Pressure dependence of Extended X-ray Absorption Fine Structure (EXAFS) cumulants of silicon and germanium have been investigated using the statistical moment method (SMM). Analytical expressions of the first and second cumulants of silicon and germanium have been derived. The equations of states for silicon and germanium semiconductors have been also obtained using which the pressure dependence of lattice constants and volume of these semiconductors have been estimated. Numerical results using the developed theories for these semiconductors are found to be in good and reasonable agreement with those of the other theories and with experiment

Hieu, Ho Khac

Hung, Nguyen Van

Hung, Vu Van

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 21, No. 3 (2011), pp. 245-252
INVESTIGATION OF THE EXAFS CUMULANTS OF SILICON
AND GERMANIUM SEMICONDUCTORS BY STATISTICAL
MOMENT METHOD: PRESSURE DEPENDENCE
HO KHAC HIEU
National University of Civil Engineering
and
Hanoi University of Science
VU VAN HUNG
Hanoi National University of Education
NGUYEN VAN HUNG
Hanoi University of Science
Abstract. Pressure dependence of Extended X-ray Absorption Fine Structure (EXAFS) cumu-
lants of silicon and germanium have been investigated using the statistical moment method (SMM).
Analytical expressions of the first and second cumulants of silicon and germanium have been de-
rived. The equations of states for silicon and germanium semiconductors have been also obtained
using which the pressure dependence of lattice constants and volume of these semiconductors have
been estimated. Numerical results using the developed theories for these semiconductors are found
to be in good and reasonable agreement with those of the other theories and with experiment.
I. INTRODUCTION
Two of the diamond-type semiconductors silicon and germanium play an important
role in technological and especially in electronic applications. The understanding of ther-
modynamic properties of these semiconductors is very useful. One of the most effective
methods for investigation of structure and thermodynamic properties of crystals is EX-
AFS [1]. The anharmonic EXAFS providing information on structure and thermodynamic
parameters of substances has been analyzed by means of cumulant expansion approach
[1, 2]. In this formulation, an EXAFS oscillation function χ (k) is given by [3]
χ (k) =
F (k)
kR2
e−2R/λ(k)Im
{
eiφ(k) exp
[
2ikR+
∑
n
(2ik)n
n!
σ(n)
]}
, (1)
where k and λ are the wave number and mean free path of emitted photoelectrons, F (k)
is the real atomic backscattering amplitude, φ (k) is the net phase shift, and σ(n) (n = 1,
2, 3, ...) are the cumulants.
The pressure dependence of the EXAFS second cumulant has been measured at
the Stanford Synchrotron Radiation Laboratory (SSRL, USA) for Cu [6], and at the
Laboratoire Pour I’Utisation du Rayonnement Electromagne´tique (LURE) (Orsay, France)
246 HO KHAC HIEU, VU VAN HUNG, AND NGUYEN VAN HUNG
for Kr [7, 8]. Such pressure effects have been calculated by correlated Debye model [6],
as well as by Monte-Carlo (MC) simulation [7] and by Loubeyre’s model [8] to interpret
experimental results.
Some EXAFS studies on crystalline and amorphous Ge under pressure have already
been presented by Kawamura et al. [9] and Freund et al. [10]. The EXAFS spectra of Ge
near K-edge in diamond-type Ge under high temperature and high pressure were measured
using a cubic-anvil-type apparatus (MAX90) with synchrotron radiation from the Photon
Factory, Tsukuba, Japan [11]. Theoretical approach has been done to estimate the second
cumulant on the basis of the isothermal equation of state of Ge up to the pressure of 10.6
GPa [11].
EXAFS is sensitive to pressure [4, 5] which can cause certain changes of cumu-
lants leading to uncertainties in physical information taken from EXAFS. Therefore, the
investigation of pressure effects of cumulants becomes very useful.
Recently, the statistical moment method (SMM) has been used for calculation of
temperature dependence of EXAFS cumulants of silicon and germanium crystals at zero
pressure [12]. The purpose of this work is to develop the SMM for calculating and ana-
lyzing the pressure dependence of cumulants of silicon and germanium crystals at a given
temperature. The equation of state has been also obtained to determine pressure depen-
dence of lattice constants and volumes of silicon and germanium crystals. The calculated
results using our derived theory are compared to experiment and to those of the other
theories [11,19,20-29] showing a good and reasonable agreement.
II. FORMALISM
II.1. General Formula of EXAFS Cumulants
Firstly, we present the SMM for calculating the cumulants of silicon and germanium
semiconductors by using the Stillinger-Weber potentials which consist of two-body and
three-body terms
ϕi =
∑
j
Φij (ri, rj) +
∑
j,k
Wijk (ri, rj, rk) , (2)
where
Φij (ri, rj) =
{
εA
[
B
( rij
σ
)
−4 − 1
]
exp
[( rij
σ − b
)
−1
]
,
rij
σ < b
0,
rij
σ ≥ b
(3)
and
Wijk (ri, rj, rk) = ελ exp
[
γ
(rij
σ
− b
)
−1
+ γ
(rik
σ
− b
)
−1
] (
cos θijk +
1
3
)2
. (4)
where θijk is the angle between bond ij and bond ik.
The effective interatomic potentials of the system is given by
U =
∑
i
ϕi =
1
2
∑
i,j
Φij (ri, rj) +
1
3
∑
i,j,k
Wijk (ri, rj, rk) , (5)
where ϕi is the internal energy associated with atom i.
EXAFS CUMULANTS OF SILICON AND GERMANIUM SEMICONDUCTORS BY SMM ... 247
Using the SMM [13], one can get power moments of the atomic displacement of
diamond-type semiconductor y0 (T ), taking into account the anharmonic effects of the
thermal lattice vibrations
y0 = y
/
0 −
β
3γ
+
1
K
(
1 +
6γ2θ2
K4
)[
1
3
− 2γθ
3k2
(x cothx− 1)− 2β
2
27γk
]
βk
γ
, (6)
where
k =
1
2
∑
i
(
∂2ϕi
∂u2ix
)
eq
≡ mω2; γ = 1
12


∑
i

(∂4ϕi
∂u4ix
)
eq
+ 6
(
∂4ϕi
∂u2ix∂u
2
iy
)
eq



 ; (7)
β =
(
∂3ϕi
∂uix∂uiy∂uiz
)
eq
; y
′
0 ≈
√
2γθ2
3K3
A ; θ = kBT ; x = ~ω/2θ; (8)
K = k − β
2
3γ
;A = a1 +
γ2θ2
K4
a2 +
γ3θ3
K6
a3 +
γ4θ4
K8
a4 +
γ5θ5
K10
a5 +
γ6θ6
K12
a6, (9)
and kB is Boltzmann constant.
The average nearest-neighbor distance (NND) of atoms in crystal at a given tem-
perature T can be determined as
r (T ) = r (0) + y0 (T ) , (10)
where r (0) denotes the NND r (T ) at the temperature 0K, which can be determined from
experiment or from the minimum condition of the potential energy of the crystal. The
lattice constant ah of the diamond-type semiconductor can be calculated easily using the
relation ah = r (T ) 4/
√
3.
Using x = r−r0 as the deviation of instantaneous bond length r from its equilibrium
value r0, we derive the first order cumulant:
σ(1) = 〈x〉 = 〈r − r0〉 ≈ r (T )− r (0) = y0 (T )
=
√
2γθ2
3K3
A− β3γ + 1K
(
1 + 6γ
2θ2
K4
) [
1
3 − 2γθ3k2 (x cothx− 1)− 2β
2
27γk
]
βk
γ ·
(11)
The second cumulant σ(2) = σ2 is an important factor in EXAFS analysis since
the thermal lattice vibrations influence sensitively the XAFS amplitudes through the
Debye-Waller factor e−W ∼ exp (−2σ2k2). The parallel mean square relative displace-
ment (MSRD) to a good approximation corresponds to the second cumulant
σ2 =
〈[
~R. (~ui − ~u0)
]2〉
=
〈
u2i
〉
+
〈
u20
〉− 2 〈uiu0〉 (12)
Using the expression of the second order moment [12, 13], we obtain the mean-square
displacement (MSD) 〈
u2i
〉
= 〈ui〉2 + θA1 + θ
k
(x cothx− 1) , (13)
where
A1 =
1
K
[
1 +
2γ2θ2
K4
(
1 +
x cothx
2
)
(x cothx+ 1)
]
. (14)
248 HO KHAC HIEU, VU VAN HUNG, AND NGUYEN VAN HUNG
For crystals that have a basic cubic structure, such as fcc, any directional dependence
of
〈
u2
〉
must have cubic symmetry. The quadratic contribution to the Debye-Waller factor
is necessarily isotropic. For crystals with a basic hexagonal structure, such as hcp,
〈
u2
〉
is
not isotropic; in general, the components along the a and c axes,
〈
u2a
〉
and
〈
u2c
〉
, for hcp
crystals, are not equal. Hence〈
u2j
〉 ≈ 〈u20〉 ; 〈uju0〉 ≈ 〈uj〉 〈u0〉 . (15)
Therefore, from Eqs. (12), (13), and (15), we derived the second cumulant expression of
the diamond-type semiconductor as
σ2 (T ) ≈ 4γ
2θ3
K5
(
1 +
x cothx
2
)
(x cothx+ 1) +
2θ
k
x cothx+ 2θ
(
1
K
− 1
k
)
. (16)
II.2. Equation of state and pressure dependence of EXAFS cumulants
From the expression for the Helmholtz free energy of system [14-16], the pressure P
of the diamond-type semiconductors can be written in the form
P = −
(
∂ψ
∂V
)
T
= − r
3v
∂ϕ0
∂r
+
3γGθ
v
; (17)
where γG is the Gru¨neisen constant, v is the atomic volume..
From the Eq. (17) one can find the NND r (P, T ) at pressure P and temperature
T . However, for numerical calculations, it is convenient to determine firstly the NND of
crystals r (P, 0) at pressure P and at absolute zero temperature T = 0K. For T = 0K
temperature, Eq. (17) is reduced to
Pv = −a
[
1
3
∂ϕ0
∂r
+
~ω
4k
∂k
∂r
]
. (18)
Eq. (18) can be solved using a computational program to find out the values of
the NND r (P, 0) of the semiconductors. From the obtained results of NND r (P, 0) one
can find the values of parameters K (P, 0), k (P, 0), γ (P, 0) and β (P, 0) at pressure P and
temperature T = 0K. Then, we can find r (P, T ) at pressure P and temperature T as
r (P, T ) = r (P, 0) + y0 (P, T ) , (19)
where y0 (P, T ) is the displacement of an atom from the equilibrium position at pressure
P and temperature T . This quantity can be determined by substituting the values of
K (P, 0), k (P, 0), γ (P, 0) and β (P, 0) into Eq. (6).
Using the above formula of NND r (P, T ), we can find the change of the crystal
volume under pressure P at a given temperature T as
V
V0
=
r3 (P, T )
r3 (0, T )
. (20)
The pressure dependence of MSD of crystals can be obtained as〈
u2i
〉
(P, T ) = y20 (P, T ) + θA1 (P, T ) +
θ
k (P, 0)
(x cothx− 1) , (21)
EXAFS CUMULANTS OF SILICON AND GERMANIUM SEMICONDUCTORS BY SMM ... 249
where
A1 (P, T ) =
1
K (P, 0)
[
1 +
2γ2 (P, 0)θ2
K4 (P, 0)
(
1 +
x cothx
2
)
(x cothx+ 1)
]
. (22)
Substituting the values of K (P, 0),k (P, 0), γ (P, 0) and β (P, 0) into Eqs. (11,16),
we can find out the values of the first, second cumulants of silicon and germanium semi-
conductors under pressure P at the given temperature T .
III. NUMERICAL RESULTS AND DISCUSSIONS
Now we apply the expressions derived in previous section to determine the pressure
dependence of lattice constant, the change of volume, the first and second cumulants of
silicon and germanium semiconductors. The interaction potential between the two inter-
mediate atoms used in this article is the Stillinger-Weber potential, where the potential
parameters of Ge and Si semiconductors are shown in Table 1 (b = 1.2, γ = 1.8) [17, 18].
Table 1. The Stillinger-Weber potential parameters [17, 18]
Potential parameters  (eV) A B σ (A˚) λ
Si 2.17 7.049556277 0.6022245584 2.0951 21.0
Ge 1.93 7.049556277 0.6022245584 2.181 31.0
In Fig. 1a we show the pressure dependence of NND of Ge crystal at room temper-
ature. The pressure dependence of the calculated NND is consistent with the one of A.
Yoshiasa et al.’s results [11]. The lattice constant of Ge crystal can be calculated using
the values of NND. The change of volume under pressure up to 11GPa of Ge crystal is
showed in Fig. 1b. Our calculation results have been compared to available experimental
data [20] as well as to the other theoretical results [11, 19] showing a good agreement.
0 2 4 6 8 10
2.32
2.34
2.36
2.38
2.4
2.42
2.44
2.46
Pressure (GPa)
N
N
D
 r 
(P
,T=
30
0K
) (
Å)
 
 
A. Yoshiasa et al. [11]
SMM
0 2 4 6 8 10
0.88
0.9
0.92
0.94
0.96
0.98
1
Pressure (GPa)
Th
e 
ch
an
ge
 o
f v
ol
um
e 
V/
V 0
 
 
A. Yoshiasa et al. [11]
XRD [19]
Exp [20]
SMM
Fig. 1. (a) Pressure dependence of NND of Ge; (b) Pressure dependence of volume
of Ge.
250 HO KHAC HIEU, VU VAN HUNG, AND NGUYEN VAN HUNG
0 200 400 600 800 1000
0
0.002
0.004
0.006
0.008
0.01
0.012
T (K)
σ
2 G
e 
(P
=0
,T)
 (Å
2 )
 
 
G. Dalba et al. [21]
A. Yoshiasa et al. [11]
SMM
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9
10
x 10−3
Pressure (GPa)
σ
2 G
e 
(P
,T=
30
0K
) (
Å2 )
 
 
A. Yoshiasa et al. [11]
SMM
Fig. 2. (a) Temperature dependence of DWF of Ge; (b) Pressure dependence of
DWF of Ge.
Fig. 2a. shows the temperature dependence of second cumulant or Debye-Waller
factor of germanium crystal at zero pressure. Our calculated results of second cumulant
have been compared to the values of A. Yoshiasa et al. [11] and G. Dalba et al. [21]. They
are found to be in good agreement with those of G. Dalba et al. [21] and in a reasonable
agreement with the results of A. Yoshiasa et al. [11]. In higher pressure, the calculated
pressure dependence of Debye-Waller factors at temperature T = 300K does not agree
well with A. Yoshiasa et al.’s values (Fig. 2b). However, the decreasing ratio between
our calculated results and A. Yoshiasa et al.’s values is similar. It denotes that, the SMM
is still good for calculating the relative change of the Debye-Waller factor of Ge crystal
under pressure.
0 2 4 6 8 10 12
5.25
5.3
5.35
5.4
5.45
5.5
Pressure (GPa)
La
tti
ce
 c
on
st
an
t a
h 
(P
,T=
30
0K
) (
Å)
 
 
SMM
XRD [23]
MC simulation [24]
(a)
0 2 4 6 8 10 12
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Pressure (GPa)
Th
e 
ch
an
ge
 o
f v
ol
um
e 
V/
V 0
 
 
SMM
XRD [23]
S. J. Duclos et al. [25]
Ab initio [26]
Bridgman [27]
Vaidya [27]
X−ray Exp [27]
(b)
Fig. 3. (a) Pressure dependent lattice constants of Si; (b) Pressure dependence
of volume of Si.
In Fig. 3a, we plot the pressure dependence of lattice constant of silicon crystal
calculated by SMM as well as the values of XRD [23] and Monte-Carlo simulations [24].
EXAFS CUMULANTS OF SILICON AND GERMANIUM SEMICONDUCTORS BY SMM ... 251
The pressure-volume relations of Si semiconductor have been showed in Fig. 3b. Our
calculated V/V0 are compared to experiment [23, 27] and to other theoretical results [25-
27] showing the good agreement.
0 100 200 300 400 500 600 700 800
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
x 10−3
T(K)
σ
2 Si
 
(P
=0
,T)
 (Å
2 )
 
 
SMM
M. Benfatto et. al. [28]
Exp [29]
(a)
0 2 4 6 8 10
−5
−4
−3
−2
−1
0
1
x 10−4
Pressure (GPa)
σ
2 Si
(P
,T=
30
0K
)−σ
2 Si
(0,
T=
30
0K
)(Å
2 )
 
 
ACEM (Morse)
SMM
(b)
Fig. 4. (a) Temperature dependence of DWF of Si; (b) Pressure dependence of
DWF of Si.
Our calculated results for the temperature dependence of Debye-Waller factor of
Si crystal at zero pressure has been showed in Fig.4a. They agree with the available
experimental data [29] and with M. Benfatto et al.’s calculated results [28].
Fig. 4b shows the pressure dependence of the change of second cumulant of Si
crystal. Because of the lack of experimental data as well as other theoretical calculations,
we compared the results calculated by our SMM with those calculated by the anharmonic
correlated Einstein model (ACEM) [30] using Morse potential. This figure shows the
agreement between results of two methods.
IV. CONCLUSIONS
In this work, the pressure effects in thermodynamic quantities of diamond-type
silicon and germanium semiconductors have been investigated by using the SMM which
has been applied to three-dimensional crystals. Moreover, the present SMM formalism
takes into account the quantum-mechanical zero-point vibrations as well as the higher-
order anharmonic terms in the atomic displacements.
Our development is establishing and solving equation of state to get the pressure
dependence of the lattice bond length, and then is the derivation of the analytical expres-
sions of pressure dependence for the first and second EXAFS cumulants, the change of
volume of diamond-type semiconductors.
The good and reasonable agreement of our calculated results with experiment and
with those of the other theories denotes the efficiency of our derived theory in the inves-
tigation of the pressure dependence of thermodynamic quantities of semiconductors.
252 HO KHAC HIEU, VU VAN HUNG, AND NGUYEN VAN HUNG
ACKNOWLEDGEMENTS
This work is supported by the research project No. 103.01.09.09 of NAFOSTED.
One of the authors (V. V. H.) acknowledges the partial support of the research project
No. 103.01.2609 of NAFOSTED.
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Received 15 September 2010.


Investigation of the EXAFS Cumulants of Silicon and Germanium Semiconductors by Statistical Moment Method: Pressure Dependence

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Investigation of the EXAFS Cumulants of Silicon and Germanium Semiconductors by Statistical Moment Method: Pressure Dependence

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