Magnon Spin Hall Magnetoresistance of a Gapped Quantum Paramagnet

Motivated by recent experimental work, we consider spin transport between a normal metal and a gapped quantum paramagnet. We model the latter as the magnonic Mott-insulating phase of an easy-plane ferromagnetic insulator. We evaluate the spin current mediated by the interface exchange coupling between the ferromagnet and the adjacent normal metal. For the strongly interacting magnons that we consider, this spin current gives rise to a spin Hall magnetoresistance that strongly depends on the magnitude of the magnetic field, rather than its direction. This Letter may motivate electrical detection of the phases of quantum magnets and the incorporation of such materials into spintronic devices.

Figure 1

Schematic of the $\mathrm{NM}|\mathrm{GQP}$ heterostructure. The field $\mathbf{B}$ is taken in the $z$ direction such that the $-z$ direction is the equilibrium axis of the spins $\mathbf{S}$. In this way, magnons in the magnetic insulator—indicated by the wavy line—carry $\hslash$ angular momentum and are excited by electron spin flips at the interface. The spin current (${\mathbf{j}}^S$) that arises via the spin Hall effect is spin polarized in the $z$ direction and flows in the $x$ direction. The charge current (${\mathbf{j}}^q$) that drives the spin Hall effect flows in the $y$ direction. The thickness of the NM is $d_N$.

Figure 2

Phase diagram of the Mott-insulating (MI) phase obtained from the decoupling approximation elaborated on in [29]. Each Mott lobe is characterized by an integer occupation $N_0$ of magnons per site. This number can be tuned, for example, by modifying the amplitude of the magnetic field. For nonzero exchange $J$, the lobes are disconnected by the superfluid (SF) phase. The dashed black line shows the values of parameters taken in Fig. 5. (Inset) Schematic phase diagram for fixed field. The temperature scale $K/k_B$ is the crossover above which the system behaves as a weakly interacting gas of magnons in the SF phase or the thermal magnon (TM) phase, whereas below this temperature, the low-lying excitations are magnonic quasiparticle and quasihole excitations.

Figure 3

Interface spin conductance as a function of spin accumulation for different values of the temperature. This calculation was performed considering $N_0=4$, $B/K=S-4$, and $J/zSK=0.05$. The temperatures are $k_{B}T/K=0.005$ (solid blue), $k_{B}T/K=0.05$ (dashed yellow), and $k_{B}T/K=0.1$ (dot-dashed green).

Figure 4

Temperature dependence of the linearized interface spin conductance normalized to the case of noninteracting magnons $G(U=0)$ in the absence of magnetic field. The different curves represent a state with a different magnon filling fraction. We considered $J/zSK=0.03$, $N_0=2$ (orange), $N_0=3$ (green), $N_0=4$ (yellow), and $N_0=5$ (blue), and the magnetic field $B/K=S-N_0$, respectively, which corresponds to the value in the middle of each of the Mott lobes. (Inset) Enlargement of the normalized conductance behavior for small temperatures (dashed rectangle in the main plot).

Figure 5

Linearized interface spin conductance as a function of magnetic field. The magnetic field takes the values shown in the dashed line on Fig. 2. As soon as the magnetic field takes values close to the edges of the Mott lobes, the conductance increases strongly. These calculations were performed considering $k_{B}T/K=0.01$ and $J/zSK=0.03$.