Effect of magnetic field on a magnetic topological insulator film with structural inversion asymmetry

The effect of magnetic field on an ultrathin magnetic topological insulator film with structural inversion asymmetry is investigated. We introduce the phase diagram, calculate the Landau level spectrum analytically, and simulate the transport behavior in Landauer-Büttiker formalism. The quantum anomalous Hall phase will survive increasing magnetic field. Due to the two spin-polarized zero modes of Landau levels, a nontrivial phase similar with the quantum spin Hall effect can be induced by a magnetic field, which is protected by structural inversion symmetry. Some exotic longitudinal and Hall resistance plateaus with fractional values are also found in a six-terminal Hall bar, arising from the coupling between edge states due to an inverted energy band and Landau levels.

Figure 1

(a1) and (b1) Phase diagrams in the plane of SIA potential $\delta V$ and Fermi energy $E_f$. Regions I–VI are in the inverted regime. Region I (IV) is the QAH (QPH) phase with Fermi level in the bulk gap, and II,V (III,VI) are the $n$($p$)-doped inverted system. Region VII is the normal insulator phase with Fermi level in the bulk gap. Regions VIII and IX are in the normal metal phase. (a2) and (b2) Hall conductance as a function of Fermi energy for several $\delta V$. (a3) and (b3) Energy dispersion of the 2D system in a certain direction. (a4) and (b4) Energy dispersion of a nanoribbon with a ribbon width 480 nm. (a1)–(a4) for $m_0/B<0$ and (b1)–(b4) for $m_0/B>0$ with the exchange field $M=-15\mathrm{meV}$. (The energy is in units of meV in this paper if not indicated.)

Figure 2

Landau level spectrum. (a/b1-4) LLs as a function of magnetic field for several $\delta V$. (a/b5) LLs as a function of $\delta V$ for $B_z=3T$. Light green lines are the zero modes $E_{0\pm }$. $E_{0-}$ is the one that lies below at small magnetic field. Red integers indicate the Chern number. (a1-5) for $m_0/B<0$, (b1-5) for $m_0/B>0$. The exchange field $M=-15\mathrm{meV}$.

Figure 3

Hall resistance $R_H$ and longitudinal resistance $R_L$ as a function of Fermi energy $E_f$ for several SIA terms in absence of magnetic field: (a1) and (a2) for $m_0/B<0$, (b1) and (b2) for $m_0/B>0$. Inset is the schematic of the Hall bar. The width of both the central region and six contacts is 480 nm.

Figure 4

Hall resistance $R_H$ and longitudinal resistance $R_L$ as a function of Fermi energy with fixed magnetic field $B_z$. $\delta V=0$, $m_0/B<0$.

Figure 5

(a) The energy dispersion for a 1D nanoribbon with a width of 480 nm for several magnetic fields. (b) Wave-function distribution along $y$ direction for the exotic edge states, (A,B), and the edge state due to LL, C. Inset of (b) is the amplification of (a). Parameters are identical to those in Fig. 4.

Figure 6

(a) and (b) Hall resistance $R_H$ and longitudinal resistance $R_L$ as a function of Fermi energy with fixed magnetic field $B_z$. $\delta V=10\mathrm{meV}$, $m_0/B<0$.

Figure 7

The energy dispersion for a nanoribbon with a width of 480 nm for several magnetic fields. Parameters are identical to those in Fig. 6.

Figure 8

(a) and (b) Hall resistance $R_H$ and longitudinal resistance $R_L$ as a function of Fermi energy with fixed magnetic field $B_z$ ($\delta V=40\mathrm{meV},m_0/B<0$).

Figure 9

The energy dispersion for a nanoribbon with a width of 480 nm for several magnetic fields. Parameters are identical to those in Fig. 8.

Figure 10

(a) and (b) Hall resistance $R_H$ and longitudinal resistance $R_L$ as a function of Fermi energy with fixed magnetic field $B_z$ ($\delta V=0$, $m_0/B>0$).

Figure 11

The energy dispersion for a nanoribbon with a width of 480 nm for several magnetic fields. Parameters are identical to those in Fig. 10.

Figure 12

(a) and (b) Hall resistance $R_H$ and longitudinal resistance $R_L$ as a function of Fermi energy with fixed magnetic field $B_z$ ($\delta V=10\mathrm{meV}$, $m_0/B>0$).

Figure 13

The energy dispersion for a nanoribbon with a width of 480 nm for several magnetic fields. Parameters are identical to those in Fig. 12.

Figure 14

Phase diagram at $B_z=3\mathrm{T}$ in the plane of Fermi energy and SIA potential for (a) $m_0/B<0$ and (b) $m_0/B>0$, respectively. Integers show the Hall conductance of the region with zero longitudinal component. Exotic plateaus with nonzero longitudinal component form in the light green region.