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Solid State Physics 2

Solid State Physics 2. Section 10-3,4. Topics. Recap Free Electron Gas in Metals Quantum Theory of Conduction Heat Capacity of Electron Gas Summary. Recap. The number of particles (here electrons) with energy in the small interval E to E+dE is given by. E+dE E. where,.

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Solid State Physics 2

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  1. Solid State Physics2 Section 10-3,4

  2. Topics • Recap • Free Electron Gas in Metals • Quantum Theory of Conduction • Heat Capacity of Electron Gas • Summary

  3. Recap The number of particles (here electrons) with energy in the small interval E to E+dE is given by E+dE E where, is the number of states in that interval and f (E) is the probability that these states are occupied

  4. Recap The electron has two spin states, so W = 2. Assuming the electron’s speed << c, we can take its energy to be in which case the number of states in the interval E to E+dE is given by

  5. 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 number of states g(E)dE

  6. Free Electron Gas in Metals From n(E) dE we can calculate many global characteristics of the electron gas. Here are a few • The Fermi energy – the maximum energy level occupied by the free electrons at absolute zero • The average energy • The equation of state; that is, the equivalent of PV = nRT for the electron gas

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

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

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

  10. Free Electron Gas in Metals At T > 0, only the electrons near the Fermi energy can be excited to higher energy states

  11. Free Electron Gas in Metals Their energy increases by about kT, which at T = 300 K is only about 1/40 eV, to be compared with EF ~ 8 eV

  12. Quantum Theory of Conduction In a very large (strictly infinite) perfect crystal, calculations show that electrons suffer no scattering. That is, their mean free path is infinite. In real crystals, electrons scatter off imperfections and the thermal vibrations of the lattice ions

  13. L A Quantum Theory of Conduction Mean Free Path – This is the average distance traveled between collisions. Consider a box of length L and cross-sectional area A that contains n particles per unit volume. Suppose each particle presents a cross sectional area a. What is the probability of a collision between a single incident particle, of negligible size, and a particle within the box?

  14. L A Quantum Theory of Conduction Mean Free Path – A collision is guaranteed to occur when Pr = 1. This occurs for a particular value of L = l, called the mean free path, where and <v>, the average speed of the incident particle, defines the relaxation timet

  15. L A Quantum Theory of Conduction Resistance – For a wire of length L and cross sectional area A, the electrical resistance can be written as where r, the resistivity, is inversely proportional to the mean free path r = C / l.

  16. Quantum Theory of Conduction Classically, lattice ions are modeled as spheres of cross-sectional area pr2. In the quantum theory, we model ions as points vibrating in three dimensions and that present a cross section of where r is the oscillation amplitude of the ions

  17. Quantum Theory of Conduction For a simple harmonic oscillator, the potential energy is Its average value is kT, assuming that the equipartition theorem holds, at least approximately, that is,

  18. Quantum Theory of Conduction The mean free path is therefore So, in this simple model, quantum theory predicts that the resistivity is proportional to the temperature

  19. 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!

  20. 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

  21. Heat Capacity of Electron Gas At T= 0, the total energy of the electron gas is At T > 0, the number of electrons that can be excited to higher energy states is roughly and each is raised in energy by roughly kT

  22. 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

  23. Summary • The energy distribution of an electron gas does not vary much until the temperature is near and above the Fermi temperature • The free electron gas model predicts a resistivity that is proportional to the temperature • The heat capacity of the electron gas is small compared with that of the ions

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