Fluctuation and higher temperature superconductivity in lighter element systems

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Physica C 468 (2008) 115–125 www.elsevier.com/locate/physc

Fluctuation and higher temperature superconductivity in lighter element systems N.W. Ashcroft

*

Laboratory of Atomic and Solid State Physics, Cornell University, 615 Clark Hall, Ithaca, NY 14853-2501, USA Accepted 20 August 2007 Available online 1 October 2007

Abstract A charged, near degenerate quantum fluid, necessarily inhomogeneous by virtue of the presence of a dynamic and more massive compensating charge distribution (itself possessing internal structure composed in part of the same particles constituting the putative quantum fluid) is found to exhibit a liquid–liquid phase transition as temperature is lowered. The ensuing low temperature phase is dissipationless and in spite of long-ranged attractions between the quantum fluid and the compensating system, no energy transfer proceeds between the two. The quantum fluid in question is electronic in origin; the emerging state is, of course, one of superconductivity or superdiamagnetism and its lack of dissipation is entirely consistent with the existence of an energy gap in the spectrum of excitations of the quantum fluid. The degeneracy temperature of the quantum fluid originates with valence electrons, and ranges from 104 to 105 K. There is vast flexibility in the choice of structure, the degree of complexity of the compensating system, and of its dynamics. Acknowledging fluctuations in both itinerant and localized charge, is it therefore possible that systems can be found where the liquid–liquid transition of interest here can be found at significant temperatures, perhaps even near room temperature? In this quest it appears that the light elements in combination continue to offer considerable promise. Ó 2007 Published by Elsevier B.V.

1. Introduction The problem of accurately predicting the onset of a new phase in any dense system of particles from a knowledge of its fundamental microscopic interactions is an especially difficult one, as is known from the fields of critical phenomena, magnetism, and so on. It appears to be particularly so for superconductivity, yet in recent years significant progress has certainly been made, via the numerically intensive Migdal–Eliashberg approach, and also more recently via functional methods. These approaches may provide a guide to the selection of compounds, particularly those drawn from the light elements in combination (and especially the intermetallics), for likely high temperature superconductors. What follows seeks to trace the origins of the superconducting instability to the underlying polarization processes, from all sources. As noted, the focus is especially

on light element and relatively weakly coupled systems (though it may be noted that the cuprate class of superconductors is replete with highly polarizable constituents). The superconducting state is found in non-uniform phases of the valence electron states in metals and alloys. It arises from metallic but inhomogeneous distributions of the latter and its emergence can clearly be viewed as the consequence of a liquid–liquid transition, the liquid itself being charged and inhomogeneous. Liquid–liquid transitions are not uncommon in condensed matter physics and provide a convenient unifying pathway to approach this topic. By way of background, therefore, consider the standard Hamiltonian for describing interacting systems of N neutral particles (say atoms or molecules) in a volume X but in the absence of external influences which is b ¼ Tb þ Vb H or

*

Tel.: +1 607 255 8613; fax: +1 607 255 6428. E-mail address: [email protected]

0921-4534/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.physc.2007.08.016

b ¼ Tb þ ð1=2Þ H

Z

Z dr

X

X

ð1Þ 0 ð2Þ

0

0

^ ðr; r Þtðr  r Þ dr q

116

Here Tb is the kinetic energy operator

N.W. Ashcroft / Physica C 468 (2008) 115–125



P h2 2  Tb ¼  j 2m rj n

^ð2Þ ðr; r0 Þ is the where mn is the mass of the particle, and q two-particle density operator which may be given entirely ^ð1Þ ðrÞ, i.e., in terms of the one-particle density operator, q ^ð1Þ ðrÞ^ ^ð2Þ ðr; r0 Þ ¼ q qð1Þ ðr0 Þ  dðr  r0 Þ^ qð1Þ ðrÞ ð2Þ q P ^ð1Þ ðrÞ ¼ j dðr  rj Þ, for instantaneous particle coorwith q dinates {rj}. In (1) t(r) is an assumed short-ranged pair potential reflecting the internal physics of the particles; the role of fluctuations associated with localized electrons in determining this is central. It may well be that the particles also have spin degrees of freedom and although these are not explicitly displayed in (1) the role of spin fluctuation in determining its states can be significant. At the more fundamental level of nuclei and electrons the Hamiltonian (1) originates with Eq. (10) below, and an inferred adiabatic separation of electronic and nuclear timescales. Not all systems describable by (1) exhibit a solid or crystalline ground-state at normal pressures. However, as temperature is progressively lowered in a gas (see AppendixA) most exhibit a spontaneous transition to a higher density b . This state, which continues to preserve the symmetry of H is a condensed phase of continuous symmetry whose density is entirely comparable to that of the crystalline broken symmetry solid, if and when it forms. In terms of the initiating Hamiltonian (1) it is characterized by the comparison h Tb i  h Vb i. For the quantum equivalent soon to follow for electrons it is pertinent to note that this liquid state, most commonly the classical fluid, can also exhibit more than one phase as temperature is lowered; the notion of a liquid–liquid transition has a clear parallel for an electron liquid. The occurrence of such transitions is entirely dependent on the form of t(r), as is readily anticipated. They have been reported in purely elemental systems (e.g., in the liquid phase of phosphorus [1]). The systems adopting a crystalline ground state under normal conditions are also characterized by the condition h Tb i  h Vb i or more instructively h Tb i > Dh Vb i, where (i) h Tb i is of quantum origin, including zero point effects at low temperatures, and (ii) Dh Vb i is the change in interaction energy that might accompany a transition in structure from one dynamic crystal to another. Of the ground state fluids found under normal conditions the key examples in nature are the heliums, 4He which is a Boson system, and 3He which is a Fermion system. Both undergo liquid–liquid transitions as temperature T is lowered. For the superconducting transition to follow, the ordering temperatures are more meaningful when expressed in terms of the physical characteristics of the interaction t that establishes the basic liquid structure, what might be called the with diagonal  order. In a phase  ^ð1Þ ðrÞ ¼ q  ¼ constant this is continuous symmetry q revealed in the pair distribution function g(r) which, for a state that is now both translationally and rotationally invariant is defined by

 ð2Þ  2 ^ ðr; r0 Þ ¼ ð qÞ gðjr  r0 jÞ q (and where a completely uncorrelated system yields g  1). In 4He and 3He the pair interactions are characterized by a depth, e, around 11 K. Normal pair-correlations in these quantum liquids result from t, and are generally well measured, at least in reciprocal space. The liquid–liquid transitions in the heliums (to states with off-diagonal long-range order) then occur at 0.2e for the Bosonic system 4He, and at a remarkable 104e for the Fermionic system (at which point the thermal de Broglie wavelength is 420ao). Observe that for two truly isolated 3He atoms with reduced mass 2754me, the constants of the Lennard-Jones representation of t are such that the system is close to possessing a bound state. It follows that pairing fluctuations are already to be anticipated, and these can be enhanced in the N-particle state by proper choice of wavefunction. Below the transition such a state is characterized by an N-particle wavefunction constructed from N/2 antisymmetrized (and hence deeply interlinked) pair states, themselves reflecting in this case p-wave symmetry. Its formation requires no other intermediaries and, importantly, raises the possibility of intrinsic pairing in itinerant electron systems; as noted the fundamental presence of electrons in this system (in wholly localized states) as well as the charged and far more massive nuclei, is entirely manifested in t. However, the 3He atom carries a spin, and spin fluctuations are crucial to the formation of the low temperature ordered state. It is to be emphasized that ordering in one set of Fermions (the 3He nuclei) is the result in part of fluctuations in another component set of Fermions (the electrons, see Eq. (14), below) which also determine via the Pauli principle the short-range repulsive character of t. Though the timescales here are very different, this general notion acquires some additional significance in the progression to follow where the two sets of Fermions originate with a single set of electrons, but traditionally separated into valence and core classes, this division having clear consequences with respect to exchange. Before turning to charged systems it should be noticed that the neutral systems described by (1) can be readily generalized to encompass the case where an external one-body potential u(1) (r) may also be present (which is required, absolutely, for the electronic case below as emphasized by Eq. (10)). The Hamiltonian is then Z Z b ¼ Tb þ ð1=2Þ dr dr0 q ^ð2Þ ðr; r0 Þvðr  r0 Þ H X X Z þ dr^ qð1Þ ðrÞuð1Þ ðrÞ: ð3Þ X

By Mermin’s extension of the Hohenberg–Kohn theorem, the internal energy E, Helmholtz energy F, and Gibbs energies G and then all functionals of the one-particle density  ð1Þ ^ ðrÞ , and under ordinary conditions they can qð1Þ ðrÞ ¼ q be obtained from the classical rendition of density functional theory for what has now become an inhomogeneous system. The approach is applicable for inhomogeneous

N.W. Ashcroft / Physica C 468 (2008) 115–125

states (typically broken symmetry states) that exist in the limit u(1)(r) ! 0, which immediately indicates a pathway to the melting problem (in fact, for either classical or quantum systems). 2. Itinerant electron (charged) systems: towards the Kohn– Luttinger question The essential question raised by the above synopsis is this: suppose a certain fraction of the electrons (conventionally, as indicated, the valence electrons) are no longer localized, but have become itinerant. They form a conducting and usually degenerate  ð1Þ  quantum fluid whose overall ^e ðrÞ reflects that of the ion system density distribution q they have departed. This is a fundamentally two-component system with long-range interactions. Given the above the issue is whether there may be liquid–liquid transitions in the itinerant electron system, again as temperature T is lowered, and again to states displaying off-diagonal longrange order, as in the heliums. In a quite basic form (for interacting electrons in a uniform compensating background) the question arises with Kohn and Luttinger [2] more than a half century after the discovery of superconductivity. Since any such transition is of a demonstrable thermodynamic character, and since electrons are charged, it is obligatory to consider a system which, at the level of the Hamiltonian, is constructed to yield well defined thermodynamic functions. The initiating system, if viewed as canonical, must be charge neutral with electronic- and counter-charge normally taken to occupy the same macroscopic domain (the volume X, or area A in a rigorously two-dimensional system, as in the two-dimensional interacting electron gas). An obvious prerequisite for a state exhibiting superconducting order is the prior existence of the metallic state of matter itself which is not guaranteed by the elementary division of electrons into core and valence classes. The early paradigm for this was a phase of continuous symmetry arising in a problem of N interacting electrons with charge e, mass me, confined to a volume X in which is present a continuum of exactly neutralizing counter-charge devoid of dynamics, and at a uniform density ðþeÞN =X ¼ e q. Both N and X assume values appropriate to an eventual thermodynamic limit. If the electrons have coordinates rj and momenta ^ pj ¼  hrj , then as is well known the Hamiltonian for this assembly, taken also as spin 1/2 Fermions, is Z Z 0 b b H e ¼ T e þ 1=2 dr dr0 tc ðr  r0 Þf^ qð2Þ e ðr; r Þ X

X

2 g qþq  2^ qeð1Þ ðrÞ

ð4Þ

where tc(r) is the fundamental Coulomb interaction (e2/r), and Tb e isthe standardPkinetic  energy operator, now for 2 h2  ^ðe1Þ ðrÞ is the electrons i:e:; Tb e ¼  j 2m r j . In Eq. (1) q e ^eð1Þ ðr; r0 Þ is one-particle density operator for electrons and q the two-particle density operator (defined in Eq. (2)). In the limit of large N and large X (where for neutrality

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Þ, (4) has full translational and rotational symmeN =X ! q try, both of which can be broken in attaining the eventual b e . Observe that as with Eq. (1), (4) has no explistates of H cit reference to electronic spin; this symmetry can also be broken in states of eventual magnetic order. In the presence of an external magnetic field there can be gauge symmetry, and this too can be broken. For this familiar system, where as emphasized the charged particles and the neutralizing background occupy the same domain, existence of the thermodynamic functions of the canonical system described by (4) but set in thermal contact with a heat bath (establishing a temperature T) was proven by Lieb and Narnhofer [3]. If the electron density is given in terms of the usual rs, ðð4p=3Þr3s a3o ¼ 1= q; where ao ¼ h2 =me e2 is Bohr’s then a transla ð1Þ radiusÞ,   ^ r ¼ constant ¼ q  , the tionally invariant state of (1) q Fermi liquid with a gapless spectrum, can arise as a proposed ground state when rs P 40 broken in the appearance of states of magnetic order, and the translation symmetry can also be broken in the appearance of the Wigner crystal or even a paired Wigner crystal [4]. These states, ordered or not, are familiar in the context of many body theory which proceeds from a development of N-particle states constructed as products of non-interacting (plane-wave) single particle states, and associated spin-functions. 3. Intrinsic superconductivity If Eq. (4) is first treated as a system of non-interacting Fermions, the familiar ground state has plane-wave single particle levels doubly occupied up to a Fermi energy, eF ¼ ðao k F Þ2 Ry ¼ ðao k F Þ2 ðe2 =2ao Þ, corresponding to a highest occupied wave-vector of magnitude kF (kFrsao = (9p/4)(1/3)). If Coulomb interactions are now restored the quite fundamental question raised above by Kohn and Luttinger concerned the physical characteristics of the very low temperature states of (4). The issue was whether for fixed rs, but for declining T, Eq. (4) could admit of a further transition, one which was exactly of the liquid–liquid type discussed for the neutral quantum liquids above but now for a manifestly charged quantum liquid. As noted, in classical fluids these transitions generally occur when effective pair interactions between particles possess sufficient complexity. In seeking a physical parallel for the system embodied by (4) the route is to proceed from the initiating N-particle system to a viewpoint of effective electron–electron interactions appropriate to a degenerate Fermionic assembly. By the standard response arguments this leads to screened electron–electron interactions, properly reflecting the degeneracy and the corresponding sharp cut-off in occupancy in momentum space, eventually manifested at lowest order in the Friedel oscillations. The necessarily attractive regions of such interactions constitute some of the complexity needed to impel instability of a liquid phase, as an appeal to the Cooper problem immediately suggests. As in 3He, the low temperature state proposed by Kohn and Luttinger for (4) is one of off-diagonal long-range

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N.W. Ashcroft / Physica C 468 (2008) 115–125

order constructed from spin-zero pairs (chronologically these arose a little earlier as Cooper pairs, but earlier still as Ogg’s dielectrons [5]). Clearly in this slightly anti-historical recounting for electrons, it will be the state universally known as the superconducting phase, and once again it is a phase constructed from highly interlinked pairs. It is apparent that a full appreciation of the true manybody physics actually fixing the effective interactions (i.e., beyond any simple static screening arguments) is of crucial importance. And the problem is manifestly dynamic, so that solution of the Eliashberg equations has been a sine qua non, though there are now interesting developments in density-functional-theory where what enters an appropriate free energy functional is not only the density but an expression of order accompanying formation of a superconducting state [6]. Over the years the occurrence of intrinsic superconductivity (i.e., pairing states devoid of the physical intervention of other intermediaries) has been discussed extensively in the literature (see Ref. [7] and references therein). The origin of the effective electron–electron interactions rests considerably with fluctuations (the time scales are typified by those of volume plasmons), and their consequences normally rise as dimensionality is lowered. It should be particularly emphasized that dimensionality plays an exceedingly important role in this issue as has been elucidated by Ginzburg and Kirzhnits [8]. Indeed, inclusion of the major fluctuational diagrams in the two-dimensional interacting electron gas appears to lead to considerable additional enhancement of the Friedel attractive regions in the effective electron–electron interactions when compared with three-dimensional equivalents at comparable densities [9]. The role of fluctuations may be especially notable for cases where the electrons might be itinerant but the requisite neutralizing charge is actually remote from them, as appears to occur in the cuprate class of superconductors. In these situations identification of the minimal grouping to ensure proper definition of the thermodynamic functions, and hence of the appropriate quantum statistical distributions, and hence even of selection of the basic one-electron states, is a matter of some interest. In a one-electron context the underlying electronic structure resulting from (4) might well be described as ‘singleband’. The Kohn–Luttinger question assumes far greater importance when posed for multi-band systems because of the possibility of additional fluctuations introduced by periodicity, particularly correlated fluctuations originating with particle and hole bands. These appear, as may be anticipated, for systems where the background charge first taken as static, and uniform, is now taken as static but periodic. 4. The Kohn–Luttinger argument and periodic backgrounds Suppose the initiating translation symmetry of the itinerant electron fluid is in fact broken by the action of sources external to the electron system; as indicated the case of most interest is a periodic system. To begin with,

and recognizing that homogeneous electron fluids can never exist in real condensed matter systems, the static background charge density in (4) is now taken as eq(r), where q(r) is periodicR on a Bravais Lattice (q(r + R) = . The system can q(r), {R} 2 BL), and X ðdr=XÞqðrÞ ¼ q have arbitrary dimensionality but for the present three dimensions will be assumed. The succeeding steps (culminating in a fundamental description of a system capable of displaying superconducting order) will endow the background charge with quantum statistics, and with dynamics, both internal and cooperative, and then proceed to examine its multipole character. Of particular concern for the emerging superconductivity problem will be the situation where the external sources physically reflect the presence of ions drawn from the lighter elements, with the immediate consequence that rather considerable ionic motion, supplementing multipole fluctuations will later become important. Note that associated with the sites {R} is a reciprocal lattice with corresponding vectors {K} (for which exp(iK Æ R = 1)). The relative scales of kf and the shortest of the vectors {K} is now of fundamental importance, and it is clear that this is set in practical terms by the standard valence assigned to the assembly of atoms which will eventually supply the itinerant electron system itself (and whose presence at this stage of the development is merely being mimicked by a static but continuous periodic charge distribution as given now by eq(r)). The system is still considered canonical, and charge neutral. With this change Hamiltonian (1) becomes Z Z 0 b b H e ¼ T e þ 1=2 dr dr0 tc ðr  r0 Þf^ qð2Þ e ðr; r Þ X

 2qðrÞ^ qeð1Þ ðrÞ þ qðrÞqðr0 Þg

ð5Þ

and in the one-electron approximation it becomes a problem in band-structure. According to the choice of average , the bands occupied in the ground state may be entirely q contained within the first Brillouin Zone, or not. The latter case is noteworthy within the context of the Kohn– Luttinger question (now raising the possibility of intrinsic superconducting order in a system with discrete translation symmetry) because of the possibility of an ensuing electron–hole character to the band-structure. This will arise when the valence is such that the Fermi surface of the non-interacting system significantly intersects the first(and even higher) Brillouin Zones, a point of considerable importance for the more familiar case (see below) where charge fluctuations arise from coherent motions of localized electronic and nuclear charge. The key physical point is that correlated fluctuations between particleand hole-bands can now occur as a consequence of the discrete translation symmetry and the evident sign change for holes means that enhanced overall effective attractions can be the net result. This matter was investigated in Ref. [7]. In this argument, no restrictions to Bravais lattices are in fact implied. Indeed, relatively simple systems but

N.W. Ashcroft / Physica C 468 (2008) 115–125

assuming structures with bases (i.e., large cells) can very much favor the formation of a multiband electronic structure and hence the correlated electron–hole excitation mechanism. It has been noted already that attractions between otherwise localized distributions of electronic charge (on separated neutral atoms, for example, and even non-neutral distributions in ions, as will be elucidated below) arise precisely because of such correlated fluctuations, in this case of multipole character. In the pursuit of sources of local polarization in the pairing problem, bound states of a molecular character can be of considerable importance. Cast in terms of effective interactions between valence electrons and contributions to these arising from additional polarizations associated with intrinsic periodicity, the Kohn–Luttinger question raises the possibility of a pairing transition at temperatures in excess of those of a uniform system with equivalent average density. 5. Fluctuating localized charge, and superconductivity But suppose now that the periodic static background introduced above is merely a physical approximation to a more fundamental initiating system which is itself of electronic origin and which possesses its own internal dynamics. The latter normally proceed in an appropriate electronic timescale within a strictly time averaged periodic framework. The physical character of the system overall is that of a set of electrons, the conduction or valence electrons, in the presence of a second set, considered localized, but also fluctuating. Framed this way, the Kohn–Luttinger question precisely parallels the one raised above for 3He: can there be intrinsic superconductivity, entirely devoid of intermediaries, where the fluctuations in one set of electrons induces pairing in a second? Electron-based mechanisms for superconductivity were raised as general possibilities by Little [10] and Ginzburg [11] in systems, such as are now being described, where the electrons leading to a possible pairing mechanism were clearly to be separated by their degree of localization from those that have been partitioned as valence electrons. These are also variously described as exciton mechanisms, and if the corresponding excitations are again coherent, at least over significant length scales, they would be immediately appealing because the associated dynamical energy scales are so high. In establishing these scales it is apparent that the presence of localized nuclear charge (Zae) now has to be explicitly recognized, and at this stage it should be emphasized P that it is still being taken as static, i.e., qð1Þ ðrÞ ¼ n R dðr  RÞ where now the {R = Rn} are taken as the static nuclear positions. It follows that the generalization of (5) requires the introduction of the full electronic P ^eð1Þ ðrÞ ¼ j dðr  rej Þ where the rj refer to density operator q the coordinates of all electrons including the substantial fraction of them that may eventually become localized around the nuclei under ordinary conditions. The evident generalization of (5) is

1 H ¼ Tb e þ 2

Z

119

Z

dr0 vc ðr  r0 Þ  fZ 2a qnð1Þ ðrÞqnð1Þ ðr0 Þ

dr X

X

0 ^ð2Þ qeð1Þ ðr0 Þ þ q  2Z a qnð1Þ ðrÞ^ e ðr; r Þg

ð6Þ

The physical character of the problem represented by (6) is similar in form to that represented by (3); in the present case charged (and highly quantum) electrons are moving in one-particle potentials established by the static charge densities ðþeÞZ a qnð1Þ ðrÞ. It is related to the problem addressed by electronic density functional theory in all-electron approaches. From the single-particle structure inherent in the Kohn–Sham equations, what will be determined by this method might again be reasonably referred to as diagonal electronic structure. This leads again to the Kohn–Luttinger question for the inhomogeneous electron fluid: in suitably chosen systems, might there be a transition as temperature is lowered to a state possessing off-diagonal long-range-order? For a system that possesses a metallic phase, the electron system is traditionally separated into Zv valence electrons per nucleus considered to be in itinerant states, and Zi = Za  Zv elections in states localized about each nucleus. Since there is penetration by the valence electrons of the space conventionally assigned to the core region, the common assumption of strictly integer valence cannot be exact. An important exception to the proposed separation is, of course, hydrogen. Correspondingly, a Hamiltonian b i may be assigned to incorporate the short-range physics H of the nuclei and the now localized (or core) electrons. Under this assumption an important and almost standard simplification emerges through the introduction of a pseudopotential which yields only the valence spectrum and which accounts for the fact that the core spectrum constitutes entirely occupied states. Formally it is obvious that the pseudopotential must depend on the structure selected, though because of the significant spread of valence and core state energies it is often taken as transferable. It is also non-local. An important point for later (and the observation of significant superconductivity in compressed lithium and calcium) is that should conditions lead to an encroachment by the core-electrons of the space initially spanned by the valence electrons, the pseudopotential can no longer be considered weak. Such conditions can be brought about by increase of density, via pressure, or by altering environments to achieve through ‘‘chemical compression,’’ the same ends. This has important general consequences for superconductivity in lighter element systems, as will be seen. Because of many-body interactions within the grouping of core electrons, charge fluctuations are anticipated. To place an ensuing multipole fluctuational concept within a somewhat more familiar context, it is useful to introduce a density operator for a single ion, (the nucleus with its localized electrons) initially situated at lattice site R. The argument is readily generalized a to system with a basis. Again, if the atomic number of the element involved is Za, then the site density operator is given by

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N.W. Ashcroft / Physica C 468 (2008) 115–125

^iR ðrÞ ¼ Z a dðr  RÞ  q

X

dðr  R  rcj ðRÞÞ

ð7Þ

j

where rcj ðRÞ denotes the coordinates of the Zi electrons considered bound, and where the superscript i denotes an ion. In terms of Fourier transforms this is " #  X iqrcj ðRÞ i iqR qR ðqÞ ¼ e Zv þ 1e ð8Þ j

where again Zv is the valence. It follows that if we proceed to the thermodynamic limit the corresponding interaction operator for a system of ions is X X ^iR ðqÞ^ Vb ii ¼ ð1=2XÞ vc ðqÞ qiR0 ðqÞ q q

R;R0

which represents the entire sum of multipole–multipole fluctuational interactions between the ions in operator form. Expansion of (8) now gives e^ qiR ðqÞ ¼ eiqR ðZ v e P c iq  d^R þ   Þ where d^R ¼ e j rj ðRÞ is the point-dipole operator at site R. Providing that lattice constants are large compared with ionic sizes, these dipoles are coupled sitewise with standard dipole–dipole interactions and they define an underlying problem whose Hamiltonian is quadratic in character. The frequency scales associated with the corresponding excitations are generally much larger than those of phonons. If xo is a typical frequency and a is a static dipole polarizability associated with the internal physics of the localized electrons, then by way of an estimate D E2 1=2  1=2  1=3 ^ ¼ hxo = e2 =a1=3 a=a3o ð9Þ d R =ao e Typical values of xo may approach an appreciable fraction of an atomic unit, and for ions with significant core-spaces or for orbitals associated with bases, a can also be appre2 1=2 ciable. It follows that ½hd^R =ao ei  can easily approach phonon displacive equivalents though, to repeat, the corresponding time scales are quite different. The associated polarization waves can provide, in principle, an additional exciton-like channel for a dynamical pairing instability in the valence electrons [12]. It is clear that the static nucleus problem represented by (6) should present an interesting test case for the more recent functional approaches to superconducting order [6]. High polarizability should be favorable but within a clear division of core and valence states. Once again the Kohn–Luttinger question is entirely pertinent but is now including the possibility that fluctuations in one set of electrons (here principally the localized or core electrons) may aid in promoting pairing in second set (the valence or itinerant electrons). It relates immediately to the propositions advanced by Little and Ginzburg, as discussed above. Up to this point the nuclei have been taken to be infinitely massive and that pairing mechanisms are completely electronic in origin (fluctuations inducing pairing arise both in core and valence electrons, as conventionally but inexactly sep-

arated). The requisite kernel of the Eliashberg equation determining the onset of superconductivity follows from inclusion of all sources of electronic fluctuation, as discussed in Refs. [7] and [12]. The static lattice assumption may finally be relaxed, thereby introducing a far lower timescale for additional fluctuations. 6. Monopole (phonon)–valence-electron interactions and superconductivity In historical terms the last in this progressive sequence of coherent fluctuation mechanisms was actually the first to be proposed as the likely underlying physical cause of superconductivity (via Fro¨hlich’s suggestion and the near contemporaneous discovery of the isotope effect). The progenitor of superconductivity was indeed the coherent charge fluctuations associated with collective harmonic lattice displacements of ions (the phonons). These normally arise when the kinetic energy operator for the nuclei is introduced into (6), yielding for an element of atomic number Za Z Z 1 b b b H ¼ Tn þ Te þ dr dr0 vc ðr  r0 Þ 2 X X

 0 ^eð2Þ ðr; r0 Þ ^nð2Þ ðr; r0 Þ  2Z a q ^ð1Þ  Z 2a q qð1Þ ð10Þ n ðrÞ^ e ðr Þ þ q and it is now clear that in addition to internal dynamics of the previously discussed localized electronic charge, Hamiltonian (10) admits of collective ionic displacements. Observe that (10) is readily extended to the elements in combination for which Z Z X 1 b ¼ Tb e þ H Tb n;a þ dr dr0 vc ðr  r0 Þ 2 X X a ( ) X X 2 ð2Þ 0 ð1Þ ð1Þ 0 ð2Þ 0 ^e ðr;r Þ ^n;a ðr;r Þ  2 ^n;a ðrÞ^ Z a;a q Z a;a q qe ðr Þ þ q  a

a

ð11Þ and the polarizability of localized electronic density associated with groupings of ions in a basis (as distinct from the Bravais lattice case) is now a matter of some interest. In particular, Hamiltonian (11) will describe the class of high temperature superconductors where the groupings lead to layered structures some (e.g., the metallic CuO2 layers) acquiring macroscopic charge transfers. Taken individually these particular layers cannot constitute thermodynamic systems (the electrostatic component of their energies is plainly not extensive) so that as noted earlier, a minimal neutral grouping, free also of dipolar terms, is required simply to define appropriate single-particle states. It is also evident that coupling between macroscopically charged sheets must ensue. In terms of the terminology used above, and recognizing the long-range character of the unscreened electron–ion interactions, the collective displacements might also be seen to originate with the monopole character of the resolution

N.W. Ashcroft / Physica C 468 (2008) 115–125

of the localized charge in which the itinerant valence electrons now find themselves. In posing the Kohn–Luttinger question for (10) it is quite evident that the mass of the nucleus or ion must be involved in determining the scale of associated fluctuations. Further, when pursuing the corresponding transitions that subsequently take place in the itinerant electron system it is again apparent, as above, that the scale of Fermi surface dimensions relative to the Brillouin zone scale is a critical issue in determining the momentum selection rules, and especially the consequent multiplicity. Suppose, as is generally the case, that the system admits of net ionic displacements these being represented in standard phonon dynamics. The ion-core originally at R is now to be found at R + uR where the displacement uR is synthesized from the normal modes of a harmonic problem, though the latter can be readily extended to a self-consistent harmonic problem. Expansion of the density attributable to the ion, which is near to R, gives eqiR ¼ expðiq  RÞ½Z v e þ iZ v eq  uR þ iq:d^R þ   

ð12Þ

where as above d^R is the point dipole operator to be associated with the localized electronic system near site R. The expansion can clearly be extended to quadrupole and further terms. The terms beyond Zve in (12) have an obvious physical identification. It is apparent that iZveq Æ uR has a long-range monopole character, the displacements uR being largely coherent when derived from harmonic traveling waves (the standard phonons, as indicated). In the ensuing treatment of (12) for the problem of superconductivity in the light elements, and especially the light elements in combination where the dipole term may be particularly relevant, it should be noted that there can be destructive interference as well as enhancement. From the form of the expansion (12) it may be observed that higher terms can lead to a coupling of polarization waves and phonons (and hence even to interference). Finally, observe that the displacements uR, originating with the dynamics of the ions are generally assumed to be quite small. There may be an exception to this, however. The electrons are crucial to ionic dynamics because of the role they play in establishing effective ionic interactions. The latter can be altered by changing the local electronic densities (through ‘doping’, a term commonly used). This can lead, especially for light elements (e.g., oxygen in the cuprate superconductors) to the possibility that harmonic wells can develop mild doping dependent tunneling barriers within them. Should this happen, the associated uR are no longer necessarily small, and should coherent tunneling states ensue the associated electronic coupling may be more favorable to superconductivity than the purely harmonic predecessor, and clearly there will still be an isotope effect. The major point in the foregoing is that framed within the Kohn–Luttinger viewpoint a pairing instability can be induced in a valence electron system through the action of fluctuations from

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all electronic sources, augmented by those originating with the countercharge situated on the much heavier nuclear component. Given that the scale of energies in the initiating problem (Hamiltonian (10)) are in a definite progression (electronic  electron–phonon  phonon) it would appear that in approaching solutions of the Eliashberg equation, inclusion of the electron–phonon terms might well come after the periodic many-body electron problem has first been treated (a perspective followed in Ref [12]). 7. Superconductivity in the light elements The pathway to the determination of Tc for superconductivity systems was extensively reviewed up to 1982 by Allen and Mitrovic [13]. The discussion above leads to additional insights guiding appropriate selections of systems that may display higher superconducting transition temperatures, especially within light element systems, which will be the focus here. The first estimate of a superconducting transition temperature Tc appears in the Cooper problem, and is succeeded by the gap in the mean-field BCS equation, and then the gap-function in Eliashberg theory. Approximate solutions to the Eliashberg equations for Tc (e.g., the MacMillan equation or the KMK equation [13]) have proven to be exceedingly useful (though not always accurate) as a guide, since the Eliashberg equations themselves have proven difficult to treat numerically. For some time it was thought, largely on the basis of the MacMillan approximate solution, that there might actually be a bound to Tc, no matter what metallic materials were selected. With the emergence of the class of high temperature superconductors this view point has clearly been subject to revision: This class, discovered by Bednorz and Mueller [14] (but with remarkable progenitors including those reported by Kenjo and Yajima [15]) contain elements, and groupings of elements with high dipole polarizabilities and with distinct layering characteristics. Because of the light masses of the lower Za elements (and hence, correspondingly, high dynamic energy scales for phonons), these appear attractive for superconductivity according to the amplitudes taken up the second term in (12). If structures lead to multiple bands (with hole bands in particular) then by the Kohn–Luttinger arguments advanced above, the prospects might be even more favorable. Starting with (10), the fundamental problems for the first four elements (hydrogen, helium, lithium, and beryllium) are contained in the statements Z Z b ¼ Tb n þ Tb e þ 1 H dr dr0 vc ðr  r0 Þ 2 X X

ð2Þ 0 ^nð1Þ ðrÞ^ ^ð2Þ ^n ðr; r0 Þ  q qeð1Þ ðr0 Þ þ q ð13Þ q e ðr; r Þg Z Z b ¼ Tb n þ Tb e þ 1 H dr dr0 vc ðr  r0 Þ 2 X X

ð2Þ 0 ^ð2Þ qnð1Þ ðrÞ^ qeð1Þ ðr0 Þ þ q ð14Þ 4^ qn ðr; r0 Þ  4^ e ðr; r Þg

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Z Z b ¼ Tb n þ Tb e þ 1 H dr dr0 vc ðr  r0 Þ 2 X X

ð2Þ 0 ð1Þ ^eð2Þ ðr; r0 Þg qn ðrÞ^ qeð1Þ ðr0 Þ þ q 9^ qn ðr; r Þ  6^ and Z Z 1 b b b H ¼ Tn þ Te þ dr dr0 vc ðr  r0 Þ 2 X X

^eð2Þ ðr; r0 Þg qnð1Þ ðrÞ^ qeð1Þ ðr0 Þ þ q 16^ qnð2Þ ðr; r0 Þ  8^

ð15Þ

ð16Þ

With the possible exception of (14), all can be considered in combination via (11), for cases where some electrons may take up itinerant (metallic) states, and others may participate in localized (molecular) states leading to the polarization waves as discussed above. Under ordinary conditions beryllium, described by (16), is a superconductor, with a transition temperature of Tc = 0.026 K. Given that the temperature To representative of its ionic dynamics (i.e., of Tb n in (16)) is so high (1000 K), the low value of Tc is traced mainly to the low value of electronic density of states at the Fermi energy (around 1/5 of the free-electron value). If this were to be increased by a factor x, all other quantities remaining approximately unchanged, then in a BCS scaling approach, the resulting Tc (say Tcx) would be roughly expected to follow T cx ¼ T o ðT c =T o Þ

1=x

so that for a threefold increase in the density of states (still well below the free electron value), the transition temperature would rise to some 30 K. Casual estimates of the superconducting transition temperature are known to be highly unreliable; this example merely illustrates the merit of pursuing low mass systems, but also in seeking combinations leading to large densities of states at the Fermi energy. The first metal in the periodic table under ordinary conditions is also a light element. Lithium, in its BCC room temperature structure has traditionally been thought of as a typical nearly free-electron metal. In the structural and transport range the electron–ion pseudopotential is relatively weak under normal conditions, and hence the interaction between electrons and the ions in motion (the phonons) is also relatively weak. It is a single band system and for simple structures the prospects for superconductivity were not thought to be particularly positive. But at low temperatures lithium takes up the d  Sm, or 9R structure, and this offers a boost in Umklapp transition possibilities. But even at ordinary conditions, lithium (Eq. (15)) represents an almost unique test case for the theory of superconductivity. At most densities plausibly accessible by experiment two of the three electrons per atom of lithium will be in states, typical of helium (Eq. (14)), and simple in form. By the traditional nomenclature the third will be in a 2s-like state, provided conditions permit a nuclear-centric view of the valence electron structure. As with 1s, this state leads to a high electron density at the lithium nucleus (either 6Li or 7Li) and hence to the possibility of a coupling between any eventual electronic and nuclear order at low tempera-

tures. Quite recent experiments confirm [16] that superconducting order is actually now observed at low temperatures, raising the number of metals attaining a superconducting state at one atmosphere from 29 to 30. The recorded transition temperature of 0.4 mK (at ordinary pressures) was in fact calculated in Ref [7] (in 1997) in an Eliashberg approach where the monopole (phonon) fluctuations were included subsequent to full treatment of the electronic terms (see above). As noted the structure taken up by lithium at low temperatures is not simple and leads to complexities in its band structure not present in the high-temperature BCC phase. But in terms of the viewpoint taken above, and in concert with experiment this can be seen as a considerable opportunity (at least via the Eliashberg approach) to refine the understanding of the purely electronic effects in this phenomenon. The situation in lithium changes dramatically at high pressure, where the bands begin to narrow in a completely non nearly-free-electron way. The reason is traced to the fact as mentioned above that under compression the core-space occupies an increasing fraction of cell volume, and the valence electrons, which must be orthogonal to, and largely excluded by, the cores states, are increasingly forced into interstitial regions. Eventually the very separation of core and valence states will become subject to challenge, but inter alia the ensuing non-uniform distribution of valence electron charge is a breakdown of the paradigm which views simple metals as mildly perturbed nearly free electron gases. The electronic response can hardly be ascribed to the standard linear methodology. The electron-ion pseudopotential becomes increasingly strong, and as a consequence of band narrowing there is the immediate expectation that the density of states at the Fermi energy increases significantly. Compressional work is ending up in part in a stiffening of the lattice. All these arguments lead to a suggestion, somewhat radical for an alkali metal: at high pressures, lithium should be considered for superconductivity particularly by the phonon mechanism for simple structures. Within a relatively short time of this suggestion being made [17] lithium was indeed found to be a superconductor, with significantly high transition temperatures [18]. Neither boron nor magnesium are superconductors at one atmosphere, to quite low temperatures. But the ordered intermetallic MgB2 is; in terms of the general guidelines outlined above it is interesting to note that once again the number of electrons per unit cell (8) and the structure cooperate in such a way that the Fermi surface is in several sheets (which can be important in determining whether more than a single gap eventuates) and Umklapp processes are again prevalent. From the perspective of boron the bands are no longer full or inactive as in the element itself; from the perspective of the magnesium, the valence electron density has been increased by a factor of 2.66 over the density for the element itself. In a sense chemical compression is assisting in the formation of a superconducting state.

N.W. Ashcroft / Physica C 468 (2008) 115–125

The phenomenon of band-narrowing at high pressures in otherwise simple metals is also revealed in calcium [19], now exhibiting superconductivity at a remarkable 25 K (and Tc appears to be still rising with pressure at the last data point). It is obviously too early to establish a pattern, but it does seem clear that further systematic investigations into intermetallic combinations of the light elements is called for. The point is that the sheer number of such compounds is extraordinary [20] and as with MgB2 for many of these the structures taken up again lead to large numbers of electrons per unit cell. This can be favorable for four reasons: first, because as indicated earlier of the concomitant rise expected in Umklapp processes by the phonon mechanism (and experimentally verifiable through the rapidity of the rise in static resistivity with temperature above Tc ), and second, because of the grouping of possibly highly polarizable ion cores or other localized orbitals and the pairing channel offered by polarization waves. Third, because of bandstructures that admit of hole bands and the subsequent weakening of the Morel–Anderson (Coulomb) pseudopotential appearing in MacMillan’s approximation for Tc. Fourth, some of the structures taken up by the light element intermetallics have a decidedly layered character, leading to electron distributions which begin to favor the intrinsic pairing tendencies found in the strict two-dimensional electron gas discussed earlier. As has recently been shown for the binary CaLi2 [21], the electronic structure of systems composed of ostensibly simple constituents can be far from simple, and this can lead to interesting insights on the physics of the Coulomb pseudopotential. This system has recently been shown to exhibit interesting superconductivity at high pressure [22]. The lightest of all elements is, of course, hydrogen. The principal vibron in the familiar hydrogen molecule has an energy of around 1/2 eV. This gives an indication of the dynamical scale that might eventually emerge when condensed hydrogen is sufficiently compressed, in fact to a metallic state. Experimentally, relative maximum compressions are at around 12.4 and it is variously estimated that a compression of about 13.6 is required to achieve metallization. Given the sheer scale of proton dynamical energies and the fact that lacking a core space (and hence also core-polarization), the electron–proton interaction suffers no pseudopotential reduction (from which it is inferred that the coupling of electron and proton dynamics can be very strong) the prospects for high temperature superconductivity [23] in metallic hydrogen continue to appear very favorable. This may include superconductivity in a fluid state of metallic hydrogen, where the system (a dual electronic/protonic Fermi fluid) is a formal two-component superconductor with properties fundamentally different from those of a one-component superconductor [24]. Yet the pressures required to achieve these compressions are undoubtedly high, and it might actually be asked whether metallic states of hydrogen rich combinations with the light elements (utilizing the notion of chemical pre-compression) could be yet another route to high temperature superconductivity.

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Exactly along these lines the group IV hydrides have been suggested as possibilities, for example SiH4 [25], which from the standpoint of testing the predictive capabilities of modern theory is especially attractive because of the systematic deuterations that are possible, i.e., SiDH3, SiD2H2, SiD3H and SiD4. Since Sn is already a superconductor at one atmosphere, the system SnH4 (stannane) is also attractive [25,26] and it may be mentioned, apropos the discussion above, that the ion has a significant polarizability. It is interesting to note that if metallic silane, for example, is regarded as a reasonably dilute alloy of metallic silicon in metallic hydrogen, and if, as proposed, metallic hydrogen is a liquid in its ground state, then the thermodynamic consequences on the pressure dependent melting point of silane (and indeed any of the metallic highhydrides) could be considerable. For in continuous alloy A–B systems, at fixed pressure, the addition of A to B commonly depresses the melting point of B (and vice versa). If B here is hydrogen which already has a zero temperature melting point, the addition of A (silicon) may be expected to lead to unusually low melting points of the stoichiometric alloy. Observation of this would lead to evidence for the existence of a ground state liquid phase for metallic hydrogen. The ambit of hydrogen rich light element systems is exceedingly wide and includes the unusual hexahydride WH6 as well as the metal ammines [5,25], which are rife with light elements, also situated in arrangements with high polarizabilities. For example, in the stoichiometric system Li(NH3)4 (which appears to have the lowest melting point of any metallic system) a small band gap separates the lithium 2s band states from the bound states, many of which are attributable to H itself. If these are brought by pressure, or further chemistry, into the conduction band then the hydrogens may well become very active players in subsequent superconductivity. It might be noted that similar arguments can be brought to bear, for example, in the diborane equivalent of lithium tetra-ammine and even in Li (BH3)2(NH3)2. To repeat, a major property of such systems is the presence of highly polarizable groupings. Finally, it should be repeated that the effects of external pressure may in some cases be mimicked by appropriate chemical substitution. The application of pressure to the original 214 cuprate superconductor had a noticeably positive effect, and the subsequent substitution of La by Y (similar valences but with differing core sizes) led to a class of superconductors with substantially larger transition temperatures [27]. In the same way it may be suggested some light element systems, promising for superconductivity at high pressures (and even perhaps including carbon [30]) might be incorporated as subsystems within even larger unit cells, the entire arrangement providing the requisite chemical pre-compressions necessary for the maintenance of superconducting states at near ambient conditions. So far as is known at present the theory of superconductivity places no especial limits on Tc. What is required as

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illustrated above are structures, dynamic energy scales and sources of polarization which combine in particularly propitious ways, and the light elements appear well suited to this. In this respect it may be seen as noteworthy that silicon in the diamond structure [28] and carbon in the diamond structure [29] can both be induced, by modest alloying with light elements to take up superconducting states. Though the transition temperatures are not high, so far, the dynamical energies can be considerable and might well be exploited in extended combinations (e.g., LiBC [30]) and in larger unit cell systems. As has been emphasized, light element intermetallics appear favored for the quest for high temperature superconductivity, but not solely because of the propitious phonon energy scales. Remarkable electronic structure, particularly of a quasitwo-dimensional character is being revealed in systems otherwise thought as ‘‘simple’’. A recent example is the Li–Be system [31] which appears to form alloys at high pressure, but again with unusual atomic groupings. In these, all of the polarization related factors favoring superconductivity as discussed above may enter. It is clear that extensions to ternary systems, especially those involving hydrogen and other light elements, are an evident next step. Finally, the light element Li (and Na also) is well known to undergo reversible transformations at low temperatures involving the coherent cooperative displacements of ions (Martensitic transformations). The transformation temperatures in Li so far exceed any superconducting transition temperatures. However, in other light element systems or even subsystems (for example oxygen in the cuprate class of superconductors), it is interesting to consider more general pathways in which the coupling of order parameters (structural and pairing) might well ensue.

where ao is Bohr’s radius and T is given in Kelvin. It follows that if equilibrium can be verifiably established in an ultra-cold and ultra-rarified gas in a demonstrable thermodynamic limit at, say, moderately massive atoms kTh  105ao, which is considerably in excess of inter-particle spacing in such systems J 103ao. Extraordinary quantum correlation is then anticipated.  ¼ N =X, a mean atomic volFrom the average density q ume can be identified and assigned a linear measure ro according to ð4p Þr3 ¼ X=N . It follows that the mean inter3 o particle spacing is 2ro, classical conditions can then be specified as approximately those for which 2ro  kTh. These usually apply for the low-density phase of (1) possessing continuous symmetry ðh^ qð1Þ ðrÞi ¼ constant ¼ N =XÞ identified as the normal gas, and for which in (1) the average of the kinetic energy h Tb i considerably exceeds that of the average of the potential energy h Vb i. As is readily confirmed the symmetries of (1) (translational and rotational) are abundantly broken when temperature declines sufficiently; the near ground state low temperature phases of most systems are solids and often periodic crystals with transitions between different crystalline orderings also frequently occurring. The crystalline solid normally conforms to the condition h Tb i  h Vb i. As noted, for crystalline phases the average of the one-particle density operator is periodic, i.e., h^ qð1Þ ðr þ RÞi ¼ h^ qð1Þ ðrÞi, for all sites {R} of a lattice. In terms of linear dimensions, the mean interparticle separations for the crystalline phases have declined by about an order of magnitude compared with gas phase. Of obvious importance in the superconductor problem viewed as a liquid–liquid transition in an electronic system is that in proceeding from gas to a condensed phase there must also be an insulator to metal transition.

Acknowledgement

References

This work was supported by the National Science Foundation, under Grant DMR-0601461.

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Appendix A It should not be assumed, of course, that the gas phase cannot support states exhibiting off-diagonal long-range order. If the system is taken as canonical and in an assumed Þ, thermodynamic limit ðN ! 1; X ! 1 and N =X ! q and also in equilibrium with a heat bath fixing a temperature T, then the quantities h, m and kB T immediately establish a length  h/{mkBT}1/2, more familiarly cast as the thermal de Broglie wavelength kTh, which, from the statistical physics of merely the kinetic part of (1) is given by  2  2 h =2m ð2p=kTh Þ ¼ pk B T  If the mass number of the atom or molecule is A (so that its mass is 1836meA) then kTh =ao ¼ 38:5=ðAT ½K Þ1=2

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