Superconducting transition temperatures of the elements related to elastic constants

For a given crystal structure, say body-centred-cubic, the many-body Hamiltonian H in which nuclear and electron motions are to be treated from the outset on the same footing, has parameters, for the elements, which can be classified as (i) atomic mass M , (ii) atomic number Z, characterizing the external potential in which electrons move, and (iii) bcc lattice spacing, or equivalently one can utilize atomic volume, Ω. Since the thermodynamic quantities can be determined from H , we conclude that Tc, the superconducting transition temperature, when it is non-zero, may be formally expressed as Tc = T (M) c (Z,Ω). One piece of evidence in support is that, in an atomic number vs atomic volume graph, the superconducting elements lie in a well defined region. Two other relevant points are that (a) Tc is related by BCS theory, though not simply, to the Debye temperature, which in turn is calculable from the elastic constants C11, C12, and C44, the atomic weight and the atomic volume, and (b) Tc for five bcc transition metals is linear in the Cauchy deviation C∗ = (C12 −C44)/(C12 +C44). Finally, via elastic constants, mass density and atomic volume, a correlation between C∗ and the Debye temperature is established for the five bcc transition elements. PACS. 74.62.-c Transition temperature variations – 74.70.Ad Metals; alloys and binary compounds 1 Background and outline We have recently been concerned with both empirical and theoretical relations between the superconducting transition temperature Tc of high-Tc cuprates and of heavy Fermion materials [1,2,3]. The generally complex crystallographic structure of such compounds makes it difficult to identify useful correlations between their superconducting properties (such as Tc) and the elastic properties of the lattice. This is not the case of several superconducting elements with a definite and relatively simple crystallographic structure, e.g. characterized by only a few non-zero components of the elastic tensor. Although any such correlation applying to the ‘simple’ superconducting elements may not be immediately generalized to other unconventional superconductors, they are anyway expected to focus on the relevant variables which would be worthwhile studying, both experimentally and theoretically, also in the new classes of superconductors. Following the Bardeen-Cooper-Schrieffer (BCS) theory [4] of the metallic elements, firmly rooted in electronphonon interaction as the basic mechanism resulting in the formation of Cooper pairs, questions have come up regarding the role of strong electron-electron interactions in both the high-Tc cuprates and heavy Fermion systems. Here, our basic philosophy will be to insist that if we were able to solve the many-body Schrödinger equation for the (considered infinite) superconducting materials, then by treating the motion of nuclei and electrons on the same footing, plus full inclusion of electron-electron interactions, such uncertainties involved in separating electronlattice and Coulomb repulsions between electrons would be bypassed. Having said that, let us take as the simplest starting point the metallic elements. Then, the input information into any computer programme to treat these elements would be as follows. First, of course, we should need to specify the structure. To be definite, below we shall single out the body-centred cubic (bcc) lattice, but everything that follows would be equally applicable to the more closely packed face-centred cubic (fcc) structure. Once the structure is specified, one would need to insert the atomic volume Ω (or, of course, essentially equivalently, the lattice parameter a). Then, the external potential created by the nuclei must be specified, which requires the atomic number Z as further input. Since one has a many-body Hamiltonian containing both electron and nuclear kinetic 2 G. G. N. Angilella et al.: Superconducting Tc of elements vs parameters in the full Hamiltonian energies, one needs also the nuclear massM . Of course, we take as obvious the input additionally of the fundamental constants h and e, plus the electronic mass m. The conclusion from the many-body Hamiltonian is therefore that, for a specified structure which we take to be bcc for reasons that will emerge below, the superconducting transition temperature Tc, given from the manybody partition function once the Schrödinger equation has been solved depends, apart from the given fundamental constants h, e and m, on M , Z and Ω, that is Tc = T (M) c (Z,Ω). (1) Of course, for all other classes of superconductors than the metallic elements we have more than one atomic number, possibly the next simplest case being the alkali-doped fullerides (see some brief comments in Section 5). With this as background, the outline of the paper is as follows. Section 2 picks out specifically five bcc superconducting transition elements, W, Mo, Ta, V and Nb. Two more elements, Cr and Fe, have low temperature bcc structures but exhibit cooperative magnetism at low temperatures (antiferroand ferro-magnetism, respectively) and are not superconductors at the lowest temperature they have yet been subjected to. The five elements listed above are considered in the (Ω,Z) plane with respect to their transition temperatures, the reduced isotope effects being taken as evidence that in Eq. (1) there is, at most, a weak and therefore relatively unimportant dependence of Tc on nuclear isotopic mass. Since even then Tc = T (M) c (Z,Ω) presents problems in its representation, Section 3 reintroduces a classification of the above five elements in which Tc is related to the Cauchy discrepancy, i.e. the departure of C12 from C44, where these are two of the three elastic constants (C11 being the other) required to characterize a cubic crystal. Section 4 then returns to an essential ingredient of BCS theory, and by using a semiempirical approach throws light on the way the Cauchy deviation relates to the Debye temperature. Section 5 constitutes a summary, plus some proposals for further studies, both theoretical and experimental, which should prove fruitful. An Appendix considers zero temperature properties, and in particular critical field Hc(0) and energy gap Eg(0), as functions of the Cauchy discrepancy. 2 Dependence of T c on atomic number Z and atomic volume Ω in bcc transition elements In Fig. 1, a plot is made of the positions of the five elements in the (Ω,Z) plane, the values of Tc being attached to these coordinates. That both Ω and Z are important variables in characterizing Tc is immediately apparent. As to the functional form Tc(Ω,Z), one can comment that: (i) For constant atomic volume, Tc markedly decreases with increasing atomic number. (ii) For constant Z, there is plainly substantial variation of Tc with atomic volume, which is proportional to the reciprocal of the concentration. Relevant to such variation is the pressure dependence of Tc 20 30 40 50 60 70 80 5 5.5 6 6.5 7 7.5 8 Z 1 / Ω [ 10 cm ] V (Tc = 5.38 K) Nb (Tc = 9.5 K) Mo (Tc = 0.92 K) Ta (Tc = 4.48 K) W (Tc = 0.01 K) Fig. 1. Shows five superconducting bcc transition elements in the (Ω,Z) plane. Actual abscissa is the reciprocal of the atomic volume, Ω. To each point, the value of the critical temperature Tc is attached. Table 1. Pressure derivatives of Tc [5], experimental bulk moduli [6], and inferred partial derivatives of Tc with respect to Ω at constant Z, Eq. (2), of the bcc superconducting transition metals at P = 0. Element W Mo Ta V Nb ∂Tc ∂P [K/GPa] — −1.4 −2.6 6.3 −2.0 B [GPa] 323 272 200 162 170 ∂Tc ∂Ω [10K ·m] — 2.4 2.9 −7.4 1.9 for a given element, provided one remains within the bcc phase. Despite high pressure can turn many elements into superconductors via an insulator-metal transition, Tc usually decreases with increasing pressure for most superconducting elements at ambient pressure (see Table 1). Within BCS theory [see also Eq. (5) below] or its extension by McMillan, this is usually justified in terms of a pressure-induced lattice stiffening, which reduces the electron-phonon constant at a more rapid rate than the electron density of states at the Fermi level is increased [7]. Pressure derivatives of Tc can then be straightforwardly related to volume derivatives (at constant Z) from the relation ∂ log Tc ∂ logΩ = −B ∂ logTc ∂P , (2) where B is the bulk modulus (see Table 1). However, even given some knowledge of these partial derivatives, the fact that Tc depends apparently in a sensitive way on these two variables for the chosen bcc structure leaves open the detailed form of the function Tc(Ω,Z) for this structure. Therefore in the following section we appeal to a known, but so far rather neglected, correlation between Tc and the Cauchy discrepancy C12−C44 between elastic constants. This is important for our present study, since it is clear that Tc can, in fact, be characterized by a single variable, rather than the pair (Ω,Z) used in Fig. 1. G. G. N. Angilella et al.: Superconducting Tc of elements vs parameters in the full Hamiltonian 3