JOURNAL OF APPLIED PHYSICS 107, 094308 共2010兲
Effects of nanoscale porosity on thermoelectric properties of SiGe
Hohyun Lee,1 Daryoosh Vashaee,2 D. Z. Wang,3 Mildred S. Dresselhaus,4,5 Z. F. Ren,3
and Gang Chen6,a兲
1
Department of Mechanical Engineering, Santa Clara University, Santa Clara, California 95053, USA
School of Electrical and Computer Engineering, Oklahoma State University, Tulsa, Oklahoma 74106, USA
3
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
4
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
5
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
6
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA
2
共Received 10 March 2009; accepted 17 March 2010; published online 6 May 2010兲
The recent achievement of the high thermoelectric figure of merit in nanograined materials is
attributed to the successful optimization of the consolidation process. Despite a thermal conductivity
reduction, it has been experimentally observed that the porous nanograined materials have lower
thermoelectric figure of merit than their bulk counterpart due to significant reduction in the electrical
conductivity. In this paper, nanoscale porosity effects on electron and phonon transport are modeled
to predict and explain thermoelectric properties in porous nanograined materials. Electron scattering
at the pores is treated quantum mechanically while phonon transport is treated using a classical
picture. The modeling results show that the charge carriers are scattered more severely in
nanograined materials than the macroscale porous materials, due to a higher number density of
scattering sites. Porous nanograined materials have enhanced Seebeck coefficient due to energy
filtering effect and low thermal conductivity, which are favorable for thermoelectric applications.
However, the benefit is not large enough to overcome the deficit in the electrical conductivity, so that
a high sample density is necessary for nanograined SiGe. © 2010 American Institute of Physics.
关doi:10.1063/1.3388076兴
I. INTRODUCTION
Direct energy conversion between heat and electricity
based on the thermoelectric effects is attractive for its potential applications in the area of waste heat recovery and
environmentally-friendly refrigeration.1,2 The energy conversion efficiency of a thermoelectric material depends on the
dimensionless figure of merit, ZT, which is proportional to
the electrical conductivity ␴, the square of the Seebeck coefficient S, and the absolute temperature T, and inversely
proportional to the thermal conductivity k. Recently, the enhancements in ZT have been achieved using the nanoscale
effects, mainly due to the thermal conductivity reduction by
phonon boundary scattering at the increased amount of
interfaces.3 Among various methods employing nanostructures, bulk samples with nanoscale grains, which can be
treated as a special case of nanocomposites,4,5 are one of the
most cost efficient ways for large scale application.6–10 The
process for nanograined materials can be easily scaled up for
batch fabrication while maintaining the nanostructures: powders are pulverized into nanometer sizes and consolidated at
high temperature. However, experimental results in nanograined SiGe have shown that the porosity can degrade the
electrical conductivity so significantly that a high figure of
merit cannot be achieved without having a high sample
density.11 Such anomalous reduction in the electrical conduca兲
Author to whom correspondence should be addressed. Tel.: 617-253-0006.
FAX: 617-258-5802. Electronic mail: gchen2@mit.edu.
0021-8979/2010/107共9兲/094308/7/$30.00
tivity cannot be explained by the traditional effective medium theories. Hence, there is a need to better understand the
effects of porosity on thermoelectric properties of nanograined materials.
There have been only a few studies on the thermoelectric
transport properties of porous media. In the 1970s,
Lidorenko et al.12 reported a 30% increase in the ratio of the
electrical conductivity to the thermal conductivity for porous
SiGe alloys, and suggested a possible enhancement in ZT
using microscale porous structures. Concurrently, the effective medium theories were exploited to explain the electrical
conductivity and the thermal conductivity in such microscale
porous materials.13,14 However, these studies cannot be applied to nanoscale porous structures due to the classical
and/or the quantum size effects: additional scattering of
phonons and electrons can happen at the pore sites or at the
grain boundaries. More recently experimental and modeling
studies on the thermal conductivity of microscale and nanoscale porous films have been made.15–17 These studies suggest that the significant reduction in the thermal conductivity
of nanoporous structures could lead to the enhancement in
ZT. Furthermore, nanoporous structures can also lead to an
improvement in the Seebeck coefficient due to the energy
filtering effect. Specifically, the Seebeck coefficient can be
enhanced by scattering only low-energy electrons when electrons pass through a finite barrier, as previous studies on
thermionic emission show.18–20 To the best of our knowl-
107, 094308-1
© 2010 American Institute of Physics
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094308-2
J. Appl. Phys. 107, 094308 共2010兲
Lee et al.
edge, however, there has been no systematic modeling on
how nanoscale porosity affects thermoelectric figure of
merit.
This paper presents a modeling study on the effect of
nanoscale pores on the thermoelectric properties. First, we
investigate why the electrical conductivity of nanoporous
materials is degraded more severely than the prediction by
the effective medium theories. Second, we explore if the
nanoscale porosity effect can lead to the enhancement in ZT.
We developed a charge scattering model caused by spherical
pore sites to calculate electron transport. For phonon transport, a modified effective medium theory is exploited to consider the classical size effect.21 Although this model can be
applied to various material systems, the result is presented
for SiGe, which is widely used as a thermoelectric material
for high temperature power generation. The present authors
also reported the experimental studies on the nanograined
SiGe, which are in line with this modeling study.22,23 The
modeling results show that the electrical conductivity degradation in nanoporous materials is due to the strong charge
carrier scattering at the large number of pore sites. Furthermore, the enhancement in the Seebeck coefficient and the
reduction in the phonon thermal conductivity are not large
enough to overcome electrical conductivity deterioration in
nanoporous SiGe. Hence, this model suggests that a high
sample density is essential for the nanograin approach to
enhance the ZT of SiGe.
II. MODEL FOR THERMOELECTRIC TRANSPORT IN
NANOPOROUS MATERIALS
A. Electron modeling
The electrical conductivity 共␴兲 and the Seebeck coefficient 共S兲 of bulk materials can be derived from the standard
formulation based on the Boltzmann transport equation under the relaxation time approximation:24
␴=−
1
3
冕
1
3
冕
q
S=
q2␶共E兲v共E兲2
⳵ f共EF,E兲
D共E兲dE,
⳵E
E − EF
⳵ f共EF,E兲
D共E兲dE/␴ ,
␶共E兲v共E兲2
T
⳵E
共1兲
共2兲
where E is the electron energy, EF is the Fermi level, q is the
charge of electrical carriers, ␶ is the energy dependent momentum relaxation time, v is the group velocity of the charge
carriers, f is the Fermi–Dirac distribution function, and D is
the density of electronic states. The validity of this approach
relies on the accurate consideration of the momentum relaxation time, or inverse of the scattering rate. In crystalline
SiGe, the major scattering mechanisms for electrons are ionized impurity scattering and acoustic phonon scattering. According to the Matthiessen’s rule, when a scattering event
happens independently from other scattering events, the total
scattering rate is obtained by taking the sum of each scattering rate corresponding to different scattering mechanisms.
Functional forms of the momentum relaxation time for various scattering mechanisms can be found in typical device
physics textbooks and are summarized in Table I.25,26
The approach used in this work is similar to that of
Vining,24 but more details are considered. In particular, we
consider the nonparabolicity of the conduction band due to a
high carrier concentration.27 The change in band structure
with varying temperature and doping concentrations is also
considered.28 The coefficients associated with band structure
are taken from the literature as summarized in Table I.24,29,30
The validity of our model for dense bulk samples was verified with the experimental results on bulk SiGe alloy by
Dismukes et al.31 While the Fermi level is used as a fitting
parameter in Vining’s work, we derived the Fermi level directly from the measured carrier concentration by Dismukes
et al.31 Without using any fitting parameter, our model can fit
the experimental data for bulk SiGe alloy for the temperature
range of 300 to 1300 K and for the various composition
ratios 共Fig. 1兲.
We develop a new scattering model to consider the effect
of nanoscale pores on the charge carriers transport properties. Figure 2 shows a schematic of charge carrier scattering
caused by a pore site. Pores are regarded as spherical sites
with a potential difference. When a charge carrier passes near
a spherical pore, it is scattered to a different wave vector due
to the potential perturbation. In order to evaluate scattering
rate caused by spherical potential, we exploit the Fermi
Golden rule as in the other scattering studies.25 In the scattering rate calculation, it is generally assumed that the band
structure is not affected by the potential perturbation. Hence,
we neglect any modification in atomic structure caused by
the pore sites. The amount of potential perturbation U0 is the
difference between the electron energy levels in the pore site
and in the environment, and is equivalent to the electron
affinity of SiGe. The mathematical form of the spherical
scattering potential with a radius a can be expressed as
U共rជ兲 = U0H共a − 兩rជ兩兲,
共3兲
where r is the distance from the center of pore and H is the
Heaviside step function. Using the Fermi’s Golden Rule, we
can obtain the transition rate SR from the original eigenstate
with the wave vector kជ to another with a wave vector kជ ⬘
SR共kជ ,kជ ⬘兲 =
2␲
兩M kជ kជ ⬘兩2␦共E⬘ − E兲,
ប
共4兲
where ប is Plank’s Constant, E and E⬘ are the energy of
charge carriers before and after scattering, and ␦ is the delta
function. M kជ kជ ⬘ is the matrix element of the scattering potential U between the eigenstates kជ and kជ ⬘
M kជ kជ ⬘ =
冕冕冕
ជ
ជ
eik·r̄U共rជ兲e−ik⬘·rជdrxdrydrz .
共5兲
As the scattering rate measures how often a charge carrier is
scattered into a different eigenstate, we need to integrate Eq.
共4兲 over all possible eigenstates kជ ⬘ that result from a scattering event. Since we assume that the scattering is elastic, Eq.
共4兲 will have nonzero value only when the energy of charge
carriers is conserved. The scattering rate or the inverse of
relaxation time, for pore sites is
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094308-3
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TABLE I. Electron modeling parameters.
Majority carrier concentration
Effective mass of electrons of Si1−xGex at X pointa
Effective mass of electrons of Si1−xGex at L pointa
Effective mass of holes of Si1−xGex at ⌫ pointa
Si energy gap at X pointa
Si energy gap at L pointa
Ge energy gap at L pointa
Ge energy gap at X pointa
Si1−xGex energy gap at each pointa
Nonparabolicityb
Electron density of states at X pointc
Electron density of states at L pointc
Hole density of states at ⌫ pointc
Dielectric constant of Si1−xGex a
Screening lengthd
Ionized impurity scattering rated
Electron-phonon scattering potentiald
Bulk moduli of Si1−xGex a
Electron-phonon scattering rated
Hole-phonon scattering rated
Lattice constanta
Alloy scattering potentialb
Alloy scattering rate 共electrons兲e
Alloy scattering rate 共holes兲e
N 共cm−3兲
mX = 关1.08共1 − x兲 + 1.41x − 0.183x共1 − x兲兴me 共kg兲
mL = 关1.08共1 − x兲 + 0.71x − 0.183x共1 − x兲兴me 共kg兲
m⌫ = 共0.81− 0.47x兲me 共kg兲
EgSi_X = 1.17– 4.73⫻ 10−4T2 / 共T + 636兲 共eV兲
EgSi_L = 1.65– 4.73⫻ 10−4T2 / 共T + 636兲 共eV兲
EgGe_L = 0.74– 4.80⫻ 10−4T2 / 共T + 235兲 共eV兲
EgGe_X = 0.85EgGe_L / 0.66 共eV兲
ⴱ
Eg = EgSi共1 − x兲 + EgGe
x − 0.4x共1 − x兲 − 0.009 ln共N / 1017兲
+ 兵关ln共N / 1017兲兴2 + 0.5其1/2 共eV兲
A = 2 共eV兲
DX = 8␲冑2E共1 + 共E / ␣兲兲冑共共mX / h2兲兲3共1 + 共2E / ␣兲兲
DL = 8␲冑2E共1 + 共E / ␣兲兲冑共共mL / h2兲兲3共1 + 共2E / ␣兲兲
D⌫ = 共8␲冑2E共m⌫兲3 / h3兲
␧ = 11.7+ 4.5x
LD = 冑4␲␧␧0kT / q2N
1 / ␶i = 2␲2N / h共q2LD2 / 4␲␧␧0兲2DX
DA = −9.5 共electron兲, 5.0 共holes兲 共eV兲
Cl = 98− 23x 共GPa兲
共1 / ␶e−p兲 = 共2␲2kT / hcl兲共DX / 6兲DA2 共s−1兲
共1 / ␶h−p兲 = 共2␲2kT / hcl兲D⌫DA2 共s−1兲
a = 0.002 733x2 + 0.019 92x + 0.5431 共nm兲
UA = 0.7 共eV兲
共1 / ␶a兲 = x共1 − x兲共3a3␲4U2A / 32h兲共DX / 6兲 共s−1兲
共1 / ␶a兲 = x共1 − x兲共3a3␲4U2A / 32h兲D⌫ 共s−1兲
a
Reference 28.
Reference 30.
c
Reference 27.
d
Reference 25.
e
Reference 26.
b
␶s−1 =
N
共2␲兲3
冕冕冕
SRkជ kជ ⬘共1 − cos ␪兲dkx⬘dk⬘y dz⬘ ,
共6兲
where ␶s is the momentum relaxation time, N is the number
density of the spherical pores, and ␪ is the angle between the
original and scattered wave vectors. Here we have simply
multiplied the scattering rate by the number density of pores
since it is assumed that each pore affects electron transport
independently and that pores are uniformly distributed. This
assumption is reasonable for the case of a high carrier concentration 共⬎1020 / cm3兲, where the electron screening length
is around 1 nm. With mathematical manipulation, Eq. 共6兲 can
be arranged as following:
␶s−1 =
N␲3U20D共E兲
兵兩2akជ 兩2关CI共兩2akជ 兩兲 − 1兴
2ប兩kជ 兩6
+ 兩2akជ 兩sin共兩2akជ 兩兲 + sin2共兩2akជ 兩兲其,
共7兲
where D共E兲 is the electron density of states and CI is an
integral function defined as
CI共x兲 ⬅
冕
The size and the number density of the pores need to be
evaluated to calculate the scattering rate in Eq. 共7兲. Once the
pore size is known, the number density N can be derived
from the definition of the porosity ⌽, which is the ratio of the
pore volume to the total sample volume
N=
0
1 − cos t
dt.
t
共8兲
The momentum relaxation time calculated from Eq. 共7兲 can
be combined with the other scattering mechanisms using
Mathiessen’s Rule, and the electrical transport properties can
be calculated using Eqs. 共1兲 and 共2兲.
共9兲
where a is the radius of a spherical pore. In the following
paragraph, we will discuss how the pore size is determined.
The pore is regarded as a vacant space among grains. As
the grain size becomes smaller, the size of the pore also
becomes smaller. To derive the relation between the size of
grains and pores, we assume that the grains are closely
packed hard spheres with diameter L as shown in Fig. 3. In
this case, a space exists inside the four spheres that touch
each other. We model the pore site as the sphere that can fit
in this space. The pore radius a is derived from the geometry
of a tetrahedron as
a=
x
3⌽
,
4␲a3
冉冑 冊
3 1
− L.
8 2
共10兲
The actual pore radius can be smaller than Eq. 共9兲 due to
adhesion forces between grains and the nonspherical geometry of grains.
While additional scattering of charge carriers by pore
sites can be determined from Eq. 共7兲, the finite volume of the
pores should not be neglected, especially when the porosity
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094308-4
J. Appl. Phys. 107, 094308 共2010兲
Lee et al.
FIG. 2. 共Color online兲 Schematic of electron scattering by a spherical potential. Pores are assumed to be a spherical site that has scattering potential.
As an electron passes by the spherical potential, it will change its momentum from the original eigenstate kជ to another kជ ⬘.
coefficient S, Eq. 共12兲 is solved by numerical iteration. The
thermal conductivity used in Eq. 共13兲 includes both the electron and phonon contributions. The electron contribution to
the thermal conductivity ke can be evaluated from
ke = −
1
3
冕
共E − EF兲2
⳵ f共EF,E兲
D共E兲dE − S2T␴ .
␶共E兲v共E兲2
T
⳵E
共14兲
FIG. 1. 共Color online兲 The model validity with SiGe for different concentrations of Ge at various temperature ranges. Lines are from the modeling,
and points are from the experimental work reported by Dismukes et al. 共Ref.
31兲. Composition ratio and carrier concentration of circle, square, triangle,
and diamond are Si73Ge27 and 1.5⫻ 1020, Si70Ge30 and 6.7⫻ 1019, Si67Ge33
and 2.3⫻ 1019, and Si74Ge26 and 2.2⫻ 1018, respectively.
becomes large. To account for the finite volume of the pores,
we treat nanoporous materials as media where one phase is
the host material and the other is vacuum. We employ the
effective medium theories to calculate the transport properties of the nanoporous media.13,14 The effective electrical
conductivity and Seebeck coefficient of porous media can be
described as
2 − 3⌽
,
2
共11兲
␥eff␴h
,
␥h␴eff
共12兲
␴eff = ␴h
Seff = Sh
Besides electronic thermal conductivity, phonon contribution
to the heat conduction in thermoelectric materials is significant. The transport analysis will not be complete without
careful consideration of the change in phonon thermal conductivity of porous nanograined structures. In the following
section, the effect of nanosized pores and nanograins on the
phonon thermal conductivity is discussed.
B. Phonon modeling
While the Seebeck coefficient and the electrical conductivity are derived by integrating the energy flux and charge
flux over all energy levels of electrons, the lattice thermal
conductivity is derived by integrating phonon energy flux
over all phonon frequencies. The simplified expression of
lattice thermal conductivity is
k=
1
3
冕
C共␻兲v共␻兲⌳共␻兲d␻ ,
共15兲
where ␻ is the phonon frequency, C is the lattice heat capacity, v is the sound velocity, and ⌳ is the phonon mean-free
path. The phonon mean-free path is given by the group ve-
and ␥ is defined as
␥ ⬅ k + T␴S2 ,
共13兲
where k is the total thermal conductivity. The subscript h
stands for the properties of the host material, which can be
calculated from Eqs. 共1兲 and 共2兲 after incorporating the scattering by the pores. Since ␥ is also dependent on the Seebeck
FIG. 3. 共Color online兲 Schematic of closely packed hard spheres. Pore site
can exist in the middle of four spheres.
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094308-5
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TABLE II. Phonon modeling parameters.
␻N = h␻ / 2␲kTD
TD = 630− 266x 共K兲
M = 28.09共1 − x兲 + 72.61x 共g / mol兲
crv = 兵关12.1共1 − x兲 + 13.6x兴 / NA其1/3
␳ = 2329+ 3493x − 499x2 共kg/ m3兲
v = 共2kTD / h兲共␲ / 6兲1/3共crv / 100兲 共m / s兲
␥ = 0.91
1 / ␶U = 3.264⫻ 10−219␥2␻N2 T / 27 Mcrv2TD 共s−1兲
1 / ␶N = 2 / ␶U 共s−1兲
1 / ␶ PD = 兵共 28.09– 72.61 / M 兲2
+ 39关 共 12.1 / NA 兲1/3 – 共 13.6 / NA 兲1/3 / crv 兴2其 ⫻ 6.17
⫻ 1011TD␻N4 共s−1兲
2
1 / ␶ p−e = 4DA v共␲m兲3 / h4␳ 共 2kT / mv2 兲 ⫻ 兵␻N TD / T − ln1
+ exp关共mv2 − 2EF兲 / 2kT + ␻N2 TD2 / 8mv2kT3 + ␻NTD / 2T兴 / 1
+ exp关共mv2 − 2EF兲 / 2kT + ␻N2 TD2 / 8mv2kT3 − ␻NTD / T兴 其 共s−1兲
1 / ␶B = v / grain
1 / ␶C = 1 / ␶U + 1 / ␶N + 1 / ␶ PD + 1 / ␶ p−e + 1 / ␶B
兰10␶C␻N2 共共␻NTD / T兲2e␻NTD/T / 共e␻NTD/T − 1兲2兲d␻N
1
2兰0共␶C / ␶U兲␻N2 共共␻NTD / T兲2e␻NTD/T / 共e␻NTD/T − 1兲2兲d␻N
2兰10共1 / ␶U兲共1 − 共2␶C / ␶U兲兲␻N2 共共␻NTD / T兲2e␻NTD/T / 共e␻NTD/T
− 1兲2兲d␻N
−2
4.67⫻ 10 共I1 + I22 / I3兲TD2 / crv
Normalized phonon frequency
Debye temperature of Si1−xGex a
Atomic mass of Si1−xGex a
Cube root volume of Si1−xGex a
Density of Si1−xGex a
Sound velocitya
Anharmonicity parameterb
3 phonon Umklapp scattering rateb
3 phonon Normal scattering rateb
Point defect 共alloy兲 scattering rateb
Phonon-electron scattering rateb
Boundary scattering ratec
Combined scattering rate
I1 b
I2 b
I3 b
Lattice thermal conductivityb
a
Reference 28.
Reference 33.
c
Reference 6.
b
locity of acoustic phonon, i.e., sound velocity, divided by
phonon scattering rate. As in the electron transport properties
calculation, we consider the various scattering mechanisms
to determine the phonon scattering rate. Three-phonon scattering, point defect scattering, and phonon-electron scattering are the major scattering mechanisms for phonons in the
bulk SiGe alloy. The functional forms of each scattering rate
and the required constants are listed in Table II. Each scattering rates calculated by the different scattering mechanisms
is superposed using Mathiessen’s Rule. In addition to these
scattering mechanisms in bulk SiGe, phonons will be scattered diffusively at grain boundaries. When the grain size is
smaller than the bulk phonon mean-free path, the grain
boundary scattering becomes dominant. In this case, the effective phonon mean-free path will be limited by the small
grain size L and given by6
1
⌳effective
=
1
⌳bulk
+
1
.
L
共16兲
The actual functional forms for the lattice thermal conductivity are slightly more complicated than Eq. 共15兲 and are
taken from studies in the 1960s 共Table II兲 共Refs. 32 and 33兲
For the lattice thermal conductivity in porous structures,
we use the modified formulation of the effective medium
theory.21 A simplified expression for the lattice thermal conductivity of porous media is described as follows:
keff = kh
2 − 2⌽
,
2+⌽
共17兲
where keff is the effective thermal conductivity, kh is the thermal conductivity of the host, and ⌽ is the porosity. The effect of small grain size is taken into consideration for the
host thermal conductivity calculation, using Eqs. 共15兲 and
共16兲. By this way, the modified effective medium theory considers both the classical size effect and the porosity effect on
the lattice thermal conductivity.
III. RESULTS AND DISCUSSIONS
Figure 4 shows our modeling results fitted on the experimental data reported by Lidorenko.12 Due to lack of information on grain size and carrier concentration, we take them
as fitting parameters. We use a grain size of 10 ␮m and
concentration of 1020 / cm3, to match the trends of both the
thermal conductivity and the electrical conductivity. For the
10 ␮m pores, electrons are not significantly scattered by the
pores due to the large separation between pores. Since the
screening length of charge carriers in the solid portion is 1
nm order of magnitude, the sparse pore sites will not affect
the charge carrier transport. Therefore, the reduction in the
electrical conductivity is mainly attributed to the lack of medium. However, when the grain size is on the nanometer
scale, the interpore spacing becomes shorter, and electron
transport is severely limited by the scattering from the pores.
Figure 5 shows the energy dependency of charge carrier
scattering by the spherical potential model for 20 nm grain
size with 1% porosity. The energy dependent scattering rate
provides the logical bases for low electrical conductivity and
enhanced Seebeck coefficient. At this pore size and porosity,
the pore scattering dominates the total momentum relaxation
time. Without the scattering by the pores, the momentum
relaxation time is an order of magnitude higher, which indicates that the electrical conductivity decreases significantly
by the nanosized pores. However, the pore scattering can
provide us with an advantage of the enhanced Seebeck coefficient, which results from the more efficient scattering of
low-energy electrons than high-energy electrons. Conse-
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094308-6
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J. Appl. Phys. 107, 094308 共2010兲
FIG. 6. 共Color online兲 Transport properties of SiGe as a function of porosity
for different grain sizes at 1300 K: 共a兲 electrical conductivity, 共b兲 Seebeck
coefficient, 共c兲 thermal conductivity, and 共d兲 ZT. The properties are normalized to the uniform bulk values. Same symbols as in 共a兲 apply for all figures.
FIG. 4. 共Color online兲 Comparison of our model with Lidorenko’s experimental results, grain size of 10 ␮m and concentration of 1020/ cm3 are used
to fit the results.
quently, the Seebeck coefficient, which measures an average
energy of electrons that contribute to electrical conductivity,
is enhanced by reducing the contributions from low-energy
electrons.
Figures 6 and 7 show the thermoelectric properties for
the porosity range between 0% and 15% and for grain sizes
down to 20 nm. The 20 nm is the grain size that we observed
for SiGe nanograined samples with a high ZT.22,23 In the
calculations, we use a temperature of 1300 K and a doping
concentration of 1.5⫻ 1020 / cm3 that is typical for high temperature SiGe thermoelectric power generation. As demon-
FIG. 5. 共Color online兲 Relaxation times due to various scattering
mechanism.
strated in Figs. 6共b兲 and 7共b兲, the enhancement in the Seebeck coefficient due to the nanoscale pores becomes larger
for smaller grain size. However, the momentum relaxation
time of the porous materials is much smaller than that of
materials with high sample density 共Fig. 5兲, so that the electrical conductivity of the nanoscale porous media is severely
degraded compared to their bulk counterparts 关Figs. 6共a兲 and
7共a兲兴. Figure 7共a兲 also shows that the electrical conductivity
is degraded more as the grain size decreases with the porosity kept constant. While the scattering rate is roughly proportional to the square of the pore size as shown in Eq. 共7兲, the
FIG. 7. 共Color online兲 Transport properties of SiGe as a function of the
grain sizes for different porosity values: 共a兲 electrical conductivity, 共b兲 Seebeck coefficient, 共c兲 thermal conductivity, and 共d兲 ZT. The properties are
normalized to the uniform bulk values. Same symbols as in 共a兲 apply for all
figures.
Downloaded 20 Jun 2010 to 136.167.54.101. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp
094308-7
J. Appl. Phys. 107, 094308 共2010兲
Lee et al.
number density is inversely proportional to the third power
of the pore size, so that the scattering rate is higher for
smaller grain size. When the grain size gets as small as 20
nm and the porosity gets larger than 10%, the strong scattering due to porosity is not even favorable to the Seebeck
coefficient 关Fig. 6共b兲兴.
The thermal conductivity also decreases as the grain size
is reduced 关Fig. 7共c兲兴. In the limit of no porosity, this trend
can be explained by the classical size effect that reduces the
phonon contribution to the thermal conductivity. When the
porosity is not negligible, the electron contribution to the
thermal conductivity is also reduced by strong scattering at
the nanoscale pore sites. Although the electron contribution
is often considered to be small in SiGe, it becomes comparable to that of phonons in nanograined SiGe due to significant reduction in the phonon contribution. As a consequence,
the total thermal conductivity reduction is larger for smaller
grain size and higher porosity 关Figs. 6共c兲 and 7共c兲兴.
In spite of the reduction in the thermal conductivity and
the enhancement in the Seebeck coefficient, the porous nanograined SiGe cannot lead to a high ZT because the electrical
conductivity drops by orders of magnitude. There exist regions of porosity and grain size that can have a small enhancement in ZT 关Fig. 7共d兲兴. However, the enhancement in
these regions is much less than the enhancement we can
obtain for nanograined samples with no porosity and the
amount of enhancement is not significant. Moreover, it is
hard to control porosity and grain size in experiment. Therefore, the challenge is not trivial if we want to increase ZT by
introducing porosity in nanograined SiGe.
IV. CONCLUSIONS
We have explored the effect of nanosized porosity on
thermoelectric transport properties. An electron scattering
model was developed to calculate the momentum relaxation
time caused by the scattering potential at pore sites. The
model is derived in a generic form and can be considered
together with other scattering mechanisms by Mathiessen’s
rule. Our modeling work shows good agreement with the
previously reported experimental studies in the macroscale
porosity effect on the thermoelectric properties. The model is
applied to nanograined SiGe, and explains the significant reduction in the electrical conductivity when the number density of pore sites is large. Although nanosized pores are efficient in reducing the thermal conductivity and increasing the
Seebeck coefficient, the electrical conductivity drops so significantly that the porosity degrades the thermoelectric figure
of merit in nanograined SiGe. Hence, a high sample density
is required to achieve a net ZT enhancement in nanograined
SiGe. However, an enhancement in ZT by the porosity effect
may be observed in other material systems. For example,
materials with small electron effective mass and a low relaxation time of charge carriers would exhibit a smaller reduction in the electrical conductivity than for SiGe. The present
model can be used to find such materials and calls for further
studies on such potential candidate materials.
ACKNOWLEDGMENTS
The authors would like to appreciate the funding from
NASA under grant No. NNC06GA48G and NSF under grant
No. CMMI 0833084.
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