ON ANALYTICAL APPROACH FOR ANALYSIS INFLUENCE OF MISMATCH-INDUCED STRESS IN A HETEROSTRUCTURE ON DISTRIBUTIONS OF CONCENTRATIONS OF DOPANTS IN A MULTIEMITTER HETEROTRANSISTOR

Computer Applications: An International Journal (CAIJ), Vol.2, No.1, February 2015

DOI : 10.5121/caij.2015.2107 71

ON ANALYTICAL APPROACH FOR ANALYSIS INFLUENCE

OF MISMATCH-INDUCED STRESS IN A HETEROSTRUC-

TURE ON DISTRIBUTIONS OF CONCENTRATIONS OF DO-

PANTS IN A MULTIEMITTER HETEROTRANSISTOR

E.L. Pankratov1,3

, E.A. Bulaeva1,2

1 Nizhny Novgorod State University,Russia

2 Nizhny Novgorod State University of Architecture and Civil Engineering, Russia

3 Nizhny Novgorod Academy of the Ministry of Internal Affairs of Russia, Russia

ABSTRACT

In this paper we analytically model technological processes of manufacturing of multiemitter heterotransis-

tors with account relaxation of mismatch-induced stress. Based on this modeling some recommendations to

increase sharpness of p-n- junctions, which included into the transistors, and increasing compactness of the

transistors have been formulated.

KEYWORDS

Multiemitter heterotransistors, distributions of concentrations, influence of mismatch-induced stress, ana-

lytical approach to model technological processes

1.INTRODUCTION In the present time integration degree of elements of integrated circuits is intensively increasing

(p-n-junctions, field-effect and bipolar transistors, ...) [1-14]. At the same time with increasing of

integration degree of elements of integrated circuits dimensions of this elements decreases. It is

attracted an interest increasing of performance of the above elements [1-7]. To increase the per-

formance it should be determine materials with higher values of charge carrier mobility for manu-

facturing elements of integrated circuits [15-18]. Another way to increase the performance is de-

velopment of new and optimization of existing technological processes. It could be used different

approaches to decrease dimensions of elements of integrated circuits. Two of them are laser and

microwave types of annealing of dopants and/or radiation defects [19-21]. Using this approaches

leads to manufacture inhomogenous distribution of temperature in the considered materials. In-

homogeneity of the distribution gives us possibility to decrease dimensions of elements of inte-

grated circuits and their discrete analogs. Another way to decrease dimensions of the elements is

doping of required parts of epitaxial layers of heterostructures by diffusion or ion implantation.

However using the approach leads to necessity to optimize annealing of dopant and /or radiation

defects [22,23]. It is also attracted an interest radiation processing of doped materials. The

processing leads to changing of distributions of concentrations of dopants in doped materials [24].

The changing could leads to decreasing dimensions of the above elements [25].

In this paper we consider manufacturing of multiemitter transistor in the heteristructure, which

consist of a substrate and three epitaxial layers (see Fig. 1). Some sections have been manufac-

tured in the epitaxial layers by using another materials (see Fig. 1). After manufacturing these

section they are should be doped by diffusion or ion implantation. The doping is necessary to

produce required type of conductivity in the sections (n or p). The first step of manufacturing of

the transistor is formation the first epitaxial layer (nearest to the substrate). After finishing of the

formation is doping of the single section by diffusion or ion implantation. Farther we consider


72

manufacturing the second epitaxial layer with own section. After that we consider doping of this

section by diffusion or ion implantation. Farther we consider manufacturing the third epitaxial

layer with several sections. After that we consider doping of these sections by diffusion or ion

implantation. Farther the dopants and/or radiation defects should be annealed. Analysis of dynam-

ics of redistribution of dopants and radiation defects in the considered heterostructure with ac-

count relaxation of mechanical stress is the main aim of the present paper.

Fig. 1a. Structure of the considered heterostructure. The heterostructure include into itself a substrate and

three epitaxial layers with sections, manufactured by using another materials. View from top

Substrate

Collector

Emitter Emitter Emitter

Fig. 1b. Heterostructure, which consist of a substrate and three epitaxial layers with sections, manufactured

by using another materials. View from one side


73

2. Method of solution To solve our aim we shall analyze spatio-temporal distribution of concentration of dopant. We

determine the spatio-temporal distribution of concentration of dopant by solving the following

boundary problems [1,3-5,26,27]

( ) ( ) ( ) ( )+

+

+

=

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxC ,,,,,,,,,,,,

( ) ( ) +

+

zL

S

S WdtWyxCtzyxTk

D

x 0,,,,,,

( ) ( )

+

zL

S

S WdtWyxCtzyxTk

D

y0

,,,,,, (1)

with initial and boundary conditions

C (x,y,z,0)=fC (x,y,z), ( )

0,,,

0

=

=xx

tzyxC,

( )0

,,,=

= xLxx

tzyxC,

( )0

,,,

0

=

=yy

tzyxC,

( )0

,,,=

= yLxy

tzyxC,

( )0

,,,

0

=

=zz

tzyxC,

( )0

,,,=

= zLxz

tzyxC.

The function C(x,y,z,t) describes the spatio-temporal distribution of concentration of dopant; is the atomic volume of the dopant; symbol S is the surficial gradient; the function

( )z

L

zdtzyxC0

,,, describes the surficial concentration of dopant in area of interface between ma-

terials of heterostructure. In this case we assume, that Z-axis is perpendicular to interface between

materials of heterostructure; (x,y,z,t) is the chemical potential; D are DS are the coefficients of volumetric and surficial diffusions (the surficial diffusion is the consequences of mismatch in-

duced stress). Values of diffusion coefficients depend on properties of materials of heterostruc-

ture, speed of heating and cooling of heterostructure, concentration of dopant. Dependences of

dopant diffusion coefficients on parameters could be approximated by the following relations

[3,28-30]

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD

LC

,

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD

SLSS

. (2)

Functions DL (x,y,z,T) and DLS (x,y,z,T) describe the spatial (due to varying of values of dopant

diffusion coefficients in the heterostruicture with varying of coordinate) and temperature (due to

Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing;

the function P (x,y,z,T) describe the limit of solubility of dopant; parameter could be integer in the following interval [1,3] [3] and varying with variation of properties of materials of hetero-structure; function V (x,y,z,t) describes the spatio-temporal distribution of concentration of radia-


74

tion vacancies; V* is the equilibrium distribution of vacancies. Detailed description of concentra-

tional dependence of dopant diffusion coefficient is presented in [3]. We determine the spatio-

temporal distributions of concentration of point radiation defects by solving the following system

of equations [28-30]

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIII

,,,,,,

,,,,,,

,,,

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

+ tzyxITzyxktzyxITzyxk

z

tzyxITzyxD

zVIIII

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

,,,,

2

,

( ) ( ) ( ) +

+

zL

S

IS WdtWyxItzyxTk

D

xtzyxV

0

,,,,,,,,,

( ) ( )

+

zL

S

IS WdtWyxItzyxTk

D

y 0,,,,,, (3)

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVV

,,,,,,

,,,,,,

,,,

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

+ tzyxITzyxktzyxVTzyxk

z

tzyxVTzyxD

zVIVVV

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

,,, ,2

,

( ) ( ) ( ) +

+

zL

S

VS WdtWyxVtzyxTk

D

xtzyxV

0

,,,,,,,,,

( ) ( )

+

zL

S

VS WdtWyxVtzyxTk

D

y 0,,,,,,

with boundary and initial conditions

( )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)

Function I (x,y,z,t) describes the spatio-temporal distribution of concentration of radiation intersti-

tials; I* is the equilibrium distribution of interstitials; DI(x,y,z,T), DV(x,y,z,T), DIS(x,y,z,T),

DVS(x,y,z,T) are the coefficients of volumetric and surficial diffusions of interstitials and vacan-

cies, respectively; terms V2(x,y,z,t) and I2(x,y,z,t) correspond to generation of divacancies and di-

interstitials, respectively (see, for example, [30] and appropriate references in this book);

kI,V(x,y,z,T) and kI,I(x,y,z,T) are the parameters of recombination of point radiation defects, kV,V(x,

y,z,T) is the parameter generation of their complexes.

We determine spatio-temporal distributions of concentrations of divacancies V(x,y,z,t) and diin-terstitials I(x,y,z,t) by solving the following boundary problem [28-30]


75

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxIII

II

,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) +

+

+

zI

I

L

IS

SI WdtWyxtzyxTk

D

xz

tzyxTzyxD

z 0,,,,,,

,,,,,,

( ) ( ) ( ) ( ) ++

+ tzyxITzyxkWdtWyxtzyx

Tk

D

yII

L

IS

ISz

,,,,,,,,,,,, 2,0

( ) ( )tzyxITzyxkI

,,,,,,+ (5)

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxVVV

VV

,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) +

+

+

zV

V

L

VS

SV WdtWyxtzyxTk

D

xz

tzyxTzyxD

z 0,,,,,,

,,,,,,

( ) ( ) ( ) ( ) ++

+

tzyxVTzyxkWdtWyxtzyx

Tk

D

yVV

L

VS

Sz

V ,,,,,,,,,,,, 2,0

( ) ( )tzyxVTzyxkV

,,,,,,+

with boundary and initial conditions

( )0

,,,

0

=

=x

I

x

tzyx

,

( )0

,,,=

= xLxx

tzyxI

,

( )0

,,,

0

==y

y

tzyxI

,

( )0

,,,=

= yLyy

tzyxI

,

( )0

,,,

0

=

=z

I

z

tzyx

,

( )0

,,,=

= zLzz

tzyxI

,

( )0

,,,

0

=

=x

V

x

tzyx

,

( )0

,,,=

= xLxx

tzyxV

,

( )0

,,,

0

==y

y

tzyxV

,

( )0

,,,=

= yLyy

tzyxV

,

( )0

,,,

0

==z

z

tzyxV

,

( )0

,,,=

= zLz

V

z

tzyx

,

I (x,y,z,0)=fI (x,y,z), V (x,y,z,0)=fV (x,y,z). (6)

Here DI(x,y,z,T), DV(x,y,z,T), DIS (x,y,z,T) and DVS(x,y,z,T) are the coefficients of volumetric

and surficial diffusions of complexes of radiation defects; kI(x,y,z,T) and kV (x,y,z,T) are the para-

meters of decay these complexes.

Chemical potential in the Eq.(1) could be determined by the following relation [26]

=E(z) [ij [uij(x,y,z,t)+uji(x,y,z,t)]/2, (7)

where E(z) is the Young modulus, ij 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


76

( ) ( )( ) ( ) ( )

+

+

=

i

j

j

i

i

j

j

i

x

tzyxu

x

tzyxu

x

tzyxu

x

tzyxutzyx

,,,,,,

2

1,,,,,,,,,

( )( )

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

tzyxu

z

zij

k

kij

ij2

,,,3,,,

21000

+

,

where is Poisson coefficient; 0 = (as-aEL)/aEL is the mismatch parameter; as, aEL are lattice dis-tances of the substrate and the epitaxial layer; K is the modulus of uniform compression; is the thermal expansion coefficient; Tr is the equilibrium temperature, which has been assumed as

room temperature. Components of displacement vector could be obtained by solution of the fol-

lowing equations [27]

( ) ( ) ( )( ) ( )

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz xz

xyxxx

+

+

=

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

2

( )( ) ( ) ( ) ( )

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz

yzyyyxy

+

+

=

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

2

( ) ( ) ( )( ) ( )

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz zz

zyzxz

+

+

=

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

2 ,

where ( )

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

+

+

+=

k

k

ij

k

kij

i

j

j

i

ijx

tzyxu

x

tzyxu

x

tzyxu

x

tzyxu

z

zE ,,,,,,

3

,,,,,,

12

( ) ( ) ( ) ( )[ ]r

TtzyxTzKzzK ,,, , (z) is the density of materials of heterostructure, ij Is

the Kronecker symbol. With account the relation for ij 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

( )( )[ ]

( ) ( )( )

( )( )[ ]

( )

+++

+

++

zx

tzyxu

z

zEzK

z

tzyxu

y

tzyxu

z

zE zzy ,,,

13

,,,,,,

12

2

2

2

2

2

( ) ( ) ( )x

tzyxTzzK

,,,

( )( ) ( )

( )[ ]( ) ( )

( ) ( )( )

+

+

+=

y

tzyxTzzK

yx

tzyxu

x

tzyxu

z

zE

t

tzyxuz x

yy ,,,,,,,,,

12

,,, 2

2

2

2

2

( )( )[ ]

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

( ) +

++

+

+

+

+ zK

z

zE

y

tzyxu

y

tzyxu

z

tzyxu

z

zE

z

yzy

112

5,,,,,,,,,

12 2

2

( )( )

( )[ ]( )

( )( )

yx

tzyxuzK

zy

tzyxu

z

zEzK

yy

+

++

,,,,,,

16

22

(8)

( )( ) ( )

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

+

+

+

+

+=

zy

tzyxu

zx

tzyxu

y

tzyxu

x

tzyxu

z

zE

t

tzyxuz

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

12

,,,22

2

2

2

2

2

2

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

+

+

+

z

tzyxTzzK

z

tzyxu

y

tzyxu

x

tzyxuzK

z

xyx,,,,,,,,,,,,


77

( )( )

( ) ( ) ( ) ( )

+

+

z

tzyxu

y

tzyxu

x

tzyxu

z

tzyxu

z

zE

z

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

616

1

.

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 determine spatio-temporal distributions of concentrations of dopant and radiati-on defects by solving the Eqs.(1), (3) and (5) framework standard method of averaging of function corrections

[25,31]. Previously we transform the Eqs.(1), (3) and (5) to the following form with account ini-

tial distributions of the considered concentrations

( ) ( ) ( ) ( )+

+

+

=

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxC ,,,,,,,,,,,,

( ) ( ) ( ) ( ) +

++

zL

S

S

CWdtWyxCtzyx

Tk

D

xtzyxf

0

,,,,,,,,

( ) ( )

+

zL

S

S WdtWyxCtzyxTk

D

y 0,,,,,, (1a)

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIII

,,,,,,

,,,,,,

,,,

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

+ tzyxftzyxVtzyxITzyxk

z

tzyxITzyxD

zIVII

,,,,,,,,,,,

,,,,,,

,

( ) ( ) ( ) ( ) +

+

zL

S

IS

IIWdtWyxItzyx

Tk

D

xtzyxITzyxk

0

2

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

( ) ( )

+

zL

S

IS WdtWyxItzyxTk

D

y 0,,,,,, (3a)

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVV

,,,,,,

,,,,,,

,,,

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

+ tzyxftzyxVtzyxITzyxk

z

tzyxVTzyxD

zVVIV

,,,,,,,,,,,

,,,,,,

,

( ) ( ) ( ) ( ) +

+

zL

S

VS

VVWdtWyxVtzyx

Tk

D

xtzyxVTzyxk

0

2

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

( ) ( )

+

zL

S

VS WdtWyxVtzyxTk

D

y 0,,,,,,

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

I

I

I

I

,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) +

+

+

zI

I

L

IS

SI WdtWyxtzyxTk

D

xz

tzyxTzyxD

z 0,,,,,,

,,,,,,


78

( ) ( ) ( ) ( ) ++

+

tzyxITzyxkWdtWyxtzyx

Tk

D

yI

L

IS

Sz

I ,,,,,,,,,,,,0

( ) ( ) ( ) ( )tzyxftzyxITzyxkIII

,,,,,,,, 2, ++ (5a)

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

V

V

V

V

,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) +

+

+

zV

V

L

VS

SV WdtWyxtzyxTk

D

xz

tzyxTzyxD

z 0,,,,,,

,,,,,,

( ) ( ) ( ) ( ) ++

+


Tk

D

yV

L

VS

Sz

V ,,,,,,,,,,,,0

( ) ( ) ( ) ( )tzyxftzyxVTzyxkVVV

,,,,,,,, 2, ++ .

Farther we replace the required functions in right sides of Eqs. (1a), (3a) and (5a) on their not yet

known average values 1. In this situation we obtain equations for the first-order approximations of the considered concentrations in the following form

( )( ) ( ) +

+

=

tzyx

Tk

Dz

ytzyx

Tk

Dz

xt

tzyxCS

S

CS

S

C,,,,,,

,,,11

1

( ) ( )tzyxfC

,,+ (1b)

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

+

+

+=

+

+

+=

TzyxkTzyxktzyxTk

Dz

y

tzyxTk

D

xztzyxf

t

tzyxV

TzyxkTzyxktzyxTk

Dz

y

tzyxTk

D

xztzyxf

t

tzyxI

VIVIVVVS

VS

V

S

VS

VV

VIVIIIIS

IS

I

S

IS

II

,,,,,,,,,

,,,,,,,,

,,,,,,,,,

,,,,,,,,

,11,

2

11

1

1

,11,

2

11

1

1

(3b)

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

+

+

+

+=

+

+

+

+=

tzyxVTzyxktzyxTk

D

yztzyxV

TzyxktzyxTk

D

xztzyxf

t

tzyx

tzyxITzyxktzyxTk

D

yztzyxI

TzyxktzyxTk

D

xztzyxf

t

tzyx

VVS

S

VS

SV

IIS

S

IS

SI

V

V

V

VV

I

I

I

II

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

,,,,,,,,,,,

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

,,,,,,,,,,,

2

,1

1

1

2

,1

1

1

(5b)

Integration of the left and right sides of Eqs. (1b), (3b) and (5b) gives us possibility to obtain rela-

tions for above first-order approximations in the following form


79

( ) ( ) ( )( ) ( )

( )

++

=

t

SLSC

V

zyxV

V

zyxVzyxTzyxD

xtzyxC

02*

2

2*111

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

( )( )

( ) ( )

+

+

+

+

tCSCS

LSC

CS

TzyxPTzyxPTzyxD

yd

TzyxPTk

z

0

11

1

1

,,,1

,,,1,,,

,,,1

( ) ( )( )

( )zyxfdV

zyxV

V

zyxV

Tkz

C,,

,,,,,,1

2*

2

2*1+

++

(1c)

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

+

+

+=

+

+

=

t

S

IS

V

t

S

IS

V

t

VIVI

t

VVVV

t

S

IS

I

t

S

IS

I

t

VIVI

t

IIII

dzyxTk

D

yzdzyx

Tk

D

xz

dTzyxkdTzyxkzyxftzyxV

dzyxTk

D

yzdzyx

Tk

D

xz

dTzyxkdTzyxkzyxftzyxI

01

01

0,11

0,

2

11

01

01

0,11

0,

2

11

,,,,,,

,,,,,,,,,,,

,,,,,,

,,,,,,,,,,,

(3c)

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

+

++

+

+

+=

+

++

+

+

+=

t

VV

t

V

t

S

S

t

S

S

V

t

II

t

I

t

S

S

t

S

S

I

dzyxVTzyxk

dzyxVTzyxkdzyxTk

D

yz

dzyxTk

D

xzzyxftzyx

dzyxITzyxk

dzyxITzyxkdzyxTk

D

yz

dzyxTk

D

xzzyxftzyx

V

V

V

VV

I

I

I

II

0

2

,

001

011

0

2

,

001

011

,,,,,,

,,,,,,,,,

,,,,,,,,

,,,,,,

,,,,,,,,,

,,,,,,,,

(5c)

We determine average values of the first-order approximations of required functions by using the

following relations [25,31]

( )

=

0 0 0 011 ,,,

1 x y zL L L

zyx

tdxdydzdtzyxLLL

. (9)

Substitution of these relations (1c), (3c) and (5c) into relation (9) gives us possibility to obtain the

following relations

( ) =x y zL L L

C

zyx

Cxdydzdzyxf

LLL 0 0 01 ,,

1 ,

( )

4

3

4

1

2

3

2

4

2

3

14

44 a

Aa

a

aLLLBaB

a

Aa zyxI

+

++

+= ,


80

( )

=

zyxIII

L L L

I

IIV

VLLLSxdydzdzyxf

S

x y z

0010 0 0100

1,,

1

,

where ( ) ( ) ( ) ( ) =

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

x y zL L Lji

ijtdxdydzdtzyxVtzyxITzyxktS , = 004 IISa

( )0000

2

00 VVIIIVSSS ,

0000

2

0000003 VVIIIVIIIV SSSSSa += , ( ) =x y zL L L

Vxdydzdzyxfa

0 0 02 ,,

( ) ( ) +x y zx y z LL L

IIV

L L L

IIIVVIVIVxdydzdzyxfSxdydzdzyxfSSSS

0 0 0

2

000 0 0

0000

2

0000 ,,,,2

222

0000

222

zyxIVVVzyxLLLSSLLL + , ( ) =

x y zL L L

IIVxdydzdzyxfSa

0 0 0001 ,, ,

4

2

2

4

2

32 48a

a

a

ayA += ,

( )2

0 0 0000 ,,

=x y zL L L

IVVxdydzdzyxfSa , 3

323 32

4

2

6qpqqpq

a

aB ++++

= ,

2

4

2

1

4

222

3

4

3

2

3

4

2

32

22

4

02

4

31

02

4

2

3

8544

84

24 a

aLLL

a

a

a

aa

a

a

a

aaLLLa

a

aq

zyxzyx

= ,

4

2

2

4

31402

1812

4

a

a

a

aaLLLaap

zyx

= ,

( ) +

+

= x y z

II

L L L

zyxzyx

II

zyx

I xdydzdzyxfLLLLLL

S

LLL

R

0 0 0

201

1 ,,1

( ) +

+

= x y z

VV

L L L

zyxzyx

VV

zyx

V xdydzdzyxfLLLLLL

S

LLL

R

0 0 0

201

1 ,,1

,

where ( ) ( ) ( ) =

0 0 0 01 ,,,,,,

x y zL L Li

Ii tdxdydzdtzyxITzyxktR .

We obtain the second-order approximations and approximations with higher orders of concentra-

tions of dopant and radiation defects by using standard iteration procedure of method of averag-

ing of function corrections [25,31]. Framework this procedure to calculate n-th-order approxima-

tions of concentrations of dopant and radiation defects we replace the required concentrations

C(x,y,z,t), I(x,y,z,t), V(x,y,z,t), I(x,y,z,t) and V(x,y,z,t) in the right sides of Eqs. (1a), (3a), (5a) on the following sums n+n-1(x, y,z,t). This substitution gives us possibility to obtain equations for the second-order approximations of the required concentrations

( ) ( )[ ]( )

( ) ( )( )

++

+

+

=

2*

2

2*1

122 ,,,,,,1,,,

,,,1

,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxC

xt

tzyxC C

( )( ) ( ) ( )

( )( )

++

+

y

tzyxC

V

tzyxV

V

tzyxV

yx

tzyxCTzyxD L

,,,,,,,,,1

,,,,,, 1

2*

2

2*1

1

( ) ( )[ ]( )

( ) ( )( )

++

+

+

+2*

2

2*1

12 ,,,,,,1,,,

,,,1,,,

V

tzyxV

V

tzyxV

zTzyxP

tzyxCTzyxD CL

( ) ( )( )[ ]

( )( ) ( )

++

+

+

Tk

D

xtzyxf

TzyxP

tzyxC

z

tzyxCTzyxD S

C

C

L

,,,,,

,,,1

,,,,,, 121 (1d)


81

( ) ( ) ( ) ( )

+

zz L

S

SL

SWdtWyxCtzyx

Tk

D

yWdtWyxCtzyx

00

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

( ) ( ) ( ) ( ) ( )

+

=

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIII

,,,,,,

,,,,,,

,,,112

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

+ tzyxVtzyxITzyxk

z

tzyxITzyxD

zVIVII

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

,,,1111,

1

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

+

++

zL

IS

IS

IIIWdtWyxItzyx

Tk

D

xtzyxITzyxk

012

2

11, ,,,,,,,,,,,,

( ) ( )[ ]

+

+

zL

IS

IS WdtWyxItzyxTk

D

y 012

,,,,,, (3d)

( ) ( ) ( ) ( ) ( )

+

=

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVV

,,,,,,

,,,,,,

,,,112

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

+ tzyxVtzyxITzyxk

z

tzyxVTzyxD

zVIVIV

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

,,,1111,

1

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

+

++

zL

VS

VS

VVVWdtWyxVtzyx

Tk

D

xtzyxVTzyxk

012

2

11, ,,,,,,,,,,,,

( ) ( )[ ]

+

+

zL

VS

VS WdtWyxVtzyxTk

D

y 012

,,,,,,

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxIII

II

,,,,,,

,,,,,,

,,, 112

( ) ( ) ( ) ( )[ ] +

+

++

z

I

I

L

IS

S

IIWdtWyxtzyx

Tk

D

xtzyxITzyxk

012

2

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

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

+

+

tzyxITzyxkWdtWyxtzyx

Tk

D

yI

L

IS

Sz

I

I ,,,,,,,,,,,,0

12

( ) ( ) ( ) ( )tzyxfz

tzyxTzyxD

z III

,,

,,,,,, 1 +

+ (5d)

( )( )

( )( )

( )+

+

=

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxVVV

VV

,,,,,,

,,,,,,

,,,112

( ) ( ) ( ) ( )[ ] +

+

++

z

V

V

L

VS

S

VVWdtWyxtzyx

Tk

D

xtzyxVTzyxk

012

2

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

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

+

+


Tk

D

yV

L

VS

Sz

V

V ,,,,,,,,,,,,0

12

( ) ( ) ( ) ( )tzyxfz

tzyxTzyxD

z VVV

,,

,,,,,, 1 +

+ .

Integration of left and right sides of Eqs. (1d), (3d) and (5d) gives us possibility to obtain final

relations for the second-order approximations of the required concentrations of dopant and radia-

tion defects in the following form


82

( ) ( )[ ]( )

( ) ( )( )

++

+

+

=

tC

V

zyxV

V

zyxV

TzyxP

zyxC

xtzyxC

02*

2

2*1

12

2

,,,,,,1

,,,

,,,1,,,

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

++

+

t

LL

V

zyxV

V

zyxVTzyxD

yd

x

zyxCTzyxD

02*

2

2*1

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

,,,

( ) ( )[ ]( )

( ) ( )( )

++

+

+

+

tC

V

zyxV

V

zyxV

zTzyxP

tzyxC

y

zyxC

02*

2

2*1

121 ,,,,,,1,,,

,,,1

,,,

( ) ( ) ( )[ ]( )

( )

++

+

+

tS

C

C

LTk

D

xzyxfd

TzyxP

zyxC

z

zyxCTzyxD

0

121 ,,,,,

,,,1

,,,,,,

( ) ( )[ ] ( )[ ] +

+ +

t L

C

SL

CS

zz

WdWyxCTk

D

ydWdWyxCzyx

0 012

012 ,,,,,,,,,

( ) dzyxS

,,, (1e)

( ) ( ) ( ) ( ) ( ) ++=t

I

t

Id

y

zyxITzyxD

yd

x

zyxITzyxD

xtzyxI

0

1

0

1

2

,,,,,,

,,,,,,,,,

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

III

t

IdzyxITzyxkd

z

zyxITzyxD

z 0

2

12,0

1 ,,,,,,,,,

,,,

( ) ( )[ ] ( )

+ +

+

t

S

ISt L

I

IS

Szyx

Tk

D

ydWdWyxI

Tk

Dzyx

x

z

00 012 ,,,,,,,,,

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

VIVI

L

IdzyxVzyxITzyxkdWdWyxI

z

01212,

012

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

( )zyxfI

,,+ (3e)

( ) ( ) ( ) ( ) ( ) ++=t

V

t

Vd

y

zyxVTzyxD

yd

x

zyxVTzyxD

xtzyxV

0

1

0

1

2

,,,,,,

,,,,,,,,,

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

VVV

t

VdzyxVTzyxkd

z

zyxVTzyxD

z 0

2

12,0

1 ,,,,,,,,,

,,,

( ) ( )[ ] ( )

+ +

+

t

S

VSt L

V

VS

Szyx

Tk

D

ydWdWyxV

Tk

Dzyx

x

z

00 012 ,,,,,,,,,

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

VIVI

L

VdzyxVzyxITzyxkdWdWyxV

z

01212,

012

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

( )zyxfV

,,+

( ) ( ) ( ) ( )

+

= t

It

I

Iy

zyx

yd

x

zyxTzyxD

xtzyx

I0

1

0

1

2

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

( ) ( ) ( ) ( )

+

+

t

S

St

I zyxTk

D

xd

z

zyxTzyxD

zdTzyxD I

II00

1 ,,,,,,

,,,,,,

( )[ ] ( ) ( )[ ] +

+ +

t L

IS

L

I

z

I

z

IWdWyxzyx

ydWdWyx

0 012

012

,,,,,,,,,

( ) ( ) ( ) ( ) ( )zyxfdzyxITzyxkdzyxITzyxkdTk

D

I

It

I

t

II

S,,,,,,,,,,,,,,

00

2

,

+++ (5e)

( ) ( ) ( ) ( )

+

= t

Vt

V

Vy

zyx

yd

x

zyxTzyxD

xtzyx

V0

1

0

1

2

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


83

( ) ( )( )

( )

+

+

t

S

St

V zyxTk

D

xd

z

zyxTzyxD

zdTzyxD V

VV00

1 ,,,,,,

,,,,,,

( )[ ] ( ) ( )[ ] +

+ +

t L

VS

L

V

z

V

z

VWdWyxzyx

ydWdWyx

0 012

012

,,,,,,,,,

( ) ( ) ( ) ( ) ( )zyxfdzyxVTzyxkdzyxVTzyxkdTk

D

V

Vt

V

t

VV

S,,,,,,,,,,,,,,

00

2

,

+++ .

We determine average values of the second-order approximations of the considered concentra-

tions by using standard relations [25,31]

( ) ( )[ ]

=

0 0 0 0122 ,,,,,,

1 x y zL L L

zyx

tdxdydzdtzyxtzyxLLL

. (10)

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

relations for the required relations 2

2C=0, 2I =0, 2V =0, ( )

4

3

4

1

2

3

2

4

2

3

24

44 b

Eb

b

bLLLFaF

b

Eb zyxV

+

++

+= ,

( )00201

11021001200

2

2

2

2

IVVIV

VIVVzyxVIVVVVVVV

ISS

SSLLLSSSC

+

++= ,

where 00

2

0000

2

004

11IIVV

zyx

VVIV

zyx

SSLLL

SSLLL

b

= , ( ++

=100100

00

32

IVVVVV

zyx

II SSSLLL

Sb

) ( ) (

++++

++yxIV

zyx

VVIVzyxIVIIIV

zyx

zyxLLS

LLLSSLLLSSS

LLLLLL

2

000000011001

12

1

) 333310

2

0010012

zyxIVIVIVVVzLLLSSSSL ++ , ( ) ( ++

=

zyxVIVVV

zyx

VVII LLLCSSLLL

SSb 1102

0000

2

) ( ) ( +++

+++

++1001

00

0110

00012

1001222

IIIVzyx

zyx

IV

IVIIzyx

zyx

VVIV

IVVVSSLLL

LLL

SSSLLL

LLL

SSSS

)( ) ( ) +

+++ 010010

2

00

1102100101

222

IVIVIV

zyxzyx

IV

IVVVVIVzyxVVIVSSS

LLLLLL

SSSCSLLLSS

2222

2

00

zyx

IVI

LLL

SC

+ , ( )( ) (

+++++

=

x

zyx

IV

zyxIVVVVVVIV

zyx

II LLLL

SLLLSSCSS

LLL

Sb 01

10010211

00

12

)( ) ( )( ++

++++VzyxIIIV

zyx

IV

zyxIVVVIVIIzyCLLLSS

LLL

SLLLSSSSLL 1001

00

10010110 2322

)zyx

IVIV

IVIVIIVVVLLL

SSSSCSS

+

2

0110

01001102 2 , ( ) ( +++

= 102

0200

00

0 2 IIzyxVVIVzyx

II SLLLSSLLL

Sb

) ( ) ( )( ) +++

+ 1102011001

0211

01

01 2 IVVVVIVIIzyxzyx

IV

VVIVV

zyx

IV

IVSSCSSLLL

LLL

SSSC

LLL

SS

2

012 IVI SC+ , zyx

IV

zyx

IIII

zyx

III

zy

IV

x

I

VILLL

S

LLL

SS

LLL

S

LL

S

LC

+

= 11202000

2

1001

1

, += 020011 VVIVVIV SSC

1100

2

1 IVVVV SS + , 4

2

2

4

2

32 48a

a

a

ayE += , 3 323 32

4

2

6rsrrsr

a

aF ++++

= ,

=

2

4

2

3

24b

br


84

2

4

2

1

4

222

3

4

3

2

3

4

2

32

22

4

2

0

4

31

0854

48

4b

bLLL

b

b

b

bb

bb

b

bbLLLb

zyxzyx

, ( =

402

4

2

412

bbb

s

)4231

18bbbbLLLzyx

.

Farther we determine solutions of the system of Eqs.(8). The solutions physically correspond to

components of displacement vector. To determine components of displacement vector we used

method of averaging of function corrections in standard form. Framework the approach we re-

place the above components in right sides of Eqs.(8) on their not yet known average values i. The substitution leads to the following results

( ) ( ) ( ) ( ) ( )x

tzyxTzzK

t

tzyxuz x

=

,,,,,,2

1

2

, ( )( )

( ) ( ) ( )y

tzyxTzzK

t

tzyxuz

y

=

,,,,,,2

1

2

,

( ) ( ) ( ) ( ) ( )z

tzyxTzzK

t

tzyxuz z

=

,,,,,,2

1

2

.

Integration of the left and right sides of the above equations on time t leads to final relations for

the first-order approximations of components of displacement vector in the following form

( ) ( ) ( )( )

( ) ( ) ( )( )

( )x

t

xuddzyxT

xz

zzKddzyxT

xz

zzKtzyxu 0

0 00 01 ,,,,,,,,, +

=

,

( ) ( ) ( )( )

( ) ( ) ( )( )

( )y

t

yuddzyxT

yz

zzKddzyxT

yz

zzKtzyxu

00 00 0

1,,,,,,,,, +

=

,

( ) ( ) ( )( )

( ) ( ) ( )( )

( )z

t

zuddzyxT

zz

zzKddzyxT

zz

zzKtzyxu 0

0 00 01 ,,,,,,,,, +

=

.

The second-order approximations and approximations with higher orders of components of dis-

placement vector could be calculated by replacement of the required components in the Eqs. (8)

on the following sums i+ui(x,y,z,t) [25,31]. This replacement leads to the following result

( )( )

( ) ( )( )[ ]

( )( ) ( )

( )[ ]

++

++=

z

zEzK

x

tzyxu

z

zEzK

t

tzyxuz xx

13

,,,

16

5,,,2

1

2

2

2

2

( ) ( )( )[ ]

( ) ( )( )

( )( )[ ]

+++

+

++

z

zEzK

z

tzyxu

y

tzyxu

z

zE

yx

tzyxuzyy

13

,,,,,,

12

,,,2

1

2

2

1

2

1

2

( )( ) ( ) ( )

x

tzyxTzzK

zx

tzyxuz

,,,,,,12

( )( ) ( )

( )[ ]( ) ( )

( ) ( ) ( ) +

+

+=

y

tzyxTzzK

yx

tzyxu

x

tzyxu

z

zE

t

tzyxuz

xyy ,,,,,,,,,

12

,,,1

2

2

1

2

2

2

2

( ) ( )( )[ ]

( ) ( )( )[ ]

( ) ( )+

+

+

+

++

+

y

tzyxu

z

tzyxu

z

zE

zzK

z

zE

y

tzyxuzyy ,,,,,,

12112

5,,, 112

1

2

( ) ( )( )[ ]

( )( )

( )yx

tzyxuzK

zy

tzyxu

z

zEzK

yy

+

++

,,,,,,

16

1

2

1

2


85

( ) ( ) ( )( )[ ]

( ) ( ) ( ) ( )

+

+

+

+=

zy

tzyxu

zx

tzyxu

y

tzyxu

x

tzyxu

z

zE

t

tzyxuz

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

12

,,, 12

1

2

2

1

2

2

1

2

2

2

2

( ) ( )( )

( )( ) ( ) ( )

+

+

+

+

z

tzyxu

y

tzyxu

x

tzyxuzK

zz

tzyxTzzK x

yx ,,,,,,,,,,,, 111

( )( )[ ]

( ) ( ) ( ) ( )

++

z

tzyxu

y

tzyxu

x

tzyxu

z

tzyxu

zz

zE zyxz ,,,,,,,,,,,,616

1111

.

Integration of left and right sides of the above equations on time t leads to the following results

( )( )

( )( )

( )[ ]( )

( )( )

( )( )[ ]

++

++=

z

zEzK

zddzyxu

xz

zEzK

ztzyxu

t

xx

13

1,,,

16

51,,,

0 012

2

2

( ) ( ) ( )

+

+

t

z

t

y

t

y ddzyxuz

ddzyxuy

ddzyxuyx 0 0

12

2

0 012

2

0 01

2

,,,,,,,,,

( )( ) ( )[ ] ( )

( ) ( ) ( )( )[ ]

( ) ( )( )

++

+

+

z

zzK

z

zEzKddzyxu

zxzzz

zE t

z

13,,,

1

12 0 01

2

( )( )

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

( )

++

zKzz

zEzKddzyxu

xzddzyxT

xx

t

1

16

5,,,

1,,,

0 012

2

0 0

( )( )[ ]

( ) ( ) ( )

+

+

0 012

2

0 012

2

0 01

2

,,,,,,,,,13

ddzyxuz

ddzyxuy

ddzyxuyxz

zEzyy

( )( ) ( )[ ]

( ) ( )( )

( )( )

( ) ( ){ +

+

+

zKddzyxuzxz

ddzyxTxz

zzK

zz

zEz

0 01

2

0 0

,,,1

,,,12

( ) ( )[ ]} xuzzE 013 +++

( ) ( )( ) ( )[ ]

( ) ( ) +

+

+=

t

x

t

xy ddzyxuyx

ddzyxuxzz

zEtzyxu

0 01

2

0 012

2

2 ,,,,,,12

,,,

( )( )

( )( )

( )( )[ ]

( ) ( ) +

++

+

+

t

x

t

y ddzyxuy

zKz

zE

zddzyxu

yxz

zK

0 012

2

0 01

2

,,,112

51,,,

( )( )

( )( ) ( ) ( ) ( )

( )

+

+

+

z

zzKddzyxu

yddzyxu

zz

zE

zz

t

z

t

y

0 01

0 01 ,,,,,,

12

1

( )( )

( )( )

( )[ ]( )

( )( )

++

z

zEddzyxu

zyz

zEzK

zddzyxT

t

y

t

2,,,

16

1,,,

0 01

2

0 0

( )( ) ( ) ( )

+

+

0 00 01

2

0 012

2

,,,,,,,,,1

1

ddzyxTddzyxu

yxddzyxu

xzxx

( )( )( )

( )( )

( )( )

( ) ( )

+

zKddzyxuyz

ddzyxuyxz

zK

z

zzK xy

0 012

2

0 01

2

,,,1

,,,

( )( )[ ] ( )

( )( )

( ) ( )

+

+

++

0 01

0 01 ,,,,,,

12

1

112

5

ddzyxu

yddzyxu

zz

zE

zzz

zEzy

( )( )

( )( )[ ]

( ) yy uddzyxuzyz

zEzK

z0

0 01

2

,,,16

1+

+

( ) ( ) ( ) ( )

+

+

+

=

0 01

2

0 012

2

0 012

2

2

2

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

ddzyxuzx

ddzyxuy

ddzyxuxt

tzyxuxzz

z


86

( )( )

( ) ( )[ ] ( )( )

+

+

+

+

0 01

0 01

2

,,,1

12,,,

ddzyxuxzzzz

zEddzyxu

zyxy

( ) ( ) ( )( )

( )( )

+

+

+

+

z

zE

zzzKddzyxu

zddzyxu

yxx

16

1,,,,,,

0 01

0 01

( ) ( ) ( )

0 01

0 01

0 01 ,,,,,,,,,6

ddzyxuy

ddzyxux

ddzyxuz

yxz

( ) ( ) ( )( )

( ) zz uddzyxTzz

zzKddzyxu

z0

0 00 01 ,,,,,, +

.

Framework this paper we determine concentration of dopant, concentrations of radiation defects

and components of displacement vector by using the second-order approximation framework me-

thod of averaging of function corrections. This approximation is usually enough good approxima-

tion to make qualitative analysis and to obtain some quantitative results. All obtained results have

been checked by comparison with results of numerical simulations.

3. Discussion

In this section we analyzed influence of mismatch-induced stress on redistributions of dopant and

radiation defects during annealing. The Figs. 2 and 3 show typical distributions of concentrations

of dopant in heterostructures for diffusion and ion types of doping. All distributions in Figs. 2 and 3 have been calculated for larger value of dopant diffusion coefficient in doped area is larger in

comparison with dopant diffusion coefficient in nearest areas. The figures show, that using inho-

mogeneity of heterostructure gives us possibility to increase sharpness of p-n- junctions with in-

creasing homogeneity of dopant distribution in doped part of epitaxial layer. Increasing of sharp-

ness of p-n-junction leads to decreasing of switching time. Increasing of homogeneity of dopant

distribution gives us possibility to decrease local overheats of materials during functioning of p-n-junction or decreasing of dimensions of the p-n-junction for fixed maximal value of local over-

heat. It should be noted, that using the considered approach for manufacture a bipolar transistor

leads to necessity to optimize annealing of dopant and/or radiation defects. Reason of this optimi-zation is following. If annealing time is small, the dopant did not achieves any interfaces between

materials of heterostructure. In this situation one can not find any modifications of distribution of

concentration of dopant. Increasing of annealing time leads to increasing of homogeneity of do-pant distribution. We optimize annealing time framework recently introduces approach

[22,23,25,32-36]. Framework this criterion we approximate real distribution of concentration of

dopant by step-wise function (see Figs. 4 and 5). Farther we determine optimal values of anneal-

ing time by minimization of the following mean-squared error


87

Fig.2. Typical distributions of concentration of dopant, which has been infused in heterostructure from

Figs. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. Increasing of

number of curve corresponds to increasing of difference between values of dopant diffusion coefficient in

layers of heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is

larger, than value of dopant diffusion coefficient in substrate

x

0.0

0.5

1.0

1.5

2.0

C(x

,)

23

4

1

0 L/4 L/2 3L/4 L

Epitaxial layer Substrate

Fig.3. Typical distributions of concentration of dopant, which has been implanted in heterostructure from

Figs. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3

corresponds to annealing time = 0.0048(Lx2+Ly

2+Lz

2)/D0. Curves 2 and 4 corresponds to annealing time

= 0.0057(Lx2+ Ly

2+Lz

2)/D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corres-

ponds to heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is

larger, than value of dopant diffusion coefficient in substrate

C(x

,)

0 Lx

2

13

4

Fig. 4. Spatial distributions of dopant in heterostructure after dopant infusion. Curve 1 is idealized distribu-

tion of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. Increasing

of number of curve corresponds to increasing of annealing time

x

C(x

,)

1

23

4

0 L

Fig. 5. Spatial distributions of dopant in heterostructure after ion implantation. Curve 1 is idealized distri-

bution of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. Increas-

ing of number of curve corresponds to increasing of annealing time


88

( ) ( )[ ] =x y zL

L L

zyx

xdydzdzyxzyxCLLL

U0 0 0

,,,,,1

, (15)

where (x,y,z) is the approximation function. Dependences of optimal values of annealing time on parame-ters are presented on Figs. 6 and 7 for diffusion and ion types of doping, respectively. It should be noted,

that it is necessary to anneal radiation defects after ion implantation. One could find spreading of concen-

tration of distribution of dopant during this annealing. In the ideal case distribution of dopant achieves ap-

propriate interfaces between materials of heterostructure during annealing of radiation defects. If dopant

did not achieves any interfaces during annealing of radiation defects, it is practicably to additionally anneal

the dopant. In this situation optimal value of additional annealing time of implanted dopant is smaller, than

annealing time of infused dopant.

0.0 0.1 0.2 0.3 0.4 0.5

a/L, , ,

0.0

0.1

0.2

0.3

0.4

0.5

D

0 L

-2

3

2

4

1

Fig.6. Dependences of dimensionless optimal annealing time for doping by diffusion on several parameters.

Curve 1 is the dependence of dimensionless optimal annealing time on the relation a/L and = = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the de-

pendence of dimensionless optimal annealing time on value of parameter for a/L=1/2 and = = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter for a/L=1/2 and =

= 0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter for a/L=1/2 and = = 0

0.0 0.1 0.2 0.3 0.4 0.5a/L, , ,

0.00

0.04

0.08

0.12

D

0 L

-2

3

2

4

1

Fig.7. Dependences of dimensionless optimal annealing time for doping by ion implantation on several

parameters. Curve 1 is the dependence of dimensionless optimal annealing time on the relation a/L and = = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optimal annealing time on value of parameter for a/L=1/2 and = = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter for a/L=1/2

and = = 0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter for a/L=1/2 and = = 0

Farther we analyzed influence of relaxation of mechanical stress on distribution of dopant in doped areas of

heterostructure. Under following condition 0< 0 one can find compression of distribution of concentration


89

of dopant near interface between materials of heterostructure. Contrary (at 0>0) one can find spreading of distribution of concentration of dopant in this area. This changing of distribution of concentration of dopant

could be at least partially compensated by using laser annealing [33]. This type of annealing gives us possi-

bility to accelerate diffusion of dopant and another processes in annealed area due to inhomogenous distri-

bution of temperature and Arrhenius law. Accounting relaxation of mismatch-induced stress in heterostruc-

ture could leads to changing of optimal values of annealing time.

4. CONCLUSIONS

In this paper we analyzed influence of relaxation of mismatch-induced stress in heterostructure on distribu-

tions of concentrations of dopants in manufactured in this heterostructure transistors. At the same time in

this paper we introduce an approach to increase compactness of multiemitter transistor. The approach based

on doping by diffusion or ion implantation of required parts of dopants and/or radiation defects.

Acknowledgments

This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, grant of

Scientific School of Russia, the agreement of August 27, 2013 02..49.21.0003 between The Ministry

of education and science of the Russian Federation and Lobachevsky State University of Nizhni Novgorod

and educational fellowship for scientific research of Nizhny Novgorod State University of Architecture 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. Now E.L. Pankratov is in his Full Doctor

course in Radiophysical Department of Nizhny Novgorod State University. He has 105 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


91

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. E.A. Bulaeva was a contributor of grant of President of Russia (grant MK-

548.2010.2). She has 52 published papers in area of her researches.

Documents

ON ANALYTICAL APPROACH FOR ANALYSIS INFLUENCE OF MISMATCH-INDUCED STRESS IN A HETEROSTRUCTURE ON DISTRIBUTIONS OF CONCENTRATIONS OF DOPANTS IN A MULTIEMITTER HETEROTRANSISTOR