Mechanism of paramagnetic spin Seebeck effect

We have theoretically investigated the spin Seebeck effect (SSE) in a normal metal (NM)/paramagnetic insulator (PI) bilayer system. Through a linear response approach, we calculated the thermal spin pumping from PI to NM and backflow spin current from NM to PI, where the spin-flip scattering via the interfacial exchange coupling between conduction-electron spin in NM and localized spin in PI is taken into account. We found a finite spin current appears at the interface under the difference in the effective temperatures between spins in NM and PI, and its intensity increases by increasing the density of the localized spin $S$. Our model well reproduces the magnetic-field-induced reduction of the paramagnetic SSE in Pt/${\mathrm{Gd}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ experimentally observed when the Zeeman energy is comparable to the thermal energy, which can be interpreted as the suppression of the interfacial spin-flip scattering. The present finding provides an insight into the mechanism of paramagnetic SSEs and the thermally induced spin-current generation in magnetic materials.

Figure 1

(a) A schematic illustration of Zeeman splitting of a spin-$S$ system. When the magnetic field $B$ is applied, the degeneracy is lifted to create $2S+1$ energy levels with the energy gap $\mathrm{\Delta }E=g{\mu }_{B}B$. (b) A schematic illustration of spin exchange at the NM/PI interface. Upper (lower) spin-flip scattering corresponds to the spin absorption (injection) process from NM (PI) to PI (NM).

Figure 2

A schematic illustration of the NM/PI junction system. $\mathbf{B},\mathbf{S},\mathbf{s},{\mathbf{J}}_{\mathrm{s}}$, and ${\mathbf{E}}_{\mathrm{ISHE}}$ indicate the applied magnetic field, spin of the localized magnetic ion in PI, spin of the conduction electron, interfacial spin current, and electric field due to the ISHE in NM, respectively.

Figure 3

The field-induced magnetization $\langle m\rangle$ (a) and normalized interfacial spin current density $j_{\mathrm{s}}/j_{\mathrm{s}}^0$ (b) with various $S$ as a function of $\xi$. The results in (b) are obtained with $T_{\mathrm{NM}}/T_{\mathrm{PI}}=0.8$.

Figure 4

A schematic illustration of the measurement setup of the paramagnetic spin Seebeck effect in the Pt/GGG system (a) and the magnified view of the Pt/GGG interface (b). ${\mathbf{J}}_{\mathrm{c}}$, $\mathbf{B}$, ${\mathbf{J}}_{\mathrm{s}}$, and $\mathbf{\nabla }T$ show the applied charge current, external magnetic field, interface spin current, and Joule-heating induced temperature gradient, respectively. (c) $M\left(B\right)$ of GGG at various $T$. (d) Experimental results of the $B$ dependence of $V_{2\omega }$ in the Pt/GGG system at various $T$. (d) Theoretical calculation of the $B$ dependence of the paramagnetic spin Seebeck voltage $V_{\mathrm{cal}}$ at various $T$ with $\mathrm{\Delta }T=16$ mK and the material parameters summarized in Table 1.

Figure 5

Calculation of the paramagnetic spin Seebeck voltage $V_{\mathrm{cal}}\left(T\right)$ for Pt/GGG (a) and bulk magnetization $M\left(T\right)$ of GGG (b) with the bulk Curie-Weiss temperature of $-2$ K at selected $B$.

Figure 6

Atomic force microscope images of the polished (111) surface of GGG (a) and the Pt film on the GGG substrate (b).