Current fluctuations in noncollinear single-electron spin-valve transistors

We present a theoretical framework to analyze fluctuations of electric current through a noncollinear single-electron spin-valve transistor in the limit of weak tunnel coupling. The system under consideration consists of two tunnel junctions that connect a small, nonmagnetic metallic island to two ferromagnetic leads with noncollinear magnetization. We study the current noise spectrum as a function of bias voltage, frequency, and the relative angle between the leads’ magnetization directions and find that both the zero- and the finite-frequency current noise are strongly affected by charging energy and spin accumulation in the island.

Figure 1

A metallic island is tunnel coupled to two adjacent ferromagnetic leads with noncollinear magnetization directions enclosing an arbitrary angle $\phi$. This system is called single-electron spin-valve transistor.

Figure 2

Scheme of the relative orientation of the polarization directions of the two leads ${\widehat{\mathbf{n}}}_L,{\widehat{\mathbf{n}}}_R$ and the accumulated spin on the island $\mathbf{S}$, parametrized by the angles $\alpha$ and $\beta$. For clarity reasons, the island spin $\mathbf{S}$ is decomposed into two parts, $\mathbf{S}={\mathbf{S}}_{\mathrm{in}}+{\mathbf{S}}_{\mathrm{pp}}$, where ${\mathbf{S}}_{\mathrm{in}}$ and ${\mathbf{S}}_{\mathrm{pp}}$ are the contributions in the $({\widehat{\mathbf{n}}}_L,{\widehat{\mathbf{n}}}_R)$ plane and perpendicular to it, respectively.

Figure 3

Antiparallel setup ($\phi =\pi$) of the single-electron spin-valve transistor: (a) average current, (b) Fano factor, and (c) second derivative of the current as a function of the applied bias voltage $V$ for different lead polarizations $p$. For all three plots, the remaining parameters were chosen to be $C_G{V}_G=3e/10$, $E_C=50k_{B}T$, and $a=10$.

Figure 4

Current through the noncollinear ($\phi =\pi /2$) single-electron spin-valve transistor with (solid) and without (dashed) the effect of the exchange field. The dashed vertical lines represent the threshold voltages of different charging states. The remaining parameters were chosen to be $p=9/10$, $C_G{V}_G=3e/10$, $E_C=50k_{B}T$, and $a=1$.

Figure 5

Noncollinear single-electron spin-valve transistor: (a) charge current, (b) zero-frequency noise, and (c) Fano factor as a function of the angle between the lead polarization directions $\phi$ with (solid) and without (dashed) the effect of the exchange field. The parameters were chosen to be $p=9/10$, $eV=2E_C$, $C_G{V}_G=3e/10$, $E_C=50k_{B}T$, and $a=1$.

Figure 6

Current fluctuations of the noncollinear ($\phi =\pi /2$) single-electron spin-valve transistor. (a) Zero-frequency Fano factor as a function of bias voltage $V$. (b) Frequency-dependent Fano factor as a function of $\omega$ for different $V$. For both plots the parameters were chosen to be $p=2/10$, $C_G{V}_G=3e/10$, $E_C=50k_{B}T$, and $a=40$.

Figure 7

Half width of the frequency-dependent Fano factor as a function of $V$ for different polarizations $p$. The frequency is measured in units $E_C{\rho }_I{N}_c$ and the remaining parameters were chosen to be $\phi =\pi /2$, $C_G{V}_G=3e/10$, $E_C=50k_{B}T$, and $a=40$.