Tunneling anisotropic thermopower and Seebeck effects in magnetic tunnel junctions

The tunneling anisotropic magnetothermopower (TAMT) and the tunneling anisotropic spin-Seebeck (TASS) effects are studied for a magnetic tunnel junction (MTJ) composed of a ferromagnetic electrode, a zinc-blende semiconductor, and a normal metal. We develop a theoretical model for describing the dependence of a thermally induced tunneling current across the MTJ on the in-plane orientation of the magnetization in the ferromagnetic layer. The model accounts for the specific Bychkov-Rashba and Dresselhaus spin-orbit interactions present in these systems, which are responsible for the $C_{2v}$ symmetry we find in the TAMT and the TASS.

Figure 1

Scheme of a three-layer magnetic tunnel junction. A thermally induced current tunnels across the semiconductor from the ferromagnet electrode into the normal metal. Spin-orbit interaction is responsible for an anisotropic dependence of the (effective) thermopower, and the spin-dependent Seebeck and spin-Seebeck coefficients on the in-plane magnetization orientation $\mathbf{n}$ (green arrow) with respect to a reference crystallographic axis, which in the present case has been taken as the GaAs $\left[110\right]$ direction.

Figure 2

Schematics of the two models used for describing the tunneling barrier (Table 2). The solid (dashed) light (dark) blue line represents the rectangular barrier (delta function) model, where $d$ and $V_0$ correspond to the thickness and the height of the GaAs layer, respectively. Additionally, the figure shows the relative orientation $\theta$ of the in-plane magnetization $\mathbf{n}$ (green arrow) with respect to the $\left[100\right]$ crystallographic axis of the GaAs layer.

Figure 3

Tunneling anisotropic magnetothermopower [Eqs. (6a) and (6b)] dependence on the angle $\varphi$ spanned between the magnetization and the GaAs $\left[110\right]$ direction. (a), (b) Results of the Dirac-delta model. (c), (d) Results of the rectangular barrier model. Solid (dashed) lines were computed by using the values $\alpha =33.52$ eV Å${}^2$ $(\alpha =-23.35$ eV Å${}^2)$ and $\eta =1.5\times {10}^{-4}{\text{K}}^{-3/2}$ $\left(\eta =0.66\times {10}^{-4}{\text{K}}^{-3/2}\right)$. The average temperature between left and right leads has been set to the value $T=4\text{K}$ and the width of the rectangular barrier is $d=80\text{Å}$.

Figure 4

Tunneling anisotropic spin-Seebeck [Eqs. (7a) and (7b)] dependence on the angle $\varphi$ spanned between the magnetization and the GaAs [110] crystallographic axis. Description as in Fig. 3.

Figure 5

Amplitude of the tunneling anisotropic magnetothermopower [Eqs. (6a) and (6b)] as a function of temperature for magnetization oriented along the $\left[\overline{1}10\right]$ crystallographic direction $\left(\varphi ={90}^{\circ }\right)$. (a), (b) Results of the Dirac-delta model. (c), (d) Results of the rectangular barrier model. Solid (dashed) lines were computed by using the values $\alpha =33.52$ eV Å${}^2$ $(\alpha =-23.35$ eV Å${}^2)$, and $\eta =1.5\times {10}^{-4}{\text{K}}^{-3/2}$ $\left(\eta =0.66\times {10}^{-4}{\text{K}}^{-3/2}\right)$. In (c) and (d) the pairs of solid and dashed lines with the largest, middle, and smallest absolute values of the TAMT and ${\mathrm{TAMT}}_{[+]}$ at $T=0\text{K}$ correspond to barrier thicknesses $d=60,80,120\text{Å}$, respectively.

Figure 6

Amplitude of the tunneling anisotropic spin-Seebeck ratio [Eqs. (7b) and (7a)] as a function of temperature for magnetization oriented along the $\left[\overline{1}10\right]$ crystallographic direction $\left(\varphi ={90}^{\circ }\right)$. (a), (b) Results of the Dirac-delta model. (c), (d) Results of the rectangular barrier model. Solid (dashed) lines were computed by using the values $\alpha =33.52$ eV Å${}^2$ $(\alpha =-23.35$ eV Å${}^2)$ and $\eta =1.5\times {10}^{-4}{\text{K}}^{-3/2}$ $\left(\eta =0.66\times {10}^{-4}{\text{K}}^{-3/2}\right)$. In (c) and (d) the pairs of solid and dashed lines with the smallest, middle, and largest absolute values of the TASS and ${\mathrm{TASS}}_{[-]}$ at $T=0\text{K}$ correspond to barrier thicknesses $d=60,80,120\text{Å}$, respectively.

Figure 7

Amplitudes of the TAMT [(a) and (b)] and the TASS [(c) and (d)] as functions of the BR-SOI parameters ${\alpha }_l$ and ${\alpha }_r$. The results correspond to an 80 Å thick rectangular barrier.

Figure 8

Same as in Fig. 7 but for the ${\mathrm{TAMT}}_{[+]}$ [(a) and (b)] and the ${\mathrm{TASS}}_{[-]}$ [(c) and (d)].