Dugdale, S. B. (2016). Life on the edge: A beginner's guide to the Fermisurface. Physica Scripta, 91(5), [053009]. DOI: 10.1088/0031-8949/91/5/053009

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Life on the edge: a beginners guide to the Fermi surface

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Invited Comment

Life on the edge: a beginners guide to theFermi surface

S B Dugdale

H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK

E-mail: s.b.dugdale@bristol.ac.uk

Received 30 November 2015, revised 22 March 2016Accepted for publication 30 March 2016Published 18 April 2016

AbstractThe concept of the Fermi surface is at the very heart of our understanding of the metallic state.Displaying intricate and often complicated shapes, the Fermi surfaces of real metals are bothaesthetically beautiful and subtly powerful. A range of examples is presented of the startlingarray of physical phenomena whose origin can be traced to the shape of the Fermi surface,together with experimental observations of the particular Fermi surface features.

Keywords: electronic structure, Fermi surface, metals, positron annihilation, Compton scattering,nesting, quantum oscillations

(Some figures may appear in colour only in the online journal)

1. Introduction

For most of us, our first contact with a Fermi surface isthrough the Sommerfeld free electron model, a model inwhich it has a spherical shape. In real metals, however, theFermi surface can be (and normally is) very different from asphere (for example, the Fermi surface of Pb is shown infigure 1). Indeed, fantasies of a modern artist is how Lifshitzand Kaganov describe the diverse forms that the Fermi sur-face has been shown to exhibit [1, 2], and the lexicon is full ofexotic sounding names such as superegg and tetracube [3]. Inthe 50 years since Allan Mackintosh suggested that defining ametal as a solid with a Fermi surface might be the mostmeaningful description that one can give [4], the concept ofthe Fermi surface has deservedly achieved great prominencein undergraduate physics courses, often with substantial focuson methods which can reveal its shape in real metals (e.g. themeasurement of quantum oscillations [5]). While there is anappreciation that electrons at (or within an energy ~k TB of)the Fermi surface are special because of the Pauli exclusion

principle preventing electrons from deep in the Fermi seafrom being easily excited (and thus making any contributionto the transport properties), many students (and even seniorcolleagues) are left wondering about the physical significanceof particular Fermi surface shapes or features.

The importance of the Fermi surface can be extended tothe influence its shape can have on the ability of electrons toscreen perturbations. From the oscillatory exchange couplingat the heart of giant magnetoresistance through to shapememory phenomena and even superconductivity, the Fermisurface can influence a range of fascinating physical phe-nomena. As Kaganov and Lifshitz put it, the Fermi surface isthe stage on which the drama of the life of the electron isplayed out [2]. The purpose of this article is to showcasesome of the drama that can result from particular shapes of theFermi surface.

1.1. A potted history of the Fermi surface

Soon after the emergence of quantum mechanics in the 1920s,the following decade saw its application to the problem ofunderstanding the behaviour of electrons in solids [7]. Theresult, the so-called band theory of solids, stands proudly asone of the theorys great early achievements. Beginning withSommerfelds successful free electron model, a quantum

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version of the earlier classical Drude theory, the next advancewas Blochs discovery, obtained by straightforward Fourieranalysis, that letting the electrons feel a periodic potentialresulted in wavefunctions which were still delocalised, butdiffered from free-electron plane waves by a modulatingfunction which had the periodicity of the lattice. These Blochelectrons, whose wavefunctions were just modulated planewaves extending over the whole crystal, were described asbeing nearly free. A further consequence of the Bloch theorywas the opening up of energy gaps (due to Bragg reflection ofthe electron waves) at the Brillouin zone boundary. It was atthis point that Wilson had the insight to explain the differencebetween metals and insulators [8, 9]. Although both containelectrons which are nearly free, in insulators the electrons arein fully filled bands, and a fully filled band cannot carry acurrent. Wilson was also to explain a semiconductor as aninsulator in which the band gap was comparable to k TB . Asolid with partially filled bands, on the other hand, will be aconductorand have a Fermi surface. Thus, while the Blochtheory explained why electrons feeling a periodic potentialcould remain delocalised (nearly free), Wilson could explainwhy not all solids are metals by showing that insulators arequalitatively different from metals, and not merely badmetals.

The Fermi surface is the surface in reciprocal spacewhich separates occupied from unoccupied electron states atzero temperature. The dynamical properties of an electron onthe Fermi surface largely depend on where it is on the Fermisurface, and the shape of the Fermi surface with respect to theBrillouin zone can be a guide to the electrical properties of ametal [10].

A comprehensive and highly influential review bySommerfeld and Bethe on the electron theory of metals waspublished in the Handbuch der Physik [11]. This article,described by Mott as astonishingly complete [12] containeda number of sketches of Flchen konstanter Energie in Raumder Wellenzahlen, surfaces of constant energy in the space ofwavevector components. Bethe later recalled [13]: It wasclear to me K that it made a great difference whether theFermi surfaces were nearly a sphere or were some interestingsurface. At the University of Bristol, Mott had recently beenappointed to the chair of theoretical physics, and considered ithis job to apply quantum mechanics to the experimental workthat was in progress there [12]. Mott, and his Bristol collea-gues, Jones and Skinner, began to wonder about those energysurfaces, and whether they were merely a mathematical fic-tion given that the nearly free electron theory ignored theCoulomb interaction between electrons [12]. In a real metal,would these energy surfaces be smeared out? Skinnersmeasurements of x-ray emission, performed with OBryan,answered that question by showing very sharp cut-offs in theemission intensity at high energy, indicating that the energysurface did remain sharp [14]. At the same time, the termi-nology in a paper by Jones and Zener [15] evolves from apurely mathematical concept (the surface of the Fermi dis-tribution function) at the beginning of the article to the morephysically tangible Fermi surface by the end. The termFermi surface stuck. It would, however, take another twenty

years before Landaus theory of the Fermi Liquid and theexperimental observation of the Fermi surface of Cu wouldput the existence of a sharp Fermi surface on firm ground.Today, the observation of a F