1. Band Theory of Solids 1

2. Electrons in Solids
Considering electrons in metals as free particles, electron gas in a box explains many experimental results which?
Still, why are some solids metals and others insulators?
For metals we assumed that electrons are free
We know, however, that there are no free electrons in insulators
QM give us the answer!
Need a more realistic potential for electrons reflecting periodic ionic structure of solids 2

3. “Realistic” Potential in Solids
ni are integers
Example: 2D Lattice 3

4. “Realistic” Potential in Solids
For one dimensional case where atoms (ions) are separated by distance d, we can write condition of periodicity as 4

5. “Realistic” Potential in Solids
Multi-electron atomic potentials are complex
Even for hydrogen atom with a “simple” Coulomb potential solutions are quite complex
So we use a model one-dimensional periodic potential to get insight into the problem 5

6. Bloch’s Theorem
Bloch’s Theorem states that for a particle moving in the periodic potential, the wavefunctions ψ(x) are of the form
uk(x) is a periodic function with the periodicity of the potential
The exact form depends on the potential associated with atoms (ions) that form the solid 6

7. Bloch’s Theorem
Bloch’s Theorem imposes very special conditions on any solution of the Schrödinger equation, independent of the form of the periodic potential
The wave vector k has a two-fold role:
It is still a wave vector in the plane wave part of the solution
It is also an index to uk(x) because it contains all the quantum numbers, which enumerate the wavefunction 7

8. Bloch’s Theorem
What is probability density of finding particle at coordinate x?
But |uk(x)|2 is periodic, so P(x) is as well 8

9. Bloch’s Theorem
The probability of finding an electron at any atom in the solid is the same!!!
Each electron in a crystalline solid “belongs” to each and every atom forming the solid 9

10. Covalent Bonding Revisited
When atoms are covalently bonded electrons supplied by atoms are shared by these atoms since pull of each atom is the same or nearly so
H2, F2, CO,
Example: the ground state of the hydrogen atoms forming a molecule
If the atoms are far apart there is very little overlap between their wavefunctions
If atoms are brought together the wavefunctions overlap and form the compound wavefunction, ψ1(r)+ψ2(r), increasing the probability for electrons to exist between the atoms 10

11. Covalent Bonding Revisited11

12. Schrödinger Equation Revisited
If a wavefunctions ψ1(x) and ψ2(x) are solutions for the Schrödinger equation for energy E, then functions
-ψ1(x), -ψ2(x), and ψ1(x)±ψ2(x) are also solutions of this equations
the probability density of -ψ1(x) is the same as for ψ1(x) 12

13. Band Theory of Solids
Consider an atom with only one electron in s-state outside of a closed shell
Both of the wavefunctions below are valid and the choice of each is equivalent
If the atoms are far apart, as before, the wavefunctions are the same as for the isolated atoms 13

14. Band Theory of Solids
Once the atoms are brought together the wavefunctions begin to overlap
There are two possibilities
Overlapping wavefunctions are the same (e.g., ψs+ (r))
Overlapping wavefunctions are different
The sum of them is shown in the figure
These two possible combinations represent two possible states of two atoms system with different energies 14

15. Tight-Binding Band Theory of Solids
Garcia Chapter 24.4 and 24.5 15

16. Electron in Two Separated Potential Wells16

17. Potential Wells Moved Closer17

18. Tight-Binding Approximation first two states in infinite and finite potential wells18

19. Symmetric and Anti-symmetric Combinations of Ground State Eigenfunctions19

20. Six States for Six Atom Solid20

21. Splitting of 1s State of Six Atoms21

22. Atoms and Band Structure
Consider multi-electron atoms:
The outer electrons (large n and l) are “closer” to each other than the inner electrons
Thus, the overlap of the wave- functions of the outer electrons is stronger than overlap of those of inner electrons
Therefore, the bands formed from outer electrons are wider than the bands formed from inner electrons
Bands with higher energies are therefore wider! 22

23. Splitting of Atomic Levels in Sodium23

24. Occupation of Bands Sodium24

25. Splitting of Atomic Levels in Carbon25

26. Occupation in Carbon at Large Atomic Separation26

27. Actual Occupation of Energy bands in Diamond27

28. 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) 28

29. 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 29

30. 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) 30

31. 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 31

32. 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) 32

33. 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 33

34. 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 34

35. 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) 35

36. 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 36

37. 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 37

38. 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 38

39. 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 39

40. 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 40

41. 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 41

42. 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 42

43. 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” 43

44. 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 44

45. 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 45

46. Conduction
Electrical current for holes and electrons in the same direction 46