Theory of thermal spin-charge coupling in electronic systems

The interplay between spin transport and thermoelectricity offers several novel ways of generating, manipulating, and detecting nonequilibrium spin in a wide range of materials. Here, we formulate a phenomenological model in the spirit of the standard model of electrical spin injection to describe the electronic mechanism coupling charge, spin, and heat transport and employ the model to analyze several different geometries containing ferromagnetic (F) and nonmagnetic (N) regions: F, F/N, and F/N/F junctions, which are subject to thermal gradients. We present analytical formulas for the spin-accumulation and spin-current profiles in those junctions that are valid for both tunnel and transparent (as well as intermediate) contacts. For F/N junctions, we calculate the thermal spin-injection efficiency and the spin-accumulation-induced nonequilibrium thermopower. We find conditions for countering thermal spin effects in the N region with electrical spin injection. This compensating effect should be particularly useful for distinguishing electronic from other mechanisms of spin injection by thermal gradients. For F/N/F junctions, we analyze the differences in the nonequilibrium thermopower (and chemical potentials) for parallel and antiparallel orientations of the F magnetizations, as evidence and a quantitative measure of the spin accumulation in N. Furthermore, we study the Peltier and spin Peltier effects in F/N and F/N/F junctions and present analytical formulas for the heat evolution at the interfaces of isothermal junctions.

Figure 1

Schematic illustrations of the (a) Seebeck and (b) spin Seebeck effects. Here, $\Delta T$ is the temperature difference, $V$ the voltage, $j$ the charge current, $j_s$ the spin current, and vertical arrows denote up and down spin projections.

Figure 2

Schematic illustrations of the (a) Peltier and (b) spin Peltier effects, where $j$ and $j_s$ denote the charge and spin currents. The thermal current $j_q$ is different in each region. Small vertical arrows denote up and down spin projections.

Figure 3

A schematic illustration of a ferromagnet metal placed in a thermal gradient, which leads to the generation of a spin current.

Figure 4

Profiles of the (a) spin accumulation, (b) the total chemical potential, and (c) the spin current for Ni${}_{81}$Fe${}_{19}$ at $T=300$ K with $L=100$ nm and $\Delta T=100$ mK. The solid lines show the results obtained if a constant temperature gradient $\nabla T=\Delta T/L_{\mathrm{F}}$ is assumed, while the dashed lines (fully overlapping with the solid ones) show the results obtained if the temperature profile is determined by $\nabla j_u=0$.

Figure 5

A schematic illustration of a F/N junction placed in a thermal gradient.

Figure 6

Profiles of the (a) spin accumulation and the (b) total chemical potential for a Ni${}_{81}$Fe${}_{19}$/Cu junction at $T=300$ K with $L_{\mathrm{F}}=L_{\mathrm{N}}=50$ nm and $\Delta T=-100$ mK. The solid lines show the results for $R_c=1\times {10}^{-16}\Omega {\mathrm{m}}^2$, the dashed lines for $R_c=1\times {10}^{-14}\Omega {\mathrm{m}}^2$.

Figure 7

Profiles of the (a) spin current and the (b) heat current for a Ni${}_{81}$Fe${}_{19}$/Cu junction at $T=300$ K with $L_{\mathrm{F}}=L_{\mathrm{N}}=50$ nm and $\Delta T=-100$ mK. The solid lines show the results for $R_c=1\times {10}^{-16}\Omega {\mathrm{m}}^2$, the dashed lines for $R_c=1\times {10}^{-14}\Omega {\mathrm{m}}^2$.

Figure 8

A schematic illustration of a F/N junction placed in a thermal gradient with a charge current being simultaneously driven through the junction.

Figure 9

Profiles of the (a) spin accumulation and the (b) total chemical potential for a Ni${}_{81}$Fe${}_{19}$/Cu junction at $T=300$ K with $L_{\mathrm{F}}=L_{\mathrm{N}}=50$ nm and $\Delta T=-100$ mK if an electric current compensates the spin accumulation due to the thermal gradient. The solid lines show the results for $R_c=1\times {10}^{-16}\Omega {\mathrm{m}}^2$, the dashed lines for $R_c=1\times {10}^{-14}\Omega {\mathrm{m}}^2$.

Figure 10

Profiles of the (a) spin current and the (b) heat current for a Ni${}_{81}$Fe${}_{19}$/Cu junction at $T=300$ K with $L_{\mathrm{F}}=L_{\mathrm{N}}=50$ nm and $\Delta T=-100$ mK if an electric current compensates the spin accumulation due to the thermal gradient. The solid lines show the results for $R_c=1\times {10}^{-16}\Omega {\mathrm{m}}^2$, the dashed lines for $R_c=1\times {10}^{-14}\Omega {\mathrm{m}}^2$.

Figure 11

A schematic illustration of a F/N junction in the electrical spin-injection setup, where (a) refers to an isothermal junction and (b) to the situation where $j_q\left(x\right)=0$. The fact that in (b) the temperature at one end of the junction is not given as an external boundary condition, but has to be calculated from the model is implied by “?”.

Figure 12

Profiles of the spin accumulation (a) and the spin current (b) for a Ni${}_{81}$Fe${}_{19}$/Cu junction with $L_{\mathrm{F}}=L_{\mathrm{N}}=1\mu \mathrm{m}$, $R_c=1\times {10}^{-16}\Omega {\mathrm{m}}^2$, and $j={10}^{11}$ A/m${}^2$. The solid lines show the results obtained for an isothermal junction at $T=300$ K, whereas the dashed lines show the results obtained for a junction with $j_q\left(x\right)=0$ and $T(-L_1)=300$ K.

Figure 13

A schematic illustration of a F/N junction in the Silsbee-Johnson spin-charge coupling setup, where (a) refers to an isothermal junction and (b) to the situation where $j_q\left(x\right)=0$. The fact that in (b) the temperature at one end of the junction is not given as an external boundary condition, but has to be calculated from the model is implied by “?”.

Figure 14

A schematic illustration of a F/N/F junction placed in a thermal gradient.

Figure 15

Profiles of the (a) spin potential, (b) the total chemical potential, and (c) the spin current for a Ni${}_{81}$Fe${}_{19}$/Cu/Ni${}_{81}$Fe${}_{19}$ junction at $T=300$ K with $L_1=L_2=100$ nm, $L_{\mathrm{N}}=50$ nm, and $\Delta T=-100$ mK. The solid lines show the profiles for the parallel configuration, the dashed lines for the antiparallel configuration.

Figure 16

Difference between the chemical potential drops of the parallel and antiparallel configurations $\Delta {\mu }^{\uparrow \uparrow }-\Delta {\mu }^{\uparrow \downarrow }$ as a function of the length of the N region $L_{\mathrm{N}}$ for a Ni${}_{81}$Fe${}_{19}$/Cu/Ni${}_{81}$Fe${}_{19}$ junction at $T=300$ K with $L_1=L_2=100$ nm and $\Delta T=-100$ mK.

Figure 17

A schematic illustration of a F/N/F junction in the electrical spin-injection setup, where (a) refers to an isothermal junction and (b) to the situation where $j_q\left(x\right)=0$. The fact that in (b) the temperature at one end of the junction is not given as an external boundary condition, but has to be calculated from the model, is implied by “?”.

Figure 18

Profiles of the heat current for an isothermal Ni${}_{81}$Fe${}_{19}$/Cu/Ni${}_{81}$Fe${}_{19}$ junction at $T=300$ K with $L_1=L_2=100$ nm, $L_{\mathrm{N}}=50$ nm, and $j={10}^7$ A/m${}^2$. The solid line shows the profile for the parallel configuration, the dashed line for the antiparallel configuration.

Figure 19

(a) Temperature profile of a Ni${}_{81}$Fe${}_{19}$/Cu/Ni${}_{81}$Fe${}_{19}$ junction with with $j_q\left(x\right)=0$, $T_1=300$ K, $L_1=L_2=100$ nm, $L_{\mathrm{N}}=50$ nm, and $j={10}^{11}$ A/m${}^2$. The solid line shows the profile for the parallel configuration, the dashed line for the antiparallel configuration. (b) The profile of the temperature difference between the parallel and antiparallel configurations is shown in the inset.