Topological Magnon Bands in a Kagome Lattice Ferromagnet

There is great interest in finding materials possessing quasiparticles with topological properties. Such materials may have novel excitations that exist on their boundaries which are protected against disorder. We report experimental evidence that magnons in an insulating kagome ferromagnet can have a topological band structure. Our neutron scattering measurements further reveal that one of the bands is flat due to the unique geometry of the kagome lattice. Spin wave calculations show that the measured band structure follows from a simple Heisenberg Hamiltonian with a Dzyaloshinkii-Moriya interaction. This serves as the first realization of an effectively two-dimensional topological magnon insulator—a new class of magnetic material that should display both a magnon Hall effect and protected chiral edge modes.

Figure 1

Spin wave excitations in the kagome lattice ferromagnet. (a) A localized spin wave excitation supported by the kagome lattice. Because of the kagome geometry, excited (red) spins are decoupled from the rest of the lattice because a rotation of the neighboring ground-state (black) spins does not lower the overall energy of the spin configuration. Dashed lines indicate the excited spin precession path. (b) The spin wave dispersion of the kagome lattice ferromagnet, which consists of two dispersive modes and a perfectly flat mode, corresponding to the localized excitation shown in (a). Arrows show the locations of touch points between neighboring bands along $[-K2K0]$. (c) Out-of-plane component of the Dzyaloshinskii-Moriya (DM) vectors ($D_z$) on the kagome lattice, which have the same direction and magnitude for every bond because of the lattice symmetries. (d) Dispersion of the Heisenberg ferromagnet with DM interaction on the kagome lattice with $D_z/J=0.15$. Arrows are at the same locations as in (b) and show the gaps that open between bands due to the DM interaction. (e) Topologically protected chiral magnon edge modes. Magnons propagate around the edge of the lattice in a single direction.

Figure 2

Ground-state spin configuration of Cu(1,3-bdc). (a),(b) Elastic neutron scattering scans through the magnetic Bragg peaks measured at $T=70\mathrm{mK}$ at the (a) (0 0 1) and (b) (0 0 4) Bragg positions. Application of a magnetic field parallel to the kagome plane quickly suppresses the Bragg peak at (0 0 1) and increases intensity of the Bragg peak at (0 0 4). (c),(d) Ground-state spin configuration of Cu(1,3-bdc), where spin-$1/2$ ${\mathrm{Cu}}^{2+}$ ions form a kagome lattice. (c) Zero magnetic field configuration. Spins within each plane are ordered ferromagnetically, and neighboring planes ordered antiferromagnetically. Spins point parallel to the kagome plane. (d) A small magnetic field can polarize and reorient all spins in the direction of the applied field. Here the field is applied perpendicular to the kagome plane.

Figure 3

Magnon dispersion in Cu(1,3-bdc). (a)–(c) Inelastic neutron scattering spectrum measured on LET with incident neutron energy 6.01 meV at $T=1\mathrm{K}$ and (a) zero field and applied fields (b) 2 T and (c) 7 T with the field applied perpendicular to the kagome plane. Black stripes are due to missing detectors. (d) Calculated inelastic structure factor for the Heisenberg ferromagnet with DM interaction convolved with the instrumental resolution function with parameters $J=0.6\mathrm{meV}$, $g_z=2.2$, $D_z=0.09\mathrm{meV}$, and ${\mu }_{0}H=7\mathrm{T}$, and displayed over the measurable range for the LET spectrometer. $|Q|$ is the magnitude of the momentum transfer vector parallel to the kagome plane.

Figure 4

Dependence of the magnon dispersion on the applied field direction. (a) Magnetic field applied perpendicular to the kagome plane. Data were taken on LET and integrated over the momentum transfer range $0.72{\AA }^{-1}\le \overrightarrow{|Q|}\le 0.88{\AA }^{-1}$. (b) Magnetic field applied parallel to the kagome plane. Data were taken on SPINS at $\overrightarrow{|Q|}=0.8{\AA }^{-1}$ with a $\overrightarrow{|Q|}$ resolution of $\sim 0.15{\AA }^{-1}$. (c),(d) Quantitative comparison to the deduced spin Hamiltonian. The data points in (c) correspond to the difference in intensities at ${\mu }_{0}H=7\mathrm{T}$ and ${\mu }_{0}H=2\mathrm{T}$ with the field perpendicular to the plane, and the data points in (d) correspond to the difference in intensities at ${\mu }_{0}H=7\mathrm{T}$ and ${\mu }_{0}H=0\mathrm{T}$ with the field parallel to the plane. The lines denote the calculated structure factor for the Heisenberg ferromagnet with DM interaction integrated over the same values of $\overrightarrow{|Q|}$ as the data and convolved with the instrumental resolution function, assuming $\delta$-function spin waves (blue, dashed) and damped spin waves with damping $\mathrm{\Gamma }=0.03\mathrm{meV}$ (red, solid). $\overrightarrow{|Q|}$ is the magnitude of the momentum transfer vector parallel to the kagome plane. Error bars and uncertainty values throughout this manuscript are statistical in nature and represent 1 standard deviation.