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Three-phonon phase space and lattice thermal conductivity in semiconductors
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2008 J. Phys.: Condens. Matter 20 165209
(http://iopscience.iop.org/0953-8984/20/16/165209)
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IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 20 (2008) 165209 (6pp) doi:10.1088/0953-8984/20/16/165209
Three-phonon phase space and lattice
thermal conductivity in semiconductors
L Lindsay and D A Broido
Department of Physics, Boston College, Chestnut Hill, MA 02467, USA
Received 25 January 2008, in final form 10 March 2008
Published 31 March 2008
Online at stacks.iop.org/JPhysCM/20/165209
Abstract
We present calculations of the phase space for three-phonon scattering events in several group
IV, III–V and II–VI semiconductors employing an adiabatic bond charge model to accurately
represent the phonon dispersions. We demonstrate that this phase space varies inversely with
the measured lattice thermal conductivities of these materials over a wide range of temperatures
where three-phonon scattering is the dominant mechanism for scattering phonons. We find that
this qualitative relationship is robust in spite of variations in material parameters between the
semiconductors. Anomalous behavior occurs in three III–V materials that have large mass
differences between constituent elements, which we explain in terms of the severely restricted
three-phonon phase space arising from the large gap between acoustic and optic phonon
branches.
1. Introduction
A material’s lattice thermal conductivity is an essential com-
ponent in determining its potential utility for thermal manage-
ment applications. Among these are the development of ma-
terials that facilitate heat dissipation in microelectronics and
nanoelectronics [1] and that provide efficient thermoelectric re-
frigeration and power generation [2].
Above a few tens of degrees kelvin, the lattice thermal
conductivity of high-quality crystalline semiconductors is
determined primarily by phonon–phonon scattering [3, 4]
which arises when atoms in the lattice deviate sufficiently from
their equilibrium positions that they sample the anharmonicity
of the interatomic potential. Other extrinsic scattering
mechanisms, such as due to isotopic impurities and boundaries,
are significantly less important in this temperature range [4].
The lowest order anharmonic terms correspond to three-
phonon scattering processes. Around room temperature,
these processes are stronger than the higher order anharmonic
scattering processes [5] and thus dominate the behavior of the
lattice thermal conductivity.
It has been noted some time ago that, in the temperature
range where three-phonon scattering is dominant, an inverse
relationship should exist between the phase space for
three-phonon scattering (defined below) and the intrinsic
lattice thermal conductivity of a material [3]. While
other material parameters such as averaged acoustic phonon
velocity, Gru¨neisen constant, heat capacity and density, are
often used in simple-model estimates of the lattice thermal
conductivity [3, 4], to date, no calculations have been
performed that compare a material’s three-phonon phase space
to measured lattice thermal conductivities. In this paper, we
perform such calculations. Specifically, we calculate the three-
phonon phase space for several group IV, III–V and II–VI
semiconductors.
The three-phonon phase space is determined entirely from
a material’s phonon dispersion, and these dispersions have
been precisely measured for many materials. We calculate
these phonon dispersions accurately using an adiabatic bond
charge model [6, 7]. We show that the three-phonon phase
space indeed varies inversely with the measured thermal
conductivities of the studied materials over a wide range
of temperatures. Furthermore, we find that this qualitative
relationship is maintained in spite of the dependence of the
thermal conductivity on other material parameters. Finally,
we find that several III–V materials having large mass ratios
of the constituent atoms possess anomalously small phase
spaces for three-phonon scattering. This results from the large
frequency gaps between acoustic and optic phonon branches,
which freezes out three-phonon scattering processes involving
acoustic and optic phonons. We connect this anomalously
small phase space to a corresponding enhancement in the
thermal conductivity, behavior that has been noted qualitatively
many years ago [8].
Section 2 presents the theory for calculating the three-
phonon phase space. In section 3, the calculated three-
phonon phase spaces for different materials are compared
to corresponding measured thermal conductivities and an
0953-8984/08/165209+06$30.00 © 2008 IOP Publishing Ltd Printed in the UK1
J. Phys.: Condens. Matter 20 (2008) 165209 L Lindsay and D A Broido
Table 1. Shown here are the semiconductors examined in this paper
as well as their corresponding lattice thermal conductivities for
temperatures T = 300 K and 150 K, mass ratios, and gaps between
acoustic and optic branches as determined by the adiabatic bond
charge model [6, 7]. The total phase space for three-phonon
scattering, P3, is also given here for comparison.
κexp,T =300 K
(W cm−1 K−1)
κexp,T =150 K
(W cm−1 K−1) m>/m<
Gap
(THz) P3 (10−2)
Si 1.53a 4.78a 1 0 0.3536
Ge 0.70b 1.71b 1 0 0.5795
C 32c 149c 1 0 0.0796
GaAs 0.45d 1.06d 1.07 0.04 0.6015
GaSb 0.36d 0.78d 1.75 0.59 0.6605
GaP 1.10e 2.13e 2.25 3.14 0.3360
InP 0.68f 1.90f 3.71 3.80 0.3822
InSb 0.166d 0.40d 1.06 0.07 0.7230
AlSb 0.60e 1.05e 4.51 4.30 0.2641
CdTe 0.075g 0.184g 1.14 0.31 0.9424
ZnTe 0.18h 0.45h 1.95 0.38 0.8528
a Reference [10]. b Reference [11].
c Reference [12]. d Reference [13].
e Reference [14]. f Reference [15].
g Reference [16]. h Reference [17].
understanding of the observed behavior is developed. Section 4
provides a summary and conclusions.
2. Theory
We consider scattering between three phonons in modes,
( j, q), ( j ′, q′) and ( j ′′, q′′). Here, q, q′ and q′′ are the
momenta of the three phonons and j , j ′ and j ′′ designate the
phonon branches. The three-phonon scattering processes are
constrained to satisfy energy and momentum conservation:
ω j (q) ± ω j ′(q′) = ω j ′′(q′′), q ± q′ = q′′ + G. (1)
Here ω j (q) is the phonon frequency of mode ( j, q), G is a
reciprocal lattice vector that is zero for normal processes and
non-zero for umklapp processes, and the ± signs correspond
to the two types of possible three-phonon processes [3, 4].
Equation (1) impose severe constraints on the phase space
available for three-phonon scattering.
We consider a reciprocal lattice with n j phonon branches
whose Brillouin zone has volume, VBZ. We define the total
phase space available for three-phonon processes, normal and
umklapp, as the sum over all possible modes for three phonons
subject to the constraints of equation (1):
P3 = 23�
(
P(+)3 +
1
2
P(−)3
)
(2)
where
P(±)3 =
∑
j
∫
dqD(±)j (q) (3)
and
D(±)j (q) =
∑
j ′, j ′′
∫
dq′δ(ω j (q) ± ω j ′(q′) − ω j ′′(q ± q′ − G))
(4)
Figure 1. Phonon dispersion of GaAs. Solid lines are calculated
results using the abcm. Open squares are given by neutron scattering
data from [18].
where the integration in equations (3) and (4) is taken over
the first Brillouin zone. The quantity, � = n3j V 2BZ, is a
normalization factor equal to the unrestricted phase space for
each type of process. This choice along with the additional 2/3
factor gives P3 = 1 when the energy-conserving delta function
in equation (4) is removed. The factor of 1/2 in equation (2)
arises to prevent double-counting of scattering events [3, 4].
We note that using the symmetry of the delta function, it is
straightforward to show that P(+)3 = P(−)3 , so only one of
these need be calculated explicitly and, from equation (2),
P3 = P(+)3 /�. In equations (3) and (4), D(±)j (q) is the two-
phonon density of states [9] and momentum conservation has
already been imposed on q′′. For umklapp processes the G
vectors that enter equation (4) are the eight shortest and six
second-shortest vectors for the truncated octahedron Brillouin
zone corresponding to the real space FCC lattice.
We focused our study on three-phonon scattering in group
IV, III–V and II–VI bulk semiconductors with diamond or
zincblende structure and in a temperature range between
150 K and room temperature, T = 300 K, where three-
phonon scattering dominates the thermal resistivity. A list
of the materials examined here with corresponding properties
relevant to later discussion is given in table 1. To accurately
describe the phonon frequencies that enter the calculation
of the allowed three-phonon scattering phase space in each
material, we employed an adiabatic bond charge model (abcm),
introduced by Weber [6, 7]. This model accurately reproduces
experimental phonon dispersion data including anisotropies
and the characteristic flattening of the transverse acoustic
branches near the Brillouin zone boundary observed in many
semiconductors. An example of this is provided for GaAs in
figure 1. For the II–VI materials, we used a modification of the
abcm as described in [19].
To obtain the best set of abcm parameters, we
implemented a χ2 minimization procedure for each material,
with χ2 defined as [20]:
χ2 =
∑
j ,q
(ωabcmj (q) − ωexpj (q))2. (5)
2
J. Phys.: Condens. Matter 20 (2008) 165209 L Lindsay and D A Broido
Table 2. Best fit parameters to experimental dispersion data for the
adiabatic bond charge model. These choices for parameters were
guided solely by their fit to the available data so that the calculated
P3 in each material is as accurate as possible.
1
3 φ
′′
i−i
1
3 φ
′′
i1−bc
1
3 φ
′′
i2−bc β1 β2 Z
2/ε
Si 6.21 6.47 6.47 8.60 8.60 0.180
Ge 6.25 9.57 9.57 7.88 7.88 0.279
Ca −10.1 51.0 51.0 12.6 12.6 0.885
GaAsa 5.24 5.85 32.5 2.48 10.4 0.392
GaSba 6.95 2.60 16.2 4.34 7.57 0.192
GaPa 3.76 8.67 65.5 −2.02 33.7 0.589
InPa 1.32 19.6 102 0.409 13.3 1.06
InSb 7.32 2.15 14.7 3.59 10.3 0.173
AlSb 5.13 7.92 37.5 −0.756 28.0 0.462
CdTeb 6.85 0.77 23.3 0.39 15.4 0.183
ZnTeb 5.51 1.06 22.9 1.07 17.0 0.180
a Additional parameters, a′ and 12 (ψ
′′
1 − ψ ′′2 ), defined in [6]
and [7], were used for these materials in order to maximize the fit
to experimental data: a′ = (0.49, 0.89, 2.49) in (C, GaP, InP);
1
2 (ψ
′′
1 − ψ ′′2 ) = (3.52, 1.08, 1.62, 3.35) in (GaAs, InP, GaSb,
InSb), respectively.
b Parameters from [19].
Here, ωabcmj (q) and ω
exp
j (q) are the calculated and measured
phonon frequencies, and the sum is over the set of ( j, q)
obtained from available experimental data. The parameter sets
providing the minimum χ2 for each material are shown in
table 2.
The phase space calculation was performed numerically
using two separate methods, direct integration and the
tetrahedron method [21]. The first involved direct integration
over q and q′ in equations (3), and (4). For each q and
set of branch indices, ( j, j ′, j ′′), the conservation conditions,
equation (1), represent four constraint equations in a 6-
dimensional (q′, q′′) momentum space. The remaining two
degrees of freedom define a two-dimensional surface in q′ for
each type of process (+ or −). The direct integration method
utilized a root-finding scheme to approximate these surfaces
for each, ( j, j ′, j ′′, q). In the tetrahedron method [21], which
is often used in density of states calculations, equation (4) is
evaluated by dividing the Brillouin zone into tetrahedra that fill
the entire reciprocal space. Linearly interpolated expressions
for the frequencies within each tetrahedron are determined
from the exact frequencies calculated at the vertices using the
abcm. Both methods gave results for the total three-phonon
scattering phase space within 4% of each other for every
material, though the tetrahedron method converged with much
less computational effort.
3. Results and discussion
Figure 2 shows the calculated phase space for three-phonon
scattering, P3, plotted against the measured room temperature
(300 K) lattice thermal conductivity for each group IV, III–
V and II–VI semiconductor listed in table 1. It is evident
from this figure that P3 varies inversely with the lattice thermal
conductivity. This makes physical sense: as P3 decreases
from material to material, the reduced phonon scattering
corresponds to an increase in the thermal conductivity.
Figure 2. Three-phonon scattering phase space versus measured
lattice thermal conductivity at T = 300 K (references in table 1) for
a host of semiconductors. Triangles correspond to hypothetical
values for anomalous materials, InP, AlSb, and GaP.
Figure 3. Three-phonon scattering phase space versus measured
lattice thermal conductivity for the same materials as in figure 2
(references in table 1) but with T = 150 K. Note the different scale
along the horizontal axis as compared to figure 2.
Materials such as cadmium telluride with a relatively large
P3 have a low lattice thermal conductivity while silicon and
diamond with small P3 have correspondingly high lattice
thermal conductivities.
Figure 3 shows the three-phonon phase space versus
measured lattice thermal conductivity values for 150 K. It
is evident that an inverse relationship between three-phonon
phase space and thermal conductivity also occurs for the lower
temperature case, but with all points shifted to the right (to
higher thermal conductivity). We have investigated results for
other temperatures between 150 and 300 K and find the same
qualitative behavior, verifying that the inverse relationship is
robust to changes in temperature when three-phonon scattering
dominates the thermal conductivity. We would expect the
inverse relationship to hold for temperatures higher than 300 K
as well, but high temperature thermal conductivity data is not
available for the full set of materials we have considered here.
3
J. Phys.: Condens. Matter 20 (2008) 165209 L Lindsay and D A Broido
Figure 4. Allowed three-phonon phase space versus frequency scale
given by highest calculated optical frequency in a host of
semiconductors.
We note that diamond has been left off of figures 2
and 3 because of its exceedingly large thermal conductivity
values. Also, unlike all other materials in table 1, diamond
has appreciable isotopic impurity scattering in the 150–300 K
temperature range; its thermal conductivity is not dominated
by phonon–phonon scattering until higher temperature. We
also note that the qualitative behavior shown in figures 2 and 3
is also obtained when the phase space due to only umklapp
scattering processes is plotted instead of the total three-phonon
phase space, P3. We find that the phase space for umklapp
scattering is roughly 55–60% of P3 in each semiconductor.
It is interesting that the inverse relationship exhibited
in figures 2 and 3 occurs in spite of variations in material
parameters that are often correlated with the lattice thermal
conductivity. For example, the variations in the strength of the
anharmonicity, sometimes represented through the Gru¨neisen
constant, are not sufficient to destroy the qualitative trend seen
in figures 2 and 3. Only knowledge of the phonon dispersion is
required.
The three-phonon phase space depends on the overall
frequency scale of the phonon dispersions. To see this, we
divide the scattering phase space into four groups of processes
according to the number of acoustic (a) and optic (o) phonons
involved: (a, a, a), (a, a, o), (a, o, o) and (o, o, o). The
(o, o, o) processes are forbidden by the energy conservation
condition, while the (a, o, o) processes are severely limited
by equation (1) due to the restriction of the momentum
space of the acoustic phonon to a small region near the
center of the Brillouin zone. The dominant scattering type
which contributes most to the three-phonon phase space in
the majority of materials is the (a, a, o) group. The (a, a, a)
processes also contribute significantly to the phase space, but
to a lesser extent than the (a, a, o) processes.
The effect of the frequency scale on the phase space can be
seen by examining analytically the three-phonon phase space
from the dominant (a, a, o) processes within a Debye–Einstein
model. In this model, the acoustic phonons have an isotropic,
linear dispersion, ωac(q) = qvD, characterized by a constant
Figure 5. Phonon dispersion curves for AlSb and GaAs illustrating
the gap between optic and acoustic modes due to a large mass ratio;
neutron scattering data from [18] and [22].
velocity, vD, and Debye cutoff wavevector of magnitude qD,
while the optical phonons have constant frequency, ωopt(q) =
ωE. The allowed phase space in the Debye–Einstein model
for (a, a, o) scattering, obtained by integrating analytically
equations (3), and (4) over the Debye sphere of radius qD with
j = j ′ = a and j ′′ = o, is:
P(+)3 (a, a, o) =
9
ωE
[
2ωE
5qDvD
− ω
2
E
q2Dv2D
+ 2ω
3
E
3q3Dv3D
− ω
6
E
30q6Dv6D
]
.
(6)
When the dispersion is scaled by a factor β , ωE → βωE
and vD → βvD, the phase space is inversely proportional to
this scaling factor: P(+)3 (a, a, o) → P(+)3 (a, a, o)/β . We
find similar scaling behavior occurs for the actual dispersions
obtained in the adiabatic bond charge model, which is
illustrated in figure 4. As the overall frequency scale,
determined by the highest optical frequency, increases from
material to material the three-phonon phase space decreases.
The same behavior occurs when the scaling is determined by
the zone-center acoustic velocities.
For the three III–V materials, AlSb, InP, and GaP, P3
falls below the general trends in figures 2–4. This anomalous
behavior results from the large gap between their acoustic
and optic branches, which severely restricts the normally
dominant (a, a, o) scattering processes. Figure 5 contrasts
the phonon dispersions for the small-gap GaAs and the large-
gap AlSb. The large-gap results in part from the substantial
mass mismatch between anion and cation, as can be seen from
table 1. The phase space reduction can be seen in the two-
phonon density of states, D(±)j (q), given in equation (4). The
two-phonon density of states, D(+)LA (q), in which a longitudinal
acoustic (LA) phonon combines with another acoustic phonon
to produce an optic phonon, is shown in figure 6 for two
large-gap materials, InP and AlSb, and a small-gap material,
GaAs. In GaAs, LA phonons throughout most of the frequency
range can contribute to D(+)LA (q). Conversely, in InP and
AlSb, only the high-frequency LA phonons have enough
energy to combine with another acoustic phonon to create an
4
J. Phys.: Condens. Matter 20 (2008) 165209 L Lindsay and D A Broido
Figure 6. Two-phonon density of states for an LA phonon in the
(q, 0, 0) direction for GaAs, InP, and AlSb for which the scattering
involves the combination of two acoustic phonons into a higher
energy optical phonon. The horizontal axis is the LA phonon
frequency scaled by the maximum frequency.
optic phonon, thus severely restricting this scattering channel
and reducing the allowed three-phonon phase space. This
restriction is reflected in the ratio of (a, a, o) processes to
(a, a, a) processes, which in AlSb, InP, GaP and GaAs is 0.03,
0.67, 0.57 and 3.16 respectively. For comparison, values for the
energy gap and three-phonon phase space are listed in table 1.
We now consider hypothetically the effect of reducing
the energy gap in InP, GaP and AlSb. In this case, the
(a, a, o) processes would increase, driving P3 up, and at the
same time this increased scattering would decrease the thermal
conductivity. We indicate this hypothetical behavior with t