Spin Currents in Rough Graphene Nanoribbons: Universal Fluctuations and Spin Injection

We investigate spin conductance in zigzag graphene nanoribbons and propose a spin injection mechanism based only on graphitic nanostructures. We find that nanoribbons with atomically straight, symmetric edges show zero spin conductance but nonzero spin Hall conductance. Only nanoribbons with asymmetrically shaped edges give rise to a finite spin conductance and can be used for spin injection into graphene. Furthermore, nanoribbons with rough edges exhibit mesoscopic spin conductance fluctuations with a universal value of $\mathrm{rms}{G}_s\approx 0.4e/4\pi$.

Figure 1

Ground state spin density for (a) an ideal and (b) an imperfect zigzag GNR. Blue (red) corresponds to up (down) spin density. (c) Band structures of an ideal GNR obtained from DFT and tight-binding approaches.

Figure 2

Spin injection profile from (a) an ideal GNR and (b) a GNR with a distorted edge into a region of $n$-doped graphene. Nonequilibrium densities for spin up (down) electrons are shown in blue (red).

Figure 3

Step disorder: Edge disorder created by a random walk, where the width of the nanoribbon is changed by one hexagon at every step. Steps are made with probability $a/d$, and the maximum deviation of the width is $\le s$ hexagons. Single vacancies: Edge atoms are removed randomly with the probability $a/d$. Extended vacancies: Similar to single vacancies, but also neighboring edge atoms are removed.

Figure 4

Average total conductance $⟨G_{\mathrm{tot}}⟩$ (solid blue line), rms of the total conductance $\mathrm{rms}{G}_{\mathrm{tot}}$ (dashed black line), and rms of the spin conductance $\mathrm{rms}{G}_s$ (solid red line) as a function of $E_F$ ($E_F=0$ is chosen to correspond to zero gate voltage). The data were averaged over 1000 configurations of single vacancies with $d=40a$ and $L=800a$. For comparison, the inset shows the same quantities for the singular case of $t^{\prime }=0$. In this situation, the spin conductance and its fluctuations vanish completely.

Figure 5

Spin conductance fluctuations: (a) $\mathrm{rms}{G}_s$ as a function of $\xi /L$ for $n$- and $p$-doped graphene: step disorder for $n$-type, $d=20a$, $s=3$ (black), single vacancies for $n$- and $p$-type, $d=40a$ (red and blue, respectively) and extended vacancies for $n$-type, $d=30a$ (green). Inset: $\xi /a$ as a function of $E_F$ for different disorder models (colors as in the main panel). (b) $\mathrm{rms}{G}_s$ as a function of $\xi /L$ for step disorder in $n$-doped graphene: $d=20a$ and $s=3$ (red; orange for $W=92a/\sqrt{3}$), $d=35a$ and $s=2$ (black), $d=35a$ and $s=6$ (blue; violet for $W=92a/\sqrt{3}$), $d=20a$ and $s=6$ (green). Inset: Maximum value of $\mathrm{rms}{G}_s$ as a function of $d/a$ for the step disorder models. In both (a) and (b), the solid line corresponds to the DMPK prediction. The data are shown for GNR lengths $L=800a$ (○), $1000a$ (□), $1200a$ (▵), $1400a$ (+), and $1600a$ (×) and width $W=32/\sqrt{3}a$ unless specified otherwise. The $\mathrm{rms}{G}_s$ is estimated from 1000 ($W=32a/\sqrt{3}$) and 750 ($W=92a/\sqrt{3}$) disorder configurations.