Magnetic-field-induced topological phase transition in Fe-doped ${\left(\mathrm{Bi},\mathrm{Sb}\right)}_2\mathrm{S}{\mathrm{e}}_3$ heterostructures

Three-dimensional topological insulators (3D TIs) possess a specific topological order of electronic bands, resulting in gapless surface states via bulk-edge correspondence. Exotic phenomena have been realized in ferromagnetic TIs, such as the quantum anomalous Hall (QAH) effect with a chiral-edge conduction and a quantized value of the Hall resistance $R_{yx}$. Here, we report on the emergence of distinct topological phases in paramagnetic Fe-doped ${\left(\mathrm{Bi},\mathrm{Sb}\right)}_2\mathrm{S}{\mathrm{e}}_3$ heterostructures with varying structure architecture, doping, and magnetic and electric fields. Starting from a 3D TI, a two-dimensional insulator appears at layer thicknesses below a critical value, which turns into an Anderson insulator for Fe concentrations sufficiently large to produce localization by magnetic disorder. With applying a magnetic field, a topological transition from the Anderson insulator to the QAH state occurs, which is driven by the formation of an exchange gap owing to a giant Zeeman splitting and reduced magnetic disorder. A topological phase diagram of ${\left(\mathrm{Bi},\mathrm{Sb}\right)}_2\mathrm{S}{\mathrm{e}}_3$ allows exploration of intricate interplay of topological protection, magnetic disorder, and exchange splitting.

Figure 1

(a)–(c) Evolution of surface spin subband structure in paramagnetic Fe-doped ${\left(\mathrm{Bi},\mathrm{Sb}\right)}_2\mathrm{S}{\mathrm{e}}_3$ thin films upon thickness reduction and external magnetic field application. (a) Surface states of spin-up (magenta) and spin-down (cyan) subbands with gap ${\mathrm{\Delta }}_{\mathrm{hy}}$ driven by intersurface hybridization, leading to 2D insulator phase. (b) Under an external perpendicular magnetic field, the spin subband degeneracy is lifted by the Zeeman effect, producing band crossing when the Zeeman energy ${\mathrm{\Delta }}_{\mathrm{Zeeman}}$ exceeds ${\mathrm{\Delta }}_{\mathrm{hy}}$. (c) At crossings of spin-up (magenta) and spin-down (cyan) bands, a gap is generated by the spin-orbit interaction, leaving a one-dimensional chiral-edge channel in the bulk band gap (red solid line). (d)–(f) Calculated electronic band structure of three-dimensional topological insulator thin films. The band structure is calculated using the 3D-TI slab for cases in which (d) ${\mathrm{\Delta }}_{\mathrm{hy}}$ is larger than ${\mathrm{\Delta }}_{\mathrm{Zeeman}}$, (e) ${\mathrm{\Delta }}_{\mathrm{hy}}$ is comparable to ${\mathrm{\Delta }}_{\mathrm{Zeeman}}$, and (f) ${\mathrm{\Delta }}_{\mathrm{Zeeman}}$ is larger than ${\mathrm{\Delta }}_{\mathrm{hy}}$. (g) ARPES intensity of the 30-nm-thick $\mathrm{F}{\mathrm{e}}_{0.05}{\left(\mathrm{B}{\mathrm{i}}_{0.34}\mathrm{S}{\mathrm{b}}_{0.66}\right)}_{1.95}\mathrm{S}{\mathrm{e}}_3$/3-nm-thick $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ film around the $\overline{\mathrm{\Gamma }}$ point measured at $T=40\mathrm{K}$. SS and VB, respectively, represent surface state and bulk valence band. $E_{\mathrm{F}}$ is indicated by a white dashed line. (h) Longitudinal resistance $R_{xx}$ of $\mathrm{F}{\mathrm{e}}_{0.05}{\left(\mathrm{B}{\mathrm{i}}_{0.34}\mathrm{S}{\mathrm{b}}_{0.66}\right)}_{1.95}\mathrm{S}{\mathrm{e}}_3/\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ at $B=0$ as a function of the magnetic layer thickness $d$ and temperature $T$. Color bar scale is in units of $h/e^2$. (i) Magnetoresistance of $\mathrm{F}{\mathrm{e}}_{0.05}{\left(\mathrm{B}{\mathrm{i}}_{0.34}\mathrm{S}{\mathrm{b}}_{0.66}\right)}_{1.95}\mathrm{S}{\mathrm{e}}_3/\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ at 2 K as a function of $d$ and $B$. Color scale bar corresponds to +20% (blue) to −100% (red).

Figure 2

Hall resistance of Fe-doped and nonmagnetic heterostructures against Sb composition $y$ at 9 T and 2 K. Hall resistances $R_{yx}$ of 14-nm-thick $\mathrm{F}{\mathrm{e}}_x{\left(\mathrm{B}{\mathrm{i}}_{1-y}\mathrm{S}{\mathrm{b}}_y\right)}_{2-x}\mathrm{S}{\mathrm{e}}_3$/3-nm-thick $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ (red circles) dominated by the anomalous component and those of 14–20-nm-thick nonmagnetic ${\left(\mathrm{B}{\mathrm{i}}_{1-y}\mathrm{S}{\mathrm{b}}_y\right)}_2\mathrm{S}{\mathrm{e}}_3$/3-nm-thick $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ films (black squares) are shown against Sb content $y$. CNP denotes the charge neutral point of nonmagnetic samples. The inset shows the Hall effect measurement for $\mathrm{F}{\mathrm{e}}_x{\left(\mathrm{B}{\mathrm{i}}_{0.33}\mathrm{S}{\mathrm{b}}_{0.67}\right)}_{2-x}\mathrm{S}{\mathrm{e}}_3$/3-nm-thick $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ films for Fe content $x=0$ (solid black line), 0.04 (orange), and 0.05 (red).

Figure 3

Magnetic-field-induced insulator-metal transition and quantum anomalous Hall effect. (a) Longitudinal sheet resistance $R_{xx}$($T$) of 14-nm-thick $\mathrm{F}{\mathrm{e}}_{0.05}{\left(\mathrm{B}{\mathrm{i}}_{0.33}\mathrm{S}{\mathrm{b}}_{0.67}\right)}_{1.95}\mathrm{S}{\mathrm{e}}_3/\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ heterostructure (inset) in various magnetic fields $B$. (b) Hall and sheet resistances, $R_{yx}$($B$) (solid red line) and $R_{xx}$($B$) (solid black line), measured for a FET device (shown in the inset) at $V_{\mathrm{G}}=-60\mathrm{V}$ and $T=1.6\mathrm{K}$. The $R_{yx}$ data below 1 T were not obtained owing to the large value of $R_{xx}$.

Figure 4

Magnetic-field-induced insulator-QAH phase transition. (a),(b) Temperature dependences of the longitudinal and Hall conductivities (${\sigma }_{xx}$ and ${\sigma }_{xy}$) at $V_{\mathrm{G}}=-60\mathrm{V}$ for various magnetic fields $B$. (c) Renormalization group flow in the $[{\sigma }_{xx}\left(T\right),{\sigma }_{xy}\left(T\right)]$ plane in various $B$ from 3 to 15 T. Each curve is extracted from data in (a),(b) employing the same color code. Empty and filled circles, respectively, present data obtained at $T=15$ and 1.6 K. The $[{\sigma }_{xx},{\sigma }_{xy}]$ flow extracted from the data as a function of the magnetic field at $T=1.6\mathrm{K}$ is also shown (solid black line).