Enhancement of Curie Temperature in Ferromagnetic Insulator-Topological Insulator Heterostructures

TL;DR

This work develops a minimal Dirac-fermion model for TI surface states coupled to a dense 2D FMI lattice and shows that Bloembergen-Rowland exchange from interband virtual transitions dominates over RKKY at short distances, producing an out-of-plane ferromagnetic order and a Curie-temperature enhancement proportional to the uniform Van Vleck susceptibility $\chi_{VV}$. The analysis yields explicit forms for the exchange couplings, demonstrates a gapped spin-wave spectrum, and provides a mean-field estimate $\delta T_c = \frac{J_0 S(S+1)}{3k_B} \chi_{VV}$, with $\chi_{VV}$ computed from BR processes. Thickness effects due to hybridization between TI surfaces are quantified, showing that 4 QL BST enhances BR interactions relative to 3 QL BST but that hybridization weakens the BR strength compared to thick films; these trends are consistent with experiments on Cr$_2$Te$_3$-(Bi,Sb)$_2$Te$_3$. The results supply a simple, quantitative framework for understanding and tuning Tc enhancements in FMI-TI heterostructures with potential relevance for spintronics and topological magnetism.

Abstract

We theoretically analyze the topological insulator (TI) surface state mediated interactions between local moments in a proximate 2D ferromagnetic insulator (FMI) motivated by recent experiments that show a significant increase in the Curie temperature Tc of FMI-TI heterostructures. Such interactions have been investigated earlier with a focus on dilute magnetic dopants in TIs. Our problem involves a dense set of moments for which we find that the short range Bloembergen-Rowland interaction, arising from virtual particle-hole transitions between the valence and conduction bands, dominates over the oscillatory Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. We show that the Tc enhancement is proportional to the Van Vleck susceptibility and that the spin-momentum locking of surface states leads to out-of-plane ferromagnetic order in the FMI. We investigate how the hybridization between top and bottom surfaces in a thin TI film impacts Tc enhancement, and show how our results can help understand recent experiments on atomically thin Cr2Te3-(Bi,Sb)2Te3.

Figures (5)

• Figure 1: Schematic illustration of the effective exchange interaction between local moments mediated by Dirac electrons. (Left) When the chemical potential $\mu$ lies in the conduction band, the resulting interaction is the oscillatory RKKY type, decaying as a power law at large distances $k_F R \gg 1$. (Right) When the chemical potential lies inside the gap ($\mu=0$), the exchange is short-ranged, corresponding to the Bloembergen--Rowland (BR) interaction that decays exponentially with the characteristic length scale $R_\Delta = \hbar v_F / \Delta$. Here, $W$ denotes the bandwidth, and $\Delta$ is the exchange gap.
• Figure 2: Coupling strengths in units of $J_0=J^2/W$ vs $R/a$. Here, $\Delta=30$ meV in panels (b) & (d) and $\mu=60$ meV in panels (c) & (d). We note that in panels (a) & (c) the red and blue curves coincide since $J_{zz}=J_{\parallel}$ for $\Delta=0$.
• Figure 3: (a) $J^{tt}_{zz}(R)$ in units of $J_0$ between two local moments on the top surface for 4 QL of BST (red, $\hbar v_F=2.43$ eV$\cdot$Å, $m_0=29$ meV, $m_1=12.9$ eV$\cdot$Å$^2$) and 3 QL of BST (black, $\hbar v_F=2.43$ eV$\cdot$Å, $m_0=-44$ meV, $m_1=37.3$ eV$\cdot$Å$^2$). (b) Dependence of $\chi^{tt}_{zz}(\boldsymbol{q}=0)\equiv\chi^{tt}_{VV}$ on $m_0$, $\hbar v_F=2.43$ eV$\cdot$Å, $m_1=12.9$ eV$\cdot$Å$^2$. The black line at $m_0=0$ is the point of band inversion. Parameters from ref. Wang2015.
• Figure 4: (a) Spin-wave spectrum $\omega/\omega_0$ (with $\omega_0 = S J_0$) as a function of $qa$ near $\boldsymbol{q}=0$ for different gap values: $\Delta = 0~\text{meV}$ (red circles), $\Delta = 10~\text{meV}$ (gray squares), $\Delta = 20~\text{meV}$ (blue triangles), and $\Delta = 30~\text{meV}$ (purple diamonds). The dispersion follows $\omega(q) \!\approx\! K + A q^2$ with $K = f(\Delta/W)\,\omega_0$. (b) The dimensionless parameter $f(\Delta/W) = K/\omega_0$ as a function of $\Delta$.
• Figure 5: $J_{zz}(R=a)$ versus $T$ for a thick TI ($\Delta=0$, gapless Dirac surface) (shown in red), 4 QL BST (shown in cyan) and 3 QL BST (shown in black). The temperature variation of the exchange coupling is less than a percent and not visible on this plot.