Abstract
Negative diagonal magnetoconductivity/resistivity is a spectacular and thought provokingproperty of driven, farfromequilibrium, low dimensional electronic systems. The physical response of this exotic electronic state is not yet fully understood since it is rarely encountered in experiment. The microwaveradiationinduced zeroresistance state in the high mobility GaAs/AlGaAs 2D electron system is believed to be an example where negative magnetoconductivity/resistivity is responsible for the observed phenomena. Here, we examine the magnetotransport characteristics of this negative conductivity/resistivity state in the microwave photoexcited twodimensional electron system (2DES) through a numerical solution of the associated boundary value problem. The results suggest, surprisingly, that a bare negative diagonal conductivity/resistivity state in the 2DES under photoexcitation should yield a positive diagonal resistance, with a concomitant sign reversal in the Hall voltage.
Introduction
Negative magnetoconductivity/resistivity is a spectacular and thought provoking theoretical property of microwave photoexcited, farfromequilibrium, twodimensional electronic systems. This property has been utilized to understand the experimental observation of the microwaveradiationinduced zeroresistance states in the GaAs/AlGaAs system^{1,2}. Yet, the negative conductivity/resistivity state remains an enigmatic and open topic for investigation, although, over the past decade, photoexcited transport has been the subject of a broad and intense experimental^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36} and theoretical^{37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66} study in the 2D electron system (2DES).
In experiment, the microwaveinduced zeroresistance states arise from “1/4cycleshifted” microwave radiationinduced magnetoresistance oscillations in the high mobility GaAs/AlGaAs system^{1,4,33} 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 microwaveinduced magnetoresistance oscillations becomes large enough that the deepest oscillatory minima approach zeroresistance. Further reduction in T then leads to the saturation of the resistance at zero, leading to the zeroresistance states that empirically look similar to the zeroresistance states observed under quantized Hall effect conditions^{1,2,7}. Similar to the situation in the quantized Hall effect, these radiationinduced zero resistance states exhibit activated transport^{1,2,7,10}. A difference with respect to the quantized Hall situation, however, is that the Hall resistance, R_{xy}, does not exhibit plateaus or quantization in this instance where the zeroresistance state is obtained by photoexcitation^{1,2,7}.
Some theories have utilized a two step approach to explain the microwaveradiationinduced zeroresistance states. In the first step, theory identifies a mechanism that helps to realize oscillations in the diagonal magnetophotoconductivity/resistivity, and provides for the possibility that the minima of the oscillatory diagonal conductivity/resistivity can even take on negative values^{37,39,41,45,48,59}. The next step in the two step approach invokes the theory of Andreev et al.^{38}, who suggest that the zerocurrentstate at negative resistivity (and conductivity) is unstable, and that this favors the appearance of current domains with a nonvanishing current density^{38,56}, followed by the experimentally observed zeroresistance states.
There exist alternate approaches which directly realize zeroresistance states without a detour through negative conductivity/resistivity states. Such theories include the radiationdriven electronorbit model^{49}, the radiationinducedcontactcarrieraccumulation/depletion model^{62}, and the synchronization model^{66}. Thus far, however, experiment has been unable to clarify the underlying mechanism(s), so far as the zeroresistance states are concerned.
The negative magnetoconductivity/resistivity state suggested theoretically in this problem^{37,39,41,45,48,59} has been a puzzle for experiment since it had not been encountered before in magnetotransport. Naively, one believes that negative magnetoresistivity/conductivity should lead to observable negative magnetoresistance/conductance, based on expectations for the zeromagneticfield 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 zeromagneticfield analogy, for an instability in a negative magnetoconductivity/resistivity state? If an instability does occur for the negative magnetoconductivity/resistivity state, what is the reason for the instability? Could negative conductivity/resistivity lead to observable negative conductance/resistance at least in some short timescale transient situation where current domains have not yet formed? Indeed, one might ask: what are the magnetotransport 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 photoexcited 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 photoexcitation should generally yield a positive diagonal resistance, i.e., R_{xx} > 0, except at singular points where R_{xx} = 0 when the diagonal conductivity σ_{xx} = 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 zeromagneticfield analogy, for a negative conductivity/resistivity state in a nonzero magnetic field, need not necessarily follow, and that experimental observations of zeroresistance and a linear Hall effect in the photoexcited GaAs/AlGaAs system could be signatures of vanishing conductivity/resistivity.
Results
Experiment
Figure 1(a) exhibits measurements of R_{xx} and R_{xy} over the magnetic field span −0.15 ≤ B ≤ 0.15 Tesla at T = 0.5 K. The blue curve, which exhibits Shubnikovde Haas oscillations at B ≥ 0.1 Tesla, represents the R_{xx} in the absence of photoexcitation (w/o radiation). Microwave photoexcitation of this GaAs/AlGaAs specimen at 50 GHz, see red traces in Fig. 1, produces radiationinduced magnetoresistance oscillations in R_{xx}, and these oscillations grow in amplitude with increasing B. At the deepest minimum, near B = (4/5)B_{f}, where B_{f} = 2πfm*/e ^{1}, the R_{xx} saturates at zeroresistance. Note also the close approach to zeroresistance near B = (4/9)B_{f}. Although R_{xx} exhibits zeroresistance, the Hall resistance R_{xy} exhibits a roughly linear variation over the Bspan of the zeroresistance states, see Fig. 1(a)^{1,2}.
Negative magnetoresistivity and zeroresistance
Both the displacement theory for the radiationinduced magnetoresistivity oscillations^{37}, and the inelastic model^{48}, suggest that the magnetoresistivity can take on negative values over the Bspans where experiment indicates zeroresistance states. For illustrative purposes, such theoretical expectations for negative resistivity are sketched in Fig. 1(b), which presents the simulated ρ_{xx} at f = 100 GHz. This curve was obtained on the basis of extrapolating, without placing a lower bound, the results of fits^{6}, which have suggested that the radiationinduced oscillatory magnetoresistivity, , where , with W/L the device widthtolength ratio, follows . Here, F = 2πfm*/e, with f = 100 GHz, the microwave frequency, m* = 0.065m_{e}, 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 ρ_{xx} minima at B ≈ 0.19 Tesla and B ≈ 0.105 Tesla exhibit negative resistivity, similar to theoretical predictions^{37,39,41,45,48,59}.
Andreev et al.^{38} have reasoned that the only timeindependent state of a system with negative resistivity/conductivity is characterized by a current which almost everywhere has a magnitude j_{0} fixed by the condition that nonlinear dissipative resistivity equals zero. This prediction implies that the ρ_{xx} curve of Fig. 1(b) is transformed into the magnetoresistance, R_{xx}, trace shown in Fig. 1(c), where the striking feature is the zeroresistance over the Bdomains 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., R_{xx} = ρ_{xx}(L/W), and placing a lower bound of zero on the resulting R_{xx}.
Device configuration
As mentioned, a question of interest is: what are the transport characteristics of a bare negative magnetoconductivity/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 (V_{xx}) and Hall (V_{xy}) 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 R_{xx} = V_{xx}/I and R_{xy} = V_{xy}/I.
Simulations
Hall effect devices can be numerically simulated on a grid/mesh^{67,68,69}, see Fig. 2, by solving the boundary value problem corresponding to enforcing the local requirement , where is the 2D current density with components j_{x} and j_{y}, , and is the conductivity tensor^{67,68}. Enforcing within the homogeneous device is equivalent to solving the Laplace equation ∇^{2}V = 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, θ_{H} = tan^{−1}(σ_{xy}/σ_{xx}). Fig. 3(a) shows a color plot of the potential profile with equipotential contours within the device at θ_{H} = 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(d) shows the potential profile at θ_{H} = 60°, which corresponds to the situation where σ_{xx} = 0.577σ_{xy}. 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 θ_{H} = 88.5°, which corresponds to the situation where σ_{xx} = 0.026σ_{xy}. 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 V_{xx} 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, σ_{xx}, or resistivity, ρ_{xx}^{37,48}, and this sets θ_{H} via θ_{H} = tan^{−1}(σ_{xy}/σ_{xx}). In the next figure, we examine the results of such simulations when the diagonal conductivity, σ_{xx}, 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 (σ_{xx} = +0.026σ_{xy}) and negative (σ_{xx} = −0.026σ_{xy}) values of the conductivity.
Fig. 4(a) shows the potential profile at σ_{xx} = +0.026σ_{xy}. 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 σ_{xx} = −0.026σ_{xy}, 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 σ_{xx} shifts from a positive (σ_{xx} = +0.026σ_{xy}) to a negative (σ_{xx} = −0.026σ_{xy}) value. Fig. 4(e) shows, remarkably, that in the negative σ_{xx} condition, the potential still decreases from left to right, implying V_{xx} > 0 and R_{xx} > 0 even in this σ_{xx} ≤ 0 condition. Fig. 4(f) shows that for σ_{xx} = −0.026σ_{xy}, 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 σ_{xx} < 0, although the diagonal voltage (and resistance) exhibits positive values.
Discussion
Existing theory indicates that photoexcitation of the high mobility 2D electron system can drive the ρ_{xx} and σ_{xx} to negative values at the minima of the radiationinduced oscillatory magnetoresistivity^{37,39,41,45,48,59}. Andreev et al.^{38}, have argued that “σ_{xx} < 0 by itself suffices to explain the zerodcresistance state” because “negative linear response conductance implies that the zerocurrent 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 zerocurrent 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 (R_{xx}) and Hall (R_{xy}) resistances in a 2D system driven periodically to negative diagonal conductivity by photoexcitation. Fig. 5(a) shows that the microwaveinduced magnetoresistance oscillations in R_{xx} grow in amplitude with increasing B. When the oscillations in the magnetoresistivity/conductivity are so large that the oscillatory minima would be expected to cross into the regime of σ_{xx} < 0 at the oscillatory minima, the R_{xx} exhibits positive values. Here, vanishing R_{xx} occurs only at singular values of the magnetic field where σ_{xx} = 0. Fig. 5(b) shows that the Hall resistance R_{xy} shows sign reversal over the same span of B where σ_{xx} < 0.
It appears that if there were an instability, it should be related to the signreversal in the Hall effect. Yet, note that sign reversal in the Hall effect is not a manifestly unphysical 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, θ_{H}, where tan(θ_{H}) = σ_{xy}/σ_{xx}. Since this single parameter depends on the ratio of the offdiagonal and diagonal conductivities, sign change in σ_{xx} produces the same physical effect as sign reversal in σ_{xy} so far as the solution to the boundary value problem is concerned. That is, one might change the sign of σ_{xy} or one might change the sign of σ_{xx}, 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 σ_{xx} 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 zeroresistance region about (4/5)B_{f}, not a sign reversal in the Hall effect, and experiment shows zeroresistance, 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 photoexcitation 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 magnetoconductivity/resistivity is forced into the regime of negative values.
Methods
Samples
The GaAs/AlGaAs material utilized in our experiments exhibit electron mobility μ ≈ 10^{7} cm^{2}/V s and electron density in the range 2.4 × 10^{11} ≤ n ≤ 3 × 10^{11} cm^{−2}. Utilized devices include cleaved specimens with alloyed indium contacts and Hall bars fabricated by optical lithography with alloyed AuGe/Ni contacts. Standard low frequency lockin techniques yield the electrical measurements of R_{xx} and R_{xy}^{1,3,4,7,12,13,14,18,22,27,29,31,32,35,36}.
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 sourcepower 0.1 ≤ P ≤ 10 mW, as in the usual microwaveirradiated transport experiment^{1}. The applied external magnetic field was oriented along the solenoid and waveguide axis.
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Acknowledgements
The basic research at Georgia State University is primarily supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Material Sciences and Engineering Division under DESC0001762. Additional support is provided by the ARO under W911NF0701015.
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Affiliations
Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303
 R. G. Mani
Department of Physics, Emory University, 400 Dowman Drive, Atlanta, GA 30322
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Contributions
The microwave transport experiments, the modelling, and the manuscript are due to R.G.M. A.K. assisted with the simulations and plots.
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The authors declare no competing financial interests.
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Correspondence to R. G. Mani.
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