Unidirectional spin-wave channeling along magnetic domain walls of Bloch type

From the pioneering work of Winter [Phys. Rev. 124, 452 (1961)], a magnetic domain wall of Bloch type is known to host a special wall-bound spin-wave mode, which corresponds to spin waves being channeled along the magnetic texture. Using micromagnetic simulations, we investigate spin waves traveling inside Bloch walls formed in thin magnetic media with perpendicular-to-plane magnetic anisotropy and we show that their propagation is actually strongly nonreciprocal, as a result of dynamic dipolar interactions. We investigate spin-wave nonreciprocity effects in single Bloch walls, which allows us to clearly pinpoint their origin, as well as in arrays of parallel walls in stripe domain configurations. For such arrays, a complex domain-wall-bound spin-wave band structure develops, some aspects of which can be understood qualitatively from the single-wall picture by considering that a wall array consists of a sequence of up/down and down/up walls with opposite nonreciprocities. Circumstances are identified in which the nonreciprocity is so extreme that spin-wave propagation inside individual walls becomes unidirectional.

Figure 1

(a) Schematic of the magnetic medium containing two stripe domains separated by a Bloch wall ($W=2L=256\mathrm{nm}$, $T=15\mathrm{nm}$). (b) Width profile of the variable magnetization in the lowest frequency spin-wave eigenmode, which propagates as a plane wave $\mathbf{m}(u,w,t)={\mathbf{m}}_0\left(w\right)e^{i(\omega t-ku)}$ with $k=+50\mathrm{rad}/\mu \mathrm{m}$, along the medium ($u$). (c) Dispersion relation of the DWCSW mode. The thin red line and the thick gray line correspond to results of our numerical simulations and to a fit of the numerical data with $\left|k\right|\le 30\mathrm{rad}/\mu \mathrm{m}$ to Eq. (1) ($v_1=340\mathrm{m}/\mathrm{s}$, $v_2=400\mathrm{m}/\mathrm{s}$, ${\omega }_0/2\pi =0.77\mathrm{GHz}$, $a=50\mathrm{nm}$), respectively. (d) Map of the transverse component of the magnetization, $M_w$, as generated by an excitation having the form of a 5 mT, vertically oriented, sinusoidal magnetic field with frequency $\omega /2\pi =2\mathrm{GHz}$, localized in $u=0$ (yellow line). Note the different scales along the $u$ and $w$ directions. (e) Line profile of $M_w$ at the equilibrium position of the domain-wall center ($w=L$). The data (symbols) can be fitted (lines) to the expression $M_w\left(u\right)=M_{0,w}sin\left[k(u-u_0)\right]e^{-(u-u_0)/L_{\text{att}}}$, with $k=16.7\mathrm{rad}/\mu \mathrm{m}$ and $L_{\text{att}}=1.00\mu \mathrm{m}$ (resp. $k=-53.1\mathrm{rad}/\mu \mathrm{m}$, $L_{\text{att}}=-2.42\mu \mathrm{m}$) for channeled spin waves propagating to the right (resp. to the left).

Figure 2

(a),(b) Close views of the magnetic configurations (arrows) and normalized magnetic charge distributions $\rho =-\mathbf{\nabla }·\mathbf{z}$ (color map), where $\mathbf{z}=\mathbf{M}/M_{\text{S}}$, for (a) leftward propagating ($\omega /2\pi =0.61\mathrm{GHz}$) and (b) rightward propagating ($\omega /2\pi =3.66\mathrm{GHz}$) channeled spin waves with wave vector $\left|k\right|=30\mathrm{rad}/\mu \mathrm{m}$. Different (large) strengths of the localized excitation field have been used in order to obtain (clearly visible) domain-wall flexures of comparable amplitudes in the two cases: (a) 100 mT, (b) 300 mT. (c),(d) Map of the transverse component ($m_w$) of the variable magnetization $\mathbf{m}$ in the magnetic configurations shown in (a) and (b). The arrows indicate schematically the location of the extrema of the three components of $\mathbf{m}$ in the DWCSW mode. Note that, for the sake of clarity, the vertical distance between the extrema of $m_u$ has been exaggerated, as a comparison with Fig. 1 reveals. In all panels (a)–(d), the area shown has a width of 64 nm and a length equal to the spin-wave wavelength $2\pi /\left|k\right|=209\mathrm{nm}$. (e) Illustration of the right hand thumb rule for determining in which direction the channeled spin waves have the lowest frequency.

Figure 3

(a) Schematic of the magnetic medium containing three stripe domains separated by two Bloch walls ($W=3L=384\mathrm{nm}$, $T=15\mathrm{nm}$). (b) Dispersion relations of the two collective DWCSW eigenmodes and (c),(d) width profiles of the variable magnetization in those modes, for $k=0$. The mode with higher frequency (red lines) has an optic character while that with lower frequency (green lines) has an acoustic nature. For the purpose of comparison, the dispersion relations of DWCSW modes hosted by single down/up (solid gray line) and up/down (dashed gray line) walls in a $2L$-wide medium are also shown in panel (b). (e),(f) Color maps of the amplitude of the vertical component of the variable magnetization, $\text{Re}\left(m_{0,v}\right)$, in the opticlike (e) and acousticlike (f) collective DWCSW eigenmodes as a function of the transverse coordinate $w$ and wave vector $k$.

Figure 4

(a) Map of the transverse component of the magnetization, $M_w$, as produced upon application of a localized (yellow line), vertically oriented, 5 mT alternating magnetic field with frequency $\omega /2\pi =2\mathrm{GHz}$ to the three-stripe-domain medium sketched in Fig. 3. (b)–(e) Width profiles of the variable magnetization in the DWCSW eigenmodes involved in (a). The same color code is used as in Figs. 3–3. (f) Map of the transverse component of magnetization as generated upon application of a 5 mT alternating magnetic field with frequency $\omega /2\pi =0.8\mathrm{GHz}$ (green line). (g) Line profile of $M_w$ at the equilibrium position of the up/down domain wall ($w\simeq 2L$). Beyond a certain distance from the excitation source ($u=0$), the data (symbols) can be fitted (lines) to the expressions $M_w\left(u\right)=M_{0,w}sin\left[k(u-u_0)\right]e^{-(u-u_0)/L_{\text{att}}}$ with $k=33.6\mathrm{rad}/\mu \mathrm{m}$ and $L_{\text{att}}=2.56\mu \mathrm{m}$, for $u>0$, and $M_w\left(u\right)=M_{0,w}{e}^{-(u-u_0)/L_{\text{att}}}$ with $L_{\text{att}}=0.21\mu \mathrm{m}$, for $u<0$.

Figure 5

(a) Dispersion relations of the DWCSW modes in a magnetic medium containing $N=16$ Bloch walls ($W=2176\mathrm{nm}$). The eight lowest (resp. highest) frequency modes, whose $\omega \left(k\right)$ curves are shown as blue lines (resp. red lines) of varying shade, have an acoustic (resp. optic) character. (b)–(d) Width profiles of the out-of-plane component of the dynamic magnetization in the sixteen DWCSW modes, for three different values of the wave vector: (b) $k=-30\mathrm{rad}/\mu \mathrm{m}$, (c) $k=0\mathrm{rad}/\mu \mathrm{m}$, and (d) $k=+30\mathrm{rad}/\mu \mathrm{m}$. For clarity, the profiles are offset vertically according to the mode indices. (e)–(g) Maps of the transverse magnetization component ($M_w$) upon application of a localized (green line), vertically oriented, 5 mT alternating magnetic field to a magnetic medium with $N=16$: (e) Full domain wall array for $\omega /2\pi =0.9\mathrm{GHz}$ and zoom in the region containing the two innermost domain walls for (f) $\omega /2\pi =0.7\mathrm{GHz}$ and (g) $\omega /2\pi =1.1\mathrm{GHz}$.