Quantum oscillations and Berry's phase in topological insulator surface states with broken particle-hole symmetry

Quantum oscillations can be used to determine properties of the Fermi surface of metals by varying the magnitude and orientation of an external magnetic field. Topological insulator surface states are an unusual mix of normal and Dirac fermions. Unlike in graphene and simple metals, Berry's geometric phase in topological insulator surface states is not necessarily quantized. We show that reliably extracting this geometric phase from the phase offset associated with the quantum oscillations is subtle. This is especially so in the presence of a Dirac gap such as that associated with the Zeeman splitting or interlayer tunneling. We develop a semiclassical theory for general mixed normal-Dirac systems in the presence of a gap, and in doing so clarify the role of topology and broken particle-hole symmetry. We propose a systematic procedure of fitting Landau-level index plots at large filling factors to reliably extract the phase offset associated with Berry's phase.

Figure 1

Model band structure with particle-hole asymmetry, highlighting the effect of the $p^2/2m$ term relative to the chemical potential. In a particle-hole symmetric system, ${\epsilon }_{+}\left(k\right)=-{\epsilon }_{-}\left(k\right)$. At finite $m$, away from $k=0$, the chemical potential is shifted up for either band with respect to the symmetric case, by an amount $\delta \mu /\mu \approx \mu /\left(2m{v}_F^2\right)=(1/\eta -1)/2$. In the limit $\mu \to 0$ or $m\to \infty$, the shift $\delta \mu /\mu \to 0$, restoring particle-hole symmetry.

Figure 2

For all energies, there are two distinct cyclotron orbits (Fermi surfaces). However, for real materials the Hamiltonian Eq. (16) is an effective one for low energies, and the outer orbit may not actually exist. Nevertheless, for systems with small $\eta$, such as Rashba systems,[37] there are two possible orbits, indexed by $\beta =\pm 1$ for outer and inner orbits. In the figure, $\mu >0$, so the inner orbit ($\beta =-1$) circulates around the $\alpha =1$ band, and the outer ($\beta =1$) orbit circulates around the $\alpha =-1$ band. For $\mu <0$, both orbits circulate around the $\alpha =-1$ band.

Figure 3

The phase offset ${\gamma }_{+}$ as a function of $\Delta$, the gap parameter, and $\eta$, the dimensionless parameter that tunes between normal fermions ($\eta =0$) and Dirac fermions ($\eta =1$). It is this quantity, and not simply the Berry's phase, that determines the phase offset in quantum oscillation experiments. If either particle-hole symmetry is maintained ($\eta =0,1$), or the system is gapless, the phase offset is a constant. If neither condition applies, the phase offset is a continuous function of $\eta$ and $\Delta$. The solid lines denote constant values of ${\gamma }_{+}$.

Figure 4

The relative cyclotron mass $m^{*}/m$ as a function of $\Delta /\mu$ and band bending. At $\eta =0$, $m^{*}=m$. At $\eta =1$, $m\to \infty$, so $m^{*}/m\to 0$. As the gap is decreased, or the chemical potential increased, the system becomes more Dirac-like. At $\eta =0$ or $\eta =1$, the cyclotron mass is independent of the gap/chemical potential, however in between the two this is not the case.

Figure 5

Robust determination of ${\gamma }_{B\to 0}$ from an index plot. The crosses represent the extremum of the resistivity or magnetization from Eq. (1) or Eq. (14) corresponding to filled Landau levels. The solid line is the small field fit, Eq. (36). It is clear that even for a system with pronounced Zeeman splitting (here $g_s=50$), the fit is reliable at all filling factors. The dashed line is the linearized, asymptotic fit [$d\gamma /dB=0$ in Eq. (36)], which is asymptotically exact as $n\to \infty$. Extrapolating this line back to $1/B\to 0$ gives the topologically relevant phase offset, ${\gamma }_{B\to 0}$, which shows the Dirac nature of the surface states. System parameters are relevant to Bi${}_2$Te${}_2$Se:[15] $v_F=3.4\times {10}^5{\mathrm{ms}}^{-1}$, $m=0.13m_e$, $g_s=50$.

Figure 6

Determination of the zero-field phase offset ${\gamma }_{B\to 0}$ using the fitting function Eq. (40) (solid lines) on experimental data for Bi${}_2$Te${}_2$Se,[15] and Bi${}_2$Se${}_3$.[10] For Bi${}_2$Te${}_2$Se, we have shifted $n$ index by $1/2$, following Xiong et al.,[20] and obtain ${\gamma }_{B\to 0}=-0.1$, whereas a linear fit to the data yields ${\gamma }_{B\to 0}=-0.78$. For Bi${}_2$Se${}_3$, we obtain ${\gamma }_{B\to 0}=0.1$, whereas a linear fit yields ${\gamma }_{B\to 0}=-0.36$. The dashed lines are the asymptotic plots of Eq. (36), with $A_2=0$, whose $n\text{−}\Lambda$ intercept gives the topologically important zero-field phase offset.