xi, xj are the coordinate x, y, z. The Eq. (3) could be transform to the following form
- 5. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 27 ( ) ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ = i j j i i j j i x tzyxu x tzyxu x tzyxu x tzyxu tzyx ,,,,,, 2 1,,,,,, ,,,µ ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )zETtzyxTzzK x tzyxu z z ij k kij ij 2 ,,,3 ,,, 21 000 Ω −− − ∂ ∂ − +− δβε σ δσ δε , where σ is Poisson coefficient; the mismatch parameter could be determined as ε0 =(as-aEL)/aEL, where as is the lattice distances of the substrate, aEL and the epitaxial layer; K is the modulus of uniform compression; parameter β describes thermal expansion coefficient, parameter Tr describes the equilibrium temperature. The equilibrium temperature coincides in our case with room tem- perature. Components of displacement vector could be obtained by solution of the following equ- ations [12] ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z xzxyxxx ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z yzyyyxy ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z zzzyzxz ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ , where ( ) ( )[ ] ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ + = k k ij k kij i j j i ij x tzyxu x tzyxu x tzyxu x tzyxu z zE ,,,,,, 3 ,,,,,, 12 δ δ σ σ ( ) ( ) ( ) ( )[ ]rTtzyxTzKzzK −−× ,,,β , ρ (z) is the density of materials of heterostructure, δij is the Kronecker symbol. With account the relation for σij last system of equation could be written as ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) + ∂∂ ∂ + −+ ∂ ∂ + += ∂ ∂ yx tzyxu z zE zK x tzyxu z zE zK t tzyxu z yxx ,,, 13 ,,, 16 5,,, 2 2 2 2 2 σσ ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) − ∂∂ ∂ + ++ ∂ ∂ + ∂ ∂ + + zx tzyxu z zE zK z tzyxu y tzyxu z zE zzy ,,, 13 ,,,,,, 12 2 2 2 2 2 σσ ( ) ( ) ( ) x tzyxT zzK ∂ ∂ − ,,, β ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) + ∂ ∂ − ∂∂ ∂ + ∂ ∂ + = ∂ ∂ y tzyxT zzK yx tzyxu x tzyxu z zE t tzyxu z xyy ,,,,,,,,, 12 ,,, 2 2 2 2 2 β σ ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) + + +∂ ∂ + ∂ ∂ + ∂ ∂ +∂ ∂ + zK z zE y tzyxu y tzyxu z tzyxu z zE z yzy σσ 112 5,,,,,,,,, 12 2 2 ( ) ( ) ( )[ ] ( ) ( ) ( ) yx tzyxu zK zy tzyxu z zE zK yy ∂∂ ∂ + ∂∂ ∂ + −+ ,,,,,, 16 22 σ (8) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) + ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ + ∂ ∂ + = ∂ ∂ zy tzyxu zx tzyxu y tzyxu x tzyxu z zE t tzyxu z yxzzz ,,,,,,,,,,,, 12 ,,, 22 2 2 2 2 2 2 σ ρ
- 6. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 28 ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ∂ ∂ − ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + z tzyxT zzK z tzyxu y tzyxu x tzyxu zK z xyx ,,,,,,,,,,,, β ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ +∂ ∂ + z tzyxu y tzyxu x tzyxu z tzyxu z zE z zyxz ,,,,,,,,,,,, 6 16 1 σ . Conditions for the system of Eq. (8) could be written in the form ( ) 0 ,,,0 = ∂ ∂ x tzyu r ; ( ) 0 ,,, = ∂ ∂ x tzyLu x r ; ( ) 0 ,,0, = ∂ ∂ y tzxu r ; ( ) 0 ,,, = ∂ ∂ y tzLxu y r ; ( ) 0 ,0,, = ∂ ∂ z tyxu r ; ( ) 0 ,,, = ∂ ∂ z tLyxu z r ; ( ) 00,,, uzyxu rr = ; ( ) 0,,, uzyxu rr =∞ . We solve the Eqs.(1), (3) and (5) by using standard method of averaging of function corrections [17]. Previously we transform the Eqs.(1), (3) and (5) to the following form with account initial distributions of the considered concentrations ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, (1a) ( ) ( ) ( ) ( ) + ∫∇ ∂ ∂ Ω++ zL S S C WdtWyxCtzyx Tk D x tzyxf 0 ,,,,,,,, µδ ( ) ( ) ∫∇ ∂ ∂ Ω+ zL S S WdtWyxCtzyx Tk D y 0 ,,,,,,µ ( ) ( ) ( ) ( ) ( ) ( ) ( )++ + = tzyxf y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI III δ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,, ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( )×− ∫∇ ∂ ∂ Ω+ + TzyxkWdtWyxItzyx Tk D xz tzyxI TzyxD z VI zL S IS I ,,,,,,,,, ,,, ,,, , 0 1µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxkWdtWyxItzyx Tk D y tzyxVtzyxI II zL S IS ,,,,,,,,,,,,,,,,,, 2 , 0 1 − ∫∇ ∂ ∂ Ω+× µ (3a) ( ) ( ) ( ) ( ) ( ) ( ) ( )++ + = tzyxf y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VVV δ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,, ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ×− ∫∇ ∂ ∂ Ω+ + TzyxkWdtWyxVtzyx Tk D xz tzyxV TzyxD z VI zL S VS V ,,,,,,,,, ,,, ,,, , 0 1µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxkWdtWyxVtzyx Tk D y tzyxVtzyxI VV zL S VS ,,,,,,,,,,,,,,,,,, 2 , 0 1 − ∫∇ ∂ ∂ Ω+× µ ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,,
- 7. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 29 ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ + Φ Φ zL IS SII I WdtWyxtzyx Tk D xz tzyx TzyxD z 0 1 ,,,,,, ,,, ,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )++ ∫ Φ∇ ∂ ∂ Ω+ Φ tzyxITzyxkWdtWyxtzyx Tk D y I zL IS SI ,,,,,,,,,,,, 0 1µ ( ) ( ) ( ) ( )tzyxftzyxITzyxk III δ,,,,,,,, 2 , Φ++ (5a) ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ + Φ Φ zL VS SVV V WdtWyxtzyx Tk D xz tzyx TzyxD z 0 1 ,,,,,, ,,, ,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )++ ∫Φ∇ ∂ ∂ Ω+ Φ tzyxITzyxkWdtWyxtzyx Tk D y I zL IS SI ,,,,,,,,,,,, 0 1µ ( ) ( ) ( ) ( )tzyxftzyxVTzyxk VVV δ,,,,,,,, 2 , Φ++ . Farther we replace concentrations of dopant and radiation defects in right sides of Eqs. (1a), (3a) and (5a) on their average values α1ρ. The average values α1ρ are not yet known. In this situation we obtain equations for the first-order approximations of the required concentrations in the fol- lowing form ( ) ( ) ( ) ( ) ( )tzyxftzyx Tk D z y tzyx Tk D z xt tzyxC CS S CS S C δµαµα ,,,,,,,, ,,, 1111 1 + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= ∂ ∂ (1b) ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= tzyx Tk D z y tzyx Tk D x z t tzyxI S IS IS IS I ,,,,,, ,,, 11 1 µαµα ∂ ∂ ( ) ( ) ( ) ( )TzyxkTzyxktzyxf VIVIIIII ,,,,,,,, ,11, 2 1 αααδ −−+ (3b) ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= tzyx Tk D z y tzyx Tk D x z t tzyxV S VS VS VS V ,,,,,, ,,, 1111 1 µαµα ∂ ∂ ( ) ( ) ( ) ( )TzyxkTzyxktzyxf VIVIVVVV ,,,,,,,, ,11, 2 1 αααδ −−+ ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= Φ Φ Φ Φ Φ tzyx Tk D y ztzyx Tk D x z t tzyx S SI IS SI I I ,,,,,, ,,, 1111 1 µαµα ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxktzyxf IIII ,,,,,,,,,,,,,, 2 ,+++ Φ δ (5b) ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= Φ Φ Φ Φ Φ tzyx Tk D y ztzyx Tk D x z t tzyx S SV VS SV V V ,,,,,, ,,, 1111 1 µαµα ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxktzyxf VVVV ,,,,,,,,,,,,,, 2 ,+++ Φ δ . Integration of the left and right sides of the Eqs. (1b), (3b) and (5b) on time gives us possibility to obtain relations for above approximation in the final form
- 8. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 30 ( ) ( ) ( ) ( ) ( ) ( ) ∫ × ++∇ ∂ ∂ Ω= t SLSC V zyxV V zyxV zyxTzyxD x tzyxC 0 2* 2 2*1111 ,,,,,, 1,,,,,,,,, τ ς τ ςτµα ( ) ( ) ( ) ( )∫ × +∇ ∂ ∂ Ω+ +× t CS SLSC CS TzyxPTk z zyxTzyxD y d TzyxPTk z 0 1 11 1 ,,, 1,,,,,, ,,, 1 γ γ γ γ αξ τµατ αξ ( ) ( ) ( ) ( )zyxfd V zyxV V zyxV C ,, ,,,,,, 1 2* 2 2*1 + ++× τ τ ς τ ς (1c) ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω= t S IS I t S IS I dzyx Tk D y zdzyx Tk D x ztzyxI 0 11 0 111 ,,,,,,,,, ττµαττµα ( ) ( ) ( )∫−∫−+ t VIVI t IIII dTzyxkdTzyxkzyxf 0 ,11 0 , 2 1 ,,,,,,,, ταατα (3c) ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω= t S IS V t S IS V dzyx Tk D y zdzyx Tk D x ztzyxV 0 11 0 111 ,,,,,,,,, ττµαττµα ( ) ( ) ( )∫−∫−+ t VIVI t VVVV dTzyxkdTzyxkzyxf 0 ,11 0 , 2 1 ,,,,,,,, ταατα ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω=Φ Φ Φ Φ Φ t S SI I t S SI II dzyx Tk D x zdzyx Tk D x ztzyx 0 11 0 111 ,,,,,,,,, ττµαττµα ( ) ( ) ( ) ( ) ( )∫+∫++ Φ t II t II dzyxITzyxkdzyxITzyxkzyxf 0 2 , 0 ,,,,,,,,,,,,,, ττττ (5c) ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω=Φ Φ Φ Φ Φ t S SV V t S SV VV dzyx Tk D x zdzyx Tk D x ztzyx 0 11 0 111 ,,,,,,,,, ττµαττµα ( ) ( ) ( ) ( ) ( )∫+∫++ Φ t VV t VV dzyxVTzyxkdzyxVTzyxkzyxf 0 2 , 0 ,,,,,,,,,,,,,, ττττ . We determine average values of the first-order approximations of concentrations of dopant and radiation defects by the following standard relation [17] ( )∫ ∫ ∫ ∫ Θ = Θ 0 0 0 0 11 ,,, 1 xL yL zL zyx tdxdydzdtzyx LLL ρα ρ . (9) We obtain required average values by substitution of the relations (1c), (3c) and (5c) into relation (9). The obtained relations could be written as ( )∫ ∫ ∫= xL yL zL C zyx C xdydzdzyxf LLL 0 0 0 1 ,, 1 α , ( ) 4 3 4 1 2 3 2 4 2 3 1 4 4 4 a Aa a aLLLBa B a Aa zyx I + − Θ+Θ +− + =α , ( ) Θ−−∫ ∫ ∫ Θ = zyxIII xL yL zL I IIV V LLLSxdydzdzyxf S 001 0 0 0100 1 ,, 1 α α α . Here ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−Θ= Θ 0 0 0 0 11, ,,,,,,,,, xL yL zL ji ij tdxdydzdtzyxVtzyxITzyxktS ρρρρ , ( )×−= 0000 2 004 VVIIIV SSSa
- 9. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 31 00IIS× , 0000 2 0000003 VVIIIVIIIV SSSSSa −+= , ( ) ×Θ+∫ ∫ ∫= 222 0 0 0 2 00002 ,, zyx xL yL zL VIVIV LLLxdydzdzyxfSSa ( ) ( ) ×∫ ∫ ∫−Θ−∫ ∫ ∫+× xL yL zL IVVzyx xL yL zL IIIVVIV xdydzdzyxfSLLLxdydzdzyxfSSS 0 0 0 00 222 0 0 0 000000 ,,,,2 2 00IVS× , ( )∫ ∫ ∫= xL yL zL IIV xdydzdzyxfSa 0 0 0 001 ,, , ( ) 2 0 0 0 000 ,, ∫ ∫ ∫= xL yL zL IVV xdydzdzyxfSa , 4 2 2 4 2 32 48 a a a a yA Θ−Θ+= , 3 323 32 4 2 6 qpqqpq a a B ++−−++ Θ = , × Θ−= 4 31 04 a aa LLLaq zyx 2 4 2 1 4 222 3 4 3 2 3 4 2 32 22 4 02 2 4 2 3 854 4 824 a a LLL a a a a a a a a a zyx Θ − Θ − Θ−ΘΘ− Θ × , − Θ− Θ= 2 4 31402 12 4 a aaLLLaa p zyx 42 18aaΘ− , ( )∫ ∫ ∫+ Θ + Θ = ΦΦ xL yL zL I zyxzyx II zyx I I xdydzdzyxf LLLLLL S LLL R 0 0 0 201 1 ,, 1 α ( )∫ ∫ ∫+ Θ + Θ = ΦΦ xL yL zL V zyxzyx VV zyx V V xdydzdzyxf LLLLLL S LLL R 0 0 0 201 1 ,, 1 α , Here ( ) ( ) ( )∫ ∫ ∫ ∫−Θ= Θ 0 0 0 0 1 ,,,,,, xL yL zL i Ii tdxdydzdtzyxITzyxktRρ . The second-order approximations and approximations with higher orders of the considered con- centrations have been calculated framework standard iterative procedure of method of averaging of function corrections [17]. The considered approximations could be obtained by replacement the required concentrations in the Eqs. (1c), (3c), (5c) on the following sum αnρ+ρ n-1(x,y,z,t). The replacement leads to following results ( ) ( )[ ] ( ) ( ) ( ) ( ) × ++ + + ∂ ∂ = ∂ ∂ 2* 2 2*1 122 ,,,,,, 1 ,,, ,,, 1 ,,, V tzyxV V tzyxV TzyxP tzyxC xt tzyxC C ςς α ξ γ γ ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ ++ ∂ ∂ + ∂ ∂ × y tzyxC V tzyxV V tzyxV yx tzyxC TzyxD L ,,,,,,,,, 1 ,,, ,,, 1 2* 2 2*1 1 ςς ( ) ( )[ ] ( ) ( ) ( ) ( ) × ++ ∂ ∂ + + +× 2* 2 2*1 12 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV zTzyxP tzyxC TzyxD C L ςς α ξ γ γ ( ) ( ) ( )[ ] ( ) ( ) ( )++ + + ∂ ∂ × tzyxf TzyxP tzyxC z tzyxC TzyxD C C L δ α ξ γ γ ,, ,,, ,,, 1 ,,, ,,, 121 ( ) ( )[ ] + ∫ +∇ ∂ ∂ Ω+ zL CS S WdtWyxCtzyx Tk D x 0 21 ,,,,,, αµ ( ) ( )[ ] ∫ +∇ ∂ ∂ Ω+ zL CS S WdtWyxCtzyx Tk D y 0 21 ,,,,,, αµ (1d)
- 10. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 32 ( ) ( ) ( ) ( ) ( ) + + = y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, 112 ( ) ( ) ( ) ( )[ ] ( )×−+− + TzyxktzyxITzyxk z tzyxI TzyxD z VIIIII ,,,,,,,,, ,,, ,,, , 2 11, 1 α ∂ ∂ ∂ ∂ ( )[ ] ( )[ ] ( ) ( )[ ] ×∫ +∇ ∂ ∂ Ω+++× zL ISVI WdtWyxItzyx x tzyxVtzyxI 0 121111 ,,,,,,,,,,,, αµαα ( ) ( )[ ] ∫ +∇ ∂ ∂ Ω+ × zL IS ISIS WdtWyxItzyx Tk D yTk D 0 12 ,,,,,, αµ (3d) ( ) ( ) ( ) ( ) ( ) + + = y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, 112 ( ) ( ) ( ) ( )[ ] ( )×−+− + TzyxktzyxVTzyxk z tzyxV TzyxD z VIVVVV ,,,,,,,,, ,,, ,,, , 2 11, 1 α ∂ ∂ ∂ ∂ ( )[ ] ( )[ ] ( ) ( )[ ] ×∫ +∇ ∂ ∂ Ω+++× zL VSVI WdtWyxVtzyx x tzyxVtzyxI 0 121111 ,,,,,,,,,,,, αµαα ( ) ( )[ ] ∫ +∇ ∂ ∂ Ω+ × zL VS VSVS WdtWyxVtzyx Tk D yTk D 0 12 ,,,,,, αµ ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, 112 ( ) ( )[ ] ( ) ( )++ ∫ Φ+∇ ∂ ∂ Ω+ Φ Φ tzyxITzyxkWdtWyxtzyx Tk D x II zL IIS SI ,,,,,,,,,,,, 2 , 0 12αµ ( ) ( )[ ] ( ) ( )++ ∫ Φ+∇ ∂ ∂ Ω+ Φ Φ tzyxITzyxkWdtWyxtzyx Tk D y I zL IIS SI ,,,,,,,,,,,, 0 12αµ ( ) ( ) ( ) ( )tzyxf z tzyx TzyxD z I I I δ ∂ ∂ ∂ ∂ ,, ,,, ,,, 1 ΦΦ + Φ + (5d) ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, 112 ( ) ( )[ ] ( ) ( )++ ∫ Φ+∇ ∂ ∂ Ω+ Φ Φ tzyxVTzyxkWdtWyxtzyx Tk D x VV zL VVS SV ,,,,,,,,,,,, 2 , 0 12αµ ( ) ( )[ ] ( ) ( )++ ∫ Φ+∇ ∂ ∂ Ω+ Φ Φ tzyxVTzyxkWdtWyxtzyx Tk D y V zL VVS SV ,,,,,,,,,,,, 0 12αµ ( ) ( ) ( ) ( )tzyxf z tzyx TzyxD z V V V δ ∂ ∂ ∂ ∂ ,, ,,, ,,, 1 ΦΦ + Φ + . Final relations for the second-order approximations could be obtained by integration of the left and the right sides of Eqs. (1d), (3d) and (5d). The final relations could be written as
- 11. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 33 ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ∫ × ++ + + ∂ ∂ = t C L V zyxV V zyxV TzyxP zyxC TzyxD x tzyxC 0 2* 2 2*1 12 2 ,,,,,, 1 ,,, ,,, 1,,,,,, τ ς τ ς τα ξ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ∫ × ∂ ∂ ++ ∂ ∂ + ∂ ∂ × t L y zyxC V zyxV V zyxV TzyxD y d x zyxC 0 1 2* 2 2*1 1 ,,,,,,,,, 1,,, ,,, ττ ς τ ςτ τ ( )[ ] ( ) ( ) ( ) ( ) ( ) ∫ × ∂ ∂ ++ ∂ ∂ + + +× t C z zyxC V zyxV V zyxV zTzyxP tzyxC 0 1 2* 2 2*1 12 ,,,,,,,,, 1 ,,, ,,, 1 ττ ς τ ς α ξ γ γ ( ) ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫ +∇ ∂ ∂ + + +× t zL CS C L WyxCzyx x d TzyxP zyxC TzyxD 0 0 12 12 ,,,,,, ,,, ,,, 1,,, τατµτ τα ξ γ γ ( ) ( )[ ] ( )zyxfdWdWyxC Tk D zyx y dWd Tk D C t zL C S S S ,,,,,,,, 0 0 12 +∫ ∫ +∇ ∂ ∂ Ω+× ττατµτ (1e) ( ) ( ) ( ) ( ) ( ) +∫+∫= t I t I d y zyxI TzyxD y d x zyxI TzyxD x tzyxI 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( )[ ] −∫ +−+∫+ t IIII t I dzyxITzyxkzyxfd z zyxI TzyxD z 0 2 12, 0 1 ,,,,,,,, ,,, ,,, ττατ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )[ ] ( )×∫ ∇ ∂ ∂ Ω+∫ ++− t S IS t VIVI zyx Tk D x dzyxVzyxITzyxk 00 1212, ,,,,,,,,,,,, τµττατα ( )[ ] ( ) ( )[ ]∫ ∫ +∇ ∂ ∂ Ω+∫ +× t zL I IS S zL I dWdWyxI Tk D zyx y dWdWyxI 0 0 12 0 12 ,,,,,,,,, ττατµττα (3e) ( ) ( ) ( ) ( ) ( ) +∫+∫= t V t V d y zyxV TzyxD y d x zyxV TzyxD x tzyxV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( )[ ] −∫ +−+∫+ t VVVV t V dzyxVTzyxkzyxfd z zyxV TzyxD z 0 2 12, 0 1 ,,,,,,,, ,,, ,,, ττατ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )[ ] ( )×∫ ∇ ∂ ∂ Ω+∫ ++− t S VS t VIVI zyx Tk D x dzyxVzyxITzyxk 00 1212, ,,,,,,,,,,,, τµττατα ( )[ ] ( ) ( )[ ]∫ ∫ +∇ ∂ ∂ Ω+∫ +× t zL I VS S zL I dWdWyxV Tk D zyx y dWdWyxV 0 0 12 0 12 ,,,,,,,,, ττατµττα ( ) ( ) ( ) ( ) ∫ × Φ +∫ Φ =Φ Φ t I t I II y zyx y d x zyx TzyxD x tzyx 0 1 0 1 2 ,,,,,, ,,,,,, ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( ) ×∫ ∇ ∂ ∂ Ω+∫ Φ +× Φ ΦΦ t SI S t I II Tk D zyx x d z zyx TzyxD z dTzyxD 00 1 ,,, ,,, ,,,,,, τµτ ∂ τ∂ ∂ ∂ τ ( )[ ] ( ) ( )[ ] ×∫ ∫ Φ+∇ ∂ ∂ Ω+∫ Φ+× ΦΦ t zL IIS zL II WdWyxzyx y dWdWyx 0 0 12 0 12 ,,,,,,,,, τατµττα ( ) ( ) ( ) ( ) ( )zyxfdzyxITzyxkdzyxITzyxkd Tk D I t I t II SI ,,,,,,,,,,,,,, 00 2 , Φ Φ +∫+∫+× τττττ (5e) ( ) ( ) ( ) ( ) ∫ × Φ +∫ Φ =Φ Φ t V t V VV y zyx y d x zyx TzyxD x tzyx 0 1 0 1 2 ,,,,,, ,,,,,, ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂
- 12. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 34 ( ) ( ) ( ) ( ) ×∫ ∇ ∂ ∂ Ω+∫ Φ +× Φ ΦΦ t SV S t V VV Tk D zyx x d z zyx TzyxD z dTzyxD 00 1 ,,, ,,, ,,,,,, τµτ ∂ τ∂ ∂ ∂ τ ( )[ ] ( ) ( )[ ] ×∫ ∫ Φ+∇ ∂ ∂ Ω+∫ Φ+× ΦΦ t zL VVS zL VV WdWyxzyx y dWdWyx 0 0 12 0 12 ,,,,,,,,, τατµττα ( ) ( ) ( ) ( ) ( )zyxfdzyxVTzyxkdzyxVTzyxkd Tk D V t V t VV SV ,,,,,,,,,,,,,, 00 2 , Φ Φ +∫+∫+× τττττ . Average values of the second-order approximations of required approximations by using the fol- lowing standard relation [17] ( ) ( )[ ]∫ ∫ ∫ ∫ − Θ = Θ 0 0 0 0 122 ,,,,,, 1 xL yL zL zyx tdxdydzdtzyxtzyx LLL ρρα ρ . (10) Substitution of the relations (1e), (3e), (5e) into relation (10) gives us possibility to obtain rela- tions for required average values α2ρ α2C=0, α2ΦI =0, α2ΦV =0, ( ) 4 3 4 1 2 3 2 4 2 3 2 4 4 4 b Eb b bLLLFa F b Eb zyx V + − Θ+Θ +− + =α , ( ) 00201 11021001200 2 2 2 2 IVVIV VIVVzyxVIVVVVVVV I SS SSLLLSSSC α αα α + −−Θ++−− = , where 00 2 0000 2 004 11 IIVV zyx VVIV zyx SS LLL SS LLL b Θ − Θ = , ( )+Θ++ Θ −= zyxIVVV zyx VVII LLLSS LLL SS b 1001 0000 3 2 ( ) ( ) 3333 10 2 00 1001 2 00 011001 0000 22 zyx IVIV zyxIVVV zyx IV zyxIVIIIV zyx VVIV LLL SS LLLSS LLL S LLLSSS LLL SS Θ −Θ++ Θ +Θ+++ Θ + , ( ) ( ) ( )×++Θ+Θ+−−++ Θ = 0110 2 01101102 0000 2 22 IVIIzyxzyxVVIVVIVVV zyx VVII SSLLLLLLSSCSS LLL SS b ( ) ( ) + Θ ++Θ+Θ+++ Θ + Θ × zyx VVIV IVIIzyxzyxIVIIIV zyx IV zyx VVIV LLL SS SSLLLLLLSSS LLL S LLL SS 0001 0110 2 011001 000001 222 ( )( ) ( ) ×−−−+Θ+Θ+++ Θ + 11021001011001 00 222 IVVVVIVzyxVVzyxIVIIIV zyx IV SSCSLLLSLLLSSS LLL S zyx IVIV IV zyx IVI zyx IV LLL SS S LLL SC LLL S Θ − Θ + Θ × 0010 012222 2 00 2 00 2 , ( ++ Θ ++ = 1001 0211 001 2 IVVV zyx VVVIV II SS LLL CSS Sb ) ( )( ) − Θ −Θ++++Θ Θ +Θ+ zyx IVIV zyIVVVIVIIzyx zyx IV zyx LLL SS LLLSSSSLLL LLL S LLL 2 0110 10010110 01 22 ( )( ) 010011021001 00 223 IVIVIIVVVVzyxIIIV zyx IV SSCSSCLLLSS LLL S +−−Θ++ Θ − , × Θ = zyx II LLL S b 00 0 ( ) ( )( ) × Θ −− −−−++Θ Θ −+× zyx IVVVV IVIVVVVIVIIzyx zyx IV VVIV LLL SSC SSSCSSLLL LLL S SS 1102 0111020110 012 0200 2 ( ) 2 01011001 22 IVIIVIIzyxIV SCSSLLLS +++Θ× , − Θ − Θ + Θ = zyx IIII zyx III IV zyx VI I LLL SS LLL S S LLL C 202000 2 1 00 11 ααα , zyx IV LLL S Θ − 11 , 110200 2 10011 IVVVVVVIVVIV SSSSC −−+= ααα , 4 2 2 4 2 32 48 a a a a yE Θ−Θ+= , + Θ = 4 2 6a a F
- 13. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 35 3 323 32 rsrrsr ++−−++ , × Θ−Θ− Θ − Θ− Θ = 4 2 32 23 4 3 2 3 4 31 02 4 2 3 4 54 4 24 b b b b b b bb LLLb b b r zyx 2 4 2 1 4 222 2 4 2 0 88 b b LLL b b zyx Θ − Θ × , 4 2 2 4 31402 1812 4 b b b bbLLLbb s zyx Θ − Θ− Θ= . Farther we determine solutions of Eqs.(8), i.e. components of displacement vector. To determine the first-order approximations of the considered components framework method of averaging of function corrections we replace the required functions in the right sides of the equations by their not yet known average values αi. The substitution leads to the following result ( ) ( ) ( ) ( ) ( ) x tzyxT zzK t tzyxu z x ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ , ( ) ( ) ( ) ( ) ( ) y tzyxT zzK t tzyxu z y ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ , ( ) ( ) ( ) ( ) ( ) z tzyxT zzK t tzyxu z z ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ . Integration of the left and the right sides of the above relations on time t leads to the following result ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ += ∞ 0 00 0 01 ,,,,,,,,, ϑϑ ϑττ ρ β ϑττ ρ β ddzyxT xz z zKddzyxT xz z zKutzyxu t xx , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ += ∞ 0 00 0 01 ,,,,,,,,, ϑϑ ϑττ ρ β ϑττ ρ β ddzyxT yz z zKddzyxT yz z zKutzyxu t yy , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ += ∞ 0 00 0 01 ,,,,,,,,, ϑϑ ϑττ ρ β ϑττ ρ β ddzyxT zz z zKddzyxT zz z zKutzyxu t zz . Approximations of the second and higher orders of components of displacement vector could be determined by using standard replacement of the required components on the following sums αi+ui(x,y,z,t) [17]. The replacement leads to the following result ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) + ∂∂ ∂ + −+ ∂ ∂ + += ∂ ∂ yx tzyxu z zE zK x tzyxu z zE zK t tzyxu z yxx ,,, 13 ,,, 16 5,,, 1 2 2 1 2 2 2 2 σσ ρ ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) + ∂ ∂ − ∂ ∂ + ∂ ∂ + + ∂∂ ∂ × x tzyxT zzK z tzyxu y tzyxu z zE yx tzyxu zyy ,,,,,,,,, 12 ,,, 2 1 2 2 1 2 1 2 β σ ( ) ( ) ( )[ ] ( ) zx tzyxu z zE zK z ∂∂ ∂ + ++ ,,, 13 1 2 σ ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) + ∂ ∂ − ∂∂ ∂ + ∂ ∂ + = ∂ ∂ y tzyxT zzK yx tzyxu x tzyxu z zE t tzyxu z xyy ,,,,,,,,, 12 ,,, 1 2 2 1 2 2 2 2 β σ ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) + + +∂ ∂ + ∂ ∂ + ∂ ∂ +∂ ∂ + zK z zE y tzyxu y tzyxu z tzyxu z zE z yzy σσ 112 5,,,,,,,,, 12 2 1 2 11 ( ) ( ) ( )[ ] ( ) ( ) ( ) yx tzyxu zK zy tzyxu z zE zK yy ∂∂ ∂ + ∂∂ ∂ + −+ ,,,,,, 16 1 2 1 2 σ
- 14. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 36 ( ) ( ) ( ) ( ) ( ) ( ) × ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ zy tzyxu zx tzyxu y tzyxu x tzyxu t tzyxu z yxzzz ,,,,,,,,,,,,,,, 1 2 1 2 2 1 2 2 1 2 2 2 2 ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ] × + + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + + × z zE z tzyxu y tzyxu x tzyxu zK zz zE xyx σσ 16 ,,,,,,,,, 12 111 ( ) ( ) ( ) ( ) ( ) × ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ ∂ ∂ × z tzyxT z tzyxu y tzyxu x tzyxu z tzyxu z zyxz ,,,,,,,,,,,,,,, 6 1111 ( ) ( )zzK β× . Integration of the left and right sides of the above relations on time t leads to the following result ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] × + −+∫ ∫ ∂ ∂ + += z zE zKddzyxu xz zE zK z tzyxu t xx σ ϑττ σρ ϑ 13 ,,, 16 51 ,,, 0 0 12 2 2 ( ) ( ) ( ) ( ) × ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ +∫ ∫ ∂∂ ∂ × t z t y t y ddzyxu z ddzyxu y ddzyxu yxz 0 0 12 2 0 0 12 2 0 0 1 2 ,,,,,,,,, 1 ϑϑϑ ϑττϑττϑττ ρ ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ×− + +∫ ∫ ∂∂ ∂ + + × z z zK z zE zKddzyxu zxzzz zE t z ρ β σ ϑττ ρσρ ϑ 13 ,,, 1 12 0 0 1 2 ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ×−∫ ∫ ∂ ∂ + +−∫ ∫ ∂ ∂ × ∞ z ddzyxu xz zE zK z ddzyxT x x t ρ ϑττ σρ ϑττ ϑϑ 1 ,,, 16 51 ,,, 0 0 12 2 0 0 ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) +∫ ∫ ∂ ∂ + −∫ ∫ ∂∂ ∂ + −× ∞∞ 0 0 12 2 0 0 1 2 ,,, 1 ,,, 13 ϑϑ ϑττ σ ϑττ σ ddzyxu yz zE ddzyxu yxz zE zK yy ( ) ( ) ( ) ( ) ( )[ ] ( ) ×∫ ∫ ∂∂ ∂ + +− ∫ ∫ ∂ ∂ + ∞∞ 0 0 1 2 0 0 12 2 ,,, 132 1 ,,, ϑϑ ϑττ σρ ϑττ ddzyxu zxz zE zK z ddzyxu z zz ( ) ( ) ( ) ( ) ( )∫ ∫ ∂ ∂ ++× ∞ 0 0 0 ,,, 1 ϑ ϑττ ρ β ρ ddzyxT xz z zKu z x ( ) ( ) ( ) ( )[ ] ( ) ( ) + ∫ ∫ ∂∂ ∂ +∫ ∫ ∂ ∂ + = t x t xy ddzyxu yx ddzyxu xzz zE tzyxu 0 0 1 2 0 0 12 2 2 ,,,,,, 12 ,,, ϑϑ ϑττϑττ σρ ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] + + ∫ ∫ ∂ ∂ +∫ ∫ ∂∂ ∂ + + × z zE ddzyxu y ddzyxu yxz zK z t x t y σ ϑττϑττ ρσ ϑϑ 112 5 ,,,,,, 1 1 0 0 12 2 0 0 1 2 ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ +∂ ∂ + + t z t y ddzyxu y ddzyxu zz zE zz zK 0 0 1 0 0 1 ,,,,,, 1 1 ϑϑ ϑττϑττ σρ ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ×∫ ∫ ∂∂ ∂ − + −∫ ∫−× t y t ddzyxu zy zK z zE ddzyxT z z zK z 0 0 1 2 0 0 ,,, 16 ,,, 2 1 ϑϑ ϑττ σ ϑττ ρ β ρ ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ×− ∫ ∫ ∂∂ ∂ +∫ ∫ ∂ ∂ + −× ∞∞ zKddzyxu yx ddzyxu xzz zE z xx 0 0 1 2 0 0 12 2 ,,,,,, 12 1 ϑϑ ϑττϑττ σρρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] + +∂ ∂ −∫ ∫ ∂∂ ∂ −∫ ∫× ∞∞ z zE yz ddzyxu yxz zK ddzyxT z z y σρ ϑττ ρ ϑττ ρ β ϑϑ 112 51 ,,,,,, 2 2 0 0 1 2 0 0
- 15. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 37 ( ) ( ) ( ) ( ) × ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ ∂ ∂ −∫ ∫ + ∞∞∞ 0 0 1 0 0 1 0 0 1 ,,,,,,,,, ϑϑϑ ϑττϑττϑττ ddzyxu y ddzyxu zz ddzyxuzK zyx ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) yy uddzyxu zyz zE zK zzz zE 0 0 0 1 2 ,,, 16 1 2 1 1 +∫ ∫ ∂∂ ∂ + −− + × ∞ ϑ ϑττ σρρσ ( ) ( ) ( ) ( )[ ] ( ) ( ) +∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ + = ∞∞ 0 0 12 2 0 0 12 2 ,,,,,, 12 ,,, ϑϑ ϑττϑττ σρ ddzyxu y ddzyxu xzz zE tzyxu zzz ( ) ( ) ( ) +∫ ∫ ∂ ∂ ∂ ∂ + ∫ ∫ ∂∂ ∂ +∫ ∫ ∂∂ ∂ + ∞∞∞ 0 0 1 0 0 1 2 0 0 1 2 ,,,,,,,,, ϑϑϑ ϑττϑττϑττ ddzyxu xz ddzyxu zy ddzyxu zx xyx ( ) ( ) ( ) ( ) ( ) ( ) ( ) × +∂ ∂ + ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ + ∞∞ z zE zzz zKddzyxu z ddzyxu y xx σρρ ϑττϑττ ϑϑ 16 11 ,,,,,, 0 0 1 0 0 1 ( ) ( ) ( ) −∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ × ∞∞∞ 0 0 1 0 0 1 0 0 1 ,,,,,,,,,6 ϑϑϑ ϑττϑττϑττ ddzyxu y ddzyxu x ddzyxu z yxz ( ) ( ) ( ) ( ) ( ) zz uddzyxT zz z zKddzyxu z 0 0 00 0 1 ,,,,,, +∫ ∫ ∂ ∂ − ∫ ∫ ∂ ∂ − ∞∞ ϑϑ ϑττ ρ β ϑττ . In this paper we calculate distributions of concentration of dopant, concentrations of radiation defects and components of displacement vector in space and time by using the second-order ap- proximation framework method of averaging of function corrections. This approximation is usually enough good approximation to make qualitative analysis and to obtain some quantitative results. All obtained results have been checked by comparison with re- sults of numerical simulations. 3. DISCUSSION Now we analyzed changing concentrations of dopant and radiation defects in space and time in a heterostructure with account mismatch-induced stress. Several spatial distributions of dopant con- centration are presented on Fig. 2 for diffusion type of doping and on Fig. 3 for ion type of dop- ing. In the both cases dopant diffusion coefficient in the epitaxial layer is larger, than in the sub- strate. One can find from these figures, that using in homogeneity of heterostructure leads to in- creasing of sharpness of p-n-junctions. In this situation the accompanying effect is increasing homogeneity of dopant distribution in doped part of heterostructure. Switching time of p-n- junc- tion correlated with the junction sharpness and could be increased with increasing of the sharp- ness. Increasing homogeneity of dopant distribution leads to decreasing local overheat leads to decreasing local heating during functioning of p-n-junction or to decrease dimensions of the p-n- junction for fixed maximal value of local overheat. However framework this approach of manu- facturing of p-n-junction and their systems (bipolar transistor and thyristors) it is necessary to op- timize annealing of dopant and/or radiation defects. Reason of this optimization is following. If annealing time is small, the dopant did not achieve any interfaces between materials of hetero- structure. In this situation one cannot find any modifications of distribution of concentration of dopant. Increasing of annealing time leads to increasing of homogeneity of distribution of con- centration of dopant. We optimize annealing time framework recently introduces approach [16,18-24]. To use the criterion one shall to approximate real distribution of concentration of do- pant by step-wise function (see Figs. 4 for diffusion doping and 5 for ion doping). The optimal values of annealing time have been calculated by minimization of the following mean-squared error
- 16. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 38 ( ) ( )[ ]∫ ∫ ∫ −Θ= xL yL zL zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (15) Fig.2. Distributions of concentration dopant in heterostructure from Fig. 1 for diffusion type of doping. Direction of diffusion is perpendicular to interface between layers of heterostructure. Do- pant diffusion coefficient in the epitaxial layer is larger, than in the substrate and increasing 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. Distributions of concentration dopant in heterostructure from Fig. 1 for ion type of doping. Direction of diffusion is perpendicular to interface between layers of heterostructure. Curves 1 and 3 corresponds to annealing time Θ= 0.0048(Lx 2 +Ly 2 +Lz 2 )/D0. Curves 2 and 4 corresponds to annealing time Θ=0.0057(Lx 2 +Ly 2 +Lz 2 )/D0. Curves 1 and 2 are the distributions of concentrations of dopant in homogenous sample. Curves 3 and 4 are the distributions of concentrations of dopant in the heterostructure. Dopant diffusion coefficient in the epitaxial layer is larger, than in the sub- strate and increasing with increasing of number of curves. C(x,Θ) 0 Lx 2 1 3 4
- 17. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 39 Fig. 4. Distributions of concentration dopant in heterostructure from Fig. 1 for diffusion type of doping. Curve 1 is idealized distribution of dopant. Curves 2-4 are the calculated distributions of dopant for several values of annealing time. Annealing time increased with increasing of number of curves. x C(x,Θ) 1 2 3 4 0 L Fig. 5. Distributions of concentration dopant in heterostructure from Fig. 1 for ion type of doping. Curve 1 is idealized distribution of dopant. Curves 2-4 are the calculated distributions of dopant for several values of annealing time. Annealing time increased with increasing of number of curves. 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 ΘD0 L -2 3 2 4 1 Fig.6. Dimensionless optimal annealing time of infused dopant. Curve 1 describes dependence of the annealing time on normalized thickness of epitaxial layer, ξ= γ = 0 and equal to each other values of dopant diffusion coefficient in all layers of heterostructure. Curve 2 describes depen- dence of the annealing time on normalized difference between diffusion coefficients ε =(DEL-DS)/ DS, a/L=1/2 and ξ=γ=0. Curve 3 describes dependence of the annealing time on value of parame- ter ξ for a/L=1/2 and ε = γ = 0. Curve 3 describes dependence of the annealing time on value of parameter γ for a/L=1/2 and ε=ξ=0. 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 ΘD0 L -2 3 2 4 1
- 18. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 40 Fig.7. Dimensionless optimal annealing time of implanted dopant. Curve 1 describes dependence of the annealing time on normalized thickness of epitaxial layer, ξ=γ =0 and equal to each other values of dopant diffusion coefficient in all layers of heterostructure. Curve 2 describes depen- dence of the annealing time on normalized difference between diffusion coefficients ε =(DEL-DS)/ DS, a/L=1/2 and ξ=γ=0. Curve 3 describes dependence of the annealing time on value of parame- ter ξ for a/L=1/2 and ε = γ = 0. Curve 3 describes dependence of the annealing time on value of parameter γ for a/L=1/2 and ε=ξ=0. Here ψ(x,y,z) is the step-wise approximation function. We calculate dependences of optimal val- ues of annealing time on parameters. These dependences are presented on Figs. 6 (for diffusion type of doping) and 7 (for ion types of doping). It is known, that radiation defects should be an- nealed after ion implantation. Distribution of concentration of these defects will be spreads during the annealing. If distribution of dopant concentration achieves nearest interface, than we obtain rare ideal case. If the dopant has not enough time for the achievement, it is practicably to addi- tionally anneal the dopant. In this situation the considered the additional annealing time became smaller, than annealing time for diffusion type of doping. Now we analyzed correlation between relaxations of mismatch-induced stress and distribution of concentration of dopant. If ε0< 0, one obtain compression of distribution of concentration of do- pant. Contrary one obtain spreading of distribution of concentration of dopant. Laser annealing gives a possibility to decrease influence of mismatch-induced stress on distribution of concentra- tion of dopant [22]. Using the annealing leads to acceleration of dopant diffusion in the irradiated area. Taking into account mismatch-induced stress leads to changing of optimal values of anneal- ing time. Mismatch-induced stress could be used to increase density of elements of integrated circuits. On the other hand could leads to generation dislocations of the discrepancy. Fig. 8 shows distributions of component of displacement vector in direction, which is perpendicular to epitaxial layer and substrate. z 0.0 0.2 0.4 0.6 0.8 1.0 Uz 1 2 0.0 a Fig. 8. Normalized dependences of component uz of displacement vector on coordinate z for nonporous (curve 1) and porous (curve 2) epitaxial layers Fig. 9. Normalized distributions of charge carrier mobility in the considered heterostructure.
- 19. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 41 Curve 1 corresponds to the heterostructure, which has been considered in Fig. 1. Curve 2 corres- pond to a homogenous material with averaged parameters of heterostructure from Fig. 1 Now we consider the changing of charge carrier mobility with changing of distribution of concen- tration of dopant. It has been recently shown, that the mobility is a function of concentration of dopant: µ~C1/3 [25]. The situation is illustrated by Fig. 9. The figure shows normalized distribu- tions of charge carrier mobility in the considered heterostructure. Curve 1 corresponds to the hete- rostructure, which has been considered in Fig. 1. Curve 2 correspond to material after averaging parameters of heterostructure from Fig. 1. 4. CONCLUSIONS In this paper we consider a possibility to increase sharpness of diffusion-junction and implanted- junction heterorectifiers. At the same time homogeneity of distribution of concentration of dopant in enriched by the dopant increases. We also consider an approach to decrease mismatch-induced stress in the considered heterostructure. We analyzed influence of changing of concentration of dopant on value of charge carrier mobility. ACKNOWLEDGEMENTS This work is supported by the agreement of August 27, 2013 № 02.В.49.21.0003 between The Ministry of education and science of the Russian Federation and Lobachevsky State University of Nizhni Novgorod, educational fellowship for scientific research of Government of Russian, edu- cational fellowship for scientific research of Government of Nizhny Novgorod region of Russia and educational fellowship for scientific research of Nizhny Novgorod State University of Archi- tecture and Civil Engineering. REFERENCES 1. V.I. Lachin, N.S. Savelov. Electronics. Rostov-on-Don: Phoenix, 2001. 2. A. Polishscuk. Modern models of integrated operating amplifier. Modern Electronics. Issue 12. P. 8-11 (2004). 3. G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Is- sue 2. P. 10-17 (2006). 4. A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Elec- tronics. Issue 1. P. 34-38 (2008). 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: the features of current flow in rectifying and nonrectifying contacts. Semiconductors. Vol. 43 (7). P. 897-903 (2009). 6. Jung-Hui Tsai, Shao-Yen Chiu, Wen-Shiung Lour, Der-Feng Guo. High-performance In- GaP/GaAs pnp δ-doped heterojunction bipolar transistor. Semiconductors. Vol. 43 (7). P. 971-974 (2009). 7. O.V. Alexandrov, A.O. Zakhar'in, N.A. Sobolev, E.I. Shek, M.M. Makoviychuk, E.O. Parshin. Formation of donor centers after annealing of dysprosium and holmium implanted silicon. Semi- conductors. Vol. 32 (9). P. 1029-1032 (1998). 8. I.B. Ermolovich, V.V. Milenin, R.A. Red'ko, S.M. Red'ko. Features of recombination processes in CdTe films, preparing at different temperature conditions and further annealing. Semiconductors. Vol. 43 (8). P. 1016-1020 (2009). 9. P. Sinsermsuksakul, K. Hartman, S.B. Kim, J. Heo, L. Sun, H.H. Park, R. Chakraborty, T. Buo- nassisi, R.G. Gordon. Enhancing the efficiency of SnS solar cells via band-offset engineering with a zinc oxysulfide buffer layer. Appl. Phys. Lett. Vol. 102 (5). P. 053901-053905 (2013). 10. J.G. Reynolds, C.L. Reynolds, Jr.A. Mohanta, J.F. Muth, J.E. Rowe, H.O. Everitt, D.E. Aspnes. Shallow acceptor complexes in p-type ZnO. Appl. Phys. Lett. Vol. 102 (15). P. 152114-152118 (2013).
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