Weak localization, spin relaxation, and spin diffusion: Crossover between weak and strong Rashba coupling limits

Disorder scattering and spin-orbit coupling are together responsible for the diffusion and relaxation of spin density in time-reversal invariant systems. We study spin relaxation and diffusion in a two-dimensional electron gas with Rashba spin-orbit coupling and spin-independent disorder, focusing on the role of Rashba spin-orbit coupling in transport. Spin-orbit coupling contributes to spin relaxation, transforming the quantum interference contribution to conductivity from a negative weak localization (WL) correction to a positive weak antilocalization (WAL) correction. The importance of spin channel mixing in transport is largest in the regime where the Bloch state energy uncertainty $\hslash /\tau$ and the Rashba spin-orbit splitting ${\Delta }_{\mathrm{SO}}$ are comparable. We find that as a consequence of this spin channel mixing, the WL-WAL crossover is nonmonotonic in this intermediate regime, which can be related to recent experimental studies of transport at two-dimensional oxide interfaces.

Figure 1

A schematic illustration of the the quantum correction to conductivity. Quantum interference between a closed electron path (red) and a nearly time-reversed counterpart (blue) alters the backscattering rate when $\mathbf{q}$, the sum of the two incoming momenta, is close to zero. The interference is constructive in the absence of spin-orbit coupling, enhancing backscattering and suppressing the diffusion constant and the conductivity, but it can be destructive and enhance the conductivity when spin orbit is present.

Figure 2

Feynman diagrams for the dominant quantum correction to the conductivity. (a) The upper line and lower lines represent the retarded and advanced Green's functions ${\widehat{G}}^{\pm }$, respectively. The maximally crossed diagrams can be reorganized into a particle-particle ladder-diagram sum. (b) Graphical representation of the ladder diagram sum for the Cooperon, which can be performed by solving a Bethe-Salpeter equation.

Figure 3

Matrix elements of ${\check{\Gamma }}^{-1}\left(\mathbf{q}\right)={\gamma }^{-1}-\check{P}\left(\mathbf{q}\right)$ at zeroth, first, and second order in a wave-vector magnitude $\left(q\right)$ expansion. The illustrated calculation was for $\eta /{\epsilon }_F=0.01$. The horizontal axis is the Rashba band splitting ${\Delta }_{\mathrm{SO}}=2\alpha k_F$ normalized by $\eta$. The plotted quantities are defined in Eq. (18).

Figure 4

Upper panel: Inverse relaxation length $|{\lambda }_i|$ for each channel, normalized by the mean free path $l$. The triplet channels 1, 2, and 4 belong to the eigenstates of $\Gamma \left(q\right)$ in Eq. (17), which are linear combinations of the triplet basis in Eq. (16). Lower panel: The argument of ${\lambda }_1^{-2}$ normalized by $\pi$. The values plotted in this figure were calculated for $\eta /{\epsilon }_F=0.01$. It should be noted that ${\lambda }_{1,2}$ is complex for ${\Delta }_{\mathrm{SO}}\lesssim 0.8\eta$, where the $O\left(q^1\right)$ channel mixing effect is dominant.

Figure 5

Behavior of the weight factors $W_{1,2,3}$ as a function of ${\Delta }_{\mathrm{SO}}/\eta$. The results illustrated here were calculated at $\eta /{\epsilon }_F=0.01$.

Figure 6

The quantum correction ratio $r(\alpha ,L)$ as a function of the band splitting ${\Delta }_{\mathrm{SO}}=2\alpha {\overline{k}}_F$, where the scattering amplitude and the phase-coherence length are fixed at $\eta /{\epsilon }_F=0.01$ and $L/l=100$, respectively. The dashed lines show the contributions from the singlet and the three triplet channels $X_{ii}$, while the black bold line shows the total contribution. The triplet channels 1, 2, and 4 belong to the eigenstates of $\Gamma \left(q\right)$ in Eq. (17), which are linear combinations of the triplet basis in Eq. (16). Note that the WL initially strengthens, then weakens and changes to WAL. There is a plateau in the ${\Delta }_{\text{SO}}$ dependence of $r$ at intermediate values.

Figure 7

The quantum correction ratio $r(\alpha ,L)$ vs the band splitting ${\Delta }_{\mathrm{SO}}=2\alpha {\overline{k}}_F$ and the scattering lifetime energy uncertainty $\eta$. The phase-coherence length is fixed at $L=2\times {10}^4$, which is about 10–100 times larger than the mean free path $l$ (depending on $\eta )$. The black dashed line is ${\Delta }_{\mathrm{SO}}=\eta$. Note that weak localization is initially enhanced $\left(r<-2\right)$ by very weak spin-orbit coupling.

Figure 8

The quantum correction ratio $r(\alpha ,L)$ as a function of the band splitting ${\Delta }_{\mathrm{SO}}=2\alpha {\overline{k}}_F$ and the coherence length $L$. In this plot, the scattering energy uncertainty is fixed at $\eta /{\epsilon }_F=0.01$. All quantities are made dimensionless by setting ${\epsilon }_F=2m=1$.

Figure 9

The behavior of the quantum correction amplitude $\Delta (\alpha ,L)$ as a function of the coherence length $L$, with the spin-orbit coupling taken at ${\Delta }_{\mathrm{SO}}/\eta =0.08,0.16,0.24,\cdots ,1.60$. The scattering amplitude is fixed at $\eta /{\epsilon }_F=0.01$. All quantities are made dimensionless by setting ${\epsilon }_F=2m=1$.