We have observed electrical-field induced magneto-conductivity oscillations measured in Corbino samples made of a GaAs high-mobility two-dimensional electron gas, and found a consistent interpretation based on a semiclassical model of 2k_{F}-selected orbital transitions between N and N + 1, 2, 3,…, Landau levels at respective local potentials, where k_{F} is the Fermi wavevector. From the oscillation period, we deduce an effective mass value, which is consistent with the bare electron band mass of GaAs. In the same devices but with a vanishing electrical field and at elevated temperatures, we observed additional oscillation features, which can be attributed to cyclotron resonance by population of acoustic phonons. We thus demonstrate a method to determine the carrier effective mass and the sound velocity of host crystals by standard electrical transport.

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 2*k _{F}* between different Landau orbits,

^{1–9}where $ k F = 2 \pi n e$ is the Fermi wave number and

*n*is the electron density, respectively. The required 2

_{e}*k*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.

_{F}^{1–9}Energy conservation requires $ \u210f \omega = j \u210f \omega c$, where

*j*= 1, 2, 3,… integers, representing N to N + j LL transition, $ \omega c = e B / m *$ is cyclotron frequency with $ m *$ being the effective mass of electrons, $ \u210f \omega $ 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 $ \sigma 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*= 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,

_{y}^{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 (>10

^{7}cm

^{2}/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_{0.325}Ga_{0.675}As quantum well (QW) wafer grown by molecular-beam epitaxy, with the radius of inner (outer) ring being 900 *μ*m (1000 *μ*m). The QW is 25 nm wide, and the spacer distance is *d _{s}* = 80 nm. Typically, a high-mobility

*μ ∼*2 × 10

^{7}cm

^{2}/Vs at an electron density

*n*= 4.2 × 10

_{e}^{11}/cm

^{2}was obtained after an illumination by red light-emitting diode at

*T*= 4 K. The Zener tunneling oscillations were measured in a

^{3}He/

^{4}He 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*. We first show in Fig. 2(a) the measured direct current

_{dc}/dV_{dc}*I*passing through the source and drain contacts, as measured by sweeping direct bias

_{dc}*V*at a series of fixed, small perpendicular magnetic field

_{dc}*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*, which have not been reported in previous work using lower mobility samples.

_{dc}vs V_{dc}In order to extend our analysis, we calculated differential conductance *dI _{dc}/dV_{dc}* by numerical differentiation on the raw data

*I*(

_{dc}*V*). The position labeled as

_{dc}*j*in the Fig. 1(a) is corresponding to the jth maxima of differential conductance

*dI*. The

_{dc}/dV_{dc}*dI*versus

_{dc}/dV_{dc}*V*is shown in the Fig. 2(b) in color map, with each horizontal trace representing a fixed

_{dc}*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*, with the same

_{dc}*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*at different

_{dc}*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 \u210f k F v H = j \u210f \omega C$,

^{5}and it is equivalent to

where $ E applied$ is the Hall field induced by the applied current bias and $ R c = \u210f 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 $ \gamma = 2$. Alternatively, in the Corbino sample, the relevant field is along the radial direction: $ E applied = I r 2 \pi \sigma r r r$, where *r* is measured from the ring center. However, in the limit of a narrow ring (width $ \Delta r$ = 100 *μ*m), we can reasonably assume a uniform radial electric field: $ E = V d c / \Delta r$. We thus arrive at a resonance condition for the Corbino sample

From Eq. (2), we can obtain $ B \u221d 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 \u221d 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 \u221d ( V d c / j ) 1 / 2$. From the fits of $ B \u221d 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 ^{3}He 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 × 10

^{11}/cm

^{2}in this cool-down). Fig. 4(a) exhibits magneto-conductance

*G*measured in a sweeping

_{xx}*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*are symmetrical with respect to

_{xx}*B*= 0 for all traces. We can further find that the onset field of SdH is

*B*∼ 3.5 kG at 1.7 K (the lowest temperature in this experiment). With increasing temperatures,

_{0}*B*moves to a larger

_{0}*B*, in the mean time PIRO begin to appear below

*B*To further analyze the data, we obtain |

_{0}.*d*/

^{2}G_{xx}*dB*| versus

^{2}*B*by numerical differentiation. Traces of |

*d*/

^{2}G_{xx}*dB*| versus

^{2}*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 2

*k*momentum. In the Corbino geometry, the momentum of electrons change by 2

_{F}*k*along the tangential direction, and the 2

_{F}*k*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.

_{F}^{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 2

*k*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.

_{F}^{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*) from the trace |

_{m}^{−1}*d*/

^{2}G_{xx}*dB*| versus

^{2}*B*at T = 5.2 K where PIRO caused by these two branches co-exist;

*B*refers to the corresponding magnetic field of PIRO peaks marked by integers

_{m}*j*,

*l*. Since the resonance condition $ 2 \u210f k F s = j ( l ) \u210f ( e B / m * )$ is equivalent to $ j ( l ) = ( 2 k F s m * / e ) \xd7 B \u2212 1$, these discrete points (

*j, B*) and (

_{m}^{−1}*l*,

*B*) 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

_{m}^{−1}*s*as 3.43 km/s and higher velocity

_{L}*s*as 4.70 km/s, which are in complete agreement with the velocity values of transverse and longitudinal acoustic phonons reported for GaAs.

_{H}^{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).