Interaction of gapless spin waves and a domain wall in an easy-cone ferromagnet

We consider a quasi-one-dimensional ferromagnet along the $x$-axis with the potential energy given by

where $A>0$ is the exchange coefficient, $K>0$ is the effective first-order uniaxial anisotropy, and $K^{\prime}>0$ is the second-order uniaxial anisotropy [20, 21, 22, 23]. Here, $K=K_{\text{u}}-\mu_{0}M_{\text{s}}^{2}/2$ includes the dipolar-induced shape anisotropy as well as the magnetocrystalline anisotropy $K_{u}$. Note that the first-order anisotropy favors the magnetization along the $z$-axis, while the second-order anisotropy tends to tilt the magnetization away from the $z$-axis. To parametrize the competition of these two effects, we define a dimensionless number $\kappa=K/(2K^{\prime})$. The condition for the system to be an easy-cone magnet is given by $\kappa<1$ [21], which will be assumed throughout the paper. The potential energy possesses two distinct symmetries. Firstly, it is invariant under time-reversal $\mathbf{m}\mapsto-\mathbf{m}$, showing the Z₂ symmetry of the system. Secondly, it is invariant under global rotations of the magnetization $\mathbf{m}(x)\mapsto\hat{R}_{z}(\varphi)\mathbf{m}(x)$, where $\hat{R}_{z}(\varphi)$ is a three-dimensional rotation matrix about the $z$-axis with angle $\varphi$, showing the U(1) spin-rotational symmetry of the system.

Given the potential energy, the couple of cones of the ground-state manifold [Fig. 1(a)] are determined as follows. Their angle about the high-symmetry axis ($z$-axis) are $\theta_{\mathrm{c}}=\arccos\sqrt{\kappa}\left(<\pi/2\right)$ for the upper blue cone and $\pi-\theta_{\mathrm{c}}$ for the lower red cone, respectively. Note that the angles are determined by $\kappa$, the relative strength of the first- and the second-order anisotropies, as expected. A ground state is a uniform array of the magnetization $\mathbf{m}$ which belongs to one of the given manifolds. It is well described by the spherical coordinates $\theta$ and $\phi$ where $\mathbf{m}=(\sin\theta\cos\phi,\,\sin\theta\sin\phi,\,\cos\theta)$, and is given by

in which each coordinate breaks one of two symmetries: $\theta$ breaks Z₂ symmetry and $\phi$ breaks U(1) spin-rotational symmetry.

For the following discussions, it is convenient to use the following natural units of length, time and energy:

where $s$ is the spin density.

The easy-cone system supports another stable state referred to as a DW, which connects two uniform ground states in different manifolds while minimizing the potential energy. That is, a DW is a stationary solution satisfying $\delta U/\delta\theta_{0}=0$ and $\delta U/\delta\phi_{0}=0$, with boundary conditions $\theta_{0}(x\rightarrow\mp\infty)=\theta_{\mathrm{c}}$ and $\theta_{0}(x\rightarrow\pm\infty)=\pi-\theta_{\mathrm{c}}\,$. The exact solution is available [32, 33] which is given by

where $\lambda_{\text{d}}=\sqrt{2\kappa}\,\lambda_{0}$ is the DW width. See Fig. 1(c) for the schematic illustration of the DW described by the Eqs. (5) and (6) with plus sign and $\Phi=0$. Here, the polar angle changes from $\theta_{\mathrm{c}}$ to $\pi-\theta_{\mathrm{c}}$ as $x$ varies from $-\infty$ to $\infty$ while the azimuthal angle is uniform. In Eq. (5), $X$ represents the DW center at which the polar angle is $\pi/2$. Due to the translational invariance of our system, $X$ is arbitrary and thus represents the zero-energy mode associated with spontaneous breaking of the translational invariance by the DW. In this paper, we are interested in spin waves on top of this DW, which we turn to below.

In order to consider a spin wave on top of a DW, we divide the magnetization $\mathbf{m}(x,t)$ into the static DW profile $\mathbf{m}_{0}(x\,;X,\Phi)$ and a small perturbation $\delta\mathbf{m}$. In the local frame, the small perturbation can be written as

where $\mathbf{\hat{\theta}}=\partial\mathbf{m}_{0}/\partial\theta$, $\mathbf{\hat{\phi}}=(1/\sin\theta_{0})(\partial\mathbf{m}_{0}/\partial\phi),$ and $\mathbf{m}_{0}$ form the local orthonormal frame.

By linearizing the Landau-Lifshitz equation [34] about the DW solution while neglecting the damping

in the local coordinate system, we have a set of first-order equations in the small field $\delta\mathbf{m}$ given by

with

See Fig. 2(a) for the exemplary plots of $p(x)$ and $q(x)$ for $\theta_{\mathrm{c}}=\pi/4$. Note that $p(x)$ approaches zero as $x\rightarrow\pm\infty$, which stems from the spontaneous breaking of the U(1) spin-rotational symmetry in ground states and is associated with the gapless nature of spin waves therein. However, $q(x)$ approaches a finite value $q_{0}=2(1-\kappa)$ as $x\rightarrow\pm\infty$, representing the finite energy cost for the magnetization to be tilted away from the easy-cone manifolds.

Far away from the DW where $p(x)$ and $q(x)$ are uniform, the spin-wave equation has plane-wave solutions $\delta_{n}(x,t)=A_{n}\exp(ikx-i\omega t)$ with the dispersion relation given by

See Fig. 1(b) for the plot and note that the gap is zero.

The incident spin wave is found to be partially reflected from the DW. The probability of reflection and transmission were obtained by numerically solving Eqs. (9) within the Green’s function formalism detailed in the Appendix A. See Fig. 2(b) for the plots of the probabilities for $\theta_{\mathrm{c}}=\pi/4$, which represents one of our main results. Note the finite reflection probability at the whole energy ranges, which is in contrast to the reflectionless spin waves in the easy-axis counterpart. The transmission probability increases as the spin-wave energy increases, since the effect of the energy barrier on its transmission becomes weaker at high energies.

When a spin wave is quantized, a quasiparticle referred to as a magnon emerges. The reflection probability and the transmission probability of magnons are the same as the ones for the spin waves, which we will invoke below when discussing the thermal-magnon-driven DW motion.

Thermally populated magnons move between the two thermal reservoirs. When traveling magnons are transmitted through a DW, they exert a torque on it by changing their spin. The torque by the right-moving magnons, which come out of the left reservoir, is given by

with polarization along the $z$-direction, where $n_{\text{B}}$ is Bose-Einstein distribution function and $k_{\text{B}}$ is Boltzmann constant. Here, the factor $2\hbar\cos{\theta_{\mathrm{c}}}=\hbar\cos{\theta(x\rightarrow-\infty)}-\hbar\cos{\theta(x\rightarrow\infty)}$ represents the angular-momentum transfer from a single magnon to the DW, $T(\epsilon)$ is the transmission probability of magnons with energy $\epsilon$, and the last factor $1/(2\pi\hbar)$ comes from the product of a density of states per unit length and the group velocity $d\omega/dk$ [38]. Similarly, the torque $\tau_{\mathrm{R}}$ exerted on the DW by the magnons moving from the right reservoir to the left reservoir can be obtained. The net torque $\tau=\tau_{\mathrm{L}}+\tau_{\mathrm{R}}$ can be approximated by

for $\Delta T\ll T_{\mathrm{avg}}$, where $\Delta T=T_{\mathrm{L}}-T_{\mathrm{R}}$ is the temperature difference between the two reservoirs.

A reflected magnon, however, exerts a force on the DW. By the derivation analogous to the above torque case, the net force exerted on the DW by magnons coming out of the two reservoirs can be approximated by

where $k/\pi$ is a product of $2\hbar k$—the linear-momentum transfer by a single magnon— and $1/(2\pi\hbar)$—the product of a density of states per unit length and the group velocity. Here, $R(\epsilon)$ is the reflection probability of magnons with energy $\epsilon$ by the scattering with the DW. Again, the approximation is valid for $\Delta T\ll T_{\mathrm{avg}}\,$.

The torque and the force on the DW by the scattering with thermal magnons give rise to the DW motion as follows. The equations of motion for the DW parameters, the position $X$ and the angle $\Phi$, are given by (in natural units defined in Eq. (4))

where $g=-2\cos\theta_{\mathrm{c}}$ is the gyrotropic coupling constant between $\dot{X}$ and $\dot{\Phi}$  [40]. For the simplicity, we neglect the effects of the damping in the equations of motion. Here, the left-hand sides are the time derivatives of the spin angular momentum and the conserved linear momentum of a DW, respectively, which are derived through the Noether’s theorem [39, 41]. The former $g\dot{X}$ can be easily understood by considering the dependence of the total spin on the DW position, which determines the lengths of the spin-up and spin-down regions: $S^{z}_{\text{tot}}\propto\pm X$ (Minus sign for increasing $S^{z}(x)\,,$ which is the case of Fig.3(a)). For the torque in Eq. (13), the DW velocity is given by

in natural unit [Eq. (4)]. We numerically obtained the velocity by using the material parameters of $\text{NdCo}_{5}$ given by lattice constant $a=0.5\,\mathrm{nm},\,$ saturation magnetization $M_{\mathrm{s}}=1.1\times 10^{6}\,\mathrm{A/m},A=1.1\times 10^{-11}\,\mathrm{J/m},\,K=2.4\times 10^{6}\,\mathrm{J/m^{3}},\,K^{\prime}=1.6\times 10^{6}\,\mathrm{J/m^{3}}$, and thereby $\kappa=0.75$ [28, 29, 30, 31]. Figure  3(b) shows the velocity of the thermally driven DWs in $\mathrm{NdCo_{5}}$ as a function of $T_{\text{avg}}$ with $\Delta T=0.1T_{\text{avg}}$. Note that there is no exponential suppression of the DW velocity as the average temperature decreases, which can be attributed to the gapless nature of spin waves in easy-cone magnets.