1. Electronic Properties of Metals A.The Free Electron Gas (FEG) Model B.Properties of the FEG: Fermi Energy E F and Density of States N(E) C.Plasma Oscillations and Plasmons D.Heat Capacity of the FEG

2. A. The Free Electron Gas Model Having studied the structural arrangements of atoms in solids, and the thermal and vibrational properties of the lattice, we now consider the electronic properties of solids in terms of a very simple model. Plot U(x) for a 1-D crystal lattice: Simple and crude finite- square-well model: Can we justify this model? How can one replace the entire lattice by a constant (zero) potential? U U = 0

3. Assumptions of the FEG Model 1. Metals have high electrical conductivity and no apparent activation energy, so at least some of their electrons are “free” and not bound to atoms 4. Therefore model the behavior of the free electrons with U = 0 inside the volume of the metal and a finite potential step at the surface. Assume each atom has n 0 free electrons, where n 0 = chemical valence. Assume that resistance comes from electrons interacting with lattice through occasional collisions: 2. Coulomb potential energy of positive ions U  1/r is screened by bound electrons and is weaker at large distances from nucleus 3. Electrons would have lowest U (highest K) near nuclei, so they spend less time near nuclei and more time far from nuclei where U is not changing rapidly.

4. B. Properties of the FEG: Fermi Energy and Density of States Solutions have 1. Wave functions: 2. Energies: Time-independent Schrödinger Equation: With U = 0: Traveling waves (plane waves) Parabolic energy “bands” E kxkx

5. Properties of the FEG This shows that the volume in k-space per solution is: By using periodic boundary conditions for a cubic solid with edge L and volume V = L 3, we define the set of allowed wave vectors: And thus the density of states in k-space is: Just as in the case of phonons! Since the FEG is isotropic, the surface of constant E in k-space is a sphere. Thus for a metal with N electrons we can calculate the maximum k value (k F ) and the maximum energy (E F ). kxkx kzkz kyky Fermi sphere

6. Fermi Wave Vector and Energy And the maximum energy is easily found: Taking into account the spin degeneracy of the electron, N electrons will be accommodated by N/2 energy states, so: These quantities are called the Fermi wavevector and Fermi energy in honor of Enrico Fermi, who (along with Arnold Sommerfeld) did the most to apply quantum mechanics to calculate the properties of solids in the late 1920s.

7. Density of States N(E) We often need to know the density of electron states, which is the number of states per unit energy, so we can quickly calculate it: Now using the general relation: we get: The differential number of electron states in a range of energy dE or wavevector dk is: This allows:

8. Reality of the Fermi Energy The valence bandwidth is in reasonable agreement with the FEG prediction of E F = 11.7 eV There are several spectroscopic techniques that allow the measurement of the distribution of valence electron states in a metal. The simplest is soft x-ray spectroscopy, in which the highest-lying core core level in a sample is ionized. Only higher-lying valence electrons can fall down to occupy the core level, and the spectrum of emitted x-rays can be measured: E F  13 eV

9. Utility of the Density of States We can simplify by using the relation: With N(E) we can immediately calculate the average energy per electron in the 3-D FEG system: Why the factor 3/5? A look at the density of states curve should give the answer: N(E) E EFEF

10. C. Plasma Oscillations and Plasmons If we take seriously the existence of the FEG, we might expect it to exhibit collective oscillations if it is disturbed. Consider a cylindrical metal sample with an induced polarization in its FEG: Induced dipole moment x polarized: FEG displaced by x x Induced polarization We can now find the induced E field: L unpolarized An E field provides a restoring force for displaced electrons

11. Plasma Oscillations and Plasmons Now for a single free electron we can write Newton’s second law: harmonic oscillator! To find the oscillation frequency, compare to the equation of motion of the mass-spring system: This reveals that the plasma frequency is given by: The quantum of “plasma energy” ħ  p is called a plasmon. Experiments with electron beams passing through thin metal foils shows that they lose energy in integer multiples of this energy quantum.

12. Experimental Evidence for Plasmons Metal Expt.  E (eV) Calc. ħ  p (eV) Li7.128.02 Na5.715.95 K3.724.29 Mg10.610.9 Al15.315.8 What trend do you see? Can you think of an explanation? Hint: look at equation for  p !

13. D. Heat Capacity of the FEG So the electronic contribution to the molar heat capacity would be expected to be or expressed per mole: This is half of the 3R we found for the lattice heat capacity at high T. But experiments show that the total C for metals is only slightly higher than for insulators—which conflicts with the classical theory! 19 th century puzzle: each monatomic gas molecule in sample at temperature T has energy, so if the N free electrons in a metal make up a classical “gas” they should behave similarly.

14. Heat Capacity of the Quantum-Mechanical FEG where  = chemical potential  E F for kT << E F Quantum mechanics showed that the occupation of electron states is governed by the Pauli exclusion principle, and that the probability of occupation of a state with energy E at temperature T is: +

15. Heat Capacity of the Quantum-Mechanical FEG So at temperature T the total energy is: The exact answer to this complicated integral is derived in more advanced texts: And the electronic heat capacity is: C el  T ! + +

16. A Rough and Ready Estimate # electrons that can absorb thermal energy We can estimate C el in just a few lines in order to confirm the linear dependence on temperature: N(E) E EFEF N(E)f(E)  2kT total thermal energy of electrons at T FEG heat capacity at T Remarkably close to the exact result! But this linear dependence is impossible to measure directly, since the heat capacity of a metal has two contributions. Now for a metal at low temperatures we can write the total heat capacity: Assuming we can measure C(T) for a metal, how can we test this relationship?

17. Heat Capacity of Metals: Theory vs. Expt. at low T Very low temperature measurements reveal: Metal  expt  FEG  expt /  FEG = m*/m Li1.630.7492.18 Na1.381.0941.26 K2.081.6681.25 Cu0.6950.5051.38 Ag0.6460.6451.00 Au0.7290.6421.14 Al1.350.9121.48 Results for simple metals (in units mJ/mol K) show that the FEG values are in reasonable agreement with experiment, but are always too high: The discrepancy is “accounted for” by defining an effective electron mass m* that is due to the neglected electron-ion interactions