1
Effects of the Narrow Band Gap on the Properties of InN
J. Wu
1,2
, W. Walukiewicz
2
, W. Shan
2
, K.M. Yu
2
, J.W. Ager III
2
,
E.E. Haller
2,3
, Hai Lu
4
and William J. Schaff
4
1. Applied Science and Technology Graduate Group, University of
California, Berkeley, CA 94720.
2. Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720.
3. Department of Materials Science and Engineering, University of
California, Berkeley, CA 94720.
4. Department of Electrical and Computer Engineering, Cornell
University, Ithaca, NY 14853.
Infrared reflection experiments were performed on wurtzite InN films with a
range of free electron concentrations grown by molecular beam epitaxy. Measurements of
the plasma edge frequencies were used to determine electron effective masses. The
results show a pronounced increase in the electron effective mass with increasing electron
concentration, indicating a non-parabolic conduction band in InN. We have also found a
large Burstein-Moss shift of the fundamental bandgap. The observed effects are
quantitatively described by the k⋅p interaction within the two band Kane model of narrow
gap semiconductors.
Electronic Mail: w_walukiewicz@lbl.gov
PACS numbers: 78.66.Fd, 78.30.-j
2
Wurtzite-structured indium nitride forms an alloy with GaN. The alloy on the Ga-
rich side is the key component of blue light emitting devices [1]. Early studies of InN
films grown by the sputtering method have suggested a direct bandgap of ~ 2 eV [2, 3].
Very recently, optical characterizations of InN crystals grown by molecular beam epitaxy
(MBE) [4, 5] or metal-organic vapor phase epitaxy [6] have provided convincing
evidence showing that the real bandgap energy of InN is actually about 0.7 eV at room
temperature. This value is close to the gap energy of 0.8 eV recently obtained by
pseudopotantial calculations [7]. The gap energy of InGaN ternary alloys has been shown
to cover a wide, continuous spectral range from the near infrared for InN to the near
ultraviolet for GaN [8].
Early measurements of the free electron effective mass in heavily doped (n > 10
20
cm
-3
) InN films have given a range of values ranging from 0.11m
0
[2] to 0.24m
0
[9]. A
recent measurement of MBE-grown InN layers (n = 2.8×10
19
cm
-3
) using infrared
spectroscopic ellipsometry has led to the value of m
*
= 0.14m
0
[10]. The recently
determined low value of the energy gap of InN puts in question and calls for a re-
evaluation of the previously determined effective mass. In addition, in the determination
of the free carrier effective mass from plasma resonance experiments, the effective mass
is inversely proportional to the optical dielectric constant (ε
∞
) as the plasma frequency is
measured [11]. Therefore, it is important to be careful with the choice of ε
∞
to compare
m
*
obtained by different groups.
In this paper we report the results of our studies of the effective mass and the
optical absorption edge of wurtzite InN with different free electron concentrations. We
find that the lowest conduction band of InN is non-parabolic with the electron effective
mass strongly dependent on the electron energy. The optical absorption edge shows a
concentration-dependent blue shift resulting from the Burstein-Moss effect. All these
observed effects are well described by the two-band Kane model for narrow gap
semiconductors.
InN films were grown on (0001) sapphire substrates with an AlN or GaN buffer
layer by molecular-beam epitaxy [12]. The thickness of the buffer layer ranges from 70
nm to 200 nm. The InN layer thickness is between 200 nm and 7.5 µm. Although most of
the samples were not intentionally doped, free electron concentrations ranging from
3
3.5×10
17
cm
-3
to 5.5×10
18
cm
-3
have been found in these samples by Hall Effect
measurements. Even higher free electron concentrations were achieved by doping InN
with Si. The free electron concentration of these doped samples varies between 1.0×10
19
cm
-3
and 4.5×10
19
cm
-3
. The details of the growth process have been published elsewhere
[12]. X-ray diffraction studies have shown that these InN epitaxial layers had high quality
and wurtzite structure with their c-axis perpendicular to the substrate surface. Hall
mobilities ranged from several hundred up to 2050 cm
2
/Vs. The samples were
characterized by conventional optical absorption and infrared reflection experiments. The
optical absorption measurements were performed using a CARY-2390 NIR-VIS-UV
spectrophotometer. The infrared reflection experiments were done on a BOMEM DA8
Fourier transform infrared spectrometer with a resolution of 4 cm
-1
. All experiments were
carried out at room temperature.
The surface reflection of extrinsic semiconductors in the infrared region by the
free carrier plasma is frequently used to determine the effective mass of the free carriers
[11]. Similar to the behavior of metals, extrinsic semiconductors strongly reflect infrared
light below the plasma frequency. The free carrier effective mass on the Fermi surface
can be calculated from the plasma frequency (ω
P
) if the carrier concentration and the
optical dielectric constant are known [10, 11].
2
0
2
*
P
ne
m
ωεε
∞
= . (1)
Figure 1 shows, from the left to the right, the infrared reflection curves of three
InN samples with free electron concentrations (Hall mobility) of 5.5×10
18
cm
-3
(615
cm
2
/Vs), 1.2×10
19
cm
-3
(1070 cm
2
/Vs) and 4.5×10
19
cm
-3
(600 cm
2
/Vs), respectively. The
plasma reflection edge is clearly resolved due to the high electron mobilities in these
samples. It can be seen that the plasma reflection edge shifts by as much as 1000 cm
-1
between these samples. We have used a standard complex dielectric function model that
includes finite lifetime broadening to fit the reflection curves [11]. The results are shown
as solid curves in Fig. 1. For thick samples with Febry-Perot oscillations, we have also
taken into account the optical interference occurring in the epilayer. The plasma
frequency and the damping parameter obtained from the fits are 950 cm
-1
and 100 cm
-1
,
1240 cm
-1
and 60 cm
-1
, and 1980 cm
-1
and 250 cm
-1
for these three samples, respectively.
4
The electron effective mass calculated from these plasma frequencies are plotted
as a function of the electron concentration in Fig. 2. In the calculation, a recently reported
isotropically averaged value of the optical dielectric constant, ε
∞
= 6.7, was used [10].
Also shown in Fig. 2 is the result from Ref.[10] measured with a MBE-grown InN film.
The three points with concentration above 10
20
cm
-3
are effective mass values calculated
from the plasma frequencies reported in Ref. [2], [9] and [13] using ε
∞
= 6.7. In contrast
to most semiconductors, the effective mass exhibits a strong dependence on the free
electron concentration.
The electron concentration dependent effective mass suggests a non-parabolic
conduction band. The conduction band non-parabolicity in other narrow bandgap
semiconductors, such as InSb [14] and InAs [15], has been recognized and studied
intensively. In these semiconductors the strong conduction band non-parabolicity is
attributed to the k⋅p interaction across the narrow direct gap between the conduction and
valence bands. The non-parabolic dispersion relation has been calculated by Kane using a
k⋅p perturbation approach [14]. We have applied Kane’s method to InN with a narrow
bandgap of 0.7 eV. Since the spin-orbit splitting (∆
so
) and crystal field splitting (∆
cr
) in
the valence bands are extremely small for group III-nitrides [16], we assume them to be
practically zero in InN. This approximation is used to estimate the perturbation of the
lowest conduction band by the valence bands. Neglecting further perturbations from
remote bands, an analytical form of the conduction band dispersion is obtained by solving
Kane’s two-band k⋅p model [14],
( )
,
2
4
2
1
2
0
22
2
0
22
−⋅+++=
GPGGC
E
m
k
EE
m
k
EkE
hh
(2)
where E
G
is the direct bandgap energy, and E
P
is an energy parameter related to the
momentum matrix element,
2
0
2
XPS
m
E
xP
= . (3)
The density of states effective mass is then k-dependent and given by [17]
( )
( )
dkkdE
k
km
C
2
*
h
= . (4)
5
The Fermi level is given by the dispersion energy in Eq.(2) evaluated at the Fermi
wavevector
(
)
3/1
2
3 nk
F
π= , neglecting the thermal broadening of the Fermi distribution.
We have calculated the effective mass as a function of electron concentration in this
model using a bandgap energy E
G
= 0.7 eV and E
P
equal to 7.5, 10 and 15 eV
respectively. The results are compared with experimental data in Fig. 2. It can be seen
that although the effective mass data were reported by different groups on InN films
grown by different methods, the calculation using E
P
= 10 eV shows reasonably good
agreement with all the measured data points. For comparison, we note that E
P
values for
wurtzite GaN as low as 7.7 eV [18] and as high as 14 eV [19] have been reported in the
literature. The extrapolation of the curve leads to an effective mass of 0.07m
0
at the
bottom of the conduction band. This value is much smaller than the effective mass of
0.22m
0
for GaN [11], but is close to 0.08m
0
for InP [19] that has an almost two times
larger band gap.
The non-parabolic dispersion relation of the conduction band also results in an
increase of the density of states from that of a parabolic band. It means that the Fermi
level rises less rapidly as the lowest conduction band is filled up with electrons. To
explore this effect, we have measured the optical absorption edge of InN films with a
wide range of free electron concentrations. Typical absorption curves (absorption
coefficient squared) are shown in the inset of Fig. 3. It can be seen that although the
conduction band is non-parabolic, the absorption can still be approximated by a square-
root dependence within ~ 0.05 eV above the onset of the absorption. We extrapolate the
linear part of the squared absorption down to the photon energy axis to determine the
absorption edge. A strong Burstein-Moss shift of the absorption edge with increasing
carrier concentration is observed. The dependence is plotted as solid circles in Fig. 3.
The absorption edge corresponds to the energy of the photons that make vertical
transitions from the upper valence band to the Fermi surface in the conduction band. The
electron concentration dependence of the absorption edge was calculated based on the
non-parabolic dispersion relation given by Eq.(2). In the calculation, the upper valance
band involved in the transition is assumed to be parabolic with the hole effective mass
equal to the free electron mass in the vacuum [20].
