Superconductivity and the Van Hove Scenario
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- Bok, J. & Bouvier, J. J Supercond Nov Magn (2012) 25: 657. doi:10.1007/s10948-012-1434-3
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We give a review of the role of the Van Hove singularities in superconductivity. Van Hove singularities (VHs) are a general feature of low-dimensional systems. They appear as divergences of the electronic density of states (DOS). Jacques Friedel and Jacques Labbé were the first to propose this scenario for the A15 compounds. In NbTi, for example, Nb chains give a quasi-1D electronic structure for the d-band, leading to a VHs. They developed this model and explained the high TC and the many structural transformations occurring in these compounds. This model was later applied by Jacques Labbé and Julien Bok to the cuprates and developed by Jacqueline Bouvier and Julien Bok. The high TC superconductors cuprates are quasi-bidimensional (2D) and thus lead to the existence of Van Hove singularities in the band structure. The presence of VHs near the Fermi level in the cuprates is now well established. In this context we show that many physical properties of these materials can be explained, in particular the high critical temperature TC, the anomalous isotope effect, the superconducting gap and its anisotropy, and the marginal Fermi liquid properties, they studied these properties in the optimum and overdoped regime. These compounds present a topological transition for a critical hole doping p≈0.21 hole per CuO2 plane.
KeywordsSuperconductivityHigh TCVan Hove scenarioDensity of states
After World War II, Solid State Physics was practically non-existent in France. Two places were at the origin of a revival of the field, Ecole Normale Supérieure (ENS) in Paris with Pierre Aigrain and the University of Orsay with Jacques Friedel, André Guinier and Raymond Castaing.
A postgraduate course was also created in the Parisian areas called “troisième cycle” in Solid State Physics. There were two poles, one in Orsay with J. Friedel and A. Guinier later joined by Pierre-Gilles de Gennes and another at ENS with P. Aigrain, later replaced by J. Bok.
The great majority of solid state physicists working in France are former students of this DEA (Diplôme d’Etudes Approfondies). One of the authors (J. Bok) is very grateful to J. Friedel for his constant confidence. We collaborated for more than 30 years in maintaining a very high level in this research school in solid state physics.
In this paper we shall review the work of Labbé, Friedel and Barišić on the A15 and our work on cuprates in the framework of the Van Hove scenario.
2 Friedel et al. Work on Supraconductivity and Van Hove Singularity
The original approach of Friedel et al. is the following: The elastic properties of most metals and metallic compounds are generally weakly dependent on temperature. In the A3B (V3Si, Nb3Sn), called A15 compounds, on the other hand relatively high superconducting critical temperatures TC (16–18 K) are observed simultaneously with structural instabilities. This fact, coupled especially with the occurrence among these same materials of structural phase transformations known to involve lattice vibrational instabilities, has created a lively interest in the lattice dynamics and in particular the nature of electron–phonon interaction in materials with this structure. The basic idea, first proposed by Labbé and Friedel and since elaborated by various authors, is that one is dealing with a nearly empty set of d-electron bands whose degeneracy is removed by the elastic strain. The re-equilibration of the electrons within the split bands of the strained structure can in favorable conditions balance the normal increase in elastic energy. This destabilizing d-electron contribution is one, however, which increases with decreasing temperature as the Fermi surface becomes sharper.
From the 1960s to the 1980s Jacques Friedel was collaborating with Jacques Labbé, and Slaven Barišić [1–15, 19]. They studied the instability of the cubic phase of intermetallic compounds of the type V3Si at very low temperature. There idea was that a Jahn–Teller type of effect for the d-band electrons can explain this instability. When the temperature is raised the effect is reduced and above a certain temperature TM, it is compensated by the action of the conduction electrons, which can reasonably be supposed to stabilize the cubic phase. It is at this temperature TM that the change of structure from tetragonal-to-cubic phase occurs. They explained the martensitic transition observed at low temperature in experiment.
At absolute zero temperature the Fermi level is supposed to lie in a region of quickly changing density of states. The deformation of the crystal reduces the degeneracy of the electron spectrum and a large change of the DOS occurs at the Fermi level. For a certain range of initial Fermi level positions this can be accompanied by the decrease of the free energy and thus by a phase transition. Furthermore such a situation will be temperature sensitive; thus one can expect a strong temperature dependence of the associated quantities, in particular of the coefficients in the series expansion of the free energy in terms of the strains of the cubic lattice.
In their model they used arbitrary values for the DOS at the Fermi level and for the bandwidth, but the lattice parameters, the shear modulus, and the variation of the elastic constant with temperature were taken from experiments [16, 17]. They concluded in favor of a first order transition.
In the 1980s, the discovery of high critical superconducting temperature in layered-metal oxides TC=35 K, as La2−xAxCuO4 (A=Ba, Sr), by Bednorz and Müller  raised once again the questions concerning the interplay between the structural and the superconducting instabilities. These new superconductors are tetragonal. But La2CuO4 is orthorhombic at ambient temperature and shows a metal-to-insulator transition below 100 K, and with the doping x. The properties of these materials come essentially from the CuO2 plane. Friedel et al.  proposed a simple electron-phonon model for these new compounds. The predictions of the model are the tetragonal-to-orthorhombic instability and separately from it, the bond dimerization within the layers. The high TC in those materials is associated with the soft shear branch in their tetragonal state, the main contribution to the electron-phonon coupling in ionic metals is attributed to the deformation-induced variation of the crystal field on the ionic sites which are involved in conduction.
The work of Friedel et al. is very unique and profound. It gave a good description of the properties of the A15 and it opened a new approach of superconductivity which was later developed by a new generation of researchers.
3 VHs Physics in 1, 2, 3 Dimensions—Electronic Structure of the Cuprates
It is now well accepted that the origin of cuprate superconductivity is to be found in the CuO2 planes, which are weakly coupled together in the c direction, so that their electronic properties are nearly two dimensional. There were examples of superconducting cuprates with high TC to which we applied our model: Bi2Sr2CaCu2O8 (Bi 2212), YBa2Cu3O7 (Y123), YBa2Cu4O8 (Y124) and Nd2−xCexCuO4+δ (NCCO). For low oxygen content (no doping) all copper ions in this plane are Cu++ ions, the material is an anti-ferromagnetic insulator due to strong electron-electron repulsion on the same copper site. With additional oxygenation or doping, holes are introduced in the CuO2 planes and the compound becomes conducting and superconducting for T<TC. The maximum TC is achieved when the hole content is around 16% per Cu atom.
This approach is not valid for the underdoped region. The strong Coulomb repulsion U between two electrons on a same site is responsible for the fact that with p=0 the cuprates are Mott insulators with antiferromagnetic (AF) order. The AF order disappears rather rapidly with doping, but AF fluctuations remain, and decrease, until the optimum doping. This region of strong correlations is present and the valid approach is that of a doped Mott insulator . This is also seen in ARPES; some points of the Fermi surface disappear for underdoped samples.
4 Calculation of TC with Electron–Phonon Interaction
4.1 Calculation of TC Using the BCS Approach
The Fermi level lies at the Van Hove singularity.
- 2.The BCS approach is valid:
The electron–phonon interaction is isotropic and so is the superconducting gap Δ.
The attractive interaction Vp between electrons is non-zero only in an interval of energy ±ħω0 around the Fermi level where it is constant. When this attraction is mediated by emission and absorption of phonons, ω0 is a typical phonon frequency.
As it is, however, this approach already explains many of the properties of the high TC cuprates near optimum doping.
4.2 The Variation of TC with Doping
4.3 Influence of the Coulomb Repulsion
Cohen and Anderson  assumed that for stability reasons μ is always greater than λ. Ginzburg  gave arguments that in some special circumstances μ can be smaller than λ. Nevertheless if we take μ≥λ, superconductivity only exists because μ∗ is of the order of μ/3 to μ/5 for a Fermi energy EF of the order of 100 ħω0. It is useless to reduce the width of the band W, because λ and μ vary simultaneously and μ∗ becomes greater if EF is reduced, thus giving a lower TC. Superconductivity can even disappear in a very narrow band if λ−μ∗ becomes negative.
(λ−μ∗) is replaced by the square root (λ−μ∗)1/2.
μ∗ is reduced compared to μ because the renormalization is controlled by the width W of the broad band and not the singularity.
The prefactor before the exponential in the formula giving TC is the width of the singularity D instead of the phonon energy ħω0.
5 Anomalous Isotope Effect
6 Non-Fermi Liquid Properties
In a classical Fermi liquid, the lifetime broadening 1/τ of an excited quasiparticle goes as ε2 and the resistivity ρ varies as T2. The marginal Fermi liquid situation is the case where 1/τ goes as ε (electronic energy) and ρ is linear in T. In the half-filled nearest-neighbor coupled Hubbard model on a square lattice, Newns et al. [38, 39] have shown that this can also occur when EF is close to ES. This calculation was, however, contradicted by Hlubina and Rice .
6.2 Hall coefficient
At low temperature T, RH≈1/ph0e, where ph0 is the hole doping, when T increases RH decreases, and for highly overdoped samples becomes even negative.
These authors are also able to define a temperature T0, where RH changes its temperature behavior, and they found that RH(T)/RH(T0) versus T/T0 is a universal curve for a large doping domain (from ph0=0.10 to ph0=0.27).
6.3 Specific Heat and Electronic Susceptibility
Several experiments on photoemission, NMR and specific heat have been analyzed using a normal state pseudogap . In fact, all that is needed to interpret these data is a density of state showing a peak above the Fermi energy. To obtain the desired DOS several authors  introduce a pseudogap in the normal state. This seems to us rather artificial, the above authors themselves write that the physical origin of this pseudogap is not understood. We have shown that by using a band structure of the form of formula (2), we may interpret the results obtained in the normal metallic state.
The variation with the doping is linked to the distance of the FL from the singularity level (EF−ES), so does the variation with the temperature due to the Fermi–Dirac distribution.
We find a characteristic temperature T0 where the variation of the electronic susceptibility χp versus T goes through a maximum. We may express De=EF−ES as a variation of doping δp=p−p0, p0 being the doping for which EF=ES, p0=0.20 hole/copper atom in the CuO2 plane.
We also explain the shift between the observed experimental optimum TC, where p=0.16 instead of 0.20, and the expected optimum TC from our theory, i.e. where De=0, by the fact that at first in our gaps calculations we have not taken into account the variation of the 3D screening parameter q0a in function of De. These calculations show the competition between the effect of the position of the VHs and the value of q0a for getting the optimum TC, this competition depends on the compound. When the overdoping is increased, i.e. the density of free carriers increases, then q0a increases too, and in our model this leads to a decrease in TC. It is why for De=0, or p=0.20, we do not have the optimum TC, and why the logarithmic law for χp is found in the overdoped range . In the underdoped range with respect to the observed optimum TC, (i.e. the density of free carriers decreases), q0a decreases too, but the Fermi level goes too far away from the singularity to obtain high TC. Our results agree completely with the experimental observations.
7 Gap Anisotropy
7.1 The Calculation
Bouvier and Bok  have shown that using a weakly screening electron–phonon interaction, and the band structure of the CuO2 planes, an anisotropic superconducting gap is found.
From our theoretical results, we find an effective coupling constant λeff in agreement with the hypothesis of the BCS weak electron-phonon coupling.
This may explain the various values of 2Δ/kBTC observed in various experiments. The critical temperature found is TC=90.75 K as for HTSC cuprates as Bi2Sr2CaCu2O8 (Bi 2212), YBa2Cu3O7−δ (Y123).
We observe of course that TC, Fig. 10, and the gaps, Fig. 18, decrease with the doping from the optimum doping to the overdoped region in these calculations. We obtain also an interesting result, which is the decrease of the anisotropy ratio α with doping [22, 23, 52]. This is confirmed by ARPES results.
7.3 Effect of the Screening on the Gap Anisotropy and TC
We show that the effect of increasing q0a is to transform the system in a metallic and more isotropic one.
8 Evidence of Lattice Involvement
Labbé and Friedel [3–5] gave an explanation for the martensitic phase transformation from the cubic to the tetragonal structure observed at low temperature in the A15 compounds of formula V3X (X=Si, Ga, Ge,…) or Nb3Sn. This change of structure occurs at a temperature TM greater than TC. The vanadium (V) atoms form a linear chain and an almost one dimensional approximation can be used for the d-electrons. In these conditions a VHs appears at the bottom of the band and can explain high TC [4, 6, 7]. The electronic energy is reduced when the lattice is deformed and leads to a band type Jahn–Teller effect. This effect can explain the observed cubic to tetragonal transition at low temperature. This effect does not change TC very much in these A15 compounds, because the role of the high DOS due to the VHs is important only for small doping (low concentration of d-electrons).
The situation is more favorable in the cuprates, which are almost bidimensional and where the VHs lies near the middle of the band. Far from or near TC, lattice deformations, tetragonal-to-orthorhombic phase transformations, deformation of the orthorhombic phase, even martensitic phase transformations, have been observed in the cuprates in function of temperature, doping, substitution, or under strain [27, 53–57]. This leads to a competition between electronic and elastic energies. Evidence of the role of phonon in the physics of cuprates has been seen experimentally, see for example the paper of Graf et al. .
When the Fermi level lies close to a VHs, of energy ES, as is the case for cuprates near optimum doping, the situation could be unstable and a small distortion increases the distance EF−ES and decreases strongly the electronic energy.
The goal for experimentalists will be to find the optimal parameters (doping, strain, temperature,…) to lead the sample to such situation that it condensates when EF is pinned in its dip in order to obtain a very high TC.
We want to indicate in favor of the electron-lattice interaction that de Gennes and Deutscher  proposed a model valid in the underdoped regime based on the idea that if two holes occupy two adjacent copper sites, a contraction of the Cu–O–Cu band occurs. This increases significantly the transfer integral between the Cu and this can lead to the formation of bound hole pairs.
Strong correlations are probably the dominant factor in the underdoped region. But in the optimum and overdoped regions, we have shown that the experimental observations may be explained by electron–phonon or electron–lattice interaction coupled with the Van Hove scenario, both in the normal and superconducting states. The existence of VHs close to the Fermi level is now well established experimentally and this fact must be taken into account in any physical description of the properties of high TC superconducting cuprates.
Jacqueline Bouvier personally met Professor Friedel for the first time the day of her thesis defense. He came, invited by one of my researcher friends, and he stayed during the cocktail. She adds: I was very impressed by his long and slim figure and his glittering eyes with strength emerging from him. After this first meeting, we met sometimes in his home to discuss our respective work. He was always very interested and always took into account my own words and capacities. I was very touched by his gentleman behavior regarding me. Until now, we are corresponding to exchange the ideas submitted in our papers. I want to deeply thank him for his intellectual interest in our work and express how I am admiring his lifelong work.