Selected Papers of J Robert Schrieffer

The following sections are included:

• PREFACE

• Contents

The following sections are included:

• Introduction

• The BCS Theory

• Strong Coupling Superconductors

• Spin Fluctuations and Superconductivity

• Conclusion

• References

SINCE the discovery of the isotope effect, it has been known that superconductivity arises from the interaction between electrons and lattice vibrations, but it has proved difficult to construct an adequate theory based on this concept. As has been shown by Fröhlich,¹ and in a more complete analysis by Bardeen and Pines² in which Coulomb effects were included, interactions between electrons and the phonon field lead to an interaction between electrons which may be expressed in the form

where |M_κ|² is the matrix element for the electronphonon interaction for the phonon wave vector κ, calculated for the zero-point amplitude of the vibrations, the c's are creation and destruction operators for the electrons in the Bloch states specified by the wave vector k and spin s, and H_Coul represents the screened Coulomb interaction…

A theory of superconductivity is presented, based on the fact that the interaction between electrons resulting from virtual exchange of phonons is attractive when the energy difference between the electrons states involved is less than the phonon energy, ℏω. It is favorable to form a superconducting phase when this attractive interaction dominates the repulsive screened Coulomb interaction. The normal phase is described by the Bloch individual-particle model. The ground state of a superconductor, formed from a linear combination of normal state configurations in which electrons are virtually excited in pairs of opposite spin and momentum, is lower in energy than the normal state by amount proportional to an average (ℏω)², consistent with the isotope effect. A mutually orthogonal set of excited states in one-to-one correspondence with those of the normal phase is obtained by specifying occupation of certain Bloch states and by using the rest to form a linear combination of virtual pair configurations. The theory yields a second-order phase transition and a Meissner effect in the form suggested by Pippard. Calculated values of specific heats and penetration depths and their temperature variation are in good agreement with experiment. There is an energy gap for individual-particle excitations which decreases from about 3.5kT_c at T = 0°K to zero at T_c. Tables of matrix elements of single-particle operators between the excited-state superconducting wave functions, useful for perturbation expansions and calculations of transition probabilities, are given.

In order to obtain an explicitly gauge-invariant description of the Meissner effect in superconductors, Wentzel¹ has recently suggested an approach which differs in several essential respects from that developed by Bardeen, Cooper, and Schrieffer.² Wentzel finds in the long-wavelength limit a relation between the current density and the magnetic vector potential which differs from the London value given by the BCS theory. We wish to point out that this discrepancy is due to Wentzel's assumption that corrections to his approach possess a convergent expansion in powers of the phonon-electron coupling constant, g. It is our belief that this expansion does not exist…

The Anderson-Rickayzen equations of motion for a superconductor derived within the random-phase approximation (RPA) are used to investigate the collective excitations of superconductors. A spherical harmonic expansion is made of the two-body interaction potential V(k,k') and a spectrum of excitations whose energies lie within the energy gap 2Δ is obtained. These excitations may be characterized by the quantum numbers L and M involved in the potential expansion. For an L-state exciton to exist, the L-wave part of the potential must be attractive at the Fermi surface. Odd-L excitons have unit spin and may be considered as spin waves. For s-state pairing in the superconducting ground state, the plasmon mode corresponds to the L = 0 exciton whose energy is strongly modified by the long-range Coulomb interaction. For a general potential several bound states may exist for given L and M. If the L-wave potential is stronger than the s-wave part of the potential, the system is unstable with respect to formation of L-state excitons. In this case, the ground state is formed with L-state pairing, special cases of which are the p-state pairing Considered by Fisher and the d-state pairing proposed recently by several authors for the ground state of He³ and nuclear matter. Corrections to the Anderson-Rickayzen equations are discussed which lead to a new set of exciton states if the L-wave potential is repulsive. These excitons are interpreted as bound electron-hole pairs, as opposed to the particle-particle excitons present with an attractive L-wave potential.

The previous Letter¹ gives an experimental upper limit for the average time required for quasiparticle recombination in a superconductor. Burstein, Langenberg, and Taylor² have calculated the contribution to the recombination rate from photon emission, and obtain a result which is much too small to account for the experimental results.¹ It is therefore of interest to calculate the recombination rate due to phonon emission…

In deriving the collective excitation spectra of nuclei it is often assumed that the collective modes have negligible effect on the quasi-particle spectrum. An expression is derived for the self-energy of the quasi-particle states including pairing interactions as well as virtual excitation of collective modes, from which the validity of the above assumption can be investigated.

Recent tunneling experiments^1,2 involving superconducting metals exhibit structure in the I - V characteristic which has been interpreted in terms of electron-phonon processes. In the preceding Letter³ Rowell, Anderson, and Thomas present the results of improved experiments which more clearly resolve this structure. Below we summarize the results of a theoretical determination of the tunneling characteristic which is in good agreement with these experiments…

The pairing theory of superconductivity is extended to treat systems having strong electron-phonon coupling. In this regime the Landau quasiparticle approximation is invalid. In the theory we treat phonon and Coulomb interactions on the same basis and carry out the analysis using the nonzero-temperature Green's functions of the Nambu formalism. The generalized energy-gap equation thus obtained is solved (at T = 0°K) for a model which closely represents lead and the complex energy-gap parameter Δ(ω)) is plotted as a function of energy for several choices of phonon and Coulomb interaction strengths. An expression for the single-particle tunneling density of states is derived, which, when combined with Δ(ω), gives excellent agreement with experiment, if the phonon interaction strength is chosen to give the observed energy gap Δ₀ at zero temperature. The tunneling experiments therefore give a detailed justification of the phonon mechanism of superconductivity and of the validity of the strong-coupling theory. In addition, by combining theory and the tunneling experiments, much can be learned about the electron-phon interaction and the phonon density of states. The theory is accurate to terms of order the square root of the electron-ion mass ratio, ~10^-2-10^-3.

If the superconducting transition temperatures of the transition metals are fit by the BCS relation¹

where ϴ_D is the Debye temperature and N(0) is the density of states at the Fermi surface, then the pairing interaction V_BCS is found to be roughly constant, except for metals at the beginning and end of each series, where it rapidly decreases.² Traditionally, one writes where V_ph represents the phonon interaction and V_C* is the Coulomb pseudopotential.³ While at present one cannot rule out the possibility that V_ph-V_C* is small or negative for metals near the ends of these series, the fact that their paramagnetic susceptibility is anomalously large⁴ suggests that there are strong ferromagnetic exchange forces which enhance the spin susceptibility and presumably also suppress superconductivity for singlet spin pairing, as first discussed by Doniach⁵ on the basis of a phenomenological exchange interaction…

The relation between the Fermi liquid and spin fluctuation descriptions of a nearly ferromagnetic system of fermiors is discussed. It is argued that the spin fluctuation theory provides a more suitable framework for describing such systems.

The role of a finite-width Riedel peak in the interference conductivity of a Josephson tunnel junction is treated phenomenologically. It is shown that allowance for the finite width of the peak can lead to a change in sign of this conductivity in the region ω < Δ from positive, as given by standard theory, to negative. Separate mechanisms responsible for finite peak width are analyzed. It is found that only some of these mechanisms lead to the change in the interference conductivity mentioned above. The results obtained are compared with experiment.

The limits of angular momentum in rotating nuclei may permit superfluidity with nonzero angular momentum, but the excitation energy appears too high to have observable consequences.

When one speaks with friends of John Bardeen they frequently use words such as brilliant, profound, practical, quiet, devoted, family, friends, golf, humor, wise, determined, generous and a man for all seasons…

The consequences of localized, classical magnetic moments in superconductors are explored and their effect on the spectral properties of the intragap bound states is studied. Above a critical moment, a localized quasiparticle excitation in an s-wave superconductor is spontaneously created near a magnetic impurity, inducing a zero-temperature quantum transition. In this transition, the spin quantum number of the ground state changes from zero to , while the total charge remains the same. In contrast, the spin-unpolarized ground state of a d-wave superconductor is found to be stable for any value of the magnetic moment when the normal-state energy spectrum possesses particle-hole symmetry. The effect of impurity scattering on the quasiparticle states is interpreted in the spirit of relevant symmetries of the clean superconductor. The results obtained by the non-self-consistent (T matrix) and the self-consistent mean-field approximations are compared and qualitative agreement between the two schemes is found in the regime where the coherence length is longer than the Fermi length.

Cuprale superconducting thin films are being used to make compact low power microwave devices. In recent years, with improved materials and designs, there has been a steady improvement in device performance, notably an increase in resonator Q and a decrease in the nonlinear intermodulation. It is important to understand how much improvement can be expected. Here we discuss the intrinsic limiting behavior that one might achieve with a perfect film, review the present status, and discuss what the limiting behavior implies for high temperature superconducting filters. Our analysis indicates that increases in unloaded Q to ~ 10⁶ and decreases in intermodulation by a factor of ~ 10⁴, compared with today's values, might be achieved.

The following sections are included:

• The Secret of Fractional Charge

• Polyacetylene and the Schrieffer Counting Argument

• Field Theory Models of Fractional Charge
  • Zero Modes
  • Vacuum Polarization and Induced Currents

• Varieties of Charge
  • Bookkeeping Charges
  • Topological Charges
  • Space-Time and Identity Charges

• Fractional Quantum Numbers with Abstract Vortices

• Fractional Quantum Numbers in the Quantum Hall Effect

• Acknowledgments

• References

We have studied thermodynamic and some dynamic properties of a one-dimensional-model system whose displacement field Hamiltonian is strongly anharmonic, and is representative of those used to study displacive phase transitions. By studying the classical equations of motion, we find important solutions (domain walls) which cannot be represented effectively by the usual phonon perturbation expansions. The thermodynamic properties of this system can be calculated exactly by functional integral methods. No Hartree or decoupling approximations are made nor is a temperature dependence of the Hamiltonian introduced artificially. At low temperature, the thermodynamic behavior agrees with that found from a phenomenological model in which both phonons and domain walls are included as elementary excitations. We then show that equal-time correlation functions calculated by both functional-integral and phenomenological methods agree, and that the dynamic correlation functions (calculated only phenomenologically) exhibit a spectrum with both phonon peaks and a central peak due to domain-wall motion.

We have studied interactions between a domain wall and phonons in a one-dimensional-model system of a structurally unstable lattice with a double-well local potential and nearest-neighbor coupling. We find that a nonlinear effect in the interacting phonon amplitudes gives rise to a Brownian-like motion of isolated domain walls at low temperatures as well as higher harmonic generation of transmitted and reflected phonons. When there is a domain wall at rest, it was known that the linearized equation of motion has three types of independent solutions: "translation mode," "amplitude oscillation" of the domain wall, and propagating "phonons." In the second-order approximation, these modes interact. An incoming phonon produces a translation of the wall, giving rise to its Brownian motion. The magnitude of the translation is computed together with the amplitude and phase shift of the higher harmonics. We estimate the diffusion constant of walls, using the fluctuation-dissipation theorem and the thermal average over the phonons, to be , where l is the lattice spacing and ω₀ is the frequency of small oscillation of the ion (with mass m) around u₀, a minimum point of an isolated double-well potential.

A theoretical analysis of the excitation spectrum of long-chain polyenes is presented. Because of the twofold degeneracy of the ground state of the dimerized chain, elementary excitations corresponding to topological solitons are obtained. The solitons can have three charge states Q = 0. ± e. The neutral soliton has spin one-half while the charged solitons have spin zero. One electronic state is localized at the gap center for each soliton or antisoliton present. The soliton's energy of formation, length, mass, activation energy for motion, and electronic properties are calculated. These results are compared with experiment.

The equations of motion of the coupled electron–phonon system are integrated in real time for the model of polyacetylene recently proposed. To illustrate the physical behavior of this nonlinear system we consider the time evolution starting from three physically relevant configurations: (i) end generated soliton, (ii) electron–hole pair generation of a charged soliton–antisoliton pair, and (iii) the dressing of an injected electron. The calculations show that the system relaxes within a time of order 10^-13 sec, converting excited electron–hole pairs into soliton–antisoliton pairs.

A theoretical study of topological excitations (kinks) in a one-dimensional one-third–filled Peierls system is presented. The charges associated with the kinks are found to be fractional , . Calculations of the spatial widths and electronic structure of different types of kinks are carried out numerically. Possible applications to tetrathia-fulvalene-tetracyanoquinodimethane (TTF-TCNQ) are mentioned.

States with fractional fermion charge have been discovered in relativistic field theory and condensed matter physics. In the latter context they lead to unexpected but experimentally verified predictions for one-dimensional electron-phonon systems like polyacetylene. We examine the common basis for this fortunate convergence between condensed matter and relativistic field theories.

The magnitude of quantum fluctuations of the charge of a fractionally charged soliton is calculated. The soliton charge operator is defined as , where is the integral of the charge-density operator sampled by a function f peaked at the position of the soliton, and falling smoothly to zero on a scale L. |0〉 is the ground state of the system in the absence of solitons. It is shown that the mean-square fluctuation of taken about its fractional average value Q_S vanishes as O(ξ₀/L) for L ≫ ξ₀, where ξ₀ is the width of the soliton. Thus, as L → ∞, the soliton is an eigenfunction of the charge operator with fractional eigenvalue. We also show that the portion of the charge fluctuations that are due to the soliton falls as exp(-L/ξ₀) as L → ∞. Nonetheless, the charge of the entire system, including all solitons, is integral.

The effects of solitons on the spectrum of fermion excitations in superfluid ³He-A are investigated. It is found that there is a two-dimensional manifold of bound states with energies within the gap of the bulk superfluid. The bound-state spectrum lacks inversion symmetry parallel to the wall.

The two-soliton field configuration for the ɸ⁴ field in the absence of phonons is approximated with the soliton positions and the amplitudes of the shape oscillation mode as collective coordinates. The potential energy of the field is calculated for this configuration as a function of soliton separation and the amplitude of the shape oscillation mode. The equations of motion for these coordinates are derived, and the collision of two solitons is studied by numerically integrating the coupled equations.

We study the statistical mechanics of a two-dimensional gas of free anyons – particles which interpolate between Bose-Einstein and Fermi-Dirac character. Thermodynamic quantities are discussed in the low-density regime. In particular, the second virial coefficient is evaluated by two different methods and is found to exhibit a simple, periodic, but nonanalytic behavior as a function of the statistics determining parameter.

The effects of lattice relaxation on the midgap adsorption due to a soliton in a ring of trans-polyacetylene is studied within the adiabatic approximation. The Franck-Condon factors are calculated for processes involving the emission of up to three phonons. The absorption edge is found to be broadened by processes involving an even number of phonons. On this background appear extremely sharp peaks, due to processes involving an odd number of phonons. In the latter case, the shape oscillation mode of the soliton is found to be the dominant mode. Our results also indicate a third localized phonon mode about the soliton, in agreement with the results of Ito et al. [24].

The shape of extended objects in classically forbidden regions is shown to undergo expansion analogous to Lorentz contraction of a relativistic body of finite velocities. The problem of two interacting Dirac particles moving in one dimension is solved explicitly and the results are generalized to soliton solutions of field theories. An estimate of the effect on tunneling rates is also given, including solitons in (CH)_z.

Elementary, low-energy excitations are examined by bosonization in one-dimensional systems with quasi-long-range order. A new, independently measurable attribute is introduced to describe such excitations. It is defined as a number w which determines how many times the phase of the order parameter winds as an excitation is transposed from far left to far right. The winding number is zero for electrons and holes with conventional quantum numbers, but it acquires a nontrivial value w = 1 for neutral spin- excitations and for spinless excitations with a unit electron charge. It may even be irrational, if the charge is irrational. Thus, these excitations are topological.

We argue that electrons in liquid helium bubbles are not fractional, they are in a superposed state.

The following sections are included:

• Introduction

• Fractional Statistics and the Quantum Hall Effect

• Cooperative Ring Exchange Theory

• References

The statistics of quasiparticles entering the quantum Hall effect are deduced from the adiabatic theorem. These excitations are found to obey fractional statistics, a result closely related to their fractional charge.

A semiclassical path-integral approach is used to calculate the contribution of large–correlated-ring exchanges to the energy of a two-dimensional Wigner crystal in a strong magnetic field. This correlation energy E_c(ν) shows cusps at fractional fillings ν_c = n/m of the lowest Landau level. The uniform Wigner crystal is locally unstable for ν ≠ ν_c and the theory predicts the existence of fractionally charged quasiparticles to accommodate the extra density ν - ν_c.

We study the transition of the interacting two-dimensional electron gas at high magnetic field from a low-density Wigner crystal to a higher-density correlated state which exhibits the fractionally quantized Hall effect. The phase transition is precipitated by a condensation of ring-exchange processes, which by nature of their sensitivity to enclosed magnetic flux allow for especially low-energy exchange condensates at certain values of the Landau-level filling factor, ν = ν_i. In the condensate, the exchanges mediate a logarithmic potential between local-density fluctuations, leading to a cusp in the ground-state energy as a function of |ν-ν_i| in the vicinity of a preferred state. In addition, we derive quasiparticle excitations of sharp fractional charge which are analogous to the excitations derived by Laughlin.

In this section are the four most influential (or at least the most often cited) of Schrieffer's papers in surface physics. The first paper [1], his first major publication as a graduate student at Illinois, preceded his work on superconductivity and dealt with the effective carrier mobility in the surface space-charge layers of semiconductors. The seminal insight was that when dealing with carriers in this region, in which quantum behavior dominates in the direction normal to the surface, collisions of the carriers with the surface must be included. This "initial attack … laid the groundwork for most of the later work" [2]…

Carriers held to a region near the surface by the potential well of a space charge layer may have their mobility reduced by surface scattering, if the width of the well is of the order of a mean free path. An effective mobility, which may differ from the bulk mobility by as much as a factor of ten, has been obtained from a solution of the Boltzmann equation. Solutions have been carried out for two types of potential functions: (a) a linear potential corresponding to a constant space-charge field, and (b) a solution of Poisson's equation including an external bias applied normal to the surface. The results have been used to calculate changes in surface conductance of germanium with changes in surface potential and predict the "field effect" and "channel effect" mobilities.

The basic concepts in the theory of chemisorption of atoms on metals will be reviewed. The sharp atomic levels of the free atom are broadened into virtual levels by electron tunneling between the atom and the solid. For rapid tunneling, correlation effects on the adatom are weak and a self-consistent field molecular orbital approach is appropriate. The adatom density of states can vary from a Lorentzian level to states split above and below the valence band of the solid, corresponding to the bonding and antibonding orbitals of a surface complex formed between the adatom and its surroundings. The Anderson model treats only Coulomb interactions on the adatom, while the CNDO method of quantum chemistry has been recently employed to include Coulomb interactions in the solid. For strong intra-atomic Coulomb interaction on the adatom, correlation effects are important and the localized magnetic moment on the adatom as predicted by the unrestricted Hartree–Fock approximation is quenched. Correlations have been treated by extended valence bond methods, multiple scattering theory, and configuration mixing techniques. Semiquantitative agreement with experiment is obtained in a number of systems.

The indirect interaction between adatom pairs on the (100) surface of a simple-cubic tight-binding solid is investigated within a molecular-orbital approach. A general scheme for calculating the surface-density-of-states change and the interaction energy of one and two single-level adatoms is presented, and contact (and a correction) is made with Grimley's formulation. The method permits binding above surface atoms, at bridge sites, or at centered positions, and yields interaction energy as a function of band filling, adatom energy level, and a general hopping potential V between an adatom and the nearest surface atom(s). Calculations have been carried out for V/W_b in the range 1/12–1/2, the upper limit giving split-off states (W_b ≡ bandwidth). The single-atom interaction shows little dependence on binding type, in all three cases being most attractive when the Fermi energy equals the noninteracting adatom level, with a strongly V-dependent strength. For the pair interaction, one finds a strength at nearest-neighbor separation of about an order of magnitude smaller than the absorption energy of a single adatom. This interaction has an exponentiallike dropoff and sign alternations as one moves along the 〈10〉 direction. Under reasonable conditions, the nearest-neighbor interaction is often repulsive while the next nearest, third nearest, or fourth nearest is attractive, suggesting the patterns c(2 × 2), (2 × 2), and c(4 × 2), respectively, which are frequently observed in the adsorption of simple gases on the (100) surfaces of transition metals. On the basis of two-dimensional Ising-model calculations including second-neighbor interactions, one can estimate the strength of V from the observed disordering temperature of the adatom lattice; the result is similar to that obtained from estimates based on the heat of adsorption.

The electron scattering cross sections, both elastic and vibrationally inelastic, have been calculated using the Xα multiple-scattering method for H₂, N₂, and CO. The accuracy of the calculational scheme is tested by comparing to data from gas-phase measurements. Good agreement is found between theory and experiment. The formalism is then applied to molecules with fixed orientation by freezing out the rotational motion. The differential inelastic scattering cross sections for vibrational excitation exhibit a variety of angular patterns depending upon the molecule and the energy and direction of the exciting electrons. In all cases which have been calculated, the cross section for vibrational excitation is dominated by negative-ion resonances. The angular distribution patterns reflect the symmetry of these ionic states. These calculations indicate the possibilities of using the characteristics of the inelastic-cross-section patterns as a function of the exciting electrons' energy and direction to study molecules adsorbed on a surface.

Historically, interest in magnetic impurities was nurtured by bi-yearly reviews at the International Conference on Low Temperature Physics. Specifically, the curious resistivity anomaly seen in nominally pure transition metals — the resistance increased as the temperature decreased — was eventually traced to dilute impurities such as iron and cobalt. This was clarified in the late 1950's by the groups of Matthias and of Friedel…

We discuss qualitatively the importance of the correlation energy in determining the ground state of a metal with an impurity atom. For a single, partly occupied impurity d-state orbital, the correlation energy acts to prevent the appearance of a nonvanishing ground-state spin, so that this simple nondegenerate model actually has a complicated structure. In one dimension, we show that this model of an impurity can never lead to a localized moment. In three dimensions, if we take linear combinations of Bloch functions transforming according to the irreducible representations of the point group of the impurity+crystal, we find that most of the new wave functions are entirely decoupled from the impurity, and only a small subset interacts with it. The noninteracting majority of states determine the Fermi level, which we therefore take to be fixed. The ground state of the band states interacting with the impurity states depends on the two-body Coulomb repulsion U, and we find that for sufficiently small U the ground state has an even number of electrons with total spin S = 0. As U is increased above a certain critical value, the ground state of the interacting subsystem changes to an odd number of electrons, having total spin , and a localized moment is said to exist. The introduction of orbital degeneracy for the impurity d state, and of Hund's rule matrix elements, makes the localized moment much stabler. The results are obtained by a combination of exact energy-level ordering theorems and a Green's-function calculation in the t-matrix approximation.

A canonical transformation is used to relate the Anderson model of a localized magnetic moment in a dilute alloy to that of Kondo. In the limit of small s-d mixing, which is the most favorable case for the occurrence of a moment, the two models are shown to be equivalent. The Anderson model thus has low-temperature anomalies similar to those previously discussed for the model.

Starting with the Anderson model for the 4f¹ configuration of cerium, the transformation of Schrieffer and Wolff is performed, taking into account combined spin and orbit exchange scattering. The resultant interaction Hamiltonian differs qualitatively from the conventional s-f exchange interaction. The Kondo effect, the spin-disorder resistivity, the Ruderman-Kittel interaction, and the depression of the superconducting transition temperature with impurity concentration are worked out for alloys containing cerium impurities on the basis of this new interaction.

By using a functional integral formulation of the theory of itinerant ferromagnets above the Curie point, we show that for strong Coulomb interaction U, there are localized moments exhibiting a characteristic Curie-law susceptibility with the correct free spin- limiting value of the Curie constant. For weak U the same formulation gives a Pauli-like susceptibility, again with the proper limit, while for intermediate values the theory gives a smooth interpolation between the extreme cases.

The coupling between two magnetic centers in a band is discussed within the framework of the functional-integral scheme. The model used is the single-orbital Hubbard model with Coulomb repulsion only on the two magnetic sites. In lowest approximation, antiferromagnetic Ising coupling is obtained when the moments are nearest neighbors. When the moments are far apart, Ruderman-Kittel-Kasuya-Yosida coupling is the most important of several terms.

The understanding of ferromagnets, like iron, which exhibit localized moment behavior above the Curie point yet show itinerancy has long stood as a major theoretical problem. An account will be given of recent progress on this problem which was achieved through functional integral methods. This technique transforms the interacting electron system into an average over a system of non-interacting electrons moving in a Gaussian-weighted external "magnetic" field which acts only on the electronic spins. For a single magnetic impurity in a free electron metal, a single approximation allows one to go from Pauli paramagnetism to localized moment behavior in a smooth manner as the atomic exchange interaction is increased. The two impurity problem leads to an effective exchange coupling as in the Heisenberg model, which is antiferromagnetic for the nondegenerate orbital case studied here. Application of the technique to homogeneous systems leads to damped spin waves in the ferromagnet in lowest approximation.

In Ruderman-Kittel-Kasuya-Yosida (RKKY) perturbation theories of magnetic multilayers, realistic energy bands are used to obtain the nesting required for oscillatory behavior, but the Bloch functions are generally approximated by plane waves. To account for the observed long-period oscillations of the exchange coupling, we present a generalized RKKY theory, which includes both the Bloch character of the wave functions and the boundary scattering at the film edges. In contrast to existing theories, these two effects lead to long-period oscillations, that are robust with regard to roughness of the spin distribution and to oscillation amplitudes that sharply increase with the localized nature of the Bloch functions, in agreement with experiment.

The BCS theory of superconductivity [1] was originally derived on the basis of an early demonstration by Fröhlich [2] that the conduction electrons could attract each other due to their interaction with vibrating ions of the crystal lattice. In a more complete analysis by Bardeen and Pines [3] in which Coulomb effects and collective plasma excitations were included, interaction between electrons and the phonon field was shown to dominate over the matrix element of the Coulomb interaction near the Fermi surface. The BCS theory predicted a non-Fermi-liquid ground state with gaped fermionic single-particle excitations. No boson excitations were present in the theory, other than phonons. The BCS quasiparticle approximation appeared to be surprisingly successful in explaining thermodynamic and electromagnetic properties of superconductors including the venerable Meissner-Ochsenfeld effect. The BCS derivation of the latter was criticized because it was not strictly gauge invariant. We know that almost every revolutionary theory breaks some old rules, but the lack of the gauge invariance was taken by Bob Schrieffer [4] and others (Anderson [5], Bogoliubov, Tolmachev, and Shirkov [6], Rickayzen [7]) quite seriously. Since Gor'kov [8] introduced the gauge-invariant Green's function formulation of the BCS theory, the problem was viewed as a technical one. Notwithstanding, efforts to resolve the 'gauge' problem resulted in a remarkable series of excellent papers on plasmons and phonons in superconductors and, more generally, on the collective excitations in solid-state plasmas…

The conditions for the existence of plasma wave instabilities in the plasma formed by the electrons and holes in semiconductors are discussed. The dispersion relations for both the high-frequency optical mode in which electrons and holes move out of phase, and the low-frequency acoustic mode in which electrons and holes move in phase are calculated. Growing acoustic waves are shown to occur for a sufficiently large relative drift velocity of the electrons and holes, and the boundary between growing and damped waves is determined for various electron-hole temperature ratios. Growth rates are calculated for several cases of interest; when the influence of impurity and phonon scattering on the electron-hole behavior is taken into account it is concluded that InSb is perhaps the most promising semiconductor in which to observe such instabilities. An investigation of the hole and electron temperatures and the relative electron-hole drift velocity as a function of field strength is carried out for InSb. It is shown that moderate field strengths (~100 v/cm) suffice to produce electron-hole drifts of the required order of magnitude for the observation of plasma wave instability; however, the scattering mechanisms present are sufficiently effective that it appears marginal whether the other condition (long hole relaxation times) necessary for the observation of the plasma wave instability is achievable in practice. In an Appendix the conditions for the occurrence of similar plasma wave instabilities in semimetals are analyzed briefly.

The coupled electron-phonon system is considered for phonon spectra of Einstein and Debye forms. The single-particle electron Green's function G is calculated in a nonperturbative manner in both models, and its spectral weight function is examined to determine the validity of a quasiparticle picture. The weight function and the poles of G both lead to several branches of excitations rather than a single "dressed" electron. The asymptotic time dependence of the G is found, and the effect of multiphonon processes on the electron decay rate is discussed. The electronic polarizability, P of the interacting system is calculated with the aid of a generalized Ward's identity for the electron-phonon vertex. This identity, which is a consequence of electronic charge conservation, is derived in an Appendix. The calculation of P is carried out in the limit that the Fermi velocity is small compared with the phase velocity of the polarization field. An Appendix on the formal development of the Green's function equations is included.

Renormalization of the phonon spectrum of the Einstein form due to a weak coupling with electrons is calculated. No splitting of the phonon mode is found. A q-dependent hardening of the long-wave optical phonons contrasts with the softening of acoustic phonons in a weakly coupled electron-phonon system and with the softening of all phonons in the strong-coupling polaronic regime.

Almost immediately after the discovery of higb-T_c superconductivity (HTSC), Bob Schrieffer became actively involved in this field, and made influential contributions in a wide range of subjects. In 1987, I joined the Institute for Theoretical Physics at UC Santa Barbara as a postdoc, where he was the director. I still vividly remember Bob's tremendous excitement about the discovery, and its profound theoretical implications. "It feels like 1956 again!", he said to me at our first meeting. More than ten years have passed since then and, as we look back at Bob's contributions to the high-T_c problem, we see that his far reaching insights have always guided the community in the right directions…

Localized baglike solutions in the pairing theory of superconductivity are studied. Starting from the Bogoliubov–de Gennes equations on a two-dimensional square lattice for a half-filled negative-U Hubbard model, cigar- and star-shaped bags are numerically obtained, inside of which the order parameter is reduced, self-consistently trapping an added quasiparticle. These nonlinear excitations are important when the coherence length is small as for the new high-temperature superconductors. Several experimental consequences are discussed.

The spin-bag approach to the high-temperature superconductivity is presented in detail. The general argument that the local supression of the electronic pseudogap leads to an attractive interaction of the quasiparticles is substantiated by the detailed calculation of the pairing potential mediated by the collective modes of the spin-density-wave background. In particular, the spin-wave spectrum, the sublattice magnetization, and the spectral distribution of the collective modes are studied within the random-phase approximation. In the low-doping limit, different shapes of the Fermi surface give rise to a superconducting gap which formally has d-wave- or p-wave-like symmetry, however the gap has no nodes on the Fermi surface. Therefore, the superconducting properties of our model are analogous to those of a conventional s-wave (i.e., nodeless) BCS superconductor. We also discuss possible bag effects in the large-U Hubbard model and in charge-density-wave systems. Finally, the relation of our work with other approaches and with experiment are discussed briefly.

It is shown that antiferromagnetic spin fluctuations in a two-dimensional metal, such as heavily doped cuprate superconductors, lead to a pseudo gap in the electronic spectrum. The self-energy of spin bags and their pairing interaction in the paramagnetic metal are calculated. These results are consistent with the corresponding results in the weakly doped ordered antiferromagnet.

It is shown that antiferromagnetic spin fluctuations in a two-dimensional metal, such as heavily doped cuprate superconductors, lead to a pseudogap in the electronic spectrum. The spectral function evolves from one peak in the Fermi-liquid regime to two peaks, one for particles and one for holes. The self-energy of spin bags and their pairing interaction are calculated. These results are consistent with the corresponding results in the weakly doped ordered antiferromagnet.

In models of strongly correlated electrons the wave-function renormalization Z_h of a hole injected in a half-filled band is small compared to unity. This result reduces the accuracy of numerical and analytical predictions of superconductivity for the Hubbard and related models. A simple method is described to deduce approximate quasiparticle operators whose Z_h is closer to 1 than for the standard bare operators. The method is tested for the t-J model. We find that pairing operators constructed out of these quasiparticles are also considerably improved. For example, at J/t = 0.2 an improvement by a factor of 10 or more is expected in the signal of superconductivity in a Monte Carlo simulation if these quasiparticle operators are used rather than the standard bare operators.

We discuss the class of superconductors that have pairing correlations which are odd in frequency, as introduced originally by Berezinskii and more recently by Balatsky and Abrahams. As follows from the equations of motion, a natural definition of the thermodynamic order parameter of the odd-pairing state is the expectation value of a composite operator which couples a Cooper pair to a spin or charge fluctuation. We use a model pairing Hamiltonian to describe properties of the odd-pairing composite-operator condensate. We show that the superfluid stiffness is positive, we discuss superconductive tunneling with an ordinary superconductor, and we derive other thermodynamic and transport properties.

The following sections are included:

• Introduction

• The Ordered Antiferromagnet

• Fluctuating Antiferromagnetic Order

• Pairing Interaction

• Acknowledgements

• REFERENCES

The anomalous momentum and temperature dependence of the spectral lineshape in data from underdoped Bi₂Sr₂CaCu₂O_8+δ (Bi2212) indicates that the quasiparticles are strongly coupled to collective excitations centered near Q = (π, π). The doping dependence of the spectral lineshape and its correlation with the size of the superconducting gap indicate these collective excitations are related to the pairing interaction in high-T_c superconductors, in analogy with phonon induced structures in tunneling spectra of low T_c materials.

Superconductors whose order parameter violates time-reversal symmetry and parity have unusual properties that set them apart from conventional superconductors. In particular, if translation symmetry is also broken, the superconducting state generates spontaneously a current and a magnetic field. These phenomena are studied using a d_x²-y²+id_xy superconductor as a prototype. Some of the most striking consequences of the symmetry breaking are examined at boundaries and in the presence of pointlike impurities and quasiparticles.

Collective, low-energy excitations in quasi-two-dimensional d-wave superconductors are explored. While the long-range Coulomb interaction shifts the charge-density wave and phase modes up to the plasma energy, the spin-density wave excitation that arises due to a strong local electron-electron repulsion can propagate as a damped collective mode within the superconducting energy gap. These excitations are suggested to be relevant to high-T_c superconductors, near the antiferromagnetic phase boundary, and may lead to distinctive signatures in spectral-density and neutron-scattering data.

We develop a quasiclassical approach to the energy spectrum of an anisotropic superconductor in a magnetic field, B, such that H_c1 ≪ B ≪ H_c2. Low temperature de Haas–van Alphen oscillations are considered for two cases: (1) the extremal electron orbit may coincide with a symmetry line and (2) the orbit crosses points where the superconducting order parameter has zeros. The signal is shown to be small in both cases.