Metals I: Free Electron Model

1. Metals I: Free Electron Model Physics 355

2. + + + + + + + + + + + + + + + + + + + + + + + + + Free Electron Model Schematic model of metallic crystal, such as Na, Li, K, etc. The equilibrium positions of the atomic cores are positioned on the crystal lattice and surrounded by a sea of conduction electrons. For Na, the conduction electrons are from the 3s valence electrons of the free atoms. The atomic cores contain 10 electrons in the configuration: 1s22s2p6.

3. Free Electrons? • How do we know there are free electrons? • You apply an electric field across a metal piece and you can measure a current – a number of electrons passing through a unit area in unit time. • But not all metals have the same current for a given electric potential. Why not?

4. Paul Drude • resistivity ranges from 108 m (Ag) to 1020 m (polystyrene) • Drude (circa 1900) was asking why? He was working prior to the development of quantum mechanics, so he began with a classical model: • positive ion cores within an electron gas that follows Maxwell-Boltzmann statistics • following the kinetic theory of gases- the electrons in the gas move in straight lines and make collisions only with the ion cores – no electron-electron interactions. (1863-1906)

5. Paul Drude • He envisioned instantaneous collisions in which electrons lose any energy gained from the electric field. • The mean free path was approximately the inter-ionic core spacing. • Model successfully determined the form of Ohm’s law in terms of free electrons and a relation between electrical and thermal conduction, but failed to explain electron heat capacity and the magnetic susceptibility of conduction electrons. (1863-1906)

6. Ohm’s Law E Experimental observation:

7. Ohm’s Law: Free Electron Model Conventional current The electric field accelerates each electron for an average time  before it collides with an ion core.

8. Ohm’s Law: Free Electron Model 

9. Ohm’s Law: Free Electron Model If electrons behave like a gas… The mean free time is related to this average speed… typical value About 1014 s Then,

10. Ohm’s Law: Free Electron Model Predicted behavior High T: Resistivity limited by lattice thermal motion. Low T: Resistivity limited by lattice defects. The mean free path is actually many times the lattice spacing – due to the wave properties of electrons.

11. Wiedemann-Franz Law (1853) Electrical Thermal Conductivities where Lorentz number (Incorrect!!)

12. Wiedemann-Franz Law (1853) (Ludwig) Lorenz Number (derived via quantum mechanical treatment)

13. Free Electron Model: QM Treatment  • Assume N electrons (1 for each ion) in a cubic solid with sides of length L – particle in a box problem. • These electrons are free to move about without any influence of the ion cores, except when a collision occurs. • These electrons do not interact with one another. • What would the possible energies of these electrons be? • We’ll do the one-dimensional case first. 0 L

14. Free Electron Model: QM Treatment At x = 0 and at L, the wavefunction must be zero, since the electron is confined to the box. One solution is:

15. Free Electron Model: QM Treatment

16. Free Electron Model: QM Treatment

17. Free Electron Model: QM Treatment Chemical Potential If an electron is added, it goes into the next available energy level, which is at the Fermi energy. It has little temperature dependence. m Fermi-Dirac Distribution For lower energies, f goes to 1. For higher energies, f goes to 0.

18. Free Electron Model: QM Treatment From thermodynamics, the chemical potential, and thus the Fermi Energy, is related to the Helmholz Free Energy: where

19. Free Electron Model: QM Treatment wherenx, ny, and nzare integers

20. Free Electron Model: QM Treatment and similarly for y and z, as well

21. Free Electron Model: QM Treatment Energy Fermi Energy Velocity

22. Free Electron Model: QM Treatment • Each value of k exists within a volume • The number of states inside the sphere of radius kF is • This successfully relates the Fermi energy to the electron density.

23. Free Electron Model: QM Treatment million meters per second Fermi Temperature

24. Free Electron Model: QM Treatment Density of States

25. Free Electron Model: QM Treatment The number of orbitals per unit energy range at the Fermi energy is approximately the total number of conduction electrons divided by the Fermi energy.

26. Free Electron Model: QM Treatment This represents how many energies are occupied as a function of energy in the 3D k-sphere. As the temperature increases above T = 0 K, electrons from region 1 are excited into region 2.