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Concept# Fermi surface

Summary

In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state. The study of the Fermi surfaces of materials is called fermiology.
Theory
Consider a spin-less ideal Fermi gas of N particles. According to Fermi–Dirac statistics, the mean occupation number of a state with energy \epsilon_i is given by
:\langle n_i\rangle =\frac{1}{e^{(\epsilon_i-\mu)/k_{\rm B}T}+1},
where,
*\left\langle n_i\right\rangle is the mean occupation number of the i^{th} state
*\epsilon_i is the kinetic energy of the i^{th

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Though the high superconducting transition temperature (Tc) is the most interesting technological aspect of high temperature superconductors, the complex way in which the spin, lattice and electronic degrees of freedom interplay, makes them of the highest scientific interest. This is illustrated by their rich phase diagram characterized by a variety of exotic groundstate including; Mott insulating, pseudogap and high temperature superconductivity. One of the most serious barriers that has prevented our understanding of the mechanism of high temperature superconductors thus far is the lack of information on how low energy Landau quasiparticles result from an organization of real particles. This organization, often colloquially referred to as "dressing", is a fundamental general concept in physics that explains a variety of physical phenomena, such as exotic particle formations and phase transitions. The role of the lattice in this dressing is particularly controversial. Phonons are quanta of lattice vibration energy, and play a crucial role in conventional superconductivity. They provide an attractive interaction allowing the electrons to condensate in superconducting Cooper pairs. However, high temperature superconductivity in the cuprates in achieved through hole doping in an antiferromagnetic Mott insulator. In this case, the antiferromagnetic background, the strong coulomb repulsion and the anisotropic superconducting gap all suggest a marginal role of the phonons. In order to assess experimentally the exact role of the phonon, angle resolved photoemission spectroscopy (ARPES) is the weapon of choice given its unique ability to probe the electronic structure of solids in an energy- and momentum-resolved manner. In this thesis, I will use ARPES to characterize the various form of dressing of the quasiparticles in the high temperature superconductor from the Fermi level all the way to the valence band complex. I'll show that when combined with isotope substitution and inelastic x-rays scattering, the strong electron-phonon interaction can be probed in vivid details. This thesis presents three fundamental results. 1) Using the isotope effect, the important and unusual role of the phonon is demonstrated, as well as the strong interplay between the magnetic and phonic degrees of freedom. 2) Using inelastic scattering, the intriguing interplay between the Fermi surface and the bond stretching phonon is exposed. And 3) By exploring the ARPES data up to high energy, two new energy scales at high binding energy as well as the spectral waterfalls phenomenon are revealed. When all these new elements are considered together, they clearly show the need to consider simultaneously the spin, lattice, Fermi surface topology and electronic degrees of freedom. The spectacular progress in ARPES techniques over the past two decades was critical for these results, and I will present in this thesis our current effort in pushing the techniques even further. In particular I'll present my efforts in building a laser based ARPES setup with a hemispherical analyzer and a spin resolved time of flight analyzer.

This thesis is devoted to a theoretical study of high-temperature superconductivity from the viewpoint of a doped Mott insulator. To this end, the square-lattice t-J model is analyzed by variational and mean-field approaches. The thesis focuses on the construction of excitations and on spectral properties in the framework of Anderson's concept of resonating-valence-bond wavefunctions. The quantum dimer model as a toy model for the resonating-valence-bond phase of Mott insulators is also explored. In the first part of the thesis, the single-particle Green's functions in the superconducting phase are analyzed using Gutzwiller-projected variational wavefunctions for the t-J model. It is found that the overall spectral weight is reduced by a momentum-dependent renormalization, and that the projection produces a particle-hole asymmetry in the renormalization of the spectral weights. The second part analyzes the Green's functions in the pseudogap phase of the cuprates within an SU(2) mean-field approach where the order parameter fluctuates between the d-wave superconductor and the non-superconducting staggered-flux state. The model predicts a photoemission spectrum with an asymmetric gap structure interpolating between the superconducting gap centered at the Fermi energy and the asymmetric staggered-flux gap. This gap asymmetry changes sign at the "hot-spots" where the Fermi surface crosses the diagonal (0,π)-(π,0). In the last part of the thesis, single hole and vortex excitations in the liquid phase of the triangular-lattice Rokhsar-Kivelson quantum dimer model are considered. It is found that the motion of a hole bound to a topological excitation is strongly constrained due to interference effects.

We use an SU(2) mean-field theory approach with input from variational wave functions of the t-J model to study the electronic spectra in the pseudogap phase of cuprates. In our model, the intermediate-temperature state of underdoped cuprates is realized by classical fluctuations of the order parameter between the d-wave superconductor and the staggered-flux state. The spectral functions of the pure and the averaged states are computed and analyzed. Our model predicts a photoemission spectrum with an asymmetric gap structure interpolating between the superconducting gap centered at the Fermi energy and the asymmetric staggered-flux gap. This asymmetry of the gap changes sign at the points where the Fermi surface crosses the diagonal (pi,0)-(0,pi).

2009