2k_F-selected conductance oscillations of high-mobility two-dimensional electron gas in Corbino devices

Magneto-transport experiments in a high-mobility two-dimensional electron gas (2DEG) under a small magnetic field (B), where the Shubnikov de Haas (SdH) oscillations are absent, have revealed a number of interesting nonlinear effects, including: the microwave-induced resistance oscillations (MIRO)^1,2 and the subsequent zero-resistance states (ZRS);^3,4 current-induced Zener tunneling oscillations between Landau levels (LLs);^5,6 and phonon-induced resistance oscillations (PIRO).^7–9 Remarkably, these oscillations all involve a large momentum transfer 2k_F between different Landau orbits,^1–9 where $k F = 2 π n e$ is the Fermi wave number and n_e is the electron density, respectively. The required 2k_F momentum transfer is thought to be provided by short-range scatterers, such as residual impurities and interface roughness, or by acoustic phonons of the hosting GaAs crystal.^1–9 Energy conservation requires $ℏ ω = j ℏ ω c$⁠, where j = 1, 2, 3,… integers, representing N to N + j LL transition, $ω c = e B / m *$ is cyclotron frequency with $m *$ being the effective mass of electrons, $ℏ ω$ is the microwave photon energy, or other energy quantum involved. The case for Zener tunneling oscillations and that for PIRO are depicted schematically in Figs. 1(a) and 1(b); here, we use a general term PIRO referring to phonon-assisted oscillations regardless of sample geometry.

The nonlinear magneto-transport effects mentioned above have been extensively studied in Hall bar samples.^{1–8,10–12} While magneto-conductivity tensor elements can be obtained by inverting the magneto-resistivity tensor, a direct measurement of conductivity $σ x x$ using Corbino samples is often desirable. For example, our present work is motivated by measurements of conductance oscillations in the Corbino samples, where electrical field E can be established along the radial axis. Different geometry between Hall bar and Corbino can give rise to different boundary conditions: I_y = 0 for Hall bar, and Hall voltage V_y = 0 for Corbino, since it is shorted by contacts. Besides, whereas the electric field distribution is uniform in Hall bar, generally, it has a gradient along the radial axis in Corbino samples. MIRO in Corbino samples has been previously reported,^13,14 and, more recently, the signals of Zener tunneling oscillations in Corbino were observed in a relatively low mobility 2DEG.¹⁵ Here, we are focusing on Zener tunneling oscillations and PIRO in Corbino samples consisting of an ultra-high mobility (>10⁷ cm²/Vs) 2DEG where high-order oscillations are clearly resolved. These data have further strengthened the case for 2k_F-selected resonance in 2DEG as a universal and robust effect, which is independent of the details of boundary conditions.

Our Corbino sample was made from a symmetrically doped GaAs/Al_0.325Ga_0.675As quantum well (QW) wafer grown by molecular-beam epitaxy, with the radius of inner (outer) ring being 900 μm (1000 μm). The QW is 25 nm wide, and the spacer distance is d_s = 80 nm. Typically, a high-mobility μ ∼ 2 × 10⁷ cm²/Vs at an electron density n_e = 4.2 × 10¹¹/cm² was obtained after an illumination by red light-emitting diode at T = 4 K. The Zener tunneling oscillations were measured in a ³He/⁴He dilution refrigerator equipped with a superconducting magnet.

In Corbino experiments, two different quantities can be measured: the conductance I_dc /V_dc and the differential conductance dI_dc/dV_dc. We first show in Fig. 2(a) the measured direct current I_dc passing through the source and drain contacts, as measured by sweeping direct bias V_dc at a series of fixed, small perpendicular magnetic field B from 0.5 kG to 1 kG in 25 G increments. The inset shows the diagram of DC electrical measurement. Temperature T, which refers to the refrigerator thermometer reading, increases from roughly 50 mK to 90 mK during the measurements, indicating substantial dissipation from Joule heating. Due to the high quality of our Corbino sample, we can resolve the first three order (j = 1, 2, 3) oscillations directly from I_dc vs V_dc, which have not been reported in previous work using lower mobility samples.

In order to extend our analysis, we calculated differential conductance dI_dc/dV_dc by numerical differentiation on the raw data I_dc (V_dc). The position labeled as j in the Fig. 1(a) is corresponding to the jth maxima of differential conductance dI_dc/dV_dc. The dI_dc/dV_dc versus V_dc is shown in the Fig. 2(b) in color map, with each horizontal trace representing a fixed B value. The bright part in the map represents the peak differential conductance value; we can thus observe five bright curves representing j up to 5. The shape of each bright curve shows the relation between the resonant magnetic field B and the bias V_dc, with the same j.

To quantify this relation obtained from the experiment, we first extract points (⁠ $V d c m$⁠, $B d c m$⁠) shown as blue circles in the map from a series of traces dI_dc/dV_dc vs V_dc at different B, and here, $V d c m$⁠, $B d c m$ refers to, respectively, the bias and magnetic field at the same order of oscillation peak, j. Not surprisingly, points fall onto the middle parts of the bright curves. In the following, we fit these discrete points based on a simple model of Zener tunneling oscillations proposed for Hall bar measurement. As we have mentioned earlier, the resonance condition is $2 ℏ k F v H = j ℏ ω C$⁠,⁵ and it is equivalent to

where $E applied$ is the Hall field induced by the applied current bias and $R c = ℏ e B k F$ is the cyclotron radius. As depicted in Fig. 1(a), this relation takes into account for the fact that a jump of 2R_c along the $E applied$ in real space satisfies the 2k_F momentum selection rule, and here, we take an approximate value $γ = 2$⁠. Alternatively, in the Corbino sample, the relevant field is along the radial direction: $E applied = I r 2 π σ r r r$⁠, where r is measured from the ring center. However, in the limit of a narrow ring (width $Δ r$ = 100 μm), we can reasonably assume a uniform radial electric field: $E = V d c / Δ r$⁠. We thus arrive at a resonance condition for the Corbino sample

From Eq. (2), we can obtain $B ∝ V d c 1 / 2$ for the same j. The black curves show a fit of discrete points (⁠ $V d c m$⁠, $B d c m$⁠) to this relation; we find that such fits are satisfactory for all the five oscillation peaks.

From the theoretical resonance condition (2), it is easy to see that for a fixed B, $V d c ∝ j$⁠. It is illustrative to plot such a relation in a “fan diagram” as shown in Fig. 3, for fixed B ranging from 0.5 kG to 1 kG consecutively with an increment of 50 G. All together, we have verified experimentally that the data conform well to a general relationship: $B ∝ ( V d c / j ) 1 / 2$⁠. From the fits of $B ∝ V d c 1 / 2$ at the peak j, we determine the effective mass of electrons, which are 0.066 $m 0$⁠, 0.063 $m 0$⁠, 0.062 $m 0$⁠, 0.061 $m 0$⁠, 0.061 $m 0$ for j = 1, 2…5, respectively, where $m 0$ is the electron mass. The obtained value 0.066 $m 0$ is within 2% equal to the well-known effective mass of electrons in GaAs, while higher-order results are somewhat smaller, which may be attributed to the errors in the peak positions.

We have further examined the PIRO phenomenon in the Corbino sample. The measurement was performed in a ³He cryostat equipped with a superconducting magnet. As mentioned in the introduction (see Fig. 1(b)), here, the required 2k_F momentum and energy quanta are provided by acoustic phonons, which are populated in an elevated T, typically above 1 K for GaAs. Magneto-conductance was measured by using a standard lock-in technique with an AC voltage source of 1 mV at 17 Hz in sweeping a small, perpendicular magnetic field B, and at a fixed temperature between 1.7 K and 9.2 K in this experiment.

We present data measured in the same sample (n_e = 4.09 × 10¹¹ /cm² in this cool-down). Fig. 4(a) exhibits magneto-conductance G_xx measured in a sweeping B at different temperatures, 1.7 K, 2.3 K, and from 3.2 K to 9.2 K with the step of 1 K. The G_xx are symmetrical with respect to B = 0 for all traces. We can further find that the onset field of SdH is B₀ ∼ 3.5 kG at 1.7 K (the lowest temperature in this experiment). With increasing temperatures, B₀ moves to a larger B, in the mean time PIRO begin to appear below B₀. To further analyze the data, we obtain |d²G_xx/dB²| versus B by numerical differentiation. Traces of |d²G_xx/dB²| versus B at different temperatures are shown in the Fig. 4(b). Starting from 2.3 K, we are able to observe distinct peaks in conductance in the Corbino samples, very similar to the PIRO peaks in resistance in the Hall bar samples. Referring to the schematic shown in Fig. 1(b), electrons transit between different Landau orbits by absorbing acoustic phonons with 2k_F momentum. In the Corbino geometry, the momentum of electrons change by 2k_F along the tangential direction, and the 2k_F momentum transfer is equivalent to a hopping at the distance of a cyclotron diameter along the radial direction. We further conclude that the temperature dependence of resonance oscillations measured in Corbino sample is similar to that obtained from the Hall bar geometry.^7,11 The PIRO peaks marked by integers j or l will be best developed in a moderate temperature range between 4.2 K and 5.2 K, and decay at both lower and higher temperatures. The explanation of this phenomenon has been given before.^7,11 The decrease of absorption and emission of phonons with 2k_F damps the amplitude of PIRO peaks at low temperature. Electron-electron interactions may modify quantum scattering rate, which contributes to the Dingle factor, and suppresses the resonance amplitude of oscillations at high temperature.¹¹ 

As we have seen, there exist two sets of peaks, caused by two branches of acoustic phonon modes within the higher temperature range from 3.2 K to 9.2 K; the two sets of resonance oscillations are marked by j and l, respectively. The PIRO peaks marked by j can be resolved up to j = 5, while only l = 2, 3 can be resolved for the second branches. We find that the appearance of l peaks requires higher temperature than j peaks, and l peak becomes more dominant at higher temperatures above 7.2 K. We extract points (j, B_m⁻¹) and (l, B_m⁻¹) from the trace |d²G_xx/dB²| versus B at T = 5.2 K where PIRO caused by these two branches co-exist; B_m refers to the corresponding magnetic field of PIRO peaks marked by integers j, l. Since the resonance condition $2 ℏ k F s = j ( l ) ℏ ( e B / m * )$ is equivalent to $j ( l ) = ( 2 k F s m * / e ) × B − 1$⁠, these discrete points (j, B_m⁻¹) and (l, B_m⁻¹) can be fitted, respectively, with a linear relation, as shown in the inset of Fig. 4(b). From these slope of the linear fits, we obtain the lower velocity of acoustic phonons s_L as 3.43 km/s and higher velocity s_H as 4.70 km/s, which are in complete agreement with the velocity values of transverse and longitudinal acoustic phonons reported for GaAs.^7,11,16

In summary, we have studied systematically nonlinear magneto-transport of electric field-induced and acoustic phonons-induced conductance oscillations in a high-mobility 2DEG with Corbino geometry. By utilizing an approximate uniform radial electric field, we show that the resonance conditions from our experiment are in accordance with the well-known resonance conditions of Zener tunneling oscillations measured in Hall bar geometry, where the relevant energy is provided by the Hall field. We further deduce an effective mass value consistent with mass of electrons in the conduction band of GaAs. In the same Corbino samples and an elevated temperature window, we observe clear oscillations due to acoustic phonon resonance. We obtain the velocity of these two branches which are 3.43 km/s and 4.70 km/s, respectively, corresponding to transverse and longitudinal acoustic phonons reported in GaAs. These results have further strengthened the case for 2k_F-selected resonance in 2DEG as a universal and robust effect, which is independent of the details of boundary conditions.

The work at Peking University was financially supported by NBRP of China (2012CB921301). The work at Rice was funded by DOE Grant No. DE-FG02-06ER46274 (measurement), NSF Grant No. DMR-1207562 (R.R.D.), and Welch Foundation Grant No. C-1682 (L.J.D.). The work at Princeton was partially funded by the Gordon and Betty Moore Foundation as well as the NSF MRSEC Program through the Princeton Center for Complex Materials (DMR-0819860).