Domain walls in gapped graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Domain walls in gapped graphene
G. W. Semenoff, V. Semenoff and Fei Zhou
Department of Physics and Astronomy, University of British Columbia6224 Agricultural Road, Vancouver, British Columbia V6T 1Z1
The electronic properties of a particular class of domain walls in gapped graphene are investigated.We show that they can support mid-gap states which are localized in the vicinity of the domainwall and propagate along its length. With a finite density of domain walls, these states can alterthe electronic properties of gapped graphene significantly. If the mid-gap band is partially filled,thedomain wall can behave like a one-dimensional metal embedded in a semi-conductor, and couldpotentially be used as a single-channel quantum wire.
Graphene is a one-atom thick layer of carbon atomswith a hexagonal lattice structure and where electronswithin ∼ ev of the Fermi energy obey a Dirac equa-tion and have a linear dispersion relation ω = v F | ~k | withFermi velocity v F ∼ c/ A sites a different energy from those on B sites by in-troducing a staggered chemical potential [1]. It can alsoarise from deformations of bonds on the graphene lattice[9] analogous to those known from the study of carbonnanotubes [10]. A third possibility is to use multi-layergraphene where the layers can be stacked so that their in-teraction breaks the sublattice symmetry. In all cases, toretain the features of the Dirac equation the gap shouldbe much less than the nearest neighbor hopping ampli-tude t ∼ . ev .The diatomic material Boron-Nitride (BN) has thesame lattice structure and valence electrons as grapheneand a staggered chemical potential by virtue of havingdifferent atoms on the two sublattices. Monolayers havebeen made in the laboratory [11]. However, the gap istoo large ∼ . ev for Dirac electrons. An approach cur-rently being pursued is to attach a graphene monolayerto a BN substrate. The resulting gap in graphene is es-timated to be ∼ mev [12] which is in the interestingrange. Another approach is epitaxial growth of grapheneon a Silicon-Carbide substrate where a larger magnitudegap ∼ . ev has been observed [13]. In this Paper, we shall consider line-like domain walldefects in the mass pattern in graphene which is gappedby a sublattice symmetry breaking staggered chemicalpotential. We shall find that they can have significantelectronic properties. The domain walls are shown inFigs. 1(b),1(c) where the zig-zag and armchair walls formboundaries between regions where the staggered chemicalpotential is shifted between the two sublattices. Such do-main walls could be realized naturally in BN and wouldbe inherited by graphene on a BN substrate, for example.We shall show that they can give rise to a band of mid-gap states. These states are localized in the vicinity ofthe wall and propagate along its length. If the mid-gapband is partially filled, the domain wall can behave likea one-dimensional metal embedded in a semi-conductor,and could potentially be used as a single-channel quan-tum wire. One might imagine that, once techniques fordeposition of graphene monolayers on substrates are bet-ter developed, the conditions for existence of these do-main wall wires could be created and manipulated to thepoint where they could be used to print electric circuitson graphene sheets.Mid-gap states already play an important role ingraphene. It was pointed out long ago [1] that an in-dex theorem governs the degeneracy of the E = 0 Lan-dau level in graphene in a magnetic field and this level ishalf-filled in the neutral material. This observation hasspectacular experimental confirmation in the half-odd-integer anomalous quantum Hall effect [5][14]-[15]. In ad-dition, theoretical studies of point-like vortex defects in amass condensate due to a Kekule distortion of graphenefind mid-gap electron states which can give the vorticesfractional charge [9][16]-[19], thus giving a two dimen-sional realization of a phenomenon previously known tooccur in one-dimensional linearly conjugated polymerssuch as polyacetylene [20]-[22]. Similar states bound tovortices in a proximity-induced superconducting conden-sate in graphene could lead to anyonic statistics withpotential applications to quantum computing [23]. Anessential common feature of these examples is the exis-tence of “zero-mode” mid-gap states in the spectrum ofthe Dirac Hamiltonian which arise from the interactionwith fields that have a non-trivial topology. In the caseof the vortex, this topology is due to the vorticity. Letus consider a simple example to show that a related phe-nomenon takes place for a domain-wall. Consider the4 × H = ~ v F mv F ~ i ddx + ddy i ddx − ddy − mv F ~ mv F ~ i ddx − ddy i ddx + ddy − mv F ~ (1)The two diagonal blocks correspond to the two graphenevalleys, which transform into each other under parity andtime reversal. A domain wall is described by replacingthe mass m in Eq. (1) by a function m ( x ) which dependson one of the coordinates, x with a soliton profilelim x →−∞ m ( x ) = − m < , lim x →∞ m ( x ) = m > m ( x )in Eq. (1) were a constant with the asymptotic value ofthe mass, m : E = ± v F q ~k + m v F . These describeelectrons in the bulk semi-conductor away from the wall.As well, there is a gapless mid-gap branch whose wave-functions have support near the wall. Explicitly, the (un-normalized) wave-functions and eigenvalues are ψ L ( x, y ) = e ik y y/ ~ − vF ~ R x dx ′ m ( x ′ ) i E = v F k y (3) ψ R ( x, y ) = e ik y y/ ~ − vF ~ R x dx ′ m ( x ′ ) i E = − v F k y (4)Note that this solution exists and is continuum normal-izable for whatever the profile of the position-dependentmass term, it only needs to have the asymptotic behaviorof a topological soliton as in Eq. (2) [24]. In particular, itshould be applicable to a one lattice spacing thick domainwall such as those drawn in Figs.1(b),1(c).What we have found are two bands of mid-gap statescorresponding to one left- and one right-moving one-dimensional massless fermion (for each spin degree offreedom) traveling along the length of the domain wall.An effective Lagrangian describing them would be L = i X s h ψ † Ls ( ∂ t − v F ∂ x ) ψ Ls + ψ † Rs ( ∂ t + v F ∂ x ) ψ Rs i (5)where s labels the two spin states. Effects of impuri-ties and local interactions can be important in one di-mension and should be taken into account. Four-fermionoperators are perturbatively marginal and adding thosewhich do not implement umklapp processes yields theTomonaga-Luttinger model which is a solvable confor-mal field theory with well-known properties. BA (a)(b)(c)
FIG. 1: a)A hexagonal graphene lattice with triangular sub-lattices A (black dots) and B (white dots) connected by vec-tors ~s = (0 , − a, ~s = ` √ / , / ´ a, ~s = ` −√ / , / ´ a with lattice constant a = 1 . A . b) A zig-zag domain wall.c) An armchair domain wall. In b) and c), the chemical po-tentials are − µ on black dots and + µ on white dots; thesublattices which have higher/lower chemical potentials areinterchanged at a domain wall creating a line of miss-matchedneighbors, denoted by a dashed line. To understand the structure of the bands in more de-tail, we must take a closer look at the tight-binding latticemodel of gapped graphene. We shall find features whichare not reflected in the continuum analysis which is onlyvalid in a small region near E = 0. The Hamiltonian is H = X A,i t b † A + ~s i a A + X B,i t a † B − ~s i b B + X A µa † A a A − X B µb † B b B (6)where a † A , a A , b † B , b B are the quantum amplitudes for anelectron to occupy sites labeled A and B on the sublat-tices A and B , respectively. The lattice and sublatticesare depicted in Fig. 1(a). The first terms in Eq. (6) de-scribe electron tunneling between nearest neighbor sites.The terms proportional to µ are on-site energies. Theybreak the sublattice symmetry and generate a gap. TheSchr¨odinger equation is( E − µ ) a A = t X i b A + ~s i , ( E + µ ) b B = t X i a B − ~s i (7)To study the zig-zag domain wall in Fig. 1(b), we solveEq. (7) with µ replaced by µ sign( A y ) and µ sign ( B y ).The spectrum has branches corresponding to electronspropagating in the bulk of the gapped graphene awayfrom the wall, E = ± s µ + t „ √ a k x + cos 3 a k y « + t sin a k y (8) Here, ( k x , k y ) are wave-vectors. This bulk spectrum hasa gap 2 µ and is symmetric about E = 0. Then, there aretwo branches with wave-functions which fall off exponen-tially with transverse distance | A y | , | B y | from the walland are oscillating functions of the longitudinal A x , B x -coordinates with wave-vector k x : E = − t − s µ + 4 t cos √ a k x , cos √ a k x ≤ E = t − s µ + 4 t cos √ a k x , cos √ a k x ≥ t − p µ + 4 t , t − µ ].If t > µ , this band crosses zero energy ( E = 0) at twovalues of k x , and thus agrees with the continuum analysiswhich is only valid when µ << t and which predicts theexistence of two zero energy modes – one for each cross-ing. The spectrum and density of states for µ = 0 . t are depicted in Fig. 2. Note that, unlike the spectrum ofbulk graphene Eq. (8) the zig-zag domain wall spectrumis not symmetric about E = 0. This is evident from itsstructure displayed in Fig. 1(b), where the miss-matchedsites along the wall are entirely black dots with chemi-cal potential − µ . The zig-zag domain wall violates thesymmetry which reflects the sign of the energy. There isan anti-wall where the miss-matched bonds are entirelywhite dots - with energy + µ . Its domain wall spectrumwould have opposite sign to Eqs. (9) and (10).We can get an intuitive understanding of the spectrumin Eqs. (9) and (10) in the limit where µ is large. Initially,neglecting t , there are two energy levels, µ for an elec-tron sitting on a white dot and − µ for an electron sittingon a black dotin Fig. 1(b). Then, if we turn on small t ,the largest effect is for the black dots on each side of thedomain wall which have a nearest neighbor at the samezeroth order energy, − µ . Turning on the hopping wouldsplit the degeneracy of these sites to − µ + t and − µ − t .Note that this does not happen for sites in the bulk awayfrom the domain wall, since they are not degenerate withtheir neighbors – corrections to their spectrum would beat the next higher order in t . The energies − µ − t and − µ + t are identical to the Taylor expansions of Eqs. (9)and (10), respectively, to first order in t. The next orderin the hopping amplitide, second order perturbation the-ory, would take into account hopping to an adjacent sitewith energy + µ and back and would be of order t /µ ,also what one would expect from expanding Eqs. (9) and . D.O.S.-4-2024 E (a) .D.O.S.-4-2024 (b) FIG. 2: (color online) Density of states (DoS in arbitraryunits, colored blue) versus energy E (in units of t) for a zig-zag domain wall a) and an armchair wall b) when µ = 0 . t .Also shown is the DoS for bulk bands (red). (10) as well as (8) to second order in t. The order t /µ contributions are momentum dependent and the energylevels become bands. Then t is made larger than µ , theyspread out into the bands depicted in Fig. 2.In the neutral ground state, the half of the electronstates with lowest energy will be filled. For the zig-zagwall there is a profound difference between two cases -when the midgap band Eq. (10) overlaps the negativeenergy bulk band ( µ/t < /
2) and when it doesn’t ( µ/t > / the domain wallwill borrow some electrons from the bulk . It will then havea finite charge density and a partially filled upper bandwhich will behave like a one-dimensional metal, even inneutral graphene.The armchair domain wall is depicted in Fig. 1(c). Itis oriented along the y -axis. It has the same gapped bulkbranches Eq. (8) as the zig-zag wall. We look for wave-functions which decay exponentially in distance | A x | , | B x | from the wall. They are superpositions of two plane-waves propagating along the wall with wave-vectors k y and k y + 2 π/
3. (This corresponds to mixing of thegraphene valleys.) The spectrum has four bands depictedin Fig. 2. It is E = ± q t sin a k y + µ K ( k y ) where K · (1 − e −√ ak (1) x )(1 − e −√ ak (2) x ) = (cid:16) e − √ a ( k (1) x + k (2) x ) (cid:17) k (1) x and k (2) x must be determined by solving two equa-tions: K − t µ (cid:16) cosh √ a k (1) x + cosh √ a k (2) x (cid:17) andcosh √ a k (2) x = cosh √ a k (1) x + cos a k y , | k y | ≤ π a . Wecan find explicit solutions in the large and small µ/t lim-its. When µ/t is large the spectrum is concentrated atfour values µ >> t : E ≈ (cid:26) ± ( µ + t + . . . ) ± ( µ − t + . . . ) (11)Two of these are inside the bulk spectrum and two arein the gap. They agree with what we would expect when µ is large, where there are two energy levels, − µ and+ µ corresponding to electrons sitting on the black orwhite dots, respectively in Fig. 1(c). Then, the leadingeffect of turning on a small t is that the pairs of adjacentdegenerate states that exist at the location of the domainwall are split by tunneling. Now, unlike for the zig-zag,there are degenerate pairs with both zeroth order energies+ µ and − µ . The splitting produces four domain wallenergies in Eq. (11). Further corrections are of order t /µ which, when taken into account spread the four levels intofour bands. which then get wider as t gets larger.In the limit µ << t we also find four bands, µ << t : E ≈ ± q µ + 4 t (cid:0) a k y (cid:1) ± q t sin a k y + µ t − cos a k y ) (12)As depicted in Fig. 2, the upper and lower band areentirely within the upper and lower bulk bands. Themiddle two overlap the bulk energy gap and themselveshave a gap which is much smaller than the bulk gap,∆ E = 4 µ / t << µ . For a typical small µ ∼ mev and t ∼ . ev , the gap in the mid-band states is tiny, less than 1 mev . This existence of a gap in the spectrum ofstates bound to the domain wall is compatible with thecontinuum analysis since the gap is vanishingly small inthe continuum limit, scaling to zero with the lattice spac-ing a , so it is not visible to the continuum Dirac Hamil-tonian. Intuition for the gap in the armchair spectrumcan also be gained by studying Fig. 1(c). Because of thealternating pattern of pairs of black and pairs of whitedots as one follows the domain wall, the translation sym-metry along the domain wall is by two lattice spacings,rather than one. This reduced translation symmetry willgap the domain wall spectrum, analogous to gapping cre-ated by a Peierls instability. What is surprising here isthat the gap is so small.In summary, we have shown that the simplest domainwalls in gapped graphene can have interesting electronicproperties. A partially filled domain wall band will be-have like a one-dimensional metal. The continuum anal-ysis suggests that similar behavior can be expected forother types of domain walls, such as those arising fromreversing the pattern of a lattice distortion. Analysisof the details of the spectrum in those cases is left tofuture work. They could also occur in other materialswhich have a Dirac spectrum, such as the hypotheticalflux phases of a square lattice where vortices have re-cently been discussed [26] and where domain walls, someof which would support zero modes should exist.G.W.S. thanks Peter Orland for discussions and the In-stitute Galileo Galilei where part of the work was done;we also thank Jun Liang Song for help with Fig. 2. Thiswork was supported in part by NSERC (Canada), Cana-dian Institute for Advanced research and the A. P. SloanFoundation. [1] G. W. Semenoff, Phys. Rev. Lett. , 2449 (1984).[2] A. J. Niemi, G. W. Semenoff, Phys. Rev. Lett. , 2077(1983).[3] R. Jackiw, Phys. Rev. D29 , 2375 (1984)[4] K. S. Novoselov et. al., Science, Vol 306, 666 (2004)[5] K. S. Novoselov et al. , Nature , 197 (2005)[6] A. K. Geim, K. S. Novoselov, Nat. Mater. , 183 (2007).[7] M. I. Katsnelson, Materials Today , 20 (2007).[8] K. Novoselov, Nature Materials , 720 - 721 (2007)[9] C.-Y. Hou, C. Chamon, C. Mudry, Phys. Rev. Lett. ,186809 (2007)[10] C. Chamon, Phys. Rev. B62 , 2806 (2000).[11] K. S. Novoselov et. al., PNAS , 30, 10451 (2005).[12] G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly,J. van den Brink, Phys. Rev.
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