Fluctuation effects, two-dimensional domain morphology, and global stability of the domain state in the voltage-biased zero-resistance state

The zero-resistance state (ZRS) of a microwave-irradiated two-dimensional electron gas in a perpendicular magnetic field is understood as a static domain state that forms when the dissipative conductivity becomes negative. The mean-field theory of Dmitriev, Khodas, Mirlin, and Polyakov [Phys. Rev. Lett. 111, 206801 (2013)] for a voltage-biased finite-width stripe with nonlocal two-dimensional electrostatics produces a phase diagram in the voltage-size plane with second-order and first-order transitions between the homogeneous and domain states, a tricritical point, and hysteresis, but explicitly leaves three issues open: the effect of fluctuations beyond the Lyapunov-functional mean field on the nature and order of the ZRS transition; the structure of the fully two-dimensional domain state expected to be the global minimum for stripe widths $l>l_{c+}\simeq7$ (in units of the nonequilibrium screening length), together with the global-stability boundary $V^*_{c1}$ of the one-dimensional domain solution; and the stability of that solution at still larger $l$ beyond the numerically verified $l\lesssim40$. The problem remains open.

Background

When a high-mobility two-dimensional electron system (2DES) in a moderate perpendicular magnetic field $B$ is irradiated with microwaves, its dissipative conductivity oscillates with $B$ (microwave-induced resistance oscillations, MIRO); at sufficiently high microwave power the linear-response dissipative conductivity $\sigma(0)$ becomes negative in certain field intervals. A state with negative absolute conductivity is unstable against charge-density fluctuations, and experiments instead show a state with (almost) vanishing longitudinal resistance — the zero-resistance state (ZRS). The standard explanation, going back to Andreev, Aleiner, and Millis (2003), is dynamical symmetry breaking: the system breaks up into static domains of electric field, arranged so that the spatially averaged dissipative current vanishes. Because a magnetic field makes the current strongly Hall (perpendicular to the field), the domain problem in a 2DES differs qualitatively from classic one-dimensional instabilities such as the Gunn effect (the instability of a semiconductor with negative differential conductivity, which produces traveling electric-field domains).

Most theory before 2013 treated the bulk (thermodynamic) limit. Alicea, Balents, Fisher, Paramekanti, and Radzihovsky [Phys. Rev. B 71, 235322 (2005)] analyzed the nonequilibrium transition into the ZRS with phenomenological fluctuating hydrodynamics and a dynamical renormalization group (a method that tracks how fluctuation effects grow or shrink as the dynamics is coarse-grained in space and time), finding that with long-range (Coulomb) interactions fluctuations can drive the transition first order in the unbiased, infinite-system limit. Finkler, Halperin, Auerbach, and Yacoby [J. Stat. Phys. 125, 1093 (2006)] used a Lyapunov-functional approach — a functional of the field configuration that decreases along the dissipative dynamics and plays the role of a nonequilibrium free energy — to study two-dimensional domain patterns at zero bias, including the effects of disorder and boundaries, and Finkler and Halperin [Phys. Rev. B 79, 085315 (2009)] showed that such states need not be static. Dorozhkin, Dmitriev, and Mirlin [Phys. Rev. B 84, 125448 (2011)] showed that in a finite stripe the instability threshold is shifted, parameterized by the ratio $l=L/\pi\lambda$ of the stripe width $L$ to the nonequilibrium screening length $\lambda=\epsilon D/2\pi|\sigma(0)|$ ($\epsilon$ the dielectric constant, $D$ the diffusion coefficient), which sets the domain-wall width and acts as the critical parameter.

Dmitriev, Khodas, Mirlin, and Polyakov [Phys. Rev. Lett. 111, 206801 (2013)] (DKMP) solved, analytically and at mean-field level, the problem of the voltage-biased ZRS stripe of arbitrary width $d/L$. Because 2D electrostatics is nonlocal, the field obeys a nonlinear integral equation involving the Hilbert transform $H$ (the transform converting the in-plane field to the normal field generated by a given charge density), $E(x)\sigma[E(x)]+(\epsilon D/2\pi)\,\partial_x H\{E(x)\}=j_x$, with a model conductivity $\sigma(E)=\sigma(0)(E_c/\pi E)\sin(\pi E/E_c)$ that is negative at small field and crosses zero at $E_c$. The exact one-dimensional (1D) domain solution, the current-voltage characteristic of the domain state (which carries a negative dissipative conductance), and the full phase diagram in the $V$–$L$ plane follow: a second-order transition line between the homogeneous and domain states for $1<l<\sqrt{2}$, terminating at a tricritical point — the point where the line of continuous transitions meets the first-order line — at ($l=\sqrt{2}$, $V=E_cL/4$), beyond which, for larger voltages and widths, the transition is first order with a hysteretic interval; the domain solution is linearly stable up to $V_{c1}$, checked numerically for $l<40$.

The authors explicitly flag what remains unresolved. First, the treatment is mean-field (minimization of the Lyapunov functional); the impact of fluctuations on the nature and order of the ZRS transition in this finite-size, voltage-biased problem is unstudied — fluctuations may shift the phase boundaries and, as Alicea et al. found in the unbiased thermodynamic limit, may force the transition first order. Second, the linear-stability analysis against transverse fluctuations (wavevectors along the stripe, $q_y\neq0$) shows they are irrelevant for $l<l_{c\perp}\simeq1.76$, but for $l>l_{c+}\simeq7$ the transverse-stability line of the homogeneous state crosses the first-order line $V_{c1}$, so in the region between the two lines the global minimum of the Lyapunov functional should be a fully two-dimensional (2D) domain state with broken translational invariance along the stripe; neither the nature of this 2D state nor the position of the global-stability boundary $V^*_{c1}$ of the 1D domain solution is known. Third, the numerical linear-stability verification of the 1D domain solution stops at $l\lesssim40$, leaving its stability at still larger $l$, where nonlinearity and fluctuations become more important, undetermined.

Problem Statement

Within the model of a voltage-biased two-dimensional electron stripe under microwave irradiation and a perpendicular magnetic field, with negative absolute dissipative conductivity $\sigma(E)$ crossing zero at the critical field $E_c$, positive diffusion coefficient $D$, and nonlocal 2D electrostatics (the Hilbert-transform integral equation $E(x)\sigma[E(x)]+(\epsilon D/2\pi)\,\partial_x H\{E(x)\}=j_x$ of Dmitriev et al., Phys. Rev. Lett. 111, 206801 (2013)), the theory of the transition between the homogeneous state and the electric-field domain state is complete only at the one-dimensional mean-field level. Establish the full theory by determining:

1. Fluctuation effects on the ZRS transition. Going beyond the mean-field Lyapunov-functional treatment, determine how fluctuations (thermal and microwave-induced noise in the charge and current dynamics) modify the phase boundaries in the voltage-size plane, and establish the nature and order of the transition between the homogeneous and domain states — in particular whether the mean-field second-order line for $1<l<\sqrt{2}$, the tricritical point at $l=\sqrt{2}$ and $V=E_cL/4$, and the first-order line with its hysteretic interval for $l>\sqrt{2}$ survive, and whether fluctuations drive the transition first order in some regimes, as they do in the unbiased thermodynamic limit (Alicea et al., 2005). Here $l=L/\pi\lambda$ with $\lambda$ the nonequilibrium screening length.

2. The two-dimensional domain state and the global-stability boundary. For $l>l_{c+}\simeq7$, in the region between the transverse-stability line of the homogeneous state and the first-order line $V_{c1}$, determine the structure of the fully two-dimensional domain state with broken translational invariance along the stripe that globally minimizes the Lyapunov functional (its morphology, symmetry, and characteristic length scales), and locate the boundary $V^*_{c1}$ across which the one-dimensional domain solution loses global stability.

3. Stability at large stripe widths. Establish the stability of the one-dimensional domain solution for $l$ well beyond 40, where the existing numerical linear-stability checks stop, including the role of nonlinearity and of fluctuations in that regime.

These three questions jointly serve one objective: completing the phase diagram of the voltage-biased zero-resistance state — the transition's order and fluctuation corrections, the true (possibly two-dimensional) global-minimum domain state, and the resulting refined stability boundaries — so that the predicted current-voltage characteristics and hysteresis of finite microwave-driven samples rest on the actual stable states rather than on the one-dimensional mean-field ansatz.

References

1. I. A. Dmitriev, M. Khodas, A. D. Mirlin, D. G. Polyakov, Emergence of Domains and Nonlinear Transport in the Zero-Resistance State, Phys. Rev. Lett. 111, 206801 (2013), DOI: 10.1103/PhysRevLett.111.206801, https://arxiv.org/abs/1305.1028
2. A. V. Andreev, I. L. Aleiner, A. J. Millis, Dynamical Symmetry Breaking as the Origin of the Zero-dc-Resistance State in an ac-Driven System, Phys. Rev. Lett. 91, 056803 (2003), DOI: 10.1103/PhysRevLett.91.056803, https://doi.org/10.1103/PhysRevLett.91.056803
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