Metamagnetism of few-layer topological antiferromagnets

${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ (MBT) materials are a promising class of antiferromagnetic topological insulators whose films provide access to novel and technologically important topological phases, including quantum anomalous Hall states and axion insulators. MBT device behavior is expected to be sensitive to the various collinear and noncollinear magnetic phases that are accessible in applied magnetic fields. Here, we use classical Monte Carlo simulations and electronic structure models to calculate the ground state magnetic phase diagram as well as topological and optical properties for few-layer films with up to six septuple layers. Using magnetic interaction parameters appropriate for MBT, we find that it is possible to prepare a variety of different magnetic stacking sequences, some of which have sufficient symmetry to disallow nonreciprocal optical response and Hall transport coefficients. Other stacking arrangements do yield large Faraday and Kerr signals, even when the ground state Chern number vanishes.

Figure 1

Bulk crystal and magnetic structure of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. Red and blue arrows and planes denote the ferromagnetic layers with staggered magnetization, forming an A-type structure. Gray rectangles identify septuple layers that are separated by a van der Waals (vdW) gap. Septuple layers are staggered in a close-packed ABC stacking. Principal antiferromagnetic interlayer ($J$) and ferromagnetic intralayer ($J^{\prime }$) interactions are indicated by black arrows.

Figure 2

(a) Schematic magnetic layer structure of MBT in the crystallographic unit cell with close-packed stacking of septuple blocks. The intralayer ($J^{\prime }$) and interlayer ($J$) magnetic interactions are indicated. The gray shaded boxes indicate the full septuple layers separated by a vdW gap. (b) Monte Carlo simulations of the bulk magnetization of MBT with field applied perpendicular and parallel to the Mn layers. Inset shows the order parameter (OP) of the staggered A-type AF order, which is defined as $(s_1-s_2)/2$, with $s_i$ the magnetization in each Mn layer, as a function of temperature. (c)–(f) Monte Carlo simulations of the magnetization of 3-, 4-, 5-, and 6-layer MBT, respectively, using the nominal Heisenberg parameters. For $N=4$ and 6, the “$*$” indicates an additional phase transition.

Figure 3

Phase diagrams obtained by MC simulation showing the layer magnetization of a uniaxial layered antiferromagnet vs magnetic field applied perpendicular to the layers and single-ion anisotropy strength ($SD$) for (a) three layers, (b) four layers, (c) five layers, and (d) six layers. Simulations start in the FM phase at 10 T and the field strength is reduced in equal steps of 0.2 T to $-$10 T. The solid lines are phase boundaries and metastability limits obtained from the Mills model; the dashed lines are guides to the eye. Collinear phases are labeled as described in the text and noncollinear spin-flop-like phases are labeled as SF. For $N=6$, panel (d) indicates a metastability limit below which only the $\mathrm{M}{2}^{\prime }$ phase is stabilized out of the SF phase.

Figure 4

Possible collinear magnetic ground states for $N=3$-, 4-, 5-, and 6-layer systems, labeled as either FM, AF, or $\mathrm{M}n$, where $n$ is the uncompensated net layer magnetization. Red dashed lines indicate broken AF bonds that cost an exchange energy of $3J$ each. Shaded green rectangles enclose degenerate states with the same magnetization but different stacking sequences indicated by the prime symbol. Shaded blue rectangles enclose the states with odd-parity magnetic configuration. The integers below each state denote their Chern numbers calculated from the simplified Dirac-cone model and (e) are the calculated band gaps vs the magnetic stacking sequences of few-layer MBT thin films, here we assigned a negative sign for the phase with nonzero Chern number.

Figure 5

(a),(b) Twelve MC simulations for $N=6$ repeated under identical starting conditions near the metastability limit for formation of the $\mathrm{M}{2}^{\prime }$ phase. Curves have a slight vertical offset for clarity. (a) For $SD=0.11\mathrm{meV}$, only the $\mathrm{M}{2}^{\prime }$ phase appears. (b) For $SD=0.12\mathrm{meV}$, either the $\mathrm{M}{2}^{\prime }$ or $\mathrm{M}{2}^{\prime \prime }$ phase appears. (c)–(e) show the evolution of the magnetization angle for each layer for different simulations. (c) For $SD=0.11$ meV, which is below the metastability limit, the angle of layers 2 and 5 in the spin-flop phase approaches ${90}^{\circ }$ before flipping into the $\mathrm{M}{2}^{\prime }$ phase. For $SD=0.12$ meV, different simulations (labeled #1 and #2) result in either (d) $\mathrm{M}{2}^{\prime }$ with layers 2 and 5 flipping or (e) $\mathrm{M}{2}^{\prime \prime }$ with layers 3 and 5 flipping.

Figure 6

Nonreciprocal optical response in 6-layer MBT thin films. Panels (a)–(f) show plots of optical longitudinal (${\sigma }_{xx}$) and Hall (${\sigma }_{xy}$) conductivities vs optical frequency calculated from the Kubo-Greenwood formula using the simplified Dirac cone electronic structure model. In these plots the solid curves show the real part of a conductivity tensor element while the dashed curves show the imaginary part. In panels (g) and (h), in which the Faraday and Kerr rotation angles in $N=6$ thin films are plotted vs optical frequency, different colors represent different metamagnetic states.