[Paper Review] Hofstadter states and reentrant charge order in a semiconductor moiré lattice

Carlos R. Kometter, Jiachen Yu+8 more|arXiv (Cornell University)|Dec 9, 2022

2D Materials and Applications10 references4 citations

TL;DR

This study reports the coexistence and competition of Hofstadter states (Chern insulators) and reentrant charge-ordered states in a twisted WSe2/MoSe2 heterobilayer, where flat and dispersive moiré bands are tuned by electron density and magnetic field. Using local electronic compressibility measurements, the authors observe interpenetrating Hofstadter states and generalized Wigner crystals, with reentrant charge order driven by hole crystallization across multiple bands, revealing tunable quantum phases in a semiconductor moiré lattice.

ABSTRACT

The emergence of moiré materials with flat bands provides a platform to systematically investigate and precisely control correlated electronic phases. Here, we report local electronic compressibility measurements of a twisted WSe$_2$/MoSe$_2$ heterobilayer which reveal a rich phase diagram of interpenetrating Hofstadter states and electron solids. We show that this reflects the presence of both flat and dispersive moiré bands whose relative energies, and therefore occupations, are tuned by density and magnetic field. At low densities, competition between moiré bands leads to a transition from commensurate arrangements of singlets at doubly occupied sites to triplet configurations at high fields. Hofstadter states (i.e., Chern insulators) are generally favored at high densities as dispersive bands are populated, but are suppressed by an intervening region of reentrant charge-ordered states in which holes originating from multiple bands cooperatively crystallize. Our results reveal the key microscopic ingredients that favor distinct correlated ground states in semiconductor moiré systems, and they demonstrate an emergent lattice model system in which both interactions and band dispersion can be experimentally controlled.

Motivation & Objective

• To investigate the interplay between flat and dispersive moiré bands in a semiconductor heterobilayer and their role in stabilizing correlated electronic phases.
• To explore the competition between crystalline (charge-ordered) and fluid (Hofstadter) states in a tunable moiré superlattice.
• To determine how magnetic field and carrier density tune the energetic ordering of moiré bands and induce phase transitions.
• To identify the microscopic mechanisms behind reentrant charge order involving holes from multiple bands.
• To establish semiconductor moiré systems as a platform for studying phase transitions between distinct correlated ground states.

Proposed method

• Employed scanning single-electron transistor (SET) measurements to map local electronic compressibility as a function of gate voltage (density) and magnetic field.
• Used a twisted WSe2/MoSe2 heterobilayer with a small lattice mismatch to generate a long-period moiré superlattice with tunable flat and dispersive bands.
• Measured dμ/dn (compressibility) to identify incompressible states indicative of correlated phases, including Hofstadter states and charge-ordered crystals.
• Analyzed the dependence of incompressible states on magnetic flux per moiré unit cell and filling factor ν to identify Chern numbers and subband occupancies.
• Compared experimental observations with single-particle Hofstadter calculations and identified deviations indicating strong electron correlation effects.
• Mapped phase transitions through density and magnetic field, revealing non-monotonic band reordering and interpenetrating quantum phases.

Experimental results

Research questions

• RQ1 How do flat and dispersive moiré bands coexist and compete to stabilize distinct correlated phases in a semiconductor moiré lattice?
• RQ2 What is the role of magnetic field and carrier density in tuning the energetic ordering of moiré subbands and inducing phase transitions?
• RQ3 Can Hofstadter states and charge-ordered states coexist or interpenetrate in a single moiré system, and what are the conditions for such coexistence?
• RQ4 What causes the reentrant charge order observed at moderate magnetic fields, and how do holes from multiple bands contribute to this state?
• RQ5 To what extent do electron correlations modify the single-particle Hofstadter spectrum in this system?

Key findings

• Reentrant charge-ordered states were observed at moderate magnetic fields and intermediate densities, formed by collective crystallization of holes from both flat and dispersive moiré bands.
• Interpenetrating Hofstadter states with Chern numbers t = 1, 2, ..., 6 were identified at high magnetic flux, indicating interaction-driven band reordering.
• Multiple incompressible states with s = -1, -2, -3, -4, -5 were observed at high hole doping (ν < -4), deviating significantly from single-particle predictions.
• The compressibility pattern revealed a cascade of phase transitions, with gaps closing and reopening as a function of magnetic field, especially near ν = -3, -3/2, and -4.
• Non-monotonic band filling and subband reordering were observed, indicating strong correlation effects that reshape the Hofstadter spectrum beyond single-particle expectations.
• The system exhibits a rich phase diagram with competing phases, including generalized Wigner crystals of doublons and Chern insulators, tunable via density and magnetic field.