In a high-mobility two-dimensional electron gas (2DEG) in a weak perpendicular magnetic field, microwave irradiation produces microwave-induced resistance oscillations (MIRO) that deepen, at high power, into zero-resistance states (ZRS). The accepted interpretation is that the homogeneous state acquires negative dissipative conductivity and breaks up into current domains; all conventional experiments impose a dc current. Levin et al. (Phys. Rev. Lett. 115, 206801 (2015)) showed that the phonon-drag voltage — the thermoelectric voltage generated by a flux of ballistic phonons from a local heater — oscillates in step with MIRO and exhibits sharp, complex features inside the ZRS intervals with no dc drive anywhere in the circuit, implying that current domains can form under purely thermoelectric conditions. However, the available homogeneous-transport theory cannot describe these features, and the domain structure, arrangement, and dynamics without dc driving remain undetermined. The problem is to determine the inhomogeneous (domain) state of the microwave-induced ZRS in the absence of external dc driving, to compute its spatial electric-field and phonon-drag-voltage signatures in realistic multi-terminal geometry, and to confront the predictions with the experimental phonon-drag-voltage data. Later surface-acoustic-wave work confirmed bulk domain formation under ordinary transport conditions, but the no-dc-drive thermoelectric case remains untouched.

Background

When a high-mobility two-dimensional electron gas (2DEG), for example in a GaAs quantum well, is placed in a weak perpendicular magnetic field BB and irradiated with microwaves (MW), its longitudinal resistance develops oscillations periodic in 1/B1/B — microwave-induced resistance oscillations (MIRO) — caused by microwave-photon-assisted electron transitions between Landau levels (the discrete cyclotron orbits of a 2D electron; in these high-mobility samples the levels overlap and only a weak quantum oscillation survives). The oscillations are governed by the ratio of the radiation frequency ω\omega to the cyclotron frequency ωc=eB/m∗\omega_c=eB/m^{*}, with m∗m^{*} the electron effective mass. With increasing MW power the MIRO minima deepen until the measured resistance vanishes over finite intervals of BB: the zero-resistance states (ZRS), discovered by Mani et al. and by Zudov et al. Theory for spatially homogeneous transport predicts that the dissipative resistivity ρxx\rho_{xx} in these intervals actually becomes negative. Since a negative conductivity makes the homogeneous current distribution unstable, Andreev, Aleiner, and Millis argued that the system instead breaks into a pattern of current domains — alternating regions carrying a strong internal Hall electric field and a dissipationless Hall current, separated by walls; an imposed current is then accommodated by shifting domain walls, so the terminal voltage drop and hence the resistance vanish. Several dc-biased experiments support this domain picture.

Nearly all MIRO/ZRS experiments probe dc transport with an imposed current. Levin et al. introduced a different, dc-free probe: the phonon-drag voltage (PDV). If part of the sample is heated by an on-chip heater, nonequilibrium (ballistic) phonons stream across the 2DEG and transfer momentum to the electrons, generating a thermoelectric voltage even without any impressed current. Writing the current density as j=σ^E−β^∇T\mathbf{j}=\hat{\sigma}\mathbf{E}-\hat{\beta}\nabla T with conductivity and thermoelectric tensors σ^,β^\hat{\sigma},\hat{\beta}, open-circuit conditions give E=ρ^β^∇T\mathbf{E}=\hat{\rho}\hat{\beta}\nabla T; in the classically strong field regime the transverse thermopower is normally zero because two terms compensate, but under MW irradiation the dissipative resistivity ρxx\rho_{xx} is strongly modulated while the drag coefficients are not, so a large transverse drag voltage proportional to the MW-induced part of ρxx\rho_{xx} appears. In a four-terminal star-shaped GaAs device (mobility 2×1062\times 10^{6} cm2^2/Vs) with a heater, Levin et al. observed PDV oscillations mirroring MIRO and — crucially — sharp jumps and complex, run-to-run and contact-to-contact diverse features in the PDV exactly inside the ZRS intervals, with no external dc drive anywhere in the circuit. The natural reading is that the ZRS domains form even without dc driving: the phonon flux is then the only spatial inhomogeneity, and the domain arrangement must arrange itself so that no net current flows through the contacts, with switching between domain configurations producing the voltage jumps.

The theory available then and now describes only spatially homogeneous transport. In the regime of weak Landau quantization (Dingle factor d=exp⁡(−π/∣ωc∣τ)d=\exp(-\pi/|\omega_c|\tau), with τ\tau the quantum lifetime), the drag electric fields along and across a unidirectional phonon flux are EL=F+2d2GE_L=F+2d^2G and ET=(2d2G−F δρxx/ρ0)/ωcτtrE_T=(2d^2G-F\,\delta\rho_{xx}/\rho_0)/\omega_c\tau_{tr}, where δρxx\delta\rho_{xx} is the MW-induced resistivity correction, τtr\tau_{tr} the transport time, and F,GF,G describe the magnetophonon oscillation of the drag; terminal voltages follow as Vij=γijLEL+γijTETV_{ij}=\gamma_{ij}^{L}E_L+\gamma_{ij}^{T}E_T with geometry-dependent lengths. This reproduces the MIRO-correlated PDV oscillations outside the ZRS, but inside the ZRS intervals the homogeneous state is itself unstable, the theory becomes inapplicable, and the measured PDV shows complex behavior the theory cannot capture; the authors state that the details of the no-dc-drive domain structures are not clear and require further study, and note that the MW-induced thermoelectric phenomena may show up in other 2D systems.

Since 2015 the dc-driven domain problem has advanced. Internal-field measurements in dc-biased ZRS show random flips of the spontaneously broken-symmetry field (Dorozhkin et al.), with a theory of the flips as a dynamic instability involving the dopant layer (Dmitriev). The linear dc response of a frozen domain pattern was analyzed by Breitkreiz. Most directly, Friess et al. used surface acoustic waves (SAW — Rayleigh waves whose piezoelectric field is partially screened by the 2DEG, so their velocity shift maps the local bulk conductivity) as a directional, contactless probe, showing that the ZRS is a bulk inhomogeneous domain phase whose domain orientation controls the acoustoelectric response and whose alignment varies between cool-downs. None of these works addresses the situation without dc drive or computes a thermoelectric response of a domain state. On the drag side, Raichev et al. extended the classical theory of ballistic-phonon drag in mesoscopic conductors (phonon-flux-induced current stripes, large Nernst effect, relaxation of higher angular harmonics of the distribution function), but without microwaves or negative conductivity. MIRO/ZRS themselves have since been observed in additional platforms (two-subband GaAs systems, electrons on liquid helium), but MW-induced thermoelectric phenomena have not been pursued in other 2D systems or geometries.

Problem Statement

Determine, theoretically and experimentally, the current-domain (more generally, inhomogeneous) state that a high-mobility two-dimensional electron gas forms in the microwave-induced zero-resistance regime when no external dc driving is applied, and derive its magnetothermoelectric — specifically phonon-drag-voltage — signatures. The unifying objective is to explain the observed behavior of the phonon-drag voltage inside the zero-resistance intervals. The jointly required results are:

  1. Domain state without dc drive. Determine the structure, arrangement, and dynamics of the electric-field/current domains (or alternative inhomogeneous configurations) that develop when microwave irradiation drives the dissipative conductivity negative and the only spatial inhomogeneity and drive is a ballistic-phonon flux, with the constraint that no net current flows through any contact. Establish which domain configurations are selected and whether they are static, metastable, or dynamically active (e.g., flipping or oscillating), and whether the sharp jumps of the drag voltages at the boundaries of the ZRS intervals and the diverse features within them follow from switching between domain configurations, as conjectured.

  2. Thermoelectric response of the domain state. From the domain description, compute the spatial distribution of the in-plane electric field and the measurable phonon-drag voltages Vij(B)V_{ij}(B) in a realistic multi-terminal geometry, beyond the homogeneous weak-quantization theory, and state under what conditions the complex and diverse PDV behavior observed inside the ZRS intervals is reproduced or must be attributed to a different mechanism.

  3. Quantitative confrontation with the experimental data. Confront the domain-state theory quantitatively with the existing PDV data, including whatever refinement of the ballistic-phonon source characterization (effective phonon temperature, mean free paths, angular distribution from the heater) is required to make the comparison quantitative. New discriminating measurements — e.g., contactless directional probes such as surface acoustic waves, or spatially resolved field sensors, operated under purely thermoelectric drive — are a suggested route for sharpening the comparison, not a mandatory deliverable.

Extension of the magnetothermoelectric detection to other 2D material systems and device geometries is an exploratory component of the problem, without a sharp completion boundary, and is not part of the required results. A complete answer identifies the no-dc-drive domain state and shows quantitatively that its predicted thermoelectric response reproduces the sharp jumps of the PDV at the boundaries of the ZRS intervals, while giving an account of the run-to-run and contact-to-contact diversity of the features inside those intervals; alternatively, a complete answer refutes the domain interpretation by a different mechanism that accounts quantitatively for both the boundary jumps and the diverse internal features, in which case the domain-state results (items 1-2) are replaced by that demonstration.

References

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