Origin and limiting mechanism of induced nonequilibrium currents in gated two-dimensional electron systems

We have studied experimentally the nonequilibrium currents (NECs) induced by sweeping either the magnetic field $B$ or the carrier density $n_S$ of a two-dimensional electron system (2DES). The gated 2DES resided in a modulation-doped $\text{GaAs}/{\text{Al}}_{\text{x}}{\text{Ga}}_{1-\text{x}}\text{As}$ heterostructure and was integrated into a micromechanical cantilever. The NECs provoke a magnetic moment which we have detected via torque magnetometry down to 300 mK. Additional electrical leads allowed for simultaneous magnetotransport measurements. We find a hysteretic behavior of the NECs and a striking asymmetry of the corresponding magnetic moment around integer filling factors $\nu =h{n}_S/eB$. Surprisingly, the shape of the hysteresis loops is the same for sweeps of $B$ or $n_S$ if plotted versus $\nu$. In a certain parameter regime each NEC signal exhibits a characteristic slope which is found to depend only on the filling factor at large $B$ or $n_S$. Based on a model considering capacitive coupling between 2DES and gate we attribute the slopes to the conductance quantization of the quantum Hall effect. The NECs are found to be limited by the time-dependent buildup of the radial Hall field governed by the gate capacitance. These findings are in contrast to a floating 2DES without a gate where the breakdown of the quantum Hall effect was previously reported to limit the NECs. Our model also explains the observed shape and dependence on temperature as well as sweep rate. The in situ measurement of the longitudinal resistance allows us to directly correlate the magnetic behavior with both the magnetic field and temperature-dependent resistance of the 2DES.

Figure 1

Sketch of the micromechanical cantilever magnetometer (sample A) with the fiber-optics interferometer used for readout of the cantilever deflection. The rectangular 2DES mesa is integrated at the end of the cantilever beam. The magnetization $M_{\text{NEC}}$ of the NECs is perpendicular to the 2DES plane. The cantilever normal is tilted by an angle of $15°$ with respect to $B$. Contacts CC: calibration coil, A–D: 2DES transport contacts, FE: field-effect electrode.

Figure 2

(a) DHvA oscillations and NECs measured in a gated 2DES with transport contacts (sample A) at $T=300\text{mK}$. Arrows indicate the sweep direction of $B$. $B_{\perp }$ is the component of the magnetic field $B$ normal to the 2DES. The NECs in the gated 2DES are pronouncedly asymmetric around integer filling factors $\nu$. The maxima of the NECs are shifted behind the position of integer $\nu$ with respect to the sweep direction. The NEC signal varies linearly around integer $\nu$, i.e., the slope $\partial M/\partial B$ is constant as indicated by the dashed lines. (b) Longitudinal resistance oscillations measured simultaneously to the magnetization $M$. The longitudinal resistance shows pronounced minima at integer $\nu$. The small offset from zero resistivity $\left(R_{\text{parasit}}\right)$ is due to a parasitic parallel conductance arising from the continuous laser illumination of the fiber-optics interferometer. This will be considered in the evaluation.

Figure 3

DHvA oscillations and NECs in a floating 2DES, i.e., a 2DES without gate electrode and without contacts (sample B) at $T=300\text{mK}$. Arrows indicate the sweep direction of $B$. Note that the NEC signal is much more symmetric around $\nu =2$ than in case of the gated 2DES [Fig. 2a]. These data are representative for the generic behavior of NECs observed on a large number of floating 2DESs by different authors.

Figure 4

(a) Magnetization $M$ highlighting the NECs induced by $\partial B/\partial t=\pm 0.2\text{T}/\text{min}$ for different carrier densities $n_S$ at $T=300\text{mK}$. The carrier density $n_S$ is kept constant for each measurement and is incremented by $\Delta n_S\approx 1.9\times {10}^{14}/{\text{m}}^2$ between the measurements. The definition of the NEC amplitude $\Delta IA$, i.e., the maximum difference of the NEC signal between up and down sweeps is shown exemplarily in gray. (b) $M$ for different fixed magnetic fields $B$ at $T=300\text{mK}$. Here, $n_S$ was varied with a rate of $\partial n_S/\partial t=\pm 3.4\times {10}^{12}/{\text{m}}^2\text{s}$. The magnetic field $B$ is changed by $\Delta B\approx 0.24\text{T}$ in a step-wise manner between the measurements. In (a) and (b) we focus on the hysteretic behavior of $M_{\text{NEC}}$ in the regime of $\nu =2$. Arrows indicate the sweep direction of $B$ or $n_S$, respectively. At high magnetic fields $B_{\perp }>3.8\text{T}$ the slope of $M$ as indicated by the dashed lines has a constant value. Below $B_{\perp }<3.8\text{T}$ the data fall below the indicated slope. No NECs are observed at small magnetic fields $B_{\perp }<2.7\text{T}$.

Figure 5

Minor loop measurements of $M$ around $\nu =2$ in the gated sample A. Arrows indicate the sweep direction. The extremal hysteresis is shown in black, the minor loops in gray. The minor loops have nearly the same slope at $\nu =2$. As discussed in Sec. 4 this indicates the back transfer of the redistributed charges at a constant rate. The minor loops reach their maxima at the same magnetic field $B$ (dotted line).

Figure 6

Minor loop measurements of $M$ in a floating 2DES around $\nu =4$ (sample C). Arrows indicate the sweep direction. The extremal hysteresis is shown in black, the minor loops in gray. The minor loops display a nearly vertical slope and saturate only when approaching the extremal curve. This behavior can be explained by current limiting mechanisms that are discussed in the framework of the QHE breakdown (cf. Ref. 19). Note that this floating 2DES also shows an NEC amplitude that is (i) one order of magnitude larger if compared to the gated 2DES in Fig. 2a and (ii) of the same order of magnitude if compared to the floating 2DES in Fig. 3. This underscores that the high NEC amplitude is a generic property of NECs observed on floating 2DESs.

Figure 7

(a) Magnetization $M$ of magnetic-field-dependent NECs for different temperatures $T$. For clarity only every fourth curve is shown. Arrows indicate the sweep direction. (b) Temperature dependence of the NEC amplitude $\Delta IA$. The NECs reduce for higher temperatures $T$ due to the relaxation of $E_r$ as a result of the increasing conductivity ${\sigma }_{xx}$. The fit to the data will be discussed in Sec. 5A. (c) Longitudinal resistivity minimum (black) measured simultaneously in van der Pauw geometry. A reciprocal Gaussian curve (gray) has been fitted to the data to extract the minimum $R_{xx}^0$ and the width $\Delta B_{\perp }$ of the region where $R_{xx}<R_{xx,\text{crit}}$. The background $R_{\text{parasit}}$ is caused by parasitic parallel conductivity due to the continuous laser illumination from the laser interferometer.

Figure 8

Sketch of our model for the generation of NECs. The total filling factor is $\nu =4$. The bulk is incompressible whereas on the sample edge compressible and incompressible strips (Ref. 29) are formed. In the compressible and incompressible strips alternating equilibrium currents flow, causing the equilibrium magnetization of the dHvA effect (Ref. 34). (a) Sweeping the perpendicular magnetic field induces an azimuthal electric field $E_{\varphi }$. In the QHE regime, this leads to a radial current density $j_r={\sigma }_{xy}{E}_{\varphi }$, i.e., a charge transfer (Ref. 22) between the center and the edge of the sample. The charge transfer builds up a radial electric field $E_r$ which drives the NEC $j^{\text{NEC}}$. In real samples, where ${\sigma }_{xx}=0$ does not strictly hold, a current $j^{\text{relax}}$ will reduce the electric field $E_r$ and lead to a decay of the NEC over time once the field sweep is stopped. (b) Changing the carrier density $n_S$ yields a radial electric field $E_r$ in the incompressible bulk of the sample. The NEC is induced due to this field $E_r$ analogous to (a). Reversal of the sweep direction yields reversed electric fields and therefore NECs $j^{\text{NEC}}$ with different signs in (a) and (b). (c) Sketch of electrostatic energy $V$. On the right hand side we assume compressible and incompressible strips following Refs. 29, 30. On the left the bending of the electrostatic energy $V$ in the incompressible bulk of the sample is depicted assuming a radial electric field profile. This sketch follows Refs. 17, 22.

Figure 9

Sketch: magnetization $M$ of a NEC for two different values of $C_0$. We assume ${\sigma }_{xx}=0$ and hence no relaxation. The relation between the jump in the dHvA effect and the NEC is not to scale. $C_0<C_{\text{Min}}$ (dashed line): the 2DES is floating, the NEC signal reaches the extremal curve which is limited by the QHE breakdown. $C_0>C_{\text{Min}}$ (solid line): the 2DES is gated and the NEC is limited by the charge transfer rate (constant slope) until the extremal curve is reached.

Figure 10

Magnetization $M$ for (a) sweeping the magnetic field $\partial B/\partial t$ and (b) sweeping the carrier density $\partial n_S/\partial t$ at $T=300\text{mK}$. Arrows indicate the sweep direction. Both NECs have the same shape if plotted against the total filling factor $\nu$. Slight differences in the noise level will be addressed in Sec. 5C.

Figure 11

(a) NEC amplitude $\Delta IA$ from Fig. 7b plotted as a function of the temperature-dependent longitudinal resistance $R_{xx}^0$ (at integer $\nu$). Above $R_{xx}^0>R_{xx,\text{crit}}$ no NECs are observed. A quadratic dependence on $R_{xx}^0-R_{xx,\text{crit}}$ is found. This is explained by the data in (b) and (c): the slope $\partial M/\partial B$ (b) of the NECs depends linearly on $R_{xx}^0$, due to the linear dependence on the relaxation current. The minimal slope of the signal is not zero, but corresponds to the slope of the dHvA effect. (c) The width $\Delta B_{\perp }$ of the region where $R_{xx}<R_{xx,\text{crit}}$, i.e., the width of the longitudinal resistance minimum in Fig. 7c, is also linearly dependent on $R_{xx}^0$. These two effects combined yield the quadratic dependence on $R_{xx}^0$.

Figure 12

Magnetization $M$ of NECs for different sweep rates $\partial B/\partial t$ at $T=300\text{mK}$. The slope $\partial M/\partial B$ of the increasing NEC signal at integer $\nu$ plotted against the reciprocal sweep rate shows an exponential decrease (inset). The solid line is a fit to the equivalent circuit model described in the text and corresponds to a decay time of 39.8 min.

Figure 13

Comparison of the noise $\delta M$ of the NEC signal when $B$ is swept (black) and when $n_S$ is swept (gray). To extract $\delta M$ we subtracted a linear fit from the magnetization signal around $\nu =2$ in Fig. 10 (right). The noise $\delta M$ is strongly enhanced when $n_S$ is swept. To exclude possibly different noise pickup due to the different measurement setup when a voltage source is connected to the gate and 2DES we analyze $\delta M$ also in between integer $\nu$, where no NECs are present (left). Here, no difference between a $B$ sweep and a $n_S$ sweep is resolved in $\delta M$.