1 Condensed Matter Physics: Quantum Statistics & Electronic Structure in Solids Read: Chapter 10 (statistical physics) and Chapter 11 (solid-state physics) Homework due Wednesday Nov. 4 th Only 5 problems: Krane Chapter 9 8, 14, 17, 20, 22 No class on Monday, Nov. 2 nd !

2 Structural Properties of Solids l Condensed matter physics: l The study of the electronic properties of solids. l Crystal structure: l The atoms are arranged in extremely regular, periodic patterns. l The set of points in space occupied by atomic centers is called a lattice.

3 Structural Properties of Solids l Most solids are polycrystalline: they’re made up of many small crystals. l Solids lacking any significant lattice structure are called amorphous and are referred to as “glasses.” l Why do solids form as they do? l When the material changes from the liquid to the solid state, the atoms can each find a place that creates the minimum-energy configuration. In the sodium chloride crystal, the spatial symmetry results because there is no preferred direction for bonding. The fact that different atoms have different symmetries suggests why crystal lattices take so many different forms.

4 Example of an Ionic Solid: NaCl Face-centered cubic (fcc)

5 Metallic Bonds l In metals, in which electrons are very weakly bound, valence electrons are essentially free and may be shared by a number of atoms. The Drude model for a metal : a free-electron gas!

6 Free Electron Gas in Metals l Solid metals are bonded by the metallic bond l One or two of the valence electrons from each atom are free to move throughout the solid l 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

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

8 Free Electron Gas in Metals Since we are dealing with fermions, the electron gas obeys Fermi-Dirac statistics. The number of electrons with energy in the range E to E+dE is given by where, the density of states g(E) is given by

9 Free Electron Gas in Metals The electron has two spin states, so W = 2. If the electron’s speed << c, we can take its energy to be So the number of states in (E, E+dE) is given by

10 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

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

12 Free Electron Gas in Metals The total number of electrons N is given by the integral At T=0, the Fermi-Dirac distribution is equal to 1 if E E F. At T=0, all energy levels are filled up to the energy E F, called the Fermi energy.

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

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

15 Free Electron Gas in Metals The average energy of an electron is This implies E F = k T F defines the Fermi temperature

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

17 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, the total energy is given by

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

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

20 Heat Capacity of Electron Gas Therefore, the total energy can be written as where  =  2 /4, as first shown by Sommerfeld The heat capacity of the electron gas is predicted to be