Theory of transport through noncollinear single-electron spin-valve transistors

We study the electronic transport through a noncollinear single-electron spin-valve transistor. It consists of a small metallic island weakly coupled to two ferromagnetic leads with noncollinear magnetization directions. The electric current is influenced by Coulomb charging and by spin accumulation. Furthermore, the interplay of Coulomb interaction and tunnel coupling to spin-polarized leads yields a many-body exchange field in which the accumulated island spin precesses. We analyze the effects of this exchange field in both the linear and nonlinear transport regime. In particular, we find that the exchange field can give rise to a high sensitivity of the island's spin orientation on the gate voltage.

Figure 1

Noncollinear single-electron spin-valve transistor; a metallic island is tunnel coupled to ferromagnetic source and drain leads whose polarizations directions enclose an angle $\phi$.

Figure 2

Scheme of relation between defined angles $\alpha ,\beta$, the polarization directions of the two leads ${\widehat{\mathbf{n}}}_L,{\widehat{\mathbf{n}}}_R$, and the accumulated spin on the island $\mathbf{S}$.

Figure 3

Energy scheme of single-electron spin-valve transistor. The island is described as two independent electron reservoirs related to the spin species $\sigma \in \{\uparrow ,\downarrow \}$. The chemical potentials ${\mu }_{L,R}$ describe the applied bias voltage and $\Delta \mu ={\mu }_{\uparrow }-{\mu }_{\downarrow }$ determines the spin accumulation on the central electrode. The charge states $N$ are tunable via gate voltage and ${\epsilon }_F$ denotes the Fermi energy.

Figure 4

Magnitude of exchange field in linear response as a function of applied gate voltage for different angles between the lead polarization directions. Parameters $p_L=p_R=0.3$, ${\Gamma }_L={\Gamma }_R=\Gamma /2$, and $E_C=10k_{B}T$ were chosen.

Figure 5

Scheme of accumulated spin $\mathbf{S}$ in linear response regime. The spin precesses within a plane defined by ${\widehat{\mathbf{n}}}_L-{\widehat{\mathbf{n}}}_R$ and ${\widehat{\mathbf{n}}}_L\times {\widehat{\mathbf{n}}}_R$. The spin precession is accompanied by a decreased magnitude of accumulated spin.

Figure 6

(a) Angle ${\beta }^{\mathrm{lin}}$ and (b) spin accumulation $S^{\mathrm{lin}}$ for different temperatures as a function of gate voltage. The chosen parameters are $p_L=p_R=0.5$, ${\Gamma }_L={\Gamma }_R=\Gamma /2$, and $\phi =\pi /2$.

Figure 7

Linear conductance in units of $e^2\Gamma {\rho }_I{N}_c$ with (solid) and without (dashed) the effect of the exchange field: (a) $V_G$ dependence in case of $\phi =\pi /2$ and (b) $\phi$ dependence with applied gate voltage $C_G{V}_G=4e/10$. For both plots, the parameters $p_L=p_R=0.9$, ${\Gamma }_L={\Gamma }_R=\Gamma /2$, and $E_C=15k_{B}T$ were chosen.

Figure 8

(a) Current $I$ through single-electron spin-valve transistor in units of $e\Gamma {\rho }_I{N}_c$ for different polarizations $p$, (b) the corresponding island spin accumulation $S$, and (c) the relevant occupation probabilities $P_N$ in the case $p=0.3$. For all three plots, the parameters $p_L=p_R=p$, ${\Gamma }_L={\Gamma }_R=\Gamma /2$, $\phi =\pi /2$, $C_G{V}_G=3e/10$, and $E_C=50k_{B}T$ were chosen.

Figure 9

TMR of a noncollinear setup ($\phi =\pi /2$) for different polarizations $p$. Plot (a) shows the bias dependence for $C_G{V}_G=3e/10$ and (b) the gate dependence for $eV=2E_C$. The dashed-dotted lines represent the cases where the exchange field is manually set to ${\mathbf{B}}_{\mathrm{exc}}^r=0$. For both plots, the remaining parameters were chosen to be $p_L=p_R=p$, ${\Gamma }_L={\Gamma }_R=\Gamma /2$, and $E_C=50k_{B}T$.

Figure 10

Second derivative of the current ${\partial }^{2}I/\partial V^2$ in units of $e^3\Gamma {\rho }_I{N}_c$ for two different polarizations $p$ and $\phi =\pi /2$. The excitation energies of the charge/spin states represented by the peaks strongly depend on the lead polarization $p$. Parameters $p_L=p_R=p$, ${\Gamma }_L={\Gamma }_R=\Gamma /2$, $C_G{V}_G=3e/10$, and $E_C=50k_{B}T$ were chosen.

Figure 11

Polarization dependence of the excitation energies $E_{N;\sigma }$ normalized to $E_C$. The chosen parameters are $p_L=p_R=p$, $\phi =\pi /2$, ${\Gamma }_L={\Gamma }_R=\Gamma /2$, $C_G{V}_G=3e/10$, and $E_C=50k_{B}T$.