Supplementary information for
Anisotropic Thermal Conductivity in Single Crystal β-Gallium Oxide
By Zhi Guo et al.
X-ray diffraction graphs
Figure S1. On axis Omega-2theta scans of the four single crystal samples used in this study.
Calculation of Grüneisen parameter and scattering phase space
The Grüneisen parameter in the third-order phonon-phonon interactions is an important parameter
for evaluating the anharmonicity. The Grüneisen parameter for each phonon mode is defined as
k  
V  k 
 k  V
(s1)
This is approximated by
k
 k 
 k 
 
V
(s2)
V
The overall Grüneisen parameter is then given by1
 
k  k C V k 
k C V k 
(s3)


 k   
1


V
T  k Tk 
e B  1 
(s4)
and
CV (k ) 
The first-principles calculation protocol is similar to that described in the main text. The
Grüneisen parameters were calculated using the dispersion relation obtained from two lattice
constants: an equilibrium structure and one with slightly expanded lattice volume by 1%.
According to equation (s2), the relative frequency shifts due to the lattice expansion were divided
by the volumetric expansion ratio to obtain the Grüneisen parameter for each mode. To test the
validity of the calculated Grüneisen parameters, we first compared our results with experimental
values2-7 and previous computational studies2, 8-15 (Figure S2). It is seen that they compare
favorably to one another.
Figure S2. Comparison of experimental/calculated Raman frequencies (a,c) and Grüneisen
parameters (b,d) of Raman active modes in Ga2O3 and GaN.
Based on equations (s3) and (s4), the overall Gruneisen parameter obtained for Ga2O3 and GaN
are 0.4661 and 0.7517, respectively, meaning Ga2O3 is even less anharmonic than GaN. However,
when excluding optical phonons, which usually do not contribute much to the overall thermal
conductivity, the Gruneisen parameters are (-1.0577[TA1], -0.303[TA2], 0.827 [LA]) and (0.86556[TA1], -0.86556[TA2], 0.51102 [LA]) for Ga2O3 and GaN, respectively. It can be found
that, in Ga2O3, the Grüneisen parameters of three acoustic branches are comparable or slightly
larger than those of GaN, which implies a comparable or slightly larger anharmonicity.
In addition to the phonon anharmonicity, the phonon lifetimes also depend on the size of the
three-phonon scattering phase space, which measures how many three-phonon groups can satisfy
the energy and momentum conservation laws. The scattering phase space volume has been shown
to be strongly correlated to thermal conductivity.16 We calculated the scattering phase space on a
16×16×16 k-mesh by searching three phonon groups that can satisfy the scattering rules. More
details on the scattering phase space volume calculation can be found in reference. 17 Figure S3
presents the calculated scattering phase space volume from different scattering channels in GaN
and Ga2O3.
Figure S3. Contribution of different scattering channels to the total scattering phase-space volume
in GaN and Ga2O3. Symbols a and o represent acoustic and optical phonons, respectively. The
scattering phase space and phonon dispersion data are normalized by the inverse of the largest
optical phonon frequency of each material for a fair comparison. 17
It is notable that the scattering phase space volume of Ga2O3 is about 30 times larger than that of
GaN. In other words, Ga2O3 will have much higher possibility of three-phonon scatterings. This
can lead to shorter phonon relaxation times and lower thermal conductivity of Ga2O3, compared
to GaN.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
B.-L. Huang and M. Kaviany, Phys. Rev. B 77 (12), 125209 (2008).
D. Machon, P. F. McMillan, B. Xu and J. Dong, Phys. Rev. B 73 (9), 094125 (2006).
A. Cingolani, M. Ferrara, M. Lugará and G. Scamarcio, Solid State Commun. 58 (11), 823
(1986).
P. Perlin, C. Jauberthie-Carillon, J. P. Itie, A. San Miguel, I. Grzegory and A. Polian, Phys.
Rev. B 45 (1), 83 (1992).
P. Perlin, A. Polian, J. P. Itie, I. Grzegory, E. Litwin-Staszewska and T. Suski, Physica B:
Condensed Matter 185 (1–4), 426 (1993).
J. Nakahara, T. Kuroda, H. Amano, I. Akasaki, S. Minomura and I. Grzegory, (Izunagaoka,
Japan, 1990).
L. Filippidis, H. Siegle, A. Hoffmann, C. Thomsen, K. Karch and F. Bechstedt, physica
status solidi (b) 198 (2), 621 (1996).
K. Miwa and A. Fukumoto, Phys. Rev. B 48 (11), 7897 (1993).
K. Karch, J. M. Wagner and F. Bechstedt, Phys. Rev. B 57 (12), 7043 (1998).
I. Gorczyca, N. E. Christensen, E. L. Peltzer y Blancá and C. O. Rodriguez, Phys. Rev. B 51
(17), 11936 (1995).
C. Bungaro, K. Rapcewicz and J. Bernholc, Phys. Rev. B 61 (10), 6720 (2000).
K. Shimada and T. Sota, J. Appl. Phys. 84 (9), 4951 (1998).
J. M. Wagner and F. Bechstedt, Phys. Rev. B 62 (7), 4526 (2000).
S. Saib, N. Bouarissa, P. Rodríguez-Hernández and A. Muñoz, Semiconductor Science and
Technology 24 (2), 025007 (2009).
15.
16.
17.
Z. Usman, C. Cao, W. S. Khan, T. Mahmood, S. Hussain and G. Nabi, The Journal of
Physical Chemistry A 115 (50), 14502 (2011).
L. Lindsay and D. A. Broido, J. Phys.: Condens. Matter 20 (16), 165209 (2008).
S. Lee, K. Esfarjani, T. Luo, J. Zhou, Z. Tian and G. Chen, Nature communications 5 (2014).