Thermoelectric spin accumulation and long-time spin precession in a noncollinear quantum dot spin valve

We explore thermoelectric spin transport and spin dependent phenomena in a noncollinear quantum dot spin valve setup. Using this setup, we demonstrate the possibility of a thermoelectric excitation of single spin dynamics inside the quantum dot. Many-body exchange fields generated on the single spins in this setup manifest as effective magnetic fields acting on the net spin accumulation in the quantum dot. We first identify generic conditions by which a zero bias spin accumulation in the dot may be created in the thermoelectric regime. The resulting spin accumulation is then shown to be subject to a fieldlike spin torque due to the effective magnetic field associated with either contact. This spin torque that is generated may yield long-time precession effects due to the prevailing blockade conditions. The implications of these phenomena in connection with single spin manipulation and pure spin current generation are then discussed.

Figure 1

Noncollinear quantum dot spin valve transport setup. (a) The setup consists of a quantum dot weakly coupled to ferromagnetic contacts $\alpha =L,R$, each with a pinned magnetization axis ${\widehat{m}}_{\alpha }$ oriented along the majority spin and a degree of polarization $p_{\alpha }$. The contact $L\left(R\right)$ acts as the collector (injector) in the forward (reverse) bias direction. The common coordinate axis is chosen to be oriented with respect to that of the quantum dot, with the $\widehat{x}$ axis being pointing in the (longitudinal) transport dimension. The angle between the contact magnetizations is $\theta$. The setup may be spin blockaded for a certain range of bias and for certain values of $\theta$. (b) Spin dynamics comprised of spin precession around the net direction of the effective exchange field ${\vec{B}}_L+{\vec{B}}_R$ and relaxation introduced via charge tunneling to and from the contacts. (c) Transport through the single level quantum dot in the sequential tunneling regime is modeled via density matrix rate equations which may be viewed as transitions between many-electron states labeled 0 through 3. (d) Example of a transport setup that displays spin blockade when the average electron number inside the quantum dot is unity. Spin blockade results in an accumulation of spins antiparallel to the collector contact spin polarization.

Figure 2

Differential conductance $G=\frac{d{J}_q}{d{V}_{\mathrm{app}}}$ versus gate and bias voltages showing the $N=1$ sector for the unpolarized case. Here the diamond edges mark the entry of a conducting energy level of the dot within the transport window. We choose $T_R=T_L=0.7$ K, $\hslash \Gamma =0.01$ meV, and $U=40k_B{T}_L$.

Figure 3

Spin accumulation and currents. (a) Stability plot depicting the $N=1$ sector for the case of symmetric polarization $p_L=p_R=1$, and $\theta =\pi /2$. Nonequilibrium spin transport across the black cut is considered in (b) and (c). (b) Spin accumulation vs applied bias indicating an $S_x=-1/2$ blocking state in the forward bias direction and an $S_z=-1/2$ blocking state in the reverse bias direction (see text). (c) Resulting charge and spin currents depicting pronounced $\widehat{y}$-polarized spin currents $J_y^S$ in the region of Coulomb blockade. (d) Stability plot for the asymmetric polarization ($p_L=1$, $p_R=0.2$) case. In this case the spin blockade and hence spin accumulation only occurs along the forward bias direction. (e) and (f) Resulting spin accumulation and currents according to the dashed line in (d). Coulomb blockaded regions at finite bias voltages feature sizable transversely polarized spin currents with vanishingly small charge and in-plane spin currents. Remaining parameters are the same as in Fig. 2.

Figure 4

Effect of temperature gradient. (a) In the case of symmetric polarization, for a small temperature gradient, the zero bias spin accumulation is always absent. Thus, spin precession is absent at zero bias, resulting in a (b) zero $\widehat{y}$ polarized spin current $[J_y^s(V_{\mathrm{app}}=0)=0]$. (c) In the asymmetric case, however, zero bias spin accumulation occurs and the resulting spin precession causes a (d) nonzero transverse spin current $[J_y^s(V_{\mathrm{app}}=0)\ne 0]$ (blue dashes). The in-plane spin currents and charge currents are vanishingly small in this region.

Figure 5

Polarization and angular dependence. (a) Dependence of the zero bias thermal pure spin current magnitude on the degree of polarization of the right contact. For the unpolarized case and the fully polarized cases, the transverse spin current is zero. The pure spin current magnitude peaks at $p_R=0.5$. (b) Angular dependence of the magnitude of the pure spin current with $p_R=0.2$. The asymmetry is due to the fact that the situation $p_R=0.2$ corresponds to the majority up-spin case.

Figure 6

Dependence of the zero bias spin current magnitude on the relative polarization between the contacts. The spin current vanishes when $p_L=p_R$ and also when either contact is unpolarized and varies quadratically in between. For $p_L>p_R$ we notice a quasilinear variation in the spin current magnitude. The noticed trends here indicate the possibility of realizing a sizable spin current magnitude for a wide range of realistic polarization magnitudes for the two contacts.