Quantum Oscillation in Narrow-Gap Topological Insulators

The canonical understanding of quantum oscillation in metals is challenged by the observation of the de Haas–van Alphen effect in an insulator, ${\mathrm{SmB}}_6$ [Tan et al, Science 349, 287 (2015)]. Based on a two-band model with inverted band structure, we show that the periodically narrowing hybridization gap in magnetic fields can induce the oscillation of low-energy density of states in the bulk, which is observable provided that the activation energy is small and comparable to the Landau level spacing. Its temperature dependence strongly deviates from the Lifshitz-Kosevich theory. The nontrivial band topology manifests itself as a nonzero Berry phase in the oscillation pattern, which crosses over to a trivial Berry phase by increasing the temperature or the magnetic field. Further predictions to experiments are also proposed.

Figure 1

Illustration of LLs and possible low-energy DOS oscillation in (a) metals, (b) parabolic band insulators, and (c) insulators with an inverted band structure. The planes and the thin circles denote the chemical potential and LL orbits, respectively. The blue arrows indicate the flow of LLs with increasing magnetic fields. The red dashed circles denote the Fermi surface in (a) and the band edges in (c).

Figure 2

(a) Band structure, (b) LL spectrum, and (c) LEDOS oscillation of the semimetal in two dimensions. The dashed gray lines in (b) and (c) indicate whenever Eq. (1) is satisfied at the chemical potential with $\gamma =1/2$. (d) The temperature dependence of the oscillation amplitude (black dots) and the fitting by the two-component LK formula (dashed blue curve). Band structure parameters are specified in the main text with $t$ set to be unity. The abscissas in (b) and (c) are labeled with the inverse number of flux quanta per unit cell.

Figure 3

Illustration of the LL hybridization in two dimensions. In each sector, each $d$ LL hybridizes with an $f$ LL drawn in the same dashing style. LLs in red are pushed upward while the blue downward. The long gray bars denote the chemical potential. All LLs shift away from it except that in the $\{N,N+1\}$ LL pairs (solid bars), one LL out of each pair shifts toward it and may pass each other.

Figure 4

(a) LL spectrum and (b) LEDOS oscillation of the 2D model with a hybridization gap. The thick vertical lines indicate $B_c$, the critical field of gap closing. The inset of (b) highlights the phase jump at finite temperature. The LEDOS are plotted at temperatures indicated by the vertical lines in (c). (c) Temperature dependence of the oscillation amplitudes extracted in $B>B_c$ (open circles) and $B<B_c$ (closed circles, multiplied by 10 for clarity). (d) Intensity plot of LEDOS, highlighting the $\pi$ phase jump around $B\simeq B_c$ and $T\simeq \mathrm{\Delta }$, which are indicated by the dashed curves.

Figure 5

(a) Single-particle DOS of the 3D model and (b) LEDOS oscillation. The thick vertical line indicates $B_c$. The LEDOS are plotted at temperatures indicated by the vertical lines in (c). (c) Temperature dependence of the oscillation amplitudes for $B>B_c$ (open circles) and $B<B_c$ (closed circles, multiplied by 20 for clarity). (d) Schematic illustration of Berry phases in different regimes.