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ORIGINAL ARTICLE
Optimization of annealing of dopant to increase sharpness
of p–n junctions in a heterostructure with drain of dopant
E. L. Pankratov
•
E. A. Bulaeva
Received: 25 March 2013 / Accepted: 25 April 2013
Ó The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract It has been recently shown, that manufacturing of
p–n junctions by dopant diffusion or ion implantation in
heterostructures and optimization of annealing time leads to
increasing of their sharpness and homogeneity of dopant
distribution in enriched area. In this paper, we consider
influence of defects of doped structure (mismatch disloca-
tions and similar), which became as drain of atoms of dopant,
on dopant distribution in diffusive-junction rectifier.
Keywords Diffusion-heterojunction rectifier Modeling
of dopant diffusion Accounting drain of dopant
Optimization of annealing of dopant
Introduction
In the present time, elaboration of new devices of solid
state electronic devices is intensively done. The second
way for intensive elaboration is refinement of characteris-
tics of traditional solid state electronic devices (Esmaeili-
Rad et al. 2007; Huang et al. 2003; Lai et al. 2007; Kitada
et al. 2009; Lei et al. 2009; Volocobinskaya et al. 2001;
Vasil’ev et al. 2002). One of the questions of the refine-
ment is increasing of sharpness of p–n junctions and
manufacturing the devices as more shallow (Andronov
et al. 1998; Sisiyany et al. 2002). One way to increase
sharpness of p–n junction is using laser annealing (Va-
ronina et al. 1999; Pankratov 2005). The second one is
using inhomogeneity of heterostructure (H) (Kelleher et al.
2009; Pankratov 2010, 2012). It could be used as another
standard of approach. In this paper, we consider an
approach to increase sharpness of diffusive-junction recti-
fier using defects of doped structure.
In this paper, we consider a H with two layers (see
Fig. 1). One layer is a substrate (S). The second one is an
epitaxial layer (EL). A dopant has been infused in the EL.
One can obtain increasing of sharpness of the diffusive-
junction rectifier and at the same time increasing of
homogeneity of dopant distribution in the rectifier after
that, when dopant achieves the interface between layers of
H (see Fig. 2). Recently it has been shown, that after
annealing of dopant with optimal value of annealing time
H compromise between increasing of sharpness of p–
n junction and increasing of homogeneity of dopant dis-
tribution in the rectifier could be achieved (Pankratov 2005,
2010, 2012). Main aim of the present paper is taking into
account drains of atoms of dopant on defects of doped
structure (mismatch dislocations and similar).
Method of solution
Redistribution of dopant has been described by the second
Fick’s law (Shalimova 1985;Gotra
1991;Alexandrov2002)
oCx; tðÞ
ot
¼
o
ox
D
C
oCx; tðÞ
ox
k
R
x; TðÞCx; tðÞ
þ k
G
x; TðÞCx; tðÞ: ð1Þ
Here C(x,t) is the spatiotemporal distribution of dopant;
D
C
is the dopant diffusion coefficient; k
R
(x,T) is the
E. L. Pankratov (&)
Nizhny Novgorod State University, 23 Gagarin avenue,
Nizhny Novgorod 603950, Russia
e-mail: elp2004@mail.ru
E. A. Bulaeva
Nizhny Novgorod State University of Architecture and Civil
Engineering, 65 Il’insky street, Nizhny Novgorod 603950,
Russia
e-mail: hellen-bulaeva@yandex.ru
123
Appl Nanosci
DOI 10.1007/s13204-013-0228-7
parameter, which characterize speed of capturing of atoms
of dopant by drains; k
G
(x,T) is the parameter, which
characterize speed of returning of atoms of dopant from
drains. We transform the Eq. (1) to the following form by
combination of two last terms into one
oCx; tðÞ
ot
¼
o
ox
D
C
oCx; tðÞ
ox
Kx; TðÞCx; tðÞ; ð1aÞ
where K(x,T) = k
R
(x,T) - k
G
(x,T). The Eq. (1)is
complemented by the following boundary and initial
conditions
oCx; tðÞ
ox
x¼0
¼
oCx; tðÞ
ox
x¼L
¼ 0; Cx; 0ðÞ¼f
C
xðÞ: ð2Þ
The conditions have been written in the most common
form. However dopant usually did not achieves the
boundary x = L of H. In this situation, appropriate
boundary condition could be written as: C(L,t) = 0.
Value of diffusion coefficient D
C
depends on properties of
materials of H, velocities of heating and cooling (with account
Arrhenius law) of the materials and spatiotemporal distribu-
tion of dopant concentration. The last dependence could be
approximated by the following function (Gotra 1991)
D
C
¼ D
L
x; TðÞ1 þ n
C
c
x; tðÞ
P
c
x; TðÞ
: ð3Þ
Here D
L
(x,T) is spatial (due to inhomogeneity of H) and
temperature (due to Arrhenius law, where T is the
temperature) dependences of diffusion coefficient;
P(x,T) is the limit of solubility of dopant; parameter c
depends on properties of materials and could be integer in
the interval c [ [1,3] (Gotra 1991). Concentrational
dependence of diffusion coefficient is discussed in detail
in (Gotra 1991).
To determine spatiotemporal distribution of dopant
concentration let us use the method of averaging of func-
tion correction (Pankratov 2010; Alexandrov 2002). To use
the approach we transform the Eq. (1a) to the following
integral form
Cx; tðÞ¼Cx; tðÞþ
1
L
2
Z
t
0
D
L
x; TðÞ1 þ n
C
c
x; sðÞ
P
c
x; TðÞ
8
<
:
Cx; sðÞds
Z
t
0
Z
x
0
Kv; TðÞCv; sðÞx vðÞdvds
Z
t
0
Z
x
0
Cv; sðÞ1 þn
C
c
v; sðÞ
P
c
v; TðÞ
oD
L
v; TðÞ
ov
dvds
þ
Z
t
0
Z
L
0
Cx; sðÞ1 þn
C
c
x; sðÞ
P
c
x; TðÞ
oD
L
x; TðÞ
ox
dxds
þ
Z
t
0
Z
L
0
L xðÞKx; TðÞCx; sðÞdxds þ
Z
L
0
L xðÞCx; tðÞdx
Z
x
0
x vðÞCv; tðÞdv
Z
L
0
L xðÞfxðÞdx þ
Z
x
0
x vðÞfvðÞdv
9
=
;
:
ð1bÞ
To determine the first-order approximation C
1
(x,t)of
dopant concentration let us replace determining function
C(x,t) in the right side of Eq. (1b) on its average value a
1
.
After the replacement one can obtain the following relation
for the first-order approximation C
1
(x,t) of dopant
concentration in the following form
D(x),
P
(x)
C(x,0)
D
EL
P
EL
D
S
P
S
L
a
0
Substrate
Epitaxial layer
Fig. 1 Heterostructue with epitaxial layer and substrate
x
C
(
x
,
Θ
)
L
/4
L
/2
03
L
/4
L
1
2
3
4
Fig. 2 Dopant distributions for different values of annealing time.
Increasing of number of curves corresponds to increasing of value of
annealing time. Interface between layers coincides with midpoint of
the heterostructure, i.e., x = L/2
Appl Nanosci
123
C
1
x; tðÞ¼a
1
þ
1
L
2
a
1
Z
t
0
D
L
x; TðÞ1 þ
na
c
1
P
c
x; TðÞ
ds
8
<
:
a
1
Z
t
0
Z
x
0
x v
ðÞ
Kv; T
ðÞ
dvds a
1
Z
t
0
Z
x
0
1 þ
na
c
1
P
c
v; TðÞ
oD
L
v; TðÞ
ov
dvds
þ a
1
Z
t
0
Z
L
0
L xðÞKx; TðÞdxds þ
Z
t
0
Z
L
0
oD
L
x; TðÞ
ox
a
1
1 þ
na
c
1
P
c
x; TðÞ
dxds þ
a
1
2
L
2
x
2
Z
L
0
L xðÞfxðÞdx þ
Z
x
0
x vðÞfvðÞdv
9
=
;
The average value a
1
of the function C
1
(x,t) could be
determined by the following standard relation
a
i
¼
1
LH
Z
H
0
Z
L
0
C
i
x; tðÞdxdt; ð4Þ
where H is the observation time on diffusion process.
Integration of the function C
1
(x,t) with account relation (4)
leads to the following equation for the average value a
1
,
which depends on parameter c
a
1
Z
H
0
H tðÞ
Z
L
0
D
L
x; TðÞ1 þ
na
c
1
P
c
x; TðÞ
dxdt
a
1
2
Z
H
0
H tðÞ
Z
L
0
L
2
x
2
Kx; TðÞdxdt
a
1
Z
H
0
H tðÞ
Z
L
0
L xðÞ1 þ
na
c
1
P
c
x; TðÞ
oD
L
x; TðÞ
ox
dxdt
þ a
1
Z
H
0
Z
L
0
L xðÞKx; TðÞdx H tðÞds
þ a
1
Z
H
0
H tðÞ
Z
L
0
1 þ
na
c
1
P
c
x; TðÞ
oD
L
x; TðÞ
ox
dxds
þ
Z
L
0
L
2
x
2
fxðÞdx
H
2
þ L
3
H
a
1
3
HL
Z
L
0
L xðÞfxðÞdx ¼ 0:
The second-order approximation of dopant
concentration C
2
(x,t) could be determined by standard
iteration procedure of method of averaging of function
correction (Pankratov 2010; Sokolov 1955), i.e., by
replacement the determining function C(x,t) in the right
side of the Eq. (1b) on sum of average value a
2
of the
second-order approximation of dopant concentration
C
2
(x,t) and the first-order approximation of dopant
concentration C
1
(x, t). In this case, the second-order
approximation could be written as
C
2
x; tðÞ¼a
2
þ C
1
x; tðÞþ
1
L
2
Z
t
0
D
L
x; TðÞ
0
@
a
2
þ C
1
x; sðÞ½1 þ n
a
2
þ C
1
x; sðÞ½
c
P
c
x; TðÞ
ds
Z
t
0
Z
x
0
x vÞ
ð
Kv; T
ðÞ
a
2
þ C
1
v; s
ðÞ½
dvds
þ
Z
t
0
Z
L
0
Kx; TðÞL xðÞa
2
þ C
1
x; sðÞ½dxds
þ
Z
t
0
Z
L
0
C
1
x; sðÞ½þa
2
1 þ n
a
2
þ C
1
x; sðÞ½
c
P
c
x; TðÞ
oD
L
x; TðÞ
ox
dxds
Z
t
0
Z
x
0
oD
L
v; TðÞ
ov
1 þ n
a
2
þ C
1
v; sðÞ½
c
P
c
v; TðÞ
a
2
þ C
1
v; sðÞ½dvds
Z
L
0
L xðÞfxðÞdx þ
Z
L
0
L xðÞa
2
þ C
1
x; tðÞ½dx
Z
x
0
x vðÞa
2
þ C
1
v; tðÞ½dv þ
Z
x
0
x vðÞfvðÞdv
1
A
:
Average value a
2
of the second-order approximation of
dopant concentration could be determined by the standard
relation (4). After substitution of the function C
2
(x,t) in the
relation (4), we obtain the equation for the average value a
2
in the following form
Z
H
0
H tðÞ
Z
L
0
D
L
x; TðÞa
2
þ C
1
x; tðÞ½1 þn
a
2
þ C
1
x; tðÞ½
c
P
c
x; TðÞ
dxdt
1
2
Z
H
0
H tðÞ
Z
L
0
Kx; TðÞL
2
x
2
a
2
þ C
1
x; tðÞ½dxdt
þ L
Z
H
0
Z
L
0
a
2
þ C
1
x; tðÞ½1 þ n
a
2
þ C
1
x; tðÞ½
c
P
c
x; TðÞ
oD
L
x; TðÞ
ox
dx
H tðÞdt þ LH
Z
H
0
Z
L
0
L xðÞa
2
þ C
1
x; tðÞ½dxdt
Appl Nanosci
123
L
Z
H
0
H tðÞ
Z
L
0
L xðÞ1 þn
a
2
þ C
1
v; tðÞ½
c
P
c
v; TðÞ
a
2
þ C
1
v; tðÞ½
oD
L
v; TðÞ
ov
dvdt þ L
Z
H
0
H tðÞ
Z
L
0
L xðÞ
a
2
þ C
1
x; tðÞ½Kx; TðÞdxdt þ
H
2
Z
L
0x
fxðÞL
2
x
2
dx
HL
Z
L
0
L xðÞfxðÞdx
1
2
Z
H
0
Z
L
0
L xðÞa
2
þ C
1
x; tðÞ½dxdt ¼ 0:
Analysis of spatiotemporal distribution of dopant
concentration has been done analytically using the second-
order approximation framework method of averaging of
function correction and has been amended numerically.
Discussion
In this section, we analyzed influence of drains of atoms of
dopant on distribution of their concentration in diffusion-
junction rectifier. Some distributions of dopant, which
corresponds to the rectifier, are present in Fig. 3. In this
case drain of the atoms is presented on the right side of the
heterostructure. Increasing of number of curve corresponds
to increasing of value of annealing time. The figure shows,
that availability of drain leads to decreasing of sharpness of
the rectifier. Analogous conclusion could be done in the
case, when drain of atoms of dopant is presented on the
interface between layers of H. The decreasing of sharpness
could be though partially compensated using inhomoge-
neity of heterostructure and nonlinearity of diffusion pro-
cess. The nonlinearity leads to large influence, when level
of doping of materials is high (Pankratov 2005).
Conclusion
In this paper, we analyzed influence of drain of atoms of
dopant on distribution of their concentration in diffusion-
junction rectifier. It has been shown, that availability of
drain leads to decreasing sharpness of p–n junction. It has
been considered abilities of compensation of the
decreasing.
Acknowledgments This work is supported by the contract
11.G34.31.0066 of the Russian Federation Government and educa-
tional fellowship of President of Russia.
Open Access This article is distributed under the terms of the
Creative Commons Attribution License which permits any use, dis-
tribution, and reproduction in any medium, provided the original
author(s) and the source are credited.
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