Probing Fractal Magnetic Domains on Multiple Length Scales in ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$

Using small-angle neutron scattering, we demonstrate that the complex magnetic domain patterns at the surface of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$, revealed by quantitative Kerr and Faraday microscopy, propagate into the bulk and exhibit structural features with dimensions down to 6 nm, the domain wall thickness. The observed fractal nature of the domain structures provides an explanation for the anomalous increase in the bulk magnetization of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ below the spin-reorientation transition. These measurements open up a rich playground for studies of fractal structures in highly anisotropic magnetic systems.

Figure 1

Temperature dependence of the magnetization and the magnetic domain patterns in the ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}{}^{11}\mathrm{B}$ single crystal. The magnetization was measured at ${\mu }_{0}H=50\mathrm{mT}$ applied along the $\mathbf{c}$ axis. The magnetic domain patterns were imaged exploiting the magneto-optical polar Kerr effect at a surface perpendicular to the $\mathbf{c}$ direction and with the $\mathbf{a}$ direction vertical.

Figure 2

Faraday images of the ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}{}^{11}\mathrm{B}$ single crystal. In the transparent ferromagnetic Bi-doped Fe-garnet film placed on the surface of the sample, the distribution of the magnetic moments mimics and amplifies the magnetic induction on the sample surface. The height is proportional to the magnitude of the magnetic induction in $\mathbf{c}$ direction perpendicular to the surface. The spin structure is shown schematically.

Figure 3

Raw SANS area-detector images of the ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}{}^{11}\mathrm{B}$ single crystal at temperatures of 20 K and 200 K. The colors and lines in the contour map represent the neutron counts on a logarithmic scale. The black lines mark the 20° wide sectors used for the averaging to obtain the data shown in Fig. 4.

Figure 4

Scattering-vector dependence of the SANS intensity of the ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}{}^{11}\mathrm{B}$ single crystal at temperatures of 20 and 200 K. These data were determined by averaging over all four sectors marked in Fig. 3, and combining patterns measured at different sample-to-detector distances and different energies of the incident neutrons. The intensities extracted from sectors rotated by 45° to the sectors outlined in Fig. 3 are, on average, a factor of 2 larger, but show a similar dependence on the scattering vector.