Paramagnetic spin Hall magnetoresistance

We report the observation of the spin Hall magnetoresistance (SMR) in a paramagnetic insulator. By measuring the transverse resistance in a $\mathrm{Pt}/{\mathrm{Gd}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ (GGG) system at low temperatures, paramagnetic SMR is found to appear with an intensity that increases with the magnetic field aligning GGG's spins. The observed effect is well supported by a microscopic SMR theory, which provides the parameters governing the spin transport at the interface. Our findings clarify the mechanism of spin exchange at a Pt/GGG interface, and demonstrate tunable spin-transfer torque through the field-induced magnetization of GGG. In this regard, paramagnetic insulators offer a key property for future spintronic devices.

Figure 1

Sketches of (a) NM/FM and (b) NM/PI interface with spin exchange. The blue, red, and green arrows represent the directions of angular momentum related to the spin-transfer torque, fieldlike torque, and spin-flip process, respectively. (c) $M$($T$) of GGG. The inset shows 1/$M$ (blue circles) and a linear fit (red solid line). (d) Sketch of SMR at the Pt/GGG interface. ${\mathbf{J}}_{\mathbf{s}}$ with σ is generated in Pt by SHE with the application of ${\mathbf{J}}_{\mathbf{c}}$.

Figure 2

(a) $M$($B$) of GGG at 2 K. (b) $\mathrm{\Delta }{\rho }_{\mathrm{T}}\left(B\right)$ at 2 K. The deep-blue (gray) curve shows $\mathrm{\Delta }{\rho }_{\mathrm{T}}$ with $B$ at $\alpha ={45}^{\circ }$ (0°). The light-blue (gray) curve in the inset shows $\mathrm{\Delta }{\rho }_{\mathrm{T}}$ with $B$ at $\alpha ={135}^{\circ }$ (90°). (c) $S_{\mathrm{SHAHE}}\left(B\right)$ with $B\left|\right|z$ at 2 K in Pt/GGG after subtracting the OHE component. The inset shows the measurement setup for SHAHE. (d) Measurement setup for transverse SMR. (e) $\mathrm{\Delta }{\rho }_{\mathrm{L}}\left(B_i\right)$ in Pt/GGG at 2.5 K. The inset shows the measurement setup. (f) $L_{\mathrm{HMR}}\left(B\right)$ at 300 K in Pt/GGG.

Figure 3

(a) $B$ dependence of the SMR signals obtained from FDMR and ADMR measurements at 2 K. The solid curve represents $S_{\mathrm{SMR}}^{\mathrm{FDMR}}$. The purple (blue) circles show $S_{\mathrm{SMR}}^{\mathrm{ADMR}}$($L_{\mathrm{SMR}}^{\mathrm{ADMR}}$) obtained via the fitting using $S_{\mathrm{SMR}}^{\mathrm{ADMR}}cos\left(\alpha \right)sin\left(\alpha \right) {[L}_{\mathrm{SMR}}^{\mathrm{ADMR}}\mathrm{co}{\mathrm{s}}^2\left(\alpha \right)]$. (b) $\mathrm{\Delta }{\rho }_{\mathrm{T}}\left(\alpha \right)$ and (c) $\mathrm{\Delta }{\rho }_{\mathrm{L}}\left(\alpha \right)$ at 2 K with rotating $\left|B\right|=3.5\mathrm{T}$. The orange solid curves in (b) and (c) are a $S_{\mathrm{SMR}}^{\mathrm{ADMR}}cos\left(\alpha \right)sin\left(\alpha \right)$ and $L_{\mathrm{SMR}}^{\mathrm{ADMR}}\mathrm{co}{\mathrm{s}}^2\left(\alpha \right)$ fit, respectively. (d), (e) Schematic illustrations of (d) the transverse and (e) longitudinal ADMR measurement setup. (f) $\mathrm{\Delta }{\rho }_{\mathrm{T}}\left(\alpha \right)$ and (g) $\mathrm{\Delta }{\rho }_{\mathrm{L}}\left(\alpha \right)$ at 2 K for various $B$. The curves have been shifted vertically for clarity.

Figure 4

(a) Schematic illustrations of the ADMR measurement setups in three different planes with the corresponding definition of the angle. (b) $\mathrm{\Delta }{\rho }_{\mathrm{L}}\left(\alpha \right), \mathrm{\Delta }{\rho }_{\mathrm{L}}\left(\beta \right)$, and $\mathrm{\Delta }{\rho }_{\mathrm{L}}\left(\gamma \right)$ at 2 K and 3.5 T. (c) $S_{\mathrm{SMR}}^{\mathrm{ADMR}}$($T$) at 3.5 T. The inset shows the $M\left(T\right)$ curve of GGG at the same magnetic field.

Figure 5

(a) $B$ dependence of the SMR and SHAHE signals together with the fitting of Eqs. (5) and (6) at 2 K. (b) $B$ dependence of $G_r, G_i$, and $G_s$. (c) Sketches of spin-transfer and (d) fieldlike torque in a NM/PI system. (e), (f) The comparison (e) between $G_r$ and the SMR signal, and (f) between $|G_i|$ and the SHAHE signal.

Figure 6

$T$ dependence of the resistivity ${\rho }_{\mathrm{L}}$ of Pt, measured in a 5-nm-thick Pt Hall bar on GGG.

Figure 7

$B$ dependence of $\mathrm{\Delta }{\rho }_{\mathrm{L}}\left(B_i\right)$ at 2.5 K. The red, blue, and green curves indicate the $B$ dependence of $\mathrm{\Delta }{\rho }_{\mathrm{L}}\left(B_i\right)$ for $B$ in the $x$, y, and $z$ directions, respectively. The inset shows the sample with the applied current ${\mathbf{J}}_{\mathbf{c}}\left|\right|\mathbf{x}$.

Figure 8

$B$ dependence of $L_{\mathrm{HMR}}$ at 300 K.

Figure 9

(a) A schematic illustration of the longitudinal resistivity measurement. A charge current ${\mathbf{J}}_{\mathbf{c}}$ is applied in the $x$ direction and the longitudinal voltage is measured. B indicates the spatial direction of the magnetic field, and the angle between ${\mathbf{J}}_{\mathbf{c}}$ and B is defined as α. (b) Longitudinal FDMR at 2.5 K. The red (blue) curve indicates $\mathrm{\Delta }{\rho }_{\mathrm{L}}$ under $B$ in the $x$ ($y$) direction. $L_{\mathrm{SMR}}^{\mathrm{FDMR}}\left(B\right)$ is defined as $L_{\mathrm{SMR}}^{\mathrm{FDMR}}\left(B\right)=\left[{\rho }_{\mathrm{L}}\left(B_x\right)-{\rho }_{\mathrm{L}}\left(B_y\right)\right]/{\rho }_{\mathrm{L}}(B=0)$. (c) Longitudinal ADMR $\mathrm{\Delta }{\rho }_{\mathrm{L}}$ in the $x-y$ plane. The green circles show the α dependence of $\mathrm{\Delta }{\rho }_{\mathrm{L}}$ at 2.5 K at 3.5 T. The blue curve indicates a $L_{\mathrm{SMR}}^{\mathrm{ADMR}}\mathrm{co}{\mathrm{s}}^2\alpha$ fitting to the experimental result. (d) FDMR and ADMR results as a function of $B$ at 2.5 K. The green curve (circles) shows the FDMR (ADMR) result of $L_{\mathrm{SMR}}$.

Figure 10

(a) $\mathrm{\Delta }{\rho }_{\mathrm{T}}$ with $B\left|\right|z$, i.e., a the Hall measurement. ${\mathbf{J}}_{\mathbf{c}}$, B, and β denote the spatial direction of the charge current and magnetic field, and relative angle, respectively. The transverse FDMR is measured at $\beta =0$ and $T=2\mathrm{K}$. (b) Transverse ADMR $\mathrm{\Delta }{\rho }_{\mathrm{T}}\left(\beta \right)$ at $T=2\mathrm{K}$ and selected $B$. (c) $B$ dependence of $A^{3\mathrm{rd}}$. The inset shows the $B$ dependence of $A^{1\mathrm{st}}$ and $A^{3\mathrm{rd}}$. The amplitudes of $A^{1\mathrm{st}}$ and $A^{3\mathrm{rd}}$ are obtained by fitting a $A^{1\mathrm{st}}cos\beta +A^{3\mathrm{rd}}\mathrm{co}{\mathrm{s}}^3\beta$ function to the ADMR results shown in (b). (d) $B$ dependence of the first- and third-order SHAHE. The blue (red) plots show the first-order (third-order) SHAHE, $S_{\mathrm{SHAHE}}^{1\mathrm{st}}$ ($S_{\mathrm{SHAHE}}^{3\mathrm{rd}}$), obtained from the ADMR results. The deep-blue curve indicates $S_{\mathrm{SHAHE}}$ obtained from the FDMR results.

Figure 11

$B$ dependence of $\left|\langle m\rangle \right|$. The red and blue lines represent the self-consistent solution and the approximation with the effective Curie-Weiss temperature solution for Eq. (F2), respectively.

Figure 12

(a) A schematic illustration of the NM/PI or NM/FM sample structure. We apply a charge current, ${\mathbf{J}}_{\mathbf{c}}$, in the $x$ direction. (b) A schematic side view of the interface. Conduction electron spins are coupled to spins in PI in the deep-blue region $\left(0<z<b\right)$ with the thickness of $b$. $d_{\mathrm{NM}}$ is the thickness of NM.

Figure 13

(a) $T$ dependence of paramagnetic SMR and magnetization of GGG. $S_{\mathrm{SMR}}$ is estimated by fitting a $S_{\mathrm{SMR}}sin\left(\alpha \right)cos\left(\alpha \right)$ function to the transverse ADMR results at $\left|B\right|=3.5\mathrm{T}$ at various $T$. The inset shows the $T$ dependence of $M$ of GGG at $\left|B\right|=3.5\mathrm{T}$. The paramagnetic SMR dominates the signal in the shaded $T$ region. (b) FDMR result of $S_{\mathrm{SMR}}$ at selected $T$. The solid blue curves represent $S_{\mathrm{SMR}}^{\mathrm{FDMR}}=\mathrm{\Delta }{\rho }_{\mathrm{T}}\left(45{}^{\circ }\right)-\mathrm{\Delta }{\rho }_{\mathrm{T}}\left(135{}^{\circ }\right)$.

Figure 14

(a) $T$ dependence of SMR with the model result at $\left|B\right|=3.5\mathrm{T}$. We subtract $S_{\mathrm{SMR}}\left(T=50\mathrm{K}\right)$ from $S_{\mathrm{SMR}}\left(T\right)$ and obtain $U_{\mathrm{SMR}}\left(T\right)=S_{\mathrm{SMR}}\left(T\right)-S_{\mathrm{SMR}}\left(T=50\mathrm{K}\right)$. The solid curve shows the best fitting of Eq. (5) as a function of $T$. (b)–(d) $B$ dependence of $U_{\mathrm{SMR}}$, and $S_{\mathrm{SHAHE}}$ at (b) 2 K, (c) 5 K, and (d) 10 K with the model results. We fit Eqs. (5) and (6) to each experimental result plotted as open circles.

Figure 15

(a) STEM image of the Pt/GGG interface. (b) The spatial distribution of elements along the Pt/GGG interface probed by EDX. The blue, red, and green lines indicate the intensity of Pt, Ga, and Gd atoms, respectively. The inset is the image of the distribution of Pt, Ga, and Gd atoms, and the color-coded image at the interface.