Chemical potential asymmetry and quantum oscillations in insulators

We present a theory of quantum oscillations in insulators that are particle-hole symmetric and nontopological but with arbitrary band dispersion, at both zero and nonzero temperature. At temperatures $T$ less than or comparable to the gap, the dependence of oscillations on $T$ is markedly different from that in metals and depends crucially on the position of the chemical potential $\mu$ in the gap. If $\mu$ is in the middle of the gap, oscillations do not change with $T$; however, if $\mu$ is asymmetrically positioned in the gap, surprisingly, oscillations go to zero at a critical value of the inverse field determined by $T$ and $\mu$ and then change their phase by $\pi$ and grow again. Additionally, the temperature dependence is different for quantities derived from the grand canonical potential, such as magnetization and susceptibility, and those derived from the density of states, such as resistivity. However, the nontrivial features arising from asymmetric $\mu$ are present in both.

Figure 1

Alternative ways of viewing overlapping bands: in terms of (a) electron (blue, solid) and hole (red, dashed) levels and (b) occupied (red, dashed) and unoccupied (blue, solid) levels. (c) The latter picture easily extends to the case when a gap opens (gap exaggerated for clarity): $\pm$ are band indices and $e/h$ refer to electron/holelike parts. In the presence of a magnetic field, LLs are formed. The arrows show the direction in which they move as field is increased.

Figure 2

(a) ${\mathrm{\Omega }}_{\mathrm{osc}}^{-,e}$ for $\zeta /{\omega }_c=0.1$ (blue, solid) and $\zeta /{\omega }_c=1.2$ (blue, dashed) compared with the case of $\zeta =0$ (black, dotted). (b) ${\mathrm{\Omega }}_{\mathrm{osc}}^{-,e}$ (blue, dashed), ${\mathrm{\Omega }}_{\mathrm{osc}}^{-,h}$ (red, dot-dashed), and $({\mathrm{\Omega }}_{\mathrm{osc}}^{-,e}+{\mathrm{\Omega }}_{\mathrm{osc}}^{-,h})/2$ (purple, solid) for $\zeta /{\omega }_c=0.1$. Here, $\delta$ is a quantity that measures how far the last LL $N_v$ is from the lower edge of the gap in $-,e$ band (for an exact definition refer to the text); thus, $\delta =0$ and $\delta =1$ signify the crossing of two consecutive LLs across the gap edge, and corresponds to one period of oscillation. The pattern is repeated giving rise to quantum oscillations.

Figure 3

Numerical calculations on a lattice model mimicking (1) with two square lattices hybridized to open a gap at quarter filling (see Ref. [23] for details). Here, $\zeta =0.1t$, where $t$ is the hopping parameter. (a) Oscillations at $T=0$: frequency of oscillations does not change with the introduction of the gap; (b) Dependence of amplitude $A$ on $T$ for $\mu =0$: a plateau appears at $T<\zeta$ in the gapped case (both curves are normalized with their respective $T=0$ values); (c) Oscillations at two different temperatures for $\mu =-0.09t$ (i.e., $\left|\mu \right|/\zeta =0.9$). The beatlike pattern with change of phase by $\pi$ accompanied by an increase in amplitude emerges when temperature is increased. (d) Dependence of oscillations on $T$ at two different field values marked $P$ and $Q$ in (c), normalized with their respective $T=0$ values.

Figure 4

Amplitude of oscillations $A$ (arbitrary units) vs $T$ for the density of states in the gapped case obtained from numerical calculations on a lattice (same as in Fig. 3) with $\zeta =0.1t$ for (a) $\mu =0$ and (b) $\mu =-0.09t$ [$P$ and $Q$ correspond to two field values given in Fig. 3]. The figures should be compared to Figs. 3 and 3, respectively. The behavior for $T\ll \zeta$ is clearly different. Note, however, the extra nontrivial features in the case $\mu \ne 0$ found in Fig. 3 persist in Fig. 4 as well.

Figure 5

The two shaded regions differ by a phase $\pi$ in oscillations. The separatrix is a schematic variation of the critical field with temperature. (Inset) The separatrix moves to the left as $\left|\mu \right|$ increases. Numerical calculations on a lattice confirm this behavior—see Ref. [23].