Physics 355

Consider the available energies for electrons in the materials. As two atoms are brought close together, electrons must occupy different energies due to Pauli Exclusion principle. Instead of having discrete energies as in the case of free atoms, the available energy states form bands. Conductors, Insulators, and Semiconductors

Free Electron Fermi Gas

For free electrons, the wavefunctions are plane waves:

Band Gap zone boundary

“doped” “thermally excited”

 + Origin of the Band Gap To get a standing wave at the boundaries, you can take a linear combination of two plane waves:

Origin of the Band Gap Electron Density

Origin of the Band Gap

Bloch Functions Felix Bloch showed that the actual solutions to the Schrödinger equation for electrons in a periodic potential must have the special form: where u has the period of the lattice, that is

Kronig-Penney Model  (a+b) bb 0aa+b U(x) x U0U0 The wave equation can be solved when the potential is simple... such as a periodic square well.

Kronig-Penney Model Region I - where 0 < x < a and U = 0 The eigenfunction is a linear combination of plane waves traveling both left and right: The energy eignevalue is:

Region II - where  b < x < 0 and U = U 0 Within the barrier, the eigenfunction looks like this and Kronig-Penney Model

 (a+b) bb 0aa+b U(x) x U0U0 To satisfy Mr. Bloch, the solution in region III must also be related to the solution in region II. IIIIII

Kronig-Penney Model A,B,C, and D are chosen so that both the wavefunction and its derivative with respect to x are continuous at the x = 0 and a. At x = 0... At x = a...

Kronig-Penney Model Result for E < U 0 : To obtain a more convenient form Kronig and Penney considered the case where the potential barrier becomes a delta function, that is, the case where U 0 is infinitely large, over an infinitesimal distance b, but the product U 0 b remains finite and constant. and also goes to infinity as U 0. Therefore:

Kronig-Penney Model What happens to the product Qb as U 0 goes to infinity? b becomes infinitesimal as U 0 becomes infinite. However, since Q is only proportional to the square root of U 0, it does not go to infinity as fast as b goes to zero. So, the product Qb goes to zero as U 0 becomes infinite. As a results of all of this...

Kronig-Penney Model

 0  22 33 ka  0  Plot of energy versus wavenumber for the Kronig-Penney Potential, with P = 3  /2.

Crucial to the conduction process is whether or not there are electrons available for conduction. Conductors, Insulators, and Semiconductors

“doped” “thermally excited”