Interplay of Quasi-Quantum Hall Effect and Coulomb Disorder in Semimetals

TL;DR

This work investigates the 3D quasi-quantum Hall effect (QQHE) in low-density, bulk-like Cd$_3$As$_2$ thin films, demonstrating that defect-tuned carrier densities around $n\approx 2.9\times 10^{16}$ cm$^{-3}$ enable QQHE signatures at modest fields near $10$ T, but that Coulomb disorder from charged defects broadens Landau bands and induces a field-dependent background that can obscure QQHE. The authors formulate an empirical magnetotransport model that combines a QQHE scaling term with a strong-Coulomb-disorder contribution, and they fit this model across samples with varying densities to quantify the balance between QQHE and disorder. The results show a clear density-dependent transition from weak to strong Coulomb disorder, highlighting that while low density is favorable for QQHE access, increased disorder must be simultaneously controlled to realize clean QQHE features. The study provides actionable guidance for enhancing 3D QQHE in topological semimetals, such as passivation and gating strategies to suppress charged disorder while preserving low carrier densities, thereby advancing the integration of QQHE physics into practical platforms.

Abstract

Low carrier densities in topological semimetals (TSMs) enable the exploration of novel magnetotransport in the quantum limit (QL). Recent findings consistent with 3D quasi-quantum Hall effect (QQHE) have positioned TSMs as promising platforms for exploring 3D quantum Hall transport, but the lack of tunability in the Fermi level has thus far limited the ability to observe a QQHE signal. Here, we tune the defect concentrations in the Dirac semimetal Cd${}_3$As${}_2$ to achieve ultra-low carrier concentrations at 2 K around $2.9\times10^{16}$cm${}^{-3}$, giving way to QQHE signal at modest fields near 10 T. At low carrier densities, where QQHE is most accessible, we find that clear QQHE is obscured by a carrier density dependent background originating from Coulomb disorder from charged point defects and Landau level broadening. Our results highlight the interplay between QQHE and Coulomb disorder, demonstrating that clear observation of QQHE in TSMs intricately depends on Fermi level and disorder magnitudes. We find that Coulomb disorder, as theoretically predicted, is an essential ingredient for understanding the magnetoresistivity for a spectrum of Fermi levels in Cd${}_3$As${}_2$, anchoring the role of defects and charged disorder in TSM applications. We discuss future constraints and opportunities in exploring 3D QQHE and quantum Hall effects in TSMs.

Figures (4)

• Figure 1: Comparison of 2D QHE and 3D QQHE. Schematic band structures and density of states (DOS) for a field applied along the z-direction for a,b) a 2D QHE system and c,d) a 3D QQHE system with Dirac dispersion. In the 2D QHE, LLs resulting from the applied field give a delta-function DOS which broadens with disorder. Changing the applied field moves LLs through the $E_F$, resulting in Hall plateaus. c) Landau bands for a 3D DSM, showing the remaining dispersion of the bands along the applied field direction. The DOS does not drop to zero between Landau bands because of this dispersion. As in 2D, disorder broadens the 3D DSM DOS. e) Hall and f) longitudinal resistivity calculated from a $k \cdot p$ model for 3D DSM Cd${}_3$As${}_2$ in the case of weak (black traces) and moderate (blue traces, offset) disorder for field along the c-axis. The chemical potential is fixed at $20$ meV. $\rho_{xx}(H)$ exhibits peak-like features, similar to the 2D QHE ($\rho_{xx}$ traces scaled for clarity). When only the lowest LB is occupied, $\rho_{xy}$ is quasi-quantized at a value contingent on the Fermi wave vector along the applied field direction, $k_{F,c}$. Moderate disorder introduces mixing of the longitudinal and Hall conductivities which affects the quasi-quantization of $\rho_{xy}$. These calculations do not account for the effects of LB broadening on the DOS. For a full discussion of theoretical calculations, disorder treatment, and comparison to the featureless case of fixed carrier density, see Methods and Supplementary Note 1.
• Figure 2: Coulomb disorder in Cd${}_3$As${}_2$. a) Charged Cd interstitials and vacancies have concentrations of $\sim10^{18}$cm${}^{-3}$ in Cd${}_3$As${}_2$ films. Paired with Fermi-level dependent screening, they generate a disorder potential. The surface plot shows a slice of the disorder potential, $V(x,y)$, at the plane defined by the dashed line. b) Ratio of the average disorder potential strength to the Fermi level as a function of $n$, calculated using Ref. Skinner2014 and relevant values for Cd${}_3$As${}_2$ (see Supplementary Note 5). Colored circles represent estimates for Samples A-E presented here. As the carrier density decreases, Cd${}_3$As${}_2$ moves into the strong Coulomb disorder limit. c) Summary of carrier densities, Fermi levels, and mobilities for Samples A-E. Carrier densities extracted from the low field slope of $\rho_{xy}(H)$ are used in combination with the DOS to estimate $E_F$ (See Supplementary Note 1).
• Figure 3: Carrier density tunes magnetotransport in Cd${}_3$As${}_2$. a,d) Fractional magnetoresistance, b,e) Hall resistivity, and c,f) tangent of the Hall angle $\tan\theta_H\equiv\rho_{xy}/\rho_{xx}$ at several temperatures for a Cd${}_3$As${}_2$ films with low temperature electron carrier densities of (a-c) Sample E: $2.3\times10^{17}$ cm${}^{-3}$ and (d-f) Sample C: $5.8\times10^{16}$ cm${}^{-3}$. Sample E shows linear MR and Hall effect with superimposed quantum oscillations. Meanwhile, $\tan\theta_H$ saturates to a constant value. On the other hand, Sample C shows a low field quadratic and high field super-quadratic MR and a clear kink in $\rho_{xy}$ (coincident features highlighted with arrows).
• Figure 4: QQHE and Coulomb scattering in Cd${}_3$As${}_2$. a-d) Longitudinal and e-h) Hall resistivity for samples of increasing carrier density at 5 K. In panels a-d, the left half of the axes correspond to raw data while the right half show fits of the data to only the QQHE (light color, solid) and to the full Eq. 1 (gray, dotted). Eq. 1 was not fit to Sample D as it lies in the weak Coulomb disorder limit. In panel e, the quasi-quantized value of the Hall resistivity in the quantum limit is displayed, showing good agreement with experiment (calculation in Supplementary Note 7). i) Carrier density dependence of $\beta$ and $\gamma$ fit parameters from Eq. 1. j) A plot of the disorder potential strength calculation vs. the Fermi level for Cd${}_3$As${}_2$. Significant Coulomb disorder in Cd${}_3$As${}_2$ combats a clean observation of the QQHE. Meanwhile, for ZrTe${}_5$ and HfTe${}_5$, stronger screening of disorder gives weaker contributions from Coulomb disorder scattering. Estimates for the pentatelluride systems from Tang2019Galeski2021WangEvidence2018GourgoutFO2022Shahi2018Galeski2020Piva2024. Dotted lines represent calculated contours of how $eV_0$ changes with Fermi level from Ref. Skinner2014 (Supplementary Note 5).