We develop a new characteristic matrix-based method to analyze cyclotron resonance experiments in high mobility two-dimensional electron gas samples where direct interference between primary and satellite reflections has previously limited the frequency resolution. This model is used to simulate experimental data taken using terahertz time-domain spectroscopy that show multiple pulses from the substrate with a separation of 15 ps that directly interfere in the time-domain. We determine a cyclotron dephasing lifetime of 15.1±0.5 ps at 1.5 K and 5.0±0.5 ps at 75 K.
© 2012 OSA
The mitigation and control of coherence in quantum states is an area of great current interest that is critical for the development of the next generation of electronic and optoelectronic devices. The high mobility two-dimensional electron gas (2DEG) is one model system used to study the fundamental material physics that governs dephasing in semiconductors. Samples with the highest mobilities are the most desirable for device applications due to their long decoherence times, so there exists a clear need to develop and enhance experimental tools that can determine coherence lifetimes and elucidate the decoherence mechanisms under a wide variety of experimental conditions (temperature, magnetic field, mobility, carrier concentration, etc).
One powerful tool used to generate and study quantum coherence in semiconductors is cyclotron resonance spectroscopy [1–3]. This technique determines a range of material properties including the effective mass of carriers, the carrier concentration, and the scattering lifetimes of carriers near the Fermi surface. Terahertz time-domain spectroscopy has recently emerged as a promising experimental implementation of cyclotron resonance that can overcome the limitations of prior generations of these experiments in high mobility 2DEGs. This includes the saturation effect that has made the determination of the mobility from the measured transmission line width difficult . This can be overcome using the phase sensitive detection techniques that are commonly used in terahertz time-domain spectroscopy [3, 5, 6].
A significant limitation of terahertz time-domain spectroscopy, discussed in this manuscript, results from the frequency-domain analysis method developed to recover the complex permittivity and permeability from the measured time-domain waveforms . In this manuscript, we develop a characteristic matrix-based method to analyze terahertz time-domain transmission experiments in samples with broken time-reversal symmetry, which include the 2DEG, magnetic materials, and other chiral materials. To demonstrate the utility of this method, we analyze terahertz transmission data through a high mobility two-dimensional electron gas sample to determine the dephasing lifetime where there is direct interference between the primary and satellite pulses. We determine a dephasing lifetime of T2 = 15.1 ± 0.5 ps at 1.5 K and T2 = 5.0 ± 0.5 ps at 75 K in a high mobility 2DEG in an external magnetic field of ±1.25 T using time-domain data that show multiple pulses with a separation of 15 ps.
The 2DEG sample studied in our experiments (sample No. EA0745) is grown on an L = 625 μm thick undoped gallium arsenide substrate using molecular beam epitaxy. This substrate is approximately 7.5λ thick at a wavelength of 300 μm (1 THz), which leads to direct interference between the primary and satellite pulses in our terahertz time-domain spectroscopy experiments. The 2DEG layer is a d = 30 nm gallium arsenide quantum well that is modulation doped to an electron sheet carrier concentration of ne = 2 × 1011 cm−2. The electron mobility, determined using electrical transport measurements, in the 2DEG layer is μe = 3.7 × 106 cm2 V−1 s−1, which corresponds to a scattering lifetime of τ0 = 229 ps.
A diagram of the terahertz time-domain spectrometer used in our experiments is shown in Fig. 1. Near single cycle terahertz pulses are created by optical rectification of a 100 fs titanium:sapphire pulse in a (110) oriented zinc telluride (ZnTe) single crystal, which results in the emission of a coherent broadband spectrum with a bandwidth of approximately Δν = 0.8 THz that is centered near ν0 = 0.5 THz [3, 5, 6]. The terahertz electric field generated by optical rectification is linearly polarized along a direction that is defined to be the x̂ direction. While not directly measured in our experiments, the typical electric field generated using this method is |ETHz| ∼ 100 Vcm−1, which has a corresponding |BTHz| ∼ 33 mT. For the temperature range in our experiments, this terahertz electromagnetic field results in negligible sample heating.
The electrooptic detector is constructed using a second ZnTe crystal. The terahertz pulse and a weak optical gate pulse overlap in time within the ZnTe, where their relative timing can be varied using a mechanical delay stage. Through the electrooptic effect, the polarization state of the optical pulse rotates by an amount proportional to the terahertz electric field amplitude within the temporal duration of the optical pulse. The changes to the optical pulse polarization due to the terahertz pulse field are measured using a quarter wave plate, polarizing cube beam splitter and a pair of balanced photodiodes [8–10]. This detector configuration records the amplitude and phase of one linearly polarized component of the transmitted terahertz electric field. After transmission through the 2DEG and substrate, the terahertz pulse polarization state is elliptical due to the induced circular dichroism and birefringence that results from the Landau quantization. In our experiments, we align the polarization axis (ŷ) of the detector to be perpendicular to the input polarization and measure this component of the electric field in an external magnetic field at both ±Bext to remove artifacts that result in the data that are due to the low polarization extinction ratio (100:1) of this detection method.
The external magnetic field, Bext, is generated in an Oxford Instruments SpectroMag split coil magnet and aligned so that it is perpendicular to the 2DEG, which is defined to be the ẑ direction. This perpendicular magnetic field results in the formation of a spectrum of Landau levels with an energy spacing, ΔE = h̄eB/m*. Here, e is the electron charge, h̄ is the reduced Planck constant, B is the magnetic field, and m* is the electron effective mass .
An external terahertz electromagnetic field, tuned to resonance with the Landau level energy spacing between the highest filled, |n〉, and lowest unfilled, |n + 1〉, levels results in the formation of a coherent superposition state, |ψ(t)〉 = A|n〉 + B|n + 1〉. Dephasing of the ensemble of cyclotrons will result in oscillations in the transmitted terahertz electric field that decay with a lifetime that we define as T2. Figures 2(a) and 2(b) show the perpendicular component of the transmitted terahertz electric field, Ey(t), after the 2DEG at T = 75 K in part (a) and T = 1.5 K in part (b) and in an external magnetic field Bext = (1.25 T)ẑ. These data show the amplitude and phase of the electric field oscillations directly in the time-domain that result from ensemble dephasing.
The finite thickness of the eight quartz magnet windows and gallium arsenide substrate lead to the formation of a series of satellite terahertz pulses, also observed in Figs. 2(a) and 2(b), at delays determined by their thicknesses and the refractive index of quartz or gallium arsenide. These pulses directly interfere with each other and are separated by approximately 15 ps, which corresponds to multiple passes in the gallium arsenide substrate.
A common method used to reduce the effects of multiple reflections is to introduce wedged optical components to refract the primary pulse along a different direction than the satellite pulses. Since the primary and satellite pulses travel with different propagation vector directions, this permits the use of spatial filtering techniques to isolate the primary pulse and eliminate the observation of these satellite pulses at the detector. In the specific magnet used in this experiment, several windows are permanently glued in place and cannot easily be replaced. While we focus in this manuscript on the substrate multiple reflection, a series of satellite pulses from the windows exists that would directly interfere with the primary pulse, as well.
A variety of numerical methods have been proposed as alternates to wedged optical components to eliminate the secondary multiple reflections in these data. These are possible because of the subpicosecond resolution of the electrooptic detection method used in terahertz time-domain spectroscopy. One method described in Ref.  numerically windows the time-domain data to remove the satellite pulses before calculating the spectrum using a fast Fourier Transform algorithm. This numerical window, however, reduces the frequency resolution of the acquired data, which makes this analysis method problematic in samples with narrow spectral features like the high mobility 2DEG or, equivalently, when the primary and satellite pulses are not well separated in time.
Windowed time-domain fitting is a recently proposed alternate method to determine the cyclotron dephasing time in high mobility 2DEG samples . An exponentially decaying sinusoidal function is directly fitted to a time-windowed subset of the transmitted terahertz time-domain waveform, which used to extrapolate the decay beyond the window. This method may be challenging to implement, however, when the dephasing lifetime, T2, is substantially longer than the width of the temporal window since the time-domain oscillations available to be fitted may not decay within the fitting window to determine T2 without a significant uncertainty, which is limited by the finite signal-to-noise ratio of the experimental data.
3. Characteristic matrix method
Development of a model of the transmitted time-domain waveform that includes multiple reflections and does not require numerical windowing would permit the analysis of cyclotron resonance experiments in high mobility samples with long cyclotron decoherence lifetimes, T2, that are the focus of future devices applications. In addition, as has been recently shown in Ref. , the use of time-delayed satellite pulses is an experimental method of coherent control of the quantum superposition state, which is needed for future applications based on coherent manipulation of the cyclotron wavefunction and will require more sophisticated methods to properly describe electromagnetic wave propagation.
Electromagnetic wave propagation in stratified media can be modeled using the characteristic matrix method, which has been previously used to analyze multilayer isotropic materials studied using terahertz time-domain spectroscopy in Ref. . Here, we derive a matrix method to describe wave propagation in stratified materials with broken time-reversal symmetry that include the 2DEG. This will result in a formalism that is similar to the characteristic matrix method of electromagnetic wave propagation in stratified isotropic media in Ref. .
Time-dependent electromagnetic fields can be decomposed into monochromatic waves with exp (−i2πνt) harmonic time dependence using the Fourier relations . Each monochromatic component of the full electromagnetic wave is a solution of Maxwell’s equations, starting in this manuscript in SI units.Eq. (2a). This matrix can be diagonalized in the circular polarization basis [σ̂+, σ̂−, ẑ] defined using , and is represented by a matrix in this alternate basis in Eq. (2b).
Monochromatic components of the total electromagnetic field, polarized in the x–y plane, can be generally written in stratified samples in Eq. (3).Eq. (3) are the interference between the forward propagating (k̂ = +ẑ) waves as well as the reverse propagating (k̂ = −ẑ) waves, as will be discussed below.16–18] and is applicable to experiments in many superconducting magnets systems, which have limited usable solid angles.
The interfaces between these stratified layers define a set, z ∈ [z0, z1 ..., zℓ], where z0 is the input interface of this sample and zℓ is the final interface, as shown in Fig. 3. The solutions to Eq. (4) within the one layer defined by zj ≤ z ≤ zj+1 are:Eq. (4).
The complete electromagnetic field at an interface (z = zj) is a sufficient initial condition to determine these four unknown constants, which can then be propagated to the next interface, zj+1, using Eq. (5). Using characteristic matrix notation, the component of the electromagnetic wave with σ̂± polarization is represented by ℚ± (z, ν).
In the semi-infinite medium before the first interface (z ≤ z0), the full electromagnetic field in Eq. (3) is the superposition of the incident field (subscript i) and the reflected waves (subscript r), while beyond the final interface (z ≥ zℓ), the full electromagnetic wave defines the transmitted component (subscript t). With these definitions, transmission coefficients for the σ̂± components are written in Eq. (10).
4. Results and discussion
At Bext = 0, the permittivity of the undoped gallium arsenide substrate at terahertz frequencies is determined by the interband contributions, which are nonresonant at terahertz frequencies and result in a purely real permittivity, εb = 12.96ε0. In the 2DEG layer, the free carrier response is an additional contribution that can be approximated by the Drude model. The measured electron mobility in the 2DEG determines the scattering time of τ0 = 229 ps or, equivalently, a Drude line width of Δν = 2.8 GHz. Since the terahertz pulse in these experiments has a usable lower frequency much greater than 2.8 GHz, we neglect this free carrier response in the absence an external field (Bext = 0) in the 2DEG.
The external magnetic field, Bext, breaks time-reversal symmetry in the 2DEG and results in a permittivity for σ̂+ that differs from the permittivity for σ̂−. The magnetic field results in the formation of a discrete set of Landau levels whose energy spacing is the cyclotron energy, hνCR. The interaction of a weak terahertz electromagnetic wave with these states can be modeled using a dipole interaction Hamiltonian, which allows us to neglect excitation to higher Landau levels in a two-level system approximation. The parity of the wavefunctions, |n〉and |n + 1〉, permits coupling of these states with the cyclotron resonance active polarization, defined to be σ̂+, while this transition cannot occur using the orthogonal circular polarization (σ̂−). As a result, the cyclotron inactive mode (σ̂−) has a permittivity element that does not change (εmm = εb) in the external magnetic field, Bext.
We model this ensemble of cyclotrons using the two-level approximation, assuming a weak terahertz electromagnetic field. This model results in a complex susceptibility for the cyclotron resonance active (σ̂+) that is given by:20].
The imaginary part of the susceptibility (χi) has a Lorentzian line shape that describes the circular dichroism induced by the magnetic field. In the quantum limit where the number of filled Landau levels is small, T2 is a direct measure of the influence of magnetic field on scattering processes in the strongly interacting electrons in the Landau levels near the Fermi surface [21–23], which can be experimentally measured using the time-domain oscillations that result from this Lorentzian circular dichroism. The real part (χr) is the associated dispersive line shape describing the concomitant circular birefringence in the external magnetic field that determines the frequency and carrier-envelope phase of the oscillations with respect to the decay envelope. Both the circular dichroism and birefringence are centered at cyclotron resonance frequency, νCR, with an absorption line width of ΔνFWHM = 2/(πT2) in the low excitation limit. The complex permittivity tensor element for the cyclotron active polarization (σ̂+) is εpp = εb + [1 + χ̃(ν)]ε0, which describes the response of the 2DEG layer to the cyclotron active (σ̂+) component of electromagnetic field in this two-level approximation.
The 2DEG/substrate sample can be modeled using the characteristic matrix equations assuming that it is a conducting layer with a permittivity tensor, ε̄1, and thickness, d, that is grown on a substrate with an isotropic permittivity, ε2, and thickness, L, as shown in Fig. 3(b). The two layers have permeabilities given by the free space value, μ0. This two layer model is a significant simplification when compared to the ∼ 600 layers of alternating AlGaAs and GaAs in the 2DEG. This two layer model is, however, sufficient to predict the satellite pulse formation and the existence of cyclotron oscillations in the transmitted data. Since this sample was grown using molecular beam epitaxy, which has precise control over the thickness and composition of each layer, a complete model of the full structure is also possible with our analysis method and will be the focus of future work.
The four elements of the total characteristic matrix in this two layer approximation are given in Eq. (13):Eq. (11). The sample is in vacuum with a free space admittance, , in the incident, Yi, and transmitted, Yt, semi-infinite media.
The model of the excitation pulse is a Gaussian spectrum with a bandwidth of Δν = 0.8 THz centered at ν0 = 0.5 THz, which results in the formation of one single cycle terahertz pulse (not shown). We use a modeled THz pulse instead of experimental data since it is difficult to acquire the incident terahertz waveform needed for this calculation to demonstrate the validity of this characteristic matrix modeling technique, which would require temporarily removing the sample in this cryogenic magnet system without warming up system. The transmitted terahertz electric field at Bext = 0 is plotted in Fig. 2(c) and shows the formation of a sequence of satellite pulses at time intervals of 15 ps. Fig. 2(d) plots the predicted change to the ŷ electric field at Bext = (1.25 T) ẑ and T = 75 K, while Fig. 2(e) shows the same experimental conditions at T = 1.5 K assuming a longer dephasing time, presumably due to the reduced influence of phonon scattering on dephasing at low temperature.
We can determine a dephasing lifetime of T2 = 15.1 ± 0.5 ps at 1.5 K and T2 = 5.0 ± 0.5 ps at 75 K by fitting the experimental data to this model calculation. The agreement between the model and experimental data shows that this characteristic matrix approach correctly describes electromagnetic propagation where direct interference between the primary and satellite pulses occurs. It is evident from these results that the amplitudes of the satellite pulse oscillations in Fig. 2(d) and 2(e), when compared to the experimental data in Fig. 2(a) and 2(b), are not correctly predicted by this model. This is likely related to the line shape function assumed in Eq. (12) that does not correctly model the interaction of the time-delayed terahertz electric field with the partly dephased population of cyclotrons from the primary terahertz pulse .
In summary, we have modeled wave propagation in stratified media with broken time-reversal symmetry using a characteristic matrix approach. We have shown that the generation of a sequence of satellite pulses can be properly described using this method and have used the resulting calculated time-domain waveforms to determine the cyclotron dephasing lifetime in a high mobility 2DEG. Deviations between the predicted time-domain waveform and measured experimental data may be due to the simple two-level approximation used for the susceptibility. Future cyclotron resonance experiments will measure the magnetic field and temperature dependence of T2 to elucidate the decoherence mechanisms in this limit using this new analysis method. This will also focus on alternate models for χ̃(ν) that correctly model the coherent interaction of these satellite pulses over a broad range of external magnetic fields and at a range of sample temperatures.
This material is based upon work supported by the National Science Foundation under Grant No. DMR-1056827. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement No. DMR-0654118, by the State of Florida, and by the DOE. I thank Mr. Bagvanth Sangala and Mr. Jeremy Curtis for helpful discussions.
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