Superconductivity in the alkali metals

At ambient pressure there are 29 elemental superconductors in the periodic table, none of which is an alkali metal. The first alkali metal to become superconducting under high pressure is Cs followed years later by Li. Alkali metals are believed to be exemplary free-electron systems. The fact that an alkali metal becomes superconducting at all is surprising and is a result of the fact that under pressure it shows marked deviations from free-electron behaviour where, counterintuitively, bands narrow and gaps widen. For this reason the alkali metals are among the most interesting systems known to study in high-pressure experiments and superconductivity is one of their most fascinating properties.     

Electron-electron interaction and instabilities in one-dimensional metals

The possible instabilities of a 1-dimensional itinerant electron gas are discussed, assuming electron-electron interaction to play the dominant role. As is well known, in the RPA, a 1-dimensional metal is prone to spin density wave (SDW), charge density wave (CDW) and Cooper pair (CP) instabilities. The spin channel decomposition of the irreducible scattering amplitude I is made and the spin channel projections are evaluated in terms of the matrix elements of bare electron-electron interactionV(x) for momenta of interest. It is found that if the bare electron interactionV(x) is repulsive and decreases monotonically with separation, only the SDW instability will occur. If the small separation (x?(2k _F )^?1) part of the interaction is greatly reduced or is made attractive,V(x) is non-monotonic,V _q (q?2k _F ) is negative, and a CDW instability is preferred. A CP instability is possible if the electron interaction is attractive,i.e., if [V _q (0<q<k _F )+V _q (q?2k _F )]<0. The above RPA results serve only as rough indicators, since in general there are important two-electron configurations with two-electron momentum close to zero and with electron hole momentum close to 2k _F , an example being the near Fermi energy configurationk ₁?k _F ,k ₂??k _F ,k ₃??k _F k ₄?k _F . Therefore as pointed out first by Bychkov, Gorkov and Dzhyaloshinskii (BGD), cross channel coupling is especially significant. It is shown that the cross channel coupling is constructive is some cases,eg., exchange of CD fluctuations leads to an effective electron-electron spin singlet attraction and vice-versa. A formalism for studying such effects is set up, and the particular example mentioned above is discussed. An RPA-like approximation is made for the form of the reducible singlet electron hole scattering amplitudeγ _s ^d and the resulting induced Cooper pair attraction is calculated to be $$\begin{gathered} [I_s ^e ]_{ind.} \rho _{{}^\varepsilon F} = [ln(\lambda \beta \omega _c )]^{ - 1} ln\{ [1 + 2\pi ^{ - 1} ln(\lambda \beta \omega _c )^2 ]/ \hfill \\ 1 + [8\pi ^{ - 1} \gamma _s ^d (q = 2k_F )^{ - 1} )^2 ]\} \hfill \\ \end{gathered} $$ where λ=1.14,β=(k _B T)^?1 andω ₀ is an electronic energy cut-off ～ε _F . The induced electron hole attraction due to the exchange of virtual Cooper pairs has a similar expression, but with a factor of (1/4) and withγ _s ^e (q=0) replacingγ _s ^d (q=2k _F ). The induced Cooper pair attraction is seen to be quite large over a broad range of temperatures close to but aboveT _CDW [i.e., aboveT such thatγ _s ^d (q=2k _F )^?1=0]. There is no requirement thatγ _s ^d (q=2k _F ) andγ _s ^e (q=0) become singular at the same temperature, as found by BGD. The BGD prediction is seen to arise from the neglect of real particle hole and particle-particle excitations while calculatingγ _s ^d andγ _s ^e . The effect of impurities, of electron-phonon coupling, of interchain coupling and of interaction between thermal order parameter fluctuations is discussed. The results are then applied to a discussion of the properties of TTF-TCNQ, where it is suggested that a CDW instability occurs becauseV _q (q=2k _F )<0,i.e., because the small separation electron repulsion is strongly reduced by the highly polarizable TTF. Because of substantial interchain coupling, the bulk CDW instability occurs close to the RPA instability temperature. The giant conductivity observed by Colemanet al is attributed to superconductive fluctuations in a 1-dimensional system with large mean field superconductive transition temperatureT _CP ^MF of order 300°K. Such a largeT _CP ^MF is shown to result from the induced Cooper pair attraction due to CD fluctuation exchange.     

Relaxation round vacancies in the alkali metals

The long-range ionic displacements round a vacancy are shown to come from both the elastic, long-wavelength, limit and from the Kohn anomaly. The effect of the topology of the Fermi surface is stressed.Attention is then focussed on the alkali metals with almost spherical Fermi surfaces, where it can reasonably be assumed that the elastic displacements dominate. Complete relaxation around a vacancy relates formation energy E_v, bulk modulus B and atomic volume Ω through E_v = αBΩ where α is a constant. The formation volume follows as a balance between a negative harmonic contribution and a positive and larger anharmonic term.     

Saturation of the resistivity in metals with lattice instabilities

We correlate s-shape resistivity in A-15, C-15 and charge density wave compounds with structural instabilities. We propose that the exponential term of the resistivity is caused by the scattering against thermally excited molecular states. We show that this interpretation is supported by a large number of experimental data and naturally explains the apparent violation of Mathiessen's rule in irradiation experiments. Connections with recent theoretical developments on strong electron-phonon coupling are discussed.     

On the compressibility of alkali metals

An analysis of experimental P-V-T data for Na and Cs is given in terms of pseudopotential theory. It is shown that for these metals the contribution to compressibility due to the higher-order terms in electron energy practically does not depend on the position of the ions.     

Defect parameters in the alkali halides

Abstract

Methods are available which calculate the free energies of defects in ionic systems within the quasi-harmonic approximation. The Harwell SHEOL code is used to calculate the defect enthaplies and entropies for a number of alkali halides where reliable diffusion data are known. We discuss the trends in the calculations and the comparison with experiment.     

Thermal resistivity of alkali metals

The temperature dependence of the thermal resistivity of alkali metals lithium, sodium, potassium, rubidium and caesium is determined within the free electron approximation by using Krebs's model for the phonon dispersion relations. The normal and Umklapp contributions are obtained separately from an improved method following the procedure of an earlier paper. The results of calculations are compared with available experimental information. The theoretical and experimental resistivity curves are found to be of similar nature, but they show considerable disagreement at low and high temperatures.

     

Electrical resistivity of alkali metals

The temperature variations of the electrical resistivity of alkali metals lithium, sodium, potassium, rubidium and caesium are calculated in the free electron approximation and using Krebs's model for the phonon spectrum. The double average over the Fermi surface is evaluated by an improved method due toBailyn and the separation between normal and Umklapp processes is affected in a more satisfactory manner. The results of the calculations are compared with the experimental data obtained at different temperatures. The theoretical resistivity curves for sodium, potassium, rubidium and caesium show satisfactory agreement with experiment, but not in the case of lithium.     

The Fermi surfaces of the alkali metals

Long before direct experimental studies of the Fermi surfaces of the alkali metals became technically feasible, evidence had accumulated to suggest that their low-lying energy bands are almost free-electron-like. Indeed, the importance of the free-electron model in the development of the theory of metals owes much t o the coincidence that sodium, now known to be the most free-electron-like of all the alkali metals, was the subject of the pioneering cohesive energy calculations of Wigner and Seitz.¹     

Material parameters for thermoelectric performance

The thermoelectric performance of a thermoelement is ideally defined in terms of the so-called figure-of-meritZ = α²σ/λ, where α,σ and λ refer respectively to the Seebeck coefficient, electrical conductivity and thermal conductivity of the thermoelement material. However, there are other parameters which are fairly good indicators of a material’s thermoelectric ‘worth’. A simple yet useful performance indicator is possible with only two parameters — energy gap and lattice thermal conductivity. This indicator can outline all potentially useful thermoelectric materials. Thermal conductivity in place of lattice thermal conductivity can provide some additional information about the temperature range of operation. Yet another performance indicator may be based on the slope of α vs. ln σ plots. α plotted against ln σ shows a linear relationship in a simplified model, but shows a variation with temperature and carrier concentration. Assuming that such a relationship is true for a narrow range of temperature and carrier concentration, one can calculate the slope m of α vs. ln σ plots against temperature and carrier concentrations. A comparison between the variation ofZT and slopem suggests that such plots may be useful to identify potential thermoelectric materials.     

Anisotropy of electron-phonon interactions in alkali metals

Electron-phonon scattering in the solid alkalis is distinguished from that in most other metals by a combination of three circumstances: The phonon spectra and structure factors are very anisotropic, the Fermi surface in the reduced zone is simply connected and virtually spherical and important large momentum transfers (0.7<(q/2k _F)<1.0) fall within the first large peaks of the phonon structure factors. Anisotropy of microscopic contributions to the macroscopic coefficients is controlled by and is quite sensitive to values of electron-ion matrix elements at large momentum transfer, and can be explored by a realistic yet relatively simple theoretical calculation. A brief summary is presented of such calculations, for the all alkalies, of mean free paths, thermoelectric powers, and electron-phonon mass enhancements. The results show marked anisotropy only for lithium, are consonant with experimental low field Hall coefficients and in addition indicate strong anisotropy in the mass enhancement for lithium.