Multiple Quantum Phases in Graphene with Enhanced Spin-Orbit Coupling: From the Quantum Spin Hall Regime to the Spin Hall Effect and a Robust Metallic State

We report an intriguing transition from the quantum spin Hall phase to the spin Hall effect upon segregation of thallium adatoms adsorbed onto a graphene surface. Landauer-Büttiker and Kubo-Greenwood simulations are used to access both edge and bulk transport physics in disordered thallium-functionalized graphene systems of realistic sizes. Our findings not only quantify the detrimental effects of adatom clustering in the formation of the topological state, but also provide evidence for the emergence of spin accumulation at opposite sample edges driven by spin-dependent scattering induced by thallium islands, which eventually results in a minimum bulk conductivity $\sim 4e^2/h$, insensitive to localization effects.

Figure 1

(a) Ball-and-stick model of a graphene substrate with randomly adsorbed thallium atoms (concentration is 15%). (b) Same as (a) but with adatoms clustered in islands with a radius distribution varying up to 3 nm (histogram shown in (d)). (c) Zoom-in of a typical thallium island. All thallium atoms are positioned in the hollow position and equally connected to the 6 carbon atoms forming the hexagon underneath (following [8]).

Figure 2

Differential conductance for an armchair ribbon of width $W=50\mathrm{nm}$ with a concentration $n=15\%$ of thallium adatoms segregated into islands with radius $r=0.5\mathrm{nm}$ (a), $r=1.5\mathrm{nm}$ (b), and $r=2\mathrm{nm}$ (c), over a section with length $L=50\mathrm{nm}$. The potential energy on the contacts is set to $V=-2.5\mathrm{eV}$. The dotted lines indicate the conductance value $2e^2/h$.

Figure 3

Spin-resolved spectral current distribution for $r=0.5\mathrm{nm}$ and $E=-33.5\mathrm{meV}$ (a),(b); $r=1.5\mathrm{nm}$ and $E=21.5\mathrm{meV}$ (c),(d); and $r=2\mathrm{nm}$ and $E=-33.5\mathrm{meV}$ (e),(f). The corresponding energies and conductance are indicated by black dots in Fig. 2. The insets in panels (c)–(f) illustrate the local average current distribution in the regions indicated by the squares.

Figure 4

Main frame: Kubo conductivity for thallium clustering and superimposed density ($n_{\mathrm{LR}}$) of long-range impurities. Inset: diffusion coefficient for wave packet with energy $E\approx 0$, for the case $n_{\mathrm{LR}}=0.5\%$, with (solid blue line) and without SOC.