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1A. Boson structure in highly correlated metals

Long term goal:Understand the mechanism of pairing in cuprates

Short term goal:Determine the electron-boson spectral density in cuprates

This project has involved the development of mathematical tools [1-10] aimed at the analysis of experimental data to recover information on the interactions which scatter the charge carriers in these systems. This means dealing with the inelastic scattering which is both temperature and frequency dependent.  While we have mainly dealt with optical data [11-15], we have shown how similar information can be extracted from Raman [6], and angular resolved photo emission [ARPES] [5].  Optics is a bulk technique and measures an average over all the electrons involved in the charge transport. Raman is somewhat more selective and preferentially sample electrons around nodal or antinodal directions.  By contrast ARPES allows measurements for each momentum k separately, but probes only the first atomic layer, so it is never certain that it is truly representative of the bulk. Of course, as has been emphasised in particular by Anderson [16], it is not guaranteed that the effect of correlations can be modelled by a boson exchange approach. Nevertheless for the superconducting cuprates, there is a whole body of experimental evidence that boson structure is indeed involved [17]. This fact is not controversial although the origin and nature of the bosons involved remains an open question.  Two possible candidates are phonons and antiferromagnetic spin fluctuations with the preponderance of evidence in favour of the latter, possibility with a smaller contribution from phonons [18-20].  On the theoretical side, the nearly antiferromagnetic Fermi liquid of Pines and co-workers is a concrete example of a phenomenology developed for the oxides which contains explicitly a boson exchange mechanism employing the spin fluctuations. We have written a review of the entire field (Reports on Progress in Physics 74, 066501 (2011)). While I am a theorist, this work is soundly based in experimental investigations. In all this work it is assumed that the underlying electronic density of state can be treated as constant in the energy range of importance. This has restricted its application to the optimal and overdoped region of the phase diagram.

 

1B. Gap symmetry in the Fe-pnictide

Long term goal: Understand the gap symmetry in Fe pnictides

Short term goal: Calculate critical current and other transport properties

This project involves attempts to pin down the form of the superconducting gap in these materials i.e. its momentum dependence.  This problem is complicated by the fact that these systems can involve, in an important way, many different bands.  A minimal model which has enjoyed some success and is sufficiently simple to provide physical insight, is a two band model [21-26]. In its simplest formulation the gap in each of the two bands would be isotropic s-wave but with a sign as well as magnitude difference between them. Another possibility is that, one or even both gaps retains s-wave symmetry but is [26-29] anisotropic and even could have a zero on its Fermi surface.  Should this be the case, impurity scattering which has the effect of homogenizing the gap over its Fermi surface, could lift the node and produce a small gap.  There are many experiments that hint to this possibility and some of our work has dealt directly with this problem mainly within the context of the optical conductivity and the thermal conductivity [30].  Of course, the temperature dependence of the penetration depth [31] is also sensitive to gap symmetry. For isotropic s-wave, the low temperature behaviour is exponentially activated but some   experiments show instead power laws. This is more consistent with anisotropic s-wave.

Another problem that we have worked on is that of the so-called “collapse of the inelastic scattering on opening a superconducting gap”.  This is a well- known phenomenon in the cuprates and it is believed to be generic to any mechanism for superconductivity which is electronically driven [32].  The basic idea is that the excitation spectrum of a metallic system which becomes superconducting also becomes gapped and if it is involved in the inelastic scattering, this scattering will be greatly reduced at low temperatures.  This has important effects including the observation of a large peak in the microwave conductivity as a function of temperature, quite distinct from the usual coherence peak of BCS theory near Tc. The thermal conductivity also displays a similar peak.  Such peaks are present in the Fe-pnictides and they can be interpreted in the same model providing strong evidence for an electronic mechanism in this family of superconductors. Our analysis of both thermal and microwave conductivity in Ba[1-x]K[x]Fe2As2 also provides additional evidence for a large anisotropic gap in at least one of the bands. Certainly the data cannot be understood with isotropic s-wave gaps on all the bands involved.

1C. Properties of Graphene

Long term goal: Understand the exotic properties of 2-d membranes

Short term goal: calculate optical properties of graphene and MoS2 with correlation

 Graphene  is a single layer of carbon atoms arranged on a honeycomb lattice.  This arrangement has two atoms per unit cell and consequently two bands in its Brillouin zone.  Each site of the direct lattice is chemically the same but there is a topological difference between the sites.  This leads naturally to the idea of two sub lattices. The main interaction is nearest neighbour hopping between atoms on different sub lattices.  As a consequence, the band structure displays two Dirac cones, for valence and   conduction band, which meet at a point in the Brillouin zone [BZ].There are two Dirac crossings at the K and –K corners of the BZ. The dynamics of the quasiparticles is governed by the relativistic Dirac equation [33-35] rather than by the more common Schrödinger equation, and this has many very important and exciting consequences.  To mention one, the energies of the quantization of the Landau levels (LL) in a magnetic field B go like the square root of B rather than the usual linear law.  These lead to unusual magneto-optical properties [36-39]]. We have also studied the effect of the electron-phonon interaction [40-42] on the properties of graphene.  While many of its properties can be understood qualitatively within a bare band model, there are indications that many body renormalizations play a critical role in some circumstances. One expects to see in optical properties, a Drude response at low photon energy (w) (a peak centred about zero) followed by a region of low conductivity and a rapid rise towards a universal background at w = 2µ₀ [twice the chemical potential].  The Drude comes from the intraband transitions while the universal background comes from the interband optical transitions between valence and conduction Dirac cones.  The bare band picture sees the region between these two contributions as Pauli blocked, with little absorption, while experiment finds a considerable amount, about one third of the universal background value [43].  This has been interpreted to be due to electron-phonon effects as well as possibly electron-electron renormalizations.

 

1D. The Underdoped Cuprates

Long term goal: Identify the correct theory of the pseudogap

Short term goal: Compare properties of models of the pseudogap with data

This topic refers to the underdoped part of the cuprate phase diagram where a pseudogap phenomenon is observed.  The origin of these effects and how they are to be described accurately in an appropriate mathematical formulation remains controversial.  Recently, a very appealing proposal has been put forward by Yang Rice and Zhang [44] [referred to here as the YRZ model].  This model is based on a resonating valence bound spin liquid theory.  An appeal of this formulation is that the end result is an anzats for the self-energy due to correlations, which is sufficiently simple that properties can easily be  calculated.  In some cases analytic results are possible.  I like to view this model as a generalization of BCS theory which contains an additional essential element related to the formation of a pseudogap as the Mott insulating state is approached by reducing the doping [x]. In the model there is a quantum critical point at x = x_c below which the pseudogap is finite and above which it is zero.  In this second case we recover the large Fermi surface of the Fermi liquid theory with a band structure which still depends on Gutzwiller factors.  These narrow the bands as correlation effects become more pronounced, and double occupancy of an atomic site is forbidden because of a large on-site Hubbard repulsion U.  Below x = x_c, a pseudogap opens and this leads to a reconstruction of the Fermi surface into Luttinger pockets centred around the antinodal direction.  As the doping is reduced towards zero, the size of the Luttinger pockets progressively shrink and the metallic response (Drude response), which depends on the existence of ungapped excitations on the contours that define the Luttinger pockets, is also progressively reduced.  We have calculated several properties [45-49] including in-plane penetration depth, Raman response, specific heat, optical response as well as ARPES and FT-STS (Fourier transform scanning tunnelling spectroscopy).  A general conclusion of these studies, as well as parallel studies by Yang Rice and Zhang and co-workers, is that the model is able to account simply and in a compellingly fashion for many of the observed properties of the superconducting under doped cuprates.  Until now, many of these were considered anomalous and difficult to understand, even qualitatively.  As an example, two distinct gaps are observed in Raman scattering when B_1g and B_2g spectra are separately considered.  Another is that the slope of the linear low temperature dependence of the penetration depth is not much affected by underdoping, while the magnitude of its zero temperature value is strongly reduced with decreasing x.  Also, the range of the linear in T dependence persists up to much higher temperatures relative to its critical temperature T_c. For the specific heat, its low temperature behaviour remains largely unchanged by the opening of the pseudogap while the jump at T=Tc is greatly reduced.

 

 

 

 

 

 

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