Supplementary Information

Confirmation of detected thermal spin signal without electrical spin injection contribution

Supplementary Figure 1: First-harmonic spin voltage under the thermal spin injection. (a) The first-harmonic spin valve signal under the thermal spin injection together with the schematic of the measurement probe configuration for CoFeAl/Cu lateral spin valve. Here, the interval distance between ferromagnetic wires is300nm and(b) those under the reversed current injection.

Although it is clear that the second-harmonic measurement excludes the contribution from the electrical spin injection (first-harmonic signal), we investigate the influence of the first-harmonic signal in the probe configuration for the thermal spin injection. In order to evaluate the contributionfrom the electrical spin injection under the thermal spin injection shown in top of Fig.S1 (a), we measured the first-harmonic spin voltage together with the second-harmonic voltage. As shown in the bottom of Fig. S1(a), we can see a spin valve signal with the small magnitude of 0.06 m. This value corresponds to 0.0468 V much smaller than the magnitude in the second-harmonic signal. Therefore, the contribution from the electrical spin injection can be negligible. In addition, the parabolic current dependence of the second-harmonic signal also supports the thermal nature of the obtained spin signal.

The observed tiny spin signal in the first-harmonic voltage originated from the inhomogeneous current distribution around the junction. A partial current flowing in the ferromagnetic injector is injected into the Cu channel because of the high electron conductivity of Cu.According to the one-dimensional spin diffusion mode, the contribution of spin current is canceled out because the magnitudes of the incoming and outgoing currents at the interface are the same. However, in realistic case, the current has an inhomogeneous distribution. The nonlocal voltage cannot exclude such inhomogeneous contribution completely. We also confirmed that the reversed spin signal with the same magnitude was observed by reversing the current probe configuration as shown in Fig. S1(b). These experimental facts strongly support the above scenario.

Evaluation of interface resistance

The interface resistance is evaluated by measuring a local four terminal resistance with the probe configuration shown in Supplementary Figure 2. We confirmed that the value of the interface resistance both for CFA/Cu and Py/Cu junction is approximately -40 m. According to Ref. S1, the negative resistance can be understood as the corner resistance contribution because of the highly transparent interface. Therefore, we conclude that the interface resistance for the CFA/Cu is negligibly small, similarly to the Py/Cu interface.

Supplementary Figure 2: Probe configuration of theinterface resistance measurement. When the junction is highly transparent, the resistance contribution is dominated by the corner resistance of the cross-shaped metallic wire.(Ref. S1)

Introducing thespin signal underthermal spin injection

By considering the temperature gradient, the electron flow for eachspin can bedescribed by the following equation.

Here, andare the spin-chemical potential for up-spin and that for the down-spin, respectively. and are the electrical conductivitiesfor up-spin and down-spin, respectively. and are theSeebeck coefficients for up-spin and down-spin, respectively.

We consider the thermal current in homogeneous one-dimensional ferromagneticfilm with the open circuit condition. In this case, the chemicalpotentialµdoes not depend on the spin. By using the open circuit conditionwith eq. (1),can be calculated as the follows,

Here,is the effective Seebeck coefficient in the ferromagnetic systems.We then calculate the generated spin current under this situation. Sincewe know the relationship between and , the generated spin currentcan be described as follows.

(Eq. S3)

Here,is the spin-dependent Seebeck coefficient defined by. P isthe electrical spin polarization defined by.

Then, we consider the spin diffusion process in the lateral spin valvestructure under the thermal spin injection. When we introduce the temperaturegradient across the ferromagnet/nonmagnet interface, the spin currentgiven by Eq. is generated, depending on the temperature gradient in theferromagnet. The generated spin current is injected into the nonmagnet andinduces the non-equilibrium spin accumulation. This non spin accumulationdiffuses not only into the nonmagnet but also back to the ferromagnet. So,the thermally excited spin current can be treated similarly to the electricallygenerated spin current in the lateral hybrid structure. In this case, theposition dependence of the spin signal due to spin accumulation is simplyobtained by replacing the value of P on Eq. (1) by Eq. (S2). Finally, weobtain Eq. (2) for the interval dependence of the spin signal due to thermalspin injection.

Numerical simulation of temperature profile

We have numerically calculated the spatial distributions of the current and the temperature of our CFA/Cu lateral spin valve under the bias current Iac = 0.78 mA in the CFA wire by using the finite element program Comsol Multiphysics. The current flows in the CFA wire through the top and bottom contact pads. Here, we assume that the thermal conductivities of Cu, CFA, Si, and SiO2 are 400, 29.8, 130 and 1.4 Wm−1 K−1, respectively. Supplementary figures 3(a) and 3(b) show the three dimensional color plots of the simulated temperaturefor lateral and vertical profiles, respectively. As shown in Fig. S3(a), the temperature of the CFA injector becomes maximum at the center between the pad and the junction. However, from the magnified image shown in Fig. S3(b), we can confirmed that the temperature gradient ()from the CFA to the Cu is effectively induced at the junction.

Supplementary Figure 3:Numerically simulated spatial distributions of the temperature in the whole device (a) and in the vicinity of the CFA/Cu junction (b) under the bias current of 0.78 mA. The temperature gradient from CFA1 to Cu is clearly confirmed.

Reference

S1.Mu, C., Hu, S., Wang, J. & Kimura, T. Thermo-electric effect in a nano-sized crossed Permalloy/Cu junction under high bias current. Appl. Phys. Lett.103, 132408 (2013).