Simple mechanisms that impede the Berry phase identification from magneto-oscillations

The phase of quantum magneto-oscillations is often associated with the Berry phase and is widely used to argue in favor of topological nontriviality of the system (Berry phase $2\pi n+\pi )$. Nevertheless, the experimentally determined value may deviate from $2\pi n+\pi$ arbitrarily, therefore more care should be made analyzing the phase of magneto-oscillations to distinguish trivial systems from nontrivial. In this paper we suggest two simple mechanisms dramatically affecting the experimentally observed value of the phase in three-dimensional topological insulators: (i) magnetic field dependence of the chemical potential, and (ii) possible nonuniformity of the system. These mechanisms are not limited to topological insulators and can be extended to other topologically trivial and nontrivial systems.

Figure 1

(a) Ladder of Landau levels versus magnetic field for spinless massive particles, solid brown line: chemical potential versus magnetic field provided that total density is fixed, according to Eq. (2), dotted line: constant chemical potential $B_N$ values are indicated. (b) The same as (a) for two-dimensional Dirac particles with linear dispersion. (c) Fan diagram $1/B_N$ versus $N$ adopted from Ref. [4]. Top and bottom axes indicate two ways to define $N$(see text). (d) Landau level ladder in monolayer graphene for explanation of the anomalous phase of the magneto-oscillations.

Figure 2

(a) Landau levels (fifth to tenth) versus magnetic field for TSS in model 3D TI ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ thin film with an effective $g$ factor equal to 30. Chemical potential versus magnetic field provided that total density is fixed, for three values of the total density. (b) The same as (a) for two-dimensional Dirac particles with linear dispersion. (b) The corresponding fan diagrams $1/B_N$ versus $N$. The inset show schematically the dependence of the phase factor on total carrier density.

Figure 3

(a) Schematic band diagram of the low density 3D TI, adopted from Ref. [11]. (b) Energy diagrams $[E_n\left(B\right)$ dependencies] for the topological surface states Landau levels (solid lines). Dotted and dashed lines: chemical potential within two models. Gray bars and orange circles: correspond to Landau gaps. (c) Fan diagrams, corresponding to $\mu =\text{const.}$ (gray bars) and $\mu ={\mu }_0+\alpha B$ (orange circles).

Figure 4

(a) The simplest case of nonuniform sample (long rectangular sample in Hall bar geometry with crack, shown by hatching). Current flows between contacts 1 and 2. Longitudinal resistivity is measured between contacts 3 and 4 and contain contribution of cracks. Hall resistance is measured between contacts 3 and 5 and corresponds to bare material. (b) and (c) Other possibilities of nonuniformity.