Supplementary Material
Ultrafast and Band-selective Auger Recombination in InGaN Quantum Wells
Kristopher W. Williams1, Nicholas R. Monahan1, Daniel D. Koleske2, Mary H. Crawford2, X.-Y. Zhu1*
1
Department of Chemistry, Columbia University, New York, NY 10027, USA
Sandia National Laboratories, Albuquerque, NM 87123, USA
*Corresponding author. XYZ: [email protected]; MHC: [email protected]
2
Supplementary Method I: Sample Preparation and Characterization
We grew the InGaN film using metalorganic chemical vapor deposition (MOCVD) in a Veeco
D125 short-jar reactor1.
GaN templates were first grown at 500 torr and 1050 ºC using
trimethylgallium (TMGa) and NH3 in H2 and N2 on (0001) sapphire. The ~4 μm thick GaN
templates had screw- and edge-component threading-dislocation densities totaling ~9x108 cm-2 as
measured by XRD2. On the GaN template, a 160 nm thick In0.02Ga0.98N underlayer (UL) was grown
at 870 °C using 15 SLM of NH3 and 10 SLM of N2 at 300 torr. On the UL, a 2.8 nm thick
In0.15Ga0.85N quantum well was grown at 745
°C. The layer thicknesses and indium
concentration were verified on companion
samples using a Panalytical MRD-Pro x-ray
diffractometer (XRD) measurement of a ω/2θ
scan around the (0002) reflection. This XRD
scan
was
analyzed
using
X’Pert
4.0
dynamical diffraction analysis software1,2.
To protect from oxidation prior to
experiments, we removed each MOCVD
grown sample from the growth chamber
under dry nitrogen and subsequently covered
Figure S1 | LEED. Low-energy electron diffraction
pattern of the clean quantum well surface showing
the 6-fold symmetry of GaN (0001) wurzite
structure. The electron energy was 141 eV.
in Crystalbond™ part of the QW surface
layer, followed by the deposition of TiAlNiAu metal contact on the entire sample surface. Before
each sample was introduced into the ultrahigh vacuum (UHV, <5 × 10-11 mbar) chamber for
photoemission measurements, the crystal bond, along with any metal evaporated on top of the
crystal bond, was removed in an acetone bath to expose a clean InGaN surface. The TiAlNiAu
contact that remained around the exposed area was connected to ground; this was necessary to
prevent sample charging during photoemission experiments. After introduction into the UHV
chamber, each QW sample was outgassed at 100 C for 12 hours, with a final flash to 250 C prior
to photoemission measurements. Low-energy electron diffraction (LEED) showed surfaces cleaned
in this manner possessed the crystalline 6-fold symmetry of a well ordered GaN (0001) wurzite
structure3, Fig. S1.
Supplementary Method II: Ultraviolet Photoemission Spectroscopy (UPS)
UPS data of the clean InGaN quantum
7.5
well surface at room temperature taken
Fermi level (EF) of the sample was
determined by referencing to the Fermi
level of a Au(111) sample. The VB is
-1
7
The position of the VB relative to the
Counts (×10 s )
with hv = 21.2 eV is shown in Fig. S2.
5.0
2.5
located 2.48 eV below EF which matches
VBM (-2.48 eV)
well with previous UPS measurement4 on
0.0
GaN and the expected valence band offset
of InGaN with a 12% In content5. Based
on an optical gap of 2.79 eV, the CB lies
310 meV above EF, also in good
agreement with the expected separation
-10
-5
EF
0
E - EF [eV]
Figure S2 | Ultraviolet photoemission spectrum. UPS
spectrum (h = 21.2 eV) showing the energetic position
of the VBM referenced to the Fermi level of the
In0.12Ga0.88N quantum well sample.
between EF and CB at the In0.12Ga0.88N surface6.
Supplementary Method III: Carrier Density Calculations
We determine the excitation density using 𝜌 = 𝐹 ∗ 𝛼 ∗ (1 − 𝑅), where F is the photon flux of
the laser pulse (photons/cm2),  is the absorption coefficient, R is the percent reflected light from
sample surface at a given light polarization at 45 incidence. Refractive index values are taken from
ref. 7. We assume an absorption coefficient at our pump energy of 3.20 eV of 105 cm-1, consistent
with reference 8.8The laser spot diameter was determined from an 80/20 knife-edge measurement to
be 127 µm. Note that the h1 = 3.20 eV pump pulse only excites the In0.12Ga0.88N quantum well, not
the GaN substrate or the In0.02Ga0.98N underlayer.
Supplementary Method IV: Time-Resolved Photoluminescence (TRPL)
As the time window of the TR-2PPE measurements was limited to 5.5 ps, it was not appropriate
to fit the trapping and radiative recombination dynamics that occur on the nanosecond timescale.
We employed time-resolved fluorescence at different carrier concentrations to extract the A (SRH)
and B (radiative recombination) coefficients for use in the ABC Model. We carried out time
resolved
photoluminescence
(TRPL)
measurement using a time-correlatedsingle-photon-counting (TCSPC) module
(B&H, SPC130) and a SPAD detector
(IDQ, id100-50) with an instrument
response function of ~ 100 ps (FWHM).
The 402 nm excitation light was generated
from
the
second
harmonic
of
the
fundamental output (805 nm, 100 fs, 250
kHz)
of
a
regenerative
amplifier
(Coherent RegA amplifier seeded by
Coherent Mira oscillator). The light was
focused onto the sample surface by a 50X,
NA=0.5
objective
(Olympus
LMPLFLN50X) and the pulse duration
was
broadened
to
~
150
fs.
All
measurements were carried out at room
Figure S3 | Time-resolved photoluminescence. (a) Biexponential fit to experimental TRPL data. (b) Trapping
and radiative lifetimes as a function of carrier density.
temperature.
As detailed in the main text, the third-order Auger rate process occurs on the sub-picosecond to
the picosecond time scale. In the PL measurement with pulsed excitation on the nanosecond time
scale, the ultrafast Auger recombination process can be safely ignored. As shown in Fig. S3(a), the
time-dependent PL intensity is well described by a bi-exponential decay function on the nanosecond
timescale. The lifetimes extracted from bi-exponential fits for each carrier density using s-polarized
light are shown in Figure S3(b); the ~500 ps lifetime is assigned to radiative recombiantion and the
longer ~2 ns lifetimes attributed to trapping . The trapping and radiative coefficients were obtained
using the following:
𝐴=𝜏
1
(S1)
𝑡𝑟𝑎𝑝
1
𝐵 = 𝑛 ∗𝜏
𝑖
(S2)
𝑟𝑎𝑑
A plot of A and B values obtained for both s- and p-polarized light is shown in Fig. S4(a-d).
(a)
8 -1
A (x10 s )
6
4
4
s-polarization
2
8 -1
<A>s-pol= 4.75 ± 0.17 × 10 s
-8
p-polarization
8 -1
<A>p-pol = 4.44 ± 0.90 × 10 s
2
0
(c)
s-polarization
3 -1
B (x 10 cm s )
0
<B>s-pol = 3.73 ± 0.5 x 10
0.5
0.0
(b)
6
-10
18
for >1.25 x 10 cm
0
3 -1
cm s
-3
1 2 3 4 5 6 7
18
-3
Carrier Density (x10 cm )
(d)
p-polarization
<B>p-pol = 2.72 ± 0.38 x10
0.1
0.0
18
-10
for >1.25x10 cm
0
3 -1
cm s
-3
2
4
6
8
10
18
-3
Carrier Density (x10 cm )
Figure S4 | Extracted A and B coefficient values. (a-b) A coefficient values for s- and p-polarized light
repectively. (c-d) B coefficient values for s- and p-polarized light respectively.
A plot of B at each carrier density for both s- and p-polarized light is shown in Figure S4(a) and
(b) respectively. The value of A was fairly constant for each carrier density, yielding an average
value of 4.75 ± 0.17 × 108 s-1 for s-polarized light and 4.44 ± 0.90 × 108 s-1 for p-polarized light.
The value of B varied significantly at low carrier densities but was consistent over the range of
carrier densities (≥ 2.5 × 1018 cm-3) used in the TR-2PPE experiments to extract Auger
recombination rates. Averaging over this density range yields a B coefficient of 3.73 ± 0.50 × 10-10
cm3 s-1 for s-polarized light and 2.72 ± 0.38 × 10-10 cm3 s-1 for p-polarized light. These values of A
and B were fixed in the ABC model, allowing for a proper fit of C in the TR-2PPE data.
Supplementary Data I
0.40
It is also observed that there is relaxation
0.38
in the electron distribution about CB after
electron distribution for the same energy
range and timescale as used in the ABC
model fitting is shown in Figure S5. The
distribution relaxes ~60 meV with a time
constant of 366 ± 58 fs. This is slightly lower
than the ~200 fs decay rate constant of the
Auger dominated portion of the carrier loss
0.36
COG (eV)
excitation. The center-of-gravity of the hot
COG
Fit to COG
0.34
0.32
0.30
0.28
0.26
0
1
2
3
4
5
t (ps)
Figure S5| Center of gravity for hot electrons in
CB. The temporal evolution of the center of gravity
(COG) of the hot electron distribution as it relaxes to
the bandedge and areas of high InN content.
dynamics. The slower energy relaxation time
suggests the Auger process is more likely to occur in the delocalized states of the system, when the
wavefunction overlap between the electrons and holes is still significant.
Supplementary Data II
In order to analyze the dynamics of
electrons
in
the
conduction
band,
an
appropriate energetic integration window had
to be selected in order to include all electrons
undergoing the mechanisms appropriate to
the ABC model. It would be inappropriate to
include carriers from the high energy tail,
Fig. 2(B) in main text, that may have come
from Auger scattering and could artificially
inflate the carrier densities at time-zero. To
Figure S6 | Energy range integration and power
law fits. The increase of integrated intensity as a
function of power for a selection of energy ranges
integrated up to times-zero with power law fits.
avoid this situation, we plot the integrated intensities up to time-zero of several energy windows
with increasing power against a power law fit, 𝐼 = 𝑎𝑃𝑒𝑥𝑝 in order to determine which integration
windows increased linearly with power. From Fig. S6, it is clear that integrating an energy range
larger than 300 meV leads to a super linear increase with power while the 300 meV window
remains linear and includes as many of the CB electrons as possible.
The sample geometry used here is different than
those of typical LEDs. This may result in a small
difference in Auger recombination rate. To
elaborate, in a conventional LED, the QW is
sandwiched on both sides by GaN, with dielectric
Screened Coulomb Potential (eV)
Supplementary Data III
0.0
(a)
-0.1
-0.2
GaN-QW-GaN
GaN-QW-Vac
-0.3
-0.4
-0.5
constant of  = 10.0 ; for the sample used here, the
QW is on one side in contact with GaN and the
other side to vacuum ( = 1). A reduced dielectric
screening is expected to increase the Auger
recombination rate since the Auger rate constant is
proportional to the square of the Coulomb potential.
Figure S7(a) shows a plot of the screened Coulomb
potential for carriers in a GaN/InGaN/GaN QW
heterostructure
and
the
GaN/InGaN/Vacuum
Coulomb Potenial Ratio
9
1.0
(b)
0.9
0.8
0.7
0.6
0.5
0
20
40
60
80
Lateral e-h separation [nm]
100
Figure S7 | Coulomb interaction of carriers in
QW. (a) A comparison of the screened Coulomb
potential
experience
by
carriers
in
Gan/InGaN/GaN
and
GaN/InGaN/Vacuum
heterostructures. (b) Ratio of the potentials.
structure used in these experiments as a function of in-plane separation. A ratio of the potentials,
Fig. S7(b), shows the effect is strongest at large carrier separation distances when field lines are
more likely to extend beyond the QW structure, but negligible for e-h distance <10 nm, which is
typical for the InGaN quantum well.
Supplementary Data IV
To calculate IQE from the differential rates, we integrate equation (1) in the main text:
¥
IQE =
ò B× n(t)
2
× dt
0
×
2
2
3
n0 = ××
×A×n(t) + B× n(t) + Cn(t) ×
×dt
,
n0
(S3),
0
where n0 is the total carrier density generated at t = 0 by the laser pulse. Fig. S8 compares the
calculated IQE vs. carrier density (solid curves) obtained from equation S1, along with fittings of
the solid curves to equation (2) in the main text. The IQE curve generated from equation S1 for a
high Auger rate, C.n3 >> B.n2, cannot be fit by the same phenomenological rate constants applied to
equation 2 in the main text. This is evident Fig. S8A: with the microscopic kinetic parameters (A, B,
C)
obtained
in
the
text
for
the
In0.12Ga0.88N single QW from TR-2PPE
and TRPL measurements. The leastsquares fitting gives phenomenological
kinetic rate constants A’, B’, and C’.
Here, B’ is one order of magnitude
smaller than B and C’ is two orders of
magnitude
smaller
than
C.
The
phenomenological rate determined over
the lifetime of the carrier in steady-state
experiments can severely underestimate C
values of large magnitude as they provide
a lifetime-averaged C rather than the
instantaneous C value determined from
direct, non-equilibrium measurements on
conduction band carriers such as detailed
in the main text. Only when the Auger
rate is close to that of the radiative
recombination rate, C.n3 ~ B.n2, can the
phenomenological Auger and radiative
rate constants start to approach those of
the microscopic counter parts, as shown in
Fig. S8B.
Fig. S8. Simulated internal quantum efficiency (IQE).
The solid black curves are obtained from equation S1.
The dashed curves are fittings of the solid curves to
equation (2) in the main text. The rate constants (A, B, C)
used in generating the black curves and those obtained
fits (A’, B’, C’) are shown on the figures.
Fig. S9 plots the phenomenological Auger rate
constants (CFit) vs. the microscopic Auger rate
constant (CGen) over ten orders of magnitude. The
two Auger rate constants are the same for slow
Auger processes (C <10-30 cm6s-1) but increasingly
deviate from each other for faster Auger processes
by
as much as three orders of magnitude.
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2
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