Disorder-induced strong-field strong-localization in 2D systems

Technical Summary: Disorder-Induced Strong Field-Strong Localization in 2D Systems

Problem Statement
Recent scanning tunneling microscopy (STM) experiments on bilayer graphene (BLG) under strong perpendicular magnetic fields have revealed three distinct quantum phases in the lowest Landau level (LLL): the incompressible fractional quantum Hall liquid (FQHL), a crystalline compressible hexagonal Wigner crystal (WC), and a spatially random amorphous solid phase (AS) at low filling factors ($\nu \approx 0.093$). While the existence of FQHL and WC is well understood through interaction-driven energetics, the nature of the AS phase at low filling factors remains controversial. Conventional transport studies often interpret insulating states at low filling factors as pinned Wigner crystals. However, STM data suggest a structural transition from a hexagonal WC to a disordered, random solid as $\nu$ decreases, challenging the interpretation as a "pinned WC." The central problem is to determine the physical mechanism driving the transition from a crystalline (or liquid) phase to this strongly localized amorphous phase, particularly examining the role of disorder compared to electron-electron interactions.

Methodology
The authors pursue a radically alternative approach by focusing exclusively on disorder-induced localization and neglecting electron-electron interactions. This contrasts with standard theoretical treatments that prioritize interaction-driven energetics (comparing ground states of WC and FQHL) and frequently ignore disorder.

The methodology includes:

1. Heuristic Scaling Arguments: Based on prior work on the integer quantum Hall effect (IQHE), the authors propose that the transition to a strongly localized phase occurs when the disorder-induced broadening of the Landau level ($\Gamma$) exceeds the effective chemical potential (or Fermi energy) of the electrons in that level.
2. Self-Consistent Born Approximation (SCBA): To quantify the disorder-induced broadening $\Gamma$, the authors employ SCBA for short-range impurity scattering. They derive analytical expressions for the density of states (DOS) of the Landau level and the broadening width $\Gamma$ as a function of the zero-field scattering time $\tau$ and the cyclotron energy $\hbar\omega_c$.
3. Derivation of the Critical Filling Factor: By equating the disorder-induced broadening with the effective chemical potential ($\Gamma_{LLL} = \nu_c \hbar\omega_c$), the authors derive a semi-quantitative formula for the critical filling factor $\nu_c$ at which the system transitions into a strongly localized Anderson solid.
4. Comparative Analysis: The derived scaling laws are tested against experimental data from various 2D systems with different mobilities and disorder levels, including Si-MOSFETs, GaAs heterostructures, and the BLG samples from the motivating STM study.

Main Contributions and Results

• Identification of the AS Phase: The work argues that the amorphous solid phase observed at low $\nu$ in Ref. [1] is a disorder-dominated "Anderson solid" rather than a pinned Wigner crystal. In this phase, disorder drives electrons into a random, frozen pattern lacking long-range translational or rotational order, characterized by a hybrid structure factor (a diffuse central spot surrounded by a ring).
• Derivation of $\nu_c$: The authors derive a critical filling factor scaling as $\nu_c \sim (\Gamma_0 / \hbar\omega_c)^{1/2}$, where $\Gamma_0$ represents the disorder-induced broadening at zero field. This implies that the critical filling factor for the onset of strong localization is sample-dependent and scales with the square root of the disorder strength.
• Disorder as the Driving Mechanism: The results suggest that the transition to the insulating state is not an intrinsic end to the competition between FQHL and WC at a universal filling factor, but rather a disorder-induced transition. In highly disordered samples, the critical filling factor $\nu_c$ shifts to higher values and can potentially suppress all FQHL and WC phases within the LLL.
• Quantitative Estimates:
  • For dirty Si-MOS systems ($\mu \sim 10^2$ cm$^2$/Vs), the theory predicts $\nu_c > 1$, consistent with the historical absence of LLL quantum Hall effects in these samples.
  • For clean BLG ($\mu \sim 10^6$ cm$^2$/Vs), the theory estimates $\nu_c \sim 0.023$. Although this is lower than the experimentally observed $\nu_c \approx 0.093$, the authors note that their approximations (short-range disorder, neglect of LL mixing) likely provide a lower bound. They argue that the discrepancy is acceptable given the model's simplicity and the fact that the theory correctly predicts the trend that $\nu_c$ decreases with increasing sample quality.
• Proposal of a Phase Diagram: The authors propose a schematic phase diagram where the "disorder line" ($\Gamma = E_F$) cuts off the existence of both WC and FQHL phases. As disorder increases or the filling factor decreases, the system crosses this line and enters an Anderson insulator phase (AS).

Significance and Claims
The work claims that the uncritical association of insulating states at low filling factors with pinned Wigner crystals must be re-evaluated in light of new STM data. The authors argue that:

1. Disorder is Crucial: The existence of a critical filling factor $\nu_c$ for the insulating transition is fundamentally a disorder-dependent phenomenon and is not determined solely by the energetic competition between FQHL and WC.
2. Limits of Pure Energetics: Theoretical models relying exclusively on interaction-driven energetics in clean systems fail to explain the sample-dependent nature of the insulating transition observed in experiments. In a pristine system, the FQHL might remain the ground state down to very low filling factors; the insulating behavior is an artifact of disorder.
3. Nature of the AS Phase: The amorphous solid differs from a pinned crystal. While a pinned WC retains its crystalline structure (albeit distorted), the AS phase represents a complete loss of spatial order due to strong disorder and effectively acts as a highly fragmented, impurity-pinned state structurally equivalent to an amorphous solid.
4. Predictive Power: The theory predicts that a systematic increase in sample disorder should shift the critical filling factor $\nu_c$ to higher values, eventually suppressing all FQHL and WC phases within the LLL. This provides a testable prediction for future experiments.

The authors emphasize that their theory is qualitative and semi-quantitative, serving as motivation to understand the role of disorder in the LLL. They acknowledge that a complete microscopic theory incorporating both strong disorder and interactions is currently unsolvable, but their pure disorder approach successfully captures the experimental trend of $\nu_c$ varying with sample quality.