Voltage-Controlled Magnetic Reversal in Orbital Chern Insulators

Chern insulator ferromagnets are characterized by a quantized anomalous Hall effect and have so far been identified experimentally in magnetically doped topological insulator thin films and in bilayer graphene moiré superlattices. We classify Chern insulator ferromagnets as either spin or orbital, depending on whether the orbital magnetization results from spontaneous spin polarization combined with spin-orbit interactions, as in the magnetically doped topological insulator case, or directly from spontaneous orbital currents, as in the moiré superlattice case. We argue that, in a given magnetic state, characterized, for example, by the sign of the anomalous Hall effect, the magnetization of an orbital Chern insulator will often have opposite signs for weak $n$ and weak $p$ electrostatic or chemical doping. This property enables pure electrical switching of a magnetic state in the presence of a fixed magnetic field.

Figure 1

1.1°-TBG moiré band structures in valley $K$ for three hBN-induced mass choices. (a) $m_1=m_2=10\mathrm{meV}$ produces a band gap $E_g^0\sim 7.5\mathrm{meV}$ at charge neutrality. (b) $m_1=10$, $m_2=0\mathrm{meV}$ produces $E_g^0\sim 2.3\mathrm{meV}$. (c) $m_1=-m_2=-10\mathrm{meV}$ produces $E_g^0\sim 3\mathrm{meV}$. The flat bands are nontrivial with Chern numbers $C=\pm 1$ in (a),(b) and trivial in (c). The moiré bands were calculated using a low-energy continuum model [22] with interlayer tunneling strength $w^{\mathrm{AB}}=110\mathrm{meV}$ and $w^{\mathrm{AA}}/w^{\mathrm{AB}}=0.85$ to account for corrugation and strain.

Figure 2

(a) Schematic moiré flat bands with SU(4) spin and valley flavor symmetry maintained. $E_g^0$ is the single-particle band gap. “c” and “v” indicate the “conduction” and “valence” bands. (b) OM (solid line) from valleys $K$ (blue) and $K^{\prime }$ (green) as a function of $\mu$. The dotted and dash-dotted lines are, respectively, the $M^1$ and $M^2$ [48] contributions defined in Eq. (4). The single-particle gap is shaded in gray. (c)–(f) Schematic moiré flat bands in TR broken symmetry states at $\nu =3$ and $\nu =1$ in which different flavors are rigidly shifted in energy by a momentum- and flavor-independent exchange energy $U$ if the flat conduction band is filled in that flavor.

Figure 3

Plots of $M_{\mathrm{c}K}^1$, $M^{\mathrm{n}\text{−}\text{doped}}$, and $M^{\mathrm{p}\text{−}\text{doped}}$. (a)–(d) With $m_1=m_2=m$. (e)–(g) With $m_1=m$ and $m_2=0$. (a) $M_{\mathrm{c}K}^1$ increases as both $\theta$ and $m$. (b) $M^{\mathrm{n}\text{−}\text{doped}}(\theta ,m)$ is negative in most parts of the parameter range shown in the figure. (c) $M^{\mathrm{p}\text{−}\text{doped}}(U,m)$ for a typical twist angle $\theta =1.1°$. (d) Enlargement of the dashed rectangle in (c). $M^{\mathrm{p}\text{−}\text{doped}}$ is positive as long as ${\mathrm{\Delta }}_g\gtrsim 1\mathrm{meV}$. (e) Similar to (a), $M_{\mathrm{c}K}^1$ increases as both $\theta$ and $m$. (f) $M^{\mathrm{n}\text{−}\text{doped}}(\theta ,m)$. (g) Enlargement of the dashed rectangle in (f), $M^{\mathrm{n}\text{−}\text{doped}}$ is only negative for $\theta \lesssim 1.04°$. (h) Similar to (d), $M^{\mathrm{p}\text{−}\text{doped}}$ is positive for a tiny gap. In (c),(d),(h), the parameter region where $U<w$ is identified to be metallic.