A model for simulating the rapid thermal annealing of silicon structures implanted with boron and carbon is developed. The model provides a fair approximation of the process of boron diffusion in silicon, allowing for such effects as the electric field, the impact of the implanted carbon, and the clustering of boron. The migration process of interstitials is described according to their drift in the field of internal elastic stress

Komarov, A. F.

Komarov, F. F.

Makarevich, Yu. V.

Mironov, A. M.

Miskevich, S. A.

Zayats, G. M.

English

Электронная библиотека БГУ

ISSN 10628738, Bulletin of the Russian Academy of Sciences. Physics, 2012, Vol. 76, No. 5, pp. 574–576. © Allerton Press, Inc., 2012.
Original Russian Text © Yu.V. Makarevich, F.F. Komarov, A.F. Komarov, A.M. Mironov, G.M. Zayats, S.A. Miskevich, 2012, published in Izvestiya Rossiiskoi Akademii Nauk.
Seriya Fizicheskaya, 2012, Vol. 76, No. 5, pp. 643–646.
574
INTRODUCTION
Lowdepth doping by impurities must be per
formed to create submicrometer semiconductor
devices and integrated VLSI microchips. The main
difficulties are in this case related to the formation of
shallow regions with ptype conductivity. The boron
atoms predominantly used in creating pregions in sil
icon have high diffusivity. In addition, the effect of
nonequilibrium accelerated atomic diffusion of boron
in silicon is observed upon postimplantation thermal
annealing. Rapid thermal annealing is one solution to
the problem, but it does not allow us to compensate for
this effect entirely.
To suppress the effect of nonequilibrium acceler
ated atomic diffusion of boron in silicon, we can use
implantation by carbon atoms—junctions created by
the combined implantation of boron and carbon in sil
icon have a much shallower localization depth than
those formed without carbon implantation [1].
The aim of this work is to construct a diffusion
model of boron implantation in silicon with allowance
for the implanted carbon atoms. Boron diffusion is
examined simultaneously with the effect of migrating
nonequilibrium defects. The impact of elastic stress on
the behavior of point defects is also considered.
DESCRIPTION OF THE MODEL
We assume that the boron diffusion in silicon is
caused by the formation, migration and dissociation of
mobile pairs created by impurity atoms, interstitial
atoms, and by the influence of an electric field [2–4].
We also assume that boron diffusion is insignificantly
affected by such vacancies, and the carbon distribution
is stationary during the period of thermodiffusion.
The equation for the diffusion of boron atoms is in
this case
(1)
where С is the active concentration of boron; J is boron
flux; N is the donor impurity concentration;  is the
total boron concentration;  is the defect concentra
tion;  is the concentration of intrinsic carriers;
=  is the effective diffusion coef
ficient of impurity atoms in the field of elastic stress via
the mechanism of formation of the complexes with
intrinsic interstitial atoms; χ =  is the normalized
concentration of charge carriers;  is the diffusion
coefficient of boron; and  and  are coefficients
that determine the relative contribution from singly
and doubly charged defects in the process of boron
diffusion.
Let us define the boundary conditions for Eq. (1).
Flux J on the surface of the crystal is assumed to be
where  is the rate of impurity transport through the
surface, and μ is a parameter characterizing the con
stant component of the rate of the impurity evapora
tion. On the right boundary of the range of modeling
(in the bulk of the substrate), we set J = 0.
The initial conditions have the form
2 2
( )
( ) ,
( ) 4
T
I I
e
C J
t x
C C C C CD
x x xC N n
∂ ∂= −
∂ ∂
⎛ ⎛ ⎞⎞∂∂ ∂= χ +⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟∂ ∂ ∂− +⎝ ⎝ ⎠⎠
TC
IC
en
( )D χ
2
1 2
1 2
1
1
iD
+ β χ + β χ
+ β + β
en n
iD
1β 2β
,sJ K C= + µ
sK
0
0
( , ) ( ),T
t
C x t C x
=
=
Controlling Boron Diffusion during Rapid Thermal Annealing 
with CoImplantation by Amphoteric Impurity Atoms
Yu. V. Makarevicha, F. F. Komarova, A. F. Komarova, A. M. Mironova, 
G. M. Zayatsb, and S. A. Miskevicha
aSevchenko Institute of Applied Physics Problems, Minsk, 220108 Belarus
bInstitute of Mathematics, National Academy of Sciences, Minsk, 220072 Belarus
email: elionics@bsu.by, zayats@im.basnet.by
Abstract—A model for simulating the rapid thermal annealing of silicon structures implanted with boron and
carbon is developed. The model provides a fair approximation of the process of boron diffusion in silicon,
allowing for such effects as the electric field, the impact of the implanted carbon, and the clustering of boron.
The migration process of interstitials is described according to their drift in the field of internal elastic stress.
DOI: 10.3103/S1062873812050164
BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS  Vol. 76  No. 5  2012
CONTROLLING BORON DIFFUSION DURING RAPID THERMAL ANNEALING 575
where  is the implanted profile of the impurity.
Allowing for the influence of the implanted interstitial
carbon atoms, the nonstationary equation for the
defects takes the form
(2)
where d(χ) is the effective diffusion coefficient of the
interstitial silicon atoms,  is the carbon concentra
tion, k is a parameter, and  and  are the migration
length and the drift velocity of the point defects.
To describe the drift velocity in the field of elastic
stresses, we use the equation [5]
where  is the cumulative effective diffusion coef
ficient of the interstitial defects,  is the Boltzmann
constant, and  is the potential energy of point
defects in the field of elastic stresses. In our model, we
considered the simplest approximation for the effec
tive drift coefficient of interstitials:
Assuming that the potential energy in the field of
elastic stresses is proportional to the carbon concentra
tion, the resulting value for the drift velocity is given by
where B is a constant.
We assume that in the bulk of semiconductor, the
total defect concentration is equal to the concentra
tion of intrinsic defects, while on the surface we have
the condition
where β is a recombination coefficient, and γ is the
surface length of recombination. The initial condi
tions are determined by the profiles of implantation.
NUMERICAL ALGORITHM
An approximate solution to the problem is found
using the finite difference method [6]. The following
temporal and spatial meshes were used:
We take the designations for finite difference rela
tionships analogous to those used in [6]. Let y,  z be
the approximate values for C, CI and χ, respectively.
0( )C x
( ) ( )
( )
( )
( )
( )
2 2
1
,
I I
I
c I
i i
C Cd C
t x x x
kC C d
l l
⎛ ⎞∂ ∂ ∂ ∂= χ −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
+ χ
− +
v
cC
il v
1( ) ,
d
C
x
B
Ud
k T x
∂
= = − χ
∂
v v
( )Cd χ
Bk
dU
( ) const.Cd χ =
,
c
x
B
B C
k T x
∂
= −
∂
v
( ) ,
I
ICd C
x
∂
χ = β + γ
∂
0 0{ , 0, 1, ..., , },j ft j j j j tτω = = τ = τ =
ω { , 0, 1, ..., , }.ix ih i N Nh l= = = =h
,y
Using the mesh  = , the conservative
advanced finitedifference scheme for Eq. (1) is given by
 
The difference condition at x = 0 has the form
 =  +  
At x = l, we assume
 =  +  
At t = 0 we assume
 
Equation (2) for the defects is also approximated
on mesh  using finitedifference schemes of the
second and first kinds over space and time, respec
tively:
 
where  = 
Condition   holds at x = l. The
finite difference condition for the surface is given by
The solution to the system of nonlinear difference
equations is obtained iteratively.
MODELING RESULTS
To assess the model’s applicability for calculating
the diffusion redistribution of boron in silicon, we per
formed modeling for different initial conditions and
parameters of the processes. The distribution profiles
of boron in silicon, obtained by numerical calculations
using our model, were compared with the correspond
ing experimental data from the literature. The results
showed that our model allows us to obtain a reasonably
good description of the boron distribution in silicon
after lowenergy implantation of boron ions and rapid
thermal annealing, with allowance for the influence of
implanted carbon on the diffusion of boron atoms.
Let us compare the results of modeling with the
experimental data from [7]. The presented boron pro
files were obtained at different stages of the following
hτω h τω ×ω
( )( ) ( )1 2( ) ( , , ) ,
j
j
i
i
j
t xx xx i
y a z yy a z y y y= + 
1,2,..., 1,i N= − 01,2, ..., .j j=
( ) ( )( )1 1( ) 0.5 ,i ia z D z D z −= +
( ) ( )
2 2
1 1 1
2
1
( , , ) 0.5 .
4 4e e
i i i i i i
i i
D z y y D z y y
a z y y
y N n y N n
− − −
−
⎛ ⎞
= +⎜ ⎟
− + − +⎝ ⎠
 

0
0.5
j
ithy = 11( )( )
j
ixa z yy = )2 01 0( , , ) ,
j
i
j
x s i
a z y y y K y
= =
− − µ
01,2,..., .j j=
0.5
j
i Nthy =− 1( )( )
j
i Nxa z yy = )2( , , ) ,
j
i Nxa z y y y =
01,2,..., .j j=
0( ,0) ( ),i iy x C x= 0,1,..., .i N=
hτω
( ) ( )
2 23
( )1( ) ,
i i
jc
j
t xi x x
i
dkCy a z y y y
χ⎛ ⎞+= − − +⎜ ⎟
⎝ ⎠
v

   
l l
1,2,..., 1,i N= − 01, 2,..., .j j=
3( )a z ( ) ( )( )10.5 .i id z d z −+
1,jNy = 01, 2,...,j j=
( )
( ) ( )
( )
0
0 1 2 2
0 0
3 01
1
0.5
,
j jc j
jj
t i i
i i
i i
j j
x ii
kC y d z
h y y
a z y y
= =
= =
==
⎛ ⎞+
⎜ ⎟+ + −
⎜ ⎟
⎝ ⎠
= −β − γ
v

 
 
l l
01,2,..., .j j=
576
BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS  Vol. 76  No. 5  2012
MAKAREVICH et al.
process. Ion implantation was performed at an angle
of 7° in Si(100) substrates. Some of the substrates were
first implanted with carbon at a dose of 5 × 1014 cm–2
and an energy of 85 keV. Boron was then implanted in
all of the substrates (at a dose of 2.5 × 1014 cm–2 and an
energy of 40 keV). Finally, the samples were subjected
to rapid annealing at a temperature of 900°С for 20 s.
The profiles of boron distribution were measured at
each stage of the process by means of secondion mass
spectrometry.
Figures 1 and 2 show the experimental results cor
responding to boron and carbon implantation in sili
con wafers, along with the results of calculations using
our model and Silvaco’s ATHENA software. Figure 1
presents the boron distributions in silicon at different
stages of the aforementioned process. We can see that
the proposed model adequately explains boron diffu
sion upon rapid thermal annealing both with and with
out carbon implantation.
For comparison, Fig. 2 shows the data obtained
using Silvaco’s ATHENA software, version 5.8.4.R
(the CNET model of diffusion [8]). We can see that
the model developed in this work gives a better esti
mate of the boron distribution profile in the depth of
the substrate (at a depth of more than 0.3 µm in this
case) than the ATHENA software.
CONCLUSIONS
We have developed a physicomathematical model
and computer software for numerically modeling the
rapid thermal annealing of silicon structures
implanted by boron and carbon ions. The migration of
impurity atoms is examined in the form of diffusing
“impurity atom–selfinterstitial atom” complexes.
The impact of an electric field, internal elastic stress,
the bonding of interstitial silicon atoms with the car
bon atoms, and the clustering of boron atoms on the
diffusion of implanted impurities were examined and
allowed for.
ACKNOWLEDGMENTS
This work was supported in part by the Belarusian
Republican Foundation for Fundamental Research,
project no. T11159.
REFERENCES
1. Vanderpool, A. and Taylor, M., Nucl. Instrum. Methods B,
2005, vol. 237, p. 142 .
2. Komarov, F.F., Velichko, O.I., Dobrushkin, V.A., et al.,
Phys. Rev., 2006, vol. 74, no. 3, p. 035205.
3. Komarov, F.F., Mironov, A.M., Zayats, G.M., et al.,
Vacuum, 2007, vol. 8, p. 1184.
4. Fedotov, A.K., Velichko, O.I., and Dobrushkin, V.A.,
J. Alloys Comp., 2004, vol. 382, p. 283.
5. Pezoldt, J., Rybin, P.V., Kulikov, D.V., et al.,
Nucl. Instrum. Methods. B, 2000, vol. 166/167, p. 758.
6. Samarskii, A.A., Teoriya raznostnykh skhem (Theory of
Difference Schemes), Moscow; Nauka, 1977.
7. Cowern, N.E.B., Cacciato, A., Custer, J.S., et al.,
Appl. Phys. Lett., 1996, vol. 68, no. 8, p. 1150.
8. Nelaev, V.V. and Stempitsii, V.R., Osnovy SAPR v mik
roelektronike. Fizicheskoe modelirovanie tekhnologii i
pribora: uchebnoe posobie (Foundations of Computer–
Aided Design Systems in Microelectronics. Physical
Simulation of Technologies and Devices: Student’s
Book), Minsk: BSUIR, 2006.
1020
1016
0.70.20.10 0.3 0.4 0.5 0.6
x, µm
C, cm–3
1019
1018
1017
1
2
3
4
5
Fig. 1. Distributions of boron atoms implanted in silicon:
(1) directly after implantation; (2) after thermal annealing
of plates preliminarily implanted by carbon; (3) after ther
mal annealing of plates not implanted by carbon; (4) mod
eling the diffusion process with no allowance for the cap
ture of interstitial silicon atoms by carbon atoms; (5) mod
eling the diffusion process with allowance for carbon.
1020
1016
0.70.20.10 0.3 0.4 0.5 0.6
x, µm
C, cm–3
1020
1018
1017
1
2
3
Fig. 2. Comparison of the results from modeling rapid
boron thermal annealing in the presence of carbon:
(1) the data obtained using SIMS; (2) the results from our
own modeling; (3) the results from modeling using the
ATHENA software.


Controlling Boron Diffusion during Rapid Thermal Annealing with CoImplantation by Amphoteric Impurity Atoms

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Controlling Boron Diffusion during Rapid Thermal Annealing with CoImplantation by Amphoteric Impurity Atoms

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