Driven flow with exclusion and spin-dependent transport in graphenelike structures

We present a simplified description for spin-dependent electronic transport in honeycomb-lattice structures with spin-orbit interactions, using generalizations of the stochastic nonequilibrium model known as the totally asymmetric simple exclusion process. Mean field theory and numerical simulations are used to study currents, density profiles, and current polarization in quasi-one-dimensional systems with open boundaries, and externally imposed particle injection ($\alpha$) and ejection ($\beta$) rates. We investigate the influence of allowing for double site occupancy, according to Pauli's exclusion principle, on the behavior of the quantities of interest. We find that double occupancy shows strong signatures for specific combinations of rates, namely high $\alpha$ and low $\beta$, but otherwise its effects are quantitatively suppressed. Comments are made on the possible relevance of the present results to experiments on suitably doped graphenelike structures.

Figure 1

Log-linear plot of probability density function for polarization at the right end (exit) of a nanotube, for a current injected with ${\mathcal{P}}_{\mathrm{in}}=1$ at its left end. Here $\alpha =\beta =1/2, N_w=14, N_r=20$, and $x_i=0.01$. (A) and (B) refer, respectively, to models with single or conditional double site occupation; in both cases, samples were taken over $N_q=1\times {10}^6$ distinct realizations of impurity configurations. EM refers to the effective-medium description given in Sec. 2.

Figure 2

Log-linear plot of polarization against position $x$ along average flow direction of a nanotube, for a current injected with ${\mathcal{P}}_{\mathrm{in}}=1$ at its left end. Here $\alpha =\beta =1/2, N_w=14, N_r=20$, and $x_i=0.01$. Each of the polarization profiles denoted $N_q=1$ (blue and red) corresponds to a distinct, fixed realization of the impurity distribution.

Figure 3

Log-linear plot of probability density function for polarization at the right end (exit) of a nanotube, for a current injected with ${\mathcal{P}}_{\mathrm{in}}=1$ at its left end. Here $\alpha =3/4, \beta =1/4, N_w=14, N_r=10$, and $x_i=0.01$. (A) and (B) refer, respectively, to models with single or conditional double site occupation; in both cases, samples were taken over $N_q=1\times {10}^6$ distinct realizations of impurity configurations. EM refers to the effective-medium description given in Sec. 2.

Figure 4

(a) Polarizations $\mathcal{P}\left(x\right)$ and (b) full densities $\rho \left(x\right)$ (spin -up plus spin -down) against position $x$ along average flow direction of a nanotube with $N_w=14, N_r=20, x_i=0.01$, and $(\alpha ,\beta )=(1/2,1/2)$ for a current injected with ${\mathcal{P}}_{\mathrm{in}}=1 [\text{A},(\text{B},1\left)\right]$ or ${\mathcal{P}}_{\mathrm{in}}=0 \left[\right(\text{B},0\left)\right]$ at its left end. In all cases, the same fixed impurity realization has been used ($N_q=1$); for each of models A and B, the two sequences of distinct sublattice densities [1, 2] are plotted separately for ease of visualization.

Figure 5

(a) Polarizations $\mathcal{P}\left(x\right)$ and (b) full densities $\rho \left(x\right)$ (spin up plus spin down) against position $x$ along average flow direction of a nanotube with $N_w=14, N_r=20, x_i=0.01$, and $(\alpha ,\beta )=(3/4,1/4)$ for a current injected with ${\mathcal{P}}_{\mathrm{in}}=1 [\text{A},(\text{B},1\left)\right]$ or ${\mathcal{P}}_{\mathrm{in}}=0 \left[\right(\text{B},0\left)\right]$ at its left end. In all cases, the same fixed impurity realization has been used ($N_q=1$); for each of models A and B, the two sequences of distinct sublattice densities [1, 2] are plotted separately for ease of visualization.

Figure 6

Polarization against position $x$ along average flow direction of a nanotube, for a current injected with ${\mathcal{P}}_{\mathrm{in}}=1$ at its left end. Here $\alpha =3/4, \beta =1/4, N_w=14, N_r=20$, and $x_i=0.01$. Average over $N_q=1000$ distinct realizations of the impurity distribution. The curves correspond to fits of central estimates to a form $\mathcal{P}\left(x\right)=exp\left[-{(x/x_0)}^{\delta }\right]$ (see text).

Figure 7

(a) Polarization at right end and (b) steady-state current $J$ for a nanotube with $N_w=14, N_r=20$, and $x_i=0.01$, for a current injected with ${\mathcal{P}}_{\mathrm{in}}=1 [\text{A},(\text{B},1\left)\right]$ or ${\mathcal{P}}_{\mathrm{in}}=0 \left[\right(\text{B},0\left)\right]$ at its left end, against ejection rate $\beta$. Injection rate $\alpha =1/2$ (fixed). Averages over $N_q=100$ distinct realizations of the impurity distribution.

Figure 8

Points (${\rho }_u$ and ${\rho }_v$) show steady-state sublattice full (spin-up plus spin-down) densities against position along flow direction, for model B on a nanotube with $N_w=14$ and $N_r=20$ at $(\alpha ,\beta )=(1/4,1/4)$. ${\mathcal{P}}_{\mathrm{in}}=0$, so impurities are statistically irrelevant in establishing total current and density profiles. Horizontal dashed lines ($U_{\mathrm{th}}$ and $V_{\mathrm{th}}$) show mean field predictions derived from Eqs. (20) and (21). See text.

Figure 9

Steady-state current $J$ as a function of impurity concentration $x_i$ for model B at $(\alpha ,\beta )=(1,1)$ with a fully polarized current injected at the left end (${\mathcal{P}}_{\mathrm{in}}=1$). $N_w=14, N_r=20$. The solid line is an ad hoc exponential fit to the data. The inset shows details of the main figure close to the vertical axis.