Abstract. The effects of strain on the diffusion of impurities in silicon crystal are
investigated using the statistical moment method (SMM). The influence of tensile
strain on diffusion coefficient D is characterized by relaxation volume V r and
migration volume V m. The numerical results for B and P diffusion in silicon that
are performed and compared to experimental data show good agreement.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 71-78
This paper is available online at
EFFECT OF STRAIN ON IMPURITY DIFFUSION IN SILICON
Vu Van Hung1 and Phan Thi Thanh Hong2
1Faculty of Physics, Hanoi National University of Education
2Faculty of Physics, Hanoi University of Education No.2
Abstract. The effects of strain on the diffusion of impurities in silicon crystal are
investigated using the statistical moment method (SMM). The influence of tensile
strain on diffusion coefficient D is characterized by relaxation volume V r and
migration volume V m. The numerical results for B and P diffusion in silicon that
are performed and compared to experimental data show good agreement.
Keywords: Diffusion coefficient, tensile strain, relaxation volume, migration
volume, moment method.
1. Introduction
When a material body is deformed, it’s shape and size are transformed, the crystal
lattice will undergo constant change and properties of the system, such as thermal,
electrical, and magnetic properties will be changed. In particular, the deformation of an
object will greatly effect the diffusion of atoms in the system.
In recent years, the effect of strain on the diffusion coefficient has attracted the
attention of both theoretical and experimental scientists. Among other things, they are
looking at the diffusion of normal impurities such as B, P, As, Sb in Si or SiGe alloy. A
series of experimental observations and theoretical calculations performed by Aziz [1],
Christensen [2], Johansson [3] and Dunham [4] indicate that the diffusion of B and P in
silicon, mediated by an interstitialcy mechanism and their diffusion coefficients, increase
with tensile strain and decrease with compression strain. And, within the same range of
temperatures, the slope of the diffusion-strain curves is also different. Thus, the diffusion
of impurities in Si under the influence of strain is a problem that bears looking into.
Received October 15, 2011. Accepted September 10, 2012.
Physics Subject Classification: 60 44 01.
Contact Phan Thi Thanh Hong, e-mail address: thanhhongdhsp_th@yahoo.com
71
Vu Van Hung and Phan Thi Thanh Hong
In this paper, we used the moment method in statistical dynamics with four order
approximation expansion of interaction potential energy, i.e. the effects of anharmonic
lattice vibration. We calculated diffusion coefficient D(0,T ) of B and P in silicon at
temperature T. Then we used results of the strain theory (presented in section II) to
determine strain dependence , ε, of diffusion coefficient D (ε) at temperature T. The
numerical results obtained using this method will be compared to the experimental data
and previous theoretical calculations.
2. Content
2.1. Theory
It is known that [5], the diffusion coefficient depends on the concentration of defects
and defect mobility. In general, defect mobilityM, in directions parallel and perpendicular
to the direction of an applied stress, will differ. Here, we concerned only with M33,
mobility in a direction which is perpendicular to the free surface (Figure 1), and which is
a measured quantity.
Figure 1. Sample suffered the effects of biaxial stress
As the diffusion is proportional to the product of the concentration and the mobility
of point defects, the effect of stress σ on diffusivity in the normal direction to the free
surface is obtained by [5]:
kBT ln
D (σ)
D (0)
= σ V (2.1)
with kB is the Boltzmann constant, T is the absolute temperature and V is the sum of the
relaxation volume, V r, and the migration volume, V m,
72
Effect of strain on impurity diffusion in silicon
V = V r + V m. (2.2)
Relaxation volume V r is the amount of outward relaxation of the sample surfaces (if
the relaxation is inward, V r is negative) due to the newly created point defect. Migration
volume V m is the additional volume change when the defect reaches the saddle point in
its migration path (Figure 2).
V fi = −Ω + V ri V mi
Figure 2. Schematic volume changes (see dashed lines) upon point defect formation
and migration for the interstitialcy mechanism
From Eqs. 2.1 and 2.2, we have
D (σ) = D (0) exp
{
σ (V r + V m)
kBT
}
(2.3)
here, D(0) is the diffusion coefficient when the absence of external stress (σ = 0) at
temperature T . In the case of zero external stress (σ = 0) the diffusion coefficient, D(0),
is given by [6]
D (0) = D (0, T ) = D0 exp
{
−Q (0, T )
kBT
}
(2.4)
with D0 being the pre-exponential factor and Q(0, T ) the activation energy for an atom
diffuse at temperature T .
From Eqs. 2.3 and 2.4, the diffusion coefficient of deformed crystal can be rewritten
as
D (σ) = D0 (σ) exp
{
−Q (0, T )
kBT
}
(2.5)
where D0(σ) is the pre-exponential factor which depends on stress σ
73
Vu Van Hung and Phan Thi Thanh Hong
D0 (σ) = D0 exp
{
σ (V r + V m)
kBT
}
. (2.6)
Thus, stress dependence σ of diffusion coefficient D(σ) is written via
pre-exponential factorD0(σ) and activation energyQ(0, T ) is not stress dependent. These
results are consistent with a molecular dynamics study of J. Johansson and S.Toxvaerd [3].
On the other hand, stress σ and strain ε of the elasticity strain are solidly related to
each other according to Hooke’ s law
σ = Eε (2.7)
with E being the Young modulus.
From Eqs. 2.3 and 2.7 we can express the diffusion coefficient depends on strain ε
according to formula
D (ε) = D (0) . exp
{
Eε (V r + V m)
kBT
}
. (2.8)
In order to determine the tensile strain influence on diffusion coefficient D (ε) at
temperature T , we must determine diffusion coefficient D(0) according to formula Eq.
2.4, and relaxation volume V r and migration volume V m.
Relaxation volume V r, in units of Ω (Ω is the atomic volume at temperature T ), is
given by [7]
V rI,V =
l3I,V − l3eq
l3eq/N
(2.9)
where, lI,V is the box length for interstitial (I) and vacancy (V) defects, respectively; leq is
the original box length without defect; and N is the sum of atoms in the box. In the case
where a box is a cubic lattice cell, the Eq. 2.9 can be rewritten as
V r =
a3d − a3p
a3p/8
(2.10)
where ap or ad denote the lattice constants of the silicon crystal, perfect or with defect,
respectively.
In knowing the schematic volume changes (see dashed lines) upon point defect
formation and migration for the interstitialcy mechanism (see Figure 2), we found that
migration volume V m has a form analogous to Eq. 2.10
V m =
a,3d − a3p
a3p/8
(2.11)
74
Effect of strain on impurity diffusion in silicon
with a′d being the lattice constant of the silicon crystal when the defect is moving.
The lattice constant a of the silicon crystal is determined according to the formula
a =
4√
3
r1 (2.12)
where r1 is the shortest distance between two atoms at temperature T and can be written
as
r1 = r10 + y0 (2.13)
with y0 being the displacement of atoms from an equilibrium position at temperature T ,
as determined by the SMM. r10 is the shortest distance between two atoms at absolute
zero temperature (T = 0K) and is determined from the equation of state [8]
pv = −r
[
1
6
∂u0
∂r
+
~ω
4k
∂k
∂r
]
(2.14)
where, p denotes hydrostatic pressure, v is the atomic volume, k denotes the vibrational
constant and u0 represents the sum of the effective pair interaction energies between the
ith and 0th atoms
u0 =
∑
i
ϕi0 (|−→ri |) (2.15)
k = mω2 =
1
2
∑
i
(
∂2ϕi0
∂u2i
)
eq
(2.16)
when p = 0, solving Eq. 2.14 with u0 and k applied to the three following cases:
(i) The perfect silicon crystal, we obtained r10p
(ii) The self- interstitial defect silicon crystal , we obtained r10d
(iii) The dopant-interstitial defect silicon crystal, we obtained r′10d.
Replacing the values of r10p; r10d; r
′
10d to Eq. 2.13, then using Eq. 2.12 we can find
the lattice constants ap; ad; a
′
d, respectively.
2.2. Numerical results and discussions
To calculate tensile strain influence on the diffusion coefficients of B and P atoms
in Si crystal, we used the empirical many-body potential developed for silicon [9] that is
presented in the following equations:
ϕ =
∑
i〈j
Φij +
∑
i〈j〈k
Wijk (2.17)
75
Vu Van Hung and Phan Thi Thanh Hong
Φij = ε
[(
r0
rij
)12
− 2
(
r0
rij
)6]
(2.18)
Wijk = Z
(1 + 3 cos θi cos θj cos θk)
(rijrjkrki)
3 (2.19)
here rij , rjk and rki denote the distances between the i-th and j-th atoms, the j-th and k-th
atoms, and the k-th and i-th atoms in crystal; θi, θj , θk are the inside angles of a triangle
created from the three atoms i, j and k; and ε, r0 and Z are the potential parameters taken
from ref. 9 (Table 1). These parameters are determined so as to fit the experimental lattice
constants and cohesive properties of Si crystal.
Table 1. Potential parameters of the empirical many - body potentials for Si [9]
ε(eV ) r0 (A˚) z (eV.A˚
9)
2.817 2.295 3484
With the impure atoms, we used the Pak-Doyam pair potential developed for boron
and phosphorus [10]
ϕij =
{
a (rij + b)
4 + c (rij + d)
2 + e , rij < r0
0 , rij ≥ r0 (2.20)
The parameters (a, b, c, d, e, r0) of these potentials are presented in Table 2.
Table 2. Potential parameters of the Pak-Doyam pair potential for B and P [10]
a b c d e r0 (A˚)
ϕB−B(eV ) -0.08772 -2.17709 0.79028 -2.85849 -0.09208 3.79
ϕP−P (eV ) -0.07435 -2.60709 0.64791 -3.27885 -0.07531 4.21
Using the interaction potentials of Eqs. 2.17 and 2.20 with Eqs. 2.15 and ??, it is
straightforward to obtain expressions of interaction energy u0 and vibrational constant
k. Using the potential parameters for Si and impurities (Tables 1 and 2) and the Maple
program, Eq. 2.14 can be solved. Then we can find the values of the nearest neighbor
distances r10p; r10d; r
′
10d at absolute zero temperature and, subsequently, the temperature
dependence of lattice constants ap; ad; a
′
d can be calculated. Using the calculated values
of ap; ad; a
′
d, the Eqs. 2.10 and 2.11, relaxation volume V
r and migration volume V m at
temperature T are calculated. Diffusion coefficients D(0) and Young modulus E can be
calculated analogous to our works [6, 8]. Therefore, with the aid of Eq. 2.8, the tensile
strain influence on diffusion coefficients D(ε) of B and P in Si at temperature T can be
calculated. The calculated SMM values ofD(ε) at temperature T = 1113K for both B and
P atoms are presented in Table 3.
76
Effect of strain on impurity diffusion in silicon
Table 3. Tensile strain dependence of the diffusion coefficient
of B and P in Si at temperature 1113K
ε (%) DB(ε) (cm2/s) DP (ε) (cm2/s)
0.0 3.4056*10−16 2.2704*10−16
0.2 4.4115*10−16 2.7841*10−16
0.4 5.7144*10−16 3.4140*10−16
0.6 7.4022*10−16 4.1865*10−16
0.8 9.5884*10−16 5.1337*10−16
1.0 12.420*.10−16 6.2953*10−16
Our numerical results show that the diffusion coefficient of both B and P in Si
increase with tensile strain. These results are in agreement with predictions made using
the density function theory of Dunham [4], the experimental observations by Aziz [1] and
conclusions by Christensen [2].
(a) B and P in Si at 1113K (b) B in Si at 1073K
Figure 3. The tensile strain dependence of the diffusion coefficient
of B and P in Si
In Figure 3.a, we show the tensile strain dependence of diffusion coefficient D(ε)
of both B and P in Si crystal at the same temperature T = 1113K. We see that diffusion
coefficients D(ε) of both B and P increase as the tensile strain increases but the diffusion
coefficient of B increases faster because the calculated values of D(0), V r and V m for B
is greater than for P at the same temperature.
In Figure 3.b, we compared the tensile strain dependence of diffusion coefficient
D(ε) of B in Si with experimental data by Aziz [1] at the temperature of 1073K. The
calculated results are in good agreement with the experimental data.
77
Vu Van Hung and Phan Thi Thanh Hong
3. Conclusion
In this paper, we have performed the statistical moment method to study the
tensile strain dependence of diffusion coefficient D when B and P in silicon obey an
interstitialcy mechanism. The SMM calculated results for diffusion coefficient D are in
good agreement with the experimental data.
Acknowledgment. This work is supported by Research Project No. 103.01 2011.16 of
NAFOSTED.
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