1. Energy Band 7
In free electron model, electrons occupy positive energy levels from E=0 to higher values of energy. They are valence electron so called “Valence band”.
The free electrons and other electrons can be thought as belonging to the whole crystal → band rather than the discrete energy level
There are three ways to look at bands.
1. Bands resulting from a periodic potential.
2. Bands resulting from interacting atoms. <tight binding interaction>
3. Bands resulting from periodic perturbation of free electrons.<weak binding approx.>
(a) bound electrons (b) free electrons (c) electrons in solid

2. One-Dimensional Periodic Potential

3. One-Dimensional Periodic Potential
In a periodic lattice with V(x+a) = V(x), it is expected that the wavefunction solution will also show this periodicity. Since yk has the form of exp(ikx), we expect,
Apply this requirement to values of x=-b and x=(a-b),

4. Kronig-Penney approximation:
One-Dimensional Periodic Potential
Kronig-Penney approximation:

5. Kronig-Penney approximation
One-Dimensional Periodic Potential
Kronig-Penney approximation

6. Kronig-Penney approximation
One-Dimensional Periodic Potential
Kronig-Penney approximation

7. Kronig-Penney approximation
One-Dimensional Periodic Potential
Kronig-Penney approximation
2. Width of energy band increases with energy
3. P is a measure of tightness of binding.
Width of bands decreases as P↑
P= : free electron → no band (continuum)
P → ∞ : discrete levels → no bands
higher bands are wider
energy bands for more tightly bound (ionic atom) electrons are narrower
allowed energy bands end at

8. The Tight-binding Approximation
As atoms
come
together
Isolated atom a
Discrete level
Electrons : identified with a particular atom
→ effectively belong to the whole crystal (solid)
No two electrons can have same energy
→ N atoms : energy level breaks up into slightly different levels

9. The Tight-binding Approximation
N atoms
Less tight → wider
More tight → narrower
Extend of split depends on the binding tightness
N atoms → n different levels
Belong to the whole crystal

10. The Tight-binding Approximation
Na :
3s and 3p energy overlapping gives continuous band E
0.3 Rydberg
2.8 Rydberg
5.1 Rydberg (-63.4 ev)
81 Rydberg (-1041 ev) E a r
Equilibrium position

11. Effect of Periodic Potential on Free Electrons
Free electrons superimposed by a small periodic potential on the free electrons (introduce a small periodic potential as a perturbation in the Schroedinger equation)
+ small periodic potential
→ particular value of k can not propagate through the crystal because of the Bragg reflection
Waves with is represented by Standing Wave
→ Open up the energy gaps
Start from C.B.
Purturbation → band
(7.14)
(7.15)

12. Effect of Periodic Potential on Free Electrons
Periodic nature of the crystalline structure
Translate to
“Extended” representation
“Reduced” representation
: The first Brillouin Zone

13. Effect of Periodic Potential on Free Electrons
Free electron like behavior
Point : extrema in energy band

14. 3-D Brillouin Zones
Simple cubic
bcc
fcc

15. Density of States in a Band
For free electrons, define the density of states N(E) such that N(E)dE as the number of orbital states lying between E and E+dE, as
Nearly free electron model: → nearly parabolic upward at the bottom of the band approximately parabolic downward at the top.
near the bottom of the band
Near the top of the band
: effective mass (b : bottom, t : top)
: maximum energy of the band
: corresponding value of

16. Density of States in a Band
Consider the example of a typical band describing an s-state in a one-dimensional simple cubic crystal as calculate by the tight-binding model

17. Density of States in a Band
(Fig. 7.6)
Near the bottom of the band
Near the top of the band
In the middle equal energy surface is not spherical → distortion

18. Equal energy surfaces
2-dimensional crystal
FCC

19. Density of States in a Band
Metal
Semiconductor or Insulator or
Partially occupied
Bands overlap
The upper most band is filled

20. Summary of different band representation
spherical

21. Electron Velocity
E vs k curves correspond to the dispersion relationship for electron waves, the group velocity can be calculated,
(7.21) + -
(7.22)
(7.23)
(7.24)
Velocity is zero at band extrema
In free electron system, goes to infinite or increase as
In thermal equilibrium, there are equal number of electron occupied states with positive velocity as there are with negative velocity;
In completely filled band, equal number of
→ no net charge transport under an electric field
→ Insulator

22. Effective mass Effect of external force on electrons in solids
↑ what is the proportionality constant?
If for 1-D
The effective mass of an electron is the reciprocal of the curvature of the E vs. k plot.
For 3-D,

23. Effective mass E ② ③ ①
Greater the curvature, smaller the effective mass E E + - -

24. Effective mass For a free electron,
(electrons at the extrema have effective mass like the inertia mass)
at the bottom of a band
at the top of a band
A negative mass implies that the induced acceleration is on the opposite direction to the force that caused it
The existence of an negative effective mass is the result of Bragg reflection effects coming from the crystal potential in which an electron acted on by force in one direction is actually accelerated in the opposite direction because it undergoes reflection at the zone face.

25. Holes
Holes: missing electrons in a nearly filled band, have a positive effective mass and a positive charge.
In a semiconductor, the band structure looks like
In the presence of an electric field, electrons in the bottom of the conduction band and holes at the top of the valence band move in the opposite directions in real space (same sign mass but different sign charge), whereas electrons and holes both at the top of the valence band move in the same direction (different sign mass cancels different sign charge) E
C.B
V.B
Holes
(+m*, +q)
“Bubble in water”

26. Holes Apply an electric field E, then the force is
Initially located at in conduction band (CB) moves toward the negative direction, where the velocity ( dE/dk) is negative.
Initially located at in the valence band (VB) moves toward the direction, where the velocity is positive.
The holes at the top of the valence band have their k changed toward positive values of the hole E vs. k diagram, and hence they acquire velocity toward +x direction in the presence of the an electric field in the +x direction. -