Numerical study of spin relaxation in a quantum wire with spin-orbit interaction

The spin-relaxation length in quantum wires with spin splitting due to the lack of the structure inversion symmetry is numerically studied using a tight-binding model. The spin-relaxation length is sensitive to the spin-split subband structure in quantum wires. When the spin splitting is smaller than the subband separations, the quantized one-dimensional subbands have a well-defined spin and the spin-relaxation length is considerably enhanced with the decrease in the width.

Figure 1

Calculated subband dispersion relation for $\delta =0.05$ and ${\lambda }_F/a=7$. (a) $W/{\lambda }_F=1.57$, (b) 2.29, (c) 2.86, (d) 3.43, and (e) 4.00. The horizontal dashed line represents the Fermi level. Subband anticrossing starts to take place at the Fermi level when $W>W_1\approx 2.8{\lambda }_F$.

Figure 2

Calculated $y$-spin expectation values of low-lying subbands as function of the width for (a) $\delta =0.02$, (b) 0.03, and (c) 0.05. The top panels show ${⟨{\sigma }_y⟩}_j$, the middle panels show $S_y^2$, and the bottom panels show $S_y^2/N^{\prime }$. The arrows indicate the anticrossing of the spin-split lowest and first-excited subbands.

Figure 3

Calculated spin-correlation function $⟨F_{yy}\left(L\right)⟩$ for $W/{\lambda }_F=3.86$ and $\delta =0.05$. The straight dotted lines show Eq. (18) for corresponding ${\Lambda }_S$.

Figure 4

Calculated spin-relaxation length ${\Lambda }_S$ versus wire width $W$ for (a) $\delta =0.02$, (b) 0.03, and (c) 0.05. The values of mean free path $\Lambda$ are listed together with corresponding symbols. The width where a large dip occurs in $S_y^2/N^{\prime }$ is denoted by the downward arrows. Typical error bars are shown only for the data with $N=2$. The vertical dashed lines represent the wire width corresponding to the Fermi-level crossing of a subband bottom and the numbers denote channel number $N$. The dotted curves show the power-law dependence ${\Lambda }_S\propto W^{-3/2}$ and the dashed straight lines show the exponential dependence.

Figure 5

${\Lambda }_S$ averaged over the same channel number as a function of $\Lambda$ for (a) $\delta =0.02$ and (b) 0.05. The dotted lines show ${\Lambda }_S=0.5\times \Lambda /{\delta }^2$ and the shaded regions correspond to $\Lambda \sim W$.