Anomalous Spin Response and Virtual-Carrier-Mediated Magnetism in a Topological Insulator

We present a comprehensive theoretical study of the static spin response in HgTe quantum wells, revealing distinctive behavior for the topologically nontrivial inverted structure. Most strikingly, the $\mathbf{q}=0$ (long-wavelength) spin susceptibility of the undoped topological-insulator system is constant and equal to the value found for the gapless Dirac-like structure, whereas the same quantity shows the typical decrease with increasing band gap in the normal-insulator regime. We discuss ramifications for the ordering of localized magnetic moments present in the quantum well, both in the insulating and electron-doped situations. The spin response of edge states is also considered, and we extract effective Landé $g$ factors for the bulk and edge electrons. The variety of counterintuitive spin-response properties revealed in our study arises from the system’s versatility in accessing situations where the charge-carrier dynamics can be governed by ordinary Schrödinger-type physics; it mimics the behavior of chiral Dirac fermions or reflects the material’s symmetry-protected topological order.

Popular Summary

Ordinary materials are classified as either metals, semiconductors, or insulators. But a recently discovered new materials class called topological insulators intriguingly combines aspects of all three: They have metallic surfaces and are insulating in the bulk, and the most paradigmatic realizations of such systems occur in semiconductor nanostructures. Topological insulators are known to exhibit very unusual electric properties. Here, we reveal their magnetism to be of an exotic type as well, opening up avenues for creating novel types of magnets.

In addition to carrying electric charge, electrons also have an internal magnetic degree of freedom called spin that causes them to behave like tiny permanent magnets. Like compass needles, the spins of electrons tend to align with an applied magnetic field and also with each other. These effects give rise to paramagnetism and ferromagnetism, two well-understood spin-related phenomena exhibited by electrons in ordinary materials. Investigating the analogous magnetic properties of topological insulators, we find unconventional magnetism originating from a peculiar quantum-physical effect. Hiding under the blanket of the uncertainty principle, electrons can make transitions across the narrow band gap in the topological insulator that would normally be forbidden by energy conservation, and the special character of these virtual transitions in a topological insulator gives rise to the material’s unusual magnetic behavior.

Our findings are obtained from calculating the static spin susceptibility of the paradigmatic topological insulator formed in mercury-telluride quantum wells. Besides describing how electrons in the material will mediate the alignment between deliberately introduced spin-carrying defects, we also characterize paramagnetism of conduction electrons in the doped topological insulator by deriving effective gyromagnetic ratios, revealing opposite trends in their dependence on physical parameters to those seen in ordinary materials.

Figure 1

(a) Mean-field critical temperature $T_{\mathrm{C}(\mathrm{i})}$ for virtual-carrier-mediated magnetism in a HgTe quantum well plotted as a function of the band-gap parameter ${\xi }_{\mathrm{M}}$. Note the striking asymmetry between the topological (${\xi }_M<0$) and normal (${\xi }_M>0$) regimes. An explicit expression for $T_0$ is given in the text; it contains the local exchange coupling between magnetic-impurity orbitals and HgTe quantum-well basis states, which is roughly constant across the transition [21]. (b) Critical temperature $T_{\mathrm{C}(\mathrm{d})}$ for the electron-doped HgTe quantum well plotted as a function of ${\xi }_{\mathrm{M}}$ and for various charge-carrier densities [corresponding to indicated values for the Fermi wave vector $k_{\mathrm{F}}$, where $k_{\mathrm{F}}=q_0$ corresponds to a charge density $n_0\equiv q_0^2/(2\pi )=4.25\times {10}^{12}{\mathrm{cm}}^{-2}$ for a typical structure 22]. We used ${\xi }_{\mathrm{D}}=-0.7$ for the BHZ-model electron-hole-asymmetry parameter in both plots, and $\gamma =-2.22$ as relevant for the (Hg,Cd,Mn)Te system [21].

Figure 2

Prefactors $C_x^2$ and $C_y^2$ determining the magnitude of intrinsic in-plane and out-of-plane spin response, respectively [see Eq. (1)], plotted as a function of the parameter $\gamma$ from the definition of the effective Kane-model spin operator (7).

Figure 3

(a) Intrinsic real-space spin susceptibility ${\overline{\chi }}_{zz}(R)$ plotted as a function of $R{q}_0$ for several values of the gap parameter ${\xi }_{\mathrm{M}}$, with fixed ${\xi }_{\mathrm{D}}=-0.7$ and $\gamma =-2.22$. (b) Parameter domains of preferential alignments between localized Mn spins having a distance $R$, mediated by the intrinsic spin response of an HgTe quantum-well system. The blue (white) domain favors a ferromagnetic (antiferromagnetic) out-of-plane spin alignment. In the yellow region, the two localized spins tend to align ferromagnetically in-plane.

Figure 4

Effective $g$ factors associated with (a) the in-plane and (b) the out-of-plane response of an electron-doped HgTe quantum well, plotted as a function of the gap parameter ${\xi }_{\mathrm{M}}$ for various levels of doping and with ${\xi }_{\mathrm{D}}=-0.7$, $2\kappa /g_{*}=-0.5$. (Note that $k_{\mathrm{F}}=q_0$ corresponds to a charge density $n_0\equiv q_0^2/(2\pi )=4.25\times {10}^{12}{\mathrm{cm}}^{-2}$ for a typical structure [22].)

Figure 5

Effective $g$ factor characterizing the response of helical edge states to an in-plane magnetic field.

Figure 6

Intrinsic spin susceptibilities (a) ${\overline{\chi }}_{zz}(\gamma ;q)$ and (b) ${\overline{\chi }}_{xx}(\gamma ;q)$ as a function of $q/q_0$ for various values of ${\xi }_{\mathrm{M}}=(-0.3,-0.2,-0.1,0,0.1,0.2,0.3)$ (from top to bottom) with ${\xi }_{\mathrm{D}}=-0.7$, ${\mathcal{C}}_{\mathrm{LH}}=0.4$, and $\gamma =-2.22$. The inset in (a) displays the normalized spin susceptibility ${\overline{\chi }}_{zz}(\gamma ;q)/{\overline{\chi }}_{zz}(\gamma ;0)$ in the small $q/q_0$ region for ${\xi }_{\mathrm{M}}=(0.1,0.2,0.3)$, which shows more clearly that ${\overline{\chi }}_{zz}(\gamma ;q)>{\overline{\chi }}_{zz}(\gamma ;0)$ in the small-$q$ limit in the normal regime also.

Figure 7

Local intrinsic spin susceptibilities ${\overline{\chi }}_{zz}(R)$ (solid orange curve) as a function of $R{q}_0$ for (a) ${\xi }_{\mathrm{M}}=0.1$ and (b) ${\xi }_{\mathrm{M}}=-0.1$ with ${\xi }_{\mathrm{D}}=-0.7$, ${\mathcal{C}}_{\mathrm{LH}}=0.4$, and $\gamma =-2.22$. The dashed green curve shows the modeled behavior in Eq. (b1) with coefficients $c=1$ for (a) and $c=1.2$ for (b). For comparison, we also show the $R^{-3}$ decay for a gapless Dirac system represented by the dot-dashed blue curve. The sharp drop of the orange curve in (a) is due to a sign change of ${\overline{\chi }}_{zz}(R)$ at about $R{q}_0\approx 8$.

Figure 8

Normalized spin susceptibilities ${\overline{\chi }}_{zz}(\gamma ;q)$ (a,c) and ${\overline{\chi }}_{xx}(\gamma ;q)$ (b,d) as a function of $q/(2k_{\mathrm{F}})$ for various levels of doping. (a,b) are the results for ${\xi }_{\mathrm{M}}=-0.2$ and (c,d) for ${\xi }_{\mathrm{M}}=0.2$, with ${\xi }_{\mathrm{D}}=-0.7$, ${\mathcal{C}}_{\mathrm{LH}}=0.4$, and $\gamma =-2.22$. The insets of (a,c) show the quadratic $\tilde{q}=q/q_0$ dependence close to $q=0$.