condensation and coherence in condensed matter · condensation and coherence in condensed matter...

Condensation and Coherence in Condensed Matter Proceedings of the Nobel Jubilee Symposium Goteborg, Sweden, December 4 - 7 , 2001

Editors

T. Claeson P. Delsing

Physica Scripta The Royal Swedish Academy of Sciences /

World Scientific

Condensation andCoherence inCondensed Matter

Proceedings of the Nobel Jubilee SymposiumGoteborg , Sweden,December 4 - 7, 2001

Editors

T. ClaesonP. Delsing

Recognized by the European Physical Society

Physica ScriptaThe Royal Swedish Academy of Sciences

KUNGL. World ScientificVETENSKAPSAKADEMIENTHE ROYAL SWEDISH ACADEMY OF SCIENCES `1 New Jersey. London •Singapore -Hong Kong

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CONDENSATION AND COHERENCE IN CONDENSED MATTER- Nobel Jubilee Symposium

Copyright © 2003 Royal Swedish Academy of Sciences

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I

Physica Scripta , Vol. T102, 2002

Contents

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Quantum Coherence Between States with Even and Odd Numbers of Electrons. A. F . Andreev. . . . . . 7

Superconductivity in High Tc Cuprates: The Cause is No Longer a Mystery. Philip W. Anderson. . . . . . 10

Nanoelectromechanics of Coulomb Blockade Nanostructures. R. I. Shekhter, L. Y. Gorelik, A. /sacsson,Y. M. Galperin and M. Jonson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Electron Spin in Single Wall Carbon Nanotubes. P. E. Linde/of, J. Borggreen, A. Jensen, J. Nygardand P. R. Poulsen .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

The Critical Regime of the Metal - Insulator Transition in Conducting Polymers : Experimental Studies.Alan J. Heeger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Superfluidity and Coherence in Bose -Einstein Condensates. Wolfgang Ketterle .. . . . . . . . . . . . 36

Jahn-Teller Bipolarons and their Condensation. K. A. M01ler . . . . . . . . . . . . . . . . . . . 39

Materials and Physics of High Temperature Superconductors: A Summary, Two Recent Experimentsand a Comment. C. W. Chu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Consequences of Unconventional Order Parameter Symmetry-High Critical Temperature Structures-.A. Barone, J. R. Kirtley, F. Tafuri and C. C. Tsuei . . . . . . . . . . . . . . . . . . . . . . . . 51

Magneto Oscillations in Unconventional Superconductors well below Hc2. J. R. Schrieffer. . . . . . . . 59

Quantum Complementarity for the Superconducting Condensate and the Resulting ElectrodynamicDuality. D. B. Havi/and, M. Watanabe, P. Agren and K. Andersson . . . . . . . . . . . . . . . . . 62

Probing Quantum Mechanics Towards the Everyday World: Where do we Stand? A. J. Leggett. . . . . . 69

Ultracold Dipolar Gases - a Challenge for Experiments and Theory. M. Baranov, L. Dobrek, K. G6ral,L. Santos and M. Lewenstein .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A Preliminary Measurement of the Fine Structure Constant Based on Atom Interferometry. Andreas Wicht,Joel M. Hensley, Edina Sarajlic and Steven Chu . . . . . . . . . . . . . . . . . . . . . . . . 82

The Question of Phase in a Bose -Einstein Condensate. S. Stenholm .. . . . . . . . . . . . . . . . 89

Observation of Qubit State with a dc-SQUID and Dissipation Effect in the SQUID. Hideaki Takayanagi,Hirotaka Tanaka, Shiro Saito and Hayato Nakano .. . . . . . . . . . . . . . . . . . . . . . . 95

Enhancement of Magnetic Ordering by the Stress Fields of Grain Boundaries in Ferromagnets.A. Kadigrobov, Z. Ivanov, R. I. Shekhter and M. Jonson . . . . . . . . . . . . . . . . . . . . 103

Experiments with d-wave Superconductors. J. Mannhart, H. Hilgenkamp, G. Hammer) and C. W. Schneider. . 107

Bose - Einstein Condensation of Metastable Helium . C. Cohen-Tannoudji .. . . . . . . . . . . . . . 111

DC Transformer and DC Josephson(-like) Effects in Quantum Hall Bilayers. S. M. Girvin. . . . . . . . . 112

Impurity- Helium Solids - Quantum Gels? V. V. Khmelenko, S. I. Kiselev, D. M. Lee and C. Y. Lee. . . . . . 118

Quantum Information with Atoms and Photons in a Cavity: Entanglement, Complementarity andDecoherence Studies. S. Haroche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

A Theory of the Relative Phase and Number Difference of Two Quantized Harmonic Oscillators. GunnarBjork, Jonas Soderho/m, Alexei Trifonov and Tedros Tsegaye . . . . . . . . . . . . . . . . . . . 133

Noise and Decoherence in Quantum Two-Level Systems. Alexander Shnirman, Yuriy Makhlinand Gerd Schon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Coherent Manipulations of Charge-Number States in a Cooper-Pair Box. V. Nakamura, Yu. A. Pashkin,T. Yamamoto and J. S. Tsai. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Ramsey Fringe Measurement of Decoherence in a Novel Superconducting Quantum Bit Based onthe Cooper Pair Box. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteveand M. H. Devoret.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Reading Out Charge Qubits with a Radio-Frequency Single-Electron-Transistor. K. Bladh, D. Gunnarsson,G. Johansson, A. Kick, G. Wendin, A. Aassime, M. Taslakov and P. Delsing . . . . . . . . . . . . . . 167

Quiet Readout of Superconducting Flux States. John Clarke, T. L. Robertson, B. L. T. Plourde,A. Garcia-Martinez, P. A. Reichardt, D. J. Van Harlingen, B. Chesca, R. Kleiner, Y. Makhlin, G. Schon,A. Shnirman and F K. Wilhelm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Initial Experiments Concerning Quantum Information Processing in Rare-Earth-Ion Doped Crystals.M. Nilsson, L. Rippe, N. Ohlsson, T. Christiansson and S. Krol% . . . . . . . . . . . . . . . . . . 178

Physica Scripta T 102, 4, 2002

List ofParticipants

Zhores I Alferov Steven Girvin Johannes E Mooijloffe Physico-Technical Inst, Yale University Delft Univ.St Petersburg [email protected] [email protected]. tudelft.nlzhores [email protected]. rssi.ru

Serge Haroche Alexander K MullerBoris Altshuler ENS, Paris Zurich Univ.Princeton Univ. & NEC [email protected] [email protected]@princeton.edu

David Haviland Yasunobu NakamuraPhilip W Anderson Royal Inst. Techn., Stockholm NEC, TsukubaPrinceton University havil [email protected]. se [email protected]@puhep l.princeton.edu

Alan J Heeger Douglas D OsheroffAlexander Andreev Univ of Calif. Santa Barbara Stanford Univ.Kapitza Institute, Moscow [email protected] [email protected]@kapitza.ras.ru

Zdravko Ivanov Mikko PaalanenAntonio Barone Chalmers, Goteborg Techn. Univ., HelsinkiUniv. Napoli "Federico II" [email protected] [email protected]@na.infn.it

Mats Jonson William D PhillipsBertram Batlogg Chalmers, Goteborg NIST, GaithersburgETH Zurich [email protected] [email protected]@solid .phys.ethz. ch

Wolfgang Ketterle Robert C RichardsonGunnar Bjork MIT Cornell Univ.Royal Inst. Techn., Stockholm [email protected] [email protected]@ele.kth.se

Klaus von Klitzing Gerd SchonSteven Chu Maz Planck Inst., Stuttgart Karlsruhe Univ.Stanford Univ, [email protected] [email protected]@stanford.edu

Matti Krusius J Robert Schrieffer,Ching-Wu (Paul) Chu Techn. Univ., Helsinki NHMFL, Tallahassee, FloridaHouston Univ. & HongKong U S & T [email protected] [email protected]@[email protected] Stefan Kroll Robert Shekhter

Univ. Lund Goteborg Univ.Tord Claeson stefan. [email protected]. se shekhter@fy. chalmers. seChalmers, Goteborgf4 atc @fy. chalmers. se Robert B Laughlin Stig Stenholm

Stanford Univ. Royal Inst. Techn., StockholmJohn Clarke [email protected] stenho lm@atom. kth. seUniv. of Calif. [email protected] David M Lee Asle Sudbe

Cornell Univ. Norwegian U S & T, TrondheimClaude Cohen-Tannoudji [email protected] [email protected] Norm. Sup., [email protected] Anthony Leggett Sune Svanberg

Univ. Illinois, Urbana Lund Univ.Eric Cornell [email protected] sune . svanberg@fysik. lth. seJILA, Univ. Colorado, [email protected] Maciej Lewenstein Hideaki Takayanagi

Hannover Univ. NTT, AtsugiPer Delsing [email protected] takayan@wil l.brl. ntt. co. j pChalmersdelsing@fy. chalmers. se Konstantin K Likharev Richard A Webb

SUNY, Stony Brook Univ. MarylandMichel Devoret [email protected] [email protected]& Yale Univ. Poul Erik Lindelof Carl [email protected] Copenhagen Univ. JILA, Univ. Colorado, [email protected] [email protected] [email protected]

Leo Esaki Jochen MannhartShibaura Inst. Techn. Augsburg [email protected] [email protected]

Physica Scripta T102 ©Physica Scripta 2002

Physica Scripta. T102, 5, 2002

Introduction

Coherence, condensation, and phase transitions are central concepts in physics. Several of the recent Nobel Prizes in Physicshave been given to pioneers of these fields. Systems can condense to condensates where particles act coherently, that is inexactly the same way. Photons in a laser is a well known example. Another example is the recent experimental realizationof a Bose Einstein condensate (BEC) in dilute gases where the wave functions of atoms overlap and form a coherent stateat low temperature, where all condensed atoms move together. BEC's are similar to superfluid and superconducting systems.

Related phenomena occur in these condensates. For example, vortices of different kinds appear under rotation in BEC andsuperfluid 3He and 4He and correspond to quantized fluxons in superconductors under magnetic fields. Macroscopic quan-tum phenomena, another token of coherence, are typical of superconductors and occur also in the superfluids, includingBEC. Coherence is of utmost importance in so called quantum computers, a new concept based upon the probabilityof a two state system to be in one or the other of the states and where a number of operations have to be performed

within a decoherence time.A Nobel Symposium provides an excellent opportunity to bring together a group of outstanding scientists for a sti-

mulating exchange of ideas and results. The Nobel symposia are small meetings and participation is by invitation only,typically 20-40 participants. In 2001, the Nobel Foundation celebrated the 100th anniversary of the first Nobel Prize and allprevious Nobel laureates were invited to attend the Nobel ceremonies in Stockholm. This gave an excellent opportunity forarranging jubilee symposia with topics that would attract several of the laureates. Our chosen subject of Condensation andCoherence in Condensed System (CoCoCo) attracted sixteen Nobel laureates and another thirty-five leading scientists whomet in Goteborg during four days before leaving for the festivities in Stockholm. The program had to be concentrated tocertain aspects and we apologize to all prominent scientists in the field that could not be invited due to space limits.

Our idea was to bring scientists together from several related sub-disciplines: atomic physics, quantum optics, condensedmatter physics, for cross breeding of ideas, concepts and experience. Subject like phase transitions in strongly coupledsystems, Bose-Einstein condensation in weakly coupled systems, macroscopic quantum phenomena, coherence inmesoscopic structures, and quantum information were intensively discussed from different points of view. Coherencephenomena in condensed systems were emphasized. A special session was devoted to the emerging field of quantumcomputing with experimental and theoretical results reported for different types of qu-bits. The 2001 Nobel Prize to EricCornell, Wolfgang Ketterle, and Carl Wieman, "for the achievement of Bose-Einstein condensation in dilute gases of alkaliatoms, and for early fundamental studies of the properties of the condensates" gave an extra flavor to the theme of theCentennial Symposium.

The Symposium was sponsored by the Nobel Foundation through its Nobel Symposium Committee. Lectures were givenat Agrenska Villan, a former merchant mansion that was donated to the Goteborg University, at the MicrotechnologyCenter of Chalmers, and at Universeum, the new science center in Goteborg. Several of the sessions were open to invitedscientists or to a broader audience, which could enjoy reviews of central topics. High school students and His Majesty theKing of Sweden had the possibility to meet and interview many of the laureates during the visit at Universeum. Receptionswere sponsored by the City of Goteborg and Chalmers University of Technology and gave participants opportunities tomeet local scientists, students, and industrialists as well as to enjoy music and a guided tour of arts. The symposium wasorganized by Sune Svanberg, Mats Jonson, and Tord Claeson. Valuable hints were given by Anders Barany, the secretary ofthe Nobel Committee of Physics. Many of the participants gave valuable comments regarding the planning of the CoCoCosymposium. Special thanks are due to our "sounding board": Anthony Leggett, Hans Mooij, Doung Osheroff, Bill Phillips,and Stig Stenholm. Per Delsing had the responsibility of editing the Proceedings. Our secretary, Ann-Marie Frykestig, andtechnician, Staffan Pehrson, did outstanding jobs organizing practical matters. Several of the members of our local universitycommunity helped with odds and ends. Mariana Ravneva ivanova and Madeline Claeson directed an appreciated

companions program.The Proceedings contain most of the material presented at the Symposium. A few contributions that summarized results

published elsewhere are exempted. We hope that these Proceedings will convey to the reader some of the excitement felt bythe participants during the Symposium. We also want to express our thanks to sponsors and contributors to the successful

scientific event.

Goteborg, May 2002Tord Claeson Per Delsing

© Physica Scripta 2002 Physica Scripta T102

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Physica Scripta. T102, 7-9, 2002

Quantum Coherence Between States with Even and Odd Numbersof ElectronsA. F. Andreev

Kapitza Institute of Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow , 117334 Russia

Received January 17 2002; accepted April 2, 2002

PACs Ref: 03

Abstract

A system with variable number of electrons is described in which the statesrepresenting coherent superpositions of states with even and odd numbersof electrons may occur . An experiment is suggested which generalizes the experi-ment of Nakamura et al. and may provide direct evidence of such coherenceand, thereby, justify the reality of a superspace.

Quantum coherence between states with even and oddnumbers of electrons is of special fundamental interest.

In 1952, Wick, Wightman, and Wigner [1] claimed that thecoherent linear superpositions of states with even and oddnumbers of fermions are incompatible with the Lorentzinvariance and introduced the superselection rule, accordingto which such linear superpositions are physically impossible.In actuality (as was pointed out in [2,3]), the superselectionrule is the alternative to the existence, along with x, y, z, and t,of additional spinor coordinates, which, in fact, are intro-duced in quantum field theory to account for supersymmetry.

In this work, it is proved theoretically that the super-selection rule is, generally, not self-consistent. Namely, asimple realistic system with variable number of electrons isconsidered, which is governed by the Hamiltonian whoseeigenvectors are all coherent superpositions of the states witheven and odd numbers of electrons. The idea of the proof is asfollows.

The number of electrons is a conserved quantity which isanalogous, in this respect, to the momentum and the angularmomentum. Physical systems with Hamiltonians whoseeigenvectors are all coherent superpositions of the states withdifferent momenta are well known. A particle in an externalpotential field depending on the particle coordinate is thesimplest system of this type. In fact, this particle is part of anisolated system consisting of the particle and a certainmassive body, the interaction with which can be described asan external field acting on the particle. As known, this isjustified only if certain requirements are fulfilled. Forexample, the states of a massive body must adiabaticallyadjust to the changes in the particle coordinate in order toprevent excitation of the body degrees of freedom.

Below, the system with variable number of electrons isconsidered which form, together with the environment, anisolated common system with a fixed number of electrons.The interaction of the system with the environment (ananalogue of a massive body in the above-mentioned example)can be described as an external field acting on the system, Inthe case considered, the interaction does not commute with

the operator of number of electrons in the system (by analogywith the fact that the potential energy of interaction betweena particle and a massive body does not commute with theparticle momentum operator). Moreover, this field has thespinor character; i.e., it changes sign under the O(2n) and R2transformation, where O(21r) is the rotation of the coordinatesystem through an angle of 2n about any axis and R2 is thedouble time reversal. All eigenfunctions of the Hamiltonianof the system are coherent superpositions of the states witheven and odd numbers of electrons.

Below, an experiment will also be proposed which gen-eralizes the Nakamura et al. experiment [4] on the obser-vation of quantum coherence between the states with different(but even in both cases) numbers of electrons. The imple-mentation of this experiment will directly demonstrate thecoherence between the states with even and odd numbers ofelectrons.

1. Let us consider a simple example where the interactionof the two parts of a total closed system can be described interms of spinor external fields.

Let there be two quantum dots and one electron which canoccur with close energies in either of the dots and has a certainspin projection, specified once and for all. The quantum dotshave gates, whose electric potentials can be varied to move theelectron energy levels. In the second quantization repre-sentation, a complete set of system states In, N) consist of two

states:

10, 1), 10, 1). (1)

Here, n and N are the numbers of electrons in the first dot(which will be referred to as system) and the second dot(referred to as environment), respectively. The characteristicfeature of states (1) is that the set of environment quantumnumbers N is uniquely determined by the quantum numbersof the system: N = 1 - n. It is this property that allowsthe interaction of the system with environment to be treatedas an external field action on the system. If we are interestedonly in the state of the system itself, we can consider any stateIn, 1 - n) of the common system as the state

In)=In,l-n)

of our system (the first dot) interacting with the environ-ment. Moreover our system in characterized by a certainHamiltonian.

With allowance made for electron tunneling between thequantum dots, the Hamiltonian of the total system is

e-mail: [email protected] H = Ho + Ht,

© Physica Scripta 2002

(2)

Physica Scripta T102

8 A. F. Andreev

where

Ho = e(n) + E(N).

In Eq . (3), e(n) and E(N) are the energies of quantum dots inthe gate potential without tunneling, and

Ht = VaA+ - V*a+A, (4)

where V is the tunneling amplitude ; a, a+ and A , A+ are theoperations of electron annihilation and creation in the systemand the environment , respectively . In the usual representation(see [5], Section 65), the action of the operators A and A+ onvectors (1) is given by the formulas (if the result in nonzero)

A10, 1) = 10, 0), A+Il, 0) = -I1, 1). (5)

The substitution of Eq. (5) into Hamiltonian (4) shows thatthe tunneling Hamiltonian involving only the systemoperators is

Hn = na + n*a+• (6)

Here, the operators a and a+ act on the vectors In) by theusual rules of an isolated system, and n is the external fieldequal to -V in the representation used. The total Ham-iltonian of the system is then

H=e(n)+E(1 -n)+Hn,

so that the total interaction energy is the sum of the secondand third terms in Eq. (7). Under the O(2n) and R2 trans-formations, the field n, as well as the operators a and otherspinor quantities, change sign so that, for a given field value,Hamiltonian (6) is not invariant about these transformations.Due to the presence of terms linear in electron operators, alleigenstates of the Hamiltonian are coherent superpositionsof the states with even and odd numbers of electrons.

2. The two-level system described by Eq. (7) obeys thetime-dependent Schrodinger equations

is = Tb, ib = cb + T*a, (8)

where T = - V and c = e(1) - e(0) + E(0) - E(l). The gen-eral state of this two-level system is a 10) +b11).

The experiment of Nakamura et al. [4] is as follows.Before the initial time (t = 0), a two-level system was in theground state with the gate potential such that c >>T.Accordingly, a = I and b = 0. At t = 0, the gate potentialrapidly changes to a value for which c = 0. Then, thepotential remains constant for a time At, after which itrapidly regains its initial value. On the time interval betweent = 0 and t = At, the system obeys Eqs (8) with c = 0 andinitial conditions a(t = 0) = 1 and b(t = 0) = 0. Thena(t) = cos ITIt, and b(t) = sin ITIt, where b = (iT/ITI)b so thatI(b)I = Ibl. At t = At + 0, one measures the excited-statepopulation

Ib(At) 12 = 2 (1 - cos 21TIAt) (9)

as a function of pulse duration At. This can be done (as in theexperiment of Nakamura et al. [4]) using a probe electrodeconnected to the box through a tunneling contact or (asin the experiment of Aassime et al. [6]) using a probe


electrometer based on a single-electron transistor. Theobserved oscillation indicate that the system coherently

oscillates between the states with electron numbers 0 and2 on the time interval (0, At). If such an experiment had beenperformed with our two -level system , it would have beenproved that the corresponding system coherently oscillatebetween the states with electron numbers 0 and 1 . It shouldbe emphasized that this interpretation of the oscillations isessentially based on the description of the system by theHamiltonian (7), which account for the interaction withenvironment by introducing the field n. However , if one inter-prets oscillations (9) as a phenomenon occurring in the totalclosed system , then they are evidence only of the oscillatoryelectron transitions between different parts of the system.According to this interpreation, the oscillation frequencyis proportional to ITI, i . e. to the tunneling amplitudes. Onemay note , in this connection , that the experiment ofNakamura et al. can be refined by passing from thesingle-pulse to two-pulse technique (See [71).

As above , let the two-level system be at t < 0 in theground state a = 1 and b = 0(c >> III). The amplitude of thefirst gate-potential rectangular pulse is the same as above(i.e. corresponds to c = 0), but its duration is fixed attl = iv/4ITI . Immediately after the pulse at t = tl + 0, thesystem is in the state with a = b = 1/.. In the intervalbetween t = t1 and t = t j + At, the potential is equal to itsinitial value corresponding to E >> ITI. Under these condi-tions , the tunneling interaction of the system with envir-onment can be ignored and it behaves as a closed system inits pure state . In this case , a(t) = l/,/ and b(t) =(l/-/2_)exp(i4(t)), with the relative phase of the groundand excited states linearly depending on time: 0(t) =-E(t - t1).

However , it would be incorrect to conceive that theenvironment is also in the pure state and the state vector ofthe total system is the product of the state vector of its parts.In this case , the total system occurs in the so-called entangledstate (see [8]).

The second gate-potential pulse with parameters of thefirst pulse is switched on at time t1 + At. Using Eq. (8), onecan see that , after completion of the second pulse at time2t1 + At (E >> ITI because At << t1), the population of theexcited state is

(3)

(7)

lb 12 = 2(1 +coscAt). (10)

The observation of oscillations (10) as a function of timedelay At between the pulses would demonstrate that the rela-tive phase of the states with different numbers of electrons in aclosed system has a definite value 4(t) linearly depending ontime. For our two-level system, this would be direct proofof the quantum coherence between the states with evenand odd numbers of electrons.

References

1. LWick, G. C., Wightman, A. S. and Wigner, E. P., Phys. Rev. 88, 101(1952).

2. Andreev, A. F., Pis'ma Zh. Eksp. Teor. Foz, 68, 638 (1998) [J. Exp.Theor. Phys. Lett. 68, 673 (1998)].

3. Andreev, A. F., Physica B 280 , 440 (2000).


Quantum Coherence Between States with Even and Odd Numbers of Electrons 9

Nakamura,(1999).

4.

5.

Y., Pashkin, Yu. A. and Tsai, J. S., Nature 398, 786 6. Aassime , A., Johansson , G., Wendin , G., Schoelkopf, R. J. andDelsing, P., Phys . Rev. Lett . 86, 3376 (2001).

7. Andreev , A. F., J. Exp . Theor . Phys . Lett . 74, 512 (2001).8. Raimond , J. M., Brune , M. and Haroche , S., Rev . Mod. Phys. 73, 565

(2001).

Landau , L. D. and Lifshitz , E. M. "Course of Theoretical Physics",Vol. 3: "Quantum Mechanics : Non-Relativistic Theory", (Nauka,Moscow , 1989, 4th ed .; Pergamon, New York, 1977, 3rd ed.).



Superconductivity in High Tc Cuprates : The Cause isNo Longer A MysteryPhilip W. Anderson*

Joseph Henry Laboratories of Physics Princeton University, Princeton, N.Z08544, USA

Received December 10, 2001; accepted in revisedform April 19, 2002

PACs Ref: 74.20.Mn

Abstract

I discuss various direct calculations of the properties of the one-band Hubbardmodel on a square lattice and conclude that these properties sufficientlyresemble those of the cuprate superconductors that no more complicated inter-actions are necessary to cause highTc superconductivity. In particular, I discussphonon effects and conclude that these may be effective in reducing Tc and thegap in electron-doped materials.

The standard preamble for all kinds of papers on theory (orexperiment, for that matter) in the field of high Tc usuallycontains the phrase "since there is no consensus on the causeof high Tc superconductivity" or words to that effect, andoften proceeds to justify thereby yet another implausible con-jecture as to some aspect of the phenomenon. Of course oneman's consensus is another man's wild disagreement, andyou can easily find those who do not agree that there is aconsensus on relativity, the quantum theory, or the StandardModel, by doing a simple web search. Not, of course, tomention theories of evolution or of the big bang, whichare unpopular in whole states and most legislatures. As a solidstater I thank my stars that the band theory of solids and theBCS model are such obscure targets that they do not garnerthis kind of disapprobation, although each had its powerfulthough somewhat irrational opponents in the past-names likeSlater and Wigner, even, for the latter.

I would agree that there is or should be no real agreementas to the cause of much of the peculiar phenomenology of thecuprates. Each of us would-be experts has his favorite list ofreally puzzling questions about them. My own favorites arethe peculiar insensitivity of Tc to disorder, and the strangetransport properties in the normal state. Another mystery isthe role of interlayer interaction effects in all kinds of ways,although we can now be reasonably certain that they are notthe main cause of superconductivity. Yet it is time, I feel, thatit should no longer be legitimate to doubt the "first cause",namely the minimal underlying model which produces,among a bewildering welter of other effects, super-conductivity at still unprecedentedly high temperatures. I canspeak with a certain lack of bias on this matter because theinterlayer tunneling theory which I advocated for five years,and wrote a book about, turns out to be one of those whichmust be consigned to the dustbin. (Although before I waswrong, I was right, at least partially, as to the correct "firstcause". [1])

My confidence is based on two things. First is the roughagreement between a number of different simulations,extrapolations, and variational calculations, and the great

*email : pwa@pupgg .princeton.edu


similarity between these calculations and the actual experi-mental data, using a "bare-bones" model for the electronsand their interactions, a model for which there is muchindependent verification. The second is the experimental datathemselves, which in many particulars seem to be trying to tellus the nature of the phenomena.

All of these successful calculations are based on the samesimple model for the underlying physics. This model is theone-band Hubbard model, or, as can be shown to beessentially equivalent in the appropriate strong-couplinglimit, the t - J model. It is interesting that there seems to beno need to include electron-phonon coupling; I will discussthe reason and possible effects of the phonons later in thisarticle. The appropriate Hamiltonian, then, is

H= E(il tI j)c * (i)c(j) + U n( i, +)n(i, -), (1)

H(t - J) = P E(i l t I j)c * (i)c(j)P + E J(ij)S(i) * S(j)• (2)

Here P is the projection operator which removes doubleoccupancy of any site. (2) is meaningless without the pro-jection. + and - of course are the spin indices [2].

The most direct calculations with (2) are the quantumMonte-Carlo simulations of Scalapino and his group [3], andusing a somewhat different method, those of Sorella andcollaborators [4]. There is considerable controversy betweenthese two groups as to the details of the results; however, whatstrikes the disinterested observer such as myself is not thesedifferences but the essential similarity of the two. Thelow-temperature phase for low doping is of course anti-ferromagnetic ((2) reduces to the Heisenberg model in thatcase) but with increasing doping one encounters first aninhomogeneous (probably striped) phase and then, atapproximately the right temperature and hole density, d-wavesuperconductivity. The parameters of (1) or (2) are notarbitrary-experiments of many types give us J, and ARPESmeasurements give us a good estimate of the band structure athigh doping, which is t-so that there is only a little freedom tomanipulate the various components of t. There are someconfirmatory simulations, particularly on ladder systems,which again give us a great deal of undersatanding of whyd-wave superconductivity arises; there are also strong indi-cations of a pseudogap phenomenon in the appropriate place.I am no real expert on this extremely tricky field, and in facthave been a pessimist about its ability to arrive at any decisiveresults, but at this point I have to confess myself impressed.Of course, the trouble with direct calculations is both thatthey do not really lead to deep understanding, nor can theysay much about such things as excitation spectra and


Superconductivity in High Tc Cuprates: The Cause is No Longer A Mystery 11

transport properties, but what they can say is: when you startout with this physics, you end up with the observed results. Iam not at all an expert in the rather involved and subtlereasoning that goes into these calculations, so will not presentthem in detail, but there is no question that these are the mostextensive direct calculations in existence.

A second approach has been taken by Randeria and Trivedi[5], by Masao Ogata [6], and to some extent by Sorella's group.This is to take seriously the original ansatz of the RVB theorythat the ground state wave function could be approximated byGutzwiller projection of an appropriate product wave functionof BCS type, and to determine variationally the energy gap andthe chemical potential of the wave function to be projected(which, they point out, need not be the physical chemicalpotential) by calculating the energy and varying it. TheRanderia group also improves the wave function somewhat byMonte Carlo methods. Again, as before the results resemblevery strongly the experimental data insofar as these can bedetermined from the calculation; in particular, the prediction ofd-wave superconductivity for an adequately doped sampleis very robust. Ogata finds that for underdoped concentra-tions he can find mixtures of antiferromagnetism and super-conductivity, as is observed in some cases.

There is still a third method of direct calculation which hasbeen applied, starting with early work by Georges andYedidia [7], and carried to an amazing degree of refinementby Bill Puttika [8] in recent years. This is direct expansion ofthe power series in 1/T for the partition function, andextrapolation of the results with Pade approximants. Thismethod has played a great role in the past in estimatingcritical behavior near phase transitions, but hasn't previouslybeen much used in quantum many-body systems. It has beenvery useful in disposing of red herrings which have appearedin various mean field approximations-showing that the t - JHamiltonian does not lead to phase separation until J isunphysically large, for instance; and also identifying the verylarge U, low doping "Nagaoka" ferromagnetic region. Butrecently Puttika has carried his series out to ten to twelveorders and has been able to calculate n(k) with sufficientaccuracy to spot the disappearance of the Fermi surface nearthe (x, 0) point which is charactristic of the pseudogap regime.The agreement of his results with those of ARPES at the sameenergy resolution is remarkable.

The model is of course not the real substance. Why does it(the model) work so well? In the first place, the question oflayer interactions: these are remarkably weak, which we cantake as an empirical fact on the basis that the Tl one-layercuprate is experimentally seen to exhibit all the phenomenathat the others do, though proven conclusively by Moler et al.[9] to have very weak interlayer interaction. The same may besaid about the gate-doped CaCuO2 sample of Kloc, Schonet al., [10] where there is only one doped layer, but perhapstotal acceptance of these results should await confirmationon more samples and in other laboratories. The simplifi-cation to the simple one-band Hubbard model is oftenjustified by the discussion of Zhang and Rice, which is moreor less correct, but I prefer the use of projective canonicaltransformations a la 0 K Anderson [11]; and actually thestrongest argument was the very early exact calculations onsmall clusters by Michael Schluter et al. [12] who showedthat an effective one-band Hubbard model worked very wellindeed.

There are two physical aspects which are not in the modeland could be important. One is long-range Coulombforces-one assumes effective screening as in a normal metal,and the only real justification is that it works. Someexperimental data bear on this: we have good data oninterlayer plasmons which show that the c-axis is a fairlyinsulating direction, actually, (except for supercurrents) butas far as we can see the intralayer plasmons are indeed high-frequency and may screen reasonably well.

The question which keeps coming up is phonons: why arethere almost no relevant phonon phenomena? The answer tothis, I would speculate, is two-fold. I believe that it is almoston the level of the calculational results above that the natureof the pseudogap is that it is a pairing phenomenon in the spinsector-i.e., that the pseudogap region can be seen as an RVBwith d-wave pairing. But the spin sector is not coupled tophonons in lowest order, because a displacement of the localpotential does not break the spin degeneracy. In the secondterm of (2) the electron-phonon interaction couples only to J.It is worth confirming this experimentally, but it is knownthat in heavy-electron materials the Kondo spins do notcouple to phonons. Therefore, the pseudogap is not affectedone way or another by phonons (or ordinary impurities) Inthe point of view that I have called "RVB redux" [13] thesuperconducting gap is caused by the kinetic energy cost ofthe opening of the pseudogap, which can only be restored bypair hopping via the anomalous "josephson" terms in thekinetic energy; thus at least for the higher doping regime theSC gap follows the pseudogap.

This is rather a speculative argument, and I am goingimmediately to say that in fact there are phonon effects on Tc.For most phonons these are reasonably small because, as hasbeen exhaustively proven for ordinary superconductors, theelectron-electron interaction due to phonons is very local. Inour Hubbard model, it acts simply as a modulation of U. Butof course the reason why we have a d-wave is that it has zeroamplitude at the origin and hence avoids the repulsive U, andneither U nor phonons are effective. But I think there is onephonon which can be expected to couple rather strongly toour d-electrons, and much more strongly to electrons than toholes (note that in our projective theory (2) there is noparticle-hole symmetry.) This is the phonon which representsthe Jahn-Teller displacement which breaks the d,,2-.2 vs d,2degeneracy (see Fig. 1). An electron site has no (x2 - y2) hole

• O • O • 0 •

0 Oxygen

• Copper

at 01 Of O+

• 0 0 • 0 •

Oi Of Oi Of

• O • 0 0 •

Fig. 1 . The phonon which is likely to interact strongly with electron around the 0points. Its displacements are those of the Jahn-Teller caused by an x2 - y2 hole,

so that it might be expected to be particularly soft for electron doping.


12 Philip W. Anderson

(n,0)

- Pseudogap

----- Superconducting gap

Fig. 2. The reduction in the superconducting gap which might result from therepulsive interaction caused by the optimal phonon of Fig. 1.

and thus tends to relax the Jahn-Teller distortion, while a holesite simply adds to the distortion a little. As is easily seen, thephonon which couples best is at wave vector n,n and hencecouples the peaks in the gap function It, 0 -> 0, it. This phononhas been the subject of considerable discussion recently butrather off the point. What it will do is to represent a repulsiveinteraction for the d-wave gap and hence lower the super-conducting Tc for electrons relative to holes., an effect whichis observed but has seemed rather puzzling. It is noteworthythat in a couple of recent measurements (see, for instance,[10]) the pseudogap and T* are about the same for electronand hole dopings at the same level, but TT is much reduced.

In fact, if the effect is rather localized in k-space we mightexpect the two gaps to behave as in Fig. 2, with the it, 0 peakdepressed for the SC gap relative to the general point alongthe Fermi surface. This seems to explain the ARPESobservations that the peak of the gap function is not at the 0,it points but rather there is a local minimum there, as we havesketched in Fig. 2 (not, of course, to scale).

One implication of this picture is that we would expect anisotope effect for the electron-doped materials, but a negativeone. A crude calculation, assuming without any justificationthat the BCS gap equation is valid, would suggest

a(ln TT)/a(ln.fopt) = -µ/(J - u) (3)

where jopt is the optical phonon frequency, i the phononcoupling constant, and J the effective antiferromagneticexchange. (3) might be valid in the overdoped regime. For

reasons mentioned but not emphasized in my book,(whichinclude a great deal of experience) I am not very convincedof the accuracy or relevance of isotope effect measurements,but perhaps the gate-doping possibility, which avoids thenecessity of making a new sample for every measurement,is a new opportunity.

In conclusion, my point here is that there is a great deal ofconsensus on the model which underlies high TT, cupratesuperconductivity, and there ought to be more: I think wehave proved our point. But there is much more to be done andspecifically, for instance, we are so far unable to give a closed-form gap equation.

References

1. Anderson, P. W. Science 235, 1196 (1987); Baskaran, G., Zou, Z. andAnderson, P. W. Sol. State Commun. 63, 973 (1987).

2. Rice, T. M., Gros. C., Joynt, R. and Sigrist , M., in Bangalore 5th Int'lConf. on Valence Fluct, (L. C. Gupta and S. K. Malik eds) (1986)p. 99.

3. Scalapino, D. J. and White, S. R., cond-mat/0007515.4. Sorella, S. et al. cond-mat/0110460, 2001.5. Paramekanti , A., Randeria, M., Trivedi, N., cond-mat/0101121 (2001);

Paramekanti , A., Randeria, M., Ramakrishnan, T. V. and Mandal, S. S.Phys. Rev. B62 , 6786 (2000).

6. Ogata, M. and Himeda, A., cond-mat/0003465, 2000.7. George, A. and Yedidia, J., private communication; see J. Yedidia,

thesis, Princeton (1990).8. Putikka, W. 0., Luchini, M. U. and Singh, R. R. P., cond-mat/9803141,

9803140, Phys. Rev. Let. 81, 2966 (1988).9. Moler, K. A., Kirtley, J. R., Hinks, D. G., Ii, T. W. and Xu, M., Science

279, 1193 (1998).10. Schon, J. H. et al., preprint (Oct 2001).11. Anderson, O. K., Jepsen, 0., Leichtenstein, A. L. and Mazim, 1. 1., Phys.

Rev. B49 , 4145 (1994).12. Schluter, M. A. and Hybertson, M. S., Physica C162 , 583 (1989).13. Anderson, P. W. "The University of Physics" (eds. R. Lhuri, J. Liu,

F. Chen, and W. Gan), (Kluwer Academic/Plenum Publishers, 2001),pp. 3-8.

Physica Scripta T102 © Physica Scripta 2002


Nanoelectromechanics of Coulomb Blockade NanostructuresR. I. Shekhtert, L. Y. Gorelik', A. Isacssont°*, Y. M. Galperin2 and M. Jonson'

' Department of Applied Physics, Chalmers University of Technology and Goteborg University, SE-412 96 Goteborg, Sweden2 Department of Physics, University of Oslo, P. O. Box 1048, N-0316 Oslo, Norway. Division of Solid State Physics,

loffe Institute of the Russian Academy of Sciences, St. Petersburg 194021, Russia

Received January 21, 2002; accepted April 17 2002

PACS Ref: 73.23.Hk, 72 .80.Le, 72.80Tm

Abstract

The aim of this paper is to emphasize the role of coupling between electronic

and mechanical degrees of freedom taking place on a nanometer length scale.

Such coupling affects significantly the electrical properties of nanocomposite

materials which are usually heteroconducting and heteroelastic by their nature.

As examples of nanoelectromechanics in normal and superconducting com-posites a self-assembled single electronic device exhibiting a dynamical insta-bility leading to shuttling of electrical charge by a movable Coulomb dot is

discussed along with an example of shuttling of Cooper pairs by a movable

Single Cooper Pair Box.

1. Introduction

The materials for future electronics must be intentionallydesigned on a nanometer length scale. Molecular manufac-turing is one of the technologies which is intensively discussedas a possible candidate to do that. It is based on the specialfeature of certain organic molecules to arrange themselvesspontaneously in ordered spatial structures (self-assembling).Metallic clusters and other small conductors can be incorpor-ated if the above mentioned molecules are attached to thesurface of those objects as is shown in Fig. 1. As a resultself-assembled ordered structures of conducting elementscan be designed on the nanometer scale. An example of such

Fig. 1. Molecular manufacturing using self-assembly. By attaching specific

organic molecules to for instance metal clusters metal organic nanocomposite

materials can be intentionally designed on a nanometer scale. Due to the dif-

ferent physical properties of the constituents, metal clusters and organic

molecules, these materials may have special properties such as being

heteroelastic and heteroconducting.

*e-mail: [email protected]

a metal organic material is the 2D array of 4 nm Au particlesfabricated by Andres et al. [1]. The mechanism responsible forelectrical charge transport in such composite metal-organicmaterials is supposed to be inter-island tunneling of electrons.

The physical properties of the above mentioned nano-composites are governed by the small size of the conductingclusters together with the significant differences in theproperties of organic and conducting components. Elec-trically such materials should be considered as hetero-conducting since the conductivities of the organic and themetallic parts can differ by many orders of magnitude.Mechanically these composite materials are heteroelastic on ananometer scale since the elastic moduli of the metallic andthe organic parts differ by three to four orders of magnitude.Further, the properties of the electronic subsystem aredominated by quantum coherence and spatial level quanti-zation from one side and electron-electron correlations fromthe other. Coulomb Blockade of single electron tunneling [2]and resonant inter-cluster tunneling are the most prominentmanifestations of the above features. The aim of the presentpaper is to emphasize that the above list should be completedby a new item which is the strong coupling between electronicand mechanical degrees of freedom.

1.1. Nanoelectromechanical coupling

The role of nanoelectromechanical coupling follows immedi-ately from the heteroelastic nature of the material taken incombination with the advanced role of Coulomb correlationscontrolling electronic inter-cluster tunneling. Tunneling of asingle electron between nanometer sized grains is accom-panied by an increase in electrostatic energy which can beof the order of 10 percent of the energy responsible forthe binding of the material. As a result significantdeformations of the soft dielectric fraction appear whichexponentially affect the tunneling of electrons. Such mutualinfluence of mechanical displacements and tunnel chargeredistribution results indeed in a dynamical coupling betweenmechanical and electronical degrees of freedom. Especially,since the typical times characterizing the dynamics of eachof them can be of the same order of magnitude formetal-organic nanocomposites this coupling cannot beneglected. Electromechanical coupling of the kind discussedhere is not only a feature of heteroelastic nanocomposites.It also has relevance for other nanoelectromechanical(NEMS) systems intentionally designed to work at thenanometer scale [3].

We will here discuss two groups of nanoelectromechanicalphenomena. The first is just the interplay between incoherenttunneling of electrons in a normal single electron tunneling


14 R. L Shekhter et al.

device, containing a movable conducting nanocluster, situ-ated in the gap between two bulk electrodes (Section 2). Thesecond is how coherent tunneling of electrons together withmechanical displacement may serve as a weak link. As anexample we consider below a superconducting tunnelingdevice with a movable superconducting island in between twobulk superconductors. Coherent tunneling of Cooper pairs isshown to be affected significantly in this case which results inmechanically assisted superconducting coupling betweenremote superconductors.

2. Nanoelectromechanics with incoherent electrons

A schematic picture of a single electronic device with a mov-able metallic cluster is presented in Fig. 2. Since electronictransport in the device is due to tunneling between the leadsand the central small size conductor, it is strongly affectedby the displacement of the latter. In this case center of massmechanical vibrations of the grain will be present (considerthe elastic springs, connecting the central electrode to theleads in Fig. 2).

A number of characteristic times determine the dynamicalevolution of the system. Electronic degrees of freedom arerepresented by frequencies corresponding to such energies asthe Fermi energy in each of the conductors and the appliedvoltage eV. In addition one has an inverse relaxation timeand inverse phase breaking time for electrons in the con-ductors and a tunneling charge relaxation time coil = RC.Mechanical degrees of freedom are characterized byvibrations with frequency (o. The condition that h(OR shouldbe much smaller than the Fermi energy is a standardcondition for a weak tunneling coupling and holds very wellin usual tunnel structures. Since a finite voltage is supposedto be applied causing a non equilibrium evolution of the

a)

V/2

b)

V/2

electrostatic field

-V/2

-V/2

Fig. 2. (a) Simple model of a soft Coulomb blockade system in which a metallicgrain (center) is linked to two electrodes by elastically deformable organic mol-ecular links. (b) A dynamic instability occurs since in the presence of a suf-ficiently large bias voltage V the grain is accelerated by the correspondingelectrostatic force towards the first one , then to the other electrode. A cyclicchange in direction is caused by the repeated "loading" of electrons nearthe negatively biased electrode and the subsequent "unloading" of the sameat the positively biased electrode . As a result the sign of the net grain chargealternates leading to an oscillatory grain motion and a novel "electron shuttle"mechanism for charge transport.

"unloading" of2N electrons

q=Ne

q=-Ne

"loading" of2N electrons)

system the question of electronic relaxation becomes rele-

vant. Two possible scenarios of electronic transfer through

the metallic cluster can be identified depending on the ratiobetween the tunneling relaxation time co' and the intra-grain electronic relaxation time To. In the case that To ismuch shorter than wR1, two sequential events of electronictunneling, necessary to transfer an electron from one lead toanother through the grain, cannot be considered as aquantum mechanical coherent process due to the relaxationand phase breaking occurring in between the events (whichare separated in time by an interval, which is approximatelyequal to coil). In this case all electronic tunneling transitionsbetween each of the leads and the grain can be consideredas incoherent independent events while fast relaxation ofelectrons in all three conductors is supposed to beresponsible for the formation of local (on each of theconductors) equilibrium electronic distribution functions.This is the approach which we will use in the present section.In the opposite limit, To much larger than coi quantumcoherence plays a dominating role in the electronic chargetransfer and all relaxation takes place in the leads far awayfrom the central part of the device. Two examples of suchbehavior can be considered. One is the resonant tunneling ofelectrons through a single quantum level, localized on avibrating molecule. This problem is out of scope of thepresent paper and is considered in [4].1 Another examplewhere the inequality to << coil is violated is the super-conducting version of the device in Fig. 2. In the latter case,redistribution of Cooper pairs between superconductorscaused by tunneling preserves phase coherence of electrons.This situation is considered in the next section of the paperbut first we turn the attention to the system in Fig. 2.

2.1. Shuttling of electrical charge by a movableCoulomb dot

The tunnel junctions between the leads and the grain in Fig. 2are modeled by tunneling resistances R1(x) and R2(x) whichare assumed to be exponential functions of the graincoordinate x. In order to avoid unimportant technical com-plications we study the symmetric case for whichR1,2 = Refx/2. When the position of the grain is fixed, theelectrical potential of the grain and its charge qst are givenby current balance between the grain and the leads [2]. Asa consequence, at a given bias voltage V the charge qt(x)is completely controlled by the ratio Ri(x)/R2(x) anddgst(x)/dx < 0. In addition the bias voltage generates anelectrostatic field E = aV in the space between the leads and,hence, a charged grain will be subjected to an electrostaticforce Fq = a Vq. The central point of our considerations isthat the grain - because of the "softness" of the organicmolecular links connecting it to the leads - may moveand change its position. The grain motion disturbs the currentbalance and as a result the grain charge will vary in time intact with the grain displacement. This variation affects thework W = aV f kq(t)dt performed on the grain during, say,one period of its oscillatory motion. It is significant that thiswork is nonzero and positive, i.e., the electrostatic force,

'In [4] an interesting interplay between mechanical vibrations and coherent elec-tronic tunneling was shown to be responsible for an electromechanical instability,producing a molecular center of mass vibration.

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