Engineering a Robust Quantum Spin Hall State in Graphene via Adatom Deposition

The 2007 discovery of quantized conductance in HgTe quantum wells delivered the field of topological insulators (TIs) its first experimental confirmation. While many three-dimensional TIs have since been identified, HgTe remains the only known two-dimensional system in this class. Difficulty fabricating HgTe quantum wells has, moreover, hampered their widespread use. With the goal of breaking this logjam, we provide a blueprint for stabilizing a robust TI state in a more readily available two-dimensional material—graphene. Using symmetry arguments, density functional theory, and tight-binding simulations, we predict that graphene endowed with certain heavy adatoms realizes a TI with substantial band gap. For indium and thallium, our most promising adatom candidates, a modest 6% coverage produces an estimated gap near 80 K and 240 K, respectively, which should be detectable in transport or spectroscopic measurements. Engineering such a robust topological phase in graphene could pave the way for a new generation of devices for spintronics, ultra-low-dissipation electronics, and quantum information processing.

Popular Summary

Topological insulators are remarkable materials: they are electrically insulating in their interior, yet they form a novel type of metallic state at their boundaries. The fundamental interest that this class of material arouses and the tantalizing promise that it holds for spin-related electronics and quantum computing have made it a white-hot topic in condensed-matter physics during the past few years. The advent of topological insulators can be traced back to theorists pondering the coupling between the spin and spatial motion (the so-called spin-orbit interaction) of electrons in graphene—a two-dimensional material that itself has earned a prominent place in condensed-matter physics. But at present only one two-dimensional topological insulator, HgTe, has been experimentally identified, whereas many three-dimensional topological insulators have now been discovered. In this theoretical paper, we revive graphene as a viable candidate for a two-dimensional topological insulator by considering adsorption of certain heavy elements on top of a single graphene sheet.

As easily accessible as chemically pure graphene is nowadays, there is practically no hope of observing topological-insulator behavior in this material because its spin-orbit coupling is exceedingly weak. Our idea is to “decorate” an ordinary graphene sheet with a dilute concentration of heavy-element atoms so that graphene “inherits” their strong spin-orbit coupling. By combining various theoretical tools—symmetry arguments, density functional theory, and transport calculations—we predict that a graphene sheet decorated by a few percent of indium or thallium atoms exhibits a robust topological-insulator phase that should be observable with existing experimental fabrication and detection methods. Experimental success in engineering such an easily accessible yet robust two-dimensional topological insulator could lead to a new generation of devices for spintronics, ultralow-dissipation electronics, and quantum computing.

Figure 1

Adatoms in graphene. (a) Depending on the element, adatoms favor either the high-symmetry “bridge” (B), “hollow” (H), or “top” (T) position in the graphene sheet. (b) Detailed view of an H-position adatom, which is best suited for inducing the intrinsic spin-orbit coupling necessary for stabilizing the topological phase. The desired spin-orbit terms mediated by the adatom are illustrated in (c). Red and yellow bonds represent the induced second-neighbor imaginary hopping, whose sign is indicated by the arrows for spin-up electrons. For spin-down electrons, the arrows are reversed.

Figure 2

Band structure and the adatom local density of states (LDOS). All data correspond to one adatom in a $4\times 4$ supercell, with the upper row corresponding to indium and the lower row corresponding to thallium. The left panels in (a) and (d) correspond to the band structure and LDOS computed using DFT without spin-orbit coupling. The horizontal dashed red lines indicate the Fermi level ($E_{\mathrm{F}}$), which shifts due to electron-doping from the adatoms. Insets zoom in on the band structure near the K point within an energy range $-35$ to 35 meV, showing that, without spin-orbit interactions, neither indium nor thallium opens a gap at the Dirac points. The central panels in (b) and (e) are the corresponding DFT results including spin-orbit coupling. Remarkably, in the indium case, a gap of 7 meV opens at the Dirac points, while, with thallium, the gap is larger still at 21 meV. Finally, the right panels in (c) and (f) were obtained using the tight-binding model described in Sec. 3.

Figure 3

Coverage effects. Spin-orbit-induced gap ${\Delta }_{\mathrm{so}}$ (circles) opened at the Dirac points and Fermi level $E_{\mathrm{F}}$ (squares) measured relative to the center of the gap for different indium and thallium adatom coverages. The open and filled symbols represent data for indium and thallium, respectively.

Figure 4

Transport. (a) Configuration used to determine the two-terminal conductance. Semi-infinite clean graphene leads are connected to the sample, which has adatoms distributed randomly across it. The length $L$ of the system is set by the number of columns (indicated by dotted lines). The width $W$ refers to the number of sites along either edge of the figure, corresponding to the number of unit cells of an armchair nanoribbon required to create an individual column. (b) Conductance $G$ as a function of the Fermi energy $E_{\mathrm{F}}$, averaged over 40 independent random-adatom realizations at different concentrations $n_i=0.1$, 0.2, 0.3 for a system of size $W=80$ and $L=40$ with ${\lambda }_{\mathrm{so}}=0.1t$. (c) Current distribution across a sample of size $W=40$ and $L=20$ at $n_i=0.2$, ${\lambda }_{\mathrm{so}}=0.1t$ and $E_{\mathrm{F}}=0.15\mathrm{eV}$. The magnitude of the current is represented by both the arrow size and color. (d) Conductance for the largest simulated system size using realistic parameters for thallium adatoms (${\lambda }_{\mathrm{so}}=0.02t$ and $\delta \mu =0.1t$) estimated from DFT data. Here $W=200$, $L=100$, and the coverage is $n_i=0.15$. In all panels we take $t=2.7\mathrm{eV}$.

Figure 5

Adatom-induced charge modulation. (a) Top and (b) side views of the charge density induced by a thallium adatom in a $4\times 4$ supercell. Yellow (blue) surfaces correspond to a positive (negative) induced charge density (relative to the case where graphene and thallium decouple completely). The charge modulations relax rather rapidly away from the adatom and occur mostly within the two innermost “rings” of carbon sites near the adatom.

Figure 6

Adiabatic continuity. Energy gap at the Dirac point as the Hamiltonian adiabatically deforms from the periodic adatom Hamiltonian $H_{4\times 4}$ at $\lambda =0$ to the Kane-Mele model at $\lambda =1$. The solid blue and dashed red lines correspond to $H_{4\times 4}$ evaluated with parameters appropriate for indium and thallium, respectively. In both cases, the gap remains finite as $\lambda$ varies from 0 to 1, indicating that the Kane-Mele model and adatom Hamiltonian are adiabatically connected and thus support the same quantum spin-Hall phase.

Figure 7

Hexagonal Rashba effect and substrate-induced disorder. Conductance $G$ as a function of the Fermi energy $E_{\mathrm{F}}$ for a system including (a) hexagonal Rashba coupling, (b) uncorrelated disorder randomly distributed over the range $[-\Delta ,\Delta ]$, and (c) correlated disorder as defined in Eq. (e1) with $\rho =0.07$ and $d=5$. In all cases, the system is of size $W=80$ and $L=40$ with adatom coverage $n_i=0.2$ and nearest-neighbor hopping strength $t=2.7\mathrm{eV}$. For panel (a), the intrinsic spin-orbit strength is ${\lambda }_{\mathrm{so}}=0.04t$; for panels (b) and (c), ${\lambda }_{\mathrm{so}}=0.1t$.