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Communication
Ultrafast Hot Carrier Dynamics in GaN and its Impact on the Efficiency Droop
Vatsal Jhalani, Jin-Jian Zhou, and Marco Bernardi
Nano Lett.
,
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• DOI: 10.1021/acs.nanolett.7b02212
• Publication Date (Web): 24 Jul 2017
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Ultrafast Hot Carrier Dynamics in GaN and its
Impact on the Efficiency Droop
Vatsal A. Jhalani,
†
,
‡
Jin-Jian Zhou,
†
,
‡
and Marco Bernardi
∗
,
†
†
Department of Applied Physics and Materials Science, Steele Laboratory, California
Institute of Technology, Pasadena, California 91125, USA.
‡
Contributed equally to this work
E-mail: bmarco@caltech.edu
Phone: +1 (0)626 3952515
Abstract
GaN is a key material for lighting technology. Yet, the carrier transport
and ul-
trafast dynamics that are central in GaN light emitting devices are not c
ompletely
understood. We present first-principles calculations of carrier
dynamics in GaN, focus-
ing on electron-phonon (e-ph) scattering and the cooling and nanoscale d
ynamics of
hot carriers. We find that e-ph scattering is significantly faster for h
oles compared to
electrons, and that for hot carriers with an initial 0.5
−
1 eV excess energy, holes take
a significantly shorter time (
∼
0.1 ps) to relax to the band edge compared to electrons,
which take
∼
1 ps. The asymmetry in the hot carrier dynamics is shown to originate
from the valence band degeneracy, the heavier effective mass of holes c
ompared to
electrons, and the details of the coupling to different phonon modes i
n the valence
and conduction bands. We show that the slow cooling of hot electrons and t
heir long
ballistic mean free paths (over 3 nm) are a possible cause of efficiency
droop in GaN
light emitting diodes. Taken together, our work sheds light on the ult
rafast dynamics
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of hot carriers in GaN and the nanoscale origin of efficiency droop.
Keywords:
Gallium nitride, light emitting diodes, ultrafast dynamics, elect
ron-phonon
scattering
Wurtzite GaN has emerged as a promising material for solid st
ate lighting
1
and power
electronics,
2,3
with potential technological benefits that are driving inte
nse research in indus-
try and academia. However, material properties essential fo
r device performance and energy
efficiency, such as carrier transport and recombination, are
not completely understood in
GaN and remain the subject of debate. Carrier transport and u
ltrafast dynamics are regu-
lated by scattering with phonons, carriers and impurities.
4
In particular, the electron-phonon
(e-ph) interaction
5,6
plays a dominant role on transport at room temperature in rel
atively
pure materials. It further regulates the energy loss (or “co
oling”) of excited carriers in-
jected at heterojunctions, a scenario of relevance in GaN li
ght emitting diodes. The excited
(so-called “hot”) carriers rapidly lose their excess energ
y with respect to the band edges,
dissipating heat by phonon emission through e-ph coupling.
Hot carriers (HCs) are also
central in degradation and current leakage in GaN transisto
rs for power electronics,
7,8
and
set the operational basis for hot electron transistors.
9
Microscopic understanding of carrier dynamics is challeng
ing in GaN since experimental
results are modulated by defects and interfaces, and are typ
ically interpreted with empiri-
cal models.
10–16
For example, the hot electron cooling times measured by diffe
rent groups
range over two orders of magnitude,
10,12–17
and reports of hot hole dynamics are scarce.
11
In addition, the efficiency decline in GaN light emitting diod
es (LEDs) at high current, a
process known as efficiency droop,
18
has been intensely investigated but its carrier dynamics
origin remains unclear. First-principles calculations foc
used on Auger recombination
19,20
as
a possible cause, though other mechanisms have been propose
d,
18
including HC effects and
electron leakage. These processes have seen less extensive
theoretical treatment compared
to Auger, leaving simplified models to guide device design.
We recently developed first-principles calculations of car
rier dynamics
6
that can obtain
2
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carrier mobility,
21,22
ultrafast dynamics,
23–26
HC relaxation times
23,24
and ballistic mean free
paths
23,25
in excellent agreement with experiment. These approaches a
re free of empirical
parameters and use the structure of the material as the only i
nput. In particular, we recently
developed a method
22
to accurately compute the e-ph relaxation times (RTs), name
ly the
average time between e-ph collisions, in polar materials, a
s is needed for GaN. These ap-
proaches are extended in this work to investigate HC dynamics
in GaN from first principles.
Here, we compute the e-ph RTs over a wide energy range, and stud
y the cooling of HCs
by numerically solving the electron Boltzmann transport eq
uation (BTE). Both the RTs and
the simulated ultrafast dynamics reveal a large asymmetry b
etween the hot electron and hole
dynamics, with hot holes relaxing to the band edges significa
ntly faster than hot electrons.
The origin of this asymmetry, the role of different phonon mod
es and the limitations and
failure of phenomenological models are analyzed in detail.
We additionally find significantly
longer mean free paths (MFPs) for electrons compared to holes
, with implications for GaN
devices. We show that the slow cooling rate of hot electrons c
an lead to inefficient light
emission at high current, thus demonstrating that the nanos
cale dynamics of HCs play a key
role in LED efficiency droop.
Electron-phonon scattering.
In polar materials like GaN, empirical models typically
assume that polar optical phonons
−
and in particular, the longitudinal optical (LO) mode
in GaN
−
dominate carrier scattering due to their long-range intera
ctions with carriers.
The empirical Fr ̈ohlich model
27
for the LO mode predicts an e-ph coupling matrix ele-
ment
g
F
(
q
) = [(
e
2
̄
hω
0
)
/
(2
q
)]
ǫ
−
1
ph
, where
q
is the phonon wavevector (and
q
its magnitude),
̄
hω
0
the LO phonon energy, and
ǫ
−
1
ph
=
ǫ
−
1
∞
−
ǫ
−
1
0
the phonon contribution to the dielec-
tric screening, with
ǫ
∞
and
ǫ
0
the high- and low-frequency dielectric constants, respect
ively.
The intra-valley e-ph scattering rate Γ
k
(where
k
is the electron crystal momentum) due to
the empirical Fr ̈ohlich coupling
g
F
can be obtained analytically for carriers in a spherical
3
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parabolic band with effective mass
m
∗
and energy
E
k
= ̄
h
2
k
2
/
2
m
∗
:
5,28
Γ
k
=
τ
−
1
0
(
E
k
̄
hω
0
)
−
1
2
[
N
0
sinh
−
1
(
E
k
̄
hω
0
)
1
2
+
(
N
0
+ 1) sinh
−
1
(
E
k
̄
hω
0
−
1
)
1
2
]
(1)
where
N
0
is the Bose-Einstein occupation factor for LO phonons and
τ
−
1
0
=
ǫ
−
1
p
[(
e
2
ω
0
)
/
(2
π
̄
h
)]
·
[
m
∗
/
(2 ̄
hω
0
)]
1
/
2
is the inverse Fr ̈ohlich time. In this work, the rate in Eq. 1 i
s used to compare
the widely employed Fr ̈ohlich empirical model with our first
-principles results.
Following an approach we recently developed,
22
we combine density functional theory
(DFT),
29
density functional perturbation theory (DFPT)
30
and
ab initio
Fr ̈ohlich
31
cal-
culations to compute the short- and long-range contributio
ns to the e-ph coupling matrix
elements, which are then interpolated on fine Brillouin zone
(BZ) grids to converge the e-ph
scattering rates Γ
n
k
for each electronic band
n
and crystal momentum
k
(see Methods). For
the polar LO mode scattering, our
ab initio
Fr ̈ohlich calculations
31
differ in important ways
from the empirical Fr ̈ohlich model as they include Born effec
tive charges and anisotropic
dielectric tensors, both computed with DFPT, and account for
the electronic bandstructure
and phonon dispersions (see Methods). Here and in the followi
ng, the carrier excess energies
are defined as the energy above the conduction band minimum (C
BM) for the electrons, and
the energy below the valence band maximum (VBM) for the holes.
The scattering rates and their inverse, the e-ph RTs
τ
n
k
= Γ
−
1
n
k
, contain rich microscopic
information on the carrier dynamics. The bandstructure cru
cially determines the e-ph scat-
tering rates. A schematic of the bandstructure of GaN near th
e band edges is given for
reference in the inset of Figure 1A. While the conduction band ex
hibits a single parabolic
valley at Γ, the valence band edge consists of a light-hole an
d two heavy-hole bands with
anisotropic effective masses and degeneracy along the Γ
−
A direction. Figure 1A shows our
computed e-ph scattering rates of electrons and holes with e
nergies within 5 eV of the band
edges. Both the total scattering rate contributed by all pho
nons and the LO mode contri-
4
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higher LO scattering rate (by roughly a factor of 2) than in th
e conduction band within 1 eV
of the CBM. The LO contribution becomes roughly constant at h
ole excess energies greater
than
∼
0.2 eV. Different from the conduction band where LO phonon emis
sion is the only
active process, the total scattering rate keeps increasing
for holes with excess energy above
0.2 eV due to intra-valley and inter-valley scattering cont
ributed, in roughly equal measure,
by all acoustic and optical phonon modes (see Figure S1). Withi
n 2 eV of the band edges,
the DOS in the valence band is greater than in the conduction b
and due to the presence
of multiple valence bands and to the higher effective masses of
holes compared to electrons.
These effects result in scattering from phonon modes other th
an the LO becoming impor-
tant at lower excess energies in the valence band due to a grea
ter phase space for large-
q
scattering. Overall, the stronger LO polar and non-polar co
ntributions in the valence band
result in significantly higher scattering rates for holes co
mpared to electrons within 2 eV of
the band edges. This asymmetry has important consequences f
or carrier dynamics.
The empirical Fr ̈ohlich model shows two major limitations i
n reproducing the first-
principles trends (see Figure 1A). The empirical and first-pri
nciples rates exhibit opposite
trends at excess energies greater than
∼
150 meV. For energies where the LO phonon emission
rate is roughly constant, the discrepancy of the empirical F
r ̈ohlich rate is as large as 30
−
50%
for both electrons and holes. Note that first-principles calc
ulations include all phonon modes
on the same footing at all energies, while empirical e-ph cal
culations would account for modes
other than the LO through mode-specific empirical deformati
on potentials.
28
Traditionally
employed empirical models are thus inadequate to compute th
e e-ph scattering rates due to
all phonon modes over a wide energy range, as is done here, and
the first-principles approach
is necessary.
As a consequence of the scattering rate asymmetry, the comput
ed e-ph RTs (see Figure
1B) of holes are overall significantly shorter than the RTs of
electrons. Within 2 eV of the
band edges, the electron RTs range between 10
−
50 fs, while the hole RTs are of order 3
−
20
fs. The electron RT above the threshold for LO phonon emissio
n in the conduction band is
6
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10
2
10
1
10
0
Γ
h
/
Γ
e
0.2
0.4
0.6
0.8
1.0
Excess Energy (eV)
0.0
Total
Non-polar
Polar LO
(
m
∗
h
DOS
/m
∗
e
)
3
/
2
Figure 2:
Origin of the scattering rate asymmetry.
Ratio of the Brillouin zone aver-
aged e-ph scattering rates of holes (Γ
h
) to those of electrons (Γ
e
) as a function of carrier
excess energy. The zeros of the excess energy are the conduct
ion and valence band edges for
electrons and holes, respectively. The data points are comp
uted using rates due to polar LO
phonons (green), non-polar phonons (red), and all phonon mo
des (blue). The dashed lines
indicate the ratios (
m
∗
h
/m
∗
e
)
1
/
2
and (
m
∗
h
DOS
/m
∗
e
)
3
/
2
discussed in the text.
∼
12 fs, in very good agreement with the LO phonon emission time
of 16 fs recently measured
for electrons by Suntrup et al.
17
The detailed energy dependence of the RTs and scattering
rates reported here is valuable for GaN device design.
Origin of the carrier relaxation asymmetry.
We address the question of whether
the asymmetry between the electron and hole scattering rate
s found here is a mere conse-
quence of the heavier effective mass of holes compared to elec
trons in GaN. In doing so, we
develop an intuition for the origin of this asymmetry by anal
yzing separately the polar and
non-polar e-ph scattering contributions. As noted above, th
e two sources of e-ph scattering
are the long-range interaction from the LO polar mode and the
short-range interactions from
all other non-polar phonons. Because e-ph processes are det
ermined by the e-ph coupling
strength and the phase space available for scattering (see M
ethods, Eq. 3), the non-polar
scattering rate Γ
(NP)
approximately follows the same energy trend as the electron
ic DOS,
D
(
E
), multiplied by an average e-ph coupling strength
h
g
2
i
, so that Γ
(NP)
(
E
)
∝ h
g
2
i
D
(
E
).
On the other hand, due to the long-range electrostatic natur
e of the polar interaction, the
strength of LO coupling is insensitive to the specific electr
onic states involved in the scat-
7
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tering process. The LO mode polar coupling behaves as
|
g
LO
(
q
)
|
2
∼
1
/q
2
at small phonon
wavevector
q
, resulting in much stronger LO scattering for small-
q
transitions. Due to this
particular phonon wavevector dependence, the scattering r
ate Γ
(P)
due to the polar LO mode
approaches the band edge with a constant trend in energy, as o
pposed to being proportional
to the DOS as in the non-polar case. In the empirical Fr ̈ohlic
h formula (see Eq. 1), the polar
scattering rate behaves as a simple function of the effective
mass, Γ
(P)
∝
(
m
∗
)
1
/
2
.
To quantify the scattering rate asymmetry, we compute in Figu
re 2 the ratio of the BZ
averaged scattering rate of holes (Γ
h
) to that of electrons (Γ
e
). The ratio Γ
h
/
Γ
e
is shown for
the average total, polar and non-polar scattering rates as a
function of excess energy. For the
LO polar contribution, the ratio closely matches the result
expected based on the empirical
Fr ̈ohlich model (see Eq. 1), namely, Γ
(P)
h
/
Γ
(P)
e
≈
(
m
∗
h
/m
∗
e
)
1
/
2
, where
m
∗
h
is the experimental
hole effective mass.
32
The agreement between the empirical and
ab initio
ratios of LO mode
scattering rates indicates that the heavier hole effective m
ass is the main source of the LO
polar scattering asymmetry, and that the empirical and
ab initio
treatments roughly factor
out in the polar scattering rate ratio when a single
m
∗
h
value is employed as a proxy of the
multiple hole bands.
Since the non-polar scattering rate is proportional to the D
OS, the ratio between the
ab initio
non-polar scattering rates is compared in Figure 2 with a heur
istic DOS ratio.
Since the DOS of a parabolic band is proportional to (
m
∗
)
3
/
2
,
34
we expect that the ratio of
the non-polar scattering rates at low energy is approximate
ly Γ
(NP)
h
/
Γ
(NP)
e
≈
(
m
∗
h
DOS
/m
∗
e
)
3
/
2
,
where
m
∗
h
DOS
is the hole DOS effective mass.
33
The
ab initio
and DOS-based empirical non-
polar ratios are in reasonable agreement. However, the inacc
uracy of approximating multiple
valence bands with a single DOS effective mass for holes, comb
ined with the stronger aver-
age e-ph coupling strength for electrons (see Figure S2 and Se
ction A in the Supplementary
Materials), both push the ratio below the heuristic predict
ion, with an energy-dependent dis-
crepancy. Therefore, the ratio between the non-polar scatt
ering rates cannot be accurately
estimated without detailed knowledge of the bandstructure
and e-ph coupling strengths.
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Finally, since the total scattering rate is determined by the
sum of the polar and non-
polar contributions, the ratio between the
total
scattering rates of holes and electrons cannot
be estimated by a simple heuristic model based on the effectiv
e masses. The average
ab initio
ratio between the total scattering rates found here is Γ
h
/
Γ
e
≈
3 within 1 eV of the band edges,
and is bracketed by the non-polar and polar ratios. Detailed
first-principles calculations, as
employed here, are necessary to quantify this asymmetry in t
he scattering rates.
Real-time hot carrier dynamics.
Simulating the HC dynamics in real time further
highlights the different behavior of electrons and holes in G
aN. To this end, we carry out
numerical simulations of the dynamics of hot electron and ho
le populations injected in GaN
with a range of initial excess energies. To represent the inj
ected carriers, we employ narrow
Gaussians distributions centered at the initial excess ene
rgy, and solve the electron BTE
5
in real time fully
ab initio
,
6
using first-principles e-ph matrix elements, bandstructur
es and
phonon dispersions (see Methods). The carrier occupations
are time-stepped using a 4
th
-
order Runge-Kutta algorithm, while the phonon occupations
are kept at 300 K, so that hot
phonon effects are neglected.
Figure 3 shows the time evolution of the electron and hole popu
lations after injection
with a 1 eV initial excess energy. At each time step, we analyz
e both the carrier occupations
(
f
n
k
for electrons and 1
−
f
n
k
for holes) along the A
−
Γ
−
M line of the BZ and the carrier
concentrations as a function of energy,
f
(
E
), obtained by integrating the occupations at
each energy over the BZ (see Methods). Following HC generatio
n, the electron and hole
distributions broaden in energy and approach a Fermi-Dirac
-like shape as they shift toward
lower excess energies, eventually reaching the band edges.
Both electrons and holes ulti-
mately thermalize to a 300 K Fermi-Dirac distribution in equ
ilibrium with the phonons,
thus reaching the correct long-time limit for our simulatio
ns.
We find that while holes reach the band edges and cool in
∼
80 fs, electrons are still
far from equilibrium at the same time. Electron cooling is ro
ughly five times slower, with
electrons relaxing to the band edges in
∼
200 fs after injection, and fully thermalizing to the
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