of 5
First-Principles Exciton Radiative Lifetimes in Wurtzite GaN
Vatsal A. Jhalani,
1
Hsiao-Yi Chen,
1, 2
Maurizia Palummo,
3
and Marco Bernardi
1,
a)
1)
Department of Applied Physics and Materials Science, California Institute of Technology, 1200 E. California Blvd.,
Pasadena, California 91125, USA.
2)
Department of Physics, California Institute of Technology, 1200 E. California Blvd., Pasadena, California 91125,
USA.
3)
Dipartimento di Fisica,Universit`a di Roma“Tor Vergata” and INFN, Via della Ricerca Scientifica 1, 00133 Roma,
Italy
Gallium nitride (GaN) is a key semiconductor for solid-state lighting, but its radiative processes are not fully
understood. Here we show a first-principles approach to accurately compute the radiative lifetimes in bulk
uniaxial crystals, focusing on wurtzite GaN. Our computed radiative lifetimes are in very good agreement with
experiment up to 100 K. We show that taking into account excitons (through the Bethe-Salpeter equation)
and spin-orbit coupling to include the exciton fine structure is essential for computing accurate radiative
lifetimes. A model for exciton dissociation into free carriers allows us to compute the radiative lifetimes
up to room temperature. Our work enables precise radiative lifetime calculations in III-nitrides and other
anisotropic solid-state emitters.
Semiconductor light-emitting diodes (LEDs) are the
preferred light source for many applications. In LEDs,
electroluminescence converts electron and hole carriers
into emitted photons. This radiative recombination pro-
cess depends on material properties such as the band
structure, dielectric screening, and optical excitations.
In a simplified picture, free electrons and holes in band
states recombine to emit light. Yet, in many solid-state
emitters electron-hole interactions are strong enough to
form excitons (bound electron-hole states)
1
, and the
main process leading to light emission is exciton radia-
tive recombination. Gallium nitride (GaN) is widely em-
ployed for efficient light emission
2
, and it has been in-
vestigated extensively in crystalline, thin film, and het-
erostructure forms, both to understand its physical prop-
erties and to improve LED devices. Even though the
exciton binding energy is rather weak in GaN (of order
20 meV
3
), accurately computing its absorption spectrum
requires taking into account excitonic effects
4
, so one ex-
pects that excitons also play a role in light emission.
The radiative properties of GaN have remained the
subject of debate
3,5–9
. Investigations of radiative pro-
cesses require photoluminescence (PL) spectroscopies or
device experiments on pure samples. Since GaN films are
typically grown epitaxially, and their doping is nontriv-
ial, these measurements are affected by sample purity and
competing non-radiative processes due to defects and in-
terfaces
10
. In addition, typical theoretical treatments of
radiative lifetimes employ simplified empirical methods
that can only qualitatively interpret, or just fit, exper-
imental data
11
. Accurate first-principles calculations of
the
intrinsic
radiative properties of a GaN crystal would
be highly desirable as they would serve as a benchmark
for interpreting PL measurements and for guiding mi-
croscopic understanding and device design. Isolated ex-
amples of
ab initio
radiative lifetime calculations in bulk
materials exist
12,13
, but they neglect key factors such as
a)
Please E-mail correspondence to: bmarco@caltech.edu
excitonic effects, the material anisotropy, and tempera-
ture dependence dictated by dimensionality.
We have recently shown a general first-principles ap-
proach that includes these physical features and can be
applied broadly to systems ranging from isolated emitters
to bulk crystals
14
. Its application to carbon nanotubes
15
,
two-dimensional materials
16–18
, and recently gas phase
molecules has shown results in very good agreement with
experiments. The method is based on the solutions of the
ab initio
Bethe-Salpeter equation (BSE)
19,20
, which can
correctly treat excitons but is computationally expensive,
plus Fermi’s golden rule to obtain the radiative lifetimes
and their temperature dependence. For comparison, the
widely used independent-particle picture (IPP) of light
emission, which does not take excitons into account, is
straightforward to compute, but as we show below it
makes large errors on the radiative lifetimes. For bulk
crystals, we are not aware of
ab initio
radiative lifetime
calculations that properly include excitons other than our
recent work, which focused on isotropic crystals
14
. How-
ever, GaN is isotropic only in the hexagonal basal plane,
while the properties along the
c
-axis are different. Its
anisotropic optical properties cannot be taken into ac-
count in current
ab initio
radiative lifetime calculations.
Here, we extend our first-principles approach
14
to the
case of a uniaxial bulk crystal, and use it to compute
intrinsic radiative lifetimes in wurtzite GaN. The results
agree well with experiment (within a factor of two) up
to 100 K, and we include thermal exciton dissociation
to retain quantitative accuracy up to room temperature.
In spite of the weak exciton binding energy in GaN, we
show that including excitons is essential for quantitative
accuracy as it improves substantially the agreement with
experiment compared to IPP calculations. We also show
that including spin-orbit coupling (SOC) and the related
exciton fine structure is important in spite of the weak
SOC in GaN. Our work advances the study of light emis-
sion in III-nitrides and anisotropic light emitters.
Our discussion in Ref. 14 forms the basis for deriving
the radiative lifetimes of a uniaxial crystal. The dielectric
arXiv:1908.09962v1 [cond-mat.mtrl-sci] 27 Aug 2019
2
tensor of a uniaxial bulk crystal,

r
= diag(

xy
,
xy
,
z
)
,
(1)
is isotropic in the basal hexagonal plane, and differ-
ent along the principal crystal axis (the
z
direction).
For a given photon wavevector
q
, there are two non-
degenerate propagating modes as solutions to Maxwell’s
equations, each corresponding to one of the two photon
polarizations
21
. We call the first solution the “in-plane”
(IP) mode since its polarization vector sits in the
xy
-
plane, and the second solution the “out-of-plane” (OOP)
mode, which sees the anisotropy of the material and as
shown below has a more complicated expression than in
the isotropic case
14
. The polarization vectors
e
and fre-
quencies
ω
of the two modes are obtained by solving the
equation of motion of the vector potential in the dielec-
tric material
14
. For the uniaxial case, we get:
ω
IP
c
=
q
2

xy
,
e
IP
=
1

xy
(
q
y
q
xy
,
q
y
q
xy
,
0
)
(2)
ω
OOP
c
=

xy
q
2
xy
+

z
q
2
z

xy

z
,
e
OOP
=
q
x
q
xy
1
/
xy
(
1 +

xy
q
2
xy

z
q
2
z
)
,
q
y
q
xy
1
/
xy
(
1 +

xy
q
2
xy

z
q
2
z
)
,
1
/
z
(
1 +

z
q
2
z

xy
q
2
xy
)
,
(3)
where
c
is the speed of light and
q
2
xy
=
q
2
x
+
q
2
y
.
The radiative recombination rate at zero temperature
for an exciton in state
S
with center-of-mass momentum
Q
can be written using Fermi’s golden rule as
14
γ
S
(
Q
) =
πe
2

0
m
2
V
λ
q
1
ω
λ
q
|
e
λ
q
·
p
S
(
Q
)
|
2
δ
(
E
S
(
Q
)
~
ω
λ
q
)
,
(4)
where V is the volume of the system and

0
the vacuum
permittivity, and the sum over
λ
adds together the contri-
butions from the IP and OOP modes. Since the values of
Q
relevant for light emission are small, we approximate
the transition dipoles
22
as
p
S
(
Q
)
p
S
(0), and obtain
them, together with the exciton energies
E
S
(0), by solv-
ing the BSE at
Q
= 0. We substitute into Eq. (4) the
two solutions in Eqs. (2) and (3), use momentum con-
servation, which fixes the emitted photon wave vector to
q
=
Q
, and obtain the radiative recombination rate at
zero temperature for each exciton state
S
in a uniaxial
bulk material (we put
Q
2
xy
=
Q
2
x
+
Q
2
y
):
γ
S
(
Q
) =
πe
2

0
m
2
V

xy
cQ
1

xy
p
Sx
Q
y
p
Sy
Q
x
Q
xy
2
IP
δ
(
E
S
(
Q
)
~
cQ

xy
)
+

xy

z
c

xy
Q
2
xy
+

z
Q
2
z
×
Q
x
p
Sx
+
Q
y
p
Sy
Q
xy
1
/
xy
1 +

xy
Q
2
xy

z
Q
2
z
p
Sz
1
/
z
1 +

z
Q
2
z

xy
Q
2
xy
2
OOP
δ
E
S
(
Q
)
~
c

xy
Q
2
xy
+

z
Q
2
z

xy

z
.
(5)
Assuming that the exciton momentum
Q
has a thermal
distribution, the radiative rate of an exciton
S
at tem-
perature
T
is written as the thermal average
γ
S
(
T
) =
d
Q
e
E
S
(
Q
)
/k
B
T
γ
S
(
Q
)
d
Q
e
E
S
(
Q
)
/k
B
T
,
(6)
where
k
B
is the Boltzmann constant. We employ an effec-
tive mass approximation for the exciton dispersion, with
IP and OOP effective masses
M
xy
and
M
z
, respectively,
obtained as the sum of the electron and hole effective
masses
23
. The exciton radiative rate at temperature
T
is
then obtained from the integral in Eq. (6):
γ
S
(
T
) =
(
E
S
(0)
2
2
M
2
3
xy
M
1
3
z
c
2
k
B
T
)
3
/
2
×
π
xy
e
2
~
[(
2

z
3

xy
+ 2
)
(
p
2
Sx
+
p
2
Sy
)
+
8
3
p
2
Sz
]

0
m
2
V E
S
(0)
2
,
(7)
where the exciton energies and transition dipoles are ob-
tained by solving the BSE. The radiative lifetime is de-
fined as the inverse radiative rate,
τ
S
=
γ
S
1
. Note
also that Eq. (7) reduces to the bulk isotropic case in
Ref. 14 if one puts

z
=

xy
and
M
xy
=
M
z
.
Finally, we take into account the fact that multiple ex-
citon states can be occupied (including dark states with
3
small transition dipoles, as is the case in GaN), and com-
pute the radiative rate assuming a thermal equilibrium
distribution:
γ
(
T
)
=
S
γ
S
e
E
S
(0)
/k
B
T
S
e
E
S
(0)
/k
B
T
.
(8)
We use this thermal average, computed with the exciton
radiative rates
γ
S
in Eq. (7), to obtain the
intrinsic
ra-
diative lifetime
γ
(
T
)
1
in bulk wurtzite GaN.
We carry out first-principles calculations on a wurtzite
GaN unit cell with relaxed lattice parameters.
The
ground state properties and electronic wave functions are
computed using density functional theory (DFT) within
the generalized gradient approximation
24,25
with the
Quantum ESPRESSO
code
26
. Fully-relativistic norm-
conserving pseudopotentials
27
generated with Pseudo
Dojo
28
are employed, in which the shells treated as va-
lence are the 3
s
, 3
p
, 3
d
, 4
s
, and 4
p
for Ga and the 2
s
and 2
p
for N. A non-linear core correction
29
is included
for all remaining core shells for both atoms. We com-
pute the quasiparticle band structure in GaN
30
with a
“one-shot”
GW
calculation
31
with the Yambo code
32,33
,
using a plasmon-pole model for the dielectric function, a
25 Ry cutoff for the dielectric matrix, 300 empty bands,
and a 14
×
14
×
10
k
-point grid
34
. The BSE is solved on a
24
×
24
×
18
k
-point grid using a 6 Ry cutoff for the static
dielectric screening and the 6 highest valence bands and
4 lowest conduction bands. These settings are sufficient
to converge the energies, transition dipoles and radiative
lifetimes of the low-energy excitons, as we have verified.
The IPP transition dipoles and energies are computed by
neglecting the electron-hole interactions in the BSE. The
exciton binding energy is converged by computing it with
several
k
-point grids from 12
×
12
×
9 to 24
×
24
×
18 and
extrapolating it to a vanishingly small
k
-point distance
(
i.e.
, to an infinitely dense grid)
35
.
Our computed radiative lifetimes between 50
150 K
are shown in Fig. 1a) along with experimental values
from Ref. 7, which are ideal for our comparison since they
were measured in a relatively pure GaN crystal. At low
temperatures up to 100 K, our first-principles radiative
lifetimes, with SOC included, are of order 200
900 ps
and are in very good agreement (within less than a fac-
tor of two) with experiment. We attribute the remain-
ing discrepancy to small uncertainties in the computed
exciton effective mass, transition dipoles, energies and
occupations, plus inherent uncertainties in the experi-
mental data. Both the computed and experimental life-
times exhibit the intrinsic
T
3
/
2
trend predicted by our
approach [see Eq. (7)]. As Fig. 1a) shows, when ne-
glecting excitons and using IPP transition dipoles and
energies, one greatly overestimates the radiative lifetime.
The IPP lifetimes are greater by nearly an order of mag-
nitude compared to our treatment, which correctly in-
cludes excitons, and by over a factor of three compared
to experiment.
As seen in Fig. 1a), including SOC when computing
the exciton states increases the radiative lifetimes by a
a)
b)
FIG. 1. a) Comparison of our radiative lifetimes computed
by including (blue) or neglecting (orange) the SOC in the
solution of the BSE, or obtained in the IPP by neglecting
excitons (red). Experimental results from Ref. 7 (purple) are
shown for comparison. The gray dashed lines show the
T
3
/
2
trend predicted by our treatment at low temperature. b)
The excitons contributing to the thermal average in Eq. (8),
along with their individual lifetimes at 100 K, computed with
(blue) and without (orange) SOC. The zero of the energy axis
is taken to be the lowest exciton energy for each case.
factor of 2
3 and significantly improves the agreement
with experiment. Though SOC is weak in GaN
the
valence band splitting at Γ is only 5 meV in our calcu-
lations
its inclusion is crucial for obtaining accurate
exciton states. Figure 1b) shows the individual radiative
lifetimes
γ
S
1
and relative energies of the low-energy
excitons contributing to the thermal average in Eq. (8),
for both the cases where SOC is included and neglected.
Without including spin and SOC, the exciton structure
consists of three bright singlet excitons, two of which
are degenerate. The lifetimes of all three excitons are
nearly identical, and their value determines the radiative
lifetime for the calculation without SOC. Including the
SOC lifts the degeneracy of the two lowest bright excitons
by
5 meV, and resolves the exciton fine structure, split-
ting each exciton into four states due to a doubling of the
number of valence and conduction states that compose
the electron-hole transitions. With SOC, we find dark
4
excitons with lifetimes roughly 3
10 orders of magni-
tude longer than the excitons found without SOC. When
included in the thermal average, these dark states are
crucial as they increase the radiative lifetime compared
to the average lifetime of the bright excitons alone. The
inclusion of SOC and the exciton fine structure are thus
important for quantitative accuracy, even though SOC
per se is weak in GaN
36
.
Due to the small exciton binding energy in GaN, at
high enough temperatures the excitons dissociate into
free electrons and holes, which mainly recombine non-
radiatively in GaN, giving rise to the lower radiative
recombination rate and quantum yield seen experimen-
tally above 100 K
7,37
. As a result of exciton dissocia-
tion, the measured radiative lifetime above
100 K in-
creases more rapidly with temperature than the intrin-
sic
T
3
/
2
trend [see Fig. 1a)]. We show a simple model
to include exciton dissociation in our first-principles ap-
proach. Assuming that excitons and free carriers are in
thermal equilibrium, we write the mass-action law for
their concentrations as
37
n
e
n
h
n
exc
=
[
n
0
+
δn
]
δp
δn
exc
=
κ
(
T
)
,
(9)
where
n
e
,
n
h
, and
n
exc
are the electron, hole, and exci-
ton densities, respectively,
n
0
is the background electron
density (from the doping), and
δn
,
δp
, and
δn
exc
are the
excited electron, hole, and exciton densities, respectively,
generated by an idealized optical pump or electrical cur-
rent. The equilibrium constant
κ
(
T
) is given by
37
κ
(
T
) = 2
(
m
red
k
B
T
2
π
~
2
)
3
/
2
e
E
b
/k
B
T
,
(10)
where
m
red
=
m
h
m
e
/
(
m
h
+
m
e
) is the reduced mass of the
exciton and
E
b
its binding energy. We find a converged
binding energy of 19.7 meV, in excellent agreement with
the experimental value of 20.4 meV
3
, and we use a typi-
cal doping of
n
0
= 2
.
5
×
10
16
cm
3
, taken from Ref. 7.
Assuming that the relative recombination probabil-
ity of free carriers and excitons is proportional to their
concentration ratio,
P
carr
/P
exc
=
δn/δn
exc
, and using
P
carr
+
P
exc
= 1, we can obtain the probabilities for exci-
ton and free carrier recombination. The measured radia-
tive rate will be a weighted average of the rates of the two
recombination processes, Γ
rad
= Γ
carr
P
carr
+ Γ
exc
P
exc
.
We assume that Γ
carr
vanishes because free carriers re-
combine mainly via non-radiative channels, such as defect
trapping, which is justified by the reported low quantum
yield seen experimentally near room temperature
7
. The
measured radiative rate due to excitons in equilibrium
with carriers becomes Γ
rad
Γ
exc
/
(1 +
κ
(
T
)
/n
0
). Using
this result, together with our computed effective masses
and converged exciton binding energy, we are able to pre-
dict the exciton radiative lifetimes also above 100 K.
Figure 2 compares the computed radiative lifetimes up
to 300 K with experimental results taken from PL mea-
surements in Refs.
7,8
. When thermal dissociation is in-
cluded, the radiative lifetime agrees with experiment even
FIG. 2. Comparison of our computed radiative lifetimes in-
cluding exciton dissociation (orange) above 100 K with exper-
imental data from Refs. 7 and 8. Also shown is our computed
intrinsic radiative lifetime (blue).
in the 100
300 K temperature range, where the experi-
mental data deviate from the intrinsic
T
3
/
2
trend. Our
ability to compute intrinsic exciton radiative lifetimes al-
lows us to conclude that the radiative lifetime increase
seen experimentally above 100 K is due to exciton ther-
mal dissociation into free carriers. This conclusion is con-
sistent with the results by Im et al.
7
, who found that a
similar exciton dissociation model could fit their experi-
mental data at high temperature.
In summary, we developed accurate first-principles ra-
diative lifetime calculations in GaN. Our method includes
the electron-hole and spin-orbit interactions, thus cor-
rectly treating excitons and their fine structure. These
advances allow us to compute intrinsic radiative lifetimes
in very good agreement with experiment, and gain mi-
croscopic insight into the excitons associated with light
emission in GaN. Future work will extend this analysis
to GaN heterostructures and quantum wells. Our results
add to the general framework we presented in Ref. 14, en-
abling precise predictions of radiative processes in solid-
state light emitting materials, while shedding light on
their ultrafast excited state dynamics.
The authors thank Davide Sangalli for fruitful discus-
sions. V.A.J. thanks the Resnick Sustainability Insti-
tute at Caltech for fellowship support. This work was
partially supported by the Department of Energy under
Grant No. de-sc0019166, which provided for theory and
method development, and by the National Science Foun-
dation under Grant No. ACI-1642443, which provided for
code development. This research used resources of the
National Energy Research Scientific Computing Center,
a DOE Office of Science User Facility supported by the
Office of Science of the U.S. Department of Energy un-
der Contract No. DE-AC02-05CH11231. M.P. thanks
CINECA for computational resources.
5
1
R. S. Knox,
Theory of Excitons (Solid State Phys. Suppl. 5)
,
Vol. 5 (Academic Press, New York, 1963).
2
S. Pimputkar, J. S. Speck, S. P. DenBaars, and S. Nakamura,
Prospects for LED lighting, Nat. Photonics
3
, 180 (2009).
3
J. F. Muth, J. H. Lee, I. K. Shmagin, R. M. Kolbas, H. C. Casey,
B. P. Keller, U. K. Mishra, and S. P. DenBaars, Absorption co-
efficient, energy gap, exciton binding energy, and recombination
lifetime of GaN obtained from transmission measurements, Appl.
Phys. Lett.
71
, 2572–2574 (1997).
4
R. Laskowski, N. E. Christensen, G. Santi, and C. Ambrosch-
Draxl, Ab initio calculations of excitons in GaN, Phys. Rev. B
72
, 035204 (2005).
5
C. I. Harris, B. Monemar, H. Amano, and I. Akasaki, Exciton life-
times in GaN and GaInN, Appl. Phys. Lett.
67
, 840–842 (1995).
6
G. Chen, M. Smith, J. Lin, H. Jiang, S.-H. Wei, M. Asif Khan,
and C. Sun, Fundamental optical transitions in GaN, Appl. Phys.
Lett.
68
, 2784–2786 (1996).
7
J. S. Im, A. Moritz, F. Steuber, V. Hrle, F. Scholz, and
A. Hangleiter, Radiative carrier lifetime, momentum matrix el-
ement, and hole effective mass in GaN, Appl. Phys. Lett.
70
,
631–633 (1997).
8
O. Brandt, J. Ringling, K. H. Ploog, H.-J. W ̈unsche, and F. Hen-
neberger, Temperature dependence of the radiative lifetime in
GaN, Phys. Rev. B
58
, R15977–R15980 (1998).
9
V. A. Jhalani, J.-J. Zhou, and M. Bernardi, Ultrafast hot carrier
dynamics in GaN and its impact on the efficiency droop, Nano
Lett.
17
, 5012–5019 (2017).
10
J. P. Wolfe, Thermodynamics of excitons in semiconductors,
Phys. Today
35
, 46–54 (1982).
11
B. K. Ridley,
Quantum Processes in Semiconductors
(Oxford
University Press, 2013).
12
X. Zhang, J.-X. Shen, W. Wang, and C. G. Van de Walle, First-
principles analysis of radiative recombination in lead-halide per-
ovskites, ACS Energy Lett.
3
, 2329–2334 (2018).
13
E. Kioupakis, Q. Yan, D. Steiauf, and C. G. Van de Walle, Tem-
perature and carrier-density dependence of auger and radiative
recombination in nitride optoelectronic devices, New J. Phys.
15
,
125006 (2013).
14
H.-Y. Chen, V. A. Jhalani, M. Palummo, and M. Bernardi, Ab
initio calculations of exciton radiative lifetimes in bulk crystals,
nanostructures, and molecules, Phys. Rev. B
100
, 075135 (2019).
15
C. D. Spataru, S. Ismail-Beigi, R. B. Capaz, and S. G. Louie,
Theory and ab initio calculation of radiative lifetime of excitons
in semiconducting carbon nanotubes, Phys. Rev. Lett.
95
, 247402
(2005).
16
M. Palummo, M. Bernardi, and J. C. Grossman, Exciton ra-
diative lifetimes in two-dimensional transition metal dichalco-
genides, Nano Lett.
15
, 2794–2800 (2015).
17
H.-Y. Chen, M. Palummo, D. Sangalli, and M. Bernardi, The-
ory and ab initio computation of the anisotropic light emission
in monolayer transition metal dichalcogenides, Nano Lett.
18
,
3839–3843 (2018).
18
S. Gao, L. Yang, and C. D. Spataru, Interlayer coupling and gate-
tunable excitons in transition metal dichalcogenide heterostruc-
tures, Nano Lett.
17
, 7809–7813 (2017).
19
G. Strinati, Effects of dynamical screening on resonances at inner-
shell thresholds in semiconductors, Phys. Rev. B
29
, 5718–5726
(1984).
20
M. Rohlfing and S. G. Louie, Electron-hole excitations and op-
tical spectra from first principles, Phys. Rev. B
62
, 4927–4944
(2000).
21
R. J. Glauber and M. Lewenstein, Quantum optics of dielectric
media, Phys. Rev. A
43
, 467–491 (1991).
22
The transition dipoles are computed using the velocity operator
to correctly include the nonlocal part of the Hamiltonian.
23
Since we find from the BSE that the lowest exciton states are
composed of transitions from the two heavy-hole bands, we ap-
proximate the hole mass as the average of the two heavy-hole
masses.
24
J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradi-
ent approximation made simple, Phys. Rev. Lett.
77
, 3865–3868
(1996).
25
J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E.
Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the
density-gradient expansion for exchange in solids and surfaces,
Phys. Rev. Lett.
100
, 136406 (2008).
26
P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,
C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni,
I. Dabo,
et al.
, QUANTUM ESPRESSO: a modular and open-
source software project for quantum simulations of materials, J.
Phys.: Condens. Matter
21
, 395502 (2009).
27
D. R. Hamann, Optimized norm-conserving Vanderbilt pseu-
dopotentials, Phys. Rev. B
88
, 085117 (2013).
28
M. van Setten, M. Giantomassi, E. Bousquet, M. Verstraete,
D. Hamann, X. Gonze, and G.-M. Rignanese, The PseudoDojo:
Training and grading a 85 element optimized norm-conserving
pseudopotential table, Comput. Phys. Commun.
226
, 39 – 54
(2018).
29
S. G. Louie, S. Froyen, and M. L. Cohen, Nonlinear ionic pseu-
dopotentials in spin-density-functional calculations, Phys. Rev.
B
26
, 1738–1742 (1982).
30
A. Rubio, J. L. Corkill, M. L. Cohen, E. L. Shirley, and S. G.
Louie, Quasiparticle band structure of AlN and GaN, Phys. Rev.
B
48
, 11810–11816 (1993).
31
G. Onida, L. Reining, and A. Rubio, Electronic excita-
tions: density-functional versus many-body Green’s-function ap-
proaches, Rev. Mod. Phys.
74
, 601–659 (2002).
32
A. Marini, C. Hogan, M. Gr ̈uning, and D. Varsano, Yambo: an
ab initio tool for excited state calculations, Comput. Phys. Com-
mun.
180
, 1392–1403 (2009).
33
D. Sangalli, A. Ferretti, H. Miranda, C. Attaccalite, I. Marri,
E. Cannuccia, P. Melo, M. Marsili, F. Paleari, A. Marrazzo,
G. Prandini, P. Bonf`a, M. O. Atambo, F. Affinito, M. Palummo,
A. Molina-S ́anchez, C. Hogan, M. Grning, D. Varsano, and
A. Marini, Many-body perturbation theory calculations using the
Yambo code, J. Phys.: Condens. Matter
31
, 325902 (2019).
34
For the
GW
band structure, we start from DFT within the local-
density approximation
38
and employ scalar-relativistic norm-
conserving pseudopotentials for both Ga and N. The 4
s
and 4
p
shells are treated as valence for Ga, and the 2
s
and 2
p
for N. A
non-linear core correction is included to account for the 3
d
core
states in Ga.
35
F. Fuchs, C. R ̈odl, A. Schleife, and F. Bechstedt, Efficient
O
(
N
2
) approach to solve the Bethe-Salpeter equation for ex-
citonic bound states, Phys. Rev. B
78
, 085103 (2008).
36
Note that spin is always important. Even in the limit of vanish-
ingly small SOC, the triplet states with ideally infinite lifetime
would still suppress the average radiative rate in Eq. (8) by a
factor of 4, and thus increase the radiative lifetimes by the same
factor compared to a calculation that does not include spin.
37
H. W. Yoon, D. R. Wake, and J. P. Wolfe, Effect of exciton-carrier
thermodynamics on the GaAs quantum well photoluminescence,
Phys. Rev. B
54
, 2763–2774 (1996).
38
D. M. Ceperley and B. J. Alder, Ground state of the electron gas
by a stochastic method, Phys. Rev. Lett.
45
, 566–569 (1980).