We report on both experiments and theory of low-terahertz frequency range (up to 400 GHz) magnetoplasmons in a gated two-dimensional electron gas at low (<4K) temperatures. The evolution of magnetoplasmon resonances was observed as a function of magnetic field at frequencies up to ∼400 GHz. Full-wave 3D simulations of the system predicted the spatial distribution of plasmon modes in the 2D channel, along with their frequency response, allowing us to distinguish those resonances caused by bulk and edge magnetoplasmons in the experiments. Our methodology is anticipated to be applicable to the low temperature (<4K) on-chip terahertz measurements of a wide range of other low-dimensional mesoscopic systems.
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The terahertz (THz) frequency plasmonic response of mesoscopic systems (with length scales ranging from 100s of nanometers to a few microns) is of increasing interest for a wide range of potential device applications, including amplitude and phase modulators [1,2], emitters , amplifiers , and detectors . To understand and optimize device characteristics fully, including the spatial distribution of plasmonic modes, various modalities of scanned near-field microscopy have recently been developed. For example, near-field THz imaging techniques have been suggested for the investigation of plasmons in 2D materials by combining sub-wavelength spatial resolution with both amplitude- and phase-sensitive detection techniques . Dean et al. demonstrated apertureless THz near-field imaging using a quantum cascade laser source, allowing coherent, high-resolution imaging of such structures at room temperature , while Mitrofanov et al. showed that scanned aperture THz near-field microscopy can reveal surface waves excited at the edges of graphene structures . While the THz response of individual mesoscopic systems can be obtained using such microscopies, accessing the low temperature response <4K is more challenging. Alternative techniques to probe the low temperature broadband (100s of GHz) response of individual mesoscopic systems are therefore required.
Magnetoplasmons are plasmons coupled to a cyclotron resonance driven by magnetic fields. Mast et al. reported magnetoplasmons in a classical two-dimensional Coulomb gas of electrons above a liquid-helium surface and observed edge magnetoplasmons (EMPs) for the first time in the frequency range of 0–200 MHz . Since then, EMPs have received significant attention as they can be used to study edge-current states in a magnetic field, which are a key factor in the formation of the quantum Hall effect [10,11], as well as being relevant to the recent study of topological insulators .
In this paper, we utilise an on-chip THz spectroscopy technique [13,14], which allows the generation and detection of THz current pulses in the plane of a two-dimensional electron gas (2DEG) using an integrated plasmonic circuit . Using this technique, on-chip THz signals are coupled from a coplanar waveguide (CPW) into a 2DEG, by combining the epitaxial of low-temperature-grown gallium arsenide (LT-GaAs) used for THz excitation and detection, with the 2DEG heterostructure . Here, we study the effect of magnetic field on the plasmonic resonances in this system, and also perform full-wave 3D electromagnetic simulations, allowing us to analyze current distributions in the 2DEG and to ascribe the observed excitations unambiguously to bulk and edge magnetoplasmons excited in different regimes of frequency and magnetic field.
2. Results and discussion
Figure 1(a) shows a schematic diagram of the experimental arrangement. The device consists of a 25-µm-wide, 73-µm-long, 120-nm-deep straight mesa etched into the substrate to define the 2DEG channel, onto which an overlaid gate forms three plasmonic coupled cavities (two ungated cavities C1 (∼8.9-µm long), and C3 (∼55-µm long), and one gated cavity C2 (∼9.2-µm long) as shown in Fig. 1(b)). The length of the gated cavity C2 was chosen to place its fundamental resonance close to 50 GHz, corresponding to the lower end of our accessible bandwidth.
The layer structure of the wafer, which was prepared by molecular beam epitaxy (MBE), is shown in Fig. 1(c). Figure 1(d) shows the device details and the electrical connections used for measurements. The mesa was first defined in the GaAs/AlGaAs heterostructure containing the 2DEG layer (see Fig. 1(c)) using a sulphuric acid: hydrogen peroxide: water wet etch (H2SO4:H2O2:H2O = 1:8:70). Ohmic contacts (200-nm-thick Au-Ge-Ni alloy, annealed at 430°C for 80 secs) were formed at each end of the 2DEG mesa to allow injection of the THz pulses into the 2DEG layer. A CPW was next patterned on the wafer surface, and metallized using electron-beam evaporation of Ti/Au (10/150nm). Pairs of LT-GaAs photoconductive (PC) switches (shown as S1/S2 and S3/S4 in Fig. 1(d)) were defined by the CPW probe arms at either side of the 2DEG mesa to generate or to detect THz pulses upon illumination by an 800 nm wavelength, mode-locked Ti:sapphire laser producing 100-fs-duration pulses at a repetition rate of 80 MHz. Finally, a gate was patterned lithographically and then metalised by electron-beam evaporation and subsequent lift-off, forming a Ti/Au (10/60-nm)-thick strip across the 2DEG mesa that was used both to control the carrier concentration in the 2DEG channel directly underneath the gate, and to modulate the propagating THz pulses. THz pulses were generated by laser pulse excitation of PC switch S1, which was biased at 10 V dc and coupled into the CPW. The signal transmitted through the 2DEG channel was then detected at PC switch S3, which was excited by a time-delayed probe beam split-off from the same laser beam which was incident on S1. To measure the transmitted signal, a gate-modulation technique was used in which an ac voltage of 100 mV RMS at 87Hz was superimposed onto the gate bias allowing lock-in detection to enhance the signal-to-noise ratio . The gate voltage Vg was swept from 0 to −3.6 V dc to investigate the dependence of the plasmon resonance on the carrier concentration. A magnetic field was applied normal to the 2DEG plane and swept from 0 to 0.8 T.
Two approaches were used to model experimental measurements. The first was to estimate the resonant frequencies of the bulk plasmon using the dispersion relation
The 2DEG was modelled by a surface current concentration coupled to Maxwell’s equations. An additional relationship between the current concentration J and the electric field strength E in the x-y plane of the 2DEG was found from the hydrodynamic equations of motion as
Here ω is the angular frequency, ωc is the angular cyclotron frequency, and γ is the effective collision frequency. A value of ∼140 GHz for the latter was used, which has previously been found to give good agreement with experiments . In the simulations, the field distribution of the CPW eigenmode was first computed at a fixed frequency and then launched at the input of one of the waveguides. The mode reflection coefficient and its transmission coefficient to the waveguide at the opposite end were then calculated, and this then repeated for a range of frequencies and, depending on the experiment modeled, either for a range of gate voltages or for a range of magnetic fields. To model the nonuniform spectrum of the experimental excitation pulse, the simulated spectra were scaled assuming an exponential decay of the excitation power with frequency. The gate-modulation measurements were modelled as numerical approximations of the derivative of the transmission coefficient with respect to voltage. Figure 1(e) shows simulation results on normalised current density of plasmonic mode excited at 130 GHz along the 2DEG channel when the electron concentration in the gated, ungated cavities are 5.2×1015 m−2, and 7.5×1015 m−2, respectively.
Figure 1(f) shows the carrier concentration in the gated and ungated 2DEG regions for Vg in the range of 0 – −4 V. To obtain these concentrations, the two-terminal resistance of the 2DEG channel was measured as a function of magnetic field at 2K. An ac voltage of 100 µV RMS at 77 Hz was applied across the 2DEG and the center conductor of the CPW, with a dc bias applied to the gate, and a lock-in amplifier was used to measure the differential conductance in a standard constant voltage configuration. The second derivative of resistance with respect to the magnetic field (d2R/dB2) shows two oscillations of different periods, corresponding to Shubnikov-de Haas oscillations in the gated and ungated cavities, which we assume to act independently as in our prior experiments . The electron concentrations, ns, in the gated and the ungated cavities were each obtained using the relationship Δ(1/B) = 2e/nsh, where e is the elementary charge and h is Planck’s constant . The electron concentration in the gated cavity was found to decrease from 7.5×1015 m−2 to 4.6×1015 m−2 as the negative gate bias was increased, Vg, from 0 V to −3.6 V, while the ns in the ungated cavity remained at a constant value of 7.5×1015 m−2; we note that in the range of Vg from - 0.2 V to – 1.4 V, the Shubnikov-de Haas oscillations in the gated cavity were indistinguishable from those of the ungated cavity. The inset of Fig. 1(f) shows the two-terminal conductance of the 2DEG channel as a function of the Vg, with the channel pinch-off voltage occurring at a gate bias of ∼ −4 V.
Figure 2(a) shows gate-modulation signals obtained at 2K for a center dc voltage of Vg = −3.1 V with ac voltage (+/- 100 mV) superimposed, when a THz pulse propagating towards PC switch S3 was generated at PC switch S1 using a dc bias of 10 V. The transmitted signals show clear periodic oscillations that satisfy the Fabry-Pérot resonance conditions as the propagating pulses pass through the 2DEG channel. Figure 2(b) shows a fast Fourier transform (FFT) spectrum of the signal in Fig. 2(a). Four resonances are observed clearly, at 57 ± 17 GHz, 132 ± 17 GHz, 215 ± 17 GHz, and 269 ± 17 GHz. The error of ± 17 GHz was estimated from the FFT frequency resolution, which was determined by the length of the time-domain scan used. The corresponding frequencies of the first four plasmon resonances calculated from Eq. (1) are f1,pred = 68 GHz, f2,pred = 135 GHz, f3,pred = 200 GHz, and f4,pred = 263 GHz.
We then studied the dependence of the plasmon resonances on carrier concentration in the gated cavity by measuring time-domain gate-modulation signals as a function of Vg in Fig. 2(c). The oscillation period is seen to increase as the gate bias is increased negatively, indicating a decrease in plasmon resonant frequencies owing to a reduction in carrier concentration beneath the gate. The gate modulation signal was strongest at Vg = −3.1 V as the carrier concentration in the gated cavity changes dramatically at this bias as shown in Fig. 1(f) . Figure 2(d) shows corresponding FFT spectra of the time-domain measurements. The plasmon resonances showed a red shift as Vg was swept from −1.5 to −3.6 V, while the amplitude of the resonances was strongest at Vg = −3.1 V, as expected from the time-domain results. The dashed white lines in Fig. 2(d) show the predicted positions of the first four resonances obtained from full-wave simulations. Agreement with measured positions is excellent for the first three resonances, whereas for the fourth resonance, the agreement is good only for Vg greater than −2.5 V. The deterioration of the agreement between experiment and theory close to the pinch-off voltage may be due to the rapid variation of the electron concentration in this region, leading to a breakdown of the relationship between the gate voltage and the electron concentration modeled from Fig. 1(f). We note that neither the experimental nor the simulated data show evidence of resonances from the ungated 2DEG regions (the lowest of which is predicted to lie around 200 GHz), since the gate-modulation technique employed is anticipated to be relatively insensitive to any resonances which are independent of gate voltage .
To investigate the formation of magnetoplasmons in the 2DEG cavity in a magnetic field, we also measured the fine structure of the gate-modulation signals at 2K as a function of magnetic field in the range of 0–0.8 T, with a step of 0.02 T. Figure 3(a) shows gate-modulated time-domain signals measured at Vg = −3.1 V for B = 0, 0.3, 0.6T as examples. A Vg of −3.1 V was chosen both to maximize the gate-modulation signal, but also to keep the plasmon resonances within our accessible bandwidth. For B = 0, periodic oscillations are observed, consistent with our earlier measurements (Fig. 2(a)); these oscillations change periodicity as magnetic field is applied. To study the evolution of the oscillating signals, Fig. 3(b) shows the gate-modulation signals as a function of magnetic field measured for Vg = −3.1 V dc, where signal was the strongest. Figure 3(c) shows FFT spectra of the time-domain measurements shown in Fig. 3(b), while Fig. 3(d) shows the results of full-wave modelling under identical conditions. Both simulation and measurement show two different regions separated by the cyclotron frequency (indicated by dashed orange lines). For any given value of magnetic field, at frequencies higher than the corresponding cyclotron frequency, several distinct features are observed: three in the measured signals (at 52, 128, and 215 GHz at B = 0 T), and five in the simulated response (at 68, 130, 190, 250, and 315 GHz at B = 0 T). The two highest simulated resonances were not observed experimentally owing to a reduction in generated THz-signal intensity at higher frequencies. The white dashed lines show the evolution of the observed above-cyclotron resonances with magnetic field, calculated according to Eq. (2). Both the measured and simulated signals follow the predicted behaviour, although there are variations in the signal strength along individual lines. As shown later, these features correspond to the excitation of bulk magnetoplasmons in the gated cavity.
Figures 3(e) and 3(f) show additional measured and simulated gate-modulation signals respectively, this time taken at Vg = −2 V; these show similar features to those observed in Figs. 3(c) and 3(d). Above the cyclotron frequency, there are several resonances that follow the behaviour predicted by Eq. (2) for bulk magnetoplasmons. At B = 0 T, three measured resonances are observed at 70, 136, and 200 GHz with the corresponding simulated resonances predicted at 75, 140, and 205 GHz. There is good agreement between the measured and simulated values above the cyclotron line.
In all measurements and simulations across a range of gate biases, a broad feature occurring at ∼100 GHz is observed below the cyclotron line. As follows from Eq. (2), bulk magnetoplasmons do not propagate below the cyclotron line. Electron motion corresponding to a bulk plasmon has a longitudinal component and a circular component, the latter being induced by the magnetic field. In the limit of high magnetic field, the circular component dominates, the electron motion is determined by the cyclotron resonance, and f = fc. Electron motion at the edges is, however, restricted, and edge magnetoplasmons may exist that are not bound by the cyclotron resonance. Noting that bulk magnetoplasmons should not therefore propagate below the cyclotron resonance , this feature tentatively suggests the presence of edge magnetoplasmons in the 2DEG channel. To further understand the origin of these features, detailed simulations of the current distribution in the 2DEG were performed under three conditions, chosen to correspond to key regions of the data, at B = 0 T and f = 130 GHz, B = 0.1 T and f = 135 GHz, and B = 0.3 T and f = 90 GHz, as indicated by white circles in Fig. 3(d).
Figures 4(a), 4(c), and 4(e) show two-dimensional representations of the normalised current density along the 2DEG as a function of position in the plane of the simulated structure. Figures 4(b), 4(d), and 4(f) show line plots of the same current density simulations, taken along the middle and the top and bottom edges of the 2DEG. The position of the gated region is shown by dashed vertical lines. In the absence of magnetic field, in Figs. 4(a) and 4(b) the current density remains similar for any value of y-position along the width of the 2DEG, with a single peak along the x-axis within the gated region, indicating excitation of a bulk, gated plasmon resonance. At B = 0.1 T and f = 135 GHz, as shown in Figs. 4(c) and 4(d), the bulk gated resonance still dominates everywhere apart from the top edge, where excitation of a higher-order edge plasmon can be seen. However, no significant bulk excitations can be seen below the cyclotron frequency, for example at B = 0.3 T and f = 90 GHz, as shown in Figs. 4(e) and 4(f). The current density here is concentrated along the edges, with the top-edge excitation being dominant. Along this top edge, a distorted one-wavelength pattern can be seen, comprising an edge plasmon resonance in the gated region. To confirm further that gated edge plasmons are responsible for the signals below the cyclotron frequency, structures were simulated with reduced loss with an effective scattering time increased to 50 ps, and for parametrically varied dimension including the 2DEG width (15–30 µm) and depth (35–75 nm), and the length of the gated region (5–10 µm). We observed that the frequency of the below-cyclotron resonance changed when the length of the gated section and the mesa height were modified, but did not depend appreciably on the 2DEG width. We note that the experimental and simulation techniques used in this work also can be adopted to probe magnetoplasmon modes predicted in other Drude-like two-dimensional materials including topological 2D systems , as well as graphene .
We have observed the evolution of magnetoplasmon resonances in a 2DEG at frequencies up to ∼400 GHz and as a function of magnetic field. The observed resonances are reproduced by with 3D full-wave simulations of the system. Analysis of the current distribution in the 2DEG allows us to firmly attribute the resonances above the cyclotron frequency to the excitation of the bulk gated magnetoplasmons, and in the case of those below the cyclotron frequency to the excitation of edge magnetoplasmons in the gated 2DEG region. We anticipate that this work will contribute to the understanding of origin and evolution of the plasmonic effects in 2DEG systems in the THz frequency range, and note that our experimental and theoretical approaches can both be extended to other low-dimensional mesoscopic systems and materials.
Engineering and Physical Sciences Research Council (EP/P021859/1, EP/R004994/1, EP/R00501X/1); National Natural Science Foundation of China (61701219).
The authors gratefully acknowledge funding from the EPSRC and from the UK National Physical Laboratory; EHL acknowledges support from the Royal Society and Wolfson Foundation. JBW acknowledges support from the NSFC. The data associated with this paper is available from University of Leeds at https://doi.org/10.5518/818
The authors declare no conflicts of interest.
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