Section 2.4 - 1

2.4 Let’s step back and look at what we’ve got so far... So, far we have been using very simple mathematical models, based on symmetry:

1. Particle-in-a-box: Simple but useful concept that helps us understand why solutions to the Schrödinger equation give us quantized energies and wavefunctions. Can be a model for electrons in short linear molecules.

2. Particle-on-a-ring: One step further; gives us a model for cyclic systems and,

as we shall see, for a free electron gas in 3D. (i.e., starting point for understanding metals!)

3. Periodic Potential: Based on symmetry alone, we can get mathematical

solutions that look like the familiar pi molecular orbitals for aromatic rings (e.g., benzene)!

4. V. Large Ring: 1D Chain of periodic potential. We can easily transition

from familiar aromatic ring M.O.s to hypothetical 1D chain crystal orbitals (C.O.s). We will now start to understand band structures using concepts familiar to chemists.

Now, let’s put all this information together and look at it in terms of understanding conductivity in solids. As chemists, we have two general goals in this area: • To be able to use familiar concepts in order to understand conductivity properties of

solids based on their atomic or molecular structure. • To be able to design a solid that will have specific desired electrical conductivity

properties, using chemical synthetic techniques.

Section 2.4 - 2

Energy Bands In order to start putting everything we’ve learned into perspective, we need to use one of the physicists’ tools: energy bands • Every solid contains electrons. • The important question for electrical conductivity is how the electrons respond to an

applied electric field. • Electrons in crystals are arranged in energy bands separated by regions in energy for

which no wavelike electron orbitals exist. • These forbidden regions are called energy gaps or band gaps. • This “energy band & band gap” model arises naturally from our familiar atomic

orbital to molecular orbital to crystal orbital model (which we’ve already derived!) • The “energy band & band gap” model can also be derived using a “nearly free

electron Fermi gas model” ... also known as the Kronig-Penny model. We’ll see this shortly.

Adapted from : “Intro. to Solid State Physics, 7th Ed.”, C. Kittel, John Wiley & Sons, Inc., N.Y., 1996.

Section 2.4 - 3

The insulator and the metal are the easiest types of conductors to understand in terms of energy bands: • A crystal behaves as a metal if one or more bands are partly filled (partly empty). • A crystal behaves as an insulator if the allowed energy bands are either completely

filled or completely empty, and the band gaps are “large”. Thus, no electrons can move in an electric field.

• The difference between an insulator and a semiconductor is the size of the band gap. • A semiconductor has completely filled or empty bands, but a small band gap:

Egap << kBT • A poor semiconductor (or insulator) can be doped in order to create a good

semiconductor.

Doping is the practice of replacing a small percentage of the atoms in the crystal lattice of a pure substance with atoms of an element that has one fewer or one more valence electrons.

• N-type semiconductors: These are semiconductors that are doped with an element

that has more valence electrons, creating an electron rich material. e.g., phosphorus-doped silicon: The “extra” electron of the phosphorus atom is loosely bound to the phosphorus ion core because it is not involved in covalent bonding to the neighboring silicon atoms. It is far easier for this electron to be thermally excited into a conduction band than it is for one of the silicon valence electrons to be so excited.

From a band model standpoint, the “extra” electrons from the phosphorus atoms occupy a discrete “donor” energy level. This is higher in energy that the valence (filled) band of crystalline silicon and lower in energy than the conduction (empty) band of crystalline silicon.

Section 2.4 - 4

Source: “Fundamentals of Physics, 3rd Ed.” Halliday & Resnick, John Wiley & Sons, N.Y., 1988.

• P-type semiconductors: These are semiconductors that are doped with an element

that has fewer valence electrons, creating an electron deficient material.

e.g., aluminum-doped silicon: In this case, there is a deficit of electrons (i.e., “holes”) introduced into the valence band of the silicon crystal because it is easy for the aluminum ion core to “steal” a valence electron (oxidize) from a neighboring silicon atom. It is far easier for electrons from the silicon crystal valence (filled) band to be thermally excited into the low-lying empty aluminum “acceptor” energy level than into the higher energy silicon conduction (empty) band.

Semiconductors doped with donor atoms are called n-type semiconductors, the “n” standing for “negative” because the negative charge carriers (electrons in the conduction band) greatly outnumber the positive charge carriers (holes in the valence band). Semiconductors doped with acceptor atoms are called p-type semiconductors, the “p” standing for “positive” because the majority carriers are the holes in the valence band. Finally, semi-metals arise when one filled and one empty energy band in the crystal overlap and electrons transfer from one the other, creating partially filled bands and metallic conduction behavior. At very low temperature, it is possible for the band structure to collapse giving rise to a metal-to-insulator transition.

Section 2.4 - 5

We have, in fact, already seen how to relate bands to the interactions of atomic orbitals in the solid state. Let’s put it all together explicitly now ...

• We all know how to create energy level diagrams for the pi M.O.s of cyclic systems,

like benzene. (above) • We know that these are discrete energy levels, each representing one molecular

orbital, and each capable of occupation by a maximum of two electrons. • We have seen that the molecular orbitals (as linear combinations of atomic orbitals)

for this type of small cyclic molecule can be derived by taking sums and differences of moving waves (solutions from the Bloch function):

• We have seen that the actual energy ordering of these M.O.s can also be derived: • We have seen that we can use a “very large ring” to derive a “1D crystal” model with

an energy level continuum:

m

n

mj u

nij

n⎟⎠

⎞⎜⎝

⎛= ∑=

πψ

2exp11

njE jπ

βα2cos2+=

∑=

−=n

mmark amki

n 1)()exp(1

φψ!!

)cos(2 akEk!!

βα +=

Section 2.4 - 6

• In both cases, the average energy for the energy distribution is given by α, which is numerically equivalent to the Valence Orbital Ionization Potential (VOIP) and therefore related to the orbital electronegativity (of the atomic orbitals used to create the M.O.s)

• The energy dispersion is 4β, where β is the resonance integral. In other words, this

is directly related to the amount of overlap between neighboring atomic orbitals.

n = 2 n = 2 n = 2 n = 2 n = infinite

BAN

DW

IDTH

4β

Section 2.4 - 7

• In both the discrete (finite) molecular orbital picture and the infinite 1D crystal orbital picture, the number of nodes increases as the energy increases.

• The result is the production of an energy band for each of the atomic orbitals located

on the atoms which make up the 1D chain.

• In 2D and 3D crystals, a similar process occurs. The atomic energy levels of each of

the atoms of the solid are broadened into bands. • The “width” of these bands (i.e., height on our energy scale) depends upon the

magnitude of the corresponding interaction integrals (the equivalent of the Hückel β for the one dimensional chain shown above) between the orbitals concerned.

• EXAMPLE:

elemental sodium Source: “Orbital Interactions in Chemistry”, Albright, Burdett & Whangbo, Wiley, N.Y. 1985.