Between waves and patterns: Spin wave freezing in films with Dzyaloshinskii-Moriya interaction

The relationship between waves and static pattern formation is an intriguing effect and remains unexplained in many areas of physics, including magnetism. We study the spin-wave-mediated spin reorientation transition (SRT) in magnetic films with uniaxial magnetic anisotropy and Dzyaloshinskii-Moriya interaction (DMI). In particular, we show that propagating spin waves can freeze in the SRT, causing periodic magnetic domains to arise, which is reminiscent of the wave amplitude distribution. This process can take place under the influence of a change in the magnetic field, but also of other parameters. Interestingly, at the SRT, DMI nonreciprocity leads to the emergence of flowing magnetization patterns, which suggests a spontaneous breaking of translational symmetry, and the formation of magnonic space-time crystals. The described phenomena are general and should take place in a large family of magnetic materials. Therefore, the results should be of great importance for the further development of spintronics and magnonics.

Figure 1

Analytical dispersion relations constructed with Eq. (1) illustrating conditions required for SRT occurrence. (a) $H$-SRT for $Q=1.1, D_{\mathrm{ex}}=0$, and $d_{\mathrm{ex}}=0.624$; (b) $Q$-SRT for $h=0, D_{\mathrm{ex}}=0$, and $d_{\mathrm{ex}}=0.624$; (c) $D$-SRT for $h=0, Q=0.6$, and $d_{\mathrm{ex}}=0.624$; (d) $d$-SRT for $h=0, Q=0.6$, and $D_{\mathrm{ex}}=0$. Black lines in (b)–(d) pertain to the same set of parameters ($h=0, Q=0.6, D_{\mathrm{ex}}=0, d_{\mathrm{ex}}=0.624$). The red dots and vertical dashed lines represent the conditions, where Eqs. (2a) and (2b) are satisfied, indicating the values of ${\kappa }^{*}$. The violet dots in (a) represent the conditions, where only Eq. (2a) is satisfied.

Figure 2

A relation between the static magnetization configuration of a stripe domain pattern and the SW profile, below and above the SRT, respectively. Simulations have been performed for an ultrathin film depending on $h$ for $d_{\mathrm{ex}}=0.624, Q=1.1$, and $D_{\mathrm{ex}}=0$. Normalized field $h$-dependencies are of the following: (a) static $\langle m_y\rangle$ and $\langle m_z^2\rangle$ (solid circles and squares); critical field $h^{*}=0.123$ is determined by simulations; maximal amplitude of $m_z$ oscillations from dynamic simulations (open squares); (b) reduced equilibrium domain period $p_{\mathrm{ex}}$ from static simulations (solid circles), SW wavelength ${\mathrm{\Lambda }}_{\mathrm{ex}}^{\min }$ from dynamic simulations (open squares), and from theory given by Eqs. (1) and (2a) (dashed green line). Solid lines are plotted as guides for the eye. The yellow background corresponds to the field range of occurrence of a magnetic state with an out-of-plane magnetization component, and the white background corresponds to an in-plane state.

Figure 3

Time evolution of the pattern formation in ultrathin Co film for $Q=1.1$. (a) $h=0.998h^{*}$ and (b) $h=1.003h^{*}$; in these simulations, $h^{*}=0.123, D_{\mathrm{ex}}=0$, and damping parameter $\alpha =0.005$. (c) $D_{\mathrm{ex}}=0.123, h=0.996h^{*}$ (where $h^{*}=0.124$), and $\alpha =0.02$. The color scales in all density maps are conserved. (d) Temporal dependencies of $\langle m_z^2\rangle$ are averaged over the whole systems, for the cases (a), (b), and (c), and additionally for the DMI case from (c) with low $\alpha =0.005$ (the orange dashed line). The same results in the form of a movie can be found in the Supplemental Material [41].

Figure 4

Simulated $h$-dependent static and dynamic properties of thin film with $Q=0.25, D_{\mathrm{ex}}=0$, and $d_{\mathrm{ex}}=11.9$. (a) SW dispersion relation for $h=1.05h^{*}$ (SW intensity is in log scale); the white dashed line represents the analytical dispersion [Eq. (1)]; (b) mode of oscillations at the point $({\kappa }_{\min },f_{\min })$ in the dispersion relation, as marked with the black dot in (a); the $m_x$ and $m_z$ dynamic magnetization components are presented by the black arrows, $m_y\approx 1$; (c),(d) static magnetization distribution at $h=0.95h^{*}$ and 0, respectively. The $m_x$ and $m_z$ static magnetization components are presented by the white arrows, while the static $m_y$ component is coded by color.

Figure 5

Calculated diagrams and SRT lines in the $(Q,D_{\mathrm{ex}})$ space for the ultrathin film case with $d_{\mathrm{ex}}=0.624$. (a),(b) Theoretical SRT [i.e., numerical solution of Eqs. (2a) and (2b)]; (a) the map of critical field $h^{*}$; (b) the map of critical period $p_{\mathrm{ex}}^{*}$ or wavelength ${\mathrm{\Lambda }}_{\mathrm{ex}}^{*}$ for the values of $h^{*}$ shown in (a); (c) the map of reduced period found in micromagnetic simulations from energy minimization at zero field; (d) a comparison of freezing (b) and zero-field (c) reduced periods for selected $Q$ values [marked with the vertical lines in (b) and (c)]. The white points denote the parameters discussed in the labeled figures.

Figure 6

Simulated SRT in the $(Q,d_{\mathrm{ex}})$ space for $D_{\mathrm{ex}}=0$. (a) The map of critical field $h^{*}$; (b) the map of critical period $p_{\mathrm{ex}}^{*}$ or wavelength ${\mathrm{\Lambda }}_{\mathrm{ex}}^{*}$ for the values of $h^{*}$ shown in (a). The open symbols in (a) denote the values of parameters discussed throughout the manuscript and presented in the indicated figures. The dashed lines in (b) are analytical and numerical approximations of zero-field SRT according to the dispersion-based theory, and the dotted line is the zero-field SRT in the $Q\ll 1$ regime. The black solid line in (a) and (b) is Hubert's theory for the zero-field SRT. The area below the scissors is filled according to theoretical dependences instead of simulations (see the text for details). The inset in (b) is a scan of $p_{\mathrm{ex}}^{*}$ or ${\mathrm{\Lambda }}_{\mathrm{ex}}^{*}$ along the zero-field SRT (the black “Hubert” line) according to simulations and applicable analytical approaches.

Figure 7

Dispersion relation in ultrathin film for $d_{\mathrm{ex}}=0.624$ (i.e., 2 nm of Co), $Q=1.1, D_{\mathrm{ex}}=0$, and increasing field, all above the critical value $h^{*}=0.123$: (a) $h=1.01h^{*}$; (b) $h=1.06h^{*}$ [this is the case presented in Fig. 1]; 1 $h=1.63h^{*}$. SW intensity is coded by color in log scale. While increasing the field, the band moves up and its intensity decreases. The dashed lines are plotted according to the dispersion-based theory [Eq. (1)].

Figure 8

A relation between the static magnetization configuration of a stripe domain pattern and the SW profile below and above the SRT, respectively. Simulations have been performed for the thin film depending on $h$ for $d_{\mathrm{ex}}=11.9, Q=0.25$, and $D_{\mathrm{ex}}=0$. Normalized field $h$-dependencies are of the following: (a) static $\langle m_y\rangle$ and $\langle m_z^2\rangle$ (solid squares and circles); critical field $h^{*}=0.04$ is determined by simulations; maximal amplitude of oscillations is determined by dynamic simulations (open squares); (b) reduced equilibrium domain period $p_{\mathrm{ex}}$ is determined by static simulations (solid circles), and wavelength of SWs ${\mathrm{\Lambda }}_{\mathrm{ex}}^{\min }$ is determined by dynamic simulations (open squares). Solid lines are plotted as guides for the eye. The yellow background corresponds to the field range of occurrence of a magnetic state with an out-of-plane magnetization component, while the white background corresponds to the in-plane state.

Figure 9

Dispersion relations and SW mode profiles in thin films, $d_{\mathrm{ex}}=11.9$ and $Q=0.6$. (a),(b) Dispersion relations for $h=1.03h^{*}$ and $1.21h^{*}$. While increasing the field above, the band moves up and its intensity decreases. SW intensity is coded by color in log scale. The white dashed lines are theoretical dependences according to Eq. (1) at the given values of the field. (c),(d) Mode profiles in $h=1.03h^{*}$, where $h^{*}=0.32$, at the points marked with dots in (a).

Figure 10

Dispersion relations and SW mode profiles in thin film with DMI, $d_{\mathrm{ex}}=11.9$ and $Q=0.6$. (a),(b) Dispersion relations for $D_{\mathrm{ex}}=0.246$ and 1.23 and $h=1.06h^{*}$ and $1.11h^{*}$, respectively. While increasing DMI, the band tilts, becoming more asymmetric. SW intensity is coded by color in log scale. The white dashed lines are theoretical dependences according to Eq. (1) at the given values of the field. (c),(d) Mode profiles for $D_{\mathrm{ex}}=0.246$ in $h=1.06h^{*}$, at the points marked with dots in (a).

Figure 11

The simulated dispersion relations in DE (solid lines) and BVMW (dashed lines) modes for the following two systems: (a) an ultrathin film, $d_{\mathrm{ex}}=0.624$ and $Q=1.1$, and in magnetic field, $h_1=1.0005h^{*}$ and $h_2=1.14h^{*}$; (b) a thick and low-anisotropy film, $Q=0.01$ and $d_{\mathrm{ex}}=100$, and in magnetic field, $h_1=1.02h^{*}$ and $h_2=7.5h^{*}$.