Comparing magnetic ground-state properties of the V- and Cr-doped topological insulator ${\left(\mathrm{Bi},\mathrm{Sb}\right)}_2{\mathrm{Te}}_3$

An insulating ferromagnetic ground state is a fundamental prerequisite for the quantum anomalous Hall (QAH) effect observed in magnetically doped topological insulators such as ${(\text{Bi},\mathrm{Sb})}_2{\mathrm{Te}}_3$. So far, the QAH effect could only be induced by V and Cr doping, with V resulting in ferromagnetism with a higher $T_C$ and a more robust QAH state. To better understand the difference between the two dopants, we have investigated epitaxial ${\mathrm{V}}_{0.1}{\mathrm{Sb}}_{1.9}{\mathrm{Te}}_3$ and ${\mathrm{Cr}}_{0.1}{\left({\mathrm{Bi}}_{0.1}{\mathrm{Sb}}_{0.9}\right)}_{1.9}{\mathrm{Te}}_3$ films using x-ray absorption spectroscopy and x-ray magnetic circular dichroism. Our analysis of the V and Cr $L_{2,3}$ spectra by multiplet ligand-field theory goes beyond existing studies by allowing several charge-transfer states. We find a strongly covalent ground state, dominated by the superposition of one and two Te-ligand-hole configurations, with a negligible contribution from ionic ${\mathrm{V}}^{3+}$ or ${\mathrm{Cr}}^{3+}$. Crucial for a comparison with theoretical models are the resulting $d$-shell fillings ($n_d^{\text{V}}=3.47$ and $n_d^{\text{Cr}}=4.33$), and spin ($m_{\text{spin}}^{\text{V}}=2.39{\mu }_{\text{B}}$ and $m_{\text{spin}}^{\text{Cr}}=3.22{\mu }_{\text{B}}$) and orbital ($m_{\text{orb}}^{\text{V}}=-0.55{\mu }_{\text{B}}$ and $m_{\text{orb}}^{\text{Cr}}=-0.03{\mu }_{\text{B}}$) magnetic moments, with the total magnetic moments being in good agreement with published magnetometry results. Our findings indicate the importance of the Te $5p$ states for the ferromagnetism in ${(\text{Bi},\mathrm{Sb})}_2{\mathrm{Te}}_3$ and favor theories involving $pd$-exchange.

Figure 1

(a) Crystal structure of V/Cr-doped ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ consisting of stacked quintuple layers along the $c$ axis (Te-Sb-Te-Sb-Te), separated by van der Waals gaps. (b) The V/Cr impurity substitutes Sb, which is surrounded by six Te atoms. The impurity atom has a slightly trigonally distorted $O_h$ symmetry. The gray sticks are shown to highlight the octahedron and are not generally representative of chemical bonds. (c) Schematics of the experimental geometry for the XMCD measurements. The polarized and monochromatized photons are absorbed by the sample in a magnetic field applied parallel to the direction of the incoming beam. The x-ray absorption spectra are obtained by measuring the drain current (TEY).

Figure 2

Schematic diagram showing the on-site energies of the various configurations for the initial state and XAS final state when the hybridization ($V_{e_g}$) is zero.

Figure 3

Experimental and calculated $L_{2,3}$ XAS and XMCD spectra. (a) ${\mathrm{Cr}}_{0.1}{\left({\mathrm{Bi}}_{0.1}{\mathrm{Sb}}_{0.9}\right)}_{1.9}{\mathrm{Te}}_3$ film. (b) ${\mathrm{V}}_{0.1}{\mathrm{Sb}}_{1.9}{\mathrm{Te}}_3$ film. The top panels show experimental x-ray absorption data, obtained as described in Sec. 2 (left-circular polarization $I_{\text{left}}$, light blue line; right-circular polarization $I_{\text{right}}$, dark red line; averaged over both polarizations $I_{\text{avg}}$, bold black line). Polarization-averaged spectra calculated by the MLFT cluster model described in the text are shown as a dashed-dotted line; to facilitate the comparison with experiment, in (a) we show Cr data (thin black line) corrected for the Te contribution using a Cr-free reference sample (${\mathrm{Te}}_{\text{ref}}$, dotted gray line). The bottom panels show the corresponding normalized experimental ($I_{\text{XMCD}}$, solid green line) and calculated XMCD spectra (dashed-dotted green line). The insets show the contributions of different configurations to the ground state. The dashed vertical lines are drawn as a guide to the eye, highlighting the position of particular features in the spectra. The peak intensity of the calculated XAS spectrum was normalized to one.

Figure 4

Temperature dependence of the remanent (10 mT) XMCD signal at the V (red circles) and Cr (blue circles) $L_3$ edges, from which the corresponding Curie temperatures can be estimates as $T_C^{\text{V}}\sim 45$ K and $T_C^{\text{Cr}}\sim 20$ K. The XMCD signal at $T=2$ K has been normalized to 1.0.

Figure 5

Sum-rule analysis for the V-doped ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ thin film. (a) Left- and right-circularly polarized XAS spectra of the V $L_{2,3}$ edges, obtained after the background correction described in Sec. 2 ($I_{\text{left}}^{\text{res}}$, solid, light blue line, and $I_{\text{right}}^{\text{res}}$, solid dark red line), along with the corresponding XMCD data (solid green line, lower panel). The dashed lines show the total integrated XAS and XMCD spectral weight, respectively. The arrows mark the values of $r, p$, and $q$ used in Eqs. (5) and (6). (b) and (c) Distribution of $m_{\text{orb}}$ and $m_{\text{spin}}$, respectively, obtained by applying the sum-rule analysis 8000 times, as described in the main text.

Figure 6

$d$-shell electron occupation $n_d$ (blue squares) and expectation value of the orbital angular momentum $\langle L_z\rangle$ (red circles) of (a) V and (b) Cr as functions of the charge-transfer energy $\mathrm{\Delta }$.

Figure 7

XAS and XMCD spectra measured around the Sb $M_{4,5}$ edges ($3d\to 5p$) of (a) ${\mathrm{Cr}}_{0.1}{\left({\mathrm{Bi}}_{0.1}{\mathrm{Sb}}_{0.9}\right)}_{1.9}{\mathrm{Te}}_3$ and (b) ${\mathrm{V}}_{0.1}{\mathrm{Sb}}_{1.9}{\mathrm{Te}}_3$. Top panel: XAS spectra for left- (light blue curve) and right- (dark red curve) circularly polarized x rays. Bottom panel: Normalized XMCD spectra (green curve).

Figure 8

Averaged XAS (black curve) measured at the Te $M_{4,5}$ edges ($3d\to 5p$) of ${\mathrm{V}}_{0.1}{\mathrm{Sb}}_{1.9}{\mathrm{Te}}_3$ film. The green curve shows the corresponding XMCD spectrum $(I_{\text{left}}-I_{\text{right}})$ measured at $T=2$ K and $H=3$ T. The inset illustrates the relative magnetic moment orientations of the TM ion and neighboring Te and Sb atoms within the one quintuple layer.