Currents and pseudomagnetic fields in strained graphene rings
CCurrents and pseudomagnetic fields in strained graphene rings
D. Faria,
1, 2, 3, ∗ A. Latg´e, S. E. Ulloa,
2, 3 and N. Sandler
2, 3 Instituto de F´ısica, Universidade Federal Fluminense, Niter´oi, Av. Litorˆanea sn 24210-340, RJ-Brazil Department of Physics and Astronomy, Nanoscale and Quantum Phenomena Institute,Ohio University, Athens, Ohio 45701-2979, USA Dahlem Center for Complex Quantum Systems and Fachbereich Physik,Freie Universit¨at Berlin, 14195 Berlin, Germany (Dated: November 2, 2018)We study the effects of strain on the electronic properties and persistent current characteristicsof a graphene ring using the Dirac representation. For a slightly deformed graphene ring flake, oneobtains sizable pseudomagnetic (gauge) fields that may effectively reduce or enhance locally theapplied magnetic flux through the ring. Flux-induced persistent currents in a flat ring have fullrotational symmetry throughout the structure; in contrast, we show that currents in the presence ofa circularly symmetric deformation are strongly inhomogeneous, due to the underlying symmetriesof graphene. This result illustrates the inherent competition between the ‘real’ magnetic field andthe ‘pseudo’ field arising from strains, and suggests an alternative way to probe the strength andsymmetries of pseudomagnetic fields on graphene systems.
PACS numbers: 73.22.Pr, 73.23.Ra, 61.48.Gh, 73.23.-b
The appearance of gauge fields in graphene is a beau-tiful and experimentally accessible example of a situ-ation where concepts of condensed matter and quan-tum field theory converge on a physical system.
Ex-perimental evidence of ‘bubble’ formation on particu-lar graphene growth processes, and controllable routesto manipulate graphene bubble morphology have mo-tivated numerous works addressing different aspects ofthese effects. In particular, the theoretical descriptionof strained graphene has been developed significantly,exploring how its electronic properties are affected oncurved and strained surfaces. It is known that elasti-cally deformed graphene can be mapped onto the Diracformalism in the continuum limit by including pseudo-magnetic fields and Fermi velocity renormalization, giving rise to local quantities that depend on strain but donot break time reversal symmetry. A recent contributionreported a space-dependent Fermi velocity leading to in-teresting experimental consequences. Lattice-correctedstrain induced vector potentials in graphene have alsobeen discussed within a tight-binding scenario, al-though these corrections do not contribute to the pseudo-magnetic field distribution.
Interesting possibilitiesfor observing the pseudomagnetic fields arise from break-ing time reversal symmetry in the system via an externalmagnetic field. This promotes a most interesting inter-play, some of which has been explored in the context ofthe quantum Hall regime.
Similar to other confined systems with periodicboundary conditions, magnetic flux-dependent persis-tent currents and conductance oscillations are ex-pected for graphene rings in an Aharonov-Bohm (AB)geometry.
Several experiments have verified thepresence of AB conductance oscillations with differentvisibility for different device geometries.
A recent re-view of quantum interference in graphene rings discussesopen questions in the field. Interestingly, the ‘infinite mass’ confinement that requires null current densityacross the boundaries results in persistent currents thatare ‘valley polarized’ in the presence of magnetic flux, suggesting that graphene quantum rings would be an ex-cellent system to analyze the effects of induced curvature.The main result of the present Rapid Communication isindeed to show that while a flat (unstrained) graphenering in the AB geometry sustains persistent currents withfull rotational symmetry, unavoidable strains in typicalsystems would result in inhomogeneous distributions ofcurrents. In other words, while the strains alone wouldresult in zero net persistent current (since time reversalsymmetry is not broken by the pseudomagnetic fields),the competition with the AB flux induces spatially in-homogeneous current distributions on the system. Thiseffect can be seen to arise both from a local rescalingof the Fermi velocity as well as by the appearance ofgauge fields that result from the elastic deformations. Assuch, the persistent currents originated by the magneticAB flux acquire a local character that follows the strainfields.Moreover, as the corresponding length scales of thecurrent inhomogeneities are given by the strain fields,one can imagine using this effect to measure the straindistribution via a scanning magnetometer that would besensitive to the induced currents. Alternatively, properlydesigned strain fields would be used to produce desiredcurrent patterns. Strained graphene . Within a tight-binding model, theeffects of lattice deformations may be incorporated intothe hopping integrals t n between nearest-neighbors, so that t n = t + δt n = t (cid:0) − β(cid:15) ij δ in δ jn /a (cid:1) , where β = | ∂ log t /∂ log a | ≈ (cid:126)δ n are the nearest-neighbor vectors of a given atom at lattice site n , and t and a are the nearest-neighbor hopping integral anddistance in the unstrained system, respectively. Indices i and j represent directions on the 2D plane, with re- a r X i v : . [ c ond - m a t . m e s - h a ll ] J un peated index summation convention throughout. Thestrain tensor, (cid:15) ij = ( ∂ j u i + ∂ i u j + ∂ i h∂ j h ), is charac-terized by u i and h , the in- and out-of-plane deforma-tions, respectively. The resulting Hamiltonian in the presence of inho-mogeneous strain and external magnetic field given by (cid:126)B = (cid:126) ∇ × (cid:126)A ext , can be written in a generalized Diracform (in the K valley) as H = − iv kj σ k (cid:16) ∂ j + i e (cid:126) A extj (cid:17) − iv σ j Γ j + v e (cid:126) σ j A δtj , (1)where σ j are Pauli matrices, and the vector potentialwith trigonal symmetry arising from strains is given by (cid:126)A δti ( (cid:126)r ) = Φ π (cid:18) − β a (cid:19) ( (cid:15) xx − (cid:15) yy , − (cid:15) xy ) , (2)with Φ = h/e . The renormalized Fermi velocity tensorin Eq. (1) is position-dependent, v ij ( (cid:126)r ) = v (cid:18) I ij − β (cid:15) ij + η ij (cid:15) kk ) (cid:19) , (3)where I is the 2 × v = 3 at / ≈ m/s (with (cid:126) = 1). Inhomogeneous strains also yieldan effective geometric vector potentialΓ i ( (cid:126)r ) = − β (cid:18) ∂ j (cid:15) ij + 12 ∂ i (cid:15) jj (cid:19) . (4)At the K (cid:48) valley, both v ij ( (cid:126)r ) and Γ i ( (cid:126)r ) are the same,while the vector potential A δtj ( (cid:126)r ) changes sign, preserv-ing overall time-reversal symmetry of the system in theabsence of (cid:126)A ext .Diagonalization of the Hamiltonian in a ring geometryresults in interesting eigenstates and persistent currentpatterns for states at and near the Dirac point (chargeneutrality point). We first summarize the results for a flatgraphene ring in a magnetic flux, to provide a suitableframework for the ring with deformation. Unstrained graphene ring . We consider a ‘flat’ ringthreaded by a magnetic flux Φ, with (cid:126)A ext = (Φ / πr )ˆ θ ,so that the Hamiltonian is given by H = − iv (cid:20) Λ ( θ ) ∂ r + Λ ( θ ) 1 r (cid:18) ∂ θ + i ΦΦ (cid:19)(cid:21) , (5)with Λ ( θ ) = σ x cos θ + σ y sin θ and Λ ( θ ) = − σ x sin θ + σ y cos θ . The wave functions for energy E are ψ ¯ m,s ( r, θ ) = e imθ (cid:18) φ ¯ m ( r ) ise iθ φ ¯ m +1 ( r ) (cid:19) , (6)with φ ¯ m ( r ) = A ¯ m J ¯ m ( kr ) + B ¯ m Y ¯ m ( kr ), where ¯ m = m + Φ / Φ , m = 0 , ± , ± , ... is the orbital angular mo-mentum, J ¯ m and Y ¯ m are Bessel functions of first andsecond kind, respectively, k = | E | / (cid:126) v , and s = sgn( E ).The upper (lower) component of the spinor correspondsto ψ A ( ψ B ) in Eq. (6) for the K valley. The energy spectrum is obtained from the transcen-dental equation that arises after imposing infinite-massboundary conditions, given by ψ B ( r, θ ) /ψ A ( r, θ ) = iτ ˆ n · ˆ re iτθ , where the normal to the boundaries is ˆ n = ± ˆ r for the inner ( − ) and outer radius (+) of the ring. The K and K (cid:48) valley eigenstates for a given m have alsoa total angular momentum j , defined for the operator J z = − i∂ φ + σ z τ /
2, where τ = ± K and K (cid:48) states with opposite momentum j (cid:48) = − j arerelated by m (cid:48) = − ( m + 1). Notice that the boundaryconditions do not mix the valleys and in fact break thevalley degeneracy for nonzero flux. - - ! F ê F o E n e r gy H m e V L m = H a L m = =- =- = =- = =- ê r y † y r H b L y A F= y B H d L ê r y † y r F ê F = H c L y A y B H e L FIG. 1. (Color online) Unstrained graphene ring. (a) Energyspectra vs magnetic flux for ring with internal and externalradii given by r = 50 and r = 100nm, for different quantumstates: m ( m (cid:48) = − ( m + 1)) integer denotes results for K ( K (cid:48) ) valley given by dashed (continuous) lines. [Thicker (red)lines near E ≈ m = 0 ( K valley); electronic probability distributionalong the ring for Φ = 0, and (c) Φ / Φ = 6. Main panelsshow the two spinor components separately; insets show totaldistribution, | ψ A | + | ψ B | . On right column, correspondingcurrent densities for K valley. (d) Notice counterpropagatingedge currents for Φ = 0, indicated by red arrows, evolve to acurrent distribution mostly on the outer radius for large Φ in(e). Figure 1(a) shows the energy spectrum of a graphenering vs magnetic flux; dashed and continuous (blue) lines(upper part of the graph) indicate results for K and K (cid:48) valleys with m and m (cid:48) values, respectively. Notice thequadratic dependence on Φ of these levels, which breaksvalley degeneracies in general. The figure also shows thelowest energy levels for a deformed ring (thicker red lines,lower part of graph), to be discussed later.The charge and current densities that satisfy the con-tinuity equation for unstrained graphene are given by ρ = ψ † ψ and J j = ( v / (cid:126) ) ψ † τ σ j ψ . Typical results forthe spinor components | Ψ A | and | Ψ B | for m = 0 aredepicted in Fig. 1 along the radial direction, for both(b) Φ / Φ = 0, and (c) 6. As expected, increasing fluxcauses the charge density to be driven to the outer edgeof the ring as the energy of the state increases. (Noticethe m = 0 state is not in general the lowest state as Φ in-creases.) Also shown in Fig. 1(d) and (e) are the currentprobability densities, highlighting the strong dependenceon the magnetic flux. Notice that the current for valley K at zero flux is given by nearly compensated counterpropagating edges, while as the flux increases, the cur-rent in the inner edge disappears. As we will see below,the interaction with the pseudomagnetic field generatedby the deformation gives rise to an intricate current pat-tern. Strained graphene rings . We now consider an out-of-plane deformation given by a circularly symmetric Gaus-sian shape described by h = Ae − r /b . The strain ten-sor is then (cid:15) = αf ( r ) (cid:18) cos θ sin θ cos θ sin θ cos θ sin θ (cid:19) , (7)where f ( r ) = 2 (cid:0) r /b (cid:1) e − r /b , with α = A /b char-acterizing the strength of the strain perturbation. Thestrain is inhomogeneous and, as a consequence, thegeometric gauge field Γ is nonzero and the renormal-ized Fermi velocity changes along the ring. The space-dependent velocity is given by v = v (cid:18) − βα (cid:19) f ( r ) (cid:20) I + 12 R (2 θ ) σ z (cid:21) , (8)where R ( θ ) = I cos θ − iσ y sin θ is the rotation matrixthrough an angle θ in the counterclockwise direction. Theresulting Dirac cone becomes elliptical, with radial andangular components v r = v (cid:18) − βα (cid:19) f ( r ) and v θ = v (cid:18) − βα (cid:19) f ( r ) , (9)while the gauge fields areΓ r = (cid:18) − βα (cid:19) f ( r ) (cid:18) r − rb (cid:19) and Γ θ = 0 . (10)These changes can be seen as perturbations of the Hamil-tonian in Eq. (5) given by V = − iv (cid:18) − βα (cid:19) f ( r ) (cid:20) Λ ( θ ) d r + Λ ( θ ) d θ r (cid:21) , (11) with d r = ∂ r + r − rb and d θ = (cid:16) ∂ θ + i ΦΦ (cid:17) . The vec-tor potential perturbation V = v (cid:16) − βα a (cid:17) f ( r )Λ ( − θ )is associated with a pseudomagnetic field (cid:126)B δt , given by (cid:126)B δt = ˆ z Φ π (cid:18) − βα a (cid:19) f ( r ) 4 rb sin (3 θ ) . (12)The eigenvalue problem with H = H + V + V can besolved using perturbation theory on the parameter α upto second order, keeping sufficient states to achieve fullconvergence of the results.We now present our main results for strained ringsconsidering the Gaussian perturbation with characteris-tic system parameters: A = 7nm and b = 70nm, witha relative deformation α = 1%, and the ring radii usedin Fig. 1(a). The two lowest states of the spectrum cor-rected by the Gaussian deformation are shown in Fig.1(a) in thick (red) curves near the bottom of the panel,both for solutions near K (dashed lines) and K (cid:48) (solid)valleys. We find that the main correction comes from the V perturbation which contains the effects of the strain-induced pseudomagnetic field, and produces energy shiftsfor the ground state as high as 10%. c J d J FIG. 2. (Color online) Deformed graphene ring. Contourplot of the local density | ψ A | + | ψ B | for m = 0 state at (a)Φ / Φ = 0, and (b) Φ = − .
5. A negative flux, as shown,pushes density towards the inner ring, while a positive fluxwould shift weights to the outer radius. (c) Current distri-bution for Φ = 0 and m = 0, for the K valley. (d) Currentvariation, ∆ (cid:126)J = (cid:126)J − (cid:126)J flat ( K -valley), where (cid:126)J flat is the cur-rent for a flat ring–shown in Fig. 1(d) –can be seen as the neteffect produced by the deformation. Notice six vortices withalternating circulation and cores near maxima/minima of thepseudomagnetic field [see Fig. 3(a)]. The pseudomagnetic field produced by the Gaussiandeformation in this system (see Fig. 3(a)) has the under-lying trigonal symmetry of the graphene lattice.
In thiscase, the field amplitudes reach (cid:39) . (cid:126)J over the strained graphene ringis displayed in Fig. 2(c), for the lowest state in the K val-ley. The current density trends are represented by a setof small (red) arrows, revealing intricate current configu-rations. The current density exhibits local maxima withtrigonal symmetry, centered near regions of largest varia-tion in the local density. The last panel, Fig. 2(d), showsthe probability current variation, ∆ (cid:126)J = (cid:126)J − (cid:126)J flat , where (cid:126)J flat is the persistent current in the flat or unstrainedring. Notice that ∆ (cid:126)J exhibits six vortices with alternat-ing circulation and cores centered on regions of extremalvalues (positive or negative) of the pseudomagnetic fieldin Eq. ( ?? ), Fig. 3(a).The spatial modulation of the persistent current in thering can also be seen from the angular profiles shown inFig. 3(b) for different flux values. The current densityachieves maximal values along nodal lines of the pseudo-magnetic field distribution. Similarly, local minima arerelated to regions of maximum pseudofield amplitude,both in positive and negative directions–see also polarplots in Fig. 3(a). As the pseudofield does not breaktime-reversal symmetry, the net current must be zero inthe absence of external magnetic flux. This indeed hap-pens when considering the contribution of the other val-ley ( K (cid:48) , coming from the state with m (cid:48) = − θ .We analyze the role of the pseudomagnetic field con-tribution to the total current (taking both valleys intoaccount) by looking at different angular values along thering. Figure 3(c) shows the total current at θ = π/ π/
3, and π/
2, in comparison with the (angle integrated)current for both the unstrained and deformed graphenering. All curves present the expected sawtooth behaviorwith flux. However, the slope of the curve and the valuenear zero flux are clearly angle and strain dependent.Notice in particular the jump reversal near Φ = 0 for θ = π/ π/
3, associated with the circulation aroundthe vortex at θ (cid:39) π/
6. The competition between the ex-ternal magnetic field and the pseudofield not only results H a L - p p p p p p ! q C u rr e n t H e u ê — ä - L H b L - - - F H F L C u rr e n t H e u ê — ä - L H c L strainedunstrained q=p ê q=p ê q=p ê FIG. 3. (Color online) (a) Map of pseudomagnetic field in Eq.( ?? ). Traces show polar plots of the current with Φ / Φ = − . K valley state and different fluxes (∆Φ / Φ = 0 . / Φ = − .
5, as indicated by the arrow.Dashed flat line near bottom shows current density for un-strained ring at Φ = 0. Notice mean value of current in-creases with strain and flux. (c) Flux dependence of the cur-rent through the ring for the lowest state : m = 0, K valleyfrom Φ / Φ = − . m = − K (cid:48) valley fromΦ / Φ = 0 to 0 .
5, at different angles along the ring. Curvesfor strained rings are shifted up for clarity; dashed lines in-dicate zero current in each case. Bottom traces show angle-integrated current for both strained (solid) and unstrained(dotted) rings; notice similar slopes, although much smallerdiscontinuity near Φ = 0 with strain. in inhomogeneous current distributions with vortices, butalso in very different total current dependence with fluxΦ. Notice that the strain changes in the total persistentcurrent, near Φ = 0, are of the same order of magni-tude as the current in the unstrained system. As such,the strain decreases the current discontinuity for positiveand negative magnetic fluxes. The total current varia-tion dependence on strain is found to be proportional to α (not shown). Conclusions . We have shown that the strain effectsarising from a Gaussian ‘bubble’ deformation of thegraphene ring result in a distribution of pseudomagnetic(gauge) fields that have trigonal symmetry, in agreementwith the underlying symmetries of graphene. While thecurrents induced by these pseudofields would identicallyvanish, an external magnetic flux makes possible the ob-servation of the full spatial distribution of currents due tostrain. As a result, strain fields change the nature of theground state and modify the amount of current presentin the device.Our discussion has focused on the infinite-mass bound-ary condition. We find also that the zig-zag boundarycondition, which does not mix valleys either and shows acompletely different spectrum, yields qualitatively simi-lar results to those presented here. It is clear that an ex-periment would typically exhibit a more complex edge. However, because the effects are produced by strainsfields, there will always be inhomogeneous current dis-tributions upon the application of a flux.Furthermore, other geometric structures with strongstrain fields would also produce complex patterns of in-duced currents reflecting the pseudomagnetic field dis-tribution. These could be explored locally by applyinga weak magnetic field, to play the role of the externalflux, assuming that the scanning current measurementdevice has spatial resolution better than the characteris- tic length scales of the gauge field distribution. A scan-ning magnetometer appears as one of the ideal instru-ments to reveal these effects. Although strong sampledisorder may increase scan noise, the trigonal symmetryof the strain signal would uniquely identify its source.
Acknowledgments . We thank F. de Juan, M. Vozmedi-ano and M. M. Asmar for useful discussions. This workwas partially supported by NSF-PIRE, and NSF andCNPq under the CIAM/MWN program. AL acknowl-edges FAPERJ support under grant E-26/101.522/2010;DF acknowledges support from CNPq (140032/2009-6)and CAPES (2412110) while visiting Ohio University,and from DAAD while at Freie Universit¨at. We are grate-ful for the welcoming environment at the Dahlem Centerand the support of the A. von Humboldt Foundation. ∗ D. Faria: [email protected] M. A. H. Vozmediano, M. I. Katsnelson, and F. Guinea,Phys. Rep. , 109 (2010). F. Guinea, Solid State Commun. , 1437 (2012). N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigui, A.Zettl, F. Guinea, A. H. Castro Neto, and M. F. Crommie,Science , 544 (2010). T. Georgiou, L. Britnell, P. Blake, R. V. Gorbachev, A.Gholinia, A. K. Geim, C. Casiraghi, and K. S. Novoselov,Appl. Phys. Lett. , 093103 (2011). N. N. Klimov, S. Jung, S. Zhu, T. Li, C. A. Wright, S. D.Solares, D. B. Newell, N. B. Zhitenev, and J. A. Stroscio,Science H. Tomori, A. Kanda, H. Goto, Y. Ootuka, K. Tsukagoshi,S. Moriyama, E. Watanabe, and D. Tsuya, Appl. Phys.Express , 075102 (2011). K.-J. Kim, Ya. M. Blanter, and K.-H. Ahn, Phys. Rev. B , 081401(R) (2011). N. Abedpour, R. Asgari, and F. Guinea, Phys. Rev. B ,115437 (2011). G. M. M. Wakker, R. P. Tiwari, and M. Blaauboer, Phys.Rev. B , 195427 (2011). J. Gonz´alez, F. Guinea, and M. A. H. Vozmediano, Nucl.Phys. B , 595 (1994). F. de Juan, M. Sturla, and M. A. H. Vozmediano, Phys.Rev. Lett. , 227205 (2012). V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres,Phys. Rev. B , 045401 (2009). A. L. Kitt, V. M. Pereira, A. K. Swan, and B. B. Goldberg,Phys. Rev. B , 115432 (2012); see important erratum atPhys. Rev. B , 159909(E) (2013). F. de Juan, J. L. Ma˜nes, M. A. H. Vozmediano, Phys. Rev.B , 165131 (2013). M. R. Masir, D. Moldovan, F. M. Peeters,arXiv:1304.0629v2 (2013). J. V. Sloan, A. A. P. Sanjuan, Z. Wang, C. Horvath, andS. Barraza-Lopez, Phys. Rev. B , 155436 (2013). B. Roy, Phys. Rev. B , 035458 (2011). B. Roy, Z. X. Hu, and K. Yang, Phys. Rev. B , 121408(R) (2013). P. Recher, B. Trauzettel, A. Rycerz, Ya. M. Blanter, C.W. J. Beenakker, and A. F. Morpurgo, Phys. Rev. B ,235404 (2007). C.-H. Yan and L.-F. Wei, J. Phys.: Condens. Matter ,295503 (2010). J. Wurm, M. Wimmer, H. U. Baranger, and K. Richter,Semicond. Sci. Technol. , 034003 (2010). S. Russo, J. B. Oostinga, D. Wehenkel, H. B. Heersche, S.S. Sobhani, L. M. K. Vandersypen, and A. F. Morpurgo,Phys. Rev. B , 085413 (2008). M. Huefner, F. Molitor, A. Jacobsen, A. Pioda, K. Ensslin,and T. Ihn, Phys. Status Solidi B , 2756 (2009). M. Huefner, F. Molitor, A. Jacobsen, A. Pioda, C.Stampfer, K. Ensslin, T. Ihn, New J. Phys. , 043054(2010). D. Smirnov, H. Schmidt, and R. J. Haug, Appl. Phys. Lett. , 203114 (2012). A. Rahman, J. W. Guikema, S. H. Lee, and N. Markovi´c,Phys. Rev. B , 081401(R) (2013). J. Schelter, P. Recher, B. Trauzettel, Solid State Commun. , 1411 (2012). M. V. Berry and R. J. Mondragon, Proc. R. Soc. A ,53 (1987). C. W. J. Beenakker, Rev. Mod. Phys. , 1337 (2008). J. Wurm, A. Rycerz, ˙I. Adagideli, M. Wimmer, K. Richter,and H. U. Baranger, Phys. Rev. Lett. , 056806 (2009). N. M. R. Peres, J. N. B. Rodrigues, T. Stauber, and J.M. B. Lopes dos Santos, J. Phys. Cond. Matt. , 344202(2009). H. Suzuura and T. Ando, Phys. Rev. B , 235412 (2002). L. D. Landau and E. M. Lifshitz,
Theory of Elasticity (Pergamon Press, Oxford, 1970). F. de Juan, A. Cortijo, and M. A. H. Vozmediano, Phys.Rev. B , 165409 (2007). M. Grujic, M. Zarenia, A. Chaves, M. Tadic, G. A. Farias,and F. M. Peeters, Phys. Rev. B84