It has been recently shown, that manufacturing p-n-junctions, field-effect and bipolar transistors, thyristors in a multilayer structure by diffusion or ion implantation under condition of optimization of dopant and/or radiation defects leads to increasing of sharpness of p-n-junctions (both single p-n-junctions and p-njunctions, which include into their system). In this situation one can also obtain increasing of homogeneity of dopant in doped area. In this paper we consider manufacturing a field-effect heterotransistor without pn-junction. Optimization of technological process with using inhomogeneity of heterostructure give us possibility to manufacture the transistors as more compact.

1. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
DOI : 10.14810/ijrap.2014.3301 1
OPTIMIZATION OF MANUFACTURE OF FIELD-
EFFECT HETEROTRANSISTORS WITHOUT P-N-
JUNCTIONS TO DECREASE THEIR DIMENSIONS
E.L. Pankratov1
, E.A. Bulaeva2
1
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
2
Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky
street, Nizhny Novgorod, 603950, Russia
ABSTRACT
It has been recently shown, that manufacturing p-n-junctions, field-effect and bipolar transistors, thyristors
in a multilayer structure by diffusion or ion implantation under condition of optimization of dopant and/or
radiation defects leads to increasing of sharpness of p-n-junctions (both single p-n-junctions and p-n-
junctions, which include into their system). In this situation one can also obtain increasing of homogeneity
of dopant in doped area. In this paper we consider manufacturing a field-effect heterotransistor without p-
n-junction. Optimization of technological process with using inhomogeneity of heterostructure give us
possibility to manufacture the transistors as more compact.
KEYWORDS
Field-effect heterotransistors; Decreasing of Dimensions of Transistors
1.INTRODUCTION
The development of electronics makes it necessary to reduce the size of integrated circuit
elements and their discrete analogs [1-7]. To reduce the required sizes formed by diffusion and
implantation of p-n-junctions and their systems (such transistors and thyristors) it could be used
several approaches. First of all it should be considered laser and microwave types of annealing [8-
14]. Using this types of annealing leads to generation inhomogenous distribution of temperature
gives us possibility to increase sharpness of p-n-junctions with simultaneous increasing of
homogeneity of distribution of dopant concentration in doped area [8-14]. In this situation one
can obtain more shallow p-n-junctions and at the same time to decrease dimensions of transistors,
which include into itself the p-n-junctions. The second way to decrease dimensions of elements of
integrated circuits is using of native inhomogeneity of heterostructure and optimization of
annealing [12-18]. In this case one can obtain increasing of sharpness of p-n-junctions and at the
same time increasing of homogeneity of distribution of dopant concentration in doped area [12-
18]. Distribution of concentration of dopant could be also changed under influence of radiation
processing [19]. In this situation radiation processing could be also used to increase sharpness of
single p-n-junction and p-n-junctions, which include into their system [20,21].
In this paper we consider a heterostructure, which consist of a substrate and a multi-section
epitaxial layer (see Figs. 1 and 2). Farther we consider manufacturing a field-effect transistor

2. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
2
without p-n-junction in this heterostructure. Appropriate contacts are presented in the Figs. 1 and
2. A dopant has been infused or implanted in areas of source and drain before manufacturing the
contacts to produce required types of conductivity. Farther we consider annealing of dopant
and/or radiation defects to infuse the dopant on required depth (in the first
case) or to decrease quantity of radiation defects (in the second case). Annealing of radiation
defects leads to spreading of distribution of concentration of dopant. To restrict the dopant
diffusion into substrate and to manufacture more thin structure it is practicably to choose
properties of heterostructure so, that dopant diffusion coefficient in the substrate should be as
smaller as it is possible, than dopant diffusion coefficient in epitaxial layer [12-18]. In this case it
is practicably to optimize annealing time [12-18]. If dopant did not achieve interface between
layers of heterostructure during annealing of radiation defects it is practicably to choose addition
annealing of dopant. Main aims of the present paper are (i) modeling of redistribution of dopant
and radiation defects; (ii) optimization of annealing time.
Fig. 1. Heterostructure with a substrate and multi-section epitaxial layer. View from top
Fig. 2. Heterostructure with a substrate and multi-section epitaxial layer. View from one side
2. Method of solution
To solve our aims we determine spatio-temporal distribution of concentration of dopant. We
determine the distribution by solving the second Fick’s law [1-4]
Substrate
Gate
Source Drain
Epitaxial layer

3. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
3
( ) ( )






=
x
t
x
C
D
x
t
t
z
y
x
C
C
∂
∂
∂
∂
∂
∂ ,
,
,
,
(1)
with boundary and initial conditions
( ) 0
,
0
=
∂
∂
=
x
x
t
x
C
,
( ) 0
,
=
∂
∂
=L
x
x
t
x
C
, C (x,0)=fС (x). (2)
Here C(x,t) is the spatio-temporal distribution of concentration of dopant; T is the temperature of
annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient depends
on properties of materials of heterostructure, speed of heating and cooling of heterostructure (with
account Arrhenius law). Dependences of dopant diffusion on parameters could be approximated
by the following function [22-25]
( ) ( )
( )
( ) ( )
( ) 







+
+






+
= 2
*
2
2
*
1
,
,
1
,
,
1
,
V
t
x
V
V
t
x
V
T
x
P
t
x
C
T
x
D
D L
C ς
ς
ξ γ
γ
, (3)
where DL (x,T) is the spatial (due to native inhomogeneity of heterostructure) and temperature
(due to Arrhenius law) dependences of dopant diffusion coefficient; P (x, T) is the limit of
solubility of dopant; parameter γ depends on properties of materials and could be integer in the
following interval γ ∈[1,3] [25]; V (x,t) is the spatio-temporal distribution of concentration of
radiation vacancies; V*
is the equilibrium distribution of vacancies. Concentrational dependence
of dopant diffusion coefficient has been described in details in [25]. It should be noted, that using
of doping of materials by diffusion leads to absents of radiation damage ζ1= ζ2= 0. We determine
spatio-temporal distributions of concentrations of radiation defects by solving the following
system of equations [23,24]
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )







−
−






=
−
−






=
t
x
V
T
x
k
t
x
V
t
x
I
T
x
k
x
t
x
V
T
x
D
x
t
t
x
V
t
x
I
T
x
k
t
x
V
t
x
I
T
x
k
x
t
x
I
T
x
D
x
t
t
x
I
V
V
V
I
V
I
I
V
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
2
,
,
2
,
,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
(4)
with initial
ρ (x,0)=fρ (x) (5a)
and boundary conditions
( ) 0
,
0
=
∂
∂
=
x
x
t
x
I
,
( ) 0
,
=
∂
∂
=L
x
x
t
x
I
,
( ) 0
,
0
=
∂
∂
=
x
x
t
x
V
,
( ) 0
,
=
∂
∂
=L
x
x
t
x
V
, I(x,0)=fI (x), V(x,0)=fV (x). (5b)

4. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
4
Here ρ =I,V; I (x,t) is the spatio-temporal distribution of concentration of interstitials; Dρ(x,T) is
the diffusion coefficient of point radiation defects (vacancies and interstitials); terms V2
(x,t) and
I2
(x,t) corresponds to generation of divacancies and diinterstitials; kI,V(x,T), kI,I(x,T) and kV,V(x,T)
are parameters of recombination of point radiation defects and generation their simplest
complexes (divacancies and diinterstitials).
We determine spatio-temporal distributions of concentrations of divacancies ΦV (x,t) and
diinterstitials ΦI (x,t) by solving of the following system of equations [23,24]
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )







−
+





 Φ
=
Φ
−
+





 Φ
=
Φ
Φ
Φ
t
x
V
T
x
k
t
x
V
T
x
k
x
t
x
T
x
D
x
t
t
x
t
x
I
T
x
k
t
x
I
T
x
k
x
t
x
T
x
D
x
t
t
x
V
V
V
V
V
V
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
2
,
2
,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
(6)
with boundary and initial conditions
( ) 0
,
0
=
∂
∂
=
x
x
t
x
I
,
( ) 0
,
=
∂
∂
=L
x
x
t
x
I
,
( ) 0
,
0
=
∂
∂
=
x
x
t
x
V
,
( ) 0
,
=
∂
∂
=L
x
x
t
x
V
, I(x,0)=fI (x), V(x,0)=fV (x). (7)
Here DΦI(x,T) and DΦV(x,T) are diffusion coefficients of divacancies and diinterstitials; kI(x,T) and
kV (x,T) are parameters of decay of complexes of radiation defects.
To determine spatio-temporal distributions of concentrations of point radiation defects we used
recently introduced approach [16,18]. Framework the approach we transform approximations of
diffusion coefficients of point radiation defects to the following form: Dρ(x,T)=D0ρ[1+ερgρ(x,T)],
where D0ρ is the average value of the diffusion coefficients, 0≤ερ< 1, |gρ(x,T)|≤1, ρ =I,V. We also
transform parameters of recombination of point defects and generation their complexes in the
same form: kI,V(x, T)=k0I,V[1+εI,V gI,V(x,T)], kI,I(x,T)=k0I,I[1+εI,I gI,I(x,T)] and kV,V(x,T) = k0V,V [1+εV,V
gV,V(x,T)], where k0ρ1,ρ2 is the appropriate average values of the above parameters, 0≤εI,V<1, 0≤εI,I
< 1, 0≤εV,V< 1, | gI,V (x,T)|≤1, | gI,I(x,T)|≤1, |gV,V(x,T)|≤1. We introduce the following dimensionless
variables: 2
0
0 L
t
D
D V
I
=
ϑ , V
I
V
I D
D
k
L 0
0
,
0
2
=
ω , ( ) ( ) *
,
,
~
I
t
x
I
t
x
I = , χ = x/Lx, η = y/Ly, φ =
z/Lz, ( ) ( ) *
,
,
~
V
t
x
V
t
x
V = , V
I
, D
D
k
L 0
0
0
2
ρ
ρ
ρ
Ω = . The introduction transforms Eq.(4) and
conditions (5) in the following form
( ) ( )
[ ] ( ) ( )
[ ] ( )
( )
[ ] ( ) ( )
( ) ( )
[ ] ( ) ( )
[ ] ( )
( )
[ ] ( ) ( )











+
−
−
+
Ω
−






∂
∂
+
∂
∂
=
∂
∂
+
−
−
+
Ω
−






∂
∂
+
∂
∂
=
∂
∂
ϑ
χ
ϑ
χ
χ
ε
ω
ϑ
χ
χ
ε
χ
ϑ
χ
χ
ε
χ
ϑ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
ω
ϑ
χ
χ
ε
χ
ϑ
χ
χ
ε
χ
ϑ
ϑ
χ
,
~
,
~
,
1
,
~
,
1
,
~
,
1
,
~
,
~
,
~
,
1
,
~
,
1
,
~
,
1
,
~
,
,
2
,
,
0
0
0
,
,
2
,
,
0
0
0
V
I
T
g
V
T
g
V
T
g
D
D
D
V
V
I
T
g
I
T
g
I
T
g
D
D
D
I
V
I
V
I
V
V
V
V
V
V
V
V
I
V
V
I
V
I
I
I
I
I
I
I
I
V
I
I
(8)

5. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
5
( ) 0
,
~
0
=
∂
∂
=
χ
χ
ϑ
χ
ρ
,
( ) 0
,
~
1
=
∂
∂
=
χ
χ
ϑ
χ
ρ
, ( )
( )
*
,
,
~
ρ
ϑ
χ
ϑ
χ
ρ ρ
f
= (9)
We determine solutions of Eqs.(8) and conditions (9) framework recently introduce [16,18]
approach as the power series
( ) ( )
∑ ∑ ∑Ω
=
∞
=
∞
=
∞
=
0 0 0
,
~
,
~
i j k
ijk
k
j
i
ϑ
χ
ρ
ω
ε
ϑ
χ
ρ ρ
ρ . (10)
Substitution of the series (10) into Eq.(8) and conditions (9) gives us possibility to obtain
equations for the zero-order approximations of concentrations of point defects ( )
ϑ
χ,
~
000
I and
( )
ϑ
χ,
~
000
V , corrections for them ( )
ϑ
χ,
~
ijk
I and ( )
ϑ
χ,
~
ijk
V (i ≥1, j ≥1, k ≥1) and conditions for all
functions ( )
ϑ
χ,
~
ijk
I and ( )
ϑ
χ,
~
ijk
V (i ≥0, j ≥0, k ≥0)
( ) ( )
( ) ( )







∂
∂
=
∂
∂
∂
∂
=
∂
∂
2
000
2
0
0
000
2
000
2
0
0
000
,
~
,
~
,
~
,
~
χ
ϑ
χ
ϑ
ϑ
χ
χ
ϑ
χ
ϑ
ϑ
χ
V
D
D
V
I
D
D
I
I
V
V
I
;
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )













∂
∂
∂
∂
+
∂
∂
=
∂
∂






∂
∂
∂
∂
+
∂
∂
=
∂
∂
−
−
χ
ϑ
χ
χ
χ
χ
ϑ
χ
ϑ
ϑ
χ
χ
ϑ
χ
χ
χ
χ
ϑ
χ
ϑ
ϑ
χ
,
~
,
,
~
,
~
,
~
,
,
~
,
~
100
0
0
2
00
2
0
0
00
100
0
0
2
00
2
0
0
00
i
V
I
V
i
I
V
i
i
I
V
I
i
V
I
i
V
T
g
D
D
V
D
D
V
I
T
g
D
D
I
D
D
I
, i ≥1;
( ) ( ) ( )
[ ] ( ) ( )
( ) ( ) ( )
[ ] ( ) ( )







+
−
∂
∂
=
∂
∂
+
−
∂
∂
=
∂
∂
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
1
,
~
,
~
000
000
,
,
2
010
2
0
0
010
000
000
,
,
2
010
2
0
0
010
V
I
T
g
V
D
D
V
V
I
T
g
I
D
D
I
V
I
V
I
I
V
V
I
V
I
V
I
;
( ) ( )
( )
[ ] ( ) ( ) ( ) ( )
[ ]
( ) ( )
( )
[ ] ( ) ( ) ( ) ( )
[ ]











+
+
−
−
∂
∂
=
∂
∂
+
+
−
−
∂
∂
=
∂
∂
ϑ
χ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
ϑ
χ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
,
~
,
~
,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
~
,
~
,
1
,
~
,
~
010
000
000
010
,
,
2
020
2
0
0
020
010
000
000
010
,
,
2
020
2
0
0
020
V
I
V
I
T
g
V
D
D
V
V
I
V
I
T
g
I
D
D
I
V
I
V
I
V
I
V
I
V
I
V
I
;

6. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
6
( ) ( ) ( )
[ ] ( )
( ) ( ) ( )
[ ] ( )







+
−
∂
∂
=
∂
∂
+
−
∂
∂
=
∂
∂
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
,
~
,
1
,
~
,
~
,
~
,
1
,
~
,
~
2
000
,
,
2
001
2
0
0
001
2
000
,
,
2
001
2
0
0
001
V
T
g
V
D
D
V
I
T
g
I
D
D
I
I
I
I
I
I
V
I
I
I
I
V
I
;
( ) ( ) ( ) ( )
( )
[ ] ( ) ( ) ( ) ( )
[ ]
( ) ( ) ( ) ( )
( )
[ ] ( ) ( ) ( ) ( )
[ ]











+
+
−
−






∂
∂
∂
∂
+
∂
∂
=
∂
∂
+
+
−
−






∂
∂
∂
∂
+
∂
∂
=
∂
∂
ϑ
χ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
χ
χ
χ
ϑ
χ
ϑ
ϑ
χ
ϑ
χ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
χ
χ
χ
ϑ
χ
ϑ
ϑ
χ
,
~
,
~
,
~
,
~
,
1
,
~
,
,
~
,
~
,
~
,
~
,
~
,
~
,
1
,
~
,
,
~
,
~
100
000
000
100
,
,
010
0
0
2
110
2
0
0
110
100
000
000
100
,
,
010
0
0
2
110
2
0
0
110
I
V
I
V
T
g
V
T
g
D
D
V
D
D
V
V
I
V
I
T
g
I
T
g
D
D
I
D
D
I
V
V
V
V
V
I
V
I
V
I
I
I
I
I
V
I
V
I
;
( ) ( ) ( )
[ ] ( ) ( )
( ) ( ) ( )
[ ] ( ) ( )







+
−
∂
∂
=
∂
∂
+
−
∂
∂
=
∂
∂
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
1
,
~
,
~
000
001
,
,
2
002
2
0
0
002
000
001
,
,
2
002
2
0
0
002
V
V
T
g
V
D
D
V
I
I
T
g
I
D
D
I
I
I
I
I
I
V
I
I
I
I
V
I
;
( ) ( ) ( ) ( )
( )
[ ] ( ) ( )
( ) ( ) ( ) ( )
( )
[ ] ( ) ( )











+
−
−








∂
∂
∂
∂
+
∂
∂
=
∂
∂
+
−
−








∂
∂
∂
∂
+
∂
∂
=
∂
∂
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
χ
χ
χ
ϑ
χ
ϑ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
χ
χ
χ
ϑ
χ
ϑ
ϑ
χ
,
~
,
~
,
1
,
~
,
,
~
,
~
,
~
,
~
,
1
,
~
,
,
~
,
~
100
000
001
0
0
2
101
2
0
0
101
000
100
001
0
0
2
101
2
0
0
101
V
I
T
g
V
T
g
D
D
V
D
D
V
V
I
T
g
I
T
g
D
D
I
D
D
I
V
V
V
I
V
I
V
I
I
I
V
I
V
I
;
( ) ( )
( )
[ ] ( ) ( )
( )
[ ] ( ) ( )
( ) ( )
( )
[ ] ( ) ( )
( )
[ ] ( ) ( )











+
−
−
+
−
∂
∂
=
∂
∂
+
−
−
+
−
∂
∂
=
∂
∂
ϑ
χ
ϑ
χ
χ
ε
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
ϑ
χ
ϑ
χ
χ
ε
ϑ
χ
ϑ
χ
χ
ε
χ
ϑ
χ
ϑ
ϑ
χ
,
~
,
~
,
1
,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
1
,
~
,
~
,
1
,
~
,
~
001
000
,
,
010
000
,
,
2
011
2
0
0
011
000
001
,
,
010
000
,
,
2
011
2
0
0
011
V
I
T
g
V
V
T
g
V
D
D
V
V
I
T
g
I
I
T
g
I
D
D
I
V
I
V
I
V
V
V
V
I
V
V
I
V
I
I
I
I
I
V
I
;
( )
0
,
~
0
=
∂
∂
=
x
ijk
χ
ϑ
χ
ρ
,
( )
0
,
~
1
=
∂
∂
=
x
ijk
χ
ϑ
χ
ρ
, (i ≥0, j ≥0, k ≥0);
( ) ( ) *
000 0
,
~ ρ
χ
χ
ρ ρ
f
= , ( ) 0
0
,
~ =
χ
ρijk (i ≥1, j ≥1, k ≥1).

7. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
7
Solutions of the obtained equations could be obtained by standard approaches (see, for example,
[26,27]). The solutions with account appropriate boundary and initial conditions could be written
as
( ) ( ) ( )
∑
+
=
∞
=1
000
2
1
,
~
n
n
n e
c
F
L
L
ϑ
χ
ϑ
χ
ρ ρ
ρ ,
where ( ) ( )
I
V
nI D
D
n
e 0
0
2
2
exp ϑ
π
ϑ −
= , ( ) ( )
V
I
nV D
D
n
e 0
0
2
2
exp ϑ
π
ϑ −
= , cn(χ) = cos (π n χ),
( ) ( )
∫
=
1
0
*
cos
1
u
d
u
f
u
n
F n
n ρ
ρ π
ρ
;
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )







∑ ∫ ∫
∂
∂
−
−
=
∑ ∫ ∫
∂
∂
−
−
=
∞
=
−
∞
=
−
1 0
1
0
100
0
0
00
1 0
1
0
100
0
0
00
,
~
,
2
,
~
,
~
,
2
,
~
n
i
V
n
nV
nV
n
I
V
i
n
i
I
n
nI
nI
n
V
I
i
d
u
d
u
u
V
T
u
g
u
s
e
e
c
n
D
D
V
d
u
d
u
u
I
T
u
g
u
s
e
e
c
n
D
D
I
ϑ
ϑ
τ
τ
τ
ϑ
χ
π
ϑ
χ
τ
τ
τ
ϑ
χ
π
ϑ
χ
, i ≥1,
where sn(χ) = sin (π n χ);
( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( )
∑ ∫ ∫ +
−
−
=
∞
=1 0
1
0
000
000
,
,
010 ,
~
,
~
,
1
,
2
,
~
n
V
I
V
I
n
n
n d
u
d
u
V
u
I
T
u
g
u
c
e
e
c
ϑ
ρ
ρ τ
τ
τ
ε
τ
ϑ
φ
χ
ϑ
χ
ρ ;
( ) ( ) ( ) ( ) ( ) ( )
[ ]×
∑ ∫ ∫ +
−
−
=
∞
=1 0
1
0
,
,
0
0
020 ,
1
2
,
~
n
V
I
V
I
n
n
n
V
I
T
u
g
u
c
e
e
c
D
D ϑ
ρ
ρ ε
τ
ϑ
χ
ϑ
χ
ρ
( ) ( ) ( ) ( )
[ ] τ
ϑ
χ
ϑ
χ
ϑ
χ
ϑ
χ d
u
d
V
I
V
I ,
~
,
~
,
~
,
~
010
000
000
010 +
× ;
( ) ( ) ( ) ( ) ( ) ( )
[ ] ( )
∑ ∫ ∫ +
−
−
=
∞
=1 0
1
0
2
000
,
,
001 ,
~
,
1
2
,
~
n
n
n
n
n d
u
d
u
T
u
g
u
c
e
e
c
ϑ
ρ
ρ
ρ
ρ
ρ
ρ τ
τ
ρ
ε
τ
ϑ
χ
ϑ
χ
ρ ;
( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( )
∑ ∫ ∫ +
−
−
=
∞
=1 0
1
0
000
001
,
,
002 ,
~
,
~
,
1
2
,
~
n
n
n
n
n d
u
d
u
u
T
u
g
u
c
e
e
c
ϑ
ρ
ρ
ρ
ρ
ρ
ρ τ
τ
ρ
τ
ρ
ε
τ
ϑ
χ
ϑ
χ
ρ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ] ( )
[ ]
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ] ( )
[ ]













∫ ∫ +
+
−
×
×
∑
−
∑ ∫ ∫
∂
∂
−
−
=
∫ ∫ +
+
−
×
×
∑
−
∑ ∫ ∫
∂
∂
−
−
=
∞
=
∞
=
∞
=
∞
=
ϑ
ϑ
ϑ
ϑ
τ
ε
τ
τ
τ
τ
τ
ϑ
χ
τ
τ
τ
ϑ
χ
π
ϑ
χ
τ
ε
τ
τ
τ
τ
τ
ϑ
χ
τ
τ
τ
ϑ
χ
π
ϑ
χ
0
1
0
,
,
000
100
100
000
1
1 0
1
0
010
0
0
110
0
1
0
,
,
100
000
000
100
1
1 0
1
0
010
0
0
110
,
1
,
~
,
~
,
~
,
~
2
,
~
,
2
,
~
,
1
,
~
,
~
,
~
,
~
2
,
~
,
2
,
~
d
u
d
T
u
g
u
V
u
I
u
V
u
I
u
c
e
e
c
d
u
d
u
u
V
T
u
g
u
s
e
e
c
n
D
D
V
d
u
d
T
u
g
u
V
u
I
u
V
u
I
u
c
e
e
c
d
u
d
u
u
I
T
u
g
u
s
e
e
c
n
D
D
I
V
I
V
I
n
nV
nV
n
n
n
V
n
nV
nV
n
I
V
V
I
V
I
n
nI
nI
n
n
n
I
n
nI
nI
n
V
I
;

8. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
8
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
[ ] ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
[ ] ( ) ( )













∑ ∫ ∫ +
−
−
−
∑ ∫ ∫
∂
∂
−
−
=
∑ ∫ ∫ +
−
−
−
∑ ∫ ∫
∂
∂
−
−
=
∞
=
∞
=
∞
=
∞
=
1 0
1
0
100
000
,
,
1 0
1
0
001
0
0
101
1 0
1
0
000
100
,
,
1 0
1
0
001
0
0
101
,
~
,
~
,
1
2
,
~
,
2
,
~
,
~
,
~
,
1
2
,
~
,
2
,
~
n
V
I
V
I
n
nV
nV
n
n
V
n
nV
nV
n
I
V
n
V
I
V
I
n
nI
nI
n
n
I
n
nI
nI
n
V
I
d
u
d
u
V
u
I
T
u
g
u
c
e
e
c
d
u
d
u
u
V
T
u
g
u
s
e
e
c
n
D
D
V
d
u
d
u
V
u
I
T
u
g
u
c
e
e
c
d
u
d
u
u
I
T
u
g
u
s
e
e
c
n
D
D
I
ϑ
ϑ
ϑ
ϑ
τ
τ
τ
ε
τ
ϑ
χ
τ
τ
τ
ϑ
χ
π
ϑ
χ
τ
τ
τ
ε
τ
ϑ
χ
τ
τ
τ
ϑ
χ
π
ϑ
χ
;
( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( )
{
( )
[ ] ( ) ( )}
( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( )
{
( )
[ ] ( ) ( )}









+
+
∑ ∫ ∫ +
+
−
−
=
+
+
∑ ∫ ∫ +
+
−
−
=
∞
=
∞
=
.
,
~
,
~
,
1
,
~
,
~
,
1
2
,
~
,
~
,
~
,
1
,
~
,
~
,
1
2
,
~
000
001
,
,
1 0
1
0
010
000
,
,
011
000
001
,
,
1 0
1
0
010
000
,
,
011
τ
τ
τ
ε
τ
τ
ε
τ
ϑ
χ
ϑ
χ
τ
τ
τ
ε
τ
τ
ε
τ
ϑ
χ
ϑ
χ
ϑ
ϑ
d
u
d
u
V
u
I
T
u
g
u
V
u
V
T
u
g
u
c
e
e
c
V
d
u
d
u
V
u
I
T
u
g
u
I
u
I
T
u
g
u
c
e
e
c
I
V
I
V
I
n
V
V
V
V
n
nV
nV
n
V
I
V
I
n
I
I
I
I
n
nI
nI
n
Farther we determine spatio-temporal distribution of concentration of complexes of point
radiation defects. To calculate the distribution we transform approximation of diffusion
coefficient to the following form: DΦρ(x,T)=D0Φρ[1+εΦρgΦρ(x,T)], where D0Φρ are the average
values of diffusion coefficient. After this transformation Eqs.(6) transforms to the following form
( ) ( )
[ ] ( ) ( ) ( ) ( ) ( )
( ) ( )
[ ] ( ) ( ) ( ) ( ) ( )







−
+





 Φ
+
=
Φ
−
+





 Φ
+
=
Φ
Φ
Φ
Φ
Φ
Φ
Φ
t
x
V
T
x
k
t
x
V
T
x
k
x
t
x
T
x
g
x
D
t
t
x
t
x
I
T
x
k
t
x
I
T
x
k
x
t
x
T
x
g
x
D
t
t
x
V
V
V
V
V
V
V
V
I
I
I
I
I
I
I
I
,
,
,
,
,
,
1
,
,
,
,
,
,
,
1
,
2
,
0
2
,
0
∂
∂
ε
∂
∂
∂
∂
∂
∂
ε
∂
∂
∂
∂
Let us determine solutions of the above equations as the following power series
( ) ( )
∑ Φ
=
Φ
∞
=
Φ
0
,
,
i
i
i
t
x
t
x ρ
ρ
ρ ε . (11)
Substitution of the series (11) into Eqs.(6) and conditions for them gives us possibility to obtain
equations for initial-order approximations of concentrations of complexes, corrections for them
and conditions for all equations in the following form
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )







−
+
Φ
=
Φ
−
+
Φ
=
Φ
Φ
Φ
t
x
V
T
x
k
t
x
V
T
x
k
x
t
x
D
t
t
x
t
x
I
T
x
k
t
x
I
T
x
k
x
t
x
D
t
t
x
V
V
V
V
V
V
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
2
,
2
0
2
0
0
2
,
2
0
2
0
0
∂
∂
∂
∂
∂
∂
∂
∂
;

9. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
9
( ) ( )
( )
( )
( ) ( )
( )
( )












 Φ
+
Φ
=
Φ





 Φ
+
Φ
=
Φ
−
Φ
Φ
Φ
−
Φ
Φ
Φ
x
t
x
T
x
g
x
D
x
t
x
D
t
t
x
x
t
x
T
x
g
x
D
x
t
x
D
t
t
x
i
V
V
V
i
V
V
i
V
i
I
I
I
i
I
I
i
I
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
,
,
,
,
,
,
,
,
1
0
2
2
0
1
0
2
2
0
, i≥1;
( )
0
,
0
=
∂
Φ
∂
=
x
i
x
t
x
ρ
,
( )
0
,
=
∂
Φ
∂
=L
x
i
x
t
x
ρ
, i≥0; Φρ0(x,0)=fΦρ (x), Φρi(x,0)=0, i≥1.
Solutions of the above equations could be written as
( ) ( ) ( )+
∑
+
=
Φ
∞
=
Φ
Φ
1
0
2
1
,
n
n
n t
e
x
c
F
L
L
t
x ρ
ρ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
∑ ∫ ∫ −
−
+
∞
=
Φ
Φ
1 0 0
2
, ,
,
,
,
2
n
t L
I
I
I
n
n
n
n d
u
d
u
I
T
u
k
u
I
T
u
k
u
c
e
t
e
x
c
n
L
τ
τ
τ
τ
ρ
ρ
,
where ( ) ( )
2
0
2
2
exp L
t
D
n
t
en ρ
ρ
π Φ
Φ −
= , ( ) ( )
∫
= Φ
Φ
L
n
n u
d
u
f
u
c
F
0
ρ
ρ
, cn(x) = cos (π n x/L);
( ) ( ) ( ) ( ) ( ) ( )
( )
∑ ∫ ∫
Φ
−
−
=
Φ
∞
=
−
Φ
Φ
Φ
1 0 0
1
2
,
,
2
,
n
t L i
I
n
n
n
n
i d
u
d
u
u
T
u
g
u
s
e
t
e
x
c
n
L
t
x τ
∂
τ
∂
τ
π ρ
ρ
ρ
ρ
ρ , i≥1,
where sn(x) = sin (π n x/L)
Spatio-temporal distribution of concentration of dopant we determine framework the recently
introduced approach. Framework the approach we transform approximation of dopant diffusion
coefficient to the following form: DL(x,T)=D0L[1+εLgL(x,T)], where D0L is the average value of
dopant diffusion coefficient, 0≤εL< 1, |gL(x,T)|≤1. Farther we determine solution of the Eq.(1) as
the following power series
( ) ( )
∑ ∑
=
∞
=
∞
=
0 1
,
,
i j
ij
j
i
L t
x
C
t
x
C ξ
ε .
Substitution of the series into Eq.(1) and conditions (2) gives us possibility to obtain zero-order
approximation of dopant concentration C00(x,t), corrections for them Cij(x,t) (i ≥1, j ≥1) and
conditions for all functions. The equations and conditions could be written as
( ) ( )
2
00
2
0
00 ,
,
x
t
x
C
D
t
t
x
C
L
∂
∂
=
∂
∂
;
( ) ( ) ( ) ( )






∂
∂
∂
∂
+
∂
∂
=
∂
∂ −
x
t
x
C
T
x
g
x
D
x
t
x
C
D
t
t
x
C i
L
L
i
L
i ,
,
,
, 10
0
2
0
2
0
0
, i ≥1;
( ) ( ) ( )
( )
( )






∂
∂
∂
∂
+
∂
∂
=
∂
∂
x
t
x
C
T
x
P
t
x
C
x
D
x
t
x
C
D
t
t
x
C
L
L
,
,
,
,
, 00
00
0
2
01
2
0
01
γ
γ
;

10. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
10
( ) ( )
( )
( )
( )
( )
+






∂
∂
∂
∂
+
∂
∂
=
∂
∂ −
x
t
x
C
T
x
P
t
x
C
t
x
C
x
D
x
t
x
C
D
t
t
x
C
L
L
,
,
,
,
,
, 00
1
00
01
0
2
02
2
0
02
γ
γ
( )
( )
( )






∂
∂
∂
∂
+
x
t
x
C
T
x
P
t
x
C
x
D L
,
,
, 01
00
0 γ
γ
;
( ) ( ) ( ) ( )
( )
( ) +






∂
∂
∂
∂
+
∂
∂
=
∂
∂ −
x
t
x
C
T
x
P
t
x
C
t
x
C
x
D
x
t
x
C
D
t
t
x
C
L
L
,
,
,
,
,
, 00
1
00
10
0
2
11
2
0
11
γ
γ
( )
( )
( ) ( ) ( )






∂
∂
∂
∂
+






∂
∂
∂
∂
+
x
t
x
C
T
x
g
x
D
x
t
x
C
T
x
P
t
x
C
x
D L
L
L
,
,
,
,
, 01
0
10
00
0 γ
γ
;
( )
0
,
0
=
=
x
ij
x
t
x
C
∂
∂
,
( )
0
,
=
=L
x
ij
x
t
x
C
∂
∂
, i ≥0, j ≥0; C00(x,0)=fC (x), Cij(x,0)=0, i ≥1, j ≥1.
Solutions of the equations with account appropriate boundary and initial conditions could be
obtained by standard approaches (see, for example, [26,27]). The solutions are
( ) ( ) ( )
∑
+
=
∞
=1
00
2
1
,
n
nC
nC t
e
x
c
F
L
L
t
x
C ,
where ( ) ( )
2
0
2
2
exp L
t
D
n
t
e C
nC π
−
= , ( ) ( )
∫
=
L
C
n
nC u
d
u
f
u
c
F
0
;
( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ∫ ∫
∂
∂
−
−
=
∞
=
−
1 0 0
10
2
0
,
,
2
,
n
t L
i
L
n
nC
nC
n
nC
i d
u
d
u
u
C
T
u
g
u
s
e
t
e
x
c
F
n
L
t
x
C τ
τ
τ
π
, i ≥1;
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
∑ ∫ ∫
∂
∂
−
−
=
∞
=1 0 0
00
00
2
01
,
,
,
2
,
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
s
e
t
e
x
c
F
n
L
t
x
C τ
τ
τ
τ
π
γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) −
∑ ∫ ∫
∂
∂
−
−
=
∞
=
−
1 0 0
00
1
00
01
2
02
,
,
,
,
2
,
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
C
u
s
e
t
e
x
c
F
n
L
t
x
C τ
τ
τ
τ
τ
π
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( )
∑ ∫ ∫
∂
∂
−
−
∞
=1 0 0
01
00
2
,
,
,
2
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
s
e
t
e
x
c
F
n
L
τ
τ
τ
τ
π
γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∑ ∫ ∫
∂
∂
−
−
=
∞
=
∞
= 1
2
1 0 0
01
2
11
2
,
,
2
,
n
nC
n
t L
L
n
nC
nC
n
nC F
n
L
d
u
d
u
u
C
T
u
g
u
s
e
t
e
x
c
F
n
L
t
x
C
π
τ
τ
τ
π
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ×
∑ ∫ −
−
∫ ∫
∂
∂
−
×
∞
=1 0
2
0 0
10
00 2
,
,
,
n
t
nC
nC
n
nC
t L
n
nC
nC
n e
t
e
x
c
F
n
L
d
u
d
u
u
C
T
u
P
u
C
u
s
e
t
e
x
c τ
π
τ
τ
τ
τ γ
γ
( ) ( )
( )
( )
( )
∫
∂
∂
×
−
L
n d
u
d
u
u
C
T
u
P
u
C
u
C
u
s
0
00
1
00
10
,
,
,
, τ
τ
τ
τ γ
γ
.
Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects has
been done analytically by using the second-order approximations framework recently introduced
power series. The approximation is enough good approximation to make qualitative analysis and
to obtain some quantitative results. All obtained analytical results have been checked by
comparison with results, calculated by numerical simulation.

11. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
11
3. Discussion
In this section based on obtained in the previous section relations we analysed dynamics of
redistribution of dopant, which was infused in heterostructure from Figs. 1 and 2 by diffusion or
ion implantation, during annealing of the dopant and/or radiation defects. We take into account
radiation damage during consideration dopant redistribution after ion doping of heterostructure.
The Fig. 3 shows typical distributions of concentrations of infused dopant in heterostructure in
direction, which is perpendicular to interface between epitaxial layer and substrate. The
distributions have been calculated under condition, when value of dopant diffusion coefficient in
epitaxial layer is larger, than value of dopant diffusion coefficient in substrate. Analogous
distributions of dopant concentrations for ion doping are presented in Fig. 4. The Figs. 3 and 4
shows, that presents of interface between layers of heterostructure under above condition gives us
possibility to obtain more thin field-effect transistor. At the same time one can find increasing of
homogeneity of dopant distribution in doped area.
Fig.3. Distributions of concentration of infused dopant in heterostructure from Figs. 1 and 2 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
Fig.4. Distributions of concentration of implanted dopant in heterostructure from Figs. 1 and 2 in
direction, which is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3
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

12. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
12
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 corresponds to homogenous sample.
Curves 3 and 4 corresponds to heterostructure under condition, when value of dopant diffusion
coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate
Fig.5. Spatial distributions of concentration of infused dopant in heterostructure from Figs. 1 and
2. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of dopant
for different values of annealing time. Increasing of number of curve correspondsincreasing of
annealing time It should be noted, that in this situation optimization of annealing time required.
Reason of this optimization is following. If annealing time is small, dopant did not achieve
interface between layers of heterostructure. In this situation inhomogeneity of dopant
concentration increases. If annealing time is large, dopant will infused too deep. Optimal
annealing time we determine framework recently introduce approach [12-18,19,20]. Framework
the criterion we approximate real distribution of dopant concentration by stepwise function (see
Figs. 5 and 6). Farther we determine the following mean-squared error between real distribution
of dopant concentration and the stepwise approximation
( ) ( )
[ ]
∫ −
Θ
=
L
x
d
x
x
C
L
U
0
,
1
ψ , (12)
where ψ (x) is the approximation function, which presented in Figs. 5 and 6 as curve 1. We
determine optimal annealing time by minimization mean-squared error (12). Dependences of
optimal annealing time on parameters for diffusion a ion types of doping are presented on Figs. 7
and 8. Optimal annealing time, which corresponds to ion doping, is smaller, than the same time
for doping by diffusion. Reason of this difference is necessity to anneal radiation defects.
Optimization of annealing time for ion doping of materials should be done only in this case, when
dopant did not achieves interface between layers of heterostructure during annealing of radiation
defects.
C(x,
Θ
)
0 Lx
2
1
3
4

13. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
13
Fig.6. Spatial distributions of concentration of implanted dopant in heterostructure from Figs. 1
and 2. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of
dopant for different values of annealing time. Increasing of number of curve corresponds to
increasing of annealing time
Fig. 7. Dependences of dimensionless optimal annealing time for doping by diffusion, which have
been obtained by minimization of mean-squared error, on 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
parameter 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
x
C(x,
Θ
)
1
2
3
4
0 L
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
Θ
D
0
L
-2
3
2
4
1

14. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
14
Fig.8. Dependences of dimensionless optimal annealing time for doping by ion implantation,
which have been obtained by minimization of mean-squared error, on 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
parameter 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
4. CONCLUSIONS
In this paper we consider an approach to manufacture more thin field-effect transistors. The
approach based on doping of heterostructure by diffusion or ion implantation and optimization of
annealing of dopant 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 SSR-339.2014.2 and educational fellowship for scientific
research.
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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
Radiophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in
Radiophysics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of
Microstructures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny
Novgorod 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 96 published papers in
area of his researches.
Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of
village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school
“Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of

16. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014
16
Architecture 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 29 published papers in area of her researches.