Since effectively the local contact vortex velocity dependent part of the Magnus force in a Josephson junction array is zero in the classical limit, we predict zero classical Hall effect. In the quantum limit because of the geometric phases due to the finite superfluid density at superconductor grains, rich and complex Hall effect is found in this quantum regime due to the Thouless-Kohmoto-Nightingale-den-Nijs effect

A. Kol

A.I. Larkin

C.D. Chen

D.J. Thouless

E. Fradkin

H.S.J. Zant van der

M. Y. Choi

P. Ao

English

arXiv

Since effectively the local contact vortex velocity dependent part of the Magnus force in a Josephson junction array is zero in the classical limit, we predict zero classical Hall effect. In the quantum limit because of the geometric phases due to the finite superfluid density at superconductor grains, rich and complex Hall effect is found in this quantum regime due to the Thouless-Kohmoto-Nightingale-den-Nijs effect.</p

Ao, Ping,

Tan, Y.,

Zhu, Xiaomei,

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Hall effect and geometric phases in Josephson junction arrays

Ao, P.

Tan, Y.

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Strathprints Institutional RepositoryAo, P. and Tan, Y. and Zhu, X.M. (1996) Hall effect and geometric phases in Josephson junctionarrays. Czechoslovak Journal of Physics, 46 (2). pp. 589-590. ISSN 0011-4626Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/  Ao, P. and Tan, Y.* and Zhu, X.-M. (1996) Hall effect and geometric phases in Josephson junction arrays. Czechoslovak Journal of Physics, 46 (2). pp. 589-590. ISSN 0011-4626      http://eprints.cdlr.strath.ac.uk/6166/   This is an author-produced version of a paper published in Czechoslovak Journal of Physics, 46 (2). pp. 589-590. ISSN 0011-4626. This version has been peer-reviewed, but does not include the final publisher proof corrections, published layout, or pagination.  Strathprints is designed to allow users to access the research output of the University of Strathclyde. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (http://eprints.cdlr.strath.ac.uk) and the content of this paper for research or study, educational, or not-for-profit purposes without prior permission or charge. You may freely distribute the url (http://eprints.cdlr.strath.ac.uk) of the Strathprints website.  Any correspondence concerning this service should be sent to The Strathprints Administrator: eprints@cis.strath.ac.uk  arXiv:cond-mat/9608090 v1   20 Aug 1996Hall effect and geometric phases in Josephson junction arraysP. Ao1, Y. Tan2, and X.-M. Zhu11Department of Theoretical Physics , Ume˚a University, S-901 87, Ume˚a, SWEDEN2Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 0NG, UKAbstractSince effectively the local contact vortex velocity dependent part of the Mag-nus force in a Josephson junction array is zero in the classical limit, we pre-dict zero classical Hall effect. In the quantum limit because of the geometricphases due to the finite superfluid density at superconductor grains, rich andcomplex Hall effect is found in this quantum regime due to the Thouless-Kohmoto-Nightingale-den-Nijs effect.1Since vortex dynamics is identical to that of an electron in the presenceof a magnetic field, numerous models for the quantum Hall effect in bothhomogeneous and inhomogeneous superconductor films have been proposed.12 Those models have fully explored the analogy to the quantum Hall effectin semiconductor heteorojunctions by treating vortices moving in a uniformmagnetic field with a homogeneous background. While the treatment can bejustified in a homogeneous superconductor film, it may not be so in inhomo-geneous cases such as in Josephson junction arrays, where both the (fractionpart) number of magnetic flux, the frustration n, and the (fraction part) ofthe fictitious magnetic flux φ0 per plaqutte are usually large, and the periodicpotential for a vortex is strong. Therefore the simple contineous limit without1This work was supported by Swedish NFR.2 to appear in Czechoslovak J. Phys., Vol.46, Suppl. S2, pp589 (1996)1the accounting of the potential is not appropriate for a Josephson junctionarray.In a Josephson junction array, because of the huge core energy, vorticescannot move into the superconducting gains. They are confined to movealong the junctions and the voids (nonsuperconducting areas), an example ofthe guided vortex motion. Since the vortex velocit part of the Magnus forceis proportional to the local superfluid density, derivable from the nonlinearSchro¨dinger Lagrangian formulation2, this force is zero for a vortex at a void,and, exponentially small at a junction. Furthermore, because of the guidemotion, even the small transverse at a junction does not produce a sidewaymotion. This implies that this local contact transverse force does not play arole in vortex dynamics in a Josephson junction array. Hence there is no Halleffect in the classical limit. This absence of the en route transverse force is inagreement with experimental observations3. The condition for the classicallimit will be given at the end of the paper.In the quantum regime, however, vortices experience geometric phasessimilar to the Aharonov-Bohm effect, due to the finite superfluid densityat superconductor grains.4 To be commensurated with the existence of thevortex inaccessible regions and the geometric phases, we consider the thetight-binding limit of vortex motion. The corresponding Hamiltonian may bewritten asH = t∑(l,j)a†l aj eiAlj +∑l,ja†l al Vlj a†jaj , (1)where al is the boson annihilation operator for a vortex at j-th void, and ( )stands for the summation over nearest neighbors. The phase Alj is definedon the links connected the nearest neighbors, and its sum around a plaquetteis equal to the geometric phase 2piφ0:∑plaquette Alj = 2piφ0. A uniformgeometric phase in a square lattice will be assumed, where the number of2‘fluxes’ φ0 is the number of Cooper pairs on a superconductor grain, whichmay be controlled by a gate voltage. The interaction between vortices isdescribed by Vlj, which is long range and repulsive. We will treat it as a shortrange repulsive interaction for a first approximation, further approximated bythe hard-core conditions. The tunneling matrix element t is,t ≃√EJEC exp{−O(1)√EJ/EC} , (2)where EJ is the Josephson junction energy and EC the junction chargingenergy5.To discuss the Hall effect of the idealized vortex problem in the quantumregime, we map the hard-core boson problem onto a fermion problem byattaching odd number of ‘fluxes’ on each vortex. The resulting Hamiltonianfor the fermion problem isH = t∑(l,j)c†l cj ei[Alj+Alj ] , (3)where cj is the corresponding the fermion annihilation operator at the j-thvoid. The number of statistical fluxes φs at the j-th void satisfys the constrainφs = −(2m+1) < c†jcj >, with∑plaquetteAlj = 2piφs, which means that 2m+1fluxes have been attached to each vortex.6 If this mapping gives a mean fieldsolution with an energy gap separated from its excitations, the statisticalfluxes can be adiabatically smeared over the lattice and effectively detachedfrom vortices. In this case φs = −(2m + 1)n, with n is the magnetic fluxfrustration, the number of vortices per plaquette. Then the resulting meanfield problem is exactly the Harper-Azbel-Wannier-Hofstadter problem, whereenergy gaps do exist. The quantum Hall behaviors of such a problem havebeen studied in detail by Thouless, Kohmoto, Nightingale, and den Nijs7. Forsuch a system the quantum Hall conductance σfH is σfH = tr , with the integertr the solution of the Diophantine equation r = srq+ trp . Here the number of3fluxes per plaquette φ = φ0−φs = p/q, with p and q coprime, n = r/q, and r,sr, tr integers with |tr| ≤ q/2. Counting the mapping generated Chern-Simonscontribution to the Hall conductance, σsH =12m+1 , the Hall conductance of theoriginal vortex system is then6,8 1/σvH = 1/σfH +1/σsH . Converting back intothe electric Hall conductance and putting back the unit, we find the electricquantum conductance of the Josephson junction array isσH =4e2hσfH + σsHσfHσsH. (4)As known in the previous study of quantum Hall effect8,6 for a given set ofthe ‘flux’ φ0 and the frustration n, there may exist several values of m, thatis, several mappings, with their mean-field solutions all corresponding to filledbands which are separated from excitations by energy gaps. If such a caseoccurs, detailed calculation is needed fo find the m with the largest energygap, which is the most stable one.One can check that following symmetries hold for the quantum Hall con-ductance σH : the periodicity, σH(φ0, n) = σH(φ0 + 1, n) ; the odd sym-metry, σH(φ0, n) = −σH(−φ0, n); the particle-hole symmetry, σH(φ0, n) =−σH(φ0, 1 − n). We note that both positive and negative Hall conductancemay be easily reached, contrast to the previous proposal of the quantum Halleffect in a Josephson junction array1. For the special mapping 2m n = φ0 themean-field solution is automatically within a gap, and the Hall conductanceis σH = 2m4e2h. With these specific sets of φ0 and n and in the zero limitof their fraction parts one can take the continuous limit of the tight-bindingmodel.We conclude by discussing of a criterion for the classical limit, in whichthere is no Hall effect. The relevant energy scale is the tunneling matrix ele-ment t. When the temperature is higher than t, thermal fluctuation will de-stroy the quantum coherence and the vortices move classically. The quantum4regime is realized for temperatures lower than t where the phase coherenceis preserved. For a Josephson junction energy EJ ∼ 1 K and the junctioncharging energy EC ∼ EJ , t ∼ 100 mK. We point out that the Hall effect inthe quantum regime may have been realized experimentally9.5REFERENCES1 M. Y. Choi, Phys. Rev. B50, 10088 (1994); A. Stern, ibid B50, 10092 (1994); A. A. Odintsovand Y. V. Nazarov, Physica B203, 513 (1994); B. Horovitz, Phys. Rev. B51, 3989 (1995); F.Gaitan and S.R. Shenoy, preprint #: cond-mat/9505088.2 X.-M. Zhu, Y. Tan, and P. Ao, preprint #: cond-mat/9507126.3 See, for example, H.S.J. van der Zant et al., Phys. Rev. B50, 340 (1994).4 P. Ao and D. J. Thouless, Phys. Rev. Lett. 70, 2158 (1993); P. Ao, Q. Niu, and D.J. Thouless,Physica B194-196, 1453 (1994).5 A.I. Larkin, Y.N. Ovichinnikov, and A. Schmid, Physica 152B, 266 (1988); U. Eckern and A.Schmid, Phys. Rev. B39, 6441 (1989); R. Fazio et al., Helv. Phys. Acta 65, 228 (1992).6 E. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, New York, 1991.7 D.J. Thouless et al, Phys. Rev. Lett. 49, 405 (1982).8 A. Kol and N. Read, Phys. Rev. B48, 8890 (1993).9 C.D. Chen et al., in Macroscopic Quantum Phenomena and Coherence in Superconducting Net-works , Edited by C. Giovannella and M. Tinkham, World Scientific, Singapore, 1995.6

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