Band Theory of Electronic Structure in Solids Read: Chapter 10 (statistical physics) and Chapter 11 (solid-state physics) Homework due Wednesday Nov. 11th Chapter 10: 1, 3, 6, 9, 11, 20, 23, 25, 29, 35 Reminder: Exam 2 on Monday, November 16th

Band Theory of Solids To understand the electronic states of a solid we must consider the presence of many N~1023 atoms The band theory of solids describes the interaction between the electrons and the lattice ions that comprise a solid

Band Theory: “Bound” Electron Approach For the total number N of atoms in a solid (1023 cm–3), N energy levels split apart within a width E. Leads to a band of energies for each initial atomic energy level (e.g. 1s energy band for 1s energy level). Two atoms Six atoms Solid of N atoms Electrons must occupy different energies due to Pauli Exclusion principle. Phys 320 - Baski

Band Theory of Solids Consider the potential energy of a 1-dimensional solid which we approximate by the Kronig-Penney Model

Band Theory of Solids The task is to compute the quantum states and associated energy levels of this simplified model by solving the Schrödinger equation 1 2 3

Band Theory of Solids For periodic potentials, Felix Bloch showed that the solution of the Schrödinger equation must be of the form and the wavefunction must reflect the periodicity of the lattice: 1 2 3

Band Theory of Solids By requiring the wavefunction and its derivative to be continuous everywhere, one finds energy levels that are grouped into bands separated by energy gaps. The gaps occur at The energy gaps are basically energy levels that cannot occur in the solid 1 2 3

Band Theory of Solids Completely free electron electron in a lattice

Band Theory of Solids When, the wavefunctions become standing waves. One wave peaks at the lattice sites, and another peaks between them. Ψ2, has lower energy than Ψ1. Moreover, there is a jump in energy between these states, hence the energy gap

Band Theory of Solids The allowed ranges of the wave vector k are called Brillouin zones. zone 1: -p/a < k < p/a The theory can explain why some substances are conductors, some insulators and others semi conductors

Fermi-Dirac “Filling” Function Probability of electrons to be found at various energy levels. Important factoid: 300K (room temp) = 25.86meV Temperature dependence of Fermi-Dirac function shown as follows:

Conductors, Insulators, Semiconductors Sodium (Na) has one electron in the 3s state, so the 3s energy level is half-filled. Consequently, the 3s band, the valence band, of solid sodium is also half-filled. Moreover, the 3p band, which for Na is the conduction band, overlaps with the 3s band. So valence electrons can easily be raised to higher energy states. Therefore, sodium is a good conductor

Conductors, Insulators, Semiconductors NaCl is an insulator, with a band gap of 2 eV, which is much larger than the thermal energy at T=300K Therefore, only a tiny fraction of electrons are in the conduction band

Conductors, Insulators, Semiconductors Silicon and germanium have band gaps of 1 eV and 0.7 eV, respectively. At room temperature, a small fraction of the electrons are in the conduction band. Si and Ge are intrinsic semiconductors

Summary In solids, the discrete energy levels of the individual atoms merge to form energy bands Energy gaps arise in solids because they contain standing wave states The size of the energy gap between the valence and conduction bands determines whether a substance is a conductor, an insulator or a semiconductor