Effect of demagnetization factors on spin current transport

The demagnetizing field associated with shape anisotropy originates from the magnetic field generated through magnetization of magnetic materials. The shape and demagnetizing effects must be considered as characteristics of a magnetic object in the presence of a magnetic field. In this study, we demonstrated the strong influence of the demagnetizing field on the spin current transports in Pt/YIG, including the spin Seebeck effect, spin Hall effect, spin Hall magnetoresistance, and planar Hall resistance. The calculated effective demagnetizing factors were closely related to the thickness- and width-dependent anomalous plateau behaviors of the spin-dependent electrical and thermal transports as well as magneto-optical measurements. Moreover, we demonstrated that the width of the plateau manifested from the shape of the YIG sample. Furthermore, through magnetic force microscopy, we directly visualized the evolution of the magnetic domains with an abrupt 90° magnetic rotation, which we further corroborated by the 90° resistance phase shift in the angular-dependent planar Hall effect.

Figure 1

Schematic diagram for (a) the thickness dependent and (b) the width dependent measurement geometry. The blue arrows indicate the grinding process to reduce (a) the thickness $t$ and (b) the width $w$ to keep the same interface between Pt and YIG. ∇$T$ and $H$ denote the temperature gradient and the external magnetic field. (c) The ISHE of Pt on single crystal YIG (blue) and the normalized M-H loop of YIG (green). (d) The SMR of Pt/YIG. (e) Normalized Kerr intensity of YIG slab. The ISHE of Pt/YIG with different (f) thickness of YIG and (g) width of YIG. The shaded areas in (c)–(g) represent the plateau width.

Figure 2

The field width plateau as a function of (a) YIG thickness from the SSE, SMR, and MOKE measurements by using both single-crystalline and polycrystalline YIG slabs with a width of 3 mm and (b) YIG width from the SSE measurement.

Figure 3

Normalized $M-H$ loops for YIG with different thicknesses under (a) in-plane $H_{\parallel }$ and (b) out-of-plane $H_{\perp }$. (c) Thickness dependence of total effective anisotropy energy per unit area $K_{\mathrm{eff}}t$ (blue). Orange line is the fitted curve of the thickness dependent $K_{\mathrm{h}}$ of Eq. (4) and the grey line is fitted by a constant $K_{\mathrm{h}}$. Normalized M-H loops for YIG with different widths under (d) in-plane $H_{\parallel }$ and (e) out-of-plane $H_{\perp }$. (f) Width dependence of total effective anisotropy energy per unit area $K_{\mathrm{eff}}w$ (blue). Orange line is the fitted curve by the width dependent $K_{\mathrm{h}}$ of Eq. (5).

Figure 4

(a) Schematic diagram for MFM measurement on YIG. (b) Magnetic field dependence of the magnetic domain structure on YIG (287μm) taken in the same $50\times 50\mu {\mathrm{m}}^2$ field of view. Cyan arrow represents the in-plane component of the magnetization in a stripe domain. Red arrow and the enclosed line indicate the general orientation and the region that rotates normal to the external field. Yellow arrow at the top left corner indicates the direction of external in-plane magnetic field with the unit of Oe. The full grey scale of MFM signal in all images is 1.5 Hz. (c) Normalized Kerr intensity with magnetic field along the in-plane magnetic field.

Figure 5

(a) The thickness and (b) width dependent magnetic potential energy ($U_{\mathrm{m}}$) at YIG surface. (c) The thickness dependent and (d) width dependent demagnetizing factors $N^{*}(=N_w-N_t)$. Inset in (c) and (d) depicts the comparison between the normalized $U_{\mathrm{m}}$ and N*.

Figure 6

(a) Schematic diagram for the planar Hall effect (PHE) measurement of Pt(5 nm)/YIG with 10-μm-wide Hall bar pattern. (b) The angular dependence of relative PHE resistance $\mathrm{\Delta }{R}_{\mathrm{XY}}$ under different in-plane magnetic field. Insets are schematic drawings of the magnetic moment under 55–60 Oe (upper drawing) or above 200 Oe (lower drawing) at field angle of 225°. (c) Field dependent $\mathrm{\Delta }{R}_{\mathrm{XY}}$ with field angle along 45° (black) and 135° (red).

Figure 7

(a) Normalized Kerr intensity with magnetic field along the in-plane (top) and out-of-plane (bottom) of YIG (287 μm). (b) Out-of-plane magnetic field dependence of the magnetic domains on YIG measured by MFM in the same field of view as that in Fig. 4.

Figure 8

The magnetic domain on YIG (943 μm) in a field of view of $32\times 32\mu {\mathrm{m}}^2$ measured by MFM under (a) in-plane (the direction is indicated by the yellow arrow and the strength in Oe at the top left corner) and (c) out-of-plane magnetic field. The full gray scale of MFM signal is 1.8 Hz in all images. Normalized Kerr intensity with (b) in-plane (d) out-of-plane magnetic field.

Figure 9

(a) The field dependent magnetization and (b) the angle dependent PHE signal for Py(5 nm)/Si.