1. Insulators, Semiconductors, Metals
The last completely filled (at least at T = 0 K) band is called the Valence Band
The next band with higher energy is the Conduction Band
The Conduction Band can be empty or partially filed
The energy difference between the bottom of the CB and the top of the VB is called the Band Gap (or Forbidden Gap)

2. Computer simulation can give exact solution in simple cases
Can be found using computer
In 1D computer simulation of light in a periodic structure, we found the frequencies and wave functions
Allowed modes fall into quasi-continuous bands separated by forbidden bands just as would be expected from the tight binding model

3. Insulators, Semiconductors, Metals
Consider a solid with the empty Conduction Band
If apply electric field to this solid, the electrons in the valence band (VB) cannot participate in transport (no current)

4. Insulators, Semiconductors, Metals
The electrons in the VB do not participate in the current, since
Classically, electrons in the electric field accelerate, so they acquire [kinetic] energy
In QM this means they must acquire slightly higher energy and jump to another quantum state
Such states must be available, i.e. empty allowed states
But no such state are available in the VB!
This solid would behave as an insulator

5. Insulators, Semiconductors, Metals
Consider a solid with the half filled Conduction Band (T = 0K)
If an electric field is applied to this solid, electrons in the CB do participate in transport, since there are plenty of empty allowed states with energies just above the Fermi energy
This solid would behave as a conductor (metal)

6. Band Overlap
Many materials are conductors (metals) due to the “band overlap” phenomenon
Often the higher energy bands become so wide that they overlap with the lower bands
additional electron energy levels are then available

7. Band Overlap Example: Magnesium (Mg; Z =12): 1s22s22p63s2
Might expect to be insulator; however, it is a metal
3s-band overlaps the 3p-band, so now the conduction band contains 8N energy levels, while only have 2N electrons
Other examples: Zn, Be, Ca, Bi

8. Band Hybridization In some cases the opposite occurs
Due to the overlap, electrons from different shells form hybrid bands, which can be separated in energy
Depending on the magnitude of the gap, solids can be insulators (Diamond); semiconductors (Si, Ge, Sn; metals (Pb)

9. Insulators, Semiconductors, Metals
There is a qualitative difference between metals and insulators (semiconductors)
the highest energy band “containing” electrons is only partially filled for Metals (sometimes due to the overlap)
Thus they are good conductors even at very low temperatures
The resisitvity arises from the electron scattering from lattice vibrations and lattice defects
Vibrations increases with temperature  higher resistivity
The concentration of carriers does not change appreciably with temperature

10. Insulators, Semiconductors, Metals
The difference between Insulators and Semiconductors is “quantitative”
The difference in the magnitude of the band gap
Semiconductors are “Insulators” with a relatively small band gap
At high enough temperatures a fraction of electrons can be found in the conduction band and therefore participate in transport

11. Insulators vs Semiconductors
There is no difference between Insulators and Semiconductors at very low temperatures
In neither material are there any electrons in the conduction band – and so conductivity vanishes in the low temperature limit

12. Insulators vs Semiconductors
Differences arises at high temperatures
A small fraction of the electrons is thermally excited into the conduction band. These electrons carry current just as in metals
The smaller the gap the more electrons in the conduction band at a given temperature
Resistivity decreases with temperature due to higher concentration of electrons in the conduction band

13. Holes
Consider an insulator (or semiconductor) with a few electrons excited from the valence band into the conduction band
Apply an electric field
Now electrons in the valence band have some energy sates into which they can move
The movement is complicated since it involves ~ 1023 electrons

14. Concept of Holes
Consider a semiconductor with a small number of electrons excited from the valence band into the conduction band
If an electric field is applied,
the conduction band electrons will participate in the electrical current
the valence band electrons can “move into” the empty states, and thus can also contribute to the current

15. Holes from the Band Structure Point of View
If we describe such changes via “movement” of the “empty” states – the picture can be significantly simplified
This “empty space” is a Hole
“Deficiency” of negative charge – holes are positively charged
Holes often have a larger effective mass (heavier) than electrons since they represent collective behavior of many electrons

16. Holes
We can “replace” electrons at the top of eth band which have “negative” mass (and travel in opposite to the “normal” direction) by positively charged particles with a positive mass, and consider all phenomena using such particles
Such particles are called Holes
Holes are positively charged and move in the same direction as electrons “they replace”

17. Hole Conduction
To understand hole motion, one requires another view of the holes, which represent them as electrons with negative effective mass
To imagine the movement of the hole think of a row of chairs occupied by people with one chair empty
To move all people rise all together and move in one direction, so the empty spot moves in the same direction

18. Concept of Holes
If we describe such changes via “movement” of the “empty” states – the picture will be significantly simplified
This “empty space” is called a Hole
“Deficiency” of negative charge can be treated as a positive charge
Holes act as charge carriers in the sense that electrons from nearby sites can “move” into the hole
Holes are usually heavier than electrons since they depict collective behavior of many electrons

19. Conduction
Electrical current for holes and electrons in the same direction