Solid State Physics 3

1. Solid State Physics3 Section 10-4,6

2. Topics • Heat Capacity of Electron Gas • Band Theory of Solids • Conductors, Insulators and Semiconductors • Summary

3. Special Extra Credit As can be seen from the graph, the prediction fails at very low temperatures. This is due, in part, to the failure of the equipartition theorem at low temperatures. Challenge: create a better model!

4. Special Extra Credit Derive the temperature dependence of R/R0 by computing the average potential energy <E> of a lattice ion assuming that the energy level of the nth vibrational state is rather than En = ne as Einstein had assumed Due: before classes end

5. 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

6. 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.

7. 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

8. 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

9. Heat Capacity of Electron Gas Consider 1 mole of copper. In this case Nk = R For copper, TF = 89,000 K. Therefore, even at room temperature, T = 300 K, the contribution of the electron gas to the heat capacity of copper is small: CV = 0.018 R

10. Band Theory of Solids So far we have neglected the lattice of positively charged ions Moreover, we have ignored the Coulomb repulsion between the electrons and the attraction between the lattice and the electrons The band theory of solids takes into account the interaction between the electrons and the lattice ions

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

12. 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

13. 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

14. 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

15. Band Theory of Solids Completely free electron electron in a lattice

16. 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

17. Band Theory of Solids The allowed ranges of the wave vector k are called Brillouin zones. zone 1: -p/a < k < p/a; zone 2: -2p/a < - p/a; zone 3: p/a < k < 2p/a etc. The theory can explain why some substances are conductors, some insulators and others semi conductors

18. 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

19. 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

20. 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

21. Summary • The heat capacity of the electron gas is small compared with that of the ions • 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