Three-dimensional quasiquantized Hall insulator phase in $\mathrm{Sr}{\mathrm{Si}}_2$

In insulators, the longitudinal resistivity becomes infinitely large at zero temperature. For classical insulators, the Hall conductivity becomes zero at the same time. However, there are special systems, such as two-dimensional quantum Hall insulators, in which a more complex scenario is observed at high magnetic fields. Here, we report experimental evidence for a quasiquantized Hall insulator in the quantum limit of the three-dimensional compound $\mathrm{Sr}{\mathrm{Si}}_2$. Our measurements reveal a magnetic-field range, in which the longitudinal resistivity diverges with decreasing temperature, while the Hall conductivity approaches a quasiquantized value that is given only by the conductance quantum and the Fermi wave vector in the field direction. The quasiquantized Hall insulator appears in a magnetic field induced insulating ground state of three-dimensional materials and is deeply rooted in quantum Hall physics.

Figure 1

(a) Crystal structure of $\mathrm{Sr}{\mathrm{Si}}_2$ with chiral space group $P{4}_{1}32$ (No. 213). (b) Trivial band structure of $\mathrm{Sr}{\mathrm{Si}}_2$ in agreement with a recent ARPES study [26]. (c) Six pairs of spin-split hole Fermi-surface pockets in the BZ along $\pm k_x$, $\pm k_y$, and $\pm k_z$. (d) Constant energy surface at $E_{\mathrm{F}}$ and $k_z=0$ measured by ARPES acquired with a photon energy $h\nu =342\mathrm{eV}$. Cross sections of a set of four near-spherical Fermi-surface pockets lying in the $k_x\text{−}{k}_y$ plane are clearly visible. The corresponding plane is highlighted in green in (c). (e) Temperature-dependent resistivity of $\mathrm{Sr}{\mathrm{Si}}_2$ with current passed along [100] in the absence of a magnetic field. (f) Temperature-dependent charge-carrier concentration $\left(n\right)$ and mobility $\left(\mu \right)$ of $\mathrm{Sr}{\mathrm{Si}}_2$ estimated using the low magnetic field slope of ${\rho }_{yx}\left(B\right)$.

Figure 2

(a) Shubnikov–de Haas (SdH) oscillations at 2 K with a magnetic field applied along important cubic crystallographic directions: [100], [110], and [111], which exhibit nearly identical quantum oscillations. (b) Fast Fourier transform (FFT) amplitudes for $\mathrm{Sr}{\mathrm{Si}}_2$ determined from SdH oscillations for $B$ along [100]. The inset shows the near-spherical Fermi-surface pockets predicted by the calculations. (c,d) Angle-dependent SdH oscillations with nearly identical oscillation frequencies that are in agreement with the nearly spherical Fermi-surface pockets in $\mathrm{Sr}{\mathrm{Si}}_2$ are revealed.

Figure 3

(a) Longitudinal resistivity ${\rho }_{xx}$ as a function of magnetic field $B$ for various temperatures $T$. The field at which the $\beta$ pocket enters the quantum limit (the lowest Landau band) is denoted by $B_{\mathrm{QL},\beta .}$ $B_{\mathrm{c}}$ marks the transition point from a metallic to an insulating state. (b) Hall conductivity ${\sigma }_{xy}$ as a function of $B$ at $T=0.6\mathrm{K}$. In the insulating phase, between $B_{\mathrm{c}}$ and $B_{\mathrm{T}}$, the experimentally determined ${\sigma }_{xy}$ (black curve) scales with the calculated 3D quasiquantized Hall conductivity value (light green curve) estimated using the experimentally extracted Fermi wave vector $k_{\mathrm{F}}$ along $B$, the electron charge $e$, and the Planck constant $h$. The thickness of the light green line represents the error of the model, originating from a statistical error of 1% in $k_{\mathrm{F}}$. $B_{\mathrm{G},\beta }$ marks the theoretical field at which $\mathrm{Sr}{\mathrm{Si}}_2$ hypothetically enters the band gap. Further detailed descriptions of the highlighted magnetic-field values can be found in the Supplemental Material [30]. (c) ${\rho }_{xx}$ as a function of $T$ for various $B$ around $B_{\mathrm{c}}$. (d) ${\sigma }_{xy}$ as a function of $T$ for various $B$ around $B_{\mathrm{c}}$. (e) Normalized resistivity, ${\rho }_{xx}\left(B\right)/{\rho }_{xx}\left(B_{\mathrm{c}}\right)$, as a function of the scaling parameter $|B\text{–}{B}_c|T^{-1/\zeta }$, with the critical exponent $\zeta =9.9$. (f) Zoomed-in view of the band structure of $\mathrm{Sr}{\mathrm{Si}}_2$, revealing two hole pockets.