Nearly Free Electron Approximation Has had success in accounting for many properties of metals:

1. Nearly Free Electron Approximation Has had success in accounting for many properties of metals: Electrical conductivity Thermal conductivity Light absorption of metals Wiedemann-Franz law (universal proportionality) However, until quantum theory was incorporated there were problems with predicting specific heat. Solids rarely have heat capacities greater than 3R, and if the electrons are freely moving, they would contribute 3/2 R to the heat capcity. This is a problem since a great deal of the heat capacity is accounted for by lattice vibrations

2. Model also fails, without Quantum Mechanics, to account for existence of band gaps and insulators Quantum Treatment Assume that the electrons in a lattice act as if they were particles in a box of length L. This is just the particle in a box model used to model translational motion in elementary quantum mechanics The solution to Hy = Ey for this model gives both the energy and the wavefunction where y2 describes the probability that the electron is in some region. For the 1 dimensional model, the solution is yn = A sin (npx/L) or A sin(kx) k = np/L Using Euler’s relation this becomes yn = A/2 [exp (ikx) – exp (-ikx)

3. The wavefunction exp (ikx) is the wavefunction of a free (unbound particle), that is one moving without a potential present, but k is not quantized as it is in the particle in a box The edge of the crystal at x=0 and x=L quantizes k. L is very large compared to the lattice spacing and this means that there will be many many quantum states that are very closely spaced in energy En = n2 h 2 / (8mL2) (Examples)

4. There are two parts of the wavefunction in yn = A/2 [exp (ikx) – exp (-ikx)] exp(ikx) relates to electrons traveling in a positive direction in the lattice exp(-ikx) relates to electrons traveling in a negative direction in the lattice This comes largely from the fact that y is an eigenfunction of the momentum operator as will with momentum p = kh/2p and E = k2 h2/ 8p2me (for a free electron) The model can be extended to three dimensions by: y (x,y,z) = exp (ik r)

5. There are many many electrons in a macroscopic crystal, but there is a vastly greater number of k states. At 0K the electrons will occupy the lowest states available, but as they are fermions, they will obey the Pauli exclusion principle meaning So they will fill up the states to some energy level, the Fermi Energy (EF) the firmi energy. At T>0 the electrons can populate higher states, higher states will be thermally populated according to the Fermi Dirac Law. This picture allows resolution of the heat capacity trouble mentioned before. As seen in figure, only those electrons near the Fermi surface can be promoted to a higher energy. SO only a small number of the electrons contribute to the heat capacity.

6. What about the band gaps? This is largely brought on by the lattice of positively charged nuclei that can interact with the electrons, so essentially the electrons are not free ie (nearly free approximation), but periodically disrupted by an attractive potential. The wavefunction for an electron in a periodic potential is given by: y(x) = exp(ikx) U(x) U(x) is a function that has the periodicity of the lattice. For an electron interacting with a potential, its wave-like properties become crucial. What tells us about the wave-like properties? DeBroglies relation: l = h/p and from before we found the momentum to be p = kh/2p So l = 2p/k Effectively, the periodic potential scatters the electrons whose wavelengths meet the Bragg condition.

7. Need to first look at the Bragg Equation William and Lawrence Bragg developed a method for treating X-ray diffraction treating scattering as reflections from parallel planes of the crystal. Constructive interference occurs when the path difference between waves reflected from the planes is equal to an integral multiple of the wavelength. 2d sin q = n l where d = plane spacing Waves not adding constructively will cancel.

8. So for the present case with lattice spacing “a” 2a sin q = n l = 2pn/k So at 90o the Bragg reflections occur when k = n p / a Again, electrons of a wavelength that satisfies the Bragg condition will be scattered and cannot pass through the lattice. That means that the energies corresponding to the values of k where k = n p / a will not be allowed. There will effectively be discontinuities in the energy. Remember the energy for a free electron depends on k E = k2 h2/ 8p2me A plot of E vs k would be in the shape of a parabola

9. But near the Bragg reflection condition, the parabola is modified, producing band gaps These gaps, are in fact zones in three dimensional k space. They are called the Brillouin Zones, zones where the electron is scattered off of the positive nuclear potential of the atoms making up the lattice. Thus in the Nearly Free Electron Approximation, the band gaps appear due to scattering of the electron waves by the atoms that make up the lattice

10. So lets review how this Model allows us to account for the different types of materials With the model, we can explain insulators as materials for which the Fermi level coincides with the beginning of the band gap Effectively to a k-value which is equal tonp/a Why can’t the electrons below the Fermi-level conduct electricity? Remember that the k values specify something beyond the energy

11. According to this model, the k values also specify both the momentum and the direction of the electrons as well If all k states in a given energy range are occupied, what does that mean regarding the direction vectors for the electrons? If all occupied ten electrons travel uniformly in all direction, and so effectively the electrons must be stationary as a whole. To move, a force must be impressed to change the momentum direction of the electrons. Can this happen?

12. Can a force be impressed on the electrons to cause them to drift in a particular direction? Below the Fermi level all states are occupied, so there cannot be a net change into any particular direction or particular set of them. So then the material cannot conduct electricity, (no net momentum in a particular direction) In a metal however, even at 0K, there is not a problem because in the case of the alkali metals and noble metals there are always states available directly above the Fermi level. Why? If the gap between the states is much more that the thermal energy, k T, then these states are not available.

13. What about the case of the alkaline earth metals? These have an even number of valence electrons. Here one has to consider whether or not the energy bands overlap with one another in energy. Expect that the alkaline earth metals might be insulators, but looking at the energy diagram in k space, we find that the bands overlap Thus there are two ways to get a metal (or metal like conductivities a. because of electron concentration b. because of band overlap

14. Semiconductors Electrical Resistivity Good Conductors 10-6 ohm-cm Semiconductors 10-2 – 10-9 Insulators 1014 – 1022 Define as an insulator in which in thermal equilibrium some charge carriers are mobile. Characteristic semiconducting properties are usually brought about by thermal agitation, impurities, lattice defects, or a lack of stoichiometry (departure from nominal chemical composition) Two types of conductivity associated with semiconductors, intrinsic conductivity and impurity conductivity

15. Intrinsic Conductivity Intrinsic temperature range – range in which the electrical properties of a semiconductor are not essentially modified by impurities within the crystal. Based on the idea of a vacant conduction band separated by energy gap from a filled valence band (the old song and dance) Both the electrons in the conduction band and the vacant states or holes left behind in the valence band will contribute At temps below the intrinsic range impurities become the dominant mechanism, at sufficiently high temps. Intrinsic takes over because there are more electrons excited than there are on the impurities.

16. The value of the intrinsic conductivity is controlled by The value of Eg relative to kT. Table of energy gap in some semiconductors. Values obtained two ways, by optically and by analysis of the dependence of conductivity on temp. Besides thermal excitation, photon absorption may also occur and the threshold to continuous optical absorption is at the frequency vg Eg = hvg The photon can be absorbed in two ways either in a direct process or in an indirect process (Direct Semiconductor or Indirect Semiconductor)

17. Direct process Lowest point in the conduction band is at the same k value as the high point in the conduction band. Here the threshold frequency vg determines the bandgap directly. Indirect process Involves both a photon and a phonon as the band edges of the conduction and valence bands are widely separated in k (momentum) space. Essentially the lowest point of the conduction band does not coincide with the highest point in valence band k(photon) = kc + K hvg = Eg + hW where hW is the phonon energy, and kcis wavevector of separation, K is the phonon wavevector

18. Photon wavevectors are negligible in magnitude at the energy range of 1eV In effect the phonon becomes an inexpensive source of crystal momentum. Energies of phonons are small 0.1 to 0.3 eV at room temp so with the photon energy this can occur. Direct absorption of phonons is also possible if the necessary band gap energy is present in the phonon. The band gap determination from the temp dependence of the conductivity or the carrier concentration in the intrinsic range can be obtained from an experiment known as the Hall Effect sometimes with additional coductivity measurements