Reentrant Landau Levels in a Dirac topological insulator

TL;DR

This work demonstrates non-$1/B$ magnetoresistance oscillations in ZrTe$_5$ that persist beyond the quantum limit, explained by reentrant Landau-level crossings arising from the competition between Zeeman and cyclotron energies in a 3D Dirac system. A minimal Dirac Hamiltonian with strong spin-orbit coupling captures the LL back-bending, field-dependent frequency $F(B)$, and LL interference that modulates the temperature dependence of the oscillations. The results unify conventional and anomalous quantum oscillations within a single framework, extract a low carrier density and a small Dirac mass gap, and reveal a 3D ellipsoidal Fermi surface with field-induced mass enhancement, providing a robust platform for exploring Dirac-electron instabilities beyond the quantum limit.

Abstract

The quantum limit, where magnetic fields confine carriers to the lowest Landau level, is predicted to host exotic quantum phases arising from strengthened electronic correlations, reduced dimensionality, and increased degeneracy. We report a novel quantization regime realized in the ultra-quantum limit of the narrow-gap Dirac insulator ZrTe5, marked by anomalous magnetoresistance oscillations. These oscillations, measured in ZrTe5 single crystals down to 700 mK and up to 60 T, are distinctly non-1/B periodic and persist for magnetic fields well beyond the quantum limit. In this regime, the competition between Zeeman and cyclotron energies drives a nonlinear evolution and back-bending of Landau levels, causing low-index levels to re-cross the Fermi energy at high fields. This mechanism departs from the standard Lifshitz-Kosevich description and provides a framework to describe how the electronic structure in topological Dirac insulators evolves beyond the quantum limit.

Figures (5)

• Figure 1: (a) Schematic of the crystal structure of ZrTe$_5$, showing chains of ZrTe$_3$ prisms aligned along the $a$ axis. These chains are interconnected along the $c$ axis via Te atoms, forming 2D layers that are stacked along the $b$ axis and held together by vdW interactions. (b) Electrical resistance of flux grown ZrTe$_5$. Resistance was measured along the $a$ axis, normalized by the room temperature resistance ($R_{300\,\text{K}}$). The sharp resistance increase at low temperature is consistent with semiconducting behavior. (c) Schematic representation of Dirac bands in ZrTe$_5$ with a gap ($\Delta$) of 10 meV, illustrating the electronic structure near the Dirac points. (d) $\log$-linear plot of $R/R_{300\text{ K}}$ vs $1/T$ of the resistance measurement shown in (b) (black line). Overlaid is an Arrhenius fit in the temperature range of 20 K-150 K (red dashed line) with fitted activation energy of 5 meV. Details of the fitting procedure are discussed in the main text.
• Figure 2: Magnetoresistance of ZrTe$_5$ with the current applied along the $a$ axis and the magnetic field applied along the $b$ axis at 4 K. (a) Magnetoresistance data up to 60 T, where the black curve represents the experimental data and the red curve indicates the background fit. The inset shows a schematic representation of the experimental configuration. Zoom in of the low field (b) and high field (c) regions of the magnetoresistance. (d) $\Delta R/R_{background}$ as a function of magnetic field to show the evolution of the magnetoresistance oscillations in field. (e) Same data shown in (b) plotted as a function of inverse magnetic field, highlighting the deviation from $1/B$ periodicity.
• Figure 3: (a) Magnetic-field positions of oscillation peaks (blue) and valleys (red), fitted using Eq. \ref{['r-fitting']} (black line). With increasing field, the oscillation index $r$ increases. To demonstrate the re-entrant LL crossings we have arbitrarily picked $r_0=-3$ to assign a nominal LL index (right axis) to each point. The gray region indicates the validity of the model within 2$\sigma$ .The inset reveals an apparent log($B$)-periodicity resulting from the restricted field range. A fit of the oscillation index to $r(B) \approx A\log B + C$ gives $A = -0.82(2)$ and $C = 4.32(4)$. (b and c) Calculated Landau level spectrum using the parameters and model discussed in the main text, shown over two magnetic-field windows for clarity. Solid lines represent spin-up states, while dashed lines denote spin-down states. Black curves correspond to conduction band LLs and red curves to valence band LLs. The Fermi energy is indicated by the blue line.
• Figure 4: (a) Magnetoresistance of ZrTe$_5$ for various temperatures between 0.7 K-57 K. Measurements were done with same configuration illustrated in the inset of Figure \ref{['Pulsed_MR']}a. The red arrows indicate the magnetic field where the temperature dependence of the $\Delta R_{\mathrm{peak/valley}} / R_{\mathrm{background}}$ was extracted. (b) Oscillations as a function of magnetic field and temperature, the vertical black lines indicate the magnetic fields where the temperature dependence of the $\Delta R_{\mathrm{peak/valley}} / R_{\mathrm{background}}$ were analyzed. (c) $\Delta R_{\mathrm{peak/valley}} / R_{\mathrm{background}}$ at the magnetic field regimes indicated in (a) and (b). The dots represent experimentally obtained $\Delta R_{\mathrm{peak/valley}} / R_{\mathrm{background}}$, while the continuous line depicts the extracted fitting using the electronic interference model described by Eq. \ref{['Interference_model']}. (d) Extracted electronic masses $m_1$ and $m_2$ related to spin-up and spin-down electrons as a function of magnetic field. (e): The phase ($\phi$) between spin-up and spin-down crossings. The dashed lines indicates the average value of low ($\mu_0 H<$ 3T ) and high ($\mu_0 H>$20T ) field.
• Figure 5: (a) MR oscillations measured at 4 K as a function of magnetic field up to 60 T for various angles $\alpha$, defined in the $bc$-plane. (I) MR after background subtraction. Curves are vertically offset for clarity. (II) Magnetic field positions of MR peaks and valleys, fit using Eq. \ref{['r-fitting']}. (b) Same as (a), but with magnetic field rotated in the $ab$-plane by angle $\beta$. (c) (I) Extracted low field oscillation frequency $F_0$ as a function of rotation angle across both planes. The error bars represent the fit uncertainties corresponding to a confidence interval of $1\sigma$. The data follow a characteristic angular dependence of an ellipsoidal Fermi surface (solid line). The inset shows the schematic of the field and current directions relative to the crystallographic axes. (II) Schematic representation of an anisotropic Fermi surface with principal axes $k_a$, $k_b$, and $k_c$ used to model the observed oscillation behavior.