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
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC
CCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,,
. (1)
Boundary and initial conditions for the equations are
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C (x,y,z,0)=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
tzyxC
TzyxDD LC ςςξ 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
tzyxI
TzyxD
yx
tzyxI
TzyxD
xt
tzyxI
IIII ,,,
,,,
,,,
,,,
,,,
,,,
,
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk
z
tzyxI
TzyxD
z
tzyxI VII ,,,,,,,,,
,,,
,,,,,, ,
2
−





∂
∂
∂
∂
+× (4)
( ) ( ) ( ) ( ) ( ) ( ) ×−





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
Tzyxk
y
tzyxV
TzyxD
yx
tzyxV
TzyxD
xt
tzyxV
VVVV ,,,
,,,
,,,
,,,
,,,
,,,
,
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk
z
tzyxV
TzyxD
z
tzyxV VIV ,,,,,,,,,
,,,
,,,,,, ,
2
−





∂
∂
∂
∂
+× .
Boundary and initial conditions for these equations are
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxρ
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxρ
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxρ
,
( ) 0
,,,
=
∂
∂
= yLy
y
tzyxρ
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxρ
,
( ) 0
,,,
=
∂
∂
= zLz
z
tzyxρ
, ρ (x,y,z,0)=fρ (x,y,z). (5)
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 I2
(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
=
Φ
ΦΦ
y
tzyx
TzyxD
yx
tzyx
TzyxD
xt
tzyx I
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk
z
tzyx
TzyxD
z
III
I
I ,,,,,,,,,,,,
,,,
,,, 2
, −+




 Φ
+ Φ
∂
∂
∂
∂
(6)
( ) ( ) ( ) ( ) ( ) +




 Φ
+




 Φ
=
Φ
ΦΦ
y
tzyx
TzyxD
yx
tzyx
TzyxD
xt
tzyx V
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk
z
tzyx
TzyxD
z
VVV
V
V ,,,,,,,,,,,,
,,,
,,, 2
, −+




 Φ
+ Φ
∂
∂
∂
∂
.
Boundary and initial conditions for these equations are
( )
0
,,,
0
=
∂
Φ∂
=x
x
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= xLx
x
tzyxρ
,
( )
0
,,,
0
=
∂
Φ∂
=y
y
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= yLy
y
tzyxρ
,
( )
0
,,,
0
=
∂
Φ∂
=z
z
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= zLz
z
tzyxρ
, ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (7)
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
zy
udvdwdtwvuC
LLL
zyx
0
2*
2
2*1
,,,,,,
1,,,
τ
ς
τ
ς
( ) ( )
( )
( ) ( ) ×∫ ∫ ∫+





+×
t x
L
z
L
L
zx
L
x z
TwyuD
LL
zx
d
x
wvxC
TwvxP
wvxC
TwvxD
0
,,,
,,,
,,,
,,,
1,,, τ
∂
τ∂τ
ξ γ
γ
( ) ( )
( )
( )
( )
( ) +





+








++× τ
∂
τ∂τ
ξ
τ
ς
τ
ς γ
γ
d
y
wyuC
TzyxP
wyuC
V
wyuV
V
wyuV ,,,
,,,
,,,
1
,,,,,,
1 2*
2
2*1
( ) ( ) ( )
( )
( )
( )
×∫ ∫ ∫ 





+








+++
t x
L
y
L
L
yx x y TzyxP
zvuC
V
zvuV
V
zvuV
TzvuD
LL
yx
0
2*
2
2*1
,,,
,,,
1
,,,,,,
1,,, γ
γ
τ
ξ
τ
ς
τ
ς
( ) ( )∫ ∫ ∫+×
x
L
y
L
z
Lzyx x y z
udvdwdwvuf
LLL
zyx
d
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
n
nCnnnnC tezcycxcatzyxC
0
0 ,,, ,
where ( ) ( )[ ]222
0
22
exp −−−
++−= 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
n
nCnnnnC
zy
N
n
nCnnn
C
y z
ewcvcxca
LL
zy
tezsysxs
n
azyx
0 11
32
1
γ
τ
π
( )
( ) ( )
( )
( ) ( ) ( )×∑








++



×
=
N
n
nnnCL vcxsaTwvxD
V
wvxV
V
wvxV
TwvxP 1
2*
2
2*1 ,,,
,,,,,,
1
,,,
τ
ς
τ
ς
ξ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )∫ ∫ ∫ ×









∑+−×
=
t x
L
z
L
N
m
mCmmmmC
zx
nCn
x z TwyuP
ewcycuca
LL
zx
dewcn
0 1 ,,,
1 γ
γ
ξ
τττ
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ×∑








++×
=
ττ
τ
ς
τ
ς dewcysucn
V
wyuV
V
wyuV
TwyuD
N
n
nCnnnL
1
2*
2
2*1
,,,,,,
1,,,
( )
( )
( ) ( ) ( ) ( ) ×∫ ∫ ∫









∑+−×
=
t x
L
y
L
N
n
nCnnnnCL
yx
nC
x y
ezcvcuca
TzvuP
TzvuD
LL
yx
a
0 1,,,
1,,,
γ
γ
τ
ξ
( ) ( )
( )
( ) ( ) ( ) ( ) ×+∑








++×
=
zyx
N
n
nCnnnnC
LLL
zyx
dezsvcucan
V
zvuV
V
zvuV
ττ
τ
ς
τ
ς
1
2*
2
2*1
,,,,,,
1
( )∫ ∫ ∫×
x
L
y
L
z
Lx y z
udvdwdwvuf ,, ,

5
where sn(χ)=sin(πnχ/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
n
nCnnnnCL
zy
N
n
nC
nCzyx
x y z
ezcycxcaTzyxD
LL
te
n
aLLL
0 0 0 0 1
2
1
65
222
1,,,
2
γ
τ
ππ
( )
( ) ( )
( )
( ) ( ) ( ) ( )×∑








++



×
=
N
n
nCnnn
nC
ezcycxs
n
a
V
zyxV
V
zyxV
TzyxP 1
2*
2
2*1 2
,,,,,,
1
,,,
τ
τ
ς
τ
ς
ξ
γ
( ) ( )[ ] ( ) ( )[ ] ( )×∫ ∫ ∫ ∫−






−+






−+×
t L L L
Ln
z
nn
y
n
x y z
TzyxDdxdydzdzc
n
L
zszyc
n
L
ysy
0 0 0 0
,,,11 τ
ππ
( ) ( ) ( ) ( ) ( )
( )
( )



++









∑+×
=
*1
1
,,,
1
,,,
1,,,
V
zyxV
TzyxP
ezcycxcaTzyxD
N
n
nCnnnnCL
τ
ς
ξ
τ γ
γ
( )
( )
( ) ( )
( )
( ) ( )[ ]∑ ×






−+








++




+
=
N
n
nC
n
x
n
n
a
xc
n
L
xsx
V
zyxV
V
zyxV
V
zyxV
1
2*
2
2*12*
2
2 1
,,,,,,
1
,,,
π
τ
ς
τ
ς
τ
ς
( ) ( ) ( ) ( ) ( ) ( )[ ] ×−






−+× 22
2
12
2 π
τ
π
τ
π
yx
n
z
nnCnnn
zx
LL
dxdydzdzc
n
L
zszezcysxc
LL
( ) ( ) ( ) ( )
( )
( )
( )∫ ∫ ∫ ∫




++









∑+×
=
t L L L N
n
nCnnnnC
x y z
V
zyxV
TzyxP
ezcycxca
0 0 0 0
2*
2
2
1
,,,
1
,,,
1
τ
ς
ξ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑






−+


+
=
N
n
n
x
nnnn
nC
L xc
n
L
xsxzsycxc
n
a
TzyxD
V
zyxV
1
*1 1,,,
,,,
π
τ
ς
( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ ×






−++






−+×
=
N
n
L
n
x
nnCn
y
n
x
xc
n
L
xsxdxdydzdeyc
n
L
ysy
1 0
11
π
ττ
π
( ) ( )[ ] ( ) ( )[ ] ( )∫ ∫






−+






−+×
y z
L L
n
z
nn
y
n xdydzdzyxfzc
n
L
zszyc
n
L
ysy
0 0
,,11
ππ
.
As an example for γ = 0 we obtain
( ) ( )[ ] ( ) ( )[ ] ( ) ( ){∫ ∫ ∫ +






−+






−+=
x y zL L L
nn
y
nn
y
nnC xsxydzdzyxfzc
n
L
zszyc
n
L
ysya
0 0 0
,,11
ππ
( )[ ] ( ) ( ) ( ) ( )[ ] ( )







∫ ∫ ∫ ∫ ×






−+



−×
t L L L
Ln
y
nnn
x
n
x y z
TzyxDyc
n
L
ysyycxs
n
xd
n
L
xc
0 0 0 0
,,,12
2
1
ππ
( ) ( )[ ] ( ) ( )
( ) ( )
×





+








++






−+×
TzyxPV
zyxV
V
zyxV
zc
n
L
zsz n
y
n
,,,
1
,,,,,,
11 2*
2
2*1 γ
ξτ
ς
τ
ς
π
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )×∫ ∫ ∫ ∫






−++×
t L L L
nnn
y
nnnCnCn
x y z
zcysxc
n
L
xsxxcedexdydzdzc
0 0 0 0
21
π
τττ
( ) ( )[ ]
( )
( ) ( )
( )
×








++





+






−+× 2*
2
2*1
,,,,,,
1
,,,
11
V
zyxV
V
zyxV
TzyxP
zc
n
L
zsz n
y
n
τ
ς
τ
ς
ξ
π γ
( ) ( ) ( ) ( ) ( )[ ] ( ) ( ){∫ ∫ ∫ ×






−++×
t L L
nnn
x
nnnCL
x y
ysycxc
n
L
xsxxcedxdydzdTzyxD
0 0 0
1,,,
π
ττ

6
( )[ ] ( ) ( )
( )
( )
( )∫




++





+



−+×
zL
Lnn
y
V
zyxV
TzyxP
TzyxDzsyc
n
L
y
0
2*
2
2
,,,
1
,,,
1,,,21
τ
ς
ξ
π γ
( ) ( )
1
65
222
*1
,,,
−




−






+ te
n
LLL
dxdydzd
V
zyxV
nC
zzz
π
τ
τ
ς .
For γ = 1 one can obtain the following relation to determine required parameters
( ) ( ) ( ) ( )∫ ∫ ∫+±−=
x y zL L L
nnnnn
n
n
nC xdydzdzyxfzcycxca
0 0 0
2
,,4
2
αβ
α
β
,
where ( ) ( ) ( ) ( )
( )
( )
( )∫ ∫ ∫ ∫ ×








++=
t L L L
L
nnC
zy
n
x y z
TzyxP
TzyxD
V
zyxV
V
zyxV
xse
n
LL
0 0 0 0
2*
2
2*12
,,,
,,,,,,,,,
12
2
τ
ς
τ
ςτ
π
ξ
α
( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∫+






−+






−+×
t
nC
zx
n
z
nn
y
nnn e
n
LL
dxdydzdzc
n
L
zszyc
n
L
ysyzcyc
0
2
2
11 τ
π
ξ
τ
ππ
( ) ( )[ ] ( ) ( )[ ] ( ) ( )
( )
×∫ ∫ ∫








++






−−






−+×
x y zL L L
n
z
nn
x
n
V
zyxV
V
zyxV
zc
n
L
zszxc
n
L
xsx
0 0 0
2*
2
2*1
,,,,,,
111
τ
ς
τ
ς
ππ
( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
( )
×∫ ∫ ∫ ∫+×
t L L L
L
nnnC
yxL
nnn
x y z
TzyxP
TzyxD
ycxce
n
LL
dxdydzd
TzyxP
TzyxD
zcysxc
0 0 0 0
2
,,,
,,,
2,,,
,,,
2 τ
π
ξ
τ
( ) ( )
( )
( ) ( )[ ] ( ) ( )[ ] ×






−+






−+








++× ydyc
n
L
ysyxc
n
L
xsx
V
zyxV
V
zyxV
n
y
nn
x
n 11
,,,,,,
1 2*
2
2*1
ππ
τ
ς
τ
ς
( ) τdxdzdzsn 2× , ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫ ×








++=
t L L L
nnnC
zy
n
x y z
V
zyxV
V
zyxV
zcxse
n
LL
0 0 0 0
2*
2
2*12
,,,,,,
12
2
τ
ς
τ
ςτ
π
β
( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ×+






−+






−+× 2
2
11,,,
π
τ
ππ n
LL
dxdydyc
n
L
ysyyczdzc
n
L
zszTzyxD zx
n
y
nnn
z
nL
( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )
( )
×∫ ∫ ∫ ∫








++






−+×
t L L L
nnn
x
nnnC
x y z
V
zyxV
V
zyxV
zcysxc
n
L
xsxxce
0 0 0 0
2*
2
2*1
,,,,,,
121
τ
ς
τ
ς
π
τ
( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫






−++






−+×
t L
n
x
nnC
yx
n
z
nL
x
xc
n
L
xsxe
n
LL
dxdydzdzc
n
L
zszTzyxD
0 0
2
1
2
1,,,
π
τ
π
τ
π
( ) ( )[ ] ( ) ( ) ( ) ( )
( )
∫ ×∫








++






−+×
y z
L L
Lnn
y
n zd
V
zyxV
V
zyxV
TzyxDzsyc
n
L
ysy
0 0
2*
2
2*1
,,,,,,
1,,,21
τ
ς
τ
ς
π
( ) ( ) ( ) 65222
nteLLLdxdxcydyc nCzyxnn πτ −× .
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
L
I
x
L
y
L
z
Lzyx y zx y z
dvdwd
x
wvxI
TwvxDudvdwdtwvuI
LLL
zyx
0
,,,
,,,,,, τ
τ
( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫
∂
∂
+×
x
L
y
L
z
L
VI
t x
L
z
L
I
zxzy x y zx z
Twvukdudwd
x
wyuI
TwyuD
LL
zx
LL
zy
,,,
,,,
,,, ,
0
τ
τ
( ) ( ) ( )×∫ ∫ ∫
∂
∂
+×
t x
L
y
Lyxzyx x y z
zvuI
LL
yx
LLL
zyx
udvdwdtwvuVtwvuI
0
,,,
,,,,,,
τ
( ) ( ) ( ) +∫ ∫ ∫−×
x
L
y
L
z
L
II
zyx
I
x y z
udvdwdtwvuITwvuk
LLL
zyx
dudvdTzvuD ,,,,,,,,, 2
,τ
( )∫ ∫ ∫+
x
L
y
L
z
L
I
zyx x y z
udvdwdwvuf
LLL
zyx
,, (4a)
( ) ( ) ( ) ×∫ ∫ ∫
∂
∂
=∫ ∫ ∫
t y
L
z
L
V
x
L
y
L
z
Lzyx y zx y z
dvdwd
x
wvxV
TwvxDudvdwdtwvuV
LLL
zyx
0
,,,
,,,,,, τ
τ
( ) ( ) ( )×∫ ∫ ∫+∫ ∫ ∫
∂
∂
+×
t x
L
y
L
V
t x
L
z
L
V
zxzy x yx z
TzvuDdudwd
x
wyuV
TwyuD
LL
zx
LL
zy
00
,,,
,,,
,,, τ
τ
( ) ( ) ( )×∫ ∫ ∫−
∂
∂
×
x
L
y
L
z
L
VI
zyxyx x y z
twvuITwvuk
LLL
zyx
LL
yx
dudvd
z
zvuV
,,,,,,
,,,
,τ
τ
( ) ( ) ( ) +∫ ∫ ∫−×
x
L
y
L
z
L
VV
zyx x y z
udvdwdtwvuVTwvuk
LLL
zyx
udvdwdtwvuV ,,,,,,,,, 2
,
( )∫ ∫ ∫+
x
L
y
L
z
L
V
zyx x y z
udvdwdwvuf
LLL
zyx
,, .
We determine solutions of the Eqs.(4a) as the following series
( ) ( ) ( ) ( ) ( )∑=
=
N
n
nnnnn tezcycxcatzyx
1
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
L
InnnI
zyx
N
n
nInnn
nI
y z
vdwdTwvxDzcyca
LLL
zy
tezsysxs
n
azyx
1 01
33
,,,
π
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−×
=
N
n
t x
L
z
L
InnnInnI
zyx
nnI
x z
dudwdTwyuDzcxceysa
LLL
zx
xsde
1 0
,,, ττ
π
ττ
( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫−∑ ∫ ∫ ∫−
=
x
L
y
L
z
L
II
N
n
t x
L
y
L
InnnInnI
zyx x y zx y
TvvukdudvdTzvuDycxcezsa
LLL
yx
,,,,,, ,
1 0
ττ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑−



∑×
==
x
L
y
L
z
L
N
n
nnnnI
zyxzyx
N
n
nInnnnI
x y z
wcvcuca
LLL
zyx
LLL
zyx
udvdwdtewcvcuca
1
2
1
( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑×
=
x
L
y
L
z
L
I
zyx
N
n
VInVnnnnVnI
x y z
udvdwdwvuf
LLL
zyx
udvdwdTvvuktewcvcucate ,,,,,
1
,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑
==
N
n
t y
L
z
L
VnnnV
zyx
N
n
nVnnn
nV
y z
vdwdTwvxDzcyca
LLL
zy
tezsysxs
n
azyx
1 01
33
,,,
π
π

8
( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−×
=
N
n
t x
L
z
L
VnnnVnnV
zyx
nnV
x z
dudwdTwyuDzcxceysa
LLL
zx
xsde
1 0
,,, ττ
π
ττ
( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫−∑ ∫ ∫ ∫−
=
x
L
y
L
z
L
VV
N
n
t x
L
y
L
VnnnVnnV
zyx x y zx y
TvvukdudvdTzvuDycxcezsa
LLL
yx
,,,,,, ,
1 0
ττ
π
( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∑−



∑×
==
x
L
y
L
z
L
N
n
nnnnI
zyxzyx
N
n
nInnnnV
x y z
wcvcuca
LLL
zyx
LLL
zyx
udvdwdtewcvcuca
1
2
1
( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑×
=
x
L
y
L
z
L
V
zyx
N
n
VInVnnnnVnI
x y z
udvdwdwvuf
LLL
zyx
udvdwdTvvuktewcvcucate ,,,,,
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
n
nI
nIzyx
x y
yc
n
L
ysyLxc
n
a
L
te
n
aLLL
1 0 0 0
2
1
65
222
12
2
221
2
1
πππ
( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫






−+×
=
N
n
t L
n
nI
y
L
nIn
z
nI
xz
xsx
n
a
L
dexdydzdzc
n
L
zszTzyxD
1 0 0
2
0
2
2
1
1
2
,,,
π
ττ
π
( )[ ] ( ) ( ) ( )[ ] ( )[ ]×∫ ∫ −






−++



−++
y z
L L
nn
z
nzIn
x
x yczdzc
n
L
zszLTzyxDxc
n
L
L
0 0
2112
2
2,,,12
ππ
( ) ( ) ( ) ( )[ ] ( ) −∫






−++×
zL
nIn
z
nzInI dexdydzdzc
n
L
zszLTzyxDdexdyd
0
12
2
2,,, ττ
π
ττ
( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫






−++






−++−
=
N
n
t L L
n
y
nyn
x
nx
nI
z
x y
yc
n
L
ysyLxc
n
L
xsxL
n
a
L 1 0 0 0
2
12
2
212
2
2
2
1
πππ
( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫



+−+−∫ −×
=
N
n
L
n
x
xnInI
L
nIIn
xz
xc
n
L
LteadexdydzdTzyxDzc
1 0
2
0
12
2
2,,,21
π
ττ
( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫



+−+






−+++
y z
L L
n
z
zIIn
y
nyn zc
n
L
LTzyxkyc
n
L
ysyLxsx
0 0
, 12
2
,,,12
2
22
ππ
( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ +






−++−+
=
N
n
L L
yn
x
nxnVnInVnIn
x y
Lxc
n
L
xsxLteteaaxdydzdzsz
1 0 0
12
2
22
π
( ) ( )[ ] ( ) ( ) ( )[ ]∫ ×






−++



−++
zL
n
z
nzVIn
y
n zdzc
n
L
zszLTzyxkyc
n
L
ysy
0
, 12
2
2,,,12
2
2
ππ
( ) ( )[ ] ( ) ( )[ ] ( )×∑ ∫ ∫ ∫






−+






−++×
=
N
n
L L L
In
y
nn
x
n
x y z
Tzyxfyc
n
L
ysyxc
n
L
xsxxdyd
1 0 0 0
,,,11
ππ
( ) ( )[ ] xdydzdzc
n
L
zszL n
z
nz






−++× 12
2
2
π
( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫






−++−−=∑−
==
N
n
t L L
n
y
nyn
nV
x
N
n
nV
nVzyx
x y
yc
n
L
ysyLxc
n
a
L
te
n
aLLL
1 0 0 0
2
1
65
222
12
2
221
2
1
πππ
( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫






−+×
=
N
n
t L
n
nV
y
L
nVn
z
nV
xz
xsx
n
a
L
dexdydzdzc
n
L
zszTzyxD
1 0 0
2
0
2
2
1
1
2
,,,
π
ττ
π

9
( )[ ] ( ) ( ) ( )[ ] ( )[ ]×∫ ∫ −






−++



−++
y z
L L
nn
z
nzVn
x
x yczdzc
n
L
zszLTzyxDxc
n
L
L
0 0
2112
2
2,,,12
ππ
( ) ( ) ( ) ( )[ ] ( ) −∫






−++×
zL
nVn
z
nzVnV dexdydzdzc
n
L
zszLTzyxDdexdyd
0
12
2
2,,, ττ
π
ττ
( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫






−++






−++−
=
N
n
t L L
n
y
nyn
x
nx
nV
z
x y
yc
n
L
ysyLxc
n
L
xsxL
n
a
L 1 0 0 0
2
12
2
212
2
2
2
1
πππ
( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫



+−+−∫ −×
=
N
n
L
n
x
xnVnV
L
nVVn
xz
xc
n
L
LteadexdydzdTzyxDzc
1 0
2
0
12
2
2,,,21
π
ττ
( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫



+−+






−+++
y z
L L
n
z
zVVn
y
nyn zc
n
L
LTzyxkyc
n
L
ysyLxsx
0 0
, 12
2
,,,12
2
22
ππ
( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ +






−++−+
=
N
n
L L
yn
x
nxnVnInVnIn
x y
Lxc
n
L
xsxLteteaaxdydzdzsz
1 0 0
12
2
22
π
( ) ( )[ ] ( ) ( ) ( )[ ]∫ ×






−++



−++
zL
n
z
nzVIn
y
n zdzc
n
L
zszLTzyxkyc
n
L
ysy
0
, 12
2
2,,,12
2
2
ππ
( ) ( )[ ] ( ) ( )[ ] ( )×∑ ∫ ∫ ∫






−+






−++×
=
N
n
L L L
Vn
y
nn
x
n
x y z
Tzyxfyc
n
L
ysyxc
n
L
xsxxdyd
1 0 0 0
,,,11
ππ
( ) ( )[ ] xdydzdzc
n
L
zszL n
z
nz






−++× 12
2
2
π
.
In the final form relations for required parameters could be written as
( )





 −
+−
+
±
+
−=
A
yb
yb
Ab
b
Ab
a nInV
nI
2
3
4
2
3
4
3
4
44
λγ
,
nInI
nInInInInI
nV
a
aa
a
χ
λδγ ++
−=
2
,
where ( ) ( ) ( ) ( )[ ] ( ){∫ ∫ ∫ ++






−++=
x y zL L L
ynn
x
nxnn Lysyxc
n
L
xsxLTzyxkte
0 0 0
, 212
2
2,,,2
π
γ ρρρρ
( )[ ] ( ) ( )[ ] xdydzdzc
n
L
zszLyc
n
L
n
z
nzn
y






−++



−+ 12
2
212
2 ππ
, ( )×∫=
t
n
x
n e
nL 0
2
2
1
τ
π
δ ρρ
( ) ( )[ ] ( ) ( )[ ] ( ) [∫ ∫ ∫ −






−+






−+×
x y zL L L
n
z
nn
y
n ydzdTzyxDzc
n
L
zszyc
n
L
ysy
0 0 0
1,,,1
2
1
2
ρ
ππ
( )] ( ) ( ) ( )[ ] ( )[ ] {∫ ∫ ∫ ∫ +−






−+++−
t L L L
znn
x
nxn
y
n
x y z
Lycxc
n
L
xsxLe
nL
dxdxc
0 0 0 0
2
21122
2
1
2
π
τ
π
τ ρ
( ) ( )[ ] ( ) ( ) ( ){∫ ∫ ++



−++
t L
nn
z
n
z
n
x
xsxe
nL
dxdydzdTzyxDzc
n
L
zsz
0 0
2
2
2
1
,,,12
2
2 τ
π
τ
π
ρρ
( )[ ] ( ) ( )[ ] ( )[ ] ( ) ×∫ ∫ −






−++



−++
y z
L L
nn
y
nyn
x
x zdTzyxDzcyc
n
L
ysyLxc
n
L
L
0 0
,,,211
2
12 ρ
ππ
( )te
n
LLL
dxdyd n
zyx
ρ
π
τ 65
222
−× , ( ) ( )[ ] ( )[ ]∫ ∫



+−+






−+=
x yL L
n
y
yn
x
nnIV yc
n
L
Lxc
n
L
xsx
0 0
12
2
1
ππ
χ

10
( )} ( ) ( ) ( )[ ] ( ) ( )tetexdydzdzc
n
L
zszLTzyxkysy nVnI
L
n
z
nzVIn
z
∫






−+++
0
, 12
2
2,,,2
π
,
( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]∫ ∫ ∫ ×






−+






−+






−+=
x y zL L L
n
z
nn
y
nn
x
nn zc
n
L
zszyc
n
L
ysyxc
n
L
xsx
0 0 0
111
πππ
λ ρ
( ) xdydzdTzyxf ,,,ρ× , 22
4 nInInInVb χγγγ −= , nInInVnInInInInVb γχδχδδγγ −−= 2
3 2 ,
2
2
3 48 bbyA −+= , ( ) 22
2 2 nInInVnInInVnInVnInInVb χλλδχδγγλδγ −+−+= , ×= nIb λ21
nInInVnInV λχδδγ −× ,
4
33 323 32
3b
b
qpqqpqy −++−−+= , 2
4
2
342
9
3
b
bbb
p
−
= ,
( ) 3
4
2
4132
3
3 542792 bbbbbbq +−= .
We determine solutions of the Eqs.(4a) as the following series
( ) ( ) ( ) ( ) ( )∑=Φ
=
Φ
N
n
nnnnn tezcycxcatzyx
1
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
L
I
zyx y zx y z
dvdwd
x
wvx
TwvxDudvdwdtwvu
LLL
zyx
0
,,,
,,,,,, τ
∂
τ∂
( ) ( ) ( )∫ ∫ ∫ ×+∫ ∫ ∫
Φ
+× ΦΦ
t x
L
y
L
I
yx
t x
L
z
L
I
I
zxzy x yx z
TzvuD
LL
yx
dudwd
y
wyu
TwyuD
LL
zx
LL
zy
00
,,,
,,,
,,, τ
∂
τ∂
( ) ( ) ( ) −∫ ∫ ∫+
Φ
×
x
L
y
L
z
L
II
zyx
I
x y z
udvdwdwvuITwvuk
LLL
zyx
dudvd
z
zvu
ττ
∂
τ∂
,,,,,,
,,, 2
, (6a)
( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− Φ
x
L
y
L
z
L
I
zyx
x
L
y
L
z
L
I
zyx x y zx y z
udvdwdwvuf
LLL
zyx
udvdwdwvuITwvuk
LLL
zyx
,,,,,,,, τ
( ) ( ) ( ) ×∫ ∫ ∫
Φ
=∫ ∫ ∫Φ Φ
t y
L
z
L
V
V
x
L
y
L
z
L
V
zyx y zx y z
dvdwd
x
wvx
TwvxDudvdwdtwvu
LLL
zyx
0
,,,
,,,,,, τ
∂
τ∂
( ) ( ) ( )∫ ∫ ∫ ×+∫ ∫ ∫
Φ
+× ΦΦ
t x
L
y
L
V
yx
t x
L
z
L
V
V
zxzy x yx z
TzvuD
LL
yx
dudwd
y
wyu
TwyuD
LL
zx
LL
zy
00
,,,
,,,
,,, τ
∂
τ∂
( ) ( ) ( ) −∫ ∫ ∫+
Φ
×
x
L
y
L
z
L
VV
zyx
V
x y z
udvdwdwvuVTwvuk
LLL
zyx
dudvd
z
zvu
ττ
∂
τ∂
,,,,,,
,,, 2
,
( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− Φ
x
L
y
L
z
L
V
zyx
x
L
y
L
z
L
V
zyx x y zx y z
udvdwdwvuf
LLL
zyx
udvdwdwvuVTwvuk
LLL
zyx
,,,,,,,, τ .
Substitution of the previously considered series in the Eqs.(6a) leads to the following form
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−=∑−
=
Φ
=
Φ
N
n
t y
L
z
L
nnnInIn
zyx
N
n
nInnn
In
y z
wcvctexsan
LLL
zy
tezsysxs
n
a
zyx
1 01
33
π
π
( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×
=
ΦΦΦ
N
n
t x
L
z
L
InnIn
zyx
I
x z
dudwdTwvuDwcuca
LLL
zx
dvdwdTwvxD
1 0
,,,,,, τ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) +∑ ∫ ∫ ∫−×
=
ΦΦΦΦ
N
n
t x
L
y
L
InnInnIn
zyx
Inn
x y
dudvdTzvuDvcuctezsan
LLL
yx
teysn
1 0
,,, τ
π

11
( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+ Φ
x
L
y
L
z
L
I
x
L
y
L
z
L
II
zyx x y zx y z
udvdwdwvufudvdwdwvuITwvuk
LLL
zyx
,,,,,,,, 2
, τ
( ) ( )∫ ∫ ∫−×
x
L
y
L
z
L
I
zyxzyx x y z
udvdwdwvuITwvuk
LLL
zyx
LLL
zyx
τ,,,,,,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−=∑−
=
Φ
=
Φ
N
n
t y
L
z
L
nnnVnVn
zyx
N
n
nVnnn
Vn
y z
wcvctexsan
LLL
zy
tezsysxs
n
a
zyx
1 01
33
π
π
( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×
=
ΦΦ
N
n
t x
L
z
L
Vnn
zyx
V
x z
dudwdTwvuDwcucn
LLL
zx
dvdwdTwvxD
1 0
,,,,,, τ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×
=
ΦΦΦΦ
N
n
t x
L
y
L
VnnVnn
zyx
VnnVn
x y
dudvdTzvuDvcuctezsn
LLL
yx
teysa
1 0
,,, τ
π
( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+× ΦΦ
x
L
y
L
z
L
V
x
L
y
L
z
L
VV
zyx
Vn
x y zx y z
udvdwdwvufudvdwdwvuVTwvuk
LLL
zyx
a ,,,,,,,, 2
, τ
( ) ( )∫ ∫ ∫−×
x
L
y
L
z
L
V
zyxzyx x y z
udvdwdwvuVTwvuk
LLL
zyx
LLL
zyx
τ,,,,,, .
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
( ) ( )[ ] ( ) ( )[ ] ×∑∫ ∫ ∫






−++−−=∑−
==
Φ
Φ
N
n
t L L
n
y
nyn
x
N
n
In
Inzyx
x y
yc
n
L
ysyLxc
L
te
n
aLLL
10 0 01
65
222
12
2
221
2
1
πππ
( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫






−+×
=
ΦΦ
Φ
N
n
t L
n
L
Inn
z
nI
In
xz
xsxdexdydzdzc
n
L
zszTzyxD
n
a
10 00
2
2
2
1
1
2
,,,
π
ττ
π
( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫






−+−



−++ Φ
y z
L L
n
z
nInn
x
x xdydzdzc
n
L
zszTzyxDycxc
n
L
L
0 0
1
2
,,,2112
2 ππ
( ) ( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫



+−+






−+−×
=
ΦΦ
Φ
N
n
t L L
n
y
nn
x
n
In
xy
In
In
x y
yc
n
L
ysyxc
n
L
xsx
n
a
L
d
Ln
e
a
1 0 0 0
22
12
2
21
22
1
πππ
τ
τ
} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫



+−+∫ −+
=
Φ
Φ
ΦΦ
N
n
t L
n
x
In
In
L
InIny
xz
xc
n
L
e
n
a
dexdydzdTzyxDycL
1 0 0
33
0
1
2
1
,,,21
π
τ
π
ττ
( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫



+−∫






−++
zy L
n
z
II
L
n
y
nn zc
n
L
TzyxktzyxIyc
n
L
ysyxsx
0
,
2
0
1
2
,,,,,,1
2 ππ
( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫



+−






−+−+
=
Φ
Φ
N
n
t L L
n
y
n
x
nIn
In
n
x y
yc
n
L
xc
n
L
xsxe
n
a
xdydzdzsz
1 0 0 0
33
1
2
1
2
1
ππ
τ
π
( )} ( ) ( )[ ] ( ) ( ) ×∑+∫






−++
=
Φ
N
n
In
L
In
z
nn
n
a
xdydzdtzyxITzyxkzc
n
L
zszysy
z
1
33
0
1
,,,,,,1
2 ππ
( ) ( ) ( )[ ] ( ) ( )[ ] ( )[ ]∫ ∫ ∫ ∫



+−






−+






−+× Φ
t L L L
n
z
n
y
nn
x
nIn
x y z
zc
n
L
yc
n
L
ysyxc
n
L
xsxe
0 0 0 0
1
2
1
2
1
2 πππ
τ
( )} ( ) xdydzdzyxfzsz In ,,Φ+

12
( ) ( )[ ] ( ) ( )[ ] ×∑∫ ∫ ∫






−++−−=∑−
==
Φ
Φ
N
n
t L L
n
y
nyn
x
N
n
Vn
Vnzyx
x y
yc
n
L
ysyLxc
L
te
n
aLLL
10 0 01
65
222
12
2
221
2
1
πππ
( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫






−+×
=
ΦΦ
Φ
N
n
t L
n
L
Vnn
z
nV
Vn
xz
xsxdexdydzdzc
n
L
zszTzyxD
n
a
10 00
2
2
2
1
1
2
,,,
π
ττ
π
( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫






−+−



−++ Φ
y z
L L
n
z
nVnn
x
x xdydzdzc
n
L
zszTzyxDycxc
n
L
L
0 0
1
2
,,,2112
2 ππ
( ) ( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫



+−+






−+−×
=
ΦΦ
Φ
N
n
t L L
n
y
nn
x
n
Vn
xy
Vn
Vn
x y
yc
n
L
ysyxc
n
L
xsx
n
a
L
d
Ln
e
a
1 0 0 0
22
12
2
21
22
1
πππ
τ
τ
} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫



+−+∫ −+
=
Φ
Φ
ΦΦ
N
n
t L
n
x
Vn
Vn
L
VnVny
xz
xc
n
L
e
n
a
dexdydzdTzyxDycL
1 0 0
33
0
1
2
1
,,,21
π
τ
π
ττ
( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫



+−∫






−++
zy L
n
z
VV
L
n
y
nn zc
n
L
TzyxktzyxVyc
n
L
ysyxsx
0
,
2
0
1
2
,,,,,,1
2 ππ
( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫



+−






−+−+
=
Φ
Φ
N
n
t L L
n
y
n
x
nVn
Vn
n
x y
yc
n
L
xc
n
L
xsxe
n
a
xdydzdzsz
1 0 0 0
33
1
2
1
2
1
ππ
τ
π
( )} ( ) ( )[ ] ( ) ( ) ×∑+∫






−++
=
Φ
N
n
Vn
L
Vn
z
nn
n
a
xdydzdtzyxVTzyxkzc
n
L
zszysy
z
1
33
0
1
,,,,,,1
2 ππ
( ) ( ) ( )[ ] ( ) ( )[ ] ( )[ ]∫ ∫ ∫ ∫



+−






−+






−+× Φ
t L L L
n
z
n
y
nn
x
nVn
x y z
zc
n
L
yc
n
L
ysyxc
n
L
xsxe
0 0 0 0
1
2
1
2
1
2 πππ
τ
( )} ( ) xdydzdzyxfzsz Vn ,,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)
2
3
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(Lx
2
+Ly
2
+
Lz
2
)/D0. Curves 2 and 4 corresponds to annealing time Θ=0.0057(Lx
2
+Ly
2
+Lz
2
)/D0. Curves 1 and 2
are 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 0.5
a/L, ξ, ε, γ
0.0
0.1
0.2
0.3
0.4
0.5
ΘD0L
-2
3
2
4
1
Fig.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 0.5
a/L, ξ, ε, γ
0.00
0.04
0.08
0.12
ΘD0L
-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.

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