Negative Longitudinal Magnetoresistance from the Anomalous $N=0$ Landau Level in Topological Materials

Negative longitudinal magnetoresistance (NLMR) is shown to occur in topological materials in the extreme quantum limit, when a magnetic field is applied parallel to the excitation current. We perform pulsed and dc field measurements on ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}\mathrm{Se}$ epilayers where the topological state can be chemically tuned. The NLMR is observed in the topological state, but is suppressed and becomes positive when the system becomes trivial. In a topological material, the lowest $N=0$ conduction Landau level disperses down in energy as a function of increasing magnetic field, while the $N=0$ valence Landau level disperses upwards. This anomalous behavior is shown to be responsible for the observed NLMR. Our work provides an explanation of the outstanding question of NLMR in topological insulators and establishes this effect as a possible hallmark of bulk conduction in topological matter.

Figure 1

Sketch of the behavior of the $N=0$ bulk LL as a function of magnetic field for a trivial (a) and a topological (b) system. The $k$ dispersion of the energy level in the direction of the applied field is shown ($k_z$). Topological surface states are shown in black in (b) at $B=0$. The $L_6^{\pm }$ bands denote the band extrema of opposite parity occurring at the $L$ point in ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}\mathrm{Se}$. (c) In-plane MR measured with $B//I$ at 10 K in two trivial ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}\mathrm{Se}$ epilayers ($x=0.10$ and $x=0.14$) and three topological ones ($x=0.19$, $x=0.23$, and $x=0.3$) up to $B=15\mathrm{T}$. ARPES dispersions and momentum distribution curves for $x=0.1$ (d),(e) and $x=0.2$ (f),(g) measured with 18 eV photons at 30 and 40 K, respectively.

Figure 2

(a) In-plane MR measured up to 60 T using pulsed magnetic field for ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}\mathrm{Se}$ with $x=0.14$ (blue) and $x=0.19$ (red). (b) Low-field Shubnikov–de-Haas oscillations and (c) Landau index versus $1/B$ shown for both samples. Arrows mark the field at which the quantum limit is reached ($B_{\mathrm{QL}}$). (d) NLMR onset extracted from Fig. 1 versus $B_{\mathrm{QL}}$ for the three topological samples considered in this work. The dashed gray line is obtained for an onset exactly equal to $B_{\mathrm{QL}}$. The Sn concentration “$x$” corresponding to each sample is shown above the data points.

Figure 3

Massive Dirac LL (black) and spin-split LLs (red and blue) of ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}\mathrm{Se}$ versus magnetic field for $x=0.19$ (a) and $x=0.14$ (b). The energy gap is $2\mathrm{\Delta }=20\mathrm{meV}$ and $v_z=4.8\times {10}^5\mathrm{m}/\mathrm{s}$ for $x=0.19$ and $5.0\times {10}^5\mathrm{m}/\mathrm{s}$ for $x=0.14$.$\tilde{m}=0.20m_0$ is used for both samples [24]. $E_f$ versus magnetic field is shown in yellow. The energy gap is shaded in green. (c) $N=0$ LLs for the 35° and 90° valleys computed using the parameters in Table 1. (d) MR calculated using Eq. (6) for parameters shown in Table 1, for $x=0.14$ and $x=0.19$ above the quantum limit.