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arXiv:2108.12367 · Physical review. B./Physical review. B

Melting phase diagram of bubble phases in high Landau levels

K. A. Villegas Rosales, Sankalp Kumar Singh, Hao Deng, Yoon Jang Chung, Loren N. Pfeiffer, K. W. West, K. W. Baldwin, MANSOUR SHAYEGAN

Published Sep 20, 2021• 1 citation
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Abstract

A low-disorder, two-dimensional electron system (2DES) subjected to a large perpendicular magnetic field and cooled to very low temperatures provides a rich platform for studies of many-body quantum phases. The magnetic field quenches the electrons' kinetic energy and quantizes the energy into a set of Landau levels, allowing the Coulomb interaction to dominate. In excited Landau levels, the fine interplay between short- and long-range interactions stabilizes bubble phases, Wigner crystals with more than one electron per unit cell. Here, we present the screening properties of bubble phases, probed via a simple capacitance technique where the 2DES is placed between a top and a bottom gate and the electric field penetrating through the 2DES is measured. The bubbles formed at very low temperatures screen the electric field poorly as they are pinned by the residual disorder potential, allowing a large electric field to reach the top gate. As the temperature is increased, the penetrating electric field decreases and, surprisingly, exhibits a pronounced minimum at a temperature that appears to coincide with the melting temperature of the bubble phase. We deduce a quantitative phase diagram, as a function of Landau level filling factor (\ensuremath{\nu}) and temperature, for the transition from the bubble to liquid phases for $4\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}5$.

DOI
10.1103/physrevb.104.l121110

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A low-disorder, two-dimensional electron system (2DES) subjected to a large perpendicular magnetic field and cooled to very low temperatures provides a rich platform for studies of many-body quantum phases. The magnetic field quenches the electrons' kinetic energy and quantizes the energy into a set of Landau levels, allowing the Coulomb interaction to dominate. In excited Landau levels, the fine interplay between short- and long-range interactions stabilizes bubble phases, Wigner crystals with more than one electron per unit cell. Here, we present the screening properties of bubble phases, probed via a simple capacitance technique where the 2DES is placed between a top and a bottom gate and the electric field penetrating through the 2DES is measured. The bubbles formed at very low temperatures screen the electric field poorly as they are pinned by the residual disorder potential, allowing a large electric field to reach the top gate. As the temperature is increased, the penetrating electric field decreases and, surprisingly, exhibits a pronounced minimum at a temperature that appears to coincide with the melting temperature of the bubble phase. We deduce a quantitative phase diagram, as a function of Landau level filling factor (\ensuremath{\nu}) and temperature, for the transition from the bubble to liquid phases for 4\ensuremath≤\ensuremathν\ensuremath≤54\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}54\ensuremath≤\ensuremathν\ensuremath≤5.
The subject of this dissertation is the experimental discovery and investigation of a new class of collective phases in two-dimensional electron systems. The experiments mainly involve magnetotransport measurements in very high quality GaAs/AlGaAs semiconductor heterostructures, where a large perpendicular magnetic field serves to resolve the electrons? energy spectrum into discrete Landau levels. The most dramatic evidence of a new many-body phase is the huge and unprecedented resistance anisotropy observed only below 150 mK and around the half-filling points of the highly excited Landau levels N > 1. Associated with these anisotropic states are other novel electron phases whose transport signature is a vanishing longitudinal conductivity occurring in the flanks of the same excited Landau levels. Although reminiscent of the well-understood integer quantum Hall states, the insulating phases are exceptional for being driven by electron interactions rather than single-particle localization. A persuasive theoretical picture based on "stripe" and "bubble" charge density wave formation in high Landau levels can account for many of the experimental results. For example, the broken orientational symmetry of the stripe state may underlie the observed transport anisotropy, while disorder-induced pinning of the bubble lattice could give rise to the insulating regions in high Landau levels. Further investigation of the anisotropic transport characteristics has elucidated possible symmetry-breaking mechanisms of the purported stripe phase and has provided evidence that the stripes may be more accurately described as a quantum electronic liquid crystal. In addition, experiments involving the breakdown of the insulating regions at high voltage biases may point to a depinning transition of the bubble phase. These results have spurred intense interest in the field of correlated electron systems in two dimensions and may be an indication of the variety of new phenomena in condensed matter systems still awaiting discovery.
We generalize a path integral Monte Carlo formula to investigate the phase boundary between the fractional quantum Hall liquid and the Wigner crystal for a two-dimensional electron (fermion) system in the presence of strong magnetic fields and at low temperatures. At very low temperature, the state of the system is found to be a solid at filling factors $\ensuremath{\nu}=1∕7$, 1∕91∕91∕9, and 1∕111∕111∕11, while the state remains liquidlike at $\ensuremath{\nu}=1∕3$ and 1∕51∕51∕5. The temperature-density phase diagrams are obtained at low filling factors. At $\ensuremath{\nu}=1∕7$ and a fixed electron density, the calculated melting temperature is $\ensuremath{\sim}220\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$, in good agreement with some experimental results. The nature of the phase transition is suggested to be continuous.
The ground-state phase diagram of 2D electrons in a high Landau level (index N=2N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2N=2) is studied by the density-matrix renormalization group method. Pair correlation functions are systematically calculated for various filling factors from $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/8$ to 1/21/21/2. It is shown that the ground-state phase diagram consists of three different charge density wave states called stripe phase, bubble phase, and Wigner crystal. The boundary between the stripe and the bubble phases is determined to be ${\ensuremath{\nu}}_{c}^{\mathrm{s}\mathrm{\ensuremath{-}}\mathrm{b}}\ensuremath{\sim}0.38$, and that for the bubble phase and Wigner crystal is ${\ensuremath{\nu}}_{c}^{\mathrm{b}\mathrm{\ensuremath{-}}\mathrm{W}}\ensuremath{\sim}0.24$. Each transition is of first order.
The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the non-perturbative fixed phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the ν−κ\nu-\kappaν−κ plane, where ν\nuν is the filling factor and κ\kappaκ is the strength of Landau level mixing. The phase boundary is seen to exhibit a striking ν\nuν dependence, with the states away from the magic filling factors ν=n/(2pn+1)\nu=n/(2pn+1)ν=n/(2pn+1) being much more susceptible to crystallization due to Landau level mixing than those at ν=n/(2pn+1)\nu=n/(2pn+1)ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n-doped and p-doped GaAs quantum wells, and, in particular, the existence of an insulating state for ν<1/3\nu<1/3ν<1/3 and also for 1/3<ν<2/51/3 <\nu< 2/51/3<ν<2/5 in low density p-doped systems. We predict that in the vicinity of ν=1/5\nu=1/5ν=1/5 and ν=2/9\nu=2/9ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal but rather into a strongly correlated crystal of composite fermions carrying two vortices.
The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the nonperturbative fixed-phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the ν-κ plane, where ν is the filling factor and κ is the strength of Landau-level (LL) mixing. The phase boundary is seen to exhibit a striking ν dependence, with the states away from the magic filling factors ν=n/(2pn+1) being much more susceptible to crystallization due to Landau-level mixing than those at ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n- and p-doped gallium arsenide quantum wells and, in particular, the existence of an insulating state for ν<1/3 and also for 1/3<ν<2/5 in low-density p-doped systems. We predict that, in the vicinity of ν=1/5 and ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal, but rather into a strongly correlated crystal of composite fermions carrying two vortices.
We report on results of finite-size numerical studies of partially filled lowest Landau level, at low electron filling factors. We find convincing evidence suggesting that electrons form Wigner crystals at sufficiently low filling factors, and the critical filling factor is ${\ensuremath{\nu}}_{c}\ensuremath{\approx}1/7.$ At $\ensuremath{\nu}=1/7$ we find the system undergoes a phase transition from the Wigner crystal to the incompressible Laughlin state when the short-range part of the Coulomb interaction is modified slightly. This transition is either continuous or very weakly first order.