Molecular Structure (Cont.)

Band Theory of solids

Readings for this section.

Petrucci: Chapter 11-7

We've seen that metallic bonding can be thought of as an overlap of orbitals from neighbouring metal atoms, allowing the electrons to delocalize along the entire dimensions of the metal.  We've also seen the concept that implies that the number of atomic orbitals must equal the number of molecular orbitals.  We resolve these two concepts in the theory of bands.  Bands are distributions of many molecular orbital energy levels, so closely spaced in energy that they seem to be continuous.  

We will resolve the discussion of solids into three types, where bands are concerned:

  1. metals, where the electrons in the highest energy levels are free to move along delocalized energy levels.  Conductivity decreases as the temperature is raised.
  2. semiconductors, where there is no conductivity at absolute zero and conductivity increases as temperature increases
  3. insulators, where there is no conductivity over a wide range of temperatures.

Formation of bands

Let's consider, a solid made up of a substance that involves only one type s of atomic orbital.  For the moment, we'll ignore the number of electrons the atom contributes.  If we look at the chart below, we'll see that at the number of such atoms with overlapping orbitals increases, so too does the number of molecular orbitals (and the MO energy levels)


The set of energy for the N = case has N energy levels, so closely spaced that they appear continuous.  This is a band of energy levels.  This band is not uniformly spaced with the central energy levels occurring more closely packed than the highest and lowest levels in each band but that's merely because of the way the orbitals overlap and is not an issue to this discussion.


If the atoms involved have p orbitals available as well then multiple bands are available.   Just as there were energy separations between the s and p orbitals on the atom that varied from atom to atom, there are band gaps that vary from solid to solid, depending on which atoms in the solid.

In each of these bands, we can consider the lowest energy levels to correspond to fully bonding molecular orbitals and the highest levels correspond to fully antibonding MOs.

 

 

Now, lets look at the band occupancy.  If the atoms we were considering (e.g. sodium) each contributes 1 electron then the band is exactly half filled.  We need only consider the s band since it will made up of N atomic orbitals overlapped to give N energy levels with only N electrons.  Thus the band occupancy can be represented by the diagram on the left.  Actually, this diagram is only valid at absolute zero. At this temperature, the highest occupied level is called the Fermi level.   As thermal energy is added to the solid, electrons become excited into some of the lower  "Unoccupied" levels, giving a distribution of level occupancy called the Fermi-Dirac distribution (shown in blue for some T > 0 K).  According to this equation, the higher the temperature above absolute zero, the higher the occupancy of the upper levels.   However, the number of electrons in or near empty orbitals remains relatively constant throughout this process hence, no increase in conductivity occurs as a result of this process.  Rather, as temp goes up, atoms vibrate more and collide with the moving electrons, creating "friction" or resistance, thus, conductivity decreases as temperature increases in conductors.

If the atoms involved in creating the gap each contribute 2 electrons then the s band would be exactly filled at T=0.  Since, in most metals the s and p bands overlap slightly, there is no band gap and so mobility of electrons is still allowed.

 

Insulators and semiconductors

If there is a non-zero band gap then we find a set of levels where the occupied levels and the unoccupied levels are separated.  With increase in temperature above absolute zero, it becomes possible to excite some electrons from the highest of the occupied band (valence band) to the lowest of the unoccupied band (conduction band).  Herein lies the distinction between insulators and semiconductors.  in insulators, the band gap is large and only a negligible number of electrons are promoted at temperatures where the solid exists.  Semiconductors, have smaller band gaps and significant promotion of electrons can occur from thermal excitation alone, causing conductivity to increase at T increases.

 


Doping

If the solid semiconductor is doped with a low concentration of atoms that have different number of electrons then different band structures occur. 

If (far left), the dopant atoms contain one less electron then there will be an extra unoccupied band slightly higher in energy than the valence band.  This is called the acceptor band and can accept electrons from the valence band with lower thermal energies than the pure crystal bands can This is called a p-type semiconductor (P is for positive since the dopant atom will appear to have one fewer electrons, it will look to be "positive").  

If (left) the dopant atoms have extra electrons then an extra band is created that can donate electrons into the conduction band.  This creates an n-type semiconductor. (N is for negative.  the dopant atom has one more electron than the rest of the lattice and appears "negative" by comparison. 

Combinations of n- and p-type semiconductors are used in the electronics industry to create diodes, transistors, etc.  But That's a whole new topic, which we won't cover in this course.

 

Prof. Michael J. Mombourquette.
Copyright © 1997
Revised: August 29, 2013.