Terahertz Time-Domain Magnetospectroscopy of a High-Mobility
  Two-Dimensional Electron Gas
Terahertz Time-Domain Magnetospectroscopy of a High-Mobility
Two-Dimensional Electron Gas
Xiangfeng Wang, David J. Hilton, Lei Ren, Daniel M. Mittleman, and Junichiro Kono∗
Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA
John L. Reno
Sandia National Laboratories, P. O. Box 5800, Albuquerque, New Mexico 87185, USA
(Dated: October 27, 2018)
We have observed cyclotron resonance in a high-mobility GaAs/AlGaAs two-dimensional electron
gas by using the techniques of terahertz time-domain spectroscopy combined with magnetic fields.
From this, we calculate the real and imaginary parts of the diagonal elements of the magnetocon-
ductivity tensor, which in turn allows us to extract the concentration, effective mass, and scattering
time of the electrons in the sample. We demonstrate the utility of ultrafast terahertz spectroscopy,
which can recover the true linewidth of cyclotron resonance in a high-mobility (> 106 cm2 V−1 s−1)
sample without being affected by the saturation effect.
PACS numbers: 300.6500 Spectroscopy, time-resolved, 320.7130 Ultrafast processes in condensed matter,
including semiconductors
Quantum coherence is an important ingredient in mod-
ern condensed matter physics as well as in emerging tech-
nologies. The creation and manipulation of a coherent
superposition of two or multiple quantum states is the
subject of many current studies. An ultrahigh-mobility
two-dimensional electron gas (2DEG) offers an ideal sys-
tem for studying novel quantum coherent phenomena in
a clean, solid-state environment. In particular, when one
applies a magnetic field perpendicular to the 2DEG, the
density of states splits into Landau levels, making a fully-
tunable, atomic-like system. In addition, a variety of phe-
nomena that occur in the 2DEG arising from carrier in-
teractions, confinement, and disorder can make quantum
coherent effects even more exotic than in atomic or molec-
ular systems. However, there has been little success in
performing coherent spectroscopy of Landau-quantized
2DEGs, although there is a long history of cyclotron res-
onance (CR) studies of 2DEGs using Fourier-transform
infrared (FTIR) spectroscopy [1, 2, 3, 4, 5, 6].
Terahertz (THz) time-domain magnetospec-
troscopy [7], which combines conventional THz
time-domain spectroscopy (THz-TDS) with a high
magnetic field, has a number of inherent advantages
compared to traditional FTIR techniques. THz-TDS
directly measures both the amplitude and phase of
the electric field E
and allows for the simultaneous
determination of the real and imaginary parts of the
conductivity without using Kramers-Kronig techniques.
Additionally, use of a temporally-gated detection
scheme, common to THz-TDS techniques, significantly
suppresses background thermal noise and results in an
enhanced signal-to-noise ratio [8, 9].
THz-TDS was used earlier [10] to observe CR in rela-
tively low-mobility (µe = 2.7 × 10
5 cm2V−1s−1) 2DEG
∗Electronic address: kono@rice.edu
samples. In addition, THz-TDS has been successfully
employed to study quantum coherent phenomena in a
wide range of systems, including the rotational transi-
tions of N2O molecules [11], intersubband transitions in
semiconductor quantum wells [12], and surface plasmons
propagating on metal-film hole arrays [13].
Here, we report the observation of long-lived,
magnetic-field-dependent coherent oscillations in a high-
mobility GaAs/AlGaAs 2DEG in a perpendicular mag-
netic field. We explain our observations in terms of a co-
herent superposition created by the incident THz pulse
between the lowest unfilled Landau level and the highest
filled Landau level. In addition, we determine elements
of the complex magnetoconductivity tensor σ̃ as a func-
tion of both frequency ν and magnetic field B, which
in turn allows us to determine the cyclotron frequency
νc, effective mass m
∗, and cyclotron resonance linewidth
∆νc (or the scattering time τ = 1/∆νc) as a function of
B. Finally, we show that THz-TDS can overcome the
“saturation effect” [14, 15] that often prevents FTIR-
based techniques from determining the true linewithds
of CR in high-mobility (µe > 10
6 cm2V−1s−1) 2DEGs.
We successfully measured the linewidth as a function of
temperature and magnetic field, which will be reported
in detail elsewhere.
Broadband, ultrashort THz pulses were generated and
detected using a standard photoconductive antenna-
receiver setup [8, 9]. Time delay was provided by an
oscillating retro-reflector operating at 3 Hz; each data
set was the averaged result of ∼ 800 scans. We used an
Oxford superconducting magnet (SM-4000-10T) to pro-
duce fields ranging from 0 to 1.4 T and temperatures from
1.5 K to 300 K. The sample studied in this experiment
was a modulation-doped GaAs/AlGaAs single quantum
well with an electron concentration of ne = 2.0 ×
1011 cm−2 and mobility of µe = 3.7 × 10
6 cm2 V−1 s−1
at 4.2 K, determined through Shubnikov-de Haas and
DC conductivity measurements.
http://arxiv.org/abs/0704.1883v1
mailto:kono@rice.edu
20100-10-20
Time (ps)
1.28T
FIG. 1: (color online) THz waveforms transmitted through
a high-mobility 2DEG at 0 T (a) and at 1.28 T (b) at 2 K.
The cyclotron oscillations induced by the magnetic field (c)
are isolated by subtracting (a) from (b).
In this experiment, the THz waveform is measured af-
ter transmission through the sample in a B field from 0 T
to 1.4 T. Figure 1 plots these waveforms at 0 T [trace (a)]
and at a finite B (1.28 T) [trace (b)]. Trace (c) is the
difference between the transmitted THz electric field at
1.28 T and 0 T highlighting the change to the THz trans-
mission due to the B field (data enlarged 10 times), which
shows the B-induced oscillations of the electric field of
the THz pulse. We verify that the observed oscillations
originate from the 2DEG and not a B-dependence of any
of the optics in the experiment by first measuring the B-
dependent THz transmission in the absence of the 2DEG
in an otherwise identical configuration. Figure 2(a) shows
similar oscillations induced in the transmitted THz wave-
form from 0.7 T to 1.4 T whose frequency and decay time
vary with B.
Figure 3(a) shows the amplitude of the Fourier-
transformed electric fields at 0 T and 1.28 T. A B-field-
induced absorption, or a dip, is clearly seen in the 1.28 T
spectrum. Figure 3(b) shows the magnitude of the com-
plex transmission coefficient (T =
∣eiφ) at 1.28 T,
while Fig. 3(c) shows the phase, φ. In our experiment,
the THz pulse is linearly polarized (x̂), and we detect
only one polarization component (x̂) after transmission
through the sample. As a result, our measurement is
dependent on the corresponding diagonal element of the
magnetoconductivity tensor, σxx [1]. In order to extract
the conductivity from the complex transmission coeffi-
cient, we model this sample as a thin conducting sheet
on a thick substrate with an index of refraction, n. In this
approximation, the ratio of the Fourier transform of the
waveforms at a finite B field, E
, to the zero-field
transmitted spectrum, E
, is given by:
2Y + σxx
) , (1)
20100-10
Time (ps)
0.8 T
0.9 T
1.0 T
1.1 T
1.2 T
1.3 T
1.4 T
0.7 T
(a) E
|N +1>
FIG. 2: (a) Time-domain cyclotron oscillations in a high-
mobility 2DEG from 0.7 T to 1.4 T at 2 K. Traces are verti-
cally offset for clarity. (b) A Landau-quantized 2DEG. DOS:
density of states.
0.80.70.60.50.40.3
Frequency (THz)
 Experiment
 Theory
1.28 T
σ''xx
0.80.70.60.50.40.3
Frequency (THz)
1.28 T
1.20.80.40.0
Frequency (THz)
 1.28 T
(a) 1.2
0.80.70.60.50.40.3
Frequency (THz)
1.28 T
FIG. 3: (color online) (a) Amplitude of the transformed elec-
tric fields at 0 T and 1.28 T. (b) Magnitude of the complex
transmission coefficient at 1.28 T. (c) Phase of the transmis-
sion coefficient. (d) Real (σ′
) and imaginary (σ′′
) parts of
the magnetoconductivity tensor element σxx at 1.28 T. The
trace is vertically offset.
where Y = n/Z0 is the admittance of the GaAs substrate
and Z0 = 377 Ω is the impedance of free space [8]. Fig-
ure 3(d) highlights real (σ′xx) and imaginary (σ
xx) parts
of the extracted conductivity tensor element at 1.28 T.
Due to the rotational symmetry of the system perpendic-
ular to the plane of the 2DEG, we would expect the same
result in the case of input polarization and detection both
along ŷ, i.e., σxx = σyy [1].
We determine the cyclotron frequency νc (s
−1), the
cyclotron resonance linewidth ∆νc = 1/τ (s
−1), and the
magnitude of the conductivity σ0 (Ω
−1) by fitting the
results shown in Figs. 3(c) and 3(d). Both the real (σ′xx)
and imaginary (σ′′xx) parts of the magnetoconductivity
tensor element are fit by
σxx = σ
xx + iσ
1 + 2πi
ν − νc
. (2)
A representative fit at 1.28 T is shown in Fig. 3(d), using
σ0 = 0.0126 Ω
−1, νc = 0.529 THz, and τ = 15.6 ps.
An applied B field perpendicular to the 2DEG results
in the formation of discrete Landau levels [see Fig. 2(b)]
with an energy separation, ∆E, between the
∣N + 1〉 levels given by
∆E = ~
= hνc (3)
where e is the electron charge and h is Planck’s constant.
An incident THz wave with a photon energy equal to
this separation coherently creates a superposition state
between the highest filled Landau level,
, and the low-
est unfilled Landau level,
∣N +1
, as shown in Fig. 2(b).
This results in an atomic-like two-level system; all other
Landau levels are either completely filled or completely
empty (as long as the THz field is sufficiently weak as in
our experiment) and do not affect the transmission of the
THz pulse. The observed damped oscillations in our ex-
perimental data can thus be viewed as the free induction
decay [16] of such coherently coupled Landau levels.
Using the extracted value of νc and Eq. (3), we obtain
a value of the effective mass of m∗ = 0.0676m0, where
m0 = 9.11 × 10
−31 kg is the free electron mass. Also,
using the extracted values of σ0 and τ , we can determine
the value of the electron concentration of ne = 1.95 ×
1011 cm−2, which is consistent with the concentration ob-
tained from transport measurements (2.0 × 1011 cm−2).
Finally, the extracted linewidth is a measure of the scat-
tering mechanisms present in the sample at this temper-
ature and B field. We have systematically studied the
temperature and B dependence of ∆νc, which would al-
low us to elucidate a detailed theoretical understanding
of the physical origins of this linewidth and will be re-
ported in detail eslewhere.
In high-mobility samples, the apparent linewidth de-
termined by FTIR measurements is much larger than the
true linewidth, a phenomenon commonly referred to as
the “saturation effect” [14, 15]. This results from the
decrease in detectable transmission of the THz radiation
over a broad spectral range; as the conductivity increases
with either increasing carrier concentration or mobility,
a spectral region exists with a finite width where, effec-
tively, no transmission is permitted. The lack of a phase
sensitive detection scheme in traditional FTIR techniques
makes the direct determination of the complex conduc-
tivity in this situation difficult; the broadened linewidth
in this case could result either from the increase in mo-
bility or concentration. Because of this saturation effect,
almost no systematic linewidth studies exist for high-
mobility 2DEG samples (> 106 cm2 V−1 s−1).
In order to overcome the lack of phase sensitive detec-
tion in FTIR measurements, different methods have been
proposed. For example, measurement of the transmission
coefficient of a 2DEG over a broad spectral range will per-
mit the use of Kramers-Kronig techniques to calculate
the phase at THz frequencies and determine the complex
conductivity [17]. A second alternate method for deter-
mining the complex conductivity assumes a Drude form
for σ̃ and fits this to the measured intensity transmission
coefficient; lack of a direct phase measurement makes this
an ambiguous determination of ne and τ [14].
THz-TDS allows for the direct determination of the
full complex conductivity of the sample without resort to
Kramers-Kronig techniques and without an a priori as-
sumption of the lineshape function. The increased signal-
to-noise ratio inherent to the gated detection scheme
allows for the determination of the lower transmitted
THz signals that result from high-mobility and high-
concentration 2DEGs. Second, the additional spectro-
scopic information determined from the phase sensitive
measurement removes the ambiguity between ne and τ .
As a result, no assumption of lineshape is necessary in or-
der to calculate the full complex conductivity. Employing
this technique allows for the simultaneous determination
of both ne and τ from the measurement of the transmit-
ted THz waveform electric field.
In summary, we have observed time-domain cyclotron
resonance oscillations in a GaAs/AlGaAs 2DEG, which
can be modeled as the decay of the coherent superpo-
sition of two coupled Landau levels induced by the in-
cident THz pulse. The real and imaginary parts of the
conductivity are determined simultaneously at different
magnetic fields without using Kramers-Kronig analysis.
We show that our THz technique has many advantages
for doing cyclotron resonance measurements, especially
for high-mobility samples.
This work was supported by the National Science
Foundation (through Grant Nos. DMR-0134058 and
DMR-0325474). Sandia is a multiprogram laboratory op-
erated by Sandia Corporation, a Lockheed Martin Com-
pany, for the National Nuclear Security Administration
under Contract DE-AC04-94AL85000.
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