Energy Bands in SolidsPhysics 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 Gapzone boundary

dopedthermallyexcited

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

Origin of the Band GapElectron Density

Origin of the Band Gap

Bloch FunctionsFelix Bloch showed that the actual solutions to the Schrdinger equation for electrons in a periodic potential must have the special form:

where u has the period of the lattice, that is

Kronig-Penney ModelThe wave equation can be solved when the potential is simple... such as a periodic square well.

Kronig-Penney ModelRegion I - where 0 < x < a and U = 0The 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 = U0Within the barrier, the eigenfunction looks like this

andKronig-Penney Model

Kronig-Penney ModelTo satisfy Mr. Bloch, the solution in region IIImust also be related to the solution in region II.IIIIII

Kronig-Penney ModelA,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 ModelResult for E < U0: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 U0 is infinitely large, over an infinitesimal distance b, but the product U0b remains finite and constant. and also goes to infinity as U0. Therefore:

Kronig-Penney ModelWhat happens to the product Qb as U0 goes to infinity?b becomes infinitesimal as U0 becomes infinite.However, since Q is only proportional to the square root of U0, it does not go to infinity as fast as b goes to zero.So, the product Qb goes to zero as U0 becomes infinite.As a results of all of this...

Kronig-Penney Model

Kronig-Penney ModelPlot of energy versuswavenumber for theKronig-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

Conductors, Insulators, and Semiconductors

Conductors, Insulators, and Semiconductors

Conductors, Insulators, and Semiconductorsdopedthermallyexcited

**Every solid contains electrons. How those electrons respond to an electric field whether they are conducting, insulating, or semi-conducting is determined by the filling of available energy bands separated by forbidden regions where no wavelike orbitals exist these are called band gaps.***plane waves of the form k(r) = exp(i kr), which are running waves of momentum p = kSo, these are particles which can have any k valueWhat happens when we add a periodic lattice of lattice constant a?

*There is an analogy between electron waves and lattice waves (phonons)For most values of k, the electrons move freely throughout the lattice (there is a group velocity that is non-zero = d/dK)At the zone boundary (K = +/-/a) , there are only standing waves (d/dK = 0).What standing wave solutions are stable with these k-values?

***

Note: only electrons that have a wavelength commensurate with the lattice (k = /a) are influenced by the periodic potential, and they form standing wave patterns

One of these (+) has electrons near the positive cores, the other has the electrons in between the cores (so they have different energies).*This is the origin of the band gap.

Suppose the potential energy of an electron in the crystal at point x is.. the gap is equal to the Fourier component of the crystal potential

*The Bloch Theorem states:The eigenfunctions of the wave equation for a periodic potential are the product of a plane wave and a function with the periodicity of the lattice.

Bloch functions can be assembled into packets to represent electrons propagating freely through the potential fied of the ion cores.****The insertion of k via the Bloch theorem lets us define the wavevector k as a means of labeling the solution.*These are the usual boundary conditions you had when you did the quantum mechanics of square well potentials.

At x=a, with the use of the Bloch requirement for the wavefunction at a under the barrier in terms of the wavefunction at b.

Thus, we have four equations and four unknowns. The solution is found by requiring that the determinant of the coefficients of A through D equals zero. The mathematical process to get this is tedious, so well avoid it.***P is sometimes referred to as the stopping power. It is a measure of the barrier strength.

This is the dispersion relationship for electrons in a periodic one-dimensional crystal. The allowed values of the energy, determined by K, are given by those ranges of

for which the function lies between +/- 1. For other values of the energy, there are no traveling wave or Bloch-like solutions to the wave equation, so that forbidden gaps in the energy spectrum are formed.

Here, the ranges of K for which the equation has solutions are plotted for the case of P = 3/2.***Apart from superconductors, the electrical resistivity of metals can be as low as 10-10 ohm-cm. The resistivity of a good insulator can be as high as 1022 ohm-cm. This range of 1032 may be the widest of any physical property.**