Massive Dirac fermions and strong Shubnikov–de Haas oscillations in single crystals of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ doped with Sm and Fe

Topological insulators (TIs) are emergent materials with unique band structure, which allow the study of quantum effect in solids, as well as contribute to high-performance quantum devices. To achieve the better performance of TIs, here, we present a codoping strategy using synergistic rare-earth (RE) Sm and transition-metal Fe dopants in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ single crystals, which combine the advantages of both a transition-metal-doped TI [high ferromagnetic ordering temperature and observed quantum anomalous Hall effect (QAHE)], and a RE doped TI (large magnetic moments and significant spin-orbit coupling). In the as-grown single crystals, clear evidences of ferromagnetic ordering were observed. The angle-resolve photoemission spectroscopy indicates the ferromagnetism opens a ∼44 meV band gap at the surface Dirac point. Moreover, the mobility of the carriers at 3 K is $\sim 7400\mathrm{c}{\mathrm{m}}^2/\mathrm{Vs}$, and we thus observed an ultra-strong Shubnikov–de Haas oscillation in the longitudinal resistivity, as well as the Hall steps in transverse resistivity <14 T. Our transport and angular-resolved photoemission spectroscopy results suggest that the RE and transition metal codoping in the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ system is a promising avenue to implement the QAHE, as well as harnessing the massive Dirac fermion in electrical devices.

Figure 1

The magnetic properties of Sm and Fe dual doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ single crystal flake. (a) The hysteresis loops measured at certain temperatures from 3 to 300 K, obtained via subtracting the diamagnetic background of original magnetic hysteresis curves [shown in the inset of (a)]. Note that the magnetization values are shown in moment of a “SmFe” unit. (b) The temperature-dependent magnetization from 3 to 300 K at different applied magnetic fields. Note that the magnetization values of 0.01 T data are plotted in three times to show the details.

Figure 2

Density functional theory (DFT) calculations for SFBS. (a) The chemical structure is shown for a supercell containing isolated Sm and Fe defects, superimposed with the spin difference electron density within a $\left(10\overline{1}\right)$ plane. Both Sm and Fe carry large magnetic moments (red), where fainter induced moments appear on the Se atoms coordinating the Fe. (b) The partial electronic density of states (PDOS) for the frozen $4f$-core calculation shows a high degree of spin polarization at the Fermi level. (b) The PDOS for the GGA + $U 4f$ valence calculation also indicates a similar half-metallic character.

Figure 3

Angle-resolved photoelectron spectroscopy (ARPES) study of SFBS. (a) Photon-energy dependence of the normal-emission ARPES intensity near $E_F$ as a function of wave vector along the $z$ direction with corresponding photon energies on top. (b) Energy distribution curves measured with $hv=60\mathrm{eV}$, allowing determination of the conduction band (CB) minimum and valence band (VB) maximum. (c) ARPES intensity measured with $hv=63\mathrm{eV}$ around the $\overline{\mathrm{\Gamma }}$ point. (d) High-resolution data of (c) but focused on the near-$E_F$ region. (e) Intensity maxima of ARPES spectra, measured from peak maxima of momentum distribution curves of CB (black) and surface state (SS; red).

Figure 4

Transport properties of SFBS single crystal. (a) The temperature-dependent resistivity of SFBS single crystal. Inset: A sketch of the measurement configurations. (b) Calculated density and mobility of carriers as a function of temperature extracted from the Hall and resistivity measurements. (c) The magnetoresistance (MR) of SFBS single crystal, displaying ultra-strong Shubnikov–de Haas (SdH) oscillations. (d) The pure SdH oscillation patterns obtained from MR data after subtracting a continuous background term. Inset: Lifshitz-Kosevich (LK) formula fitting for effective mass. (e) The oscillation patterns in calculated longitudinal conductivity and (f) Landau fan diagram. (g) The zoom-in plot of 3 K Hall effect curve and MR curve in −4 to −14 T regime, in which the steplike effect can be found in the Hall effect curve.

Figure 5

The quantum oscillations in SFBS single crystals. (a) The angular dependence of Shubnikov–de Haas (SdH) oscillations, in which the oscillation frequencies are calculated and summarized in (b).