Defect-driven ferrimagnetism and hidden magnetization in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$

${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ (MBT) materials are promising antiferromagnetic topological insulators in which field-driven ferromagnetism is predicted to cause a transition between axion insulator and Weyl semimetallic states. However, the presence of antiferromagnetic coupling between Mn/Bi antisite defects and the main Mn layer can reduce the low-field magnetization, and it has been shown that such defects are more prevalent in the structurally identical magnetic insulator ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ (MST). We use high-field magnetization measurements to show that the magnetization of MBT and MST occur in stages and full saturation requires fields of $\sim 60$ T. As a consequence, the low-field magnetization plateau state in MBT, where many determinations of the quantum anomalous Hall state are studied, actually consists of ferrimagnetic septuple blocks containing both uniform and staggered magnetization components.

Figure 1

High-field magnetization data of ${\mathrm{Mn}}_{0.99\left(1\right)}{\mathrm{Bi}}_{2.01\left(2\right)}{\mathrm{Te}}_{4.00\left(2\right)}$ at 4 and 30 K with the field applied either along the $c$ axis or in the $ab$ plane.

Figure 2

High-field magnetization data of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ at $T=4\mathrm{K}$ (black solid line) and $T=30\mathrm{K}$ (gray dashed line) compared to results from classical Monte Carlo simulations at $T=0.1\mathrm{K}$ (red solid circles) and $T=30\mathrm{K}$ (blue open circles). The field is applied along the $c$ axis in (a) and in the $ab$ plane in (b). Heisenberg model simulation parameters are described in the text with $x=0.038$ and $S{J}^{\prime }=1.2\mathrm{meV}$. The shaded red area corresponds to the range of magnetization in the MC simulations when $S{J}^{\prime }$ is varied by $\pm 10\%$.

Figure 3

High-field magnetization data for ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ (black line) measured at 50 K compared to CMC simulations (red circles). For comparison, the quantum mechanical $S=5/2$ Brillouin function (purple long-dashed line) and classical Langevin function (blue dotted line) are shown. Finally, the light green short-dashed line corresponds to the linear magnetization extrapolated from low-field susceptibility data [7], $M=\chi H$, where $\chi =0.15{\mu }_{\mathrm{B}}{\mathrm{T}}^{-1}{\mathrm{f}.\mathrm{u}.}^{-1}$ at 50 K.

Figure 4

High-field magnetization data for nominal ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ samples that adopt both MST-FM (red lines) and MST-AF (black lines) ground states with chemical formulas of ${\mathrm{Mn}}_{1.00\left(4\right)}{\mathrm{Sb}}_{2.09\left(2\right)}{\mathrm{Te}}_{3.91\left(2\right)}$ and ${\mathrm{Mn}}_{0.778\left(5\right)}{\mathrm{Sb}}_{2.222\left(6\right)}{\mathrm{Te}}_{4.00\left(1\right)}$, respectively. Magnetization data are normalized (a) per formula unit and (b) per Mn. Measurements are performed at 4 K and with the applied field $H\parallel c$ (solid lines) and $H\parallel ab$ (dashed lines). The inset in (a) zooms in on the low-field region.

Figure 5

High-field magnetization data of (a) MST-AF (${\mathrm{Mn}}_{0.778\left(5\right)}{\mathrm{Sb}}_{2.222\left(6\right)}{\mathrm{Te}}_{4.00\left(1\right)}$) and (b) MST-FM (${\mathrm{Mn}}_{1.00\left(4\right)}{\mathrm{Sb}}_{2.09\left(2\right)}{\mathrm{Te}}_{3.91\left(2\right)}$) samples at $T=4\mathrm{K}$ (black solid line) compared to results from classical Monte Carlo simulations at $T=0.1\mathrm{K}$ (red solid circles). The field applied along the $c$ axis. Heisenberg model simulation parameters are described in the text with $S{J}^{\prime }=2.1\mathrm{meV}$ and $x=0.13$, $y=0.22$ (AF) and $x=0.17$, $y=0$ (FM). The shaded red area corresponds to varying $S{J}^{\prime }$ by $\pm 15$%.

Figure 6

(a) Schematic representation of the $3\times 3\times 1$ supercell used in the DFT calculation for MST. One of the 18 Sb atoms is substituted by Mn. Mn, Sb, and Te atoms are indicated with blue, green, and yellow spheres, respectively. The first three nearest intrablock ${\mathrm{Mn}}_{\text{Sb}}\text{−}{\mathrm{Mn}}_{\text{Mn}}$ exchange parameters are labeled $J_1^{\prime }$, $J_2^{\prime }$, and $J_3^{\prime }$, respectively, and are also shown from the perspective along the $c$ axis in (b). (c)–(e) show the three different scenarios for the formation of antisite-defect configurations at nearest-, next-nearest-, and next-next-nearest-neighbor positions, respectively (Te atoms are not shown).