1. Introduction

Cyclotron resonance (CR) in two-dimensional electron gases (2DEGs) formed in semiconductor inversion layers, heterojunctions, and quantum wells has been studied for many years, exhibiting a number of surprising phenomena [1–4]. These phenomena (so-called “CR anomalies”), observed as unexpected line broadening, splittings, and shifts, are believed to arise from the interplay of band structure, disorder, and interaction effects. As the sample mobility becomes higher, CR lines are expected to become narrower, which should permit more systematic studies to precisely determine the origins of these effects. CR has indeed been studied in a variety of 2DEG systems with a wide range of mobilities (10⁴ to 10⁶ cm²/Vs). However, CR studies of high-mobility (>10⁶ cm²/Vs) 2DEGs using traditional Fourier-transform infrared spectroscopy have had limited success due, at least partially, to the CR saturation effect [5]. In the high mobility/high carrier density limit, the 2DEGs behave as a metallic mirror, reflecting off most of the incident light at the CR peak, and thus, only a small amount of the incident light is transmitted by the high mobility 2DEG [6,7]. This effect results in an undesirable broadening of transmittance linewidths, relative to the true CR linewidths, as shown in Section 2 below.

Here, we demonstrate how the CR saturation effect can be overcome using terahertz (THz) time-domain techniques. First, through theoretical simulations while systematically increasing the mobility, we show that, in the high mobility limit, the calculated power transmittance peak is broader than the CR peak in the calculated real part of the complex conductivity by an amount dependent on the carrier density. However, the linewidth determined from the conductivity keeps decreasing with increasing mobility, always representing the true CR linewidth. In our experiments, described in Section 4, we observed long-lived time-domain CR oscillations in a high-mobility (3.7 × 10⁶ cm²/Vs) GaAs/AlGaAs 2DEG, measured with our time-domain THz magneto-spectroscopy system [8,9]. Quantum mechanically, these time-domain oscillations are due to a resonant optical transition across the Fermi energy between the highest filled Landau level and the lowest unfilled Landau level. The observed coherent oscillations were fitted well by sinusoids with exponentially-decaying amplitudes, through which we were able to directly determine the CR decay times and CR frequencies simultaneously, thereby overcoming the CR saturation effect.

2. Cyclotron resonance saturation effect: theoretical simulations

We approximate the sample as a 2DEG layer between two GaAs substrate layers, which have an admittance, Y = n _GaAs/Z ₀, where Z ₀ = 377 Ω is the impedance of free space and n _GaAs = 3.60 is the refractive index of GaAs at THz frequencies [10]. When the wavelength and skin depth are much larger than the 2DEG thickness, the THz electric field can be considered to be uniform. Namely, the THz electric field is not a function of 2DEG thickness, and the 2DEG can be treated as an interface between the two substrates. We employ the thin conducting film approximation described in Ref. [11], in which the transmission coefficient, T′, and reflection coefficient, R′, for a left-circularly polarized (or cyclotron active) THz field are given by

 

Fig. 1. (a) Calculated absorptance A, reflectance R, and transmittance T versus CR mobility at a fixed density n = 2 × 10¹¹ cm⁻². (b) Calculated linewidths of the real part of the conductivity and transmittance at n = 2 × 10¹⁰, 2 × 10¹¹, and 2 × 10¹² cm⁻². The mobility was changed from 1 × 10⁴ to 1 × 10⁸ cm²/Vs in (a) and (b). (c, d, e, and f) Calculated real part of the conductivity σ′ (blue lines) and 1−T (red lines) for different mobilities: (c) 3 × 10⁴, (d) 3 × 10⁵, (e) 3 × 10⁶, and (f) 3 × 10⁷ cm²/Vs, respectively. The deviation of the transmittance peak linewidth from the real conductivity linewidth becomes more significant with increasing mobility.

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where E(B) is the transmitted THz field in an applied external magnetic field B and E(0) is the transmitted THz field at 0 T. Therefore, the transmittance T is given by T = |T′|², and the reflectance R is given by R = |R′|². The absorptance A can then be calculated through A = 1 − T − R. Within the Drude model, the complex conductivity, σ, of the 2DEG layer in magnetic fields is given by:

where n is the electron sheet density in the 2DEG, m* is the electron effective mass, τ _CR is the CR decay time, and f_c is the CR frequency. The CR mobility is defined as µ _CR = eτ _CR/m*. We only consider the left circularly polarized (or cyclotron active) component because the right circularly polarized component does not contribute to cyclotron resonance.

Absorptance (A), transmittance (T), and reflectance (R) at the CR frequency f_c were thus calculated with varying mobilities from 3 × 10⁴ to 3 × 10⁷ cm²/Vs at a fixed density of n = 2 × 10¹¹ cm⁻². The results are shown in Fig. 1(a). At low mobilities, the absorptance A increases with the mobility followed by a saturation at µ _CR ~ 5.66 × 10⁵ cm²/Vs and then decreases with the further increase of the mobility. The reflectance R shows a monotonic increase while the transmittance T decreases with increasing mobility. For a mobility of 5 × 10⁶ cm²/Vs, the absorption at the resonance is ~18.28%, the transmittance is ~1.04%, and the reflectance is ~80.68%. It is clear that most of the incident power is reflected by the 2DEG and only a small amount is transmitted in this regime, where the mobility is sufficiently high. A low transmittance and high reflectance in high-mobility samples make Fourier-transform infrared techniques difficult and points to the need for an experimental technique with a sufficiently high signal-to-noise ratio to isolate the small transmission signal. Time-domain THz spectroscopy is an ideal tool for measuring CR in these high-mobility samples because of its high sensitivity.

Furthermore, the CR saturation effect affects the determination of the true CR linewidth from transmittance spectra. Figure 1(b) shows the calculated linewidths both in transmittance and conductivity spectra plotted versus the mobility for several carrier densities (n = 2 × 10¹⁰, 2 × 10¹¹, and 2 × 10¹² cm⁻²). It can be seen from Eq. (3) that the linewidth determined from the conductivity peak is inversely proportional to the CR decay time. In the low density and low mobility limit, a direct relation is present between the transmittance linewidth and the conductivity linewidth; in this limit, the true conductivity linewidth can be determined from the transmittance linewidth. However, with increasing mobility, the transmittance linewidth starts deviating from the true CR linewidth, and eventually saturates. The higher the sample density, the larger the deviation. Figures 1(c)–1(f) show the real part of the conductivity (σ′) and 1−T versus frequency for four different mobilities at a fixed density of 2 × 10¹¹ cm⁻². Again, the deviation is seen to increase with increasing mobility. It is clear from Eq. (3) that the CR linewidth can be accurately determined by examining the linewidth of the peak in the real part of the conductivity or the lifetime of time-domain oscillations. Hence, in the high mobility and/or high carrier density limit, the advantage of time-domain THz spectroscopy results from its ability to determine the full complex conductivity, σ in the frequency domain, and/or a direct measurement of the free induction decay of CR in the time domain as shown in the following section.

3. Experimental methods

In order to experimentally demonstrate the utility of time-domain THz spectroscopy to determine the true CR linewidth of a high-mobility 2DEG and overcome the saturation effect, we used a time-domain THz magneto-spectroscopy system [8,9]. The system consisted of a chirped pulse amplifier (CPA-2001, Clark-MXR, Inc.) with a wavelength of 800 nm and a pulse width of ~200 fs and a pair of <110> ZnTe crystals to generate and detect coherent radiation from 0.1 to 2.6 THz through surface rectification and electro-optic sampling [12], respectively. A shaker, operating at 2 Hz, provided time delays up to 80 ps. We averaged over 1000 scans for acquiring each time-domain waveform. We used a magneto-optical cryostat to generate magnetic fields up to 10 T, and the sample temperature was kept at 1.6 K for this study. We performed our experiments in a Faraday geometry with both the incident THz wave and the external magnetic field perpendicular to the sample surface. The sample was a GaAs 2DEG (sample No.EA0745) with n = 2 × 10¹¹ cm⁻² and µ _DC = 3.7 × 10⁶ cm²/Vs at 4 K, which was physically large enough (0.6 × 10 × 15 mm³) to neglect any finite-size-induced magneto-plasmonic shift of CR frequencies (see, e.g., [1–3]).

 

Fig. 2. THz waveforms transmitted through a high-mobility 2DEG at 0 T and at 1.42 T at 1.6 K. The coherent cyclotron resonance oscillations, present only in the 1.42 T waveform, are isolated by subtracting the 0 T waveform from the 1.42 T waveform.

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Fig. 3. Coherent cyclotron resonance oscillations of a GaAs 2DEG at magnetic fields from 0.6 to 2.2 T at 1.6 K, measured by time-domain THz magneto-spectroscopy. The traces are vertically offset for clarity. These CR oscillations are isolated by subtracting the transmitted THz waveform at 0 T from the transmitted THz waveforms at different magnetic fields.

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4. Experimental results and discussion

We measured transmitted THz electric fields through the 2DEG sample in magnetic field from 0 to 2.2 T at 1.6 K. In order to deduce the magnetic-field-induced changes to the transmitted waveforms, we subtracted the 0 T waveform from each finite-field waveform. One example is shown in Fig. 2, showing clear oscillations, after subtracting the 0 T waveform from the 1.42 T waveform. The magnetic-field-induced oscillations, extracted in this manner, from 0.6 T to 2.2 T are plotted in Fig. 3. The frequency of the oscillations is clearly seen to increase with increasing magnetic field. These oscillations result from cyclotron resonance, i.e., an optical transition between the highest-filled Landau level and the lowest-unfilled Landau level.

We fit these CR oscillations with an exponentially-decaying sinusoid:

where τ _CR is the CR decay time, f_c is the cyclotron frequency, E ₀ is a DC offset, A is the initial field amplitude, and θ ₀ is the initial phase of the oscillations. We fitted the CR oscillations within a 26 ps time window, because of two multiple reflection peaks that appear at ~28.3 ps and ~44.2 ps, respectively, due to magnet windows. They induce additional coherent oscillations that overlap those induced by the main THz pulse. In addition, the oscillations from −2 to 2 ps were not used for fitting because they contain extra information about initial absorption of the incident THz pulse. The 26 ps window chosen contains enough periods for fitting accuracy.

Several fitting examples are shown in Fig. 4. Here, the data recorded at magnetic fields of 1.8 T, 1.1 T, and 0.9 T were fitted with Eq. (4), showing excellent agreement with the experimental data. The obtained fitting parameters are summarized in Table 1 with estimated uncertainties. One can see that standard deviations are less than 1 % for CR decay times (τ _CR), while the CR frequency (f_c) is determined with much smaller uncertainties (<0.01%) as expected. The extracted cyclotron frequency f_c vs. magnetic field shows a linear relationship (not shown), also as expected from the relation 2πf_c = eB/m*. From the slope of the linear curve, we obtained an effective mass of 0.068m ₀, which is the standard value for electrons in a GaAs 2DEG. Also included in Table 1 are the CR decay times that would be obtained from transmittance linewidths (as fully discussed in Section 2). Clear differences between the CR decay times obtained through time-domain CR oscillations and power transmittance are seen.

Finally, we point out that one of the main limitations of frequency-domain analysis techniques is the presence of a second time-delayed satellite pulse due to multiple reflections in the sample substrate or magnet windows. In our experiment, there exist multiple satellite pulses due to multiple reflections in the magnet windows and the sample substrate that cannot be experimentally eliminated. The ultrafast spectroscopic technique gives us direct access to the time-domain data; this is not the case in earlier Fourier-transform infrared techniques that measure intensity transmission.

 

Fig. 4. Time-domain cyclotron resonance oscillations at 1.8, 1.1, and 0.9 T obtained by subtracting the transmitted THz waveform at 0 T from the transmitted THz waveforms at these three magnetic fields (red curves). We fit these oscillations using an exponentially-decaying sinusoid [see Eq. (4)]. Fitting curves are shown as blue curves.

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Table 1. CR decay times (τ_CR) and CR frequencies (f_c) obtained through fitting time-domain CR oscillations at 1.8, 1.1, and 0.9 T. Standard deviations of the fittings are also shown. For comparison, CR decay times that would be obtained from transmittance linewithds (τ_Tr) are shown in the last column.

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The presence of a time-shifted copy of the transmitted THz pulse, E(t) = E ₀(t) + AE ₀(t − T), where E ₀(t) is the time-domain waveform in the absence of multiple reflections, results in a spectrum E(ν) = [1 + A exp(i2πTν)]×E ₀(ν), where A is the ratio of the second pulse amplitude to the first pulse amplitude and E ₀(ν) is the spectrum of the THz pulse in the absence of multiple reflections. If T < τ _CR, then the modulation term (the square brackets) modifies the spectral shape, E ₀(ν), and does not permit the direct recovery of the single pulse spectrum.

There exist two methods of overcoming this multiple pulse effect to extract the conductivity linewidth or, equivalently, the CR lifetime from the single pulse spectrum, E ₀(ν). First, the frequency-domain waveform can be directly modeled using:

where the summation over j is the summation over the N different multiple pulse reflections, T_j, that each have relative amplitudes, A_j. The delays are determined by the optical window thicknesses and refractive indices, while the amplitudes are determined by the appropriate window Fresnel reflection and transmission coefficients. This frequency-domain fitting would require the extraction of the set of T_j and A_j parameters in addition to the linewidth and cyclotron frequency that would be extracted from E ₀(ν). Second, a direct time-domain analysis provides an alternate method to extract the CR lifetime, τ _CR, from experiments where the pulse-to-pulse separation, T, is not large with respect to τ _CR. A direct fitting of the time-domain to Eq. (4) allows us to directly obtain τ _CR.

5. Conclusions

We have observed time-domain cyclotron resonance oscillations in a GaAs/AlGaAs 2DEG using a THz magneto-spectroscopy system from 0.6 to 2.2 T at 1.6 K. By fitting these oscillations with an exponentially-decaying sinusoid, we were able to directly determine the cyclotron resonance decay time and frequency. We demonstrated, both experimentally and theoretically, that using time-domain THz magneto-spectroscopy one can overcome the cyclotron resonance saturation effect and determine the true CR linewidths in high-mobility 2DEG samples. We also performed calculations for mobilities as high as 10⁷ cm²/Vs so that these results can be used to provide other researchers with guidance in applying this technique and apparatus to even higher mobility samples. In addition, we showed that our time-domain model can correctly deduce the CR lifetime even in the presence of multiple pulses due to multiple reflections.