Metamagnetic tricritical behavior of the magnetic topological insulator ${\mathrm{MnBi}}_{4}{\mathrm{Te}}_{7}$ (2024)

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  Figure 1

  Crystal structure and magnetic properties of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$. (a)The view of crystal structure of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$ from the [110] directions. Blue block: edge-sharing ${\mathrm{BiTe}}_6$ octahedra; pink block: edge-sharing ${\mathrm{MnTe}}_6$ octahedra. Red arrow: magnetic moment directions of Mn ions. $J_{\perp }$, the interlayer exchange coupling; $J_{\parallel }$, the intralayer exchange coupling. (b)The $\left(00l\right)$ x-ray diffraction peaks of cleaved $ab$ plane of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$. Inset: a piece of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$ crystal against 1mm scale. (c)The temperature dependent field-cooled and zero-field-cooled susceptibility and inverse susceptibility taken at $H=100$ Oe for $H\parallel c$. (d)Magnetic hysteresis loop of isothermal magnetization taken at $T=5.0\mathrm{K}$ and the loop with demagnetizing correction.

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  Figure 2

  Typical magnetization data and corresponding susceptibility data of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$. (a)Magnetization as a function of applied magnetic field at various temperatures approaching the tricritical point. (b)The susceptibility calculated from magnetization (a)as a function of applied magnetic field. The inset shows the typical curve at 10K. We can define critical fields of the spin-flip transition. $H_a^{-}$ and $H_a^{+}$ are the lower and upper critical fields, respectively.

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  Figure 3

  $dM/dH$ plotted as functions of $T$ and $H_a$ along the $c$ axis of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$. The mixed phase region gets narrower approaching the TCP.

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  Figure 4

  Typical ac susceptibility data of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$ as a function of applied magnetic field. Data were recorded for an excitation amplitude of 2 Oe parallel to the $c$ axis at different temperatures. (a)The real part of the ac susceptibility, $\mathrm{Re}{\chi }_{ac}$, as a function of applied magnetic field for 10Hz frequency approaching the tricritical temperature. (b)The imaginary part of the ac susceptibility, $\mathrm{Im}{\chi }_{ac}$, as a function of applied magnetic field. The inset shows the typical curve at 11K and we define the upper critical fields $H_a^{+}$ and lower critical fields $H_a^{-}$. We can determine the fields of the second order transition $H_c$ above 12.4K, where the imaginary part of the susceptibility is approximately equal to zero.

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  Figure 5

  $\mathrm{Im}{\chi }_{ac}$ plotted as functions of $T$ and $H_a$ along the $c$ axis of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$. The mixed phase region gets narrower approaching the TCP.

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  Figure 6

  (a)Magnetic phase diagram of ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$ as a function applied magnetic field and temperature along the $c$ axis. The tricritical point ($T_t,H_t$) for our measurement accuracy is (12.40K, 660 Oe) and the Néel temperature is about 12.9K. (b) $M\text{−}T$ phase diagram. $M_t$ is tricritical magnetization, $M_t=6.39\pm 0.43$ emu/${\mathrm{cm}}^3$. The magnetization values plotted in the diagram correspond to the values under $H_a^{+}$ or $H_a^{-}$ critical fields and the error bars reflect the uncertainty in locating the onset of the susceptibilities. The exponents ${\beta }_2^{+}$, ${\beta }_2^{-}$ mean normalized magnetization change along different paths, upper boundary $H^{+}$ and lower boundary $H^{-}$ of mixed phase, respectively. We can define ${\beta }_2$ according to $\mathrm{\Delta }M/M_t\sim {(1-T/T_t)}^{{\beta }_2}$, where $\mathrm{\Delta }M=M\left(H_a^{+}\right)-M\left(H_a^{-}\right)$. The inset shows data for the discontinuity close to the TCP.

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  Figure 7

  Tricritical exponent ${\delta }_2$ can be defined by $M\sim H^{1/{\delta }_2}$ approaching TCP. (a)Typical normalized magnetization versus normalized field at 11.79K. The inset shows double logarithmic plots for both the paramagnetic part $H>H_c$ and antiferromagnetic part $H<H_c$. The red line is fitted with the tricritical relation, $M\sim H^{1/2}$, and we can determine tricritical region data. (b)Double logarithmic plots of normalized magnetization and normalized field for different isotherms. Lines have slopes $0.95\pm 0.02$ (critical) and $0.52\pm 0.09$ (tricritical) corresponding to $1/{\delta }_2=0.52\pm 0.09$. The crossover from critical to tricritical regime is controlled by the crossover exponent ${\mathrm{\Phi }}_2={\delta }_2{\beta }_2$.

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  Figure 8

  (a)Scaling plot of $m/{\left|t\right|}^{{\beta }_2}$ versus $h_2/{\left|t\right|}^{{\beta }_2{\delta }_2}$ [Eq.(A19)] using mean-field exponents. (b)Scaling magnetization data vs scaling variable with the effective tricritical exponents ${\beta }_2^{\mathrm{eff}}=1.10$ and ${\delta }_2^{\mathrm{eff}}=1.65$. Using Eq.(A22), the dotted line is fitted to the data from the tricritical region as in Fig.7.

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  Figure 9

  Data of Fig.8 scaled according to $m/{\left|t\right|}^{{\beta }_2}{|ln|t\left|\right|}^{0.5}$ vs $h_2/{\left|t\right|}^{{\beta }_2{\delta }_2}$ with mean-field exponents ${\delta }_2=2,{\beta }_2=1$ and logarithmic correction. The dotted line is fitted to the tricritical data with Eq.(A24).

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  Figure 10

  Tricritical phase diagram of a metamagnet in the space of temperature $T$, staggered magnetic field $H_1$, and uniform magnetic field $H_2$. The tricritical point is ($T_t$, $H_{2t}$) [43].

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