Introduction

Negative magneto-conductivity/resistivity is a spectacular- and thought provoking- theoretical property of microwave photoexcited, far-from-equilibrium, two-dimensional electronic systems. This property has been utilized to understand the experimental observation of the microwave-radiation-induced zero-resistance states in the GaAs/AlGaAs system1,2. Yet, the negative conductivity/resistivity state remains an enigmatic and open topic for investigation, although, over the past decade, photo-excited transport has been the subject of a broad and intense experimental1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36 and theoretical37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66 study in the 2D electron system (2DES).

In experiment, the microwave-induced zero-resistance states arise from “1/4-cycle-shifted” microwave radiation-induced magnetoresistance oscillations in the high mobility GaAs/AlGaAs system1,4,33 as these oscillations become larger in amplitude with the reduction of the temperature, T, at a fixed microwave intensity. At sufficiently low T under optimal microwave intensity, the amplitude of the microwave-induced magnetoresistance oscillations becomes large enough that the deepest oscillatory minima approach zero-resistance. Further reduction in T then leads to the saturation of the resistance at zero, leading to the zero-resistance states that empirically look similar to the zero-resistance states observed under quantized Hall effect conditions1,2,7. Similar to the situation in the quantized Hall effect, these radiation-induced zero resistance states exhibit activated transport1,2,7,10. A difference with respect to the quantized Hall situation, however, is that the Hall resistance, Rxy, does not exhibit plateaus or quantization in this instance where the zero-resistance state is obtained by photo-excitation1,2,7.

Some theories have utilized a two step approach to explain the microwave-radiation-induced zero-resistance states. In the first step, theory identifies a mechanism that helps to realize oscillations in the diagonal magneto-photo-conductivity/resistivity and provides for the possibility that the minima of the oscillatory diagonal conductivity/resistivity can even take on negative values37,39,41,45,48,59. The next step in the two step approach invokes the theory of Andreev et al.38, who suggest that the zero-current-state at negative resistivity (and conductivity) is unstable and that this favors the appearance of current domains with a non-vanishing current density38,56, followed by the experimentally observed zero-resistance states.

There exist alternate approaches which directly realize zero-resistance states with-out a de-tour through negative conductivity/resistivity states. Such theories include the radiation-driven electron-orbit- model49, the radiation-induced-contact-carrier-accumulation/depletion model62 and the synchronization model66. Thus far, however, experiment has been unable to clarify the underlying mechanism(s), so far as the zero-resistance states are concerned.

The negative magneto-conductivity/resistivity state suggested theoretically in this problem37,39,41,45,48,59 has been a puzzle for experiment since it had not been encountered before in magneto-transport. Naively, one believes that negative magneto-resistivity/conductivity should lead to observable negative magneto-resistance/conductance, based on expectations for the zero-magnetic-field situation. At the same time, one feels that the existence of the magnetic field is an important additional feature and this raises several questions: Could the existence of the magnetic field be sufficiently significant to overcome nominal expectations, based on the zero-magnetic-field analogy, for an instability in a negative magneto-conductivity/resistivity state? If an instability does occur for the negative magneto-conductivity/resistivity state, what is the reason for the instability? Could negative conductivity/resistivity lead to observable negative conductance/resistance at least in some short time-scale transient situation where current domains have not yet formed? Indeed, one might ask: what are the magneto-transport characteristics of a bare negative conductivity/resistivity state? Remarkably, it turns out that an answer has not yet been formulated for this last question.

To address this last question, we examine here the transport characteristics of the photo-excited 2DES at negative diagonal conductivity/resistivity through a numerical solution of the associated boundary value problem. The results suggest, rather surprisingly, that negative conductivity/resistivity in the 2DES under photo-excitation should generally yield a positive diagonal resistance, i.e., Rxx > 0, except at singular points where Rxx = 0 when the diagonal conductivity σxx = 0. The simulations also identify an associated, unexpected sign reversal in the Hall voltage under these conditions. These features suggest that nominal expectations, based on the zero-magnetic-field analogy, for a negative conductivity/resistivity state in a non-zero magnetic field, need not necessarily follow and that experimental observations of zero-resistance and a linear Hall effect in the photo-excited GaAs/AlGaAs system could be signatures of vanishing conductivity/resistivity.

Results

Experiment

Figure 1(a) exhibits measurements of Rxx and Rxy over the magnetic field span −0.15 ≤ B ≤ 0.15 Tesla at T = 0.5 K. The blue curve, which exhibits Shubnikov-de Haas oscillations at |B| ≥ 0.1 Tesla, represents the Rxx in the absence of photo-excitation (w/o radiation). Microwave photo-excitation of this GaAs/AlGaAs specimen at 50 GHz, see red traces in Fig. 1, produces radiation-induced magnetoresistance oscillations in Rxx and these oscillations grow in amplitude with increasing |B|. At the deepest minimum, near |B| = (4/5)Bf, where Bf = 2πfm*/e 1, the Rxx saturates at zero-resistance. Note also the close approach to zero-resistance near |B| = (4/9)Bf. Although Rxx exhibits zero-resistance, the Hall resistance Rxy exhibits a roughly linear variation over the B-span of the zero-resistance states, see Fig. 1(a)1,2.

Figure 1
figure 1

(a) The dark- and photo-excited-diagonal- (Rxx) and the photo-excited-Hall-resistance (Rxy) are exhibited vs. the magnetic field B for a GaAs/AlGaAs heterostructure at T = 0.5 K. Rxx exhibits a non-vanishing resistance with Shubnikov-de Haas oscillations in the dark (blue trace) at |B| ≥ 0.1Tesla. Under photo-excitation at f = 50 GHz (red traces), Rxx exhibits large magnetoresistance oscillations with vanishing resistance in the vicinity of ±(4/5)Bf, where Bf = 2πfm*/e. Note the absence of a coincidental plateau in Rxy. (b) Theory predicts negative diagonal resistivity, i.e., ρxx < 0, under intense photoexcitation at the oscillatory minima, observable here in the vicinity of B ≈ 0.19 Tesla and B ≈ 0.105 Tesla. (c) Theory asserts that negative resistivity states are unstable to current domain formation and zero-resistance. Consequently, the B-span of negative resistivity in panel (b) corresponds to the domain of zero-resistance states (ZRS), per theory.

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Negative magneto-resistivity and zero-resistance

Both the displacement theory for the radiation-induced magnetoresistivity oscillations37 and the inelastic model48, suggest that the magnetoresistivity can take on negative values over the B-spans where experiment indicates zero-resistance states. For illustrative purposes, such theoretical expectations for negative resistivity are sketched in Fig. 1(b), which presents the simulated ρxx at f = 100 GHz. This curve was obtained on the basis of extrapolating, without placing a lower bound, the results of fits6, which have suggested that the radiation-induced oscillatory magnetoresistivity, , where , with W/L the device width-to-length ratio, follows . Here, F = 2πfm*/e, with f = 100 GHz, the microwave frequency, m* = 0.065me, the effective mass, e, the electron charge and with the dark resistivity which reflects typical material characteristics for the high mobility GaAs/AlGaAs 2DES. This figure shows that the deepest ρxx minima at B ≈ 0.19 Tesla and B ≈ 0.105 Tesla exhibit negative resistivity, similar to theoretical predictions37,39,41,45,48,59.

Andreev et al.38 have reasoned that the only time-independent state of a system with negative resistivity/conductivity is characterized by a current which almost everywhere has a magnitude j0 fixed by the condition that nonlinear dissipative resistivity equals zero. This prediction implies that the ρxx curve of Fig. 1(b) is transformed into the magnetoresistance, Rxx, trace shown in Fig. 1(c), where the striking feature is the zero-resistance over the B-domains that exhibited negative resistivity in Fig. 1(b). The curve of Fig. 1(c) follows from Fig. 1(b) upon multiplying the ordinate by the L/W ratio, i.e., Rxx = ρxx(L/W) and placing a lower bound of zero on the resulting Rxx.

Device configuration

As mentioned, a question of interest is: what are the transport characteristics of a bare negative magneto-conductivity/resistivity state? To address this issue, we reexamine the experimental measurement configuration in Fig. 2. Transport measurements are often carried out in the Hall bar geometry which includes finite size current contacts at the ends of the device. Here, a constant current is injected via the ends of the device and “voltmeters” measure the diagonal (Vxx) and Hall (Vxy) voltages between probe points as a function of a transverse magnetic field, as indicated in Fig. 2. Operationally, the resistances relate to the measured voltages by Rxx = Vxx/I and Rxy = Vxy/I.

Figure 2
figure 2

This dual purpose figure illustrates an idealized measurement configuration and the simulation mesh.

A Hall bar (blue outline) is connected via its current contacts (thick black rectangles at the ends) to a constant current source, which may be modelled as a battery with a resistor in series. For convenience, the negative pole of the battery has been grounded to set the potential of this terminal to zero. A pair of “voltmeters” are used to measure the diagonal (Vxx) and Hall (Vxy) voltages. For the numerical simulations reported in this work, the Hall bar is represented by a mesh of points (i, j), where the potential is evaluated by a relaxation method. Here, 0 ≤ i ≤ 100 and 0 ≤ j ≤ 20. The long (short) axis of the Hall bar corresponds the x (y)-direction.

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Simulations

Hall effect devices can be numerically simulated on a grid/mesh67,68,69, see Fig. 2, by solving the boundary value problem corresponding to enforcing the local requirement , where is the 2D current density with components jx and jy, and is the conductivity tensor67,68. Enforcing within the homogeneous device is equivalent to solving the Laplace equation ∇2V = 0, which may be carried out in finite difference form using a relaxation method, subject to the boundary conditions that current injected via current contacts is confined to flow within the conductor. That is, current perpendicular to edges must vanish everywhere along the boundary except at the current contacts. We have carried out simulations using a 101 × 21 point grid with current contacts at the ends that were 6 points wide. For the sake of simplicity, the negative current contact is set to ground potential, i.e., V = 0, while the positive current contact is set to V = 1. In the actual Hall bar device used in experiment, the potential at the positive current contact will vary with the magnetic field but one can always normalize this value to 1 to compare with these simulations.

Figure 3 summarizes the potential profile within the Hall device at three values of the Hall angle, θH = tan−1(σxy/σxx). Fig. 3(a) shows a color plot of the potential profile with equipotential contours within the device at θH = 0°, which corresponds to the B = 0 situation. This panel, in conjunction with Fig. 3(b), shows that the potential drops uniformly within the device from the left- to the right- ends of the Hall bar. Fig. 3(c) shows the absence of a potential difference between the top- and bottom- edges along the indicated yellow line at x = 50. This feature indicates that there is no Hall effect in this device at B = 0, as expected.

Figure 3
figure 3

This figure summarizes the potential profiles within a Hall bar device that is 100 units long and 20 units wide at three values of the Hall angle, θH, where θH = tan−1(σxy/σxx).

(a) This panel shows the potential profile at θH = 0°, which corresponds to the B = 0 situation. Current contacts are indicated by black rectangles at the left- and right- ends, midway between the top and the bottom edges. The left end of the Hall bar is at V = 1 and the right end is at V = 0. Potential (V) is indicated in normalized arbitrary units. Panel (b) shows that the potential decreases linearly along the indicated yellow line at y = 10 from the left end to the right end of the device. Panel (c) shows the absence of a potential difference between the top- and bottom- edges along the indicated yellow line at x = 50. That is, there is no Hall effect at B = 0. Panel (d) shows the potential profile at θH = 60°, which corresponds to σxx = 0.577σxy. Note that the equipotential contours develop a tilt with respect to the same in panel (a). Panel (e) shows the potential drop from the left to the right edge along the line at y = 10. Panel (f) shows a decrease in the potential from the bottom to the top edge. This potential difference is the Hall voltage at θH = 60°. Panel (g) shows the potential profile at θH = 88.5°, which corresponds to σxx = 0.026σxy. Note that in the interior of the device, the equipotential contours are nearly parallel to the long axis of the Hall bar, in sharp contrast to (a). Panel(h) shows the potential variation from the left to the right end of the device along the line at y = 10. The reduced potential variation here between the Vxx voltage probes (red and black triangles) is indicative of a reduced diagonal resistance. Panel(i) shows a large variation in the potential along the line at x = 50 between the bottom and top edges.

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Figure 3(d) shows the potential profile at θH = 60°, which corresponds to the situation where σxx = 0.577σxy. Note that, here, the equipotential contours develop a tilt with respect to the same in Fig. 3(a). Fig. 3(e) shows a mostly uniform potential drop from the left to the right edge along the line at y = 10, as Fig. 3(f) shows a decrease in the potential from the bottom to the top edge. This potential difference represents the Hall voltage under these conditions.

Figure 3(g) shows the potential profile at θH = 88.5°, which corresponds to the situation where σxx = 0.026σxy. Note that in the interior of the device, the equipotential contours are nearly parallel to the long axis of the Hall bar, in sharp contrast to Fig. 3(a). Fig. 3(h) shows the potential variation from the left to the right end of the device. The reduced change in potential between the Vxx voltage probes (red and black inverted triangles), in comparison to Fig. 3(b) and Fig. 3(e) is indicative of a reduced diagonal voltage and resistance. Fig. 3(i) shows a large potential difference between the bottom and top edges, indicative of a large Hall voltage.

The results presented in Fig. 3 display the normal expected behavior for a 2D Hall effect device with increasing Hall angle. Such simulations can also be utilized to examine the influence of microwave excitation since microwaves modify the diagonal conductivity, σxx, or resistivity, ρxx37,48 and this sets θH via θH = tan−1(σxy/σxx). In the next figure, we examine the results of such simulations when the diagonal conductivity, σxx, reverses signs and takes on negative values, as per theory, under microwave excitation. Thus, figure 4 compares the potential profile within the Hall bar device for positive (σxx = +0.026σxy) and negative (σxx = −0.026σxy) values of the conductivity.

Figure 4
figure 4

This figure compares the potential profile within the Hall bar device for positive (σxx = +0.026σxy) and negative (σxx = −0.026σxy) conductivity.

Panel (a) shows the potential profile at θH = 88.5° which corresponds to σxx = +0.026σxy. Note that in the interior of the device, the equipotential contours are nearly parallel to the long axis of the Hall bar. Panel(b) shows the potential variation from the left to the right end of the device along the line at y = 10. The small potential drop here between the Vxx voltage probes (red and black triangles) is indicative of a reduced diagonal resistance in this low σxx condition. Panel(c) suggests the development of a large Hall voltage between the bottom and top edges. Here the voltage decreases towards the top edge. Panel (d) shows the potential profile at θH = 91.5° which corresponds to σxx = −0.026σxy. The key feature here is the reflection of the potential profile with respect panel (a) about the line at y = 10 when the σxx shifts from a positive (σxx = +0.026σxy) to a negative (σxx = −0.026σxy) value. Panel (e) shows that in the negative σxx condition, the potential still decreases from left to right, implying a positive diagonal voltage Vxx and diagonal resistance Rxx. Panel (f) shows that for σxx = −0.026σxy, the potential increases from the bottom edge to the top edge, unlike in panel (c). Thus, the Hall voltage undergoes sign reversal in going from the σxx = +0.026σxy situation to the σxx = −0.026σxy condition, compare panels (c) and (f).

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Fig. 4(a) shows the potential profile at σxx = +0.026σxy. This figure is identical to Fig. 3(g). The essential features are that the equipotential contours are nearly parallel to the long axis of the Hall bar, see Fig. 4(b), signifying a reduced diagonal resistance. Concurrently, Fig. 4(c) suggests the development of a large Hall voltage between the bottom and top edges. Here the Hall voltage decreases from the bottom- to the top- edge.

Fig. 4(d) shows the potential profile at σxx = −0.026σxy, i.e., the negative conductivity case. The important feature here is the reflection of the potential profile with respect Fig. 4(a) about the line at y = 10 when the σxx shifts from a positive (σxx = +0.026σxy) to a negative (σxx = −0.026σxy) value. Fig. 4(e) shows, remarkably, that in the negative σxx condition, the potential still decreases from left to right, implying Vxx > 0 and Rxx > 0 even in this σxx ≤ 0 condition. Fig. 4(f) shows that for σxx = −0.026σxy, the potential increases from the bottom edge to the top edge, in sharp contrast to Fig. 4(c). Thus, these simulations show clearly that the Hall voltage undergoes sign reversal when σxx < 0, although the diagonal voltage (and resistance) exhibits positive values.

Discussion

Existing theory indicates that photo-excitation of the high mobility 2D electron system can drive the ρxx and σxx to negative values at the minima of the radiation-induced oscillatory magneto-resistivity37,39,41,45,48,59. Andreev et al.38, have argued that “σxx < 0 by itself suffices to explain the zero-dc-resistance state” because “negative linear response conductance implies that the zero-current state is intrinsically unstable.” Since our simulations (Fig. 4) show clearly that negative magneto conductivity/resistivity leads to positive, not negative, conductance/resistance, it looks like one cannot argue for an instability in the zero-current state based on presumed “negative linear response conductance.”

For illustrative purposes, using the understanding obtained from the simulation results shown in Fig. 4, we sketch in Fig. 5 the straightforward expectations, for the behavior of the diagonal (Rxx) and Hall (Rxy) resistances in a 2D system driven periodically to negative diagonal conductivity by photo-excitation. Fig. 5(a) shows that the microwave-induced magnetoresistance oscillations in Rxx grow in amplitude with increasing B. When the oscillations in the magneto-resistivity/conductivity are so large that the oscillatory minima would be expected to cross into the regime of σxx < 0 at the oscillatory minima, the Rxx exhibits positive values. Here, vanishing Rxx occurs only at singular values of the magnetic field where σxx = 0. Fig. 5(b) shows that the Hall resistance Rxy shows sign reversal over the same span of B where σxx < 0.

Figure 5
figure 5

This figure illustrates expectations, based on the results illustrated in Fig. 3 and 4, for the behavior of the diagonal (Rxx) and Hall (Rxy) resistances in a 2D system driven periodically to negative conductivity/resistivity by photo-excitation.

(a) The diagonal resistance Rxx exhibits microwave-induced magnetoresistance oscillations that grow in amplitude with increasing B. In the regime of negative conductivity at the oscillatory minima, the Rxx exhibits positive values. (b) Over the same span of B, the Hall resistance Rxy shows sign reversal.

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It appears that if there were an instability, it should be related to the sign-reversal in the Hall effect. Yet, note that sign reversal in the Hall effect is not a manifestly un-physical effect since it is possible to realize Hall effect sign reversal in experiment even with a fixed external bias on the sample, as in the simulations, simply by reversing the direction of the magnetic field or by changing the sign of the charge carriers. The unusual characteristic indicated by these simulations is Hall effect sign reversal even without changing the direction of the magnetic field or changing the sign of the charge carriers. This feature can be explained, however, by noting that the numerical solution of the boundary value problem depends on a single parameter, the Hall angle, θH, where tan(θH) = σxy/σxx. Since this single parameter depends on the ratio of the off-diagonal and diagonal conductivities, sign change in σxx produces the same physical effect as sign reversal in σxy so far as the solution to the boundary value problem is concerned. That is, one might change the sign of σxy or one might change the sign of σxx, the end physical result is the same: a sign reversal in the Hall effect.

One might also ask: why do the simulations indicate a positive diagonal resistances for the negative diagonal conductivity/resistivity scenario? The experimental setup shown in Fig. 2 offers an answer to this question: In the experimental setup, the Hall bar is connected to an external battery which enforces the direction of the potential drop from one end of the specimen to the other. This directionality in potential drop is also reflected in the boundary value problem. As a consequence, the red potential probe in Fig. 2, 3 or 4 would prefer to show a higher potential than the black potential probe so long as the σxx is not identically zero and this leads to the positive resistance even for negative diagonal conductivity/resistivity in the numerical simulations.

We remark that the experimental results of Fig. 1(a) are quite unlike the expectations exhibited in Fig. 5. Experiment shows an ordinary Hall effect without anomalies over the zero-resistance region about (4/5)Bf, not a sign reversal in the Hall effect and experiment shows zero-resistance, not the positive resistance expected for a system driven to negative conductivity.

In conclusion, the results presented here suggest that a bare negative magneto- conductivity/resistivity state in the 2DES under photo-excitation should yield a positive diagonal resistance with a concomitant sign reversal in the Hall effect.

We have also understood that these results could be potentially useful for understanding plateau formation in the Hall effect as, for example, in the quantum Hall situation, if new physics comes into play in precluding sign reversal in the Hall effect, when the diagonal magneto-conductivity/resistivity is forced into the regime of negative values.

Methods

Samples

The GaAs/AlGaAs material utilized in our experiments exhibit electron mobility μ ≈ 107 cm2/V s and electron density in the range 2.4 × 1011 ≤ n ≤ 3 × 1011 cm−2. Utilized devices include cleaved specimens with alloyed indium contacts and Hall bars fabricated by optical lithography with alloyed Au-Ge/Ni contacts. Standard low frequency lock-in techniques yield the electrical measurements of Rxx and Rxy1,3,4,7,12,13,14,18,22,27,29,31,32,35,36.

Microwave transport measurements

Typically, a Hall bar specimen was mounted at the end of a long straight section of a rectangular microwave waveguide. The waveguide with sample was inserted into the bore of a superconducting solenoid, immersed in pumped liquid Helium and irradiated with microwaves, at a source-power 0.1 ≤ P ≤ 10 mW, as in the usual microwave-irradiated transport experiment1. The applied external magnetic field was oriented along the solenoid and waveguide axis.