Forbidden Backscattering and Resistance Dip in the Quantum Limit as a Signature for Topological Insulators

Identifying topological insulators and semimetals often focuses on their surface states, using spectroscopic methods such as angle-resolved photoemission spectroscopy or scanning tunneling microscopy. In contrast, studying the topological properties of topological insulators from their bulk-state transport is more accessible in most labs but seldom addressed. We show that, in the quantum limit of a topological insulator, the backscattering between the only two states on the Fermi surface of the lowest Landau band can be forbidden at a critical magnetic field. The conductivity is determined solely by the backscattering between the two states, leading to a resistance dip that may serve as a signature for topological insulator phases. More importantly, this forbidden backscattering mechanism for the resistance dip is irrelevant to details of disorder scattering. Our theory can be applied to revisit the experiments on ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$, ${\mathrm{ZrTe}}_5$, and ${\mathrm{Ag}}_2\mathrm{Te}$ families, and will be particularly useful for controversial small-gap materials at the boundary between topological and normal insulators.

Figure 1

In the quantum limit of a 3D topological insulator, the backscattering between the only two states at the Fermi energy can be forbidden at a critical magnetic field, leading to a resistance dip. (a) The zero-field energy spectrum vs $k_z$ of a 3D topological insulator at $k_x=k_y=0$. (b) In a strong magnetic field, the lowest Landau energy bands of the 3D topological insulator vs $k_z$. The Fermi energy $E_F$ crosses only the $0+$ Landau band. $k_F$ and $-k_F$ stand for the only two states at the Fermi energy. (c) The magnetoresistance of ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$ adapted from Ref. [13]. (d) The calculated magnetoresistance using Eq. (14) (see Sec. S2B of Ref. [14] for details). The abbreviation “sos” means spin-orbit scattering. The parameters are $M_0=-0.01\mathrm{eV}$, $M_z=0$, ${\tilde{M}}_{\perp }=18\mathrm{eV}{\AA }^2$, ${\alpha }_1=100\mathrm{eV}\mathrm{T}$, and ${\alpha }_2=0.0025\mathrm{eV}{\mathrm{T}}^{-2}$.

Figure 2

The mass term $m$ and form factor $I_S$ as functions of the magnetic field $B$ for $M_0<0$ and different $M_{\perp }$ and $M_z$. Red, yellow, and green backgrounds indicate the quantum limit for a carrier density of $6\times {10}^{16}/{\mathrm{cm}}^3$. Without loss of generality, we have assumed $M_0<0$, so $M_{\perp }>0$ and $M_z>0$ means a strong topological insulator, $M_{\perp }\le 0$ and $M_z\le 0$ means a trivial insulator, $M_{\perp }\le 0$ and $M_z>0$ means a weak topological insulator (001), and $M_{\perp }>0$ and $M_z\le 0$ means a weak topological insulator (110). The parameters are $M_0=-0.01\mathrm{eV}$; $M_z=-27$, 0, 27, 820, $1200\mathrm{eV}{\AA }^2$ from left to right; and $M_{\perp }=-13.5$, 0, $13.5\mathrm{eV}{\AA }^2$ from bottom to top.

Figure 3

The same as Fig. 2, but with the Zeeman effect, which shifts the zero point to a lower magnetic field. The parameters are the same as those in Fig. 2, except that $M_z=-27$, 0, 27, 205, $270\mathrm{eV}{\AA }^2$ from left to right; $g_z^c=7.5$; and $g_z^v=-7.5$.

Figure 4

The same as Fig. 3, but for the resistivity ${\rho }_{zz}$ in the presence of the screened Coulomb scattering potential (in units of $2{\pi }^2\hslash n_{\mathrm{imp}}/e^2ϵ$). $C_z=1.409\mathrm{eV}{\AA }^2$. $n_{\mathrm{imp}}$ for the trivial insulator and topological insulator are $6.8\times {10}^{18}/{\mathrm{cm}}^3$ and $1.0\times {10}^{19}/{\mathrm{cm}}^3$, respectively. For $M_0>0$, the same results can be obtained by flipping ${\tilde{M}}_{\perp }$ and $M_z$ accordingly. In the noncolored regions, the magnetic field is not strong enough to put all electrons into the lowest Landau bands.