Graphene Antidot Lattices - Designed Defects and Spin Qubits
Thomas G. Pedersen, Christian Flindt, Jesper Pedersen, Niels Asger Mortensen, Antti-Pekka Jauho, Kjeld Pedersen
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Graphene Antidot Lattices – Designed Defects and Spin Qubits
Thomas G. Pedersen, Christian Flindt,
2, 3
Jesper Pedersen, Niels Asger Mortensen, Antti-Pekka Jauho,
2, 3 and Kjeld Pedersen Department of Physics and Nanotechnology, Aalborg University, DK-9220 Aalborg East, Denmark MIC – Department of Micro and Nanotechnology, NanoDTU,Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark Laboratory of Physics, Helsinki University of Technology, P. O. Box 1100, 02015 HUT, Finland (Dated: September 24, 2018)Antidot lattices, defined on a two-dimensional electron gas at a semiconductor heterostructure, area well-studied class of man-made structures with intriguing physical properties. We point out thata closely related system, graphene sheets with regularly spaced holes (“antidots”), should displaysimilar phenomenology, but within a much more favorable energy scale, a consequence of the Diracfermion nature of the states around the Fermi level. Further, by leaving out some of the holes onecan create defect states, or pairs of coupled defect states, which can function as hosts for electronspin qubits. We present a detailed study of the energetics of periodic graphene antidot lattices,analyze the level structure of a single defect, calculate the exchange coupling between a pair of spinqubits, and identify possible avenues for further developments.
PACS numbers: 73.21.La, 73.20.At, 03.67.Lx
Graphene is the rapidly rising star of low-dimensionalmaterials. Following the initial reports on fabricationby mechanical peeling [1] and epitaxial growth [2], thisexceptional material has stimulated considerable experi-mental [3] and theoretical research [4] as well as proposalsfor novel electronic devices [5]. The promising prospectsfor graphene devices are based on several remarkableproperties. Mainly, the sample quality and mobility (ex-ceeding 15000 cm /Vs [3]) can be very high. In addition,patterning of such monolayer films by e-beam lithogra-phy [3, 6] with features as small as 10 nm [3, 7] is pos-sible. Very recently, spintronics devices have been con-sidered [8]. The incentive for graphene based spintronicslies partly in the long spin coherence time that is charac-teristic of carbon-based materials. This also has obviousadvantages within the field of solid-state quantum infor-mation processing, where confined electron spins havebeen promoted as carriers of quantum information [9].Being a light element, carbon has a rather small spin-orbit coupling, and, moreover, the predominant C iso-tope has a vanishing hyperfine interaction. This makesgraphene, at least in principle, a superior material com-pared to existing quantum computing implementationsin GaAs [10, 11].Antidot lattices, defined on semiconductor het-erostructures, display many intricate transport proper-ties, in particular in magnetic fields where the compet-ing length scales lead to rich physics [12]. In this Letterwe wish to draw attention to the possibility of formingantidot lattices on graphene. As mentioned above, state-of-the-art e-beam lithography has been used to carvegraphene nanoribbons with feature sizes down to tens ofnanometers. We propose to use similar techniques to cre-ate regular holes in the graphene sheet, in order to formantidot lattices. The antidot lattice has the important
FIG. 1: Illustration of the triangular antidot lattice (a) witha unit cell characterized by side length L and hole radius R (b). In (c), several examples with corresponding { L, R } parameters are shown. consequence that it turns the semi-metallic graphene intoa gapped semiconductor, where the size of the gap can betuned via the antidot lattice parameters. As our analysisshall show, this electronic structure can be manipulatedfurther so as to create coupled electron spin qubits, thussuggesting that these perforated graphene sheets are apromising platform for a large-scale spin qubit architec-ture. Localized spin qubit states can be formed in theantidot lattice by deliberately omitting some of the an-tidots. This idea has previously been analyzed for thetwo-dimensional electron gas in e.g. GaAs heterostruc-tures [13]. As we will now argue, moving to graphenehas three major advantages: (i) increased coherence time;(ii) favorable energy scale of the defect states; and (iii)increased lateral confinement.The proposed antidot lattice is simply a triangular ar-ray of holes in a graphene sheet, as illustrated in Fig.1a. The lattice consists of hexagonal unit cells as shownin Fig. 1b, in which a roughly circular hole is created.We characterize the structure by the side length L ofthe hexagonal unit cell and the radius R of the hole,both measured in units of the graphene lattice constant a ≈ .
46 ˚A. A lattice is designated by the notation { L, R } . Note that while L is an integer, R can be non-integer. As is evident from the examples in Fig. 1c, L isequal to the number of carbon atoms in the outermostrow of the hexagon. Also of importance are the totalnumber of sites in the unit cell N Total (equal to the num-ber of atoms before the hole is made) and the number ofremoved atoms N Removed . As an example, for the { , } lattice N Total = 294 and N Removed = 60. Below, resultsfor structures with L ≤
14 and varying R have been com-piled taking care that no dangling bonds are formed, i.e.that all atoms have at least two neighbors. While thesestructures are too small for present-day lithography, re-sults for realistic structures are easily obtained by simplescaling laws, as demonstrated below.We model the structures using a tight-binding (TB) de-scription considering a single π -orbital on each site andassuming a nearest-neighbor hopping integral of − β , with β ≈ .
033 eV [14]. In this description, energy levels arealways distributed symmetrically above and below zero,which defines the Fermi level in the undoped case. TheTB approximation is necessary due to the large antidotcells. It is known to accurately reproduce the low-energypart of the density-functional (DFT) band structure ofgraphene [15]. Edges, however, require a modification ofhopping integrals near the edge to ensure agreement be-tween DFT and TB calculations [16]. We have checkedthat the computed band structures are generally robustagainst such modifications, which simply produce a mi-nor additional opening of the band gap. The electronicband structure and density of states for the { , } struc-ture are illustrated in Fig. 2. Importantly, a substantialenergy gap of approximately 0.73 eV opens around theFermi level [17]. Hence, as hinted above, the periodic per-turbation turns the semi-metal into a semiconductor. Inthe top panel of Fig. 3, band gaps E g of several structuresare plotted versus the quantity N / /N Total . Whenplotted in this manner, a roughly linear behavior is ob-served. This simple result may be rationalized withinthe linearized Hamiltonian approximation treating elec-trons as massless Dirac fermions subject to the periodicperturbation of the antidot lattice. In this description,the wave function is a two-component spinor represent-ing the two sublattices. The corresponding Hamiltonianis the 2 × H = (cid:18) V ( x, y ) v F ( p x − ip y ) v F ( p x + ip y ) V ( x, y ) (cid:19) (1)where V is the periodic antidot potential, p is themomentum operator, and the Fermi velocity v F = √ βa/ (2 ~ ) ≈ m/s. In the absence of a potential,the energy eigenvalues are simply E = ± ~ v F | k | . If the FIG. 2: Energy band structure and associated density ofstates for a { , } antidot lattice. The notation Γ, M, andK refers to high symmetry points of the Brillouin zone. potential is approximated by infinite barriers at the po-sitions of the antidots, the eigenvalue problem is reducedto the form v F ( p x + p y ) ψ = E ψ, (2)with the boundary condition that ψ vanishes in the bar-rier region. The equation is mathematically similar tothe usual effective mass equation. For an antidot lat-tice in a usual semiconductor material such as GaAs,simple scaling arguments lead to a band gap varyingas E g ∝ A − f ( A Removed /A Total ), where A Total is thearea of the unit cell and A Removed is the area removedinside each unit cell. In graphene, a similar behav-ior is expected except that the linear band structurechanges the prefactor from A − to A − / , i.e. E g ∝ A − / g ( A Removed /A Total ) ∝ N − / g ( N Removed /N Total ).The fit in Fig. 3 shows that g approximately fol-lows a square root behavior g ( N Removed /N Total ) ∝ p N Removed /N Total . Thus, the net result is a gap varyingas E g ≈ K × N / /N Total with a constant K ≈ N / /N Total is small and inthis case the linear fit is an excellent approximation. Theweaker scaling ( A − / instead of A − ) of graphene isvery favorable for the purpose of obtaining large bandgaps even for relatively large structures. The practicallimits of present day e-beam lithography probably re-strict the obtainable size of the unit cell to around 10 nmacross corresponding to a total number of carbon atomsof N Total ≈ N Removed ≈ N Total / FIG. 3: Compilation of energy gaps (upper panel) and defectstate binding energies (lower panel). When displayed versus N / /N Total , very simple scaling is observed. Note that N / /N Total is small for realistic structures. intact, i.e. without a hole. Such defects may support lo-calized electronic states and may consequently be utilizedfor electron spin qubits, as we will now demonstrate. Anexample of single and double defects for the { , } struc-ture is shown in Fig. 4. For isolated single defects, wecompute localized states by periodically replicating thesuper cell consisting of one intact and six perforated cellsillustrated in the figure. The states are sufficiently lo-calized that cross-talk between neighboring super cellsis negligible. Periodicity is not crucial for the appear-ance of bound states [13]. Defect states are identified byan energy lying in the fundamental energy gap, i.e. thegap containing the Fermi energy. In fact, other energygaps may exist as illustrated in Fig. 2; here we focussolely on states in the fundamental gap. If the gap issufficiently large (i.e. if N / /N Total is large) severaldefect states are supported. In the lower panel of Fig. 3,a compilation of binding energies for the three lowest de-fect states is shown. We define the binding energy E Bind as the downwards shift of the defect state energy mea-sured from the conduction band edge. Hence, a defectstate at the Fermi energy would have a binding energyof E g /
2. For small band gaps, only a single defect stateis supported but several defect states appear in an irreg-ular pattern as the confinement increases. Note that thescatter in the data points in the plot reflects actual vari-ations and not computational inaccuracy. Importantly,the binding energy in the limit of small band gaps is seento approach a constant fraction ≃ .
07 of the energy gap.Hence, for the 10 nm unit cell considered above, a defect
FIG. 4: Single (left) and double (right) defects for the { , } antidot lattice. To compute defect states, super cells con-taining defects surrounded by six intact units are repeatedperiodically. state would be bound by roughly 16 meV. This impliesthat liquid nitrogen cooling should be sufficient to ob-serve these states.Next, we consider two tunnel coupled defect statesin a “double defect”, illustrated in Fig. 4. With anelectron occupying a non-degenerate state in each de-fect, the spins of the two electrons couple due to theexchange interaction J S · S . If the two single-defectstates are energetically aligned, the exchange coupling isgiven as J = 4 t /U according to the Hubbard approxi-mation. Here, t is the tunnel coupling between the twodefect states, and U is the single-defect Coulomb energy.As discussed in Ref. [9], the exchange coupling consti-tutes a key element in quantum computing architecturesbased on electron spins as qubits, enabling interactionsbetween different qubits. Importantly, the exchange cou-pling can be controlled with external gate potentials.Metallic gates could be realized by lithographic methodsand placed either below or on top of the graphene sheetbut will not be considered further here. For evaluationof the exchange coupling, we calculate the single-defectCoulomb energy U by the method presented in Ref. 18(ignoring overlap between different atomic π -orbitals) us-ing the Ohno form to interpolate between the intra- andlong-range inter-atomic Coulomb coupling. A Hubbard U π for carbon π -orbitals of 20.08 eV [18] and dielectricconstant of 2.5 [19] (as appropriate for graphene on SiO )are applied. The tunnel coupling t is extracted from thesingle-particle energy spectrum.Our findings for the Coulomb energy U are illustratedin Fig. 5. In the plot, R D is the effective defect ra-dius calculated by including half the area of the sur-rounding cells and writing the total area as πR D . Thesmallest U ’s are found for the least localized states forwhich U scales as the expected R − D . The inset shows, as FIG. 5: Compilation of Coulomb energies U for single defectstates showing roughly linear scaling with R − D for delocalizedstates. Inset: Level structure for single- and double defects ina { } lattice. The Coulomb energy for this case is indicatedby the circle. an example, the single-electron level diagram for single-and double defects in a { , } lattice. This structurehas N Total = 864 and N Removed = 348 and supports twosingle-defect states. Of these, the upper one is non-degenerate and the Coulomb energy is 0.315 eV. Due tothe large double defect super cell, this is about the largeststructure that we have been able to analyze. As shown,the level splitting corresponds to a tunnel coupling of t ≈ { , } values we may esti-mate the exchange coupling to be on the order of J ≈ µ eV. Naturally, this value could be tuned by appropri-ate design of the barrier region that, for simplicity, hasbeen constructed from two intact unit cells. Also, goingto larger single defects would decrease U and, in turn,increase J . Note, however, that t depends exponentiallyon barrier width whereas U is only weakly dependent ongeometry. Hence, the geometric influence on J will bedetermined mainly through t rather than U .We believe that the approach outlined above can beextended to more complicated structures. Going from asingle pair of spin qubits in an isolated double defect toseveral coupled spins could be achieved with little addedcomplication. Similarly, a double defect could be re-placed by a linear array of defects. Hence, the number ofqubits can be increased essentially without complicatingthe fabrication procedure. In practice, excellent controlof the e-beam lithography process remains a critical issue.In summary, we have shown that antidot lattices pavethe way for controlled manipulation of the electronicproperties of graphene sheets. The material can be ren-dered semiconducting with a significant and controllableenergy gap. The magnitude of the gap is explained by a simple scaling argument and could reach several tenths ofeVs for realistic structures. Introducing defects into theantidot lattice leads to the formation of localized elec-tronic states. Combined with the extremely long spincoherence time of carbon-based materials this could leadto a practical realization of spin qubits. With a properlydesigned double defect, two-electron states derived fromdefect levels near the Fermi level are found to fulfil therequirements for such qubits.APJ and CF acknowledge the FiDiPro program of theFinnish Academy of Sciences for support during the finalstages of this work. We thank Dr. A. 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