# Robustness and universality of surface states in Dirac materials

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Edited by David Vanderbilt, Rutgers, The State University of New Jersey, Piscataway, NJ, and approved April 10, 2018 (received for review December 30, 2017)

## Significance

We predict that generic Dirac materials host ballistically propagating surface states. Despite being nontopological, these states are robust and insensitive to surface disorder, occurring in a universal manner at generic boundaries, both atomically smooth and disordered. These properties originate from the multicomponent character of the Dirac–Bloch wavefunction, which results in strong coupling to generic boundaries. Dispersion of the surface states can be tuned by external gates to exhibit a variety of different regimes, including flat bands and linear crossings within the bulk bandgap. These surface states provide a natural explanation for anomalous long-range edge transport observed recently in a variety of Dirac materials in both topological and nontopological phases.

## Abstract

Ballistically propagating topologically protected states harbor exotic transport phenomena of wide interest. Here we describe a nontopological mechanism that produces such states at the surfaces of generic Dirac materials, giving rise to propagating surface modes with energies near the bulk band crossing. The robustness of surface states originates from the unique properties of Dirac–Bloch wavefunctions which exhibit strong coupling to generic boundaries. Surface states, described by Jackiw–Rebbi-type bound states, feature a number of interesting properties. Mode dispersion is gate tunable, exhibiting a wide variety of different regimes, including nondispersing flat bands and linear crossings within the bulk bandgap. The ballistic wavelike character of these states resembles the properties of topologically protected states; however, it requires neither topological restrictions nor additional crystal symmetries. The Dirac surface states are weakly sensitive to surface disorder and can dominate edge transport at the energies near the Dirac point.

Surface states and the mechanisms allowing them to propagate along crystal boundaries—the topics of long-standing interest of the theory of solids—acquired a new dimension with the advent of topological materials (1, 2). In these materials robust surface states are made possible by nontrivial topology of the bulk bands (1, 3). Here we outline a different mechanism leading to robust surface states, realized in solids with Dirac bands that mimic relativistic particles near band crossings (2). In this scenario robust surface states originate from unusual scattering properties of Dirac particles, occurring for generic boundary conditions at the crystal boundary. As we will see, since this mechanism does not rely on band topology, it can lead to robust surface states in solids with either topological or nontopological bulk band dispersion. The surface states exist for either gapless or narrow-gapped Dirac bulk bands. Furthermore, these states are to some degree immune to surface disorder. Namely, as discussed below, surface modes can propagate coherently by diffracting around surface disorder through system bulk. This diffraction behavior suppresses backscattering and results in exceptionally long mean free paths. Since Dirac surface states require neither special topological properties of the band structure nor special symmetry, they are more generic than the topological surface states. As such, these states can shed light on recent observations of edge transport in nontopological materials.

Indeed, it is often taken for granted that an observation of edge transport signals nontrivial band topology (4⇓⇓–7). However, recent experiments on semiconducting structures, where tunable band inversion enables switching between topological and nontopological phases, indicate that current-carrying edge modes can appear regardless of the band topology (8⇓⇓–11). One piece of evidence comes from transport and scanning measurements in InAs/GaSb, which indicate that helical edge channels survive switching from a topological to a trivial band structure (8). Additionally, refs. 9 and 10 report an unexpectedly weak dependence of edge transport on the in-plane magnetic field. Namely, it is found that the edge transport is observed even when Zeeman splitting is considerably larger than the spin-orbit splitting, i.e., in the nontopological regime. A similar behavior is observed in HgTe devices (11). Furthermore, recently several groups have used Josephson interferometry to directly image long-range edge currents in graphene, a signature nontopological material (12⇓⇓⇓–16). These observations point to the existence of robust nontopological surface states.

As we will see, the Dirac surface states can arise naturally due to strong coupling of electronic waves to generic boundaries. Namely, the phase shifts of waves in the bulk that scatter off the surface have a strong energy dependence near the Dirac point where the particle and “antiparticle” bands cross (or nearly cross). The energy dependence of phase shifts, as always, leads to the formation of states behaving as plane waves confined to the surface and decaying into the material bulk as evanescent waves. The formation of these states is governed by a mechanism that resembles the seminal Jackiw–Rebbi (JR) theory (17) for the states formed at the domain walls separating regions with sign-changing Dirac mass. Unlike the JR problem, however, the Dirac surface states do not have a topological character; i.e., in general they are not protected by topological invariants. Nevertheless, these states are robust and form surface modes with the energies near the Dirac crossing of bulk bands (Fig. 1*C*).

The diffraction-based mechanism that suppresses backscattering and makes the Dirac surface states insensitive to surface disorder has an interesting analogy with the properties of the high-mobility electron gas realized in GaAs/AlAs quantum wells. In these systems an exceptionally high mobility could be achieved by adjusting the well width to reduce the overlap of the carrier mode with the well boundary and, in this way, suppress carrier scattering at the surface disorder. Scattering suppression through this mechanism results in a dramatic increase of the mean free path, growing rapidly vs. the well width,

At this point it is instructive to compare Dirac surface states to the well-known Tamm–Shockley states. These are nontopological states residing inside the bandgap that governs surface physics of many semiconductors. The Tamm–Shockley states form a surface band that splits off the bandgap edge upon varying the surface potential. The existence of these states depends on the details of the crystal structure near the surface, which makes them nonuniversal and less robust than the Dirac surface states (*SI Appendix, The Tamm-Shockley Surface States*). Indeed, unlike the Dirac states, they require fine tuning and are present only in a part of parameter space (Fig. 1*B*). Further, since these states are typically confined to the surface on the scale of a few lattice constants, they are sensitive to surface disorder potential and, unlike Dirac surface states, are easily localized by the disorder.

Dirac surface states arise in diverse fields, from high-energy to solid-states physics. Early work on Dirac surface states in a periodic potential dates back to the 1960s (20, 21). These studies have led to interesting developments in nuclear and particle physics such as the MIT bag model and neutrino billiards (22⇓⇓–25). Recently, the interest in this problem has been renewed with the advent of graphene and other Dirac materials (26, 27). However, while a number of important aspects of these states have been explored for atomically clean boundaries (28⇓⇓–31), the ease with which Dirac surface states emerge, as well as their ubiquitous character, has remained unnoticed. Below we discuss the mechanism underlying this behavior and address the key properties such as robustness, stability, and immunity to disorder. Our work complements recent studies of topological semimetals (32).

## Surface States: General Theory

We first consider the general properties of Dirac surface states in a 3D solid and then focus on the case of a graphene monolayer. We analyze a Dirac Hamiltonian in 3D with boundary conditions of a general form**1** describe the 3D Bloch band structure near the Dirac band crossing. The matrix M is a unitary Hermitian operator constrained by time-reversal symmetry and current conservation (26, 27),

The form of these boundary conditions and the constraints on M in Eq. **2** can be understood as follows. First, since the Dirac equation is first order in derivatives, the boundary condition must be stated in terms of ψ alone without invoking derivatives of ψ. The most general boundary condition can therefore be written as **2**, respectively (for a more detailed discussion see refs. 26 and 27).

The task of finding surface states from the Dirac Hamiltonian of a general form, Eq. **1**, can be simplified by transforming it to a 1D Dirac problem as follows. Without loss of generality, we take the system boundary to be a 2D plane perpendicular to the x direction. Accounting for translation invariance along y and z, we use Fourier transform, seeking the states of the form

To simplify the analysis, we use, without loss of generality, an asymmetric representation for the transformed matrices**5** (see Fig. 2*C*).

The advantage of this representation, in particular the choice of **4** greatly facilitates this analysis. In this representation the operators in Eq. **2** take the form **2** can now be resolved as follows (26, 27). The relation **2** gives**5** is invariant under unitary transformations of valley matrices **5** can now be solved for M of a general form detailed in Eq. **7**, giving states that decay into the bulk as evanescent waves *C*). The energies *SI Appendix, Dirac Mode Dispersion* and Fig. 1). Solutions confined to the surface exist only when *A* and *B* (see discussion below).

The dependence of the dispersion in Eq. **8** on the angles

## A Relation to the Jackiw–Rebbi Bound States

To better understand the unique properties of the bulk Bloch states which enable surface states we sketch a relation between our problem and the seminal JR problem of the midgap states of the 1D Dirac operator with a sign-changing mass. As a first step we perform a similarity transformation that brings M to a standardized form by moving all the complexity of the problem from the boundary conditions into the transformed Hamiltonian (*SI Appendix, Transformation to the Universal Boundary Conditions*). The transformation is generated by a **8**. The matrices

The **10** are given by the *R*-symmetric eigenstates of

This representation helps us to understand the robustness of surface states. It is instructive to treat **11** is nothing but the canonical JR problem, yielding zero-mode eigenstates which at the same time are eigenstates of **11** these states remain bound to the surface albeit with a shifted energy *k*, defines dispersion of surface states. For *A*).

## Edge States in Graphene

This general discussion has direct implications for graphene, the Dirac material best studied to date. In monolayer graphene,

Particle–hole symmetry C, if present, generates universal values *i* surface states form a flat band that touches one of the bulk bands bottom or top, *ii* there are no surface states. However, as we now show, these restrictions are lifted for realistic non-particle–hole-symmetric edges, allowing the phases

The C symmetry can be lifted by an edge potential that creates Dirac band bending near the edge. The edge potential can either occur naturally due to, e.g., edge reconstruction (34) or hydrogen passivation (35, 36) or be induced externally by a side gate, as illustrated in Fig. 3 *A* and *B*. Focusing on the first case, we consider electrostatic potential localized near the edge at a lengthscale of a few atomic spacings **13**. The transfer matrix can be obtained by integrating the Dirac equation over x,**15**. Approximating *M* that describes the boundary condition altered by

We note parenthetically that the latter condition restricts the validity of our approach to short-range edge potentials and narrow-gap Dirac band structures such that

To understand the impact of the edge potential on the edge states dispersion we consider the setup in Fig. 3*B* wherein *A* and *B* for the armchair and the zigzag edge. The armchair edge hosts a one-branch mode with relativistic dispersion**19**, despite its relativistic appearance, has no C-symmetric counterpart; i.e., it does not obey particle–hole symmetry. Furthermore, the dispersion acquires a flat-band character at

The solution for the zigzag edge features a more complex behavior. For each *B*. Modes occurring at *B*) span both positive and negative energies. Upon variation of *B*). The dispersion becomes flat at *B*). Since according to ref. 27 the zigzag boundary condition with

The contribution of surface states to spatially resolved DOS is illustrated in Fig. 2*D* (for derivation see *SI Appendix, Spatially Resolved Density of States*). Surface states give rise to an enhanced DOS near the boundary for one type of carriers, electrons or holes, depending on the *D* the contribution of surface states is seen as a high-DOS region at positive energies, embedded into the family of Friedel oscillations dispersing as

## The Role of Disorder

An interesting aspect of Dirac surface states is their weak interaction with surface disorder. Realistic crystal boundaries often feature strong disorder potential, arising due to dangling bonds and other defects, which impedes transport along the surface. Suppression of conduction by surface disorder is typically quite strong for the Tamm–Shockley states. However, Dirac surface states are to a great extent protected from surface scattering due to their small overlap with the surface disorder. This behavior is reminiscent of the carrier dynamics in GaAs/AlAs quantum wells where the mobility increases drastically with the well width due to a rapid mean free path growth

To illustrate the effect of backcattering suppression by electron-wave diffraction around surface disorder, we consider gaussian short-range correlated disorder at the graphene edge,

In the limit of a weak disorder the mean free path can be evaluated by perturbation theory (*SI Appendix, Disorder at the Edge*). Here we discuss the results for the zigzag surface state (case *i* in Eq. **12**). In this case, the mean free path is**21** predicts mean free path values much greater than the carrier wavelength; the dimensionless parameter

We note that localization effects may become important if the disorder is strong enough. In our case, since disorder is mainly at the surface, the behavior is expected to be quite different for electron energies inside and outside the bulk energy gap. In the first case, electron states with energies within the bulk gap reside near the surface. These states couple to surface disorder relatively strongly and can become localized. In the second case the states at the surface will hybridize with the states in the bulk, which suppresses localization due to surface disorder. In addition, as discussed above, the slow decay of electron states from the surface into the bulk gives the surface states a large width that allows electrons to diffract around surface disorder. Such diffraction also suppresses localization. For quasi-1D surface states, such as those in graphene, the 1D mean free path provides a good estimate for localization length at the energies in the bulk gap. For 2D surface states, on the other hand, the localization length is expected to be much longer than the mean free path estimated perturbatively. The latter in this case sets only a lower bound for localization length.

## The Effect of Magnetic Field

Experimental detection of surface states by conventional transport techniques can be challenging since the signatures of surface states are often obscured by the continuum of bulk states (the overlap of bulk and surface states contributions to the density of states is illustrated in Fig. 3*D*). Here we consider a different approach relying on the Landau-level spectroscopy in a magnetic field applied perpendicular to the surface. The signatures of Landau levels of the states in a 3D bulk are usually softened by the momentum dispersion in the direction along the field. In contrast, the spectrum of the 2D surface states will be discrete. Therefore, while both the bulk states and the surface states produce Landau levels, the spectral features such as, e.g., the tunneling density of states measured by scanning tunneling microscopy will be dominated by the surface states.

To study the effect of magnetic field, we use a simple model of electrons confined by a 2D delta-function sheet potential of the strength proportional to **18**):*SI Appendix, Surface States in Magnetic Field*). For **23**, the discrete levels exist only for one sign of energy, positive or negative.

The discrete character of the surface Landau levels as well as their striking lack of particle–hole symmetry provides a direct and simple diagnostic of the surface states. Further evidence can be obtained using the property of surface states to be tunable through changing the surface potential by side gates (Fig. 3 and Eq. **18**). Due to a periodic dependence on the potential strength, the electron–hole asymmetry can be inverted by reversing the potential sign or by applying a stronger potential.

In conclusion, our key finding is that surface states are a natural attribute of a Dirac band structure, appearing in a robust manner for generic boundary conditions. The surface states feature a number of interesting and potentially useful properties. In particular, we predict that these states are insensitive to surface imperfections: By diffracting around surface disorder electron waves can propagate ballistically with abnormally long mean free path values. These states can coexist with the bulk states or appear within the bulk bandgap; their dispersion can be tuned by gate potential or by magnetic field, giving rise to a range of unique signatures amenable to a variety of experimental probes.

## Acknowledgments

We acknowledge support of the Center for Integrated Quantum Materials under NSF Award DMR-1231319; the MIT Center for Excitonics; the Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences under Award de-sc0001088; and Army Research Office Grant W911NF-18-1-0116.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: oles{at}mit.edu.

Author contributions: O.S. and L.L. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1722663115/-/DCSupplemental.

Published under the PNAS license.

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