Band Structure Diagrams

Band Structure Diagrams

Why should chemists care about band structure diagrams?

Thus far we have discussed molecular orbital (MO) diagrams in some detail. These describe the electronic energy levels of isolated molecules. As we have discussed they also can be used to approximate the energy levels in a solid. However, in a solid intermolecular (between neighboring molecules) interactions can also be important. In fact many interesting properties of solids such as metallic conductivity, superconductivity, magnetism, etc. depend crucially upon intermolecular interactions.

The solid state equivalent to a MO diagram is called a band structure diagram. The band structure diagram contains information about both the bonding interactions within a molecule (intramolecular) and the intermolecular interactions. Chemists often refer to band structure diagrams as spaghetti plots, because or their appearance. However, the ability to understand band structure diagrams allows one to extract valuable information about a material:

Electronic conductivity
Optical properties
Stability of a compound toward oxidation or reduction (doping)

In this way knowledge of the electronic band structure provides predictive insight and understanding of certain very important physical properties of solids. Furthermore, just as the energy levels in a MO diagram are dependent upon the energy levels of the atomic orbitals as well as the symmetry and interatomic distances in a molecule, the electronic band structure is dependent upon energy levels of the constituent atomic orbitals and the crystal structure. This brings us to a very important point:

The electronic band structure of a material is the key link between its crystal structure and physical properties. Therefore, solid state chemists who hope to explore structure-property relationships in materials should be able to understand the origin of band structure in chemical terms.

An additional motivation to understand band structures is given by the fact that such knowledge is often needed to effectively communicate with condensed matter physicists, many of whom are very good at calculating band structures. A rough comparison of the way chemists and physicists view solids is given in the table below.

Chemist	Physicist
Localized electrons	Delocalized electrons
MO Diagram	Band Structure Diagram
Molecular Orbital	Wavevector
Chemical bonding	Potentials
Qualitative (Intuitive)	Quantitative (Mathematical)

Scientists who are able to see both points of view tend to have great insight into the behavior of solid state materials.

The purpose of the rest of this lecture is an attempt to understand band structure diagrams in terms of basic chemical bonding interactions. The treatment is very similar to the excellent description given by Roald Hoffmann in his book, "Solids and Surfaces: A Chemists View of Bonding in Extended Structures."

The Band Structure of a 1D chain of H atoms

Lets begin by constructing the band structure, which corresponds to the simplest system we can think of, a 1D chain of hydrogen atoms.

If there are N atoms in the chain there will be N energy levels and N electronic states ("MOs"). The wavefunction for each electronic state is

Y_k₌S e^ikna c_n

where a is the lattice constant (spacing between H atoms) and n is a counter for each atomic orbital in the chain. Notice that each Y_k is really just a symmetry adapted linear combination (SALC) of AO’s, where the symmetry operator in this case is translational symmetry.

The variable k can be thought of as an index which labels the respective SALC’s, it also tells us the phase of the orbitals as the following figure illustrates:

At k=0 all of the orbitals are in phase, while at k = p/a the orbitals are out of phase, at intermediate values the phase of the orbitals varies (the wavelength of the phase variation is given by the denominator i.e. a, 2a, etc.).

Remarkably the value of k also is related to the wavelength of the wavepacket which describes an electron in a given state, Y_k, by the relationship:

l = 2p/k

We will return to this relationship later when we discuss electronic conductivity.

It should be intuitively obvious that of all the possible states, Y_k, the bonding combination at k = 0 will have the lowest energy and the antibonding combination at k =p/a will have the highest energy. States in between such as k = p/2a will be intermediate. Keeping these things in mind we can draw an approximate band structure diagram for this very simple system.

Despite its simplicity there are features of this band structure diagram that we can learn from

(1) The band runs "uphill" (from 0 to p/a) because the in phase (at k=0) combination of orbitals is bonding and the out of phase (at k=p/a) is antibonding. If one were to consider a sigma interaction of p orbitals, the opposite situation would hold and the band would run "downhill".

(2) As the orbital overlap is increased (for example by decreasing the interatomic spacing) the bonding interaction is further stabilized while the antibonding is further destabilized. This will cause the band width to increase. Keep in mind though, that because the antibonding destabilization is larger than the bonding stabilization the top of the band will rise more than the bottom will sink.

Small overlap ® narrow bands ® localized electrons
Large overlap ® wide bands ® delocalized electrons

For a more thorough description of these effects refer to either your class notes, or to Hoffmann’s book (Solids and Surfaces).

The Fermi level – Metals, Semiconductors and Insulators

The dashed line in the hydrogen chain band structure represents the energy of the highest filled state (HOMO). This energy level is called the Fermi level, E_F. Depending upon the position of E_F we can make several important distinctions regarding the expected electrical conductivity of a material. Keep in mind that a real band structure diagram would have many different bands, but only one Fermi level

Metal – E_F cuts through a band.
Semiconductor/Insulator – E_F does not intersect a band.

The distinction between a semiconductor and an insulator is somewhat arbitrary. In either case, each band will be either completely full or completely empty (at T = 0 K anyway). Therefore, electrons cannot move from one state to another with an infinitesimal amount of energy as they can in metals, they have to be excited across an energy gap before conductivity can occur. The magnitude of this gap is the energy difference between the most energetic state among the filled bands (valence band, VB) and the lowest energy state among the empty bands (conduction band, CB). It is called the band gap, E_g. One arbitrary distinction which is sometimes used is to say that semiconductors have E_g < 3 eV, while insulators have E_g > 3 eV, although others may use a different distinction.

Peierls Distortion to Form H₂

Because the Fermi level cuts through a band we expect a linear chain of equally spaced hydrogen atoms to be a metallic conductor. However, it should be immediately obvious to a chemist that this system is not stable with respect to dimerization (to form H₂ molecules). What effect would such a distortion have on the band structure?

The unit cell constant would double, a(H₂) = 2 a(H), so that 2 hydrogen atoms are now contained within the unit cell
Since there are now 2 atomic orbitals in the unit cell, the band structure diagram will contain two bands
The size of the Brillouin zone (0 < k < p/a) in reciprocal space is reduced by a factor of 2, due to the doubling of a

The orbital interactions at various k points for each band and the band structure diagram are shown below

These figures illustrate the following points:

The relative energy and intramolecular orbital interactions associated with each band can be determined from a MO diagram. In this very simple example we have a bonding MO which leads to the valence band and an antibonding MO which leads to the conduction band.
Due to the interactions between molecules the valence band runs "uphill" and the conduction band runs "downhill".
There are two electrons per unit cell so that the valence band is full and the conduction band empty. Thus E_F occupies a position midway between the two bands and the system is now a semiconductor/insulator.
The minimum energy gap between bands occurs at k = p/a. Since the maximum in the VB and the minimum in the CB occur at the same value of k, the compound is said to be a direct gap semiconductor.
Compared with the band structure of a linear chain of hydrogen atoms the bands in this system will be narrower. This is a consequence of the reduced overlap between molecules (in the previous example the molecule was simply a hydrogen atom).
The net effect of this distortion is to lower the energy of the filled states (which originate from bonding MO’s) while raising the energy of empty states (antibonding MO’s). This leads to a net stabilization of the system (to calculate the total ground state energy we only need worry about the occupied levels). In a 1D system such a distortion will always be stable when you have a half filled band (as we did in the H band structure). Chemists call this a Jahn-Teller distortion, while physicists call it a Peierls distortion.

Summary

(1) What is being plotted?

Energy vs. k : where k is a wavevector that tells us the phase of the AO’s as well as the wavelength / crystal momentum of an electron in the Y_kstate_.

(2) How many lines are there in a band structure diagram?

The number of bands is equal to the number of atomic orbitals in the unit cell.

(3) How is the average (center of gravity) energy level of each band determined?

From the MO diagram, due to intramolecular interactions.

(4) Why are some bands wide, others narrow?

This is dependent upon the degree of intermolecular overlap. When the overlap is strong wide bands result.

(5) How do we determine whether a band runs "uphill" or "downhill"?

Uphill bands are bonding when all orbitals are in phase (k = 0), and antibonding when the orbitals are out of phase (k = p/a). The opposite holds true for downhill bands.

(6) How can we tell if a material will be a metal, an insulator or a semiconductor?

When E_F cuts through a band we expect metallic conductivity (though this is not true when overlap is negligible and bands become very narrow), otherwise we have a semiconductor / insulator. In the latter case, E_F lies in the middle of the band gap.

In summary remember that the band positions are determined by the energies of the atomic orbitals and the overlap of orbitals within a molecule. The width of the bands is determined by the overlap of orbitals with neighboring molecules (intermolecular overlap).

strong intermolecular overlap à wide bands à delocalized e^- à good conductivity

weak intermolecular overlap à narrow bands à localized e^- à poor conductivity