Magnetic-field tuning of the spin dynamics in the magnetic topological insulators $\left({\mathrm{MnBi}}_2{\mathrm{Te}}_4\right){\left({\mathrm{Bi}}_2{\mathrm{Te}}_3\right)}_n$

We report a high-frequency/high-magnetic field electron spin resonance (HF-ESR) spectroscopy study in the sub-THz frequency domain of the two representatives of the family of magnetic topological insulators $\left({\mathrm{MnBi}}_2{\mathrm{Te}}_4\right){\left({\mathrm{Bi}}_2{\mathrm{Te}}_3\right)}_n$ with $n=0$ and 1. The HF-ESR measurements in the magnetically ordered state at a low temperature of $T=4\mathrm{K}$ combined with the calculations of the resonance modes showed that the spin dynamics in ${\mathrm{MnBi}}_{\text{4}}{\mathrm{Te}}_{\text{7}}$ is typical for an anisotropic easy-axis type ferromagnet whereas ${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$ demonstrates excitations of an anisotropic easy-axis type antiferromagnet. However, by applying the field stronger than a threshold value $\sim 6\mathrm{T}$, we observed in ${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$ a crossover from the antiferromagnetic (AFM) resonance modes to the ferromagnetic (FM) modes, whose properties are very similar to the FM response of ${\mathrm{MnBi}}_{\text{4}}{\mathrm{Te}}_{\text{7}}$. We attribute this remarkably unusual effect unexpected for a canonical easy-axis antiferromagnet, which, additionally, can be accurately reproduced by numerical calculations of the excitation modes, to the closeness of the strength of the AFM exchange and magnetic anisotropy energies which appears to be a very specific feature of this compound. Our experimental data evidences that the spin dynamics of the magnetic building blocks of these compounds, the Mn-based septuple layers (SLs), is inherently ferromagnetic, featuring persisting short-range FM correlations far above the magnetic ordering temperature as soon as the SLs get decoupled either by introducing a nonmagnetic quintuple interlayer, as in ${\mathrm{MnBi}}_{\text{4}}{\mathrm{Te}}_{\text{7}}$, or by applying a moderate magnetic field, as in ${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$, which may have an effect on the surface topological band structure of these compounds.

Figure 1

${\mathrm{MnBi}}_{\text{4}}{\mathrm{Te}}_{\text{7}}$ : Frequency versus field $\nu \left(H\right)$ diagram of the resonance modes (symbols) at $T=4\mathrm{K}$ (a) and $T=30\mathrm{K}$ (c) for the external magnetic field applied parallel and perpendicular to the $c$ axis (left vertical scale) and selected HF-ESR spectra at various excitation frequencies (right vertical scale). Solid lines depict the calculated $\nu \left(H\right)$ resonance branches (see the text); (b) The excitation gap ${\mathrm{\Delta }}_{\mathrm{FM}}(H=0)$ as a function of temperature. Open symbols correspond to ${\mathrm{\Delta }}_{\mathrm{FM}}(H=0)$ evaluated from the $\nu \left(H\right)$ dependence measured at selected fixed temperatures for $\mathbf{H}\parallel c$ axis and $\mathbf{H}\parallel ab$ plane. Closed symbols are the gap estimates from the $T$ dependence of the resonance fields measured at two fixed frequencies $\nu =75.5\mathrm{GHz}$ and $\nu =219.5\mathrm{GHz}$ for the same two field geometries. (d) Resonance fields as a function of temperature measured at $\nu =75.5$ GHz for $\mathbf{H}\parallel c$ axis and $\mathbf{H}\parallel ab$ plane. Horizontal solid line indicates the paramagnetic resonance field corresponding to the $g$ factor, $g=2.02$.

Figure 2

${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$: HF-ESR spectra at selected excitation frequencies recorded at $T=4\mathrm{K}$ for two orientations of the external magnetic field $\mathbf{H}\parallel c$ axis (a) and $\mathbf{H}\parallel ab$ plane (b). For clarity, all spectra are normalized and shifted vertically. Labels L1, L2, and L4 indicate the resonance modes and $*$ marks the impurity peak.

Figure 3

${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$: Frequency versus field $\nu \left(H\right)$ diagram of the resonance modes (symbols) at $T=4\mathrm{K}$ for the external magnetic field applied parallel (a) and perpendicular (b) to the $c$ axis. Black and red solid curves in (c) are the results of the modeling of the resonance branches. The shaded area around the curves depicts the calculation uncertainty corresponding to the error bars of the anisotropy and exchange constants. The dashed lines correspond to the calculations of the modes of an easy-axis ferromagnet according to Eqs. (4) and (5). The straight green solid line represents the paramagnetic resonance condition $\nu =(g{\mu }_{\mathrm{B}}/h)H_{\mathrm{res}}$. The vertical dash-dot line indicates the threshold field $\sim 6\mathrm{T}$ for the crossover from the AFM type to FM type of the resonance modes (see the text for details).

Figure 4

${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$ : (a) HF-ESR spectra at selected temperatures recorded at $\nu =76\mathrm{GHz}$ for $\mathbf{H}\parallel c$ axis. Labels L1 and L3 indicate the resonance modes. The spectra are normalized and shifted vertically for clarity; (b) AFM zero field gap as a function of temperature (symbols). The solid line is a fit using Eq. (6).

Figure 5

${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$ : (a) HF-ESR spectra (mode L4) at selected temperatures recorded at $\nu =109.6\mathrm{GHz}$ for $\mathbf{H}\parallel ab$ plane. The spectra are normalized and shifted vertically for clarity. (b) Resonance shift of L4 $\delta H\left(T\right)=H_{\mathrm{res}}\left(T\right)-H_{\mathrm{res}}\left(150\mathrm{K}\right)$ as a function of temperature at several selected microwave frequencies for $\mathbf{H}\parallel ab$ plane. The corresponding resonance fields in the paramagnetic state at $T=150\mathrm{K}$ are indicated in the legend; (c) same as (b) but instead of the relative shifts the resonance fields $H_{\mathrm{res}}$ are plotted on the absolute scale. Horizontal dashed lines represent the $T$-independent resonance fields in the paramagnetic state whose values for the given frequency $\nu$ are listed in the legend to panel (b). The horizontal dash-dot line indicates the threshold field $\sim 6\mathrm{T}$ for the crossover from the AFM type to FM type of spin excitations in ${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$ with increasing the field strength. The color code depicts the LRO, short-range FM-correlated and paramagnetic regimes of the the spin dynamics observed in ${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$ by HF-ESR.

Figure 6

${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$: Signal traces $S_D\left(H\right)$ featuring steplike changes in the range 6–8 T recorded at $T=4\mathrm{K}$ for two frequencies of 76 and 310 GHz for the up and down sweeps of the magnetic field $\mathbf{H}\parallel ab$ plane. Resonance mode L4 is indicated by the arrow. Inset shows the temperature dependence of the jump field measured at $\nu =109.6\mathrm{GHz}$. Solid lines are the fits to the critical exponent function $\propto {[1-(T/T_{\mathrm{N}})]}^{\beta }$.

Figure 7

(a)–(d) Two-dimensional color maps of the resonance excitations measured on two single-crystalline samples of ${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$ in the frequency-sweep mode at $T=4\mathrm{K}$. The change of the color corresponds to the relative change of the signal at the detector $S_D(\nu ,H)/S_D(\nu ,H-\delta H)$ by incrementing the field by $\delta H=0.05-0.2\mathrm{T}$. Horizontal white stripes at 0–43 GHz, 67–83 GHz, and 110–140 GHz are blind frequency spots of the HF-ESR setup. Selected HF-ESR spectra measured in the field-sweep mode are shown as black traces in panels (c) and (d) (right vertical scale) for comparison. (a) and (c) depict results for sample 1 measured with $\mathbf{H}\parallel ab$ plane by increasing the field from 0 to 15 T or decreasing it from 15 to 0 T, respectively; (b) and (d) depict results for sample 2 measured by increasing the field from 0 to 15 T applied along the $c$ axis and in the $ab$ plane, respectively.

Figure 8

The calculated magnetization curves in the ordered state of ${\mathrm{MnBi}}_{\text{2}}{\mathrm{Te}}_{\text{4}}$ for $\mathbf{H}\parallel c$ axis and $\mathbf{H}\parallel ab$ plane. Vertical dashed lines indicate the spin-flop field for $\mathbf{H}\parallel c$ axis and the saturation fields for the out-of-plane and in-plane orientations of the applied field.