Aharonov-Bohm effect for a valley-polarized current in graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Aharonov-Bohm effect for a valley-polarized current in graphene
A. Rycerz
Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krak´ow, Poland
C. W. J. Beenakker
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: September, 2007)This is a numerical study of the conductance of an Aharonov-Bohm interferometer in a tight-binding model of graphene. Two single-mode ballistic point contacts with zigzag edges are connectedby two arms of a hexagonal ring enclosing a magnetic flux Φ. The point contacts function as valleyfilters, transmitting electrons from one valley of the band structure and reflecting electrons fromthe other valley. We find, in the wider rings, that the magnetoconductance oscillations with thefundamental periodicity ∆Φ = h/e are suppressed when the two valley filters have opposite polarity,while the second and higher harmonics are unaffected or enhanced. This frequency doubling isinterpreted in terms of a larger probability of intervalley scattering for electrons that travel severaltimes around the ring. In the narrowest rings the current is blocked for any polarity of the valleyfilters, with small, nearly sinusoidal magnetoconductance oscillations. Qualitatively similar resultsare obtained if the hexagonal ring is replaced by a ring with an irregular boundary.
PACS numbers: 73.23.-b, 73.23.Ad, 73.63.Rt, 85.35.Ds
I. INTRODUCTION
The isolation of single two-dimensional layers of carbon(graphene) [1] has led experimental and theoretical physi-cists to reexamine classic effects from mesoscopic physics[2]. The search is for novel features that arise from theunusual conical band structure of a carbon monolayer [3].One effect which has so far received little attention is theperiodic magnetoconductance oscillation in a ring knownas the Aharonov-Bohm effect [4, 5]. Preliminary experi-ments on the magnetoconductance of graphene rings havebeen reported by several groups [6, 7]. Theoretically, theenergy spectrum of a closed graphene ring was studiedas a function of the enclosed magnetic flux Φ by Recheret al. [8].Here we study by computer simulation the electrontransport through an open graphene ring, contacted toelectron reservoirs by ballistic point contacts. Our workbuilds on an earlier finding [9] that a single-mode pointcontact with zigzag edges operates as a valley filter. De-pending on whether the Fermi level in the point contactlies in the conduction or valence band, the transmittedelectrons occupy states in one or the other valley of theband structure. It was also shown in Ref. 9 that twoadjacent valley filters function as a highly effective val-ley valve, passing or blocking the current depending onwhether the two filters have the same or opposite polar-ity.We find that the magnetoconductance oscillations in amultimode graphene ring with valley-filtering point con-tacts show the expected ∆Φ = h/e periodicity if the twofilters have the same polarity. For opposite polarity, how-ever, a period doubling appears: The lowest harmonic issuppressed while the second and higher harmonics areunaffected or enhanced. We attribute the period dou-bling to intervalley scattering, which is more effective for electrons that have travelled more than once along thering.In few-mode rings the period doubling does not hap-pen, instead the conductance is strongly suppressed withsmall, nearly sinusoidal, magnetoconductance oscilla-tions. They appear in their clearest form in a hexago-nal ring, but we find qualitatively the same behavior ina more generic ring with irregular boundaries.
II. AHARONOV-BOHM INTERFEROMETERIN GRAPHENE
The analysis starts from the tight-binding model ofgraphene, with Hamiltonian H = X i,j τ ij | i ih j | + X i V i | i ih i | . (1)The system is coupled to the vector potential A throughthe hopping matrix element τ ij = − τ exp πi Φ Z R j R i d r · A ! , (2)with τ = 3 eV the hopping energy and Φ = h/e the fluxquantum. The orbitals | i i and | j i are nearest neighborson a honeycomb lattice (with lattice points R i ), other-wise τ ij = 0. The energy-independent velocity v near theDirac point equals v = √ τ a/ ~ ≈ m/s, with thelattice constant a = 0 .
246 nm.The electrostatic potential V i = V ( x i ) varies onlyalong the axis connecting the input and output pointcontacts (see Fig. 1). Namely, the potential equals U atthe first constriction (0 < x < l , where l is the constric-tion length), U ′ at the second constriction ( l + L C < x < Wlw xy W B = 0 B ! = 0 R xV(x)µ n N n' U U' µ'lµl L C FIG. 1: Schematic diagram of a hexagonal graphene ring at-tached to graphene leads (top panel) and corresponding po-tential profile (bottom panel). The constriction with zigzagedges between each lead and the ring allows one to controlinput and output valley polarizations by varying the electro-static potentials U and U ′ . l + L C , with L C = (4 R − w ) / √
3, where R is the ringradius and w is the constriction width), and zero every-where else. By varying U and U ′ at a fixed Fermi energy µ ∞ in the external leads, we can vary the Fermi energies µ = µ ∞ − U and µ ′ = µ ∞ − U ′ in the two constrictions.We took µ ∞ = τ / w = 10 √ a and l = 16 a are chosen to provide valley polarizations above90% [9].We denote the number of transmitted modes throughthe first constriction by n and through the second con-striction by n ′ . A positive number indicates that theFermi level lies in the conduction band, while a nega-tive number indicates that the Fermi level lies in the va-lence band. For example, as shown in Ref. [9], the case − / < µ ′ < < µ < / π ~ v/w ) cor-responds to n = 1, n ′ = − < µ, µ ′ < / n = n ′ = 1 (valley filters of the same polarity).The AB interferometer is modeled by a hexagon witha hexagonal hole in the center and zigzag edges alongthe entire perimeter. (We will consider a more genericshape in Sec. IV.) To vary the number of modes N thatcan propagate along the ring, we keep the radius fixed at R = 35 √ a and vary the inner radius. We take ringwidths W/ √ a = 5 , ,
15, corresponding to N = 1 , , N are not accessible because of the val-ley degeneracy of the second and higher modes.) Theexternal leads have a fixed width W ∞ / √ a = 70, corre-sponding to 29 propagating modes.We take the vector potential A = ( A x , ,
0) with A x = (cid:26) By, − W ∞ − w √ < x < l + L C + W ∞ − w √ , , otherwise . (3)This corresponds to a uniform perpendicular magneticfield B in the area containing the ring, the two pointcontacts, and the widening region connecting the pointcontacts to the external leads (see grey rectangle in Fig.1, top panel). For technical reasons, the magnetic fieldis set to zero in the external leads. The resulting step inthe field strength at the entrance and exit of the interfer-ometer will reduce its conductance somewhat, but thisis not expected to change the qualitative features of themagnetoconductance oscillations that we are interestedin.We calculate the transmission matrix t numerically andthen obtain the conductance from the Landauer formula G = 2 e h Tr tt † . (4)(The factor of two accounts for the spin degeneracy.) III. RESULTS
To analyze the operation of the hexagonal graphenering as a valley valve [9], we first show in Fig. 2 theconductance in zero magnetic field. We take µ = 0 . τ to keep n = 1 at the first constriction (the polarizer),and vary the Fermi energy µ ′ at the second constriction(the analyzer). For negative µ ′ ( n ′ = − G . − e /h , showing that intervalleyscattering in the system is negligible. The resonancesfor µ ′ . − / µ ′ ( n ′ = 1) we would expect a conductance of order e /h ,because the two constrictions transmit the same valleypolarization. This is indeed observed for N >
1, but re-markably enough the conductance remains . − e /h if the arms of the ring support a single propagating mode( N = 1, black line in Fig. 2). We attribute this anomalyto current blocking from mismatched valley polarizationsat the vertices of the hexagonal ring, as explained in Fig.3. In Fig. 4 the conductance G is plotted as a function ofthe flux Φ = ( ¯ S/S c )Φ c through the ring, where Φ c is theflux per unit cell of area S c = √ a , and ¯ S = ( S o + S i ) / S o and S i . The first harmonic frequency of the Fourier spectrumshown in Fig. 5 is within a few percent of the expected n = − n ′ =1 n = − n ′ =1 n = n ′ =1 n = n ′ =1 µ ′ / ∆ G × h / e
10 10 10 0.1 -2 -1 0 1 2N = 1 µ ′ / ∆ G × h / e -3-5-7 FIG. 2: Conductance of the hexagonal graphene ring in zeromagnetic field at fixed µ = 0 . τ ≈ ∆ / µ ′ / ∆ (with ∆ = πτ /
20 for w = 10 √ a ). The parameter µ ′ = µ ∞ − U ′ is varied by varying U ′ at fixed µ ∞ . Thethree curves are for different ring widths, corresponding to N = 1 , , N = 1 on an expanded (logarithmic)scale.FIG. 3: Schematic illustration of the mechanism of currentblocking by mismatched valley polarizations at a vertex ofthe hexagonal ring with zigzag edges, responsible for the sup-pressed conductance when N = 1. Shown is the first Brillouinzone at the two sides of the vertex, rotated by π/
3. Solid ar-rows indicate the direction of propagation of the lowest modein one of the two valleys (located near the three red dots in theBrillouin zone). The lowest mode in the other valley (near thethree black dots) propagates in the opposite direction (dot-ted arrow, shown only near one of the black dots for clarity).The current is blocked because the corresponding points inthe Brillouin zone propagate in opposite directions at the twosides of the vertex. (n,N,n’) = (1,1,1)(1,1,-1) 0 0.5 1 0 2 4 6 8 10 12 14 (n,N,n’) = (1,3,1)(1,3,-1) 0 0.5 1 0 2 4 6 8 10 12 14 (n,N,n’) = (1,5,1)(1,5,-1) -4-4 · · Φ (Φ ) G × h / e FIG. 4: Conductance as a function of magnetic flux throughthe ring. Red solid curves show the case n = n ′ = 1, µ = µ ′ =0 . τ of identical valley polarizations in both constrictions,and blue dashed curves show the case n = − n ′ = 1, µ = − µ ′ = 0 . τ of opposite polarizations. The number N ofpropagating modes in the ring is varied between the threepanels. value Φ − = e/h , indicating that ¯ S accurately representsthe effective area of the ring.The harmonic content of the magnetoconductanceoscillations is strikingly different when the current isblocked ( N = 1, top panel in Fig. 4) and when it is notblocked ( N = 3 ,
5, lower two panels). On the one hand,when the conductance is suppressed below 10 − e /h themagnetoconductance oscillations are nearly sinusoidal,almost without higher harmonics. This is as expectedfor transmission through evanescent modes. On the otherhand, when the conductance is of order e /h the oscil-lations are highly nonsinusoidal, with appreciable higherharmonics, as expected for transmission through propa-gating modes.
0 1 2 3 4 5 6 7 (n,N,n’) = (1,3,1)(1,3,-1) 0 1 2 3 4 5 6 7 po w e r s pe c t r u m frequency(n,N,n’) = (1,5,1)(1,5,-1) (Φ − ) FIG. 5: Fourier transform of the magnetoconductance datashown in Fig. 4 for N = 3 and 5. A feature of the nonsinusoidal magnetoconductance os-cillations shown in Fig. 4, and quantified by the Fouriertransform in Fig. 5, is the suppression of the lowest har-monic (period ∆Φ = Φ ) in the case of opposite valleypolarizations in the two constrictions ( n = − n ′ = 1).This suppression of the fundamental periodicity is dra-matic for N = 3 (top panels in Figs. 4 and 5), but itis also noticeable for N = 5 (lower panels). The second(and higher) harmonics, in contrast, are enhanced in thecase of opposite valley polarizations.The suppression of the first harmonic indicates thatelectrons which travel only once along the ring have asmall probability for intervalley scattering and can there-fore not contribute to the conductance when the twoconstrictions have opposite valley polarization. Higherharmonics correspond to electrons which travel severaltimes along the ring, with a larger probability for in-tervalley scattering and therefore a larger probability tocontribute to the conductance. IV. GENERIC GRAPHENE RING
In order to determine how generic our results for thehexagonal ring might be, we have repeated our calcula-tions for an approximately circular ring with an irregularboundary (see Fig. 6). We constructed the circular ring
R WlwW FIG. 6: Graphene ring with an irregular, approximately circu-lar, boundary. Bottom panel: A magnified section of the ringwith R = 35 √ a and W = 10 √ a used in our simulations,which shows the irregularity of the boundary. by starting from the hexagonal ring and then keepingonly lattice sites within an annulus formed by two con-centric circles. The electrostatic potential shown in thebottom panel of Fig. 1 and the vector potential given byEq. (3) remain unchanged, only the length of the cen-tral area L C is replaced now by L ′ C = √ R − w . Thepoint contacts still have a zigzag boundary, so the fil-tering property at entrance and exit should remain in-tact, but the circular ring no longer has zigzag bound-aries along its entire perimeter (as it did in the case ofthe hexagonal ring). Results are shown in Figs. 7–9.Because the boundaries of the ring are not uniform,the number of propagating modes in the ring is not well-defined. We still label the data by the same number N as in the hexagonal ring, but now this number refers onlyto the number of modes which the ring would support ifthe boundaries were of zigzag form. This uncertainty inthe definition of N might explain why we now see thecurrent blocking not only for N = 1 but also for N = 3 n = − n ′ =1 n = − n ′ =1 n = n ′ =1 n = n ′ =1 µ ′ / ∆ G × h / e
10 10 0.01 -2 -1 0 1 2N = 1N = 3 -5-8 µ ′ / ∆ G × h / e FIG. 7: Conductance of the circular graphene ring in zeromagnetic field at fixed µ = 0 . τ as a function of µ ′ / ∆. Thelower panel shows the data for N = 1 , (Fig. 7, lower panel). Only for N = 5 do we obtain anappreciable current through the ring. The conductancefor N = 5 is still well below the optimal value of 2 e /h ,but more than 100 times larger than for N = 1 , N = 1 ,
3) we see nearly sinusoidal oscillations, almostwithout higher harmonics. When the current is notblocked ( N = 5) higher harmonics do appear and thelowest harmonic is strongly suppressed when the polarityof the valley filters in the two point contacts is opposite. V. CONCLUSIONS
In conclusion, we have identified signatures of valleypolarization in the magnetoconductance of an Aharonov-Bohm interferometer in graphene. The suppression ofthe lowest harmonic of the conductance oscillations thatappears when the two point contacts have opposite val-ley polarity indicates that electrons which have travelledonly once along the ring preserve their valley polariza-tion, and therefore cannot contribute to the conductance.
010 10 0 2 4 6 8 10 12 14 (n,N,n’) = (1,1,1)(1,1,-1)010 10 0 2 4 6 8 10 12 14 (n,N,n’) = (1,3,1)(1,3,-1) 0 0.5 1 0 2 4 6 8 10 12 14 (n,N,n’) = (1,5,1)(1,5,-1) -6-6 · -2-2 · · Φ (Φ ) G × h / e FIG. 8: Conductance of the circular ring as a function ofmagnetic flux.
0 1 2 3 4 5 6 7 po w e r s pe c t r u m frequency(n,N,n’) = (1,5,1)(1,5,-1) (Φ − ) FIG. 9: Fourier transform of the magnetoconductance dataof Fig. 8 for N = 5. The special case of a single-mode hexagonal ringshows nearly sinusoidal magnetoconductance oscillationsaround a greatly suppressed average conductance, at-tributed to a current blocking effect at a vertex wheretwo zigzag edges meet at an angle of π/
3. This currentblocking is not a special feature of the hexagonal geome-try. Instead, we have found the same current blocking ina circular ring with irregularly shaped boundaries. Thisfinding is consistent with a recent theory for the bound-ary condition of a terminated honeycomb lattice at an ar-bitrary crystallographic orientation [11]. It is found thatthe zigzag boundary condition applies generically for anyangle φ = 0 (mod π/
3) of the boundary (where φ = 0labels the armchair orientation). The current blocking mechanism illustrated in Fig. 3 for the vertex betweentwo zigzag boundaries would then apply generically toany boundary near an orientation of φ = 0 (mod π/ Acknowledgment
This work was supported by the Dutch Science Foun-dation NWO/FOM. Discussions with A.R. Akhmerovand P. Recher are gratefully acknowledged. One of theauthors (A.R.) acknowledges the support from a specialgrant by the Polish Science Foundation (FNP) and by thePolish Ministry of Science (Grant No. 1–P03B–001–29). [1] A.K. Geim and K.S. Novoselov, Nature Mat. , 183(2007).[2] Y. Imry, Introduction to Mesoscopic Physics (OxfordUniversity, Oxford, 1996).[3] P.R. Wallace, Phys. Rev. , 622 (1947).[4] Y. Aharonov and D. Bohm, Phys. Rev. , 485 (1959).[5] Y. Imry and R.A. Webb, Sci. Am. (4), 36 (1989).[6] B. Oezyilmaz, D. Efetov, K. Bolotin, M.Y. Han, P.Jarillo-Herrero, and P. Kim, abstract N28.00002, APSMarch Meeting (Denver, 2007). [7] A. Morpurgo and L.M.K. Vandersypen, private commu-nication.[8] P. Recher, B. Trauzettel, A. Rycerz, Ya.M. Blanter,C.W.J. Beenakker, and A.F. Morpurgo, arXiv:0706.2103.[9] A. Rycerz, J. Tworzyd lo, and C.W.J. Beenakker, NaturePhys. , 172 (2007).[10] P.G. Silvestrov and K.B. Efetov, Phys. Rev. Lett.98