Quantum oscillations in the field-induced ferromagnetic state of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$

The intrinsic antiferromagnetic topological insulator $\mathrm{Mn}{\mathrm{Bi}}_2{\mathrm{Te}}_4$ undergoes a metamagnetic transition in a $c$-axis magnetic field. It has been predicted that ferromagnetic $\mathrm{Mn}{\mathrm{Bi}}_2{\mathrm{Te}}_4$ is an ideal Weyl semimetal with a single pair of Weyl nodes. Here we report measurements of quantum oscillations detected in the field-induced ferromagnetic phase of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$, where Sb substitution tunes the majority carriers from electrons to holes. Single-frequency Shubnikov–de Haas oscillations were observed in a wide range of Sb concentrations ($0.54\le x\le 1.21$). The evolution of the oscillation frequency and the effective mass shows reasonable agreement with the Weyl semimetal band structure of ferromagnetic $\mathrm{Mn}{\mathrm{Bi}}_2{\mathrm{Te}}_4$ predicted by density functional calculations. Intriguingly, the quantum oscillation frequency shows a strong temperature dependence, indicating that the electronic structure depends sensitively on magnetism.

Figure 1

Transport characterization of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$. (a) Crystal structure of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$. (b) Temperature dependence of zero-field in-plane resistivity ${\rho }_{xx}$. (c) Field dependence of Hall resistivity ${\rho }_{xy}$ with field along the $c$-axis at $T=2\mathrm{K}$. Dash-dotted lines denote data taken on Oak Ridge samples, while solid lines represent UW samples. (d) Carrier density vs chemical doping $x$. Carrier density is extracted by fitting the linear background of the Hall resistivity above $H_{c2}$.

Figure 2

Magnetization characterization of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$. (a) Field dependence of magnetization of the $x=0.63$ OR sample measured at various temperatures. (b) Field-temperature magnetic phase diagram of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$ for $x=0.63$. Circles and triangles denote spin-flop field $H_{c1}$ and saturation field $H_{c2}$, respectively. Dotted lines are guides for the eyes for the phase boundaries. Color in the phase diagrams maps to the logarithm of the absolute value of the second derivative of magnetization with respect to field. Note: the temperature range is 2–30 K. (c) Doping dependence of spin-flop field $H_{c1}$ (circles) and saturation field $H_{c2}$ (triangles) at 2 K. Red data are taken on UW samples, whereas black data are taken on OR samples.

Figure 3

Magnetoresistance of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$. Field dependence of ${\rho }_{xx}$ for $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$ with $c$-axis magnetic fields up to 36 T [OR samples (a)–(d)], 14 T [UW samples (e)–(j)], and 31 T [UW sample (k)]. The temperature of the measurements is color-coded based on the scale bar on the right.

Figure 4

Shubnikov–de Haas oscillations of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$. Oscillatory part of resistivity for $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$ as a function of inverse field obtained by subtracting a fifth-order polynomial background.

Figure 5

Fast Fourier transform (FFT) spectrum and oscillation frequency of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$. (a)–(k) FFT spectrum of the oscillatory part of resistivity for $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$ at various temperatures. (l) Doping dependence of the peak frequency $F_s$. Red (black) dots are data taken from OR (UW) samples. The gray dashed line denotes the doping level closest to the charge-neutral point.

Figure 6

Anomalous temperature dependence of SdH oscillations of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$. (a),(b) Oscillatory part of ${\rho }_{xx}$, second derivative of ${\rho }_{xx}$, and background subtracted ${\rho }_{xy}$ for (a) electron-doped ($x=0.63$) and (b) hole-doped ($x=0.75$) $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$ at varied temperatures as a function of inverse field $1/{\mu }_{0}H$. (c),(d) Temperature dependence of peak and valley positions in the SdH oscillations for (c) $x=0.63$ and (d) $x=0.75$. (e),(f) Oscillation frequency $F_s$ as a function of temperature for (e) electron-doped $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$ ($x=0.63$) and (f) hole-doped ($x=0.75$).

Figure 7

Effective mass $m/m_e$ of $\mathrm{Mn}{\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_4$ extracted from Lifshitz-Kosevich fitting. (a)–(k) Temperature dependence of the oscillation amplitude measured near 36 T [(a)–(d) OR samples], 14 T [(e)–(g),(i)–(k) UW samples], and 31 T [(h) UW sample]. Red solid curves are fits to the Lifshitz-Kosevich factor to obtain the effective mass. (l) Doping dependence of the effective mass for OR samples (red) and UW samples (black). The gray dashed line denotes the doping level closest to the charge-neutral point.

Figure 8

Density functional theory calculations. (a) Band structure of $\mathrm{Mn}{\mathrm{Bi}}_2{\mathrm{Te}}_4$ in the FM state with magnetic moments saturated along the $c$-axis calculated by DFT. (b) Calculated oscillation frequency vs Fermi energy for the electron (red) and hole (blue) pocket. The electron pocket has two extremal orbits. The dashed (solid) line represents the $k_z\ne 0$ ($k_z=0$) extremal orbit. (c) Evolution of the Fermi surface projected onto the $k_x-k_z$ plane passing from heavily electron-doped (rightmost panel) to heavily hole-doped (leftmost panel).

Figure 9

Comparison between experiment and theory. Theoretical predictions (dashed lines) and experimental measurements of the effective mass (a) and carrier density (b) as a function of quantum oscillation frequency. Circles are for UW samples, and squares are for OR samples. In both (a) and (b), red lines represent calculations for the conduction band, and blue lines are calculations for the valence band. In (a), the red dashed line is the calculations for the $k_z\ne 0$ extremal orbit of the conduction band, while the red solid line is for the $k_z=0$ orbit of the conduction band.

Figure 10

The phase shift of quantum oscillations extracted from the Landau fan diagram: (a)–(h) Plot of the Landau level index against the inverse field, with integer Landau level indices assigned to resistivity peaks. Red solid curves are linear fits. (i) The phase shift $\Gamma$ extracted from the $y$-intercept of the linear fits in the Landau fan diagram as a function of oscillation frequency. See the main text for more details.