Chiral magnonic edge states in ferromagnetic skyrmion crystals controlled by magnetic fields

Achieving control over magnon spin currents in insulating magnets, where dissipation due to Joule heating is highly suppressed, is an active area of research that could lead to energy-efficient spintronics applications. However, magnon spin currents supported by conventional systems with uniform magnetic order have proven hard to control. An alternative approach that relies on topologically protected magnonic edge states of spatially periodic magnetic textures has recently emerged. A prime example of such textures is the ferromagnetic skyrmion crystal which hosts chiral edge states providing a platform for magnon spin currents. Here we uncover that an external magnetic field can drive a topological phase transition in the spin-wave spectrum of a ferromagnetic skyrmion crystal. The topological phase transition is signaled by the closing of a low-energy bulk magnon gap at a critical field. In the topological phase, below the critical field, two topologically protected chiral magnonic edge states lie within this gap, but they unravel in the trivial phase, above the critical field. Remarkably, the topological phase transition involves an inversion of two magnon bands that at the $\mathrm{\Gamma }$ point correspond to the breathing and counterclockwise modes of the skyrmions in the crystal. Our findings suggest that an external magnetic field could be used as a knob to switch on and off magnon spin currents carried by topologically protected chiral magnonic edge states.

Figure 1

Ferromagnetic skyrmion crystal texture. Triangular crystal of ferromagnetic Néel skyrmions obtained from the classical ground-state texture of the spin-lattice Hamiltonian model for $D/J=1.0$ and $b=0.8$. The out-of-plane component of the moments is color coded.

Figure 2

Magnetic-field dependence of the magnon spectrum. (a)–(d) Bulk magnon spectra of the FM SkX along the first Brillouin zone loop $K\text{−}\mathrm{\Gamma }\text{−}M\text{−}K$ for increasing values of the applied magnetic field $b=g{\mu }_{\text{B}}B/JS$. Nearly flat (dispersive) bands are depicted in purple (gray). The counterclockwise (A), breathing (B), and clockwise (C) modes are respectively denoted in blue, red, and green. (e) Localized skyrmion distortions corresponding to the nearly flat bands from (a)–(d), labeled by their azimuthal number $m$.

Figure 3

Magnetic-field-driven topological phase transition. (a)–(c) Bulk magnon spectrum of the FM SkX in the vicinity of the topological phase transition for (a) $b=0.80$, (b) $b=b_c=0.88$, and (c) $b=0.95$. The counterclockwise (A) and breathing (B) modes, shown respectively in blue and red, get inverted. (d) Chern numbers of the three lowest energy bands $\{C_1,C_2,C_3\}$ as functions of the magnetic field. Here $C_3$ exhibits a sudden decrease from one to zero as the magnetic field increases past $b_c$, rendering the third band topologically trivial. (e) Size of the third gap ${\mathrm{\Delta }}_3$ as a function of the magnetic field, which vanishes at $b_c$ and grows linearly in the vicinity of the topological phase transition. (f) and (g) Snapshots of the time evolution of the out-of-plane magnetization of the counterclockwise and breathing modes, respectively. Here $T$ denotes the period of both spin-wave modes, for simplicity.

Figure 4

Robustness of magnonic edge states against disorder. One-dimensional magnon spectra are shown on an infinite strip for (a) pristine FM SkX at $b=0.80$, (b) disordered FM SkX at $b=0.80$, (c) pristine FM SkX at $b=0.95$, and (d) disordered FM SkX at $b=0.95$. Bulk subbands are shown in gray and $G_{\mathrm{s}}$ is the size of the one-dimensional Brillouin zone. (a) and (b) For $b<b_c$, the pristine FM SkX supports topologically protected chiral magnonic edge states within the third gap [right (left) moving in blue (red)] which are stable against disorder, while the topologically trivial edge states (purple) are not. (c) and (d) For $b>b_c$, the topologically trivial edge states within the third gap (purple) of the pristine FM SkX disappear upon introducing disorder. Edge states within higher-energy gaps are indicated in black.

Figure 5

Propagating nearly flat magnon modes. Snapshots of nearly flat magnon modes of the ferromagnetic skyrmion crystal with (a) elliptical distortion, $m=2$, and (b) triangular distortion, $m=3$. For both panels, $\mathit{k}=\frac{1}{2}{\mathit{k}}_M$, where ${\mathit{k}}_M$ is a vector in the first Brillouin zone directed from the $\mathrm{\Gamma }$ to the $M$ point. The phase fronts are parallel to the dashed white line and simply correspond to locally distorted skyrmions rotated about their centers by the same angle.

Figure 6

Magnetic texture reconstruction near the strip edges. (a) Pristine system with a uniform external magnetic field $b=0.80$. (b) Disordered system with an external magnetic field $b=0.80$ everywhere except along the five rows of spins closest to the edge where it was increased to $b=3.0$.

Figure 7

Probability density of chiral magnonic edge states. (a) Magnetic unit cell of the pristine ferromagnetic skyrmion crystal in the strip geometry at $b=0.80$. Only the magnetic texture near the top and bottom edges is shown. (b) Probability density $|{\mathrm{\Psi }}_{L,R}{|}^2$ of the left-moving (red) and right-moving (blue) magnonic edge states. Both states have the same energy with $k_x=0.45G_{\mathrm{s}}$ for the right-moving state and $k_x=-0.45G_{\mathrm{s}}$ for the left-moving state, where $G_{\mathrm{s}}$ is the size of the one-dimensional Brillouin zone.

Figure 8

Magnon spectrum computed for a ferromagnetic skyrmion crystal supported by a square lattice of spins at $b=0.52$. Bulk magnon bands come in partially degenerate pairs since the magnetic unit cell comprises twice as many spins as the triangular lattice considered in the main text. The counterclockwise (blue) and breathing (red) modes are naturally still present at the $\mathrm{\Gamma }$ point.

Figure 9

Magnonic edge states. One-dimensional magnon spectra of a ferromagnetic skyrmion crystal are shown on an infinite strip supported by a square lattice of spins in (a) the topological phase at $b=0.52$ and (b) the trivial phase at $b=0.60$. Topologically protected chiral magnonic edge states [right moving (blue) and left moving (red)] are present within the third gap in the topological phase (a). Topologically trivial edge states (purple) can be seen within the second magnon gap, which could be removed by disorder. Only topologically trivial edge states (purple) are present in the energy window of the trivial phase shown in (b).