Nonuniversal shot noise in quasiequilibrium spin valves

We show that the breakdown of the Wiedemann-Franz law due to electron-electron scattering in diffusive spin valves may result in a strong suppression of the Fano factor that describes the ratio between shot noise and average current. In the parallel configuration of magnetizations, we find the universal value $\sqrt{3}/4$ in the absence of a normal-metal spacer layer, but including the spacer leads to a nonmonotonous suppression of this value before reaching back to the universal value for large spacer lengths. On the other hand, in the case of an antiparallel configuration with a negligibly small spacer, the Fano factor is $\sqrt{3(1-P^2)}/4$, where $P$ denotes the polarization of the conductivities. For $P\to \pm 1$, the current through the system is almost noiseless.

Figure 1

Spin valve considered in this paper: Two ferromagnetic wires with length $L_F$ are connected to each other via a normal-metal spacer of length $L_N$ and to electrodes held at different potentials ${\mu }_L$ and ${\mu }_R={\mu }_L-eV$ and at a temperature $T_0\ll \left|eV\right|/k_B$.

Figure 2

The would-be distribution of temperature along the spin valve for spin-up and spin-down electrons in the absence of interaction between them for the antiparallel magnetizations of the electrodes in the extreme case $L_N=0$. As the higher temperature corresponds to the lower conductance, the would-be temperature profiles are highly asymmetric, and switching on the interaction results in a heat exchange whose direction is shown by the wide arrows.

Figure 3

Potential (top) and temperature (bottom) profiles in spin valves with different magnetization configurations. The shaded regions refer to the ferromagnets, where the magnetization directions are denoted by the arrows. In the top figures, the blue curves (upper on the left part of the figure) are for ${\mu }_{\uparrow }$, whereas the red (lower on the left) are for ${\mu }_{\downarrow }$. The three sets of curves are for ${\sigma }_F/{\sigma }_N=0.2$ (solid lines), ${\sigma }_F={\sigma }_N$ (dashed lines), and ${\sigma }_F=50{\sigma }_N$ (dash-dotted lines). We have chosen $L_N=L_F$ and $P=0.8$.

Figure 4

Fano factor of the spin valve for parallel magnetizations as a function of the ratio of minority electron conductance ${\sigma }_{m}A/L_F$ to the normal-metal conductance ${\sigma }_{N}A/L_N$ for different ratios ${\sigma }_M/{\sigma }_m$. Note the log scale in the horizontal axis. For ${\sigma }_{N}A/L_N\to 0$ the normal metal dominates the resistance, and the Fano factor tends to the value $\sqrt{3}/4\approx 0.433$. In the opposite limit ${\sigma }_N/L_N\gg {\sigma }_M/L_F$ a symmetric spin valve does not maintain any spin accumulation, and the same Fano factor $\sqrt{3}/4$ is retained.

Figure 5

Fano factor for antiparallel magnetizations as a function of the ratio of the conductance for minority electrons to the conductance of the normal metal, for different ratios ${\sigma }_M/{\sigma }_m$. Note the log scale in the horizontal axis. For ${\sigma }_{N}A/L_N\to 0$ the normal metal dominates the resistance, and the Fano factor tends to $\sqrt{3}/4$. In the opposite limit ${\sigma }_N/L_N\gg {\sigma }_M/L_F$ the Fano factor tends to $F_{\mathrm{AP}}\le \sqrt{3}/4$ given by Eq. (9).