Tunable thermopower in a graphene-based topological insulator

Following the recent proposal by Weeks et al., which suggested that indium (or thallium) adatoms deposited on the surface of graphene should turn the latter into a quantum spin Hall (QSH) insulator characterized by a sizable gap, we perform a systematic study of the transport properties of this system as a function of the density of randomly distributed adatoms. While the samples are, by construction, very disordered, we find that they exhibit an extremely stable QSH phase with no signature of the spatial inhomogeneities of the adatom configuration. We find that a simple rescaling of the spin-orbit coupling parameter allows us to account for the behavior of the inhomogeneous system using a homogeneous model. This robustness opens the route to a much easier experimental realization of this topological insulator. We additionally find this material to be a very promising candidate for thermopower generation with a target temperature tunable from 1 to $80\mathrm{K}$ and an efficiency $ZT\approx 1$.

Figure 1

Upper inset: schematic of our setup: a four-terminal graphene cross with armchair edges and width of the small arm $W=22$. The different colors correspond to an actual calculation of the current density for spin-up electrons upon injection from contact 0. The existence of an edge state is manifest. Main figure: Scaling of the transmissions $T_{j0}$ from contact 0 for various couplings ${\lambda }_{\text{so}}$ and adatom densities $n_{\text{ad}}$, plotted as a function of the effective SO coupling strength ${\lambda }_{\text{so}}^{\mathrm{eff}}={\lambda }_{\text{so}}{n}_{\text{ad}}$. Dashed (black) lines correspond to “longitudinal” transmission $T_{20}$ and dotted lines correspond to “Hall” transmissions $T_{10}$ (red/gray) and $T_{30}$ (green/light gray). The two sets of curves correspond to an energy $E=0$ (open symbols) and $E=0.05$ (filled symbols). Different symbol shapes correspond to different values of SO coupling, ${\lambda }_{\text{so}}=0.02$, $0.05$, $0.1$, and $0.15$ (because they all collapse on the same curve, various symbols may seem indistinguishable from each other). Note that no averaging over adatom configurations has been performed here. Lower inset: illustration of a graphene plaquette. Full lines stand for direct hopping elements while dashed lines correspond to the SO-induced hopping elements.

Figure 2

Statistics of the transmission coefficients $T_{20}$ (lower curve, black) and $T_{30}$ (upper curve, red) when averaged over 2500 adatom configurations, as a function of the effective SO coupling strength. The shaded regions represent one standard deviation from the mean value which is indicated by white symbols (circles for $T_{20}$ and squares for $T_{30}$). The corresponding density of probability for three points (green, cyan, and violet filled circles, from left to right) are given in the inset: as the SO coupling increases, the probability distribution $P\left(T_{20}\right)$ shifts from a Gaussian (right green and middle cyan) to a log-normal (left violet). Fits are indicated by dashed lines. All data points were generated for an energy $E=0$, a fixed adatom density $n_{\text{ad}}=0.1$, and the same system sizes as in Fig. 1.

Figure 3

Transmission coefficients $T_{20}$ (circles) and $T_{30}$ (squares) as a function of the effective SO coupling strength for various values of on-site disorder: $V=0$ (filled black), $V=0.4$ (empty red), and $V=0.8$ (empty dashed magenta). No qualitative change is brought about by disorder, although some deviation from the $V=0$ curve in the crossover region between the metal and the QSH phases starts to be visible at strong enough ($V=0.8$) disorder. The data were averaged over 50 distinct realizations of disorder and adatom configuration for each value of ${\lambda }_{\text{so}}^{\mathrm{eff}}$. The energy was kept at $E=0$, and the system sizes are as in Fig. 1. Inset: longitudinal transmission $T_{20}$ as a function of energy for ${\lambda }_{\text{so}}^{\mathrm{eff}}=0$ (empty symbols) and ${\lambda }_{\text{so}}^{\mathrm{eff}}=0.02$ (filled symbols). Circles are for $V=0$ and triangles for a single disorder configuration with $V=0.8$. Disorder has no effect on $T_{20}$ in the QSH phase ($E<{\Delta }_{\text{so}}\approx 0.1$).

Figure 4

Averaged longitudinal transmission as a function of $W/\xi$ for energy $E=0$. Filled symbols: fixed ${\lambda }_{\text{so}}=0.1$ with $n_{\text{ad}}=0.2$ (circles), $0.15$ (squares), $0.1$ (triangles), and $0.05$ (diamonds). Open symbols: idem but the role of ${\lambda }_{\text{so}}$ and fixed $n_{\text{ad}}$ are exchanged. Stars: ${\lambda }_{\text{so}}=0.02$. Line: $Y\propto \exp -W/\xi$, where $\xi$ is given by Eq. (8). Upper inset: $1/\xi$ (extracted from the data of the main plot) as a function of ${\lambda }_{\text{so}}^{\mathrm{eff}}$ including additional points for different energy $E=0.02$, different aspect ratio of the sample, and more values of $n_{\text{ad}}$ and ${\lambda }_{\text{so}}$ (various symbols). Dashed line: Eq. (8). Lower inset: ${\Delta }_{\text{so}}$ as a function of ${\lambda }_{\text{so}}^{\mathrm{eff}}$ for ${\lambda }_{\text{so}}=0.02$, $0.05$, and $0.1$ (various symbols). Dashed line: Eq. (7). In all cases, error bars due to sample-to-sample fluctuations are smaller than the symbol sizes.

Figure 5

Upper plot: Transmission $T$ as a function of the energy $E$ in a simple armchair ribbon of width $W$ for ${\lambda }_{\text{so}}^{\mathrm{eff}}=0.01$ (left black line: $n_{\text{ad}}=0.2$, crosses: $n_{\text{ad}}=0.5$) with $W=130$, ${\lambda }_{\text{so}}^{\mathrm{eff}}=0.02$ (middle magenta line) with $W=65.5$, and ${\lambda }_{\text{so}}^{\mathrm{eff}}=0.04$ (right violet line and pluses: two different samples) with $W=32.5$. Lower right: $ZT$ as a function of temperature $\theta$ for ${\lambda }_{\text{so}}^{\mathrm{eff}}=0.01$, $0.02$, and $0.04$ from right to left. The Fermi energy was chosen to be equal to $1.5{\Delta }_{\text{so}}$. Lower left: peak value $Z{T}_{\max }$ of $ZT\left(\theta \right)$ as a function of ${\lambda }_{\text{so}}^{\mathrm{eff}}$ for ${\lambda }_{\text{so}}=0.02$ (diamond), ${\lambda }_{\text{so}}=0.05$ (circles), and ${\lambda }_{\text{so}}=0.1$ (crosses).