Aharonov-Bohm effect and broken valley-degeneracy in graphene rings
P. Recher, B. Trauzettel, A. Rycerz, Ya. M. Blanter, C. W. J. Beenakker, A. F. Morpurgo
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Aharonov-Bohm effect and broken valley-degeneracy in graphene rings
P. Recher,
1, 2
B. Trauzettel, A. Rycerz, Ya. M. Blanter, C. W. J. Beenakker, and A. F. Morpurgo Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krak ´ o w, Poland (Dated: October 24, 2018)We analyze theoretically the electronic properties of Aharonov-Bohm rings made of graphene. Weshow that the combined effect of the ring confinement and applied magnetic flux offers a controllableway to lift the orbital degeneracy originating from the two valleys, even in the absence of intervalleyscattering. The phenomenon has observable consequences on the persistent current circulatingaround the closed graphene ring, as well as on the ring conductance. We explicitly confirm thisprediction analytically for a circular ring with a smooth boundary modelled by a space-dependentmass term in the Dirac equation. This model describes rings with zero or weak intervalley scatteringso that the valley isospin is a good quantum number. The tunable breaking of the valley degeneracyby the flux allows for the controlled manipulation of valley isospins. We compare our analyticalmodel to another type of ring with strong intervalley scattering. For the latter case, we study a ringof hexagonal form with lattice-terminated zigzag edges numerically. We find for the hexagonal ringthat the orbital degeneracy can still be controlled via the flux, similar to the ring with the massconfinement. PACS numbers: 73.23.-b, 73.23.Hk, 73.23.Ra, 81.05.Uw
I. INTRODUCTION
Graphene offers the remarkable possibility to probepredictions of quantum field theory in condensed mat-ter systems, as its low-energy spectrum is described bythe Dirac-Weyl Hamiltonian of massless fermions. How-ever, in graphene, Dirac electrons occur in two degeneratefamilies, corresponding to the presence of two differentvalleys in the band structure – a phenomenon known as“fermion doubling”. This valley degeneracy makes it dif-ficult to observe the intrinsic physics of a single valley inexperiments, because in many cases the contribution ofone valley to a measurable quantity is exactly cancelledby the contribution of the second valley. A prominentexample that single-valley physics is interesting, is theproduction of a fictitious magnetic field in a single val-ley by a lattice defect or distortion. The field has theopposite sign in the other valley, so its effect is hiddenwhen both valleys are equally populated. Another exam-ple is the existence of weak antilocalization in diffusivegraphene, which is destroyed by intervalley scattering. Therefore, from a fundamental point of view, it is de-sirable to find a feasible and controlled way to lift thevalley degeneracy in graphene. From a more practicalpoint of view, the lifting of the orbital degeneracy is es-sential for spin-based quantum computing in graphenequantum dots , which is a promising direction of futureresearch because of the superior spin coherence proper-ties expected in carbon structures.Here, we show that the confinement of electrons ingraphene in an Aharonov-Bohm (AB) ring (see Fig. 1)provides a conceptually simple way to achieve a con-trolled lifting of the valley degeneracy. We find that thering confinement – which breaks the effective time rever- aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa F aW FIG. 1: A circular graphene ring of radius a and width W subjected to a magnetic flux Φ threading the ring. sal symmetry (TRS) in a single valley in the absence ofintervalley scattering – leads generically to a lifting ofthe valley degeneracy controllable by magnetic flux. Todemonstrate this, we choose an analytical model with asmooth ring boundary described by a mass term in Sec-tion II. Such a mass term might be generated in a realsystem by the influence of the lattice of the substrateon the band structure of the sample, see Refs. 8,9 forconcrete examples of such an effect. However, our con-clusions are not restricted to the mass confinement butpotentially hold for any boundary that preserves the val-ley isospin as implied by general symmetry arguments. We show that the signature of the broken valley degen-eracy is clearly visible in the persistent current and theconductance through the ring. It is further illustratedhow to use the lifting of the valley degeneracy with fluxto manipulate and measure the valley isospin.In Section III, we compare our analytical model for asmooth boundary to a system where intervalley scatter-ing is strong. For this purpose, we study a ring of hexago-nal shape with zigzag edges where intervalley scattering isinduced at the corners of the hexagon. We calculate thespectrum numerically in a tight-binding approach andfind that the orbital degeneracy can still be tuned by themagnetic flux, similar to the analytical model. We testthis ability against a small distortion of the 6-fold sym-metry of the ring and find small avoided crossings at zeroflux.
II. RING WITH SMOOTH BOUNDARY
In this section, we analyze in detail the spectral prop-erties of a graphene ring subjected to a magnetic fluxand its signatures in persistent current and conductancethrough the ring assuming a smooth confinement inducedby a space-dependent mass term in the Dirac equation.We also discuss how to address the valley degree of free-dom in such a ring structure.
A. Spectrum
The graphene ring with valley degree of freedom τ = ± is modeled by the Hamiltonian (¯ h = c = 1) H τ = H + τ V ( r ) σ z , (1)where we use the valley isotropic form for H = v ( p + e A ) · σ with p = − i∂/∂ r , − e being the electron charge, v the Fermi velocity and σ i = x,y,z are the Pauli matri-ces. The vector potential is A = (Φ / πr ) e ϕ with Φ themagnetic flux threading the ring, see Fig. 1. The termproportional to σ z in Eq. (1) is a mass term confiningthe Dirac electrons on the ring. Introducing polar coor-dinates, the Hamiltonian H is written as H ( r, ϕ ) = − iv (cos ϕσ x + sin ϕσ y ) ∂ r − iv (cos ϕσ y − sin ϕσ x ) 1 r (cid:18) ∂ ϕ + i ΦΦ (cid:19) . (2)The angular orbital momentum in the z -direction is l z = − i∂ ϕ and Φ = 2 π/e . The two valleys τ = ± decoupleand we can solve the spectrum for each valley separately, H τ ψ τ = Eψ τ . We note the similarity of H τ to a ringwith the Rashba interaction. However, an importantdifference is that for graphene, the confinement potentialacts on the (pseudo)spin, whereas for Rashba interac-tion the confinement potential is spin-independent. Asa consequence, the confining potential in Eq. (1) couplesthe pseudo-spin components and breaks effective TRS( p → − p , σ → − σ ) even in the absence of a flux Φ. Since H τ commutes with J z = l z + σ z , its eigenspinors ψ τ are eigenstates of J z , ψ τ ( r, ϕ ) = e i ( m − / ϕ (cid:18) χ τ ( r ) χ τ ( r ) e iϕ (cid:19) (3) - 1 - 0.5 0 0.5 11.571.581.591.6 [ [ [ v / W ] E + m a) -1 - 0.5 0 0.5 11.571.581.591.6 E ± m [ v / W ] [ [ b) FIG. 2: Energy spectrum E τ m ( E >
0) as a function of mag-netic flux Φ for various total angular-momentum values m (blue m >
0, red m <
0) using Eq. (8) with a/W =10: (a)shows a single valley only τ = +1. The dotted lines are theexact numerical evaluation of E using Eq. (6). In (b) we showthe full spectrum including the other valley τ = − ± Φ symmetry in the combined spec-trum of both valleys and the lifting of the valley degeneracyat finite Φ are clearly visible. with eigenvalues m , where m is a half-odd integer, m = ± , ± , . . . The radial component χ τ ( r ) ≡ ( χ τ ( r ) , χ τ ( r )) satisfies e H τ ( r ) χ τ ( r ) = Eχ τ ( r ) with e H τ ( r ) = − ivσ x ∂ r + τ V ( r ) σ z + vσ y r (cid:18) m − m + (cid:19) , (4)where we have defined m = m + (Φ / Φ ).For V ( r ) = 0, χ τ ( r ) and χ τ ( r ) are solutions to Bessel’sdifferential equation of order m − and m + , respec-tively. Therefore, the eigenspinor for e H τ ( r ) with energy E and V ( r ) = 0 can be written as χ τ = a τ H (1) m − ( ρ ) i sgn( E ) H (1) m + ( ρ ) + b τ H (2) m − ( ρ ) i sgn( E ) H (2) m + ( ρ ) , (5)where H (1 , ν ( ρ ) are Hankel functions of the (first, sec-ond) kind and the dimensionless radial coordinate is ρ = | E | r/v . The coefficients a τ and b τ are determinedby the boundary condition of the ring induced by V ( r )(with V ( r ) → + ∞ outside the graphene ring). We usethe infinite mass boundary condition ψ τ = τ ( n ⊥ · σ ) ψ τ where n ⊥ = ( − sin ϕ, cos ϕ ) at r = a + W and with op-posite sign at r = a − W . Here a is the ring radius and W its width (see Fig. 1). Eliminating the coefficients a τ and b τ gives the energy eigenvalue equation z = z ∗ with z = H (1) m − ( ρ ) − τ sgn( E ) H (1) m + ( ρ ) H (1) m − ( ρ ) + τ sgn( E ) H (1) m + ( ρ ) , (6)which is equivalent to φ = πn with φ the phase of z and n an integer. In Eq. (6), we have abbreviated ρ ≡ | E | ( a − W ) /v and ρ ≡ | E | ( a + W ) /v . To obtainan analytical approximation of the spectrum, we use theasymptotic form of the Hankel functions for large ρ , in-cluding corrections up to order 1 /ρ . This indeed is thedesired limit as ρ = | E | r/v ∼ | E | a/v ∝ a/W ≫ and leadsto the following energy eigenvalue equation | E | = vW (cid:18) n − τ sgn( E )2 (cid:19) π + 12 (cid:16) va (cid:17) m | E |− τ sgn( E ) (cid:16) va (cid:17) vW m | E | . (7)An iteration of Eq. (7) by replacing | E | on the right-hand-side of the equation by the first (leading) term of | E | givesthe energy eigenvalues (neglecting terms of O [( W/a ) ]) E τnm = ± ε n ± λ n m m ∓ τ (cid:0) n + (cid:1) π ! . (8)In Eq. (8), ε n = v ( n + ) π/W , n = 0 , , , ... , and λ n = ( v/a ) / ε n . These energy eigenvalues are plot-ted as a function of flux for n = 0 and different values of m (its half-odd integer values reflect the π -Berry phaseof closed loops in graphene) in Fig. 2. Fig. 2(a) shows theenergy levels for one valley, τ = +1. It is clearly visiblethat E τ ( m ) = E τ ( − m ), since effective TRS is broken bythe confinement. In Fig. 2(b), the spectrum of both val-leys is shown with full lines for τ = +1, and dashed linesfor τ = −
1. At Φ = 0, E τ ( m ) = E − τ ( − m ) as it shouldbe, since real TRS is present at zero magnetic field. Cru-cially, however, at finite Φ, E + = E − in general, showingthat the valley degeneracy is indeed lifted since effectiveand real TRS are broken . If n ≫
1, the term ∝ τ m in Eq. (8) becomes suppressed and the valley degener-acy is restored, correctly predicting that the spectrumis insensitive to the boundary condition if 2 π/q n ≪ W ,where q n = π ( n + ) /W is the transverse wave number.We show next that a broken valley degeneracy resultsin observable features in the persistent current and theconductance through the ring. B. Persistent current and conductance
The persistent current in the closed ring is given at zerotemperature by j = − P τ P nm ∂E τnm /∂ Φ where the sumruns over all occupied states. In Fig. 3, we show the per-sistent current as a function of number of electrons on [ [ j N=1N=2 N=3N=4 a) - - 0.5 - 0.25 0 0.25 0.50246 [ [ j N=1N=2 N=3N=4 b) FIG. 3: Persistent current as a function particle number N (including spin) and flux Φ for n = 0 ( E > τ = +1 whereas (b) includes both valleys.Curves for different N are displaced with dashed horizontallines defining j = 0 for each curve. The broken valley degen-eracy is clearly visible in (b) via two substructures of length∆Φ = 2Φ /π and ∆Φ = (1 − (2 /π ))Φ whereas (a) predictsa non-zero persistent current at Φ = 0 due to effective TRSbreaking in a single valley. the ring N (including spin) and magnetic flux relative tothe half-filled band. (We subtract the contribution to thepersistent current that arises from all states with E < .In Fig. 3(a), only one valley, τ = +1 is considered and a finite persistent current at Φ = 0 is predicted. Therefore,a non-zero persistent current at zero flux detects valleypolarization. In Fig. 3(b), we show the case of equal pop-ulation of both valleys. Then, the persistent current as afunction of flux is zero at Φ = 0, but shows a substruc-ture (kinks at Φ = 0) within one period directly relatedto the broken valley degeneracy at finite flux. We notethat this substructure is due to the linear term in m ofthe spectrum Eq. (8) which is prominent within the firstfew transverse modes n which can host many electrons N .In Fig. 4, we plot the conductance through the ringweakly coupled to leads as a function of Fermi energy E F (or gate voltage) assuming a constant interaction model with charging energy U . At Φ = 0, the conductance ex-hibits a four-fold symmetry due to spin and valley degen-eracy. A finite flux breaks the valley degeneracy which isobservable via a splitting of the conductance peaks mov-ing with magnetic flux, see Fig. 4. C ondu c t an c e N=1,2,3,4 N=1,2 N=3,4 N=5,6 N=7,8 N=9,10 N=11,12N=5,6,7,8 N=9,10,11,12
E – N U F FIG. 4: Ring conductance assuming a constant interactionmodel with charging energy U for the first 12 electrons inthe conduction band ( E > E F in the leads due to spin- and valley-degeneracy.At finite magnetic flux (Φ / Φ = 0 .
1, full line), the conduc-tance peaks shift due to breaking of the valley degeneracy.Each peak is labelled with the filling factor N at this specificresonance. C. Valley qubit
We now turn to the question of how to make use ofthe broken valley degeneracy in order to directly addressthe valley degree of freedom in graphene experimentally(valleytronics ). The valley degree of freedom forms(in principle) a two-level system that can be representedby an isospin | + i for valley τ = +1 and |−i for valley τ = −
1. We point out that the graphene ring weaklycoupled to current leads could be used to investigate the relaxation and coherence of such valley isospins. Close -0.2 -0.1 0 0.1 0.21.5711.574 E m ± [ v / W ] | +> + | –> | +> – | –> | +> | +> | –> | –> FIG. 5: Valley qubit. The lowest four levels (for n = 0 and a/W =10) are shown as a function of flux. Blue and red(dashed) lines correspond to valleys τ = +1 and τ = − | + i to the crossing point with the new eigenstates beingsuperpositions of | + i and |−i as indicated in the figure. to a degeneracy point of two levels belonging to differentvalleys (e.g. at Φ = 0), Eq. (8) predicts a valley splitting of states with fixed m -values controllable by flux, simi-lar to the Zeeman-splitting for electron spins in a mag-netic field. In semiconductor quantum dots, such pairs ofspin-split states can be addressed via electron tunnelingfrom/to leads weakly coupled to the quantum dot andcan be used for read-out of single spins or measuringtheir relaxation ( T ) time. The graphene ring could beused in very much the same way to measure the intrinsicvalley isospin relaxation time T in graphene as well asthe valley isospin polarization.In Fig. 5, we show the situation when some small level-mixing leads to avoided crossings of valley-split statesnear the degeneracy point Φ = 0. Such valley mixing nat-urally appears through boundary roughness of the ringor atomic defects in the bulk. Using the magnetic fluxas a knob, we can sweep the system from a | + i ground-state level, filled with one electron, to a superposition( | + i + |−i ) / √ |−i groundstate, seeFig. 5. Such a situation can be used to produce Rabioscillations of the valley isospin states by tuning the sys-tem fast (non-adiabatically) from | + i to the degeneracypoint where the spin will oscillate between | + i and |−i intime: cos(∆ t ) | + i − i sin(∆ t ) |−i , where 2∆ is the energysplitting at the degeneracy point. We expect that this qubit is rather robust if intervalleyscattering is weak, since time-reversal symmetry assuresthe (approximate) degeneracy of states from different val-leys at zero flux Φ. Indeed, the Hamiltonian Eq. (1) hasthe same spectrum in both valleys at zero flux, indepen-dent on the shape of the mass potential V ( x, y ). Thismeans that we do not rely on a special symmetry of theconfining potential (like the circle discussed here). Inaddition, long-range disorder will also not lift the valleydegeneracy.
D. Valley isospin-orbit coupling
In an open ring geometry with adiabatic contacts toleads, new interesting coherent rotations of the valleyisospin occur while propagating along the ring. The lin-ear term in m in Eq. (8) can be thought of as a valleyisospin-orbit coupling term , since the valley isospin τ cou-ples to the orbital motion m . A general incoming spinoris a superposition of spinors belonging to different val-leys. Due to the valley isospin-orbit coupling, the angularmomentum m [determined by the incoming (continuous)energy E and the applied magnetic flux Φ via Eq. (8)] willbe different for the two valleys. Consequently, the spinorin Eq. (3) will pick up different phases exp( imϕ ) for thetwo valleys while propagating along the ring thereby ro-tating the valley isospin in a transport experiment. III. SPECTRUM FOR A HEXAGONAL RINGWITH ZIGZAG EDGES
Here, we compare our analytical model with the in-finite mass boundary described in Section II, to a ringwith strong intervalley scattering. We numerically in-vestigate the spectrum of a ring of hexagonal form withzigzag edges as shown in Fig. 6. (Electrical conductionthrough this geometry was studied in Ref. 21.) In a
FIG. 6: A hexagonal ring with zigzag edges of inner radius r a and outer radius r b and flux through the hole Φ. Here, r a = 3 √ l and r b = 6 √ l with l the lattice spacing zigzag nanoribbon, the valley isospin is a good quantumnumber, i.e. the zigzag boundary does not mix valleys. Since two neighboring arms of the ring are rotated by60 ◦ with respect to each other, the roles of the A andB sublattices are interchanged in subsequent segments.Explicitly, this means, that if a zigzag edge is terminatedon a A side, it will be terminated on a B side at a neigh-boring arm of the ring. Equivalently, in the reciprocal( k -) space, this implies that equivalent states of subse-quent zigzag nanoribbon segments are lying in oppositevalleys. This necessarily induces intervalley mixing atthe corners between two subsequent zigzag nanoribbonsegments. This mixing is very strong in the lowest modeof the ring, where the direction of motion and the valleyis tightly coupled in each arm of the hexagonal ring (the zigzag edge is therefore another example where ef-fective TRS in a single valley is broken). An electronwave, approaching a corner of the hexagon in one valley,is either transmitted into the next arm, or reflected backinto the same arm. In both cases, the valley index has toflip.We investigate the spectrum of such a ring numericallyin a tight-binding approach with Hamiltonian H = X i,j t ij | i ih j | + X i ǫ i | i ih i | . (9)The hopping element in the presence of a magnetic flux is t ij = − t exp[ − i (2 π/ Φ ) R r i r j d r · A ] where A is the vectorpotential and ǫ i = 0 are the on-site energies. The vectorpotential is chosen as A = ( A x , ,
0) with A x = B [ y a Θ( y ) − y a Θ( − y )] × Θ( L C / − | x | ) , (10) E [t] F [ F ]a) E [t] F [ F ]b) FIG. 7: Plot (a) shows an energy band of the hexagonal ring(see Fig. 6) with ring dimensions r a = 7 √ l and r b = 14 √ l inthe lowest mode 0 < E < ∼ . t . The levels are grouped intobands containing 6 levels. The top most and the lowest levelin each band is non-degenerate whereas the middle four levelsare two-fold degenerate at zero flux. This degeneracy is liftedby the flux through the ring. In (b) we contrast the perfectcrossing of two levels at zero flux (red dots) with anticrossedlevels (blue circles) induced by the addition of one unit cellto each of two parallel arms of the ring [we have shifted theenergy axis for the asymmetric case (blue circles) by +3 · − t for better comparison]. where y a ( x ) = min[ r a , √ L C / − | x | )], with L C =4 r a / √ L C = 4 r a / √ l for a hexagon ring with one unit celladded to the top and bottom arm. This represents auniform magnetic field B inside the ring hole and zerooutside. The spectrum as a function of magnetic fluxΦ = 2 √ r a B is shown in Fig. 7 in an energy win-dow which lies well within the lowest mode of a zigzagnanoribbon of width W = r b − r a . A band of levels inthe lowest mode is shown in Fig. 7(a). Within that mode,the spectrum follows a clear pattern which is observed forgeneric values of r a and r b . It consists of bands separatedby energy gaps. Each band hosts six levels. The top andbottom level is non-degenerate with dE/d Φ = 0 at zeroflux (corresponding to standing waves). The other fourlevels are two-fold degenerate at Φ = 0 with a broken de-generacy at finite flux. These levels correspond to rightand left-going states in the ring.We remark that this level pattern reflects the scatter-ing off a periodic array of six scatterers subjected to pe-riodic boundary conditions. It is to be noted that theorbital degeneracy of the hexagonal ring can be tuned bythe flux, similar to the ring with the smooth confinementdiscussed in Section II. If the 6-fold rotational symmetryof the ring is broken, the crossings at zero flux becomeslightly avoided as is shown in Fig. 7(b) (blue circles)where we have added one unit cell to two of the paral-lel arms of the ring (this corresponds to a length changeof the arms by about 5%). This shows that our resultsare also relevant for rings with a sligthly reduced sym-metry. Note that the sensitivity of the level crossing atzero flux to the ring geometry is consistent with strongintervalley scattering where time-reversal symmetry doesnot protect the degeneracy at zero flux.
IV. CONCLUSION
We have analyzed the Aharonov-Bohm effect ingraphene rings. We have investigated two different ringsystems – a ring with a smooth boundary (with zero orweak intervalley scattering) and a hexagonal ring withzigzag edges. For the ring with a smooth boundary,the combined effect of the effective time reversal symme-try (TRS) breaking within a single valley induced by asmooth boundary and the applied magnetic flux – break-ing the real TRS – gives us at hand a controllable toolto break the valley degeneracy in such rings. We have shown that this effect of a broken valley degeneracy byflux is revealed in the persistent current and in the ringconductance. This tool could be useful for spin-based orvalley-based quantum computing. The presence of a de-generate pair of levels from different valleys at zero fluxis assured by time-reversal symmetry in the absence ofintervalley scattering. Therefore, the proposed effect isnot sensitive to the actual geometry of the ring.We have also considered the opposite case of strong in-tervalley scattering by investigating numerically a hexag-onal ring with lattice-terminated zigzag edges. Here,strong intervalley scattering is induced by the cornersof the ring at low energies. We found that the orbital de-generacy of graphene can still be tuned by the flux similarto the ring with a smooth boundary. This effect, how-ever, relies on a certain degree of symmetry of the ringas we show by sligthly distorting the ring. We thereforeconclude that the orbital degeneracy in graphene ringscan be controlled with an Aharonov-Bohm flux in ringswith zero or weak intervalley scattering and in systemswith strong intervalley scattering if the ring possesses an(approximate) geometric symmetry.We acknowledge helpful discussions with A.R.Akhmerov and J.H. Bardarson. This work was financiallysupported by the Dutch Science Foundation NWO/FOM,the Swiss NSF, and the NCCR Nanoscience. A. Rycerzacknowledges support by the Polish Ministry of Science(Grant No. 1-P03B-001-29) and by the Polish ScienceFoundation (FNP). For a recent review on the topic, see A.K. Geim and K.S.Novoselov, Nature Materials , 183 (2007). S.V. Iordanskii and A.E. Koshelev, JETP Letters , 574(1985); S.V. Morozov et al. , Phys. Rev. Lett. , 016801(2006); A.F. Morpurgo and F. Guinea, Phys. Rev. Lett. , 166804 (2006). H. Suzuura and T. Ando, Phys. Rev. Lett. , 266603(2002); D.V. Khveshchenko, Phys. Rev. Lett. , 036802(2006); E. McCann, K. Kechedzhi, V.I. Fal’ko, H. Suzuura,T. Ando, and B.L. Altshuler, Phys. Rev. Lett. , 146805(2006). B. Trauzettel, D.V. Bulaev, D. Loss, and G. Burkard, Na-ture Physics , 192 (2007). M.V. Berry and R.J. Mondragon, Proc. R. Soc. Lond.A , 53 (1987). Any boundary condition that does not mix the valleys canbe written as ψ = M ψ , with M = n ⊥ · σ determined bya unit vector n ⊥ in the plane tangent to the boundary. It holds that [ M , T ] = 0 where T = iσ y C ( C denotes com-plex conjugation) is the time reversal operation in a singlevalley. A.R. Akhmerov and C.W.J. Beenakker, Phys. Rev. Lett. , 157003 (2007). G. Giovannetti, P.A. Khomyakov, G. Brocks, P.J. Kelly,and J. van den Brink, Phys. Rev. B , 073103 (2007). S.Y. Zhou, G.-H. Gweon, A.V. Fedorov, P.N. First, W.A.de Heer, D.-H. Lee, F. Guinea, A.H. Castro Neto, and A. Lanzara, arXiv:0709.1706 (2007). F.E. Meijer, A.F. Morpurgo, and T.M. Klapwijk, Phys.Rev. B , 033107 (2002). D. Frustaglia and K. Richter, Phys. Rev. B , 235310(2004). We use H (1) ν ( ρ ) = p / ( πρ ) exp[ i ( ρ − νπ/ − π/ { δ ν ( ρ ) } with δ ν ( ρ ) = − (4 ν − ν − / ρ + i ( ν − / / ρ + O ( ρ − ) and H (2) ν ( ρ ) = [ H (1) ν ( ρ )] ∗ . In the regime a ∼ W , the breaking of the valley degener-acy with flux is also observed as we find numerically usingEq. (6). L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S.Tarucha, R.M. Westervelt, and N.S. Wingreen, in
Meso-scopic Electron Transport , NATO ASI Series E, Vol. 345(Kluwer Academic Publishers, Dordrecht, 1997) ed. byL.L. Sohn, L.P. Kouwenhoven, and G. Sch¨on, p. 105. We estimate U ∼ a = 0 . µ m and a/W = 5 witha simple ring capacitor model with plate separation of 285nm and SiO dielectric. The single-particle levelspacing for n = 0 and small m is ∼ λ = ( v/a )( W/a ) /π ∼ µ eV andgrows linearly with m . A. Rycerz, J. Tworzyd lo, and C.W.J. Beenakker, NaturePhysics , 172 (2007). J.M. Elzerman, R. Hanson, L.H.W. van Beveren, B.Witkamp, L.M.K. Vandersypen, and L.P. Kouwenhoven,Nature , 431 (2004). R. Hanson, B. Witkamp, L.M.K. Vandersypen, L.H.W. vanBeveren, J.M. Elzerman, and L.P. Kouwenhoven, Phys.Rev. Lett. , 196802 (2003). Near Φ = 0, the spectrum is approximated by a “val-ley qubit”-Hamiltonian H m = 2 λ (Φ / Φ )[ m − (1 /π )] τ z − ∆ τ x + c ( m ) with c ( m ) being independent of Φ, and τ z and τ x act in valley space. This is shown as follows: If ψ τ is an eigenstate of H τ ,then T ψ τ with T = iσ y C the time-reversal operation ina single valley, is an (orthogonal) eigenstate of H τ withmass potential − V ( x, y ) and with the same energy. But H τ with mass potential − V ( x, y ) is identical to H − τ with mass potential V ( x, y ). A. Rycerz and C.W.J. Beenakker, arXiv:0709.3397. L. Brey and H.A. Fertig, Phys. Rev. B , 235411 (2006). For W ≫ l , the energy spacing δ between the 1st (lowest)and 2nd mode in a zigzag nanoribbon is δ = (3 / / √ πtl/W . For the ring dimensions used inFig. 7, this gives δ ∼ . t . R. Gilmore,