Supplementary Material for
Nontrivial contribution of Fröhlich electron-phonon interaction to
lattice thermal conductivity of wurtzite GaN
Jia-Yue Yang,1,† Guangzhao Qin,1,† and Ming Hu1,2,
1
Institute of Mineral Engineering, Division of Material Science and Engineering, Faculty of
Georesources and Materials Engineering, RWTH Aachen University, 52064 Aachen, Germany
2
Aachen Institute of Advanced Study in Computational Engineering Science (AICES), RWTH Aachen
University, 52062 Aachen, Germany
†

J. Y. Yang and G. Z. Qin contribute equally to this work.
Author to whom all correspondence should be addressed. E-Mail: [email protected]
1
S1. Fröhlich Electron-Phonon Interaction
Based on density functional perturbation theory, the fluctuation of ions will generate
deformed potential and thus induce the coupling of electrons and phonons. The electron-phonon
matrix is defined as [1,2]
g mn , (k , q)   mk q  qV  nk ,
(S.1)
where əV is the deformed potential induced by lattice vibration and ψnk is the electronic
wavefunction for energy band n, wavevector k. According to Fröhlich model, the electronphonon interaction matrix in polar materials takes the form [2],
i
gq 
q
12
 e 2 LO 1 1 
(  ) ,

 2 N  0    0 
(S.2)
where N is the number of unit cell in the Born-von Kármán supercell, Ω is the unit cell volume,
and ωLO is the frequency of longitudinal optical (LO) phonons at long wavelength. The
parameter e is the electron charge, ħ is the reduced Planck constant, ε0 and ε∞ is the highfrequency and static dielectric permittivity, respectively. Due to the dispersionless LO phonons,
the Fröhlich electron-phonon coupling matrix would diverge at long wavelengths as q → 0.
To eliminate the polar singularity, Verdi and Giustino proposed one method to split the
electron-phonon matrix into the short-range and long-range part, as given by [2,3]
S
L
gmn, (k, q)  gmn
, (k, q)  gmn , (k, q).
(S.3)
The great advantage of this strategy lies in that gL deals with the long-wavelength divergence
issue and gS can be treated by the regular Wannier-Fourier interpolation due to the intrinsically
localized nature of maximally localized Wannier Functions (MLWF)[4]. Besides, another key
part is to use the Born effective charge and high-frequency permittivity tensor to calculate the
singular Fröhlich coupling in gL, which is explicitly expressed by [2,3]
2
12






  2 N q M  q 
(G  q)  Z  e (q)


G  q (G  q )  ε  (G  q )
e2
L
g mn
(
k
,
q
)

i
,
 0
(S.4)
  mk q e(G q )r  nk .
The parameter G is a reciprocal lattice vector, Nq is the number of q-points in the Brillouin zone,
and eκ is the normalized vibrational eigenmode for each atom κ in the unit cell. The important
physical quantities Z* and ε∞ is the Born effective charge and high-frequency dielectric
permittivity tensor, respectively.
After splitting the electron-phonon interaction matrix into the short-range and long-range
part, one can effectively avoid the singular Fröhlich coupling by taking the following strategy: (1)
compute the full electron-phonon matrix g at coarse k- and q-mesh based on the density
functional perturbation theory; (2) calculate the long-range component gL based on Eq. (S4) and
subtract it from the full matrix g to obtain the short-range component gS; (3) interpolate the shortrange component gS at arbitrary dense k- and q-mesh by the MLWF; (4) add the long-range
component gL back to the interpolated short-range component gS at arbitrary dense k- and q-mesh.
Such strategy has been successfully implemented in the EPW program[1,3].
To verify the above strategy in the case of wurtzite GaN, the spatial decay of some
important physical quantities in the MLWF representation, such as electronic Hamiltonian Hel,
Dipole (P), dynamical matrix (Dph) and electron-phonon matrix g, are computed. In the MLWF
representation, the above physical quantities are expressed as [1,5]
H Rele ,Re   wk eik ( Re Re )U k† H kelU k ,
(S.5)
k
P  eZ  u,
3
(S.6)
DRphp ,Rp   wq e
 iq( Rp  R p )
eq Dqpheq† ,
(S.7)
q
g (R e , R p ) 
1
Np
w w e
k
 i ( k  R e  q R p )
U k†q g (k , q)U k uq1.
q
(S.8)
k ,q
The array U is mainly used to transform the Bloch functions into MLWFs and R is the real-space
position in the unit cell. The above parameter w is the weight, eq is the orthonormal eigenvectors
of the dynamical matrix, u is the displacement of atoms in the unit cell, and uq is the phonon
eigenvectors. The derivations and detailed interpretations of Eqs. S5-S8 can be found in Ref. [5].
In the case of wurtzite GaN, we computed the electronic states within a plane-wave basis
with a kinetic energy cutoff of 160 Ry. The coarse k- and q-mesh was chosen as 12×12×12 and
6×6×6, respectively, to compute the charge density, phonon dispersion and electron-phonon
interaction matrix. To transform Bloch functions into the MLWF representation, we adopted 18
MLWFs, with initial sp3 projection for N atom and d projection for Ga atom in the Wannier
calculation. Then the relevant physical quantities, i.e., electronic Hamiltonian, dynamical matrix
and electron-phonon matrix elements, are Fourier-transformed back to the Block representations
at the much denser k- and q-mesh of 35×35×35 and 15×15×15, respectively. In Fig. S1, it shows
that the above physical quantities decay rapidly in the real-space representation and verify the
reliability of interpolating them from coarse mesh to dense mesh by the MLWF-Fourier
approach.
After obtaining the full electron-phonon matrix g, it is capable to compute the important
physical quantity, such as electron-phonon coupling strength λ and Eliashberg spectral function
α2F. The above quantities are defined as [3]
q 
1
N ( F )q
2
dk
gmn, (k , q)  ( nk   F ) ( mk q   F ),
BZ 

mn
4
(S.9)
100
(a)
10
-2
10
-3
10-4
10-5
10-6
0
100
10
10
-3
5
10
15
10-3
10-4
|Re| [Angstrom]
(c)
10-5
5
10
15
10-6
0
20
10-4
10-6
0
10
-2
10-5
10-1
-2
(b)
10-1
Dipole P
10-1
20
electron-phonon matrix g
Dynamical matrix Dph
Hamilonian Hel
100
|Rp| [Angstrom]
5
10
15
|u| [Angstrom]
20
100
(d)
10-1
10
-2
10-3
10-4
10-5
10-6
0
5
10
15
20
|Re| [Angstrom]
FIG. S1. Spatial decay of the (a) electronic Hamiltonian Hel, (b) Dipole (P), (c) dynamical matrix (Dph)
and electron-phonon matrix g for wurtzite GaN.
 2 F ( ) 
1
dq
q q  (  q ).

BZ
2

(S.10)
The Eliashberg spectral function α2F is obtained via an average over the Brillouin zone
(BZ), εF is the Fermi energy level and N(εF) is the density of states per spin at the Fermi level. In
Figs. S2-S3, the computed branch-dependent electron-phonon coupling strength λqυ and
Eliashberg spectral function α2F for wurtzite GaN are presented. It observes that dominant peak
exist around the Γ point in λ in the first Brillouin zone, indicating that the electron-phonon
interaction mainly arises from electrons with phonons with long wavelength. The dominant
peaks of α2F locate around 4.2 THz and 16.9 THz, corresponding to the frequencies of transverse
5
optic (TO) and longitudinal optic (LO) phonons, respectively. It suggests the strong coupling
between electrons and TO and LO phonons.
12
10

8
6
4
2
0

KM

A
H
L A
FIG. S2. The Branch-dependent electron-phonon coupling strength λqυ for wurtzite GaN.
3
2F()
2
1
0
0
5
10
15
20
25
Phonon Frequency [THz]
FIG. S3. The averaged Eliashberg spectral function α2F over the BZ for wurtzite GaN.
6
S2. Detailed Comparison between Two Different Simulation Packages
To elucidate that the contradicting comparison between Ref. [6] and experimental data [7,8]
mainly arises from the significant electron-phonon interaction (EPI), instead of using two
different packages in calculating phonon-phonon interaction (PPI) and EPI, we compare
representative physical quantities, such as phonon dispersion curve and frequency dependent
Grüneisen parameter, using VASP [9] and QUANTUM-ESPRESSO package [10]. The phonon
dispersion curve and Grüneisen parameter can provide important information about phonon
group velocity and phonon anharmonic scattering, which is key to compute the lattice thermal
conductivity. In Figs. S4 and S5, it shows that the error produced by using the two different
packages is less than 3%, which is believed to be far smaller to induce the contradicting
comparison between Ref. [6] and experimental data [7,8].
Phonon Frequency [THz]
25
20
15
10
5
0

K M
 A
H L

FIG. S4. Comparison of the computed phonon dispersion curve of wurtzite GaN by using two different
packages: QUANTUM-ESPRESSO (solid line) and VASP (dashed line).
7
2.0
Grüneisen Parameter
1.5
1.0
0.5
0.0
-0.5
QUANTUM-ESPRESSO
VASP
-1.0
-1.5
0
5
10
15
20
25
Phonon Frequency [THz]
FIG. S5. Comparison of the computed frequency dependent Grüneisen parameter of wurtzite GaN by
using two different packages: QUANTUM-ESPRESSO (square) and VASP (circle).
S3. Convergence Test on the Cutoff of the Nth Neighbors
To accurately compute the third order interatomic force constants, we have conducted the
convergence test on the cutoff of the Nth neighbors to be considered in calculating the lattice
thermal conductivity of wurtzite GaN. In Fig. S6, it shows that both the normalized in-plane (  L )
and out-of-plane (  L ) lattice thermal conductivity of wurtzite GaN converge at the cutoff of the
5th neighbors.
8
Normalized Thermal Conductivity [W/mK]
2.0
in-plane
out-of-plane
1.8
1.6
1.4
1.2
1.0
1
2
3
4
5
Cutoff of Nth neighbors
FIG. S6. The convergence test on the cutoff of the Nth neighbors to be considered in calculating the
thermal conductivity of wurtzite GaN.
9
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