Quantum transport properties of $\beta \text{-B}{\mathrm{i}}_4{\mathrm{I}}_4$ near and well beyond the extreme quantum limit

We have investigated the magnetotransport properties of $\beta \text{-B}{\mathrm{i}}_4{\mathrm{I}}_4$ bulk crystal, which was recently theoretically proposed and experimentally demonstrated to be a topological insulator. At low temperature $T$ and magnetic field $B$ , a series of Shubnikov–De Haas (SdH) oscillations is observed on the magnetoresistivity (MR). Detailed analysis reveals a light cyclotron mass of $0.1m_e$, and the field angle dependence of MR reveals that the SdH oscillations originate from a convex Fermi surface. In the extreme quantum limit (EQL) region, there is a metal-insulator transition occurring soon after the EQL. We perform scaling analysis, and all the isotherms fall onto a universal scaling with a fitted critical exponent $\zeta \approx 6.5$. The enormous value of critical exponent $\zeta$ implies this insulating quantum phase originated from strong electron-electron interactions in high fields. However, in the far end of EQL, both the longitudinal and Hall resistivity increase exponentially with $B$ , and the temperature dependence of the MR reveals an energy gap induced by the high magnetic field, signifying a magnetic freeze-out effect. Our findings indicate that bulk $\beta \text{-B}{\mathrm{i}}_4{\mathrm{I}}_4$ is an excellent candidate for a three-dimensional topological system for exploring EQL physics and relevant exotic quantum phases.

Figure 1

(a) The $\beta \text{-B}{\mathrm{i}}_4{\mathrm{I}}_4$ crystal in space group $C12/m1$ (No.12), with lattice parameters $a=14.353\AA , b=4.423\AA , c=10.476\AA$, and $\beta =107.8^{\circ }$ between the $a$ and $c$ axes. Pink and gray spheres represent the bismuth and iodine ions, respectively, and the gray dashed box shows the unit cell. (b) The XRD pattern of $\beta \text{-B}{\mathrm{i}}_4{\mathrm{I}}_4$ single crystals with ($00l$) reflections. Inset: A scanning electron microscopy (SEM) image of a ribbon-shaped sample usually has quite irregular facets. (c) Temperature dependence of ${\rho }_{xx}$ curves normalized to $\rho$ (250 K) for sample 1 with different exposure times in the air. The fresh sample refers to the sample with the least exposure time, which was measured immediately after the contacts were made. Upper inset: Room temperature curves of sample 2 show the insulating behavior of the $\alpha$ phase. A hysteresis profile can be seen at 300 K. (d) ${\rho }_{xx}$ and ${\rho }_{xy}$ of sample 1 taken at 1.5 K; the magnetic field is parallel to the $z$ direction.

Figure 2

(a) The angle dependence of the SdH oscillations from $\theta =0^{\circ }$ to $\theta ={90}^{\circ }$; the blue and red triangles mark the $N=1$ and $N=2$ Landau level peaks, respectively. (b) The obtained oscillation frequency as a function of $\theta$ from 0° to ${180}^{\circ }$. The red line draws the angle dependence of the frequency for a 2D Fermi surface; $F_0$ corresponds to the minimal cross-section area at $\theta =67.5^{\circ }$. The inset shows the interception extracted from the Landau level (LL) fan diagram. (c) $-d^2{\rho }_{xx}/d{\mathit{B}}^2$ as a function of 1/$B$, at different temperatures ranging between 1.5 and 10 K. The inset shows the FFT transform of the data in the main panel. (d) The normalized oscillation amplitudes as a function of temperature, taken in different fields. The red line shows the fitting of the effective cyclotron mass following Lifshitz-Kosevich theory.

Figure 3

(a) The field dependence of ${\rho }_{xx}$ up to 43 T under different temperatures. The red dashed line marks the ending of the quantum oscillation, above which the system enters the EQL. The inset shows the metal to insulator transition happens at a critical field of 15.9 T. (b) Plot of the ${\rho }_{xx}$ as a function of temperature at fixed magnetic field $B$. (c) The field dependence of ${\rho }_{xy}$ up to 43 T. (d) Scaling plot of the normalized ${\rho }_{xx}$; all the isotherms fall onto a nearly single curve as a function of $|\mathit{B}-{\mathit{B}}_c|T^{-1/\zeta }$ with $\zeta \approx 6.5$; the inset shows the linear fitting of the $log\left(\mathrm{d}{\rho }_{xx}/d\mathit{B}\right)$ at $B_c$ vs log(1/$T$), which yields $1/\zeta$.

Figure 4

(a) $ln\left({\rho }_{xx}\right)$ as a function of field $B$ up to 43 T; the curves have been shifted for clarity. The linear behavior above 20 T demonstrates the exponential increase of ${\rho }_{xx}$. (b) The Arrhenius plot of the temperature dependence of ${\rho }_{xx}$ under fixed magnetic field ranging from 15.9 to 43 T. (c) The energy gap is relative to a function of $B$, and it increases almost linearly with the field. (d) The Hall resistivity up to 52 T at 4.2 K, where the field is parallel to the $z$ direction. Under high field, ${\rho }_{xx}$ also increase exponentially with $B$; the red line draws a simple $e^{\mathit{B}}$ fitting.