Geometry induced charge separation on a helicoidal ribbon
GGeometry induced charge separation on a helicoidal ribbon
Victor Atanasov ∗ Institute for Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria † Rossen Dandoloff
Laboratoire de Physique Th´eorique et Mod´elisation ,Universit´e de Cergy-Pontoise, F-95302 Cergy-Pontoise, France ‡ Avadh Saxena
Theoretical Division and Center for Nonlinear Studies,Los Alamos National Laboratory, Los Alamos, NM 87545 USA § We present an exact calculation of the effective geometry-induced quantum potential for a particleconfined on a helicoidal ribbon. This potential leads to the appearance of localized states at therim of the helicoid. In this geometry the twist of the ribbon plays the role of an effective transverseelectric field on the surface and thus this is reminiscent of the quantum Hall effect.
PACS numbers: 02.40.-k, 03.65.Ge, 73.43.Cd
The interplay of geometry and topology is a recurringtheme in physics, particularly when these effects mani-fest themselves in unusual electronic and magnetic prop-erties of materials. Specifically, helical ribbons providea fertile playground for such effects. Both the helicoid(a minimal surface) and helical ribbons are ubiquitousin nature: they occur in biology, e.g. as beta-sheetsin protein strucutres , macromolecules (such as DNA) ,and tilted chiral lipid bilayers . Many structural motifsof biomolecules result from helical arrangements : cel-lulose fibrils in cell walls of plants, chitin in arthropodcuticles, collagen protein in skeletal tissue. Condensedmatter examples include screw dislocations in smecticA liquid crystals , certain ferroelectric liquid crystals ,and recently synthesized graphene ribbons. In particu-lar, graphene M¨obius strips have been investigated fortheir unusual electronic and spin properties . A helicoidto spiral ribbon transition and geometrically inducedbifurcations from the helicoid to the catenoid have alsobeen studied.Graphene ribbons can be doped with charges. In thiscontext, our goal is to answer the following questions:what kind of an effective quantum potential does a charge(or electron) experience on a helicoid or a helical ribbondue to its geometry (i.e., curvature and twist)? If theouter edge of the helicoid is charged, how is this poten-tial modified and if there are any bound states? Ourmain findings are: the twist ω will push the electronsin vanishing angular momentum state towards the inneredge of the ribbon and the electrons in non-vanishingangular momentum states to the outer edge thus creat-ing an inhomogeneous effective electric field between theinner and outer rims of the helicoidal ribbon. This isreminiscent of the quantum Hall effect; only here it isgeometrically induced. We expect our results to lead tonew experiments on graphene ribbons and other relatedtwisted materials where the predicted effect can be ver- ified. In a related context we note that de Gennes hadexplained the buckling of a flat solid ribbon in terms ofthe ferroelectric polarization charges on the edges . FIG. 1: A helicoidal ribbon with inner radius ξ and outerradius D . For ξ = 0 it becomes a helicoid. Vertical axis isalong x and the transverse direction ξ is across the ribbon. In order to answer the questions posed above, herewe study the helicoidal surface to gain a broader un-derstanding of the interaction between quantum parti-cles and curvature and the resulting possible physical ef-fects. The properties of free electrons on this geometryhave been considered before . The results of this pa-per are based on the Schr¨odinger equation for a confinedquantum particle on a sub-manifold of R . Followingda Costa an effective potential appears in the two di-mensional Schr¨odinger equation which has the followingform: V curv = − (cid:126) m ∗ (cid:0) M − K (cid:1) , (1)where m ∗ is the effective mass of the particle, (cid:126) is thePlank’s constant; M and K are the Mean and the Gaus-sian curvature, respectively. a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t To describe the geometry we consider a strip whoseinner and outer edges follow a helix around the x -axis(see Fig. 1 with ξ = 0). The surface represents a helicoid and is given by the following equation: (cid:126)r = x (cid:126)e x + ξ [cos( ωx ) (cid:126)e y + sin( ωx ) (cid:126)e z ] , (2)where ω = πnL , L is the total length of the strip and n is the number of 2 π -twists. Here ( (cid:126)e x , (cid:126)e y , (cid:126)e z ) is the usualorthonormal triad in R and ξ ∈ [0 , D ], where D is thewidth of the strip. Let d(cid:126)r be the line element and themetric is encoded in | d(cid:126)r | = (1 + ω ξ ) dx + dξ = h dx + h dξ , where h = h ( ξ ) = (cid:112) ω ξ and h = 1 are the Lam´ecoefficients of the induced metric (from R ) on the strip.Here is an appropriate place to add a comment on the helicoidal ribbon , that is a strip defined for ξ ∈ [ ξ , D ](see Fig. 1). All the conclusions still hold true and all ofthe results can be translated using the change of variables ξ = ξ + s ( D − ξ ) , s ∈ [0 , . Here s is a dimensionless variable and one easily sees thatfor ξ → H = − (cid:126) m ∗ h (cid:20)(cid:18) ∂∂ξ h ∂∂ξ (cid:19) + ∂∂x h ∂∂x (cid:21) + V curv . (3)Let us elaborate on the curvature-induced potential V curv . Since the helicoid is a minimal surface M vanishesand we are left with the following expression V curv = (cid:126) m ∗ K = − (cid:126) m ∗ ω [1 + ω ξ ] . (4)Using Gauss’ Theorema egregium the above potentialcan also be rewritten as V curv = (cid:126) m ∗ K = − (cid:126) m ∗ h (cid:18) ∂ h ∂ξ (cid:19) . (5)After rescaling the wave function ψ (cid:55)→ √ h ψ (becausewe require the wave function to be normalized with re-spect to the area element dxdξ ) we arrive at the followingexpression for the Hamiltonian: H = − (cid:126) m ∗ (cid:18) ∂ ∂ξ + 1 h ∂ ∂x (cid:19) + V eff ( ξ ) , (6)where the effective potential in the (transverse) ξ direc-tion is given by: V eff ( ξ ) = − (cid:126) m ∗ (cid:34) h (cid:18) ∂ h ∂ξ (cid:19) + 14 1 h (cid:18) ∂h ∂ξ (cid:19) (cid:35) . (7) x1 2 3U x K0,4K0,200,20,4 FIG. 2: The behavior of the potential U ( ξ ) for ω = 1 and (cid:126) = 2 m ∗ = 1 . Here the red curve corresponds to m = 0 , thegreen curve to m = 1 , and the yellow line to the approxima-tion given by Eq. (14). Note that in bent tubular waveguides and curvedquantum strip waveguides the effective potential is lon-gitudinal. In the present case there is no longitudinal ef-fective potential. After insertion of h = (cid:112) ω ξ theeffective potential becomes: V eff ( ξ ) = − (cid:126) m ∗ ω (1 + ω ξ ) (cid:20) ω ξ (cid:21) . (8)This effective potential is of pure quantum-mechanicalorigin because it is proportional to (cid:126) . Note that thisexpression is exact and is valid not just for small ξ : hereno expansion in a small parameter has been used.Next, we write the time-independent Schr¨odingerequation as: (cid:20) − (cid:126) m ∗ ∂ ∂ξ + V eff ( ξ ) (cid:21) ψ − (cid:126) m ∗ h ∂ ψ∂x = Eψ. (9)Using the ansatz: ψ ( x, ξ ) = φ ( x ) f ( ξ ) we split the depen-dence on the variables and we get two differential equa-tions: − (cid:126) m ∗ d φ ( x ) dx = E φ ( x ) , (10)and − (cid:126) m ∗ d f ( ξ ) dξ + U ( ξ ) f ( ξ ) = Ef ( ξ ) , (11)where U ( ξ ) = V eff ( ξ ) + E h ( ξ ) . (12)With a solution φ ( x ) = e ik x x of Eq. (10) we have E = (cid:126) m ∗ k x , where k x is the partial momentum in x -direction. Let usconsider here the azimuthal angle around the x axis: ωx and the angular momentum along this axis: L x = − i (cid:126) ω ∂∂x . This operator has the same eigenfunctions L x φ ( x ) = (cid:126) mφ ( x ) as the operator in Eq. (10). The correspondingeigenvalues are (cid:126) m. We conclude that the momentum k x is quantized k x = mω, m ∈ N . This is not surprising because of the periodicity of thewave function along x . Note that the value of the an-gular momentum quantum number determines the direc-tion the electron takes along the x axis either upward m > m < . This situation is reversed fora helicoid with opposite chirality.Equation (11) represents the motion in the direction ξ with a net potential U ( ξ ) = − (cid:126) m ∗ ω (cid:26) − m (1 + ω ξ ) + 1(1 + ω ξ ) (cid:27) , (13)which is depicted in Fig. 2.This potential is a sum of two contributions, an attrac-tive part: ω ξ ) and a variable part which is repulsivefor m ≥ m = 0 (see Fig 2). The ac-tion of this part for m (cid:54) = 0 qualifies it as a centrifugal po-tential. It pushes a particle to the boundary of the strip.The finite size of the width D determines the cut-off of U ( ξ ) and hence the probability of finding the particle isgreatest near the rim of the helicoid. Since the behaviorof the potential U ( ξ ) for a particle with m = 0 qualifiesit as a quantum anti-centifugal one, it concentrates theelectrons around the central axis for a helicoid (or theinner rim for a helicoidal ribbon). Such anti-centrifugalquantum potentials have been considered before .The behavior described above can be inferred usingthe uncertainty principle. Localized states must appearaway from the central axis or the inner rim. Physically,one may understand the appearance of localized statesaway from the central axis using the following reason-ing: for greater ξ a particle on the strip will avail morespace along the corresponding helix and therefore thecorresponding momentum and hence the energy will besmaller than for a particle closer to the central axis.We note that the separability of the quantum dynamicsalong x and ξ directions with different potentials pointsto the existence of an effective mass anisotropy on thehelicoidal surface.For the sake of simplicity let us approximate the po-tential U ( ξ ) given in Eq. (13) (for m = 1) by a straightline. The sole purpose of this approximation is to pin-point the basic distribution of the probability density.Assuming it to be linear (see Fig. 2) and starting fromcertain ξ = a (cid:28) U a ( ξ ) = ( D − ξ ) U D − a , U = U ( ξ = a ) . (14)The value of a can be determined from an area preservingcondition U ( D − a ) = (cid:82) Dξ U ( x ) dx, where ξ < D is the position from which we evolve the surface. Whendealing with a helicoidal ribbon we must take ξ (cid:54) = 0 . After obtaining a result for this case we can easily obtaina result for the helicoid case by taking the limit ξ → . Next we introduce a characteristic lengthscale l in theproblem l − = 2 m ∗ | U | (cid:126) ( D − a ) , λl = 2 m ∗ (cid:126) (cid:18) E − DU D − a (cid:19) , where λ is a dimensionless energy scale. After intro-ducing the dimensionless variable ζ = − λ − ξ/l theSchr¨odinger equation for the radial part becomes d fdζ − ζf ( ζ ) = 0 , (15)with the following boundary conditions: f ( − λ − ξ /l ) = f ( − λ − D/l ) = 0 . This form of the equation is valid for U > m (cid:54) = 0 . For m = 0 we have a negative U = −| U | whichrequires the introduction of the dimensionless variable ζ = − λ + ξ/l and the corresponding equation is givenby (15), only in this case the boundary conditions are f ( − λ + ξ /l ) = f ( − λ + D/l ) = 0 . Let us assume that the ratio
D/l (cid:29) f ( ζ ) = const Ai( ζ ) , and theboundary condition f ( − λ ± ξ /l ) = 0 (the upper signcorresponds to m = 0 and the lower to m (cid:54) = 0 states)gives the quantized energies E n ( m ) = U ( m ) DD − a + (cid:18) λ n ± ξ l (cid:19) (cid:126) m ∗ l , where λ n are the zeroes of the Airy function Ai( − λ n ) =0 . Let us list the first three of them: ( λ , λ , λ ) =(2 . , . , . . Here we have taken account of thecase when the interior of the helicoid is cut at a distance ξ from the axis, that is the ribbon case. The helicoidcase is obtained after setting ξ → . For the vanishing angular momentum state we have U (0) < bound state. The prob-ability amplitude has a node at ξ in the ribbon case orat the origin for the helicoid case. The evolution along ξ starts at the corresponding zero of the Airy function andevolves in the positive direction where the Airy functionvanishes. For non-vanishing angular momentum stateswe have U ( m ) > ξ starts at the corresponding zero of theAiry function and evolves in the negative direction wherethe Airy function is oscillatory as one would expect fora confined positive energy spectrum. The observationthat the m (cid:54) = 0 states at ξ = λ n l have the same energy E n ( m ) = U ( m ) D/ ( D − a ) for all n leads us to believethat this is a particular positive energy oscillatory statewhose wavelength fits D (1 − ξ /l ) ≈ D ( l/D (cid:29) . We would like to conclude with the observation thatthe electric dipole moment for the ( m = 0 , λ ) boundstate (also the ground state for this geometrical config-uration) is non-zero due to the anisotropic distributionof the probability density along ξ . Indeed, suppose weconsider a ribbon doped with a uniform surface chargedensity σ, then the electric dipole vector (cid:126)p = p x (cid:126)e x + p ξ (cid:126)e ξ in the moving coordinate system ( (cid:126)e x , (cid:126)e ξ , (cid:126)e = (cid:126)e x × (cid:126)e ξ )will have non-vanishing x and ξ components: p x = Qπω , p ξ = 2 πω σl β n , (16)where the total charge is Q = (cid:90) π/ω dx (cid:48) (cid:90) Dξ σ | ψ ( x (cid:48) , ξ (cid:48) ) | dξ (cid:48) and β n = (cid:90) D/l (cid:29) ξ /l (cid:12)(cid:12)(cid:12)(cid:12) Ai (cid:18) − λ n ∓ ξ l ± t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) tdt. Here the upper sign corresponds to m = 0 and thelower to m (cid:54) = 0 states. For ξ /l = 0 . D/l = 10 wesummarize the values of β n in the following table n = 1 n = 2 n = 3 n = 10 m = 0 0.816 1.822 2.829 3.605 m (cid:54) = 0 2.712 2.451 2.299 1.783 Let us suppose that the outer rim of the helicoid isuniformly charged or there is a uniformly charged wiregoing through the core, then this will create an acceler-ating electric field term in the effective potential U ( ξ ) , that is U e ( ξ ) = U ( ξ ) + e E ξ. The dynamics is still separa-ble. In the cup-shaped potential U e the electrons will befound with the greatest probability where the potentialhas a minimum. This means that the extra charge onthe helicoid will concentrate in a strip around the valueof ξ min , i.e. a solution to dU e /dξ = − e E . Application of an electric or magnetic field along the x -axis would nontrivially affect the motion of electronson the surface–this problem will need to be studied nu-merically. It would be very interesting to observe thepredicted effect in graphene ribbons or helicoidal ribbonssynthesized from a semiconducting material.Our main findings can be summarized as follows: thetwist ω will push the electrons with m (cid:54) = 0 ( m = 0)towards the outer (inner) edge of the ribbon and cre-ate an effective electric field between the central axis andthe helix, the latter representing the rim of the helicoid. Instead of a helicoidal ribbon, if we consider a cylindri-cal helical ribbon then both the curvature and torsionare constant and the effective potential is quite simple.We expect our results to motivate new low temperature(
T < (cid:126) /k B m ∗ l , where k B is Boltzmann’s constant)experiments on twisted materials.This work was supported in part by the U.S. Depart-ment of Energy. ∗ Also at Laboratoire de Physique Th´eorique et Mod´elisation, Universit´e de Cergy-Pontoise, F-95302 Cergy-Pontoise,France † Electronic address: [email protected] ‡ Electronic address: rossen.dandoloff@u-cergy.fr § Electronic address: [email protected] C. W. G. Fishwick, A. J. Beevers, L. M. Carrick, C. D.Whitehouse, A. Aggeli, and N. Boden, Nano Lett. , 1475(2003). J. Crusats et al., Chem. Commun. iss. 13, 1588 (2003). O-Y. Zhong-can and L. Ji-xing, Phys. Rev. Lett. , 1679(1990); Phys. Rev. A , 6826 (1991). J. M. Garcia Ruiz, A. Carnerup, A. G. Christy, N. J. Wel-ham, and S. T. Hyde, Astrobiology , 353 (2002). R. D. Kamien and T. C. Lubensky, Phys. Rev. Lett. ,2892 (1999). D. M. Walba, E. K¨orblova, R. Shao, J. E. Maclennan, D.R. Link, M. A. Glaser, and N. A. Clark, Science , 2181(2000). A. Yamashiro, Y. Shimoi, K. Harigaya, and K. Wak-abayashi, Physica E , 688 (2004); Phys. Rev. B , 193410 (2003); De-en Jiang and Sheng Dai, J. Phys. Chem.C , 5348 (2008). R. Bruinsma and R. Ghafouri, Phys. Rev. Lett. , 138101(2005). A. Boudaoud, P. Patricio, and M. Ben Amar, Phys. Rev.Lett. , 3836 (1999). P.-G. de Gennes, C.R. Acad. Sci. Ser. 2, , 259 (1987). R. Dandoloff and T.T. Truong, Phys.Lett. A , 233(2004). R.C.T. da Costa, Phys. Rev. A , 1982 (1981). M. Spivak,
A Comprehensive introduction to differentialgeometry (Publish or Perish, Boston, 1999). J. Goldstone and R.L. Jaffe Phys. Rev.
B 45 , 14100(1992). I.J. Clark and A.J. Bracken J. Phys. A: Math.Gen. ,339 (1996); , 2103 (1998). M. A. Cirone, K. Rzazewski , W. P. Schleich , F. Straub,and J. A. Wheeler, Phys. Rev. A , 022101 (2001); V.Atanasov and R. Dandoloff, Phys. Lett. A371