Dependence of Charge Carriers Mobility in the P-N-Heterojunctions on Composition of Multilayer...

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. Bulaeva

1,2

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.


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

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxC ,,,,,,,,,,,, (1)

( ) ( ) ( ) ( )

∫∇

∂

∂Ω+

∫∇

∂

∂Ω+

zz L

SS

L

SS WdtWyxCtzyx

Tk

D

yWdtWyxCtzyx

Tk

D

x 01

01 ,,,,,,,,,,,, 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

=∂

∂

=xx

tzyxC,

( )0

,,,=

∂

∂

= xLxx

tzyxC,

( )0

,,,

0

=∂

∂

=yy

tzyxC,

( )0

,,,=

∂

∂

= yLxy

tzyxC,

( )0

,,,

0

=∂

∂

=zz

tzyxC,

( )0

,,,=

∂

∂

= zLxz

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

zdtzyxC0

,,,

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]


25

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD

LCςςξ

γ

γ

,

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD

SLSSςςξ

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

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIIIII

,,,,,,

,,,,,,

,,,,,,

,∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

( ) ( ) ( ) ( ) ( ) ( ) +−

+× tzyxVtzyxITzyxk

z

tzyxITzyxD

ztzyxI

VII,,,,,,,,,

,,,,,,,,, ,

2

∂

∂

∂

∂ (3)

( ) ( ) ( ) ( )

∫∇

∂

∂Ω+

∫∇

∂

∂Ω+

zL

S

ISzL

S

IS WdtWyxItzyxTk

D

yWdtWyxItzyx

Tk

D

x 00

,,,,,,,,,,,, µµ

( )( )

( )( )

( )( ) ×−

+

= Tzyxk

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVVVV

,,,,,,

,,,,,,

,,,,,,

,∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

( ) ( ) ( ) ( ) ( ) ( ) +−

+× tzyxVtzyxITzyxk

z

tzyxVTzyxD

ztzyxV

VIV,,,,,,,,,

,,,,,,,,,

,

2

∂

∂

∂

∂

( ) ( ) ( ) ( )

∫∇

∂

∂Ω+

∫∇

∂

∂Ω+

zL

S

VSzL

S

VS WdtWyxVtzyxTk

D

yWdtWyxVtzyx

Tk

D

x 00

,,,,,,,,,,,, µµ

( )0

,,,

0

==x

x

tzyxI

∂

∂,

( )0

,,,=

= xLxx

tzyxI

∂

∂,

( )0

,,,

0

==y

y

tzyxI

∂

∂,

( )0

,,,=

= yLyy

tzyxI

∂

∂,

( )0

,,,

0

==z

z

tzyxI

∂

∂,

( )0

,,,=

= zLzz

tzyxI

∂

∂,

( )0

,,,

0

==x

x

tzyxV

∂

∂,

( )0

,,,=

= xLxx

tzyxV

∂

∂,

( )0

,,,

0

==y

y

tzyxV

∂

∂,

( )0

,,,=

= yLyy

tzyxV

∂

∂,

( )0

,,,

0

==z

z

tzyxV

∂

∂,

( )0

,,,=

= zLzz

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


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=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

I

I

I

I

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) ( ) ×+

∫Φ∇

∂

∂Ω+

Φ+

Φ

Φ TzyxkWdtWyxtzyxTk

D

xz

tzyxTzyxD

zI

zL

IS

SII

I,,,,,,,,,

,,,,,,

01µ

∂

∂

∂

∂

( ) ( ) ( ) ( ) ( )tzyxITzyxkWdtWyxtzyxTk

D

ytzyxI

II

zL

IS

SI ,,,,,,,,,,,,,,, 2

,0

1 +

∫Φ∇

∂

∂Ω+×

Φµ (5)

( )( )

( )( )

( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

V

V

V

V

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) ( ) ×+

∫Φ∇

∂

∂Ω+

Φ+

Φ

Φ TzyxkWdtWyxtzyxTk

D

xz

tzyxTzyxD

zV

zL

VS

SVV

V,,,,,,,,,

,,,,,,

01

µ∂

∂

∂

∂

( ) ( ) ( ) ( ) ( )tzyxVTzyxkWdtWyxtzyxTk

D

ytzyxV

VV

zL

VS

SV ,,,,,,,,,,,,,,, 2

,0

1 +

∫Φ∇

∂

∂Ω+×

Φµ

( )0

,,,

0

=Φ

=x

I

x

tzyx

∂

∂,

( )0

,,,=

= xLxx

tzyxI

∂

∂,

( )0

,,,

0

==y

y

tzyxI

∂

∂,

( )0

,,,=

= yLyy

tzyxI

∂

∂,

( )0

,,,

0

=Φ

=z

I

z

tzyx

∂

∂,

( )0

,,,=

= zLzz

tzyxI

∂

∂,

( )0

,,,

0

=Φ

=x

V

x

tzyx

∂

∂,

( )0

,,,=

= xLxx

tzyxV

∂

∂,

( )0

,,,

0

==y

y

tzyxV

∂

∂,

( )0

,,,=

= yLyy

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

ijx

u

x

uu

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


27

( ) ( ) ( ) ( ) ( )

−

∂

∂+

∂

∂

∂

∂+

∂

∂=

i

j

j

i

i

j

j

i

x

tzyxu

x

tzyxu

x

tzyxu

x

tzyxutzyx

,,,,,,

2

1,,,,,,,,,µ

( )( )

( ) ( ) ( ) ( )[ ] ( )zETtzyxTzzKx

tzyxu

z

zij

k

kij

ij2

,,,3,,,

21000

Ω

−−

−

∂

∂

−+− δβε

σ

δσδε ,

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

tzyxuz xzxyxxx

∂

∂+

∂

∂+

∂

∂=

∂

∂ ,,,,,,,,,,,,2

2 σσσρ

( )( ) ( ) ( ) ( )

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz

yzyyyxy

∂

∂+

∂

∂+

∂

∂=

∂

∂ ,,,,,,,,,,,,2

2 σσσρ

( ) ( ) ( ) ( ) ( )z

tzyx

y

tzyx

x

tzyx

t

tzyxuz zzzyzxz

∂

∂+

∂

∂+

∂

∂=

∂

∂ ,,,,,,,,,,,,2

2 σσσρ ,

where ( )

( )[ ]( ) ( ) ( ) ( )

×∂

∂+

∂

∂−

∂

∂+

∂

∂

+=

k

k

ij

k

kij

i

j

j

i

ijx

tzyxu

x

tzyxu

x

tzyxu

x

tzyxu

z

zE ,,,,,,

3

,,,,,,

12δ

δ

σσ

( ) ( ) ( ) ( )[ ]r

TtzyxTzKzzK −−× ,,,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

zEzK

x

tzyxu

z

zEzK

t

tzyxuz

yxx,,,

13

,,,

16

5,,,2

2

2

2

2

σσρ

( )( )[ ]

( ) ( )( )

( )( )[ ]

( )−

∂∂

∂

+++

∂

∂+

∂

∂

++

zx

tzyxu

z

zEzK

z

tzyxu

y

tzyxu

z

zE zzy ,,,

13

,,,,,,

12

2

2

2

2

2

σσ

( ) ( ) ( )x

tzyxTzzK

∂

∂−

,,,β

( )( ) ( )

( )[ ]( ) ( )

( ) ( )( )

+∂

∂−

∂∂

∂+

∂

∂

+=

∂

∂

y

tzyxTzzK

yx

tzyxu

x

tzyxu

z

zE

t

tzyxuz xyy ,,,,,,,,,

12

,,, 2

2

2

2

2

βσ

ρ

( )( )[ ]

( ) ( ) ( ) ( )( )[ ]

( ) +

++∂

∂+

∂

∂+

∂

∂

+∂

∂+ zK

z

zE

y

tzyxu

y

tzyxu

z

tzyxu

z

zE

z

yzy

σσ 112

5,,,,,,,,,

12 2

2

( )( )

( )[ ]( )

( )( )

yx

tzyxuzK

zy

tzyxu

z

zEzK

yy

∂∂

∂+

∂∂

∂

+−+

,,,,,,

16

22

σ (8)

( )( ) ( )

( )[ ]( ) ( ) ( ) ( )

+

∂∂

∂+

∂∂

∂+

∂

∂+

∂

∂

+=

∂

∂

zy

tzyxu

zx

tzyxu

y

tzyxu

x

tzyxu

z

zE

t

tzyxuz

yxzzz,,,,,,,,,,,,

12

,,,22

2

2

2

2

2

2

σρ


28

( )( ) ( ) ( )

( ) ( )( )

+∂

∂−

∂

∂+

∂

∂+

∂

∂

∂

∂+

z

tzyxTzzK

z

tzyxu

y

tzyxu

x

tzyxuzK

z

xyx ,,,,,,,,,,,,β

( )( )

( ) ( ) ( ) ( )

∂

∂−

∂

∂−

∂

∂−

∂

∂

+∂

∂+

z

tzyxu

y

tzyxu

x

tzyxu

z

tzyxu

z

zE

z

zyxz,,,,,,,,,,,,

616

1

σ.

Conditions for the system of Eq. (8) could be written in the form

( )0

,,,0=

∂

∂

x

tzyur

; ( )

0,,,

=∂

∂

x

tzyLux

r

; ( )

0,,0,

=∂

∂

y

tzxur

; ( )

0,,,

=∂

∂

y

tzLxuy

r

;

( )0

,0,,=

∂

∂

z

tyxur

; ( )

0,,,

=∂

∂

z

tLyxuz

r

; ( )00,,, uzyxurr

= ; ( )0,,, uzyxurr

=∞ .

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

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxC ,,,,,,,,,,,, (1a)

( ) ( ) ( ) ( ) +

∫∇

∂

∂Ω++

zL

S

S

CWdtWyxCtzyx

Tk

D

xtzyxf

0

,,,,,,,, µδ

( ) ( )

∫∇

∂

∂Ω+

zL

S

S WdtWyxCtzyxTk

D

y 0

,,,,,,µ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ++

+

= tzyxf

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIIII δ

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂,,

,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) ( ) ×−

∫∇

∂

∂Ω+

+ TzyxkWdtWyxItzyx

Tk

D

xz

tzyxITzyxD

zVI

zL

S

IS

I ,,,,,,,,,,,,

,,, ,0

1µ∂

∂

∂

∂

( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxkWdtWyxItzyxTk

D

ytzyxVtzyxI

II

zL

S

IS ,,,,,,,,,,,,,,,,,, 2

,0

1−

∫∇

∂

∂Ω+× µ (3a)

( )( )

( )( )

( )( ) ( )++

+

= tzyxf

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVVV δ

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂,,

,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) ( ) ×−

∫∇

∂

∂Ω+

+ TzyxkWdtWyxVtzyx

Tk

D

xz

tzyxVTzyxD

zVI

zL

S

VS

V,,,,,,,,,

,,,,,,

,0

1µ

∂

∂

∂

∂

( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxkWdtWyxVtzyxTk

D

ytzyxVtzyxI VV

zL

S

VS ,,,,,,,,,,,,,,,,,, 2

,0

1 −

∫∇

∂

∂Ω+× µ

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

I

I

I

I

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,


29

( ) ( ) ( ) ( ) +

∫Φ∇

∂

∂Ω+

Φ+

Φ

Φ

zL

IS

SII

IWdtWyxtzyx

Tk

D

xz

tzyxTzyxD

z 01

,,,,,,,,,

,,, µ∂

∂

∂

∂

( ) ( ) ( ) ( ) ++

∫Φ∇

∂

∂Ω+

ΦtzyxITzyxkWdtWyxtzyx

Tk

D

yI

zL

IS

SI ,,,,,,,,,,,,0

1µ

( ) ( ) ( ) ( )tzyxftzyxITzyxkIII

δ,,,,,,,,2

, Φ++ (5a)

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

V

V

V

V

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) +

∫Φ∇

∂

∂Ω+

Φ+

Φ

Φ

zL

VS

SVV

VWdtWyxtzyx

Tk

D

xz

tzyxTzyxD

z 01

,,,,,,,,,

,,, µ∂

∂

∂

∂

( ) ( ) ( ) ( ) ++

∫Φ∇

∂

∂Ω+


Tk

D

yI

zL

IS

SI ,,,,,,,,,,,,0

1µ

( ) ( ) ( ) ( )tzyxftzyxVTzyxkVVV

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

Dz

ytzyx

Tk

Dz

xt

tzyxCCS

S

CS

S

C δµαµα ,,,,,,,,,,,

1111

1 +

∇

∂

∂Ω+

∇

∂

∂Ω=

∂

∂ (1b)

( ) ( ) ( ) +

∇

∂

∂Ω+

∇

∂

∂Ω= tzyx

Tk

Dz

ytzyx

Tk

D

xz

t

tzyxIS

IS

IS

IS

I,,,,,,

,,,11

1 µαµα∂

∂

( ) ( ) ( ) ( )TzyxkTzyxktzyxfVIVIIIII

,,,,,,,,,11,

2

1αααδ −−+ (3b)

( ) ( ) ( ) +

∇

∂

∂Ω+

∇

∂

∂Ω= tzyx

Tk

Dz

ytzyx

Tk

D

xz

t

tzyxVS

VS

VS

VS

V,,,,,,

,,,1111

1 µαµα∂

∂

( ) ( ) ( ) ( )TzyxkTzyxktzyxfVIVIVVVV

,,,,,,,,,11,

2

1αααδ −−+

( ) ( ) ( ) +

∇

∂

∂Ω+

∇

∂

∂Ω=

Φ Φ

Φ

Φ

Φ tzyxTk

D

yztzyx

Tk

D

xz

t

tzyxS

SI

IS

SI

I

I ,,,,,,,,,

1111

1 µαµα∂

∂

( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxktzyxfIIII

,,,,,,,,,,,,,,2

,+++ Φ δ (5b)

( ) ( ) ( ) +

∇

∂

∂Ω+

∇

∂

∂Ω=

Φ Φ

Φ

Φ

Φ tzyxTk

D

yztzyx

Tk

D

xz

t

tzyxS

SV

VS

SV

V

V ,,,,,,,,,

1111

1 µαµα∂

∂

( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxktzyxfVVVV

,,,,,,,,,,,,,, 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


30

( ) ( ) ( )( ) ( )

( )∫ ×

++∇

∂

∂Ω=

t

SLSC

V

zyxV

V

zyxVzyxTzyxD

xtzyxC

02*

2

2*1111

,,,,,,1,,,,,,,,,

τς

τςτµα

( )( ) ( )

( )∫ ×

+∇

∂

∂Ω+

+×

tCS

SLSC

CS

TzyxPTk

zzyxTzyxD

yd

TzyxPTk

z

0

1

11

1

,,,1,,,,,,

,,,1

γ

γ

γ

γ αξτµατ

αξ

( ) ( )( )

( )zyxfdV

zyxV

V

zyxVC

,,,,,,,,

12*

2

2*1 +

++× τ

τς

τς (1c)

( ) ( ) ( ) +∫ ∇∂

∂Ω+∫ ∇

∂

∂Ω=

t

S

IS

I

t

S

IS

Idzyx

Tk

D

yzdzyx

Tk

D

xztzyxI

011

0111 ,,,,,,,,, ττµαττµα

( ) ( ) ( )∫−∫−+t

VIVI

t

IIIIdTzyxkdTzyxkzyxf

0,11

0,

2

1,,,,,,,, ταατα (3c)

( ) ( ) ( ) +∫ ∇∂

∂Ω+∫ ∇

∂

∂Ω=

t

S

IS

V

t

S

IS

V dzyxTk

D

yzdzyx

Tk

D

xztzyxV

011

0111 ,,,,,,,,, ττµαττµα

( ) ( ) ( )∫−∫−+t

VIVI

t

VVVVdTzyxkdTzyxkzyxf

0,11

0,

2

1,,,,,,,, ταατα

( ) ( ) ( ) +∫ ∇∂

∂Ω+∫ ∇

∂

∂Ω=Φ

Φ

Φ

Φ

Φ

t

S

SI

I

t

S

SI

IIdzyx

Tk

D

xzdzyx

Tk

D

xztzyx

011

0111

,,,,,,,,, ττµαττµα

( ) ( ) ( ) ( ) ( )∫+∫++ Φ

t

II

t

IIdzyxITzyxkdzyxITzyxkzyxf

0

2

,0

,,,,,,,,,,,,,, ττττ (5c)

( ) ( ) ( ) +∫ ∇∂

∂Ω+∫ ∇

∂

∂Ω=Φ

Φ

Φ

Φ

Φ

t

S

SV

V

t

S

SV

VVdzyx

Tk

D

xzdzyx

Tk

D

xztzyx

011

0111

,,,,,,,,, ττµαττµα

( ) ( ) ( ) ( ) ( )∫+∫++ Φ

t

VV

t

VVdzyxVTzyxkdzyxVTzyxkzyxf

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 0 011 ,,,

1 xL yLzL

zyx

tdxdydzdtzyxLLL

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

Cxdydzdzyxf

LLL 0 0 01

,,1

α , ( )

4

3

4

1

2

3

2

4

2

3

14

44 a

Aa

a

aLLLBaB

a

Aa zyx

I

+−

Θ+Θ+−

+=α ,

( )

Θ−−∫ ∫ ∫

Θ=

zyxIII

xL yLzL

I

IIV

VLLLSxdydzdzyxf

S001

0 0 0100

1 ,,1

αα

α .

Here ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−Θ=Θ

0 0 0 011, ,,,,,,,,,

xL yLzL

ji

ijtdxdydzdtzyxVtzyxITzyxktS ρρρρ , ( )×−= 0000

2

004 VVIIIV SSSa


31

00IIS× ,

0000

2

0000003 VVIIIVIIIV SSSSSa −+= , ( ) ×Θ+∫ ∫ ∫= 222

0 0 0

2

00002,,

zyx

xL yLzL

VIVIVLLLxdydzdzyxfSSa

( ) ( ) ×∫ ∫ ∫−Θ−∫ ∫ ∫+×xL yL

zL

IVVzyx

xL yLzL

IIIVVIVxdydzdzyxfSLLLxdydzdzyxfSSS

0 0 000

222

0 0 0000000 ,,,,2

2

00IVS× , ( )∫ ∫ ∫=xL yL

zL

IIVxdydzdzyxfSa

0 0 0001 ,, , ( )

2

0 0 0000

,,

∫ ∫ ∫=xL yL

zL

IVVxdydzdzyxfSa ,

4

2

2

4

2

3248

a

a

a

ayA Θ−Θ+= , 3 323 32

4

2

6qpqqpq

a

aB ++−−++

Θ= , ×

Θ−=

4

31

04a

aaLLLaq zyx

2

4

2

1

4

222

3

4

3

2

3

4

2

32

22

4

02

2

4

2

3

8544

824 a

aLLL

a

a

a

aa

a

a

a

azyx

Θ−

Θ−

Θ−ΘΘ−

Θ× , −

Θ−Θ=

2

4

31402

12

4

a

aaLLLaap

zyx

42 18aaΘ− ,

( )∫ ∫ ∫+Θ

+Θ

= ΦΦ

xL yLzL

I

zyxzyx

II

zyx

I

Ixdydzdzyxf

LLLLLL

S

LLL

R

0 0 0

201

1 ,,1

α

( )∫ ∫ ∫+Θ

+Θ

= ΦΦ

xL yLzL

V

zyxzyx

VV

zyx

V

Vxdydzdzyxf

LLLLLL

S

LLL

R

0 0 0

201

1 ,,1

α ,

Here ( ) ( ) ( )∫ ∫ ∫ ∫−Θ=Θ

0 0 0 01 ,,,,,,

xL yLzL

i

IitdxdydzdtzyxITzyxktRρ .

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

tzyxCC ςς

αξ

γ

γ

( )( ) ( ) ( )

( )( )

×

∂

∂

++

∂

∂+

∂

∂×

y

tzyxC

V

tzyxV

V

tzyxV

yx

tzyxCTzyxD

L

,,,,,,,,,1

,,,,,, 1

2*

2

2*1

1 ςς

( ) ( )[ ]( )

( ) ( )( )

×

++

∂

∂+

+

+×2*

2

2*1

12 ,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

zTzyxP

tzyxCTzyxD C

Lςς

αξ

γ

γ

( ) ( ) ( )[ ]( )

( ) ( ) ++

+

+∂

∂× tzyxf

TzyxP

tzyxC

z

tzyxCTzyxD

C

C

Lδ

αξ

γ

γ

,,,,,

,,,1

,,,,,, 121

( ) ( )[ ] +

∫ +∇∂

∂Ω+

zL

CS

S WdtWyxCtzyxTk

D

x 021 ,,,,,, αµ

( ) ( )[ ]

∫ +∇∂

∂Ω+

zL

CS

S WdtWyxCtzyxTk

D

y 021

,,,,,, αµ (1d)


32

( ) ( ) ( ) ( ) ( )+

+

=

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIII ∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,112

( ) ( ) ( ) ( )[ ] ( ) ×−+−

+ TzyxktzyxITzyxk

z

tzyxITzyxD

zVIIIII

,,,,,,,,,,,,

,,, ,

2

11,

1 α∂

∂

∂

∂

( )[ ] ( )[ ] ( ) ( )[ ]

×∫ +∇∂

∂Ω+++×

zL

ISVIWdtWyxItzyx

xtzyxVtzyxI

0121111

,,,,,,,,,,,, αµαα

( ) ( )[ ]

∫ +∇∂

∂Ω+

×zL

IS

ISIS WdtWyxItzyxTk

D

yTk

D

012 ,,,,,, αµ (3d)

( ) ( ) ( ) ( ) ( )+

+

=

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVV ∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,, 112

( ) ( ) ( ) ( )[ ] ( ) ×−+−

+ TzyxktzyxVTzyxk

z

tzyxVTzyxD

zVIVVVV

,,,,,,,,,,,,

,,,,

2

11,

1 α∂

∂

∂

∂

( )[ ] ( )[ ] ( ) ( )[ ]

×∫ +∇∂

∂Ω+++×

zL

VSVIWdtWyxVtzyx

xtzyxVtzyxI

0121111

,,,,,,,,,,,, αµαα

( ) ( )[ ]

∫ +∇∂

∂Ω+

×zL

VS

VSVS WdtWyxVtzyxTk

D

yTk

D

012

,,,,,, αµ

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

I

I

I

I

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,112

( ) ( )[ ] ( ) ( )++

∫ Φ+∇∂

∂Ω+ Φ


Tk

D

xII

zL

IIS

SI ,,,,,,,,,,,, 2

,0

12αµ

( ) ( )[ ] ( ) ( ) ++

∫ Φ+∇∂

∂Ω+ Φ


Tk

D

yI

zL

IIS

SI ,,,,,,,,,,,,0

12αµ

( ) ( ) ( ) ( )tzyxfz

tzyxTzyxD

z I

I

Iδ

∂

∂

∂

∂,,

,,,,,, 1

ΦΦ +

Φ+ (5d)

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyx V

V

V

V

V

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,, 112

( ) ( )[ ] ( ) ( )++

∫ Φ+∇∂

∂Ω+ Φ

ΦtzyxVTzyxkWdtWyxtzyx

Tk

D

xVV

zL

VVS

SV ,,,,,,,,,,,,2

,0

12αµ

( ) ( )[ ] ( ) ( )++

∫ Φ+∇∂

∂Ω+ Φ

ΦtzyxVTzyxkWdtWyxtzyx

Tk

D

yV

zL

VVS

SV ,,,,,,,,,,,,0

12αµ

( ) ( ) ( ) ( )tzyxfz

tzyxTzyxD

z V

V

Vδ

∂

∂

∂

∂,,

,,,,,, 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


33

( ) ( )( )[ ]

( )( ) ( )

( )∫ ×

++

+

+∂

∂=

tC

L

V

zyxV

V

zyxV

TzyxP

zyxCTzyxD

xtzyxC

02*

2

2*1

12

2

,,,,,,1

,,,

,,,1,,,,,,

τς

τς

ταξ

γ

γ

( )( )

( ) ( )( )

( )∫ ×

∂

∂

++

∂

∂+

∂

∂×

t

Ly

zyxC

V

zyxV

V

zyxVTzyxD

yd

x

zyxC

0

1

2*

2

2*1

1 ,,,,,,,,,1,,,

,,, ττς

τςτ

τ

( )[ ]( )

( ) ( )( )

( )∫ ×

∂

∂

++

∂

∂+

+

+×t

C

z

zyxC

V

zyxV

V

zyxV

zTzyxP

tzyxC

0

1

2*

2

2*1

12 ,,,,,,,,,1

,,,

,,,1

ττς

τς

αξ

γ

γ

( )( )[ ]

( )( ) ( )[ ] ×∫ ∫ +∇

∂

∂+

+

+×t zL

CS

C

LWyxCzyx

xd

TzyxP

zyxCTzyxD

0 012

12 ,,,,,,,,,

,,,1,,, τατµτ

ταξ

γ

γ

( ) ( )[ ] ( )zyxfdWdWyxCTk

Dzyx

ydWd

Tk

DC

t zL

C

S

S

S ,,,,,,,,0 0

12 +∫ ∫ +∇∂

∂Ω+× ττατµτ (1e)

( ) ( ) ( ) ( ) ( )+∫+∫=

t

I

t

I dy

zyxITzyxD

yd

x

zyxITzyxD

xtzyxI

0

1

0

1

2

,,,,,,

,,,,,,,,, τ

∂

τ∂

∂

∂τ

∂

τ∂

∂

∂

( )( )

( ) ( ) ( )[ ] −∫ +−+∫+t

IIII

t

IdzyxITzyxkzyxfd

z

zyxITzyxD

z 0

2

12,0

1 ,,,,,,,,,,,

,,, ττατ∂

τ∂

∂

∂

( ) ( )[ ] ( )[ ] ( )×∫ ∇∂

∂Ω+∫ ++−

t

S

ISt

VIVIzyx

Tk

D

xdzyxVzyxITzyxk

001212, ,,,,,,,,,,,, τµττατα

( )[ ] ( ) ( )[ ]∫ ∫ +∇∂

∂Ω+∫ +×

t zL

I

IS

S

zL

IdWdWyxI

Tk

Dzyx

ydWdWyxI

0 012

012

,,,,,,,,, ττατµττα (3e)

( ) ( )( )

( )( )

+∫+∫=t

V

t

V dy

zyxVTzyxD

yd

x

zyxVTzyxD

xtzyxV

0

1

0

1

2

,,,,,,

,,,,,,,,, τ

∂

τ∂

∂

∂τ

∂

τ∂

∂

∂

( )( )

( ) ( ) ( )[ ] −∫ +−+∫+t

VVVV

t

VdzyxVTzyxkzyxfd

z

zyxVTzyxD

z 0

2

12,0

1 ,,,,,,,,,,,

,,, ττατ∂

τ∂

∂

∂

( ) ( )[ ] ( )[ ] ( )×∫ ∇∂

∂Ω+∫ ++−

t

S

VSt

VIVIzyx

Tk

D

xdzyxVzyxITzyxk

001212, ,,,,,,,,,,,, τµττατα

( )[ ] ( ) ( )[ ]∫ ∫ +∇∂

∂Ω+∫ +×

t zL

I

VS

S

zL

IdWdWyxV

Tk

Dzyx

ydWdWyxV

0 012

012 ,,,,,,,,, ττατµττα

( ) ( ) ( ) ( )∫ ×

Φ+∫

Φ=Φ Φ

tI

tI

IIy

zyx

yd

x

zyxTzyxD

xtzyx

0

1

0

1

2

,,,,,,,,,,,,

∂

τ∂

∂

∂τ

∂

τ∂

∂

∂

( ) ( )( )

( ) ×∫∇∂

∂Ω+∫

Φ+×

Φ

ΦΦ

tSI

S

tI

II Tk

Dzyx

xd

z

zyxTzyxD

zdTzyxD

00

1 ,,,,,,

,,,,,, τµτ∂

τ∂

∂

∂τ

( )[ ] ( ) ( )[ ] ×∫ ∫ Φ+∇∂

∂Ω+∫ Φ+× ΦΦ

t zL

IIS

zL

IIWdWyxzyx

ydWdWyx

0 012

012

,,,,,,,,, τατµττα

( ) ( ) ( ) ( ) ( )zyxfdzyxITzyxkdzyxITzyxkdTk

D

I

t

I

t

II

SI ,,,,,,,,,,,,,,00

2

, Φ

Φ+∫+∫+× τττττ (5e)

( ) ( ) ( ) ( )∫ ×

Φ+∫

Φ=Φ Φ

tV

tV

VVy

zyx

yd

x

zyxTzyxD

xtzyx

0

1

0

1

2

,,,,,,,,,,,,

∂

τ∂

∂

∂τ

∂

τ∂

∂

∂


34

( ) ( )( )

( ) ×∫∇∂

∂Ω+∫

Φ+×

Φ

ΦΦ

tSV

S

tV

VV Tk

Dzyx

xd

z

zyxTzyxD

zdTzyxD

00

1 ,,,,,,

,,,,,, τµτ∂

τ∂

∂

∂τ

( )[ ] ( ) ( )[ ] ×∫ ∫ Φ+∇∂

∂Ω+∫ Φ+× ΦΦ

t zL

VVS

zL

VVWdWyxzyx

ydWdWyx

0 012

012 ,,,,,,,,, τατµττα

( ) ( ) ( ) ( ) ( )zyxfdzyxVTzyxkdzyxVTzyxkdTk

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 0122 ,,,,,,

1 xL yLzL

zyx

tdxdydzdtzyxtzyxLLL

ρρα ρ . (10)

Substitution of the relations (1e), (3e), (5e) into relation (10) gives us possibility to obtain rela-

tions for required average values α 2ρ

α2C=0, α2ΦI =0, α2ΦV =0, ( )

4

3

4

1

2

3

2

4

2

3

24

44 b

Eb

b

bLLLFaF

b

Eb zyx

V

+−

Θ+Θ+−

+=α ,

( )00201

11021001200

2

2

2

2

IVVIV

VIVVzyxVIVVVVVVV

ISS

SSLLLSSSC

α

ααα

+

−−Θ++−−= ,

where 00

2

0000

2

004

11IIVV

zyx

VVIV

zyx

SSLLL

SSLLL

bΘ

−Θ

= , ( )+Θ++Θ

−=zyxIVVV

zyx

VVII LLLSSLLL

SSb

1001

0000

32

( ) ( )3333

10

2

00

1001

2

00

011001

0000 22zyx

IVIV

zyxIVVV

zyx

IV

zyxIVIIIV

zyx

VVIV

LLL

SSLLLSS

LLL

SLLLSSS

LLL

SS

Θ−Θ++

Θ+Θ+++

Θ+ ,

( ) ( ) ( ) ×++Θ+Θ+−−++Θ

= 0110

2

01101102

0000

2 22IVIIzyxzyxVVIVVIVVV

zyx

VVII SSLLLLLLSSCSSLLL

SSb

( ) ( ) +Θ

++Θ+Θ+++Θ

+Θ

×zyx

VVIV

IVIIzyxzyxIVIIIV

zyx

IV

zyx

VVIV

LLL

SSSSLLLLLLSSS

LLL

S

LLL

SS0001

0110

2

011001

000001 222

( )( ) ( ) ×−−−+Θ+Θ+++Θ

+11021001011001

00 222IVVVVIVzyxVVzyxIVIIIV

zyx

IV SSCSLLLSLLLSSSLLL

S

zyx

IVIV

IV

zyx

IVI

zyx

IV

LLL

SSS

LLL

SC

LLL

S

Θ−

Θ+

Θ× 0010

012222

2

00

2

00 2 , ( ++Θ

++= 1001

0211

001 2IVVV

zyx

VVVIV

IISS

LLL

CSSSb

) ( )( ) −Θ

−Θ++++ΘΘ

+Θ+zyx

IVIV

zyIVVVIVIIzyx

zyx

IV

zyxLLL

SSLLLSSSSLLL

LLL

SLLL

2

0110

10010110

01 22

( )( )010011021001

00 223IVIVIIVVVVzyxIIIV

zyx

IV SSCSSCLLLSSLLL

S+−−Θ++

Θ− , ×

Θ=

zyx

II

LLL

Sb

00

0

( ) ( )( ) ×Θ

−−−−−++Θ

Θ−+×

zyx

IVVVV

IVIVVVVIVIIzyx

zyx

IV

VVIVLLL

SSCSSSCSSLLL

LLL

SSS

1102

0111020110

012

02002

( ) 2

01011001 22IVIIVIIzyxIV

SCSSLLLS +++Θ× , −Θ

−Θ

+Θ

=zyx

IIII

zyx

III

IV

zyx

VI

ILLL

SS

LLL

SS

LLLC 202000

2

1

00

11ααα

,

zyx

IV

LLL

S

Θ− 11 ,

110200

2

10011 IVVVVVVIVVIV SSSSC −−+= ααα , 4

2

2

4

2

32 48a

a

a

ayE Θ−Θ+= , +

Θ=

4

2

6a

aF


35

3 323 32rsrrsr ++−−++ , ×

Θ−Θ−

Θ−

Θ−

Θ=

4

2

32

23

4

3

2

3

4

31

02

4

2

3

454

424 b

bb

b

b

b

bbLLLb

b

br

zyx

2

4

2

1

4

222

2

4

2

088 b

bLLL

bb

zyx

Θ−

Θ× ,

4

2

2

4

31402

1812

4

b

b

b

bbLLLbbs

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

tzyxTzzK

t

tzyxuz x

∂

∂−=

∂

∂ ,,,,,,2

1

2

βρ , ( )( )

( ) ( ) ( )y

tzyxTzzK

t

tzyxuz

y

∂

∂−=

∂

∂ ,,,,,,2

1

2

βρ ,

( ) ( ) ( ) ( ) ( )z

tzyxTzzK

t

tzyxuz 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 00 001 ,,,,,,,,,

ϑϑ

ϑττρ

βϑττ

ρ

βddzyxT

xz

zzKddzyxT

xz

zzKutzyxu

t

xx ,

( ) ( ) ( )( )

( ) ( ) ( )( )

( )∫ ∫∂

∂−∫ ∫

∂

∂+=

∞

0 00 001 ,,,,,,,,,

ϑϑ

ϑττρ

βϑττ

ρ

βddzyxT

yz

zzKddzyxT

yz

zzKutzyxu

t

yy ,

( ) ( ) ( )( )

( ) ( ) ( )( )

( )∫ ∫∂

∂−∫ ∫

∂

∂+=

∞

0 00 001

,,,,,,,,,ϑϑ

ϑττρ

βϑττ

ρ

βddzyxT

zz

zzKddzyxT

zz

zzKutzyxu

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

zEzK

x

tzyxu

z

zEzK

t

tzyxuz

yxx,,,

13

,,,

16

5,,, 1

2

2

1

2

2

2

2

σσρ

( ) ( )( )[ ]

( ) ( )( ) ( )

( )+

∂

∂−

∂

∂+

∂

∂

++

∂∂

∂×

x

tzyxTzzK

z

tzyxu

y

tzyxu

z

zE

yx

tzyxuzyy ,,,,,,,,,

12

,,,2

1

2

2

1

2

1

2

βσ

( ) ( )( )[ ]

( )zx

tzyxu

z

zEzK z

∂∂

∂

+++

,,,

13

1

2

σ

( )( ) ( )

( )[ ]( ) ( )

( ) ( ) ( )+

∂

∂−

∂∂

∂+

∂

∂

+=

∂

∂

y

tzyxTzzK

yx

tzyxu

x

tzyxu

z

zE

t

tzyxuz xyy ,,,,,,,,,

12

,,,1

2

2

1

2

2

2

2

βσ

ρ

( )( )[ ]

( ) ( ) ( ) ( )( )[ ]

( ) +

++∂

∂+

∂

∂+

∂

∂

+∂

∂+ zK

z

zE

y

tzyxu

y

tzyxu

z

tzyxu

z

zE

z

yzy

σσ 112

5,,,,,,,,,

12 2

1

2

11

( )( )

( )[ ]( )

( )( )

yx

tzyxuzK

zy

tzyxu

z

zEzK

yy

∂∂

∂+

∂∂

∂

+−+

,,,,,,

16

1

2

1

2

σ


36

( )( ) ( ) ( ) ( ) ( )

×

∂∂

∂+

∂∂

∂+

∂

∂+

∂

∂=

∂

∂

zy

tzyxu

zx

tzyxu

y

tzyxu

x

tzyxu

t

tzyxuz

yxzzz,,,,,,,,,,,,,,, 1

2

1

2

2

1

2

2

1

2

2

2

2

ρ

( )( )[ ]

( )( ) ( ) ( ) ( )

( )[ ]×

++

∂

∂+

∂

∂+

∂

∂

∂

∂+

+×

z

zE

z

tzyxu

y

tzyxu

x

tzyxuzK

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

zEzKddzyxu

xz

zEzK

ztzyxu

t

xxσ

ϑττσρ

ϑ

13,,,

16

51,,,

0 012

2

2

( )( ) ( ) ( ) ×

∫ ∫

∂

∂+∫ ∫

∂

∂+∫ ∫

∂∂

∂×

t

z

t

y

t

yddzyxu

zddzyxu

yddzyxu

yxz 0 012

2

0 012

2

0 01

2

,,,,,,,,,1 ϑϑϑ

ϑττϑττϑττρ

( )( ) ( )[ ] ( )

( ) ( ) ( )( )[ ]

( ) ( )( )

×−

++∫ ∫

∂∂

∂+

+×

z

zzK

z

zEzKddzyxu

zxzzz

zE t

zρ

β

σϑττ

ρσρ

ϑ

13,,,

1

12 0 01

2

( )( )

( ) ( )( )[ ]

( )( )

×−∫ ∫∂

∂

++−∫ ∫

∂

∂×

∞

zddzyxu

xz

zEzK

zddzyxT

xx

t

ρϑττ

σρϑττ

ϑϑ 1,,,

16

51,,,

0 012

2

0 0

( ) ( )( )[ ]

( ) ( )( )

( )

+∫ ∫

∂

∂

+−∫ ∫

∂∂

∂

+−×

∞∞

0 012

2

0 01

2

,,,1

,,,13

ϑϑ

ϑττσ

ϑττσ

ddzyxuyz

zEddzyxu

yxz

zEzK

yy

( )( )

( ) ( )( )[ ]

( ) ×∫ ∫∂∂

∂

++−

∫ ∫

∂

∂+

∞∞

0 01

2

0 012

2

,,,132

1,,,

ϑϑ

ϑττσρ

ϑττ ddzyxuzxz

zEzK

zddzyxu

zzz

( )( ) ( )

( )( )∫ ∫

∂

∂++×

∞

0 00 ,,,

1 ϑ

ϑττρ

β

ρddzyxT

xz

zzKu

zx

( ) ( )( ) ( )[ ]

( ) ( ) +

∫ ∫

∂∂

∂+∫ ∫

∂

∂

+=

t

x

t

xyddzyxu

yxddzyxu

xzz

zEtzyxu

0 01

2

0 012

2

2 ,,,,,,12

,,,ϑϑ

ϑττϑττσρ

( )( )( )

( ) ( ) ( )( )[ ]

++

∫ ∫∂

∂+∫ ∫

∂∂

∂+

+×

z

zEddzyxu

yddzyxu

yxz

zK

z

t

x

t

yσ

ϑττϑττρσ

ϑϑ

112

5,,,,,,

1

1

0 012

2

0 01

2

( )( )

( )( )

( ) ( ) ×

∫ ∫

∂

∂+∫ ∫

∂

∂

+∂

∂+

+t

z

t

yddzyxu

yddzyxu

zz

zE

zzzK

0 01

0 01

,,,,,,1

1 ϑϑ

ϑττϑττσρ

( )( )

( )( )

( )( )

( )[ ]( ) ( ) ×∫ ∫

∂∂

∂

−+

−∫ ∫−×t

y

t

ddzyxuzy

zKz

zEddzyxT

z

zzK

z 0 01

2

0 0

,,,16

,,,2

1 ϑϑ

ϑττσ

ϑττρ

β

ρ

( )( )

( ) ( )[ ]( ) ( ) ( ) ×−

∫ ∫

∂∂

∂+∫ ∫

∂

∂

+−×

∞∞

zKddzyxuyx

ddzyxuxzz

zE

zxx

0 01

2

0 012

2

,,,,,,12

1 ϑϑ

ϑττϑττσρρ

( )( )

( )( )( )

( )( )

( )( )[ ]

++∂

∂−∫ ∫

∂∂

∂−∫ ∫×

∞∞

z

zE

yzddzyxu

yxz

zKddzyxT

z

zy

σρϑττ

ρϑττ

ρ

β ϑϑ

112

51,,,,,,

2

2

0 01

2

0 0


37

( ) ( ) ( ) ( ) ×

∫ ∫

∂

∂+∫ ∫

∂

∂

∂

∂−∫ ∫

+∞∞∞

0 01

0 01

0 01 ,,,,,,,,,

ϑϑϑ

ϑττϑττϑττ ddzyxuy

ddzyxuzz

ddzyxuzKzyx

( )( ) ( ) ( )

( )( )

( )[ ]( )

yyuddzyxu

zyz

zEzK

zzz

zE0

0 01

2

,,,16

1

2

1

1+∫ ∫

∂∂

∂

+−−

+×

∞ ϑ

ϑττσρρσ

( ) ( )( ) ( )[ ]

( ) ( )

+∫ ∫

∂

∂+∫ ∫

∂

∂

+=

∞∞

0 012

2

0 012

2

,,,,,,12

,,,ϑϑ

ϑττϑττσρ

ddzyxuy

ddzyxuxzz

zEtzyxu

zzz

( ) ( ) ( )

+∫ ∫

∂

∂

∂

∂+

∫ ∫

∂∂

∂+∫ ∫

∂∂

∂+

∞∞∞

0 01

0 01

2

0 01

2

,,,,,,,,,ϑϑϑ

ϑττϑττϑττ ddzyxuxz

ddzyxuzy

ddzyxuzx

xyx

( ) ( ) ( )( ) ( )

( )( )

×+∂

∂+

∫ ∫

∂

∂+∫ ∫

∂

∂+

∞∞

z

zE

zzzzKddzyxu

zddzyxu

yxx

σρρϑττϑττ

ϑϑ

16

11,,,,,,

0 01

0 01

( ) ( ) ( )

−∫ ∫

∂

∂−∫ ∫

∂

∂−∫ ∫

∂

∂×

∞∞∞

0 01

0 01

0 01

,,,,,,,,,6ϑϑϑ

ϑττϑττϑττ ddzyxuy

ddzyxux

ddzyxuz

yxz

( ) ( )( )( )

( )zz

uddzyxTzz

zzKddzyxu

z0

0 00 01 ,,,,,, +∫ ∫

∂

∂−

∫ ∫

∂

∂−

∞∞ ϑϑ

ϑττρ

βϑττ .

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


38

( ) ( )[ ]∫ ∫ ∫ −Θ=xL yL

zL

zyx

xdydzdzyxzyxCLLL

U0 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)

23

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(Lx2+Ly

2+Lz2)/D0. Curves 2 and 4 corresponds to

annealing time Θ = 0.0057(Lx2+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

13

4


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

23

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 0.5a/L, ξ, ε, γ

0.0

0.1

0.2

0.3

0.4

0.5

Θ D

0 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 0.5a/L, ξ, ε, γ

0.00

0.04

0.08

0.12

Θ D

0 L

-2

3

2

4

1


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.


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.

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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.

Dependence of Charge Carriers Mobility in the P-N-Heterojunctions on Composition of Multilayer...

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