Effect of Dzyaloshinski-Moriya interaction on elastic small-angle neutron scattering

For magnetic materials containing many lattice imperfections (e.g., nanocrystalline magnets), the relativistic Dzyaloshinski-Moriya (DM) interaction may result in nonuniform spin textures due to the lack of inversion symmetry at interfaces. Within the framework of the continuum theory of micromagnetics, we explore the impact of the DM interaction on the elastic magnetic small-angle neutron scattering (SANS) cross section. It is shown that the DM interaction gives rise to a polarization-dependent asymmetric term in the spin-flip SANS cross section. Analysis of this feature may provide a means to determine the DM constant.

Figure 1

Sketch of the (a) perpendicular and (b) parallel scattering geometry. The applied-field direction ${\mathbf{H}}_0$ defines in both cases the ${\mathbf{e}}_z$ direction of a Cartesian laboratory coordinate system. The angle $\theta$ specifies the orientation of the scattering vector $\mathbf{q}$ on the two-dimensional detector: (a) $\mathbf{q}\cong \left\{0,q_y,q_z\right\}=q\left\{0,sin\theta ,cos\theta \right\}$ and (b) $\mathbf{q}\cong \left\{q_x,q_y,0\right\}=q\left\{cos\theta ,sin\theta ,0\right\}$.

Figure 2

Contour plots of the Fourier components of the magnetization at selected applied magnetic fields $\left({\mathbf{k}}_0\perp {\mathbf{H}}_0\right)$. $|{\tilde{M}}_x{|}^2$ (upper row), $|{\tilde{M}}_y{|}^2$ (middle row), and $CT=-({\tilde{M}}_y{\tilde{M}}_z^{*}+{\tilde{M}}_y^{*}{\tilde{M}}_z)$ (lower row) [Eqs. (34, 35, 36)]. ${\mathbf{H}}_0\parallel {\mathbf{e}}_z$ is horizontal in the plane. $H_i$ values (in T) from left to right column: 0.5; 1.5; 10.0. All data were normalized to unity by the respective maximum value.

Figure 3

Same as Fig. 2, but for ${\mathbf{k}}_0\parallel {\mathbf{H}}_0$. $|{\tilde{M}}_x{|}^2$ (upper row), $|{\tilde{M}}_y{|}^2$ (middle row), and $CT=-({\tilde{M}}_x{\tilde{M}}_y^{*}+{\tilde{M}}_x^{*}{\tilde{M}}_y)$ (lower row) [Eqs. (37, 38, 39)].

Figure 4

Contour plots of $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ at selected applied magnetic fields [Eq. (46)] $\left({\mathbf{k}}_0\perp {\mathbf{H}}_0\right)$. $H_i$ values (in T) from left to right: 0.4; 0.8; 4.2. All data were normalized to unity by the respective maximum value.

Figure 5

Azimuthally-averaged $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ at ${\mu }_0{H}_i=0.8\mathrm{T}$ with and without the DM term (see inset) (log-log scale) $\left({\mathbf{k}}_0\perp {\mathbf{H}}_0\right)$. Both $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ have been convoluted with the same Gaussian distribution function $(\overline{R}=8\mathrm{nm}$; $\sigma =0.7)$ for both ${\tilde{H}}_p^2$ and ${\tilde{M}}_z^2$.

Figure 6

Contour plots of the spin-flip difference cross section $-2i\chi \left(\mathbf{q}\right)$ [Eq. (49)] at selected applied magnetic fields $\left({\mathbf{k}}_0\perp {\mathbf{H}}_0\right)$. $H_i$ values (in T) from left to right: 0.3; 0.6; 1.0; 3.5. All data were normalized to unity by the respective maximum value.

Figure 7

Azimuthal average of the spin-flip difference cross section, $-2i\chi (q,H)=-2i{\int }_0^{\pi /2}\chi (q,H,\theta )d\theta$, at (a) selected applied magnetic fields and constant $D=2.0\mathrm{mJ}/{\mathrm{m}}^2$, and (b) for constant field ${\mu }_0{H}_i=0.8\mathrm{T}$ but varying DM constant $D$ $\left({\mathbf{k}}_0\perp {\mathbf{H}}_0\right)$. The field (in T) in (a) increases from top to bottom: 0.5; 0.6; 0.8; 2.0. The DM constant (in $\mathrm{mJ}/{\mathrm{m}}^2)$ in (b) increases from bottom to top: 1.5; 2.0; 2.5.