on optimization ofon optimization of doping of a heterostructure during manufacturing of p-i-ndiodes

International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015

1

ON OPTIMIZATION OF DOPING OF A HETERO-

STRUCTURE DURING MANUFACTURING OF P-I-N-

DIODES

E.L. Pankratov1,3

, 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 3

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

novskoe Shosse, Nizhny Novgorod, 603950, Russia

ABSTRACT

We introduce an approach of manufacturing of a p-i-n-heterodiodes. The approach based on using a δ-

doped heterostructure, doping by diffusion or ion implantation of several areas of the heterostructure. After

the doping the dopant and/or radiation defects have been annealed. We introduce an approach to optimize

annealing of the dopant and/or radiation defects. We determine several conditions to manufacture more

compact p-i-n-heterodiodes.

KEYWORDS

p-i-n-heterodiodes; decreasing of dimensions of p-i-n-diodes; optimization of manufacturing; δ-doping;

analytical approach for modelling

1. INTRODUCTION

Development of the solid state electronic devices leads to necessity of increasing of performance,

reliability and integration rate of elements of integrated circuits (p-n- junctions, their systems et

al) [1-7]. Increasing of integration rate of elements of integrated circuits leads to necessity to de-

crease their dimensions. One way to increase performance of the above elements is determination

new materials with high charge carrier’s motilities. The second one is elaboration new technolo-

gical processes and optimization existing one [8-21]. Dimensions of elements of integrated cir-

cuits could be also decreased by using new technological processes and optimization existing one

[8-17].

In the present paper we consider an approach to decrease dimensions of p-i-n-diodes. The ap-

proach based on using a heterostructure from Fig. 1. The heterostructure consist of a substrate

and four epitaxial layers. A δ-layer has been grown between average epitaxial layers to manufac-

ture required type of conductivity (n or p) in the doped area. Another epitaxial layers have been

manufactured with sections (see Fig. 1). The sections have been manufactured by using another

materials. We assumed, that the sections have been doped by diffusion or ion implantation to pro-

duce in this section required type of conductivity (p or n). The considered approach gives us pos-

sibility to manufacture more thin p-i-n-diodes. After doping sections dopant and/or radiation de-

fects have been annealed. Annealing of dopant gives us possibility to infuse the dopant on re-

quired depth. Annealing of radiation defects gives us possibility to decrease their quantity.

Framework this paper we analyzed dynamics of dopant and radiation defects during annealing.


2

Substrate

Epitaxial layer 1

Epitaxial layer 2

Epitaxial layer 3

Epitaxial layer 4

δ-layer

Fig. 1. A multilayer structure with a substrate and four epitaxial layers.

Heterostructure, which consist of a substrate, four epitaxial layers and sections framework the layers

2. METHOD OF SOLUTION

We determine distributions of concentrations of dopants in space and time to solve our aim. To

determine the distributions we solved the second Fick's law [1,3,17]

( ) ( ) ( ) ( )

+

+

=

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxCCCC

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,,,,,,,. (1)

Boundary and initial conditions for the equations are

( )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)=f (x,y,z). (2)

The function C(x,y,z,t) describes distribution of concentration of dopant in space and time; T is

the temperature of annealing; DС is the dopant diffusion coefficient. Dopant diffusion coefficient

has different values in different materials. Value of dopant diffusion coefficient changing with

heating and cooling of heterostructure (with account Arrhenius law). Dopant diffusion coefficient

could be approximated by the following relation [22-24]

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD

LCςςξ

γ

γ

. (3)

The function DL(x,y,z,T) describes variation of dopant diffusion coefficient in space (due inhomo-

geneity of heterostructure) and with variation of temperature (due to Arrhenius law); the function

P (x,y,z,T) describes the limit of solubility of dopant; parameter γ is integer in the following inter-

val γ ∈[1,3] [22]; the function V (x,y,z,t) describes variation of distribution of concentration of

radiation vacancies in space and time; the parameter V* describes the equilibrium distribution of

concentration of vacancies. Dependence of dopant diffusion coefficient on concentration of do-

pant has been described in details in [22]. It should be noted, that using diffusion type of doping

did not generation radiation defects. In this situation ζ1=ζ2=0. We determine distributions of con-

centrations of radiation defects in space and time by solving the following system of equations

[23,24]


3

( ) ( ) ( ) ( ) ( ) ( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂Tzyxk

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIIIII ,,,

,,,,,,

,,,,,,

,,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz

tzyxITzyxD

ztzyxI

VII,,,,,,,,,

,,,,,,,,,

,

2 −

∂

∂

∂

∂+× (4)

( ) ( ) ( ) ( ) ( ) ( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂Tzyxk

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVVVV ,,,

,,,,,,

,,,,,,

,,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz

tzyxVTzyxD

ztzyxV

VIV,,,,,,,,,

,,,,,,,,,

,

2 −

∂

∂

∂

∂+× .

Boundary and initial conditions for these equations are

( )0

,,,

0

=∂

∂

=xx

tzyxρ,

( )0

,,,=

∂

∂

= xLxx

tzyxρ,

( )0

,,,

0

=∂

∂

=yy

tzyxρ,

( )0

,,,=

∂

∂

= yLyy

tzyxρ,

( )0

,,,

0

=∂

∂

=zz

tzyxρ,

( )0

,,,=

∂

∂

= zLzz

tzyxρ, ρ (x,y,z,0)=fρ (x,y,z). (5)

Here ρ =I,V; the function I (x,y,z,t) describes distribution of concentration of radiation interstitials

in space and time; Dρ(x,y,z,T) are the diffusion coefficients of point radiation defects; terms

V2(x,y,z,t) and I

2(x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V(x,y,z,T) is the

parameter of recombination of point radiation defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parame-

ters of generation of simplest complexes of point radiation defects.

Distributions of concentrations of divacancies ΦV(x,y,z,t) and dinterstitials ΦI (x,y,z,t) in space and

time have been determined by solving the following system of equations [23,24]

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyx I

I

I

I

I

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkz

tzyxTzyxD

zIII

I

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

,,,,,, 2

, −+

Φ+ Φ

∂

∂

∂

∂ (6)

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

V

V

V

V

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkz

tzyxTzyxD

zVVV

V

V,,,,,,,,,,,,

,,,,,, 2

,−+

Φ+ Φ

∂

∂

∂

∂.

Boundary and initial conditions for these equations are

( )0

,,,

0

=∂

Φ∂

=xx

tzyxρ,

( )0

,,,=

∂

Φ∂

= xLxx

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=yy

tzyxρ,

( )0

,,,=

∂

Φ∂

= yLyy

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=zz

tzyxρ,

( )0

,,,=

∂

Φ∂

= zLzz

tzyxρ, ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (7)

Here DΦρ(x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects;

kI(x,y,z,T) and kV (x,y,z,T) are the parameters of decay of these complexes.


4

To determine spatio-temporal distribution of concentration of dopant we transform the Eq.(1) to the following integro-differential form

( ) ( ) ( )( )

×∫ ∫ ∫

++=∫ ∫ ∫

t y

L

z

Lzy

x

L

y

L

z

Lzyx y zx y z V

wvxV

V

wvxV

LL

zyudvdwdtwvuC

LLL

zyx

02*

2

2*1

,,,,,,1,,,

τς

τς

( ) ( )( )

( ) ( ) ×∫ ∫ ∫+

+×

t x

L

z

LL

zx

L

x z

TwyuDLL

zxd

x

wvxC

TwvxP

wvxCTwvxD

0

,,,,,,

,,,

,,,1,,, τ

∂

τ∂τξ

γ

γ

( ) ( )( )

( )( )

( )+

+

++× τ

∂

τ∂τξ

τς

τς

γ

γ

dy

wyuC

TzyxP

wyuC

V

wyuV

V

wyuV ,,,

,,,

,,,1

,,,,,,1

2*

2

2*1

( ) ( ) ( )( )

( )( )

×∫ ∫ ∫

+

+++

t x

L

y

LL

yx x y TzyxP

zvuC

V

zvuV

V

zvuVTzvuD

LL

yx

02*

2

2*1,,,

,,,1

,,,,,,1,,,

γ

γ τξ

τς

τς

( ) ( )∫ ∫ ∫+×x

L

y

L

z

Lzyx x y z

udvdwdwvufLLL

zyxd

z

zvuC,,

,,,τ

∂

τ∂. (1a)

We determine solution of the above equation by using Bubnov-Galerkin approach [25]. Frame-

work the approach we determine solution of the Eq.(1a) as the following series

( ) ( ) ( ) ( ) ( )∑==

N

nnCnnnnC

tezcycxcatzyxC0

0 ,,, ,

where ( ) ( )[ ]222

0

22exp −−− ++−=zyxCnC

LLLtDnte π , cn(χ)=cos(π nχ/Lχ). The above series includes

into itself finite number of terms N. The considered series is similar with solution of linear Eq.(1)

(i.e. with ξ =0) and averaged dopant diffusion coefficient D0. Substitution of the series into

Eq.(1a) leads to the following result

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫

×

∑+−=∑

==

t y

L

z

L

N

nnCnnnnC

zy

N

nnCnnn

C

y z

ewcvcxcaLL

zytezsysxs

n

azyx

0 1132

1γ

τπ

( )( ) ( )

( )( ) ( ) ( ) ×∑

++

×=

N

nnnnCL

vcxsaTwvxDV

wvxV

V

wvxV

TwvxP 12*

2

2*1,,,

,,,,,,1

,,,

τς

τς

ξγ

( ) ( ) ( ) ( ) ( ) ( )( )∫ ∫ ∫ ×

∑+−×

=

t x

L

z

L

N

mmCmmmmC

zx

nCn

x z TwyuPewcycuca

LL

zxdewcn

0 1 ,,,1

γ

γ ξτττ

( ) ( ) ( )( )

( ) ( ) ( ) ( ) ×∑

++×

=

τττ

ςτ

ς dewcysucnV

wyuV

V

wyuVTwyuD

N

nnCnnnL

12*

2

2*1

,,,,,,1,,,

( )( )

( ) ( ) ( ) ( ) ×∫ ∫ ∫

∑+−×

=

t x

L

y

L

N

nnCnnnnCL

yx

nC

x y

ezcvcucaTzvuP

TzvuDLL

yxa

0 1,,,1,,,

γ

γτ

ξ

( ) ( )( )

( ) ( ) ( ) ( ) ×+∑

++×

=zyx

N

nnCnnnnC

LLL

zyxdezsvcucan

V

zvuV

V

zvuVττ

τς

τς

12*

2

2*1

,,,,,,1

( )∫ ∫ ∫×x

L

y

L

z

Lx y z

udvdwdwvuf ,, ,


5

where sn(χ) = sin (π n χ/Lχ). We determine coefficients an by using orthogonality condition of terms

of the considered series framework scale of heterostructure. The condition gives us possibility to

obtain relations for calculation of parameters an for any quantity of terms N. In the common case

the relations could be written as

( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫

×

∑+−=∑−

==

t L L L N

nnCnnnnCL

zyN

nnC

nCzyxx y z

ezcycxcaTzyxDLL

ten

aLLL

0 0 0 0 12

165

222

1,,,2

γ

τππ

( )( ) ( )

( )( ) ( ) ( ) ( )×∑

++

×=

N

nnCnnn

nC ezcycxsn

a

V

zyxV

V

zyxV

TzyxP 12*

2

2*1 2,,,,,,

1,,,

ττ

ςτ

ςξ

γ

( ) ( )[ ] ( ) ( )[ ] ( )×∫ ∫ ∫ ∫−

−+

−+×t L L L

Ln

z

nn

y

n

x y z

TzyxDdxdydzdzcn

Lzszyc

n

Lysy

0 0 0 0

,,,11 τππ

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

( )

++

∑+×

=*1

1

,,,1

,,,1,,,

V

zyxV

TzyxPezcycxcaTzyxD

N

nnCnnnnCL

τς

ξτ

γ

γ

( )( )

( ) ( )( )

( ) ( )[ ]∑ ×

−+

++

+

=

N

n

nC

n

x

nn

axc

n

Lxsx

V

zyxV

V

zyxV

V

zyxV

12*

2

2*12*

2

21

,,,,,,1

,,,

π

τς

τς

τς

( ) ( ) ( ) ( ) ( ) ( )[ ] ×−

−+×22 2

122 π

τπ

τπ

yx

n

z

nnCnnn

zxLL

dxdydzdzcn

Lzszezcysxc

LL

( ) ( ) ( ) ( )( )

( )( )

∫ ∫ ∫ ∫

++

∑+×

=

t L L L N

nnCnnnnC

x y z

V

zyxV

TzyxPezcycxca

0 0 0 02*

2

21

,,,1

,,,1

τς

ξτ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑

−+

+

=

N

nn

x

nnnn

nC

Lxc

n

Lxsxzsycxc

n

aTzyxD

V

zyxV

1*1

1,,,,,,

π

τς

( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ ×

−++

−+×=

N

n

L

n

x

nnCn

y

n

x

xcn

Lxsxdxdydzdeyc

n

Lysy

1 0

11π

ττπ

( ) ( )[ ] ( ) ( )[ ] ( )∫ ∫

−+

−+×y z

L L

n

z

nn

y

nxdydzdzyxfzc

n

Lzszyc

n

Lysy

0 0

,,11ππ

.

As an example for γ = 0 we obtain

( ) ( )[ ] ( ) ( )[ ] ( ) ( ){∫ ∫ ∫ +

−+

−+=x y zL L L

nn

y

nn

y

nnCxsxydzdzyxfzc

n

Lzszyc

n

Lysya

0 0 0

,,11ππ

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

∫ ∫ ∫ ∫ ×

−+

−×t L L L

Ln

y

nnn

x

n

x y z

TzyxDycn

Lysyycxs

nxd

n

Lxc

0 0 0 0

,,,122

1ππ

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

×

+

++

−+×TzyxPV

zyxV

V

zyxVzc

n

Lzsz

n

y

n,,,

1,,,,,,

112*

2

2*1 γ

ξτς

τς

π

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ×∫ ∫ ∫ ∫

−++×t L L L

nnn

y

nnnCnCn

x y z

zcysxcn

Lxsxxcedexdydzdzc

0 0 0 0

21π

τττ

( ) ( )[ ]( )

( ) ( )( )

×

++

+

−+×2*

2

2*1

,,,,,,1

,,,11

V

zyxV

V

zyxV

TzyxPzc

n

Lzsz

n

y

n

τς

τς

ξ

π γ

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ){∫ ∫ ∫ ×

−++×t L L

nnn

x

nnnCL

x y

ysycxcn

LxsxxcedxdydzdTzyxD

0 0 0

1,,,π

ττ


6

( )[ ] ( ) ( )( )

( )( )

∫

++

+

−+×zL

Lnn

y

V

zyxV

TzyxPTzyxDzsyc

n

Ly

02*

2

2

,,,1

,,,1,,,21

τς

ξ

π γ

( ) ( )1

65

222

*1

,,,−

−

+ te

n

LLLdxdydzd

V

zyxVnC

zzz

πτ

τς .

For γ = 1 one can obtain the following relation to determine required parameters

( ) ( ) ( ) ( )∫ ∫ ∫+±−=x y zL L L

nnnnn

n

n

nCxdydzdzyxfzcycxca

0 0 0

2 ,,42

αβα

β,

where ( ) ( ) ( ) ( )( )

( )( )∫ ∫ ∫ ∫ ×

++=

t L L LL

nnC

zy

n

x y z

TzyxP

TzyxD

V

zyxV

V

zyxVxse

n

LL

0 0 0 02*

2

2*12 ,,,

,,,,,,,,,12

2

τς

τςτ

π

ξα

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∫+

−+

−+×t

nC

zx

n

z

nn

y

nnne

n

LLdxdydzdzc

n

Lzszyc

n

Lysyzcyc

022

11 τπ

ξτ

ππ

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

×∫ ∫ ∫

++

−−

−+×x y zL L L

n

z

nn

x

n

V

zyxV

V

zyxVzc

n

Lzszxc

n

Lxsx

0 0 02*

2

2*1

,,,,,,111

τς

τς

ππ

( ) ( ) ( )( )

( )( ) ( ) ( )

( )( )

×∫ ∫ ∫ ∫+×t L L L

L

nnnC

yxL

nnn

x y z

TzyxP

TzyxDycxce

n

LLdxdydzd

TzyxP

TzyxDzcysxc

0 0 0 02 ,,,

,,,

2,,,

,,,2 τ

π

ξτ

( ) ( )( )

( ) ( )[ ] ( ) ( )[ ] ×

−+

−+

++× ydyc

n

Lysyxc

n

Lxsx

V

zyxV

V

zyxVn

y

nn

x

n11

,,,,,,1

2*

2

2*1ππ

τς

τς

( ) τdxdzdzsn

2× , ( ) ( ) ( ) ( ) ( )( )

∫ ∫ ∫ ∫ ×

++=

t L L L

nnnC

zy

n

x y z

V

zyxV

V

zyxVzcxse

n

LL

0 0 0 02*

2

2*12

,,,,,,12

2

τς

τςτ

πβ

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ×+

−+

−+×22

11,,,π

τππ n

LLdxdydyc

n

Lysyyczdzc

n

LzszTzyxD zx

n

y

nnn

z

nL

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

×∫ ∫ ∫ ∫

++

−+×t L L L

nnn

x

nnnC

x y z

V

zyxV

V

zyxVzcysxc

n

Lxsxxce

0 0 0 02*

2

2*1

,,,,,,121

τς

τς

πτ

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−++

−+×t L

n

x

nnC

yx

n

z

nL

x

xcn

Lxsxe

n

LLdxdydzdzc

n

LzszTzyxD

0 02

12

1,,,π

τπ

τπ

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

∫ ×∫

++

−+×y z

L L

Lnn

y

nzd

V

zyxV

V

zyxVTzyxDzsyc

n

Lysy

0 02*

2

2*1

,,,,,,1,,,21

τς

τς

π

( ) ( ) ( ) 65222nteLLLdxdxcydyc

nCzyxnnπτ −× .

Analogous way could be used to calculate values of parameters an for larger values of parameter

γ. However the relations will not be present in the paper because the relations are bulky.

Equations of the system (4) have been also solved by using Bubnov-Galerkin approach. Previous-

ly we transform the differential equations to the following integro- differential form


7

( ) ( ) ( )×∫ ∫ ∫

∂

∂=∫ ∫ ∫

t y

L

z

LI

x

L

y

L

z

Lzyx y zx y z

dvdwdx

wvxITwvxDudvdwdtwvuI

LLL

zyx

0

,,,,,,,,, τ

τ

( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫∂

∂+×

x

L

y

L

z

LVI

t x

L

z

LI

zxzy x y zx z

Twvukdudwdx

wyuITwyuD

LL

zx

LL

zy,,,

,,,,,,

,0

ττ

( ) ( ) ( )×∫ ∫ ∫

∂

∂+×

t x

L

y

Lyxzyx x y z

zvuI

LL

yx

LLL

zyxudvdwdtwvuVtwvuI

0

,,,,,,,,,

τ

( ) ( ) ( ) +∫ ∫ ∫−×x

L

y

L

z

LII

zyx

I

x y z

udvdwdtwvuITwvukLLL

zyxdudvdTzvuD ,,,,,,,,, 2

,τ

( )∫ ∫ ∫+x

L

y

L

z

LI

zyx x y z

udvdwdwvufLLL

zyx,, (4a)

( ) ( ) ( )×∫ ∫ ∫

∂

∂=∫ ∫ ∫

t y

L

z

LV

x

L

y

L

z

Lzyx y zx y z

dvdwdx

wvxVTwvxDudvdwdtwvuV

LLL

zyx

0

,,,,,,,,, τ

τ

( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫∂

∂+×

t x

L

y

LV

t x

L

z

LV

zxzy x yx z

TzvuDdudwdx

wyuVTwyuD

LL

zx

LL

zy

00

,,,,,,

,,, ττ

( ) ( ) ( ) ×∫ ∫ ∫−∂

∂×

x

L

y

L

z

LVI

zyxyx x y z

twvuITwvukLLL

zyx

LL

yxdudvd

z

zvuV,,,,,,

,,,,

ττ

( ) ( ) ( ) +∫ ∫ ∫−×x

L

y

L

z

LVV

zyx x y z

udvdwdtwvuVTwvukLLL

zyxudvdwdtwvuV ,,,,,,,,,

2

,

( )∫ ∫ ∫+x

L

y

L

z

LV

zyx x y z

udvdwdwvufLLL

zyx,, .

We determine solutions of the Eqs.(4a) as the following series

( ) ( ) ( ) ( ) ( )∑==

N

nnnnnn

tezcycxcatzyx1

0 ,,, ρρρ .

Coefficients anρ are not yet known. Substitution of the series into Eqs.(4a) leads to the following

results

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑==

N

n

t y

L

z

LInnnI

zyx

N

nnInnn

nI

y z

vdwdTwvxDzcycaLLL

zytezsysxs

n

azyx

1 0133

,,,π

π

( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−×=

N

n

t x

L

z

LInnnInnI

zyx

nnI

x z

dudwdTwyuDzcxceysaLLL

zxxsde

1 0

,,, ττπ

ττ

( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∑ ∫ ∫ ∫−=

x

L

y

L

z

LII

N

n

t x

L

y

LInnnInnI

zyx x y zx y

TvvukdudvdTzvuDycxcezsaLLL

yx,,,,,, ,

1 0

ττπ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑−

∑×

==

x

L

y

L

z

L

N

nnnnnI

zyxzyx

N

nnInnnnI

x y z

wcvcucaLLL

zyx

LLL

zyxudvdwdtewcvcuca

1

2

1

( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑×=

x

L

y

L

z

LI

zyx

N

nVInVnnnnVnI

x y z

udvdwdwvufLLL

zyxudvdwdTvvuktewcvcucate ,,,,,

1,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑==

N

n

t y

L

z

LVnnnV

zyx

N

nnVnnn

nV

y z

vdwdTwvxDzcycaLLL

zytezsysxs

n

azyx

1 0133

,,,π

π


8

( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−×=

N

n

t x

L

z

LVnnnVnnV

zyx

nnV

x z

dudwdTwyuDzcxceysaLLL

zxxsde

1 0

,,, ττπ

ττ

( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∑ ∫ ∫ ∫−=

x

L

y

L

z

LVV

N

n

t x

L

y

LVnnnVnnV

zyx x y zx y

TvvukdudvdTzvuDycxcezsaLLL

yx,,,,,,

,1 0

ττπ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑−

∑×

==

x

L

y

L

z

L

N

nnnnnI

zyxzyx

N

nnInnnnV

x y z

wcvcucaLLL

zyx

LLL

zyxudvdwdtewcvcuca

1

2

1

( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑×=

x

L

y

L

z

LV

zyx

N

nVInVnnnnVnI

x y z

udvdwdwvufLLL

zyxudvdwdTvvuktewcvcucate ,,,,,

1, .

We determine coefficients anρ by using orthogonality condition on the scale of heterostructure.

The condition gives us possibility to obtain relations to calculate anρ for any quantity N of terms

of considered series. In the common case equations for the required coefficients could be written

as

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

N

n

t L L

n

y

nyn

nI

x

N

nnI

nIzyxx y

ycn

LysyLxc

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πππ

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

N

n

t L

n

nI

y

L

nIn

z

nI

xz

xsxn

a

Ldexdydzdzc

n

LzszTzyxD

1 0 02

0

22

11

2,,,

πττ

π

( )[ ] ( ) ( ) ( )[ ] ( )[ ] ×∫ ∫ −

−++

−++y z

L L

nn

z

nzIn

x

xyczdzc

n

LzszLTzyxDxc

n

LL

0 0

21122

2,,,12ππ

( ) ( ) ( ) ( )[ ] ( ) −∫

−++×zL

nIn

z

nzInIdexdydzdzc

n

LzszLTzyxDdexdyd

0

122

2,,, ττπ

ττ

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

nyn

x

nx

nI

z

x y

ycn

LysyLxc

n

LxsxL

n

a

L 1 0 0 02

122

2122

22

1

πππ

( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫

+−+−∫ −×=

N

n

L

n

x

xnInI

L

nIIn

xz

xcn

LLteadexdydzdTzyxDzc

1 0

2

0

122

2,,,21π

ττ

( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫

+−+

−+++y z

L L

n

z

zIIn

y

nynzc

n

LLTzyxkyc

n

LysyLxsx

0 0, 12

2,,,12

222

ππ

( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ +

−++−+=

N

n

L L

yn

x

nxnVnInVnIn

x y

Lxcn

LxsxLteteaaxdydzdzsz

1 0 0

122

22π

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ×

−++

−++zL

n

z

nzVIn

y

nzdzc

n

LzszLTzyxkyc

n

Lysy

0, 12

22,,,12

22

ππ

( ) ( )[ ] ( ) ( )[ ] ( ) ×∑ ∫ ∫ ∫

−+

−++×=

N

n

L L L

In

y

nn

x

n

x y z

Tzyxfycn

Lysyxc

n

Lxsxxdyd

1 0 0 0

,,,11ππ

( ) ( )[ ] xdydzdzcn

LzszL

n

z

nz

−++× 122

2π

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

N

n

t L L

n

y

nyn

nV

x

N

nnV

nVzyxx y

ycn

LysyLxc

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πππ

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

N

n

t L

n

nV

y

L

nVn

z

nV

xz

xsxn

a

Ldexdydzdzc

n

LzszTzyxD

1 0 02

0

22

11

2,,,

πττ

π


9

( )[ ] ( ) ( ) ( )[ ] ( )[ ] ×∫ ∫ −

−++

−++y z

L L

nn

z

nzVn

x

xyczdzc

n

LzszLTzyxDxc

n

LL

0 0

21122

2,,,12ππ

( ) ( ) ( ) ( )[ ] ( ) −∫

−++×zL

nVn

z

nzVnVdexdydzdzc

n

LzszLTzyxDdexdyd

0

122

2,,, ττπ

ττ

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

nyn

x

nx

nV

z

x y

ycn

LysyLxc

n

LxsxL

n

a

L 1 0 0 02

122

2122

22

1

πππ

( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫

+−+−∫ −×=

N

n

L

n

x

xnVnV

L

nVVn

xz

xcn

LLteadexdydzdTzyxDzc

1 0

2

0

122

2,,,21π

ττ

( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫

+−+

−+++y z

L L

n

z

zVVn

y

nynzc

n

LLTzyxkyc

n

LysyLxsx

0 0, 12

2,,,12

222

ππ

( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ +

−++−+=

N

n

L L

yn

x

nxnVnInVnIn

x y

Lxcn

LxsxLteteaaxdydzdzsz

1 0 0

122

22π

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ×

−++

−++zL

n

z

nzVIn

y

nzdzc

n

LzszLTzyxkyc

n

Lysy

0, 12

22,,,12

22

ππ

( ) ( )[ ] ( ) ( )[ ] ( ) ×∑ ∫ ∫ ∫

−+

−++×=

N

n

L L L

Vn

y

nn

x

n

x y z

Tzyxfycn

Lysyxc

n

Lxsxxdyd

1 0 0 0

,,,11ππ

( ) ( )[ ] xdydzdzcn

LzszL

n

z

nz

−++× 122

2π

.

In the final form relations for required parameters could be written as

( )

−+−

+±

+−=

A

ybyb

Ab

b

Aba nInV

nI

2

3

4

2

3

4

3 444

λγ,

nInI

nInInInInI

nVa

aaa

χ

λδγ ++−=

2

,

where ( ) ( ) ( ) ( )[ ] ( ){∫ ∫ ∫ ++

−++=x y zL L L

ynn

x

nxnnLysyxc

n

LxsxLTzyxkte

0 0 0, 212

22,,,2

πγ ρρρρ

( )[ ] ( ) ( )[ ] xdydzdzcn

LzszLyc

n

Ln

z

nzn

y

−++

−+ 122

2122 ππ

, ( )×∫=t

n

x

ne

nL 02

2

1τ

πδ ρρ

( ) ( )[ ] ( ) ( )[ ] ( ) [∫ ∫ ∫ −

−+

−+×x y zL L L

n

z

nn

y

nydzdTzyxDzc

n

Lzszyc

n

Lysy

0 0 0

1,,,12

12

ρππ

( )] ( ) ( ) ( )[ ] ( )[ ] {∫ ∫ ∫ ∫ +−

−+++−t L L L

znn

x

nxn

y

n

x y z

Lycxcn

LxsxLe

nLdxdxc

0 0 0 02

211222

12

πτ

πτ ρ

( ) ( )[ ] ( ) ( ) ( ){∫ ∫ ++

−++t L

nn

z

n

z

n

x

xsxenL

dxdydzdTzyxDzcn

Lzsz

0 02

22

1,,,12

22 τ

πτ

πρρ

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ×∫ ∫ −

−++

−++y z

L L

nn

y

nyn

x

xzdTzyxDzcyc

n

LysyLxc

n

LL

0 0

,,,2112

12 ρππ

( )ten

LLLdxdyd

n

zyx

ρπ

τ65

222

−× , ( ) ( )[ ] ( )[ ]∫ ∫

+−+

−+=x yL L

n

y

yn

x

nnIVyc

n

LLxc

n

Lxsx

0 0

122

1ππ

χ


10

( )} ( ) ( ) ( )[ ] ( ) ( )tetexdydzdzcn

LzszLTzyxkysy

nVnI

L

n

z

nzVIn

z

∫

−+++0

, 122

2,,,2π

,

( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]∫ ∫ ∫ ×

−+

−+

−+=x y zL L L

n

z

nn

y

nn

x

nnzc

n

Lzszyc

n

Lysyxc

n

Lxsx

0 0 0

111πππ

λ ρ

( ) xdydzdTzyxf ,,,ρ× , 22

4 nInInInVb χγγγ −= ,

nInInVnInInInInVb γχδχδδγγ −−= 2

3 2 ,

2

2

348 bbyA −+= , ( ) 22

2 2nInInVnInInVnInVnInInV

b χλλδχδγγλδγ −+−+= , ×=nI

b λ21

nInInVnInVλχδδγ −× ,

4

33 323 32

3b

bqpqqpqy −++−−+= ,

2

4

2

342

9

3

b

bbbp

−= ,

( ) 3

4

2

4132

3

3 542792 bbbbbbq +−= .

We determine solutions of the Eqs.(4a) as the following series

( ) ( ) ( ) ( ) ( )∑=Φ=

Φ

N

nnnnnn

tezcycxcatzyx1

0 ,,, ρρρ .

Coefficients anΦρ are not yet known. Let us previously transform the Eqs.(6) to the following in-tegro-differential form

( ) ( ) ( )×∫ ∫ ∫

Φ=∫ ∫ ∫Φ Φ

t y

L

z

L

I

I

x

L

y

L

z

LI

zyx y zx y z

dvdwdx

wvxTwvxDudvdwdtwvu

LLL

zyx

0

,,,,,,,,, τ

∂

τ∂

( ) ( ) ( )∫ ∫ ∫ ×+∫ ∫ ∫Φ

+× ΦΦ

t x

L

y

LI

yx

t x

L

z

L

I

I

zxzy x yx z

TzvuDLL

yxdudwd

y

wyuTwyuD

LL

zx

LL

zy

00

,,,,,,

,,, τ∂

τ∂

( ) ( ) ( ) −∫ ∫ ∫+Φ

×x

L

y

L

z

LII

zyx

I

x y z

udvdwdwvuITwvukLLL

zyxdudvd

z

zvuττ

∂

τ∂,,,,,,

,,, 2

, (6a)

( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− Φ

x

L

y

L

z

LI

zyx

x

L

y

L

z

LI

zyx x y zx y z

udvdwdwvufLLL

zyxudvdwdwvuITwvuk

LLL

zyx,,,,,,,, τ

( ) ( ) ( )×∫ ∫ ∫

Φ=∫ ∫ ∫Φ Φ

t y

L

z

L

V

V

x

L

y

L

z

LV

zyx y zx y z

dvdwdx

wvxTwvxDudvdwdtwvu

LLL

zyx

0

,,,,,,,,, τ

∂

τ∂

( )( )

( )∫ ∫ ∫ ×+∫ ∫ ∫Φ

+× ΦΦ

t x

L

y

LV

yx

t x

L

z

L

V

V

zxzy x yx z

TzvuDLL

yxdudwd

y

wyuTwyuD

LL

zx

LL

zy

00

,,,,,,

,,, τ∂

τ∂

( ) ( ) ( ) −∫ ∫ ∫+Φ

×x

L

y

L

z

LVV

zyx

V

x y z

udvdwdwvuVTwvukLLL

zyxdudvd

z

zvuττ

∂

τ∂,,,,,,

,,, 2

,

( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− Φ

x

L

y

L

z

LV

zyx

x

L

y

L

z

LV

zyx x y zx y z

udvdwdwvufLLL

zyxudvdwdwvuVTwvuk

LLL

zyx,,,,,,,, τ .

Substitution of the previously considered series in the Eqs.(6a) leads to the following form

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑−=

Φ=

ΦN

n

t y

L

z

LnnnInIn

zyx

N

nnInnn

In

y z

wcvctexsanLLL

zytezsysxs

n

azyx

1 0133

π

π

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

ΦΦΦ

N

n

t x

L

z

LInnIn

zyx

I

x z

dudwdTwvuDwcucaLLL

zxdvdwdTwvxD

1 0

,,,,,, τπ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) +∑ ∫ ∫ ∫−×=

ΦΦΦΦ

N

n

t x

L

y

LInnInnIn

zyx

Inn

x y

dudvdTzvuDvcuctezsanLLL

yxteysn

1 0

,,, τπ


11

( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+ Φ

x

L

y

L

z

LI

x

L

y

L

z

LII

zyx x y zx y z

udvdwdwvufudvdwdwvuITwvukLLL

zyx,,,,,,,,

2

, τ

( ) ( )∫ ∫ ∫−×x

L

y

L

z

LI

zyxzyx x y z

udvdwdwvuITwvukLLL

zyx

LLL

zyxτ,,,,,,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑−=

Φ=

ΦN

n

t y

L

z

LnnnVnVn

zyx

N

nnVnnn

Vn

y z

wcvctexsanLLL

zytezsysxs

n

azyx

1 0133

π

π

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

ΦΦ

N

n

t x

L

z

LVnn

zyx

V

x z

dudwdTwvuDwcucnLLL

zxdvdwdTwvxD

1 0

,,,,,, τπ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

ΦΦΦΦ

N

n

t x

L

y

LVnnVnn

zyx

VnnVn

x y

dudvdTzvuDvcuctezsnLLL

yxteysa

1 0

,,, τπ

( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+× ΦΦ

x

L

y

L

z

LV

x

L

y

L

z

LVV

zyx

Vn

x y zx y z

udvdwdwvufudvdwdwvuVTwvukLLL

zyxa ,,,,,,,, 2

,τ

( ) ( )∫ ∫ ∫−×x

L

y

L

z

LV

zyxzyx x y z

udvdwdwvuVTwvukLLL

zyx

LLL

zyxτ,,,,,, .

We determine coefficients anΦρ by using orthogonality condition on the scale of heterostructure.

The condition gives us possibility to obtain relations to calculate anΦρ for any quantity N of terms

of considered series. In the common case equations for the required coefficients could be written

as

( ) ( )[ ] ( ) ( )[ ] ×∑∫ ∫ ∫

−++−−=∑−==

ΦΦ

N

n

t L L

n

y

nyn

x

N

nIn

Inzyxx y

ycn

LysyLxc

Lte

n

aLLL

10 0 0165

222

122

2212

1

πππ

( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫

−+×=

ΦΦΦ

N

n

t L

n

L

Inn

z

nI

Inxz

xsxdexdydzdzcn

LzszTzyxD

n

a

10 002

22

11

2,,,

πττ

π

( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−+−

−++ Φ

y zL L

n

z

nInn

x

xxdydzdzc

n

LzszTzyxDycxc

n

LL

0 0

12

,,,21122 ππ

( ) ( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫

+−+

−+−×=

ΦΦΦ

N

n

t L L

n

y

nn

x

n

In

xy

In

In

x y

ycn

Lysyxc

n

Lxsx

n

a

Ld

Ln

ea

1 0 0 022

122

2122

1

πππτ

τ

} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫

+−+∫ −+=

ΦΦ

ΦΦ

N

n

t L

n

x

In

InL

InIny

xz

xcn

Le

n

adexdydzdTzyxDycL

1 0 033

0

12

1,,,21

πτ

πττ

( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫

+−∫

−++zy L

n

z

II

L

n

y

nnzc

n

LTzyxktzyxIyc

n

Lysyxsx

0,

2

0

12

,,,,,,12 ππ

( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫

+−

−+−+=

ΦΦ

N

n

t L L

n

y

n

x

nIn

In

n

x y

ycn

Lxc

n

Lxsxe

n

axdydzdzsz

1 0 0 033

12

12

1

ππτ

π

( )} ( ) ( )[ ] ( ) ( ) ×∑+∫

−++=

ΦN

n

InL

In

z

nnn

axdydzdtzyxITzyxkzc

n

Lzszysy

z

133

0

1,,,,,,1

2 ππ

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

+−

−+

−+× Φ

t L L L

n

z

n

y

nn

x

nIn

x y z

zcn

Lyc

n

Lysyxc

n

Lxsxe

0 0 0 0

12

12

12 πππ

τ

( )} ( ) xdydzdzyxfzszIn

,,Φ+


12

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

ΦΦ

N

n

t L L

n

y

nyn

x

N

nVn

Vnzyxx y

ycn

LysyLxc

Lte

n

aLLL

10 0 0165

222

122

2212

1

πππ

( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫

−+×=

ΦΦΦ

N

n

t L

n

L

Vnn

z

nV

Vnxz

xsxdexdydzdzcn

LzszTzyxD

n

a

10 002

22

11

2,,,

πττ

π

( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−+−

−++ Φ

y zL L

n

z

nVnn

x

xxdydzdzc

n

LzszTzyxDycxc

n

LL

0 0

12

,,,21122 ππ

( ) ( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫

+−+

−+−×=

ΦΦΦ

N

n

t L L

n

y

nn

x

n

Vn

xy

Vn

Vn

x y

ycn

Lysyxc

n

Lxsx

n

a

Ld

Ln

ea

1 0 0 022

122

2122

1

πππτ

τ

} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫

+−+∫ −+=

ΦΦ

ΦΦ

N

n

t L

n

x

Vn

VnL

VnVny

xz

xcn

Le

n

adexdydzdTzyxDycL

1 0 033

0

12

1,,,21

πτ

πττ

( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫

+−∫

−++zy L

n

z

VV

L

n

y

nnzc

n

LTzyxktzyxVyc

n

Lysyxsx

0,

2

0

12

,,,,,,12 ππ

( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫

+−

−+−+=

ΦΦ

N

n

t L L

n

y

n

x

nVn

Vn

n

x y

ycn

Lxc

n

Lxsxe

n

axdydzdzsz

1 0 0 033

12

12

1

ππτ

π

( )} ( ) ( )[ ] ( ) ( ) ×∑+∫

−++=

ΦN

n

VnL

Vn

z

nnn

axdydzdtzyxVTzyxkzc

n

Lzszysy

z

133

0

1,,,,,,1

2 ππ

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

+−

−+

−+× Φ

t L L L

n

z

n

y

nn

x

nVn

x y z

zcn

Lyc

n

Lysyxc

n

Lxsxe

0 0 0 0

12

12

12 πππ

τ

( )} ( ) xdydzdzyxfzszVn

,,Φ+ .

3. DISCUSSION

In this section we used relations from previous section for analysis of influence of parameters on

distributions of concentrations of infused and implanted dopants. The typical distributions are presented on Figs. 2 and 3, respectively. The figures correspond to doping of one layer of two-

layer structure with higher value of dopant diffusion coefficient in comparison with value of do-

pant diffusion coefficient in the second layer. The figures show, that using interface between lay-ers of heterostructure leads to increasing of compactness of p-i-n-heterodiode. At the same time

we homogeneity of distributions of concentrations of dopants in doped areas increases with de-

creasing of the homogeneity outside the enriched areas.

We note, that using the considered approach of manufacturing p-i-n-diodes leads to necessity to

optimize annealing of dopant and/or radiation defects. Necessity of this optimization is following.

Increasing of annealing time of dopant gives a possibility to obtain too homogenous distribution

of concentration of dopant. If the annealing time is small, the dopant has not enough time to

achieve interface between layers of heterostructure. In this situation one can not find any modifi-

cation of distribution of concentration of dopant due to existing the interface. In this situation it is required optimization of annealing time. We optimize annealing time of dopant framework re-

cently introduced criterion [26-35]. Dependences of optimal annealing time on parameters are

presented on Figs. 4 and 5. Optimal annealing time of implanted dopant is smaller, than optimal

annealing time of infused dopant. Reason of the difference is necessity of annealing of radiation

defects. One can find spreading of distribution of concentration of dopant during annealing of

radiation defects. In the ideal distribution of concentration of dopant achieves interface between layers of heterostructure during annealing of radiation defects. It is practicably to use additional

annealing of dopant in the case, one the dopant has not enough time to achieve the interface. The

Fig. 5 shows just dependences of optimal values of additional annealing time of dopant.


13

It is known, that using diffusion type of doping did not leads radiation damage of materials in comparison with ion type of doping. However ion doping of heterostructures attracted an interest

for large difference of the lattice constant. In this case radiation damage leads to mismatch-

induced stress [36].

Fig.2. Dependences of concentration of infused dopant in heterostructure from Fig. 1 near one of interfaces. Value of dopant diffusion coefficient in the left layer is larger, than value of dopant

diffusion coefficient in the right layer. Increasing of number of curve corresponds to increasing of

difference between values of dopant diffusion coefficient in layers of heterostructure

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. Dependence of concentration of implanted dopant on coordinate in heterostructure from

Fig. 1 near one of interfaces. 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(Lx

2+Ly2+Lz

2)/D0. Curves 1 and 2

are distributions of concentration of implanted dopant in a homogenous sample. Curves 3 and 4

are distributions of concentration of implanted dopant in a heterostructure near one of interfaces.

Value of dopant diffusion coefficient in the left layer is larger, than value of dopant diffusion

coefficient in the right layer. Increasing of number of curve corresponds to increasing of differ-

ence between values of dopant diffusion coefficient in layers of heterostructure


14

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.4. Optimal annealing time of infused dopant as a functions of several parameters. Curve 1 is

the dependence of dimensionless optimal annealing time on the relation a/L and ξ = γ = 0 for

equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is

the dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and

ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of para-

meter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal 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

Fig.5. Optimal annealing time for ion doping of heterostructure as a function of several parame-

ters. Curve 1 is the dependence of dimensionless optimal annealing time on the relation a/L and ξ

= γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostruc-

ture. Curve 2 is the dependence of dimensionless optimal annealing time on value of parameter ε

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

value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless op-

timal annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0


15

4. CONCLUSIONS

In this paper we introduce an approach to manufacture p-i-n-heterodiodes. The approach based on

using a δ-doped heterostructure, doping by diffusion or ion implantation of several areas of the

heterostructure. After the doping the dopant and/or radiation defects have been annealed.At the

same time we consider an analytical approach to model technological processes. Framework the

approach gives a possibility to manage without stitching decisions on the interfaces between lay-

ers heterostructures.

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 and educational fellowship for scientific research of Government of Russian

and 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.


17

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. 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.