Protected pseudohelical edge states in proximity graphene ribbons and flakes
We investigate topological properties of models that describe graphene on realistic substrates which induce proximity spin-orbit coupling in graphene. A phase diagram is calculated for the parameter space of (generally different) intrinsic spin-orbit coupling on the two graphene sublattices, in the presence of Rashba coupling. The most fascinating case is that of staggered intrinsic spin-orbit coupling which, despite being topologically trivial, , does exhibit edge states protected against time-reversal scattering for zigzag ribbons as wide as micrometers. We call these states pseudohelical as their helicity is locked to the sublattice. The spin character and robustness of the pseudohelical modes is best exhibited on a finite flake, which shows that the edge states have zero -factor, carry a finite spin current in the crossection of the flake, and exhibit spin-flip reflectionless tunneling at the armchair edges.
Graphene is an exciting material to investigate electrical transportCastro Neto et al. (2009), but it also has remarkable spin properties that make it useful for spintronics applicationsŽutić et al. (2004); Fabian (2007). One outstanding issue in graphene spintronics Han et al. (2014) is the enhancement of spin-orbit coupling (SOC) which is only about 10 eV in pristine grapheneGmitra et al. (2009). Perhaps the principal drive for enhancing SOC is our desire to realize in graphene topological effects such as the quantum spin Hall state (QSHS)Kane and Mele (2005a, b), anomalous quantum Hall effectQiao et al. (2010); Yang et al. (2011), or topological superconductivityRen et al. (2016).
The most promising way to induce large and uniform SOC in graphene is via proximity effects which allow graphene to inherit some properties from the substrate. There is, however, a trade-off. Often the substrates break the sublattice (pseudospin) symmetry, which has two important effects. First, an orbital gap opens due to the staggered potential, and, second, intrinsic spin-orbit coupling acquires a staggered term. In the original model of Kane and MeleKane and Mele (2005a), the intrinsic coupling on A and B sublattices is the same. In proximity graphene, they are in general different, such as graphene on copperFrank et al. (2016). The most extreme case is graphene on transition metal dichalcogenides (TMDCs), schematically depicted in Fig. 1(a). By these substrates spin-valley locking is induced in graphene, manifested in the appearance of the valley Zeeman coupling—opposite (in sign) intrinsic SOCs in the sublatticesGmitra and Fabian (2015); Gmitra et al. (2016); Wang et al. (2015). Rashba coupling is also induced. Valley Zeeman effect is proposed to be detected as a giant spin lifetime anisotropyCummings et al. (2017). There are already experiments on graphene on TMDCsAvsar et al. (2014); Lu et al. (2014); Larentis et al. (2014); Wang et al. (2015, 2016); Yang et al. (2016); Omar and van Wees (2017); Völkl et al. (2017). Weak localization Wang et al. (2015, 2016); Yang et al. (2016); Völkl et al. (2017) and spin transport measurementsAvsar et al. (2014); Omar and van Wees (2017); Dankert and Dash (2016) confirm the proximity induced SOC in graphene in the range of 1–10 meV. Density functional theory calculations predict SOC of about 1 meVGmitra and Fabian (2015); Gmitra et al. (2016); Wang et al. (2015); Yang et al. (2016); Kaloni et al. (2014). In the extreme case of strong SOC, as in graphene on WSe, inverted band structure arises Gmitra et al. (2016); Wang et al. (2015); Alsharari et al. (2016) indicating the possibility of topological edge states. At the moment there is no consistent picture. It was reported that the inverted structure is topologically nontrivial Wang et al. (2015), which appears in line with the claim of helical edge modes Gmitra et al. (2016), but inconsistent with the statement that the system has and there are no topologically protected edge statesWang et al. (2015). We aim to provide a unified picture of, on one hand, the topological nature of proximity models, and, on the other hand, the existence and character of protected edge states. We introduce a modified Haldane modelHaldane (1988) with staggered intrinsic spin-orbit coupling to illustrate how the edge states appear in models with spin-valley locking. There are in general two pairs of edge states formed at each edge. This makes the bulk model trivial, which we confirm by a map of the invariant, an overdue reference needed to classify proximity graphene on a specific substrate. Can protected edge states arise in such a trivial system? Yes, and the key is to gap out unwanted (valley) pair of states. This is effortlessly realized in ribbons (of micron sizes), as we show. The remaining pair is protected against time reversal scattering, just like the QSHS. But unlike helical states of the QSHS, our edge states are pseudohelical, being spin up at one zigzag edge, and spin down at the other. These states are connected by reflectionless spin-flip scattering at the armchair edges in a flake. Pseudohelical states carry spin current, and have zero -factor.
The electronic structure of a bipartite hexagonal lattice with broken sublattice and broken horizontal reflection symmetry, such as graphene on a substrate, can be described by the -symmetric HamiltonianGmitra et al. (2013, 2016); Kochan et al. (2017),
The hopping terms are schematically depicted in Fig. 1(b). The nearest neighbor hopping occurs between sites and , preserving spin . The staggered potential has signs and , for sublattice A and B, respectively. Horizontal reflection symmetry is broken by the Rashba SOC which mixes states of opposite spins and sublattices. The unit vector points from site to site and is the vector of spin Pauli matrices. The last term, the intrinsic SOC, is a next-nearest neighbor hopping. It couples same spins and depends on clockwise () or counterclockwise () paths along a hexagonal ring from site to site . This Hamiltonian distinguishes intrinsic SOC at different sublattices , where stands for A or B. This is the principal extension of the models introduced earlier by Haldane Haldane (1988) and by Kane and MeleKane and Mele (2005a) (in fact, already McClure and YafetMcClure and Yafet (1962) introduced intrinsic SOC for graphene). This extension makes the models experimentally relevant, while introducing new physics.
Fourier transformation and linearization of Hamiltonian (1) around the and points addressed by the valley index , respectively, results in the sum of the following HamiltoniansKochan et al. (2017); Gmitra and Fabian (2015)
corresponding to the order of terms in Eq. (1). The Fermi velocity is expressed as with lattice constant . The sublattice (pseudospin) degrees of freedom are described by Pauli matrices . Following Kane and Mele Kane and Mele (2005a), we use in this work for numerical examples values of , , , and if not indicated differently. In reality we expect weaker couplings from proximity effectsGmitra et al. (2016), but here our goal is to demonstrate qualitative features of the models. We will also comment on what one can expect quantitatively in real samples.
To illustrate the physics of our model, let us first look only at spin up (spinless) electrons and choose the two opposite limits as the uniform, and as the staggered intrinsic SOC model cases. The corresponding Hamiltonians are,
The energy spectrum of a zigzag ribbon for spin-up electrons is plotted in Figs. 2 (a) and (b). The two valleys with bulk-like subbands are well visible. Between valley maxima and minima, edge modes appear due to the chiral nature of grapheneHaldane (1988). While for the uniform case the two edge states have opposite velocities, in our case of spin-valley locking, the edge states have the same velocity, producing spin-up current in the ground state.
The spectra in Figs. 2 (a) and (b) can be understood from simple considerations. For electrons, the phase of the Bloch wave function on sublattice A/B rotates (increases by ) counterclockwise/clockwise. For electrons this behavior is reversed. The staggered potential leads to the following pseudospin-valley state: ; here and label the conduction and valence bands. We now add intrinsic SOC, which can be viewed as an action of a vector potential (Peierls phase), whose rotation within the sublattices is sketched in Figs. 1(b) and (c). If the Bloch phase rotation has the same sense as the rotation of the vector potential, the energy of the state increases. If the rotations are opposite, the energy decreases. (This is analogous to a system with an orbital momentum in a magnetic field.)
In the uniform case, the vector potential rotates counterclockwise [Fig. 1(b)] so that at electrons in sublattice A gain energy and B lose. The opposite is true at , with the result seen in Fig. 2(a). This establishes the connection to Eq. (6), which is a valley-pseudospin Zeeman coupling. Once the effective magnetic field overcomes the staggered potential , the sublattice occupation becomes (B, A; A, B), flipping A and B at , and a chiral state that crosses the gap develops, shown in Fig. 2(a). This is the well known case of a Chern insulatorHaldane (1988).
In the case of staggered intrinsic SOC regime, the Peierls field acts on each sublattice equally in each valley [see Fig. 1(c)]. The energy levels shift in opposite directions in the two valleys, and the sublattice expectation values remain (B, A; B, A); see Fig. 2(b). The Hamiltonian in Eq. (7) represents a valley Zeeman coupling. If the system becomes metallic, as the conduction band in the point has lower energy than the valence band in the point. Nevertheless, there are isolated propagating states, which connect states of same sublattice expectation value from the different valleys.
Let us now reinstate both spins into the picture. The complete spectra for zigzag ribbons are obtained by mirroring the spectra in Figs. 2(a) and (b) around the time reversal invariant point . If we also introduce Rashba SOC, we get additional spin mixing. The results are shown in Figs. 2(c) and (d). In the uniform case, the resulting band structure is additive, leading to two pairs of helical edge states, a manifestation of the QSHSKane and Mele (2005a). The only effect of Rashba SOC is the mixing of spins in the bulk bands, most apparent in the conduction bands.
In the staggered case, Fig. 2(d), there are also (what appears to be) helical edge modes present, as in the QSHS, which would have an energetic overlap with the bulk states if Rashba SOC is absent. Contrary to the QSHS, the edge states with same spin on different edges travel along the same direction, see Fig. 1(c), leading to a net spin current. The effect of Rashba SOC leads to the opening of a bulk gap in the valleys due to the different spin expectation values of valence and conduction bands. This gap is an inverted one, as parts of the former valence bands are now higher in energy than parts of the former conduction bands, which is called mass inversionAlsharari et al. (2016). Inside this Rashba gap two new edge states appear in each valley, with quenched spins. Each valley contributes one mode per edge with opposite velocities on the distinct boundaries [see Fig. 1(c)]. Having both valley-centered and helical states, we term this quantum valley spin Hall state (QVSHS).
In general the symmetries of a bulk Hamiltonian can be used to classify its topological propertiesRyu et al. (2010). The Hamiltonian in Eq. (1) possesses time-reversal symmetry, has broken particle-hole and sublattice symmetries, which puts it into the class of AII HamiltoniansRyu et al. (2010). In two dimensions this leads to the possibility of a classification, which for our set of models is shown in Fig. 3, in the space of the two sublattice intrinsic SOC parameters. This map shows four distinct regions separated by gap closings, where one can expect a change in topology. invariants are calculated numerically Gresch et al. (2017) (see suppl. mat.). The QSHS regions in the upper right and lower left corners exhibit nontrivial topologies, while the QVSHS, focused at diagonal, are trivial. Bulk band structure schemes of representing this phase diagram are in the supplemental material.
We find the staggered cases to have a trivial invariant, as stated already in Ref. Yang et al., 2016 for . In addition, we find that unlike zigzag ribbons, armchair ones have no edge states. The valley Chern number in the staggered case is 1 (see suppl. mat.) as found also in Ref. Alsharari et al., 2016. This Chern number characterizes the states that occur inside the valley and confirms the existence of one conducting channel per edge and valley. We note that our system regarding the valley centered states is very similar to bilayer graphene subject to a perpendicular electric field which shows a quantum valley Hall state (QVHS)Qiao et al. (2011). This system represents twice a copy of our one with a valley Chern number of 2 due to spin degeneracy, showing an absence of states in armchair ribbons as well. This absence is due to intervalley (short-range) scattering as and are mapped onto each other in the armchair geometry.
Crucial for our further analysis is the localization behavior of the edge states. To get the localization length we fit (see suppl. mat.), where is measured from the edge. We find that the spin-polarized edge states decay very fast, over half a unit cell (), whereas the valley states have a much longer localization length (). This indicates that for narrow ribbons valley states should be gapped due to hybridization. A comparison of the band structures for zigzag ribbons of width of ten unit cells for uniform and staggered case is shown in Figs. 4(a) and (b), respectively. Indeed, the valley states exhibit a gap in the QVSHS in Fig. 4(b).
The reason for the larger decay length of the valley states is that they are spectrally close to the bulk states, see Fig. 2(d). We find the relation between the value of inverted gap and the zigzag ribbon width , for which valley edge states gap out due to finite size quantization effects (see suppl. mat.). For realistic gapsGmitra et al. (2016) (in the order of meVs) valley states should not affect the edge physics of ribbons more narrow than 10000 unit cells (about m), wide enough for experimental investigations.
With the valley states gapped out, we are left with a single pair of spin-polarized states at each edge inside the gap. What are these states and how do they compare to the helical modes of the QSHS? In particular, since the spin-up modes head in one direction along the two edges, how do the states meet in a finite flake? To clarify this question, we calculated finite graphene flakes taking states from within the gap as shown in Fig. 4(a) and (b). To simulate short-range scattering we removed one orbital from the left zigzag edge (setting the onsite energy there to ). Additionally we calculated spin and site expectation values as well as probability bond currentsBoykin et al. (2010). In the QSHS, Fig. 4(c), we find as expected a true helical edge state flowing along the boundary, avoiding the short range scatterer and preserving its spin along . The time reversal partner of this state has the opposite chirality and opposite spin polarization (not shown).
Edge states appearing in the finite-size gap of QVSHS are shown in Fig. 4(d). They have several fascinating features. (a) The probability bond current navigates around the short range scatterer and does not scatter back. The reason is that there is only the time-reversal partner, of the edge state at this energy and, as for topologically protected states, backscattering is forbidden as long as the impurity is nonmagnetic and scattering is elastic (mathematically, ). (b) Spin polarization is opposite on the two edges, which are formed by the two sublattices. This is why we call these states pseudohelical—with pseudo describing either the pseudospin-spin locking or “not-really-helical” character of the states. Net spin current flows in this state along the zigzag direction. Also, we explicitly checked that the out-of-plane -factor of the pseudohelical states is zero, as expected since, although they are locally spin polarized, globally the pseudohelical states are spinless. (c) Finally, also at odds with true helical states which exist along armchair, pseudohelical states exhibit reflectionless tunneling through the armchair boundary. The tunneling is assisted by Rashba SOC providing the necessary spin flip. The coupling of the edge states is done via the continuing of velocities, not via the spin character.
To conclude, we need to enrich the parameter space of spin-orbit Hamiltonians when dealing with proximity graphene. Novel models include the extreme spin-valley locking in which intrinsic SOC is opposite on the two sublattices. We provide the full map of the invariant for this extended class of parameters and show that the spin-valley locking models are trivial. Nevertheless, we prove that robust protection against back scattering can be induced in finite ribbons of micron sizes, by gapping out unwanted states and leaving only what we call pseudohelical states in the gap which have fascinating properties. These findings are important for graphene on substrates such as TMDCs, especially with the ability of atomically precise growth of zigzag ribbonsRuffieux et al. (2016). Of interest is also the general idea that we can stabilize edge (and perhaps surface) states against elastic time-reversal scattering by gapping out an otherwise coexisting class of other states.
Acknowledgements.This work was supported by the DFG SFB Grant No. 689 and GRK Grant No. 1570, and the International Doctorate Program Topological Insulators of the Elite Network of Bavaria. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 696656. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de).
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