Condensed Matter Physics: Quantum Statistics & Electronic Structure in Solids Read: Chapter 10 (statistical physics) and Chapter 11 (solid-state physics) Homework due Wednesday Nov. 12th Chapter 10: 1, 3, 6, 9, 11, 20, 23, 25, 29, 35

Quantum Statistics In quantum physics, identical particles must be treated as indistinguishable – this affects counting of states In quantum physics there are two types of particles – fermions (half-integer spin, e.g. electrons) and bosons (integral spin, e.g. photons) Bosons and Fermions obey different statistics

Different statistics Fermions Bosons Obey the Pauli Exclusion Principle – no two particles can have the exact same quantum numbers Distribution function  

Differences fermions bosons Only one particle per quantum state allowed Cannot all “condense” to E=0 Must fill up to the “Fermi Energy” ABE does not depend strongly on temperature The only states that have any probability at low temperatures are those at E=0 for which the exponential approaches 1 This is Bose-Einstein Condensation

Free Electron Gas in Metals Solid metals are bonded by the metallic bond One or two of the valence electrons from each atom are free to move throughout the solid All atoms share all the electrons. A metal is a lattice of positive ions immersed in a gas of electrons. The binding between the electrons and the lattice is what holds the solid together

Fermi-Dirac “Filling” Function Probability of electrons to be found at various energy levels. For E – EF = 0.05 eV  f(E) = 0.12 For E – EF = 7.5 eV  f(E) = 10 –129 Exponential dependence has HUGE effect! Temperature dependence of Fermi-Dirac function shown as follows:

Free Electron Gas in Metals The number of electrons in the interval E to E+dE is therefore The first term is the Fermi-Dirac distribution and the second is the density of states g(E)dE

Free Electron Gas in Metals From n(E) dE we can calculate many global characteristics of the electron gas. Here are just a few The Fermi energy – the maximum energy level occupied by the free electrons at absolute zero The average energy The total number of electrons in the electron gas

Free Electron Gas in Metals The total number of electrons N is given by The average energy of a free electron is given by

Free Electron Gas in Metals At T = 0, the integrals are easy to do. For example, the total number of electrons is

Free Electron Gas in Metals The average energy of an electron is This implies EF = k TF defines the Fermi temperature

Summary of metallic state The ions in solids form regular lattices A metal is a lattice of positive ions immersed in a gas of electrons. All ions share all electrons The attraction between the electrons and the lattice is called a metallic bond At T = 0, all energy levels up to the Fermi energy are filled

Heat Capacity of Electron Gas By definition, the heat capacity (at constant volume) of the electron gas is given by where U is the total energy of the gas. For a gas of N electrons, each with average energy <E>, the total energy is given by

Heat Capacity of Electron Gas Total energy In general, this integral must be done numerically. However, for T << TF, we can use a reasonable approximation.

Heat Capacity of Electron Gas At T= 0, the total energy of the electron gas is For 0 < T << TF, only a small fraction kT/EF of the electrons can be excited to higher energy states Moreover, the energy of each is increased by roughly kT

Heat Capacity of Electron Gas Therefore, the total energy can be written as where a = p2/4, as first shown by Sommerfeld The heat capacity of the electron gas is predicted to be