Bulk and edge spin transport in topological magnon insulators

We investigate the spin transport properties of a topological magnon insulator, a magnetic insulator characterized by topologically nontrivial bulk magnon bands and protected magnon edge modes located in the bulk band gaps. Employing the Landau-Lifshitz-Gilbert phenomenology, we calculate the spin current driven through a normal $\mathrm{metal}|\mathrm{topological}$ magnon $\mathrm{insulator}|\mathrm{normal}$ metal heterostructure by a spin accumulation imbalance between the metals, with and without random lattice defects. We show that bulk and edge transport are characterized by different length scales. This results in a characteristic system size where the magnon transport crosses over from being bulk dominated for small systems to edge dominated for larger systems. These findings are generic and relevant for topological transport in systems of nonconserved bosons.

Figure 1

Spin transport in a topological magnon insulator (TMI) of length $l$, driven by the difference in spin accumulation $\mathbf{\mu }$ in the attached normal metal (NM) leads. Apart from the spin current $I^{\mathrm{bulk}}$ carried by bulk magnons, in the TMI there is also a current $I^{\mathrm{edge}}$ carried by topologically protected edge excitations. This edge current circles the system boundaries in a chiral fashion and is robust to perturbations [1].

Figure 2

(a) The honeycomb lattice structure, consisting of two interpenetrating triangular sublattices with lattice constant $a$. The DMI changes sign depending on whether there is a right or left turn between next-nearest neighbors. (b) A typical magnon edge state with frequency $\omega =2.73J$ in the bulk band gap, obtained by diagonalizing the inverse magnon propagator (4) for vanishing damping and spin pumping (${\alpha }_i=0={\mu }_i$) on a finite lattice with 930 lattice sites. Spin, magnetic field, and DMI are fixed to $S=1$, $H=0.1J$, and $D=0.2J$ in the plot.

Figure 3

Spin currents ejected at the right lead as a function of the system length $l$, for temperature $T=0.8J$, bulk Gilbert damping $\alpha =2.5\times {10}^{-3}$, and different average defect concentrations $w$. Bulk and edge contributions are depicted as blue squares and yellow circles, respectively. The corresponding straight lines are exponential fits according to Eq. (7), with the spin diffusion lengths denoted in the plots. The defect concentrations are (a) $w=0$ (clean system), (b) $w=0.05$, (c) $w=0.1$, and (d) $w=0.15$. Deviations from the exponential fit for $l\lesssim 25a$ signal the crossover to the thin-film regime in which the spin currents decay algebraically [35, 41].

Figure 4

Ratio of edge to bulk current injected at the left lead as a function of temperature $T$, for bulk Gilbert damping $\alpha =5\times {10}^{-3}$ and average defect concentrations $w=0$ (green squares) and $w=0.1$ (red circles). The corresponding lines are fits with Eq. (8).

Figure 5

(a) Schematic depiction of the contribution of edge and bulk magnons to the total spin current, with the crossover length scale $l^{*}$. Note that $l^{*}\to \infty$ for clean systems [see Fig. 3]. (b) Numerically calculated crossing points $l^{*}$ (points) and the order of magnitude estimate presented in the text (solid line). Parameters are the same as in Fig. 3.