1. Wide bands  Good orbital overlap  High carrier mobility
Effect of Orbital Overlap band width or dispersion=the difference in energy between the highest and lowest energy levels in the band
If we reduce the lattice parameter a (bring closer together) it has the following effects:
The spatial overlap of the orbitals increases
The band becomes more bonding (energy reduces) at k=0
The band becomes more antibonding (energy up) k=p/a.
The increased antibonding is larger than the increased bonding.
The bandwidth increases.
The electron mobility increases.
Wide bands  Good orbital overlap  High carrier mobility

2. Group: Linear Chain of F
F: 1s22s22px22py22pz1 F F F F F F F F F F
(a)
(b)
(c)
(d)
p/a k
E(k) EF
p/a k EF
p/a k EF
p/a k EF
How many total bands should there be? 9/2 rounded up=5. We are not showing the 1s band
Which of the following is the correct band structure for a linear chain of F atoms (atomic #=9)?

3. Linear F Chain
There are 4 n=2 orbitals in the unit cell (a single F atom with 1 2s + 3 2p orbitals)
There is a lower 1s band (not shown)
p/a k
E(k) EF
p/a k EF
p/a k EF
p/a k EF

4. Band Structure: Linear Chain of F
Bonding 2s s
Antibonding 2s s*
A sigma bond shares only one electron
Antibonding 2pz s*
Bonding 2pz s
Which one of these has lower energy?

5. Linear F Chain
There are 4 n=2 orbitals in the unit cell (a single F atom with 1 2s + 3 2p orbitals)
The fact that the wavefunction corresponding to a p-orbital changes sign at the nucleus causes a 2p s band to run downhill (opposite of the 2s s band).
Are there other kinds of bonds?

6. Band Structure: Linear Chain of F
Bonding 2s s
Antibonding 2s s*
A sigma bond shares only one electron
Antibonding 2pz s*
Bonding 2pz s
Which pi bond type is lower energy?
Which of these two p orbitals is doubly degenerate?
A pi bond shares two electrons
Bonding 2px/2py p
Antibonding 2px/2py p*

7. Group: Linear Chain of F F: 1s22s22px22py22pz1
(b)
(c)
(d)
p/a k
E(k) EF
p/a k EF
p/a k EF
p/a k EF
Based on these arguments, we are down to c or d
How do we determine between the final remaining two?

8. Linear F Chain
Do the sigma and pi p orbitals have the same band width? What affects bandwidth?
For the same lattice parameter, the reduced spatial overlap of the p interaction causes the p bands to be narrower than the s bands.

9. Band Structure: Linear Chain of F
Antibonding 2pz s*
Doubly degenerate EF
Antibonding 2px/2py p*
Doubly degenerate
Would naively expect the bonding of the sigma band to be lower than other p bands, as discussed in band overlap slide.
Antibonding 2s s*
Bonding 2px/2py p
Bonding 2pz s
Bonding 2s s k
p/a

10. Band Structure: Linear Chain of F
A more accurate treatment of the band structure would show an avoided crossing between the 2pz s and 2s s * interactions at k=p/a. There would be mixing between these two bands (creating sp-hybrid like states).
Antibonding 2pz s* EF
Doubly degenerate
Bonding 2px/2py p
Bonding 2s s k
p/a

11. Band Overlap Many materials are metals due to band overlap
Flat band diagrams
Many materials are metals due to band overlap
Often the higher energy bands become so wide that they overlap with the lower bands
additional electron energy levels are then available
Depends on how Fermi level lines up with these bands
Overlap here too

12. 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)

13. Learning Objectives for Today
After today’s class you should be able to:
Calculate effective mass and cyclotron frequency
Understand the difference between an electron and an electron quasiparticle
Discuss other quasiparticles
Distinguish between two types of excitons

14. - Flat band structure in semiconductors
The ground state is sometimes called the vacuum state
Conduction
band
Electron energy
Energy
gap
Forbidden
band -
Valence Band = Dirac Sea of Electrons
Eph>Eg
The ground state is called the vacuum state since any filled bands have no effect on conductivity. Thus, a filled band is the same as a vacuum (no states).
If the photon energy is higher than the energy gap the electron can be excited

15. Quasiparticles
Quasiparticles occur when a solid behaves as if it contained free particles.
Example: as an electron travels through a semiconductor, its scatters with electrons and nuclei; however it ~behaves like an electron with a different mass traveling unperturbed through free space. This "electron" with a m* is called an "electron quasiparticle". (m* =Effective mass)
Aggregate motion of electrons in the valence band of a semiconductor is the same as if the semiconductor contained positively charged quasiparticles called holes or positrons.

16. E - + Electron energy band structure in semiconductor Conduction band
gap
Forbidden
band +
Valence
band
Quasiparticles: excitations of the ground state (vacuum state), behave as if particles in free space
Excited electron leaves in the valence band a positive hole/positron.
Electron and positron are quasiparticles and charge carriers (equal)
Positively charged hole interacts with negatively charged electron by Coulomb interaction.
Can alternatively think about it as the other electrons rather not be close to that excited electron due to charge of the same sign.
Sometimes easier to keep track of the holes (rather than electrons). Equivalent to bubble rising in water.
Electrons and holes should be treated as equals, just as the math suggests.

17. Excitons (quasiparticle) are bound electron-hole states
A free electron and a free hole (empty electronic state in the valence band) exert Coulomb force on each other: hydrogen-like bound states possible: excitonic states
Eb is the exciton binding energy =
energy released upon exciton formation, or
energy required for exciton breakup
On board: Draw first BZ from –pi/a to pi/a with a few bands to explain what this energy band is showing
Note: exciton can move through crystal, i.e. not bound to specific atom!
OSE5312 Spr Class 12: Optical properties of semiconductors 1

18. Two Types of Excitons Wannier – Matt excitons:
Wannier – Matt excitons (free exciton): mainly exist in semiconductors, have a large radius, are delocalized states that can move freely throughout the crystal, the binding energy ~ 0.01 eV
Frenkel excitons (tight bound excitons): found in insulator and molecular crystals, bound to specific atoms or molecules and have to move by hopping from one atom to another, the binding energy ~ eV.
Max thermal energy (lattice vibrations called phonons) ~ kBT = eV (RT)
Wannier – Matt excitons:
stable at cryogenic temperature
Frenkel excitons:
stable at room temperature
A phonon is another quasiparticle

19. Light Hole and Heavy Hole bands (holes with different m*)
Excitons in most of semiconductors are not observable at room temperature, because of the low binding energy
Quasiparticle in a box
Light Hole and Heavy Hole bands
(holes with different m*)
Transport energy without transporting charge
meV energies
LH=light hole band
HH=heavy hole band, refers to relative effective masses of the holes (two energy bands have different effective masses)
On the other hand, excitonic emission is very important for opto-electronic applications, as it is narrow and highly energetic & can transport energy

20. Let’s focus on Effective Mass
Confused electron
Confused hole
Let’s focus on Effective Mass

21. Comparing free electrons and the electron quasiparticle
Free electron Electron quasiparticle
While electrons scatter, we treat electron quasiparticles like free electrons with m*.
The renormalized mass effectively takes into account all interactions (with crystal, electrons, phonons)
Thus, electron quasiparticles don’t scatter.
Negative effective mass just means that the band is flipped the other way (E proportional to – k^2).

22. EFFECTIVE MASS
Real metals: electrons still behave like
free particles, but with “renormalized” effective mass m*
In potassium (a metal), assuming m* =1.25m gets the correct
(measured) electronic heat capacity
Physical intuition: m* > m, due to “cloud” of phonons and other excited electrons that slow it down (add mass)
Interactions with the periodic crystal, electron-electron interactions and electron-phonon interactions renormalize the elementary excitation to an “electron-like quasiparticle” of mass m*
Fermi Surface

23. Physical Meaning of the Band Effective Mass
The effective mass is inversely proportional to the curvature of the energy band.
If our bands were perfectly quadratic, this would give a constant effective mass. However, we know that’s not true.
Plot both second derivative and inverse of second derivative.
Effective mass picture breaks down at zone boundary where dE/dk=0.
Negative mass, means go in opposite direction to force applied
Near the bottom of a nearly-free electron band m* is approximately constant, but it changes dramatically near the inflection point and even becomes negative(!) near the zone edge. What does that mean for electron near BZE?

24. Because the energy bands are different along different directions, the effective mass depends on which direction in k-space we are “looking”
Effective Mass
Fcc shown
Split off band predicted by k dot p theory. Due to spin orbit coupling
Heavy and light holes have different bands so different effective masses (also different from electrons)

25. Group: Find the effective mass tensor for electrons in a simple cubic tight-binding band at the center , face center X, and at the corner R of the Brillouin zone.
See page of notes (write on board)

26. Besides Getting the Energy Bands, How Could You Measure the Effective Mass?

27. Deflection of Electrons in a Uniform Magnetic Field (1)
The force F acting on an electron in a uniform magnetic field is given by
Since the magnetic force F is at a right angle to the velocity direction, the electron moves round a circular path.

28. Deflection of Electrons in a Uniform Magnetic Field (2)
The centripetal acceleration of the electrons is
Hence
which gives
Cyclotron frequency

29. Effective Mass Measure effective mass:
cyclotron resonance v. crystallographic direction
-Measure the absorption of radio frequency energy v. magnetic field strength.
Put sample in a microwave resonance cavity at <40 K and adjust the rf frequency until it matches the cyclotron frequency. Will see a resonant peak in the energy absorption.

30. Effective Masses in Semiconductors
Note that the mass of the light hole band approaches the heavy hole band away from k=0
This is Germanium. =
m* determined by cyclotron resonance (rf) at low carrier concentration = =

31. Indirect Bandgap Direct Bandgap
Spin-orbit split band energy ~ 10 – 1000 meV, increases with decreasing Eg.

32. m*  Eg for direct-gap crystals
For InSb, InAs, InP

33. Dynamics of Bloch Electrons (electrons in a periodic potential)
Metals have partially-filled upper bands, while semiconductors and insulators have filled upper bands. What is the qualitative difference between filled and partially-filled bands?
Current density for single electron:
For a collection of electrons:
But the symmetry of the energy bands requires:
Thus we conclude:
n is electron density
So for a filled band, which has an equal number of electrons with k positive and negative,
Filled energy bands carry no current! We will see that this is true even when an electric field is applied.
Note: the electrons in filled bands are not stationary…there are just the same number moving in each direction, so the net current is zero.

34. Electron in an Electric Field
An external electric field causes a change in the k vectors of all electrons:
If the electrons are in a partially filled band, this will break the symmetry of electron states in the 1st BZ and produce a net current. But if they are in a filled band, even though all electrons change k vectors, the symmetry remains, so J = 0. E kx
When an electron reaches the 1st BZ edge (at k = /a) it immediately reappears at the opposite edge (k = -/a) and continues to increase its k value. v kx
As an electron’s k value increases, its velocity increases, then decreases to zero and then becomes negative when it re-emerges at k = -/a!!
Thus, an AC current is predicted to result from a DC field! (Bloch oscillations)

35. Do We Observe This?
Not until fairly recently, due to the effect of collisions on electrons in a periodic but vibrating lattice.
In both experiments the periodic potential was fabricated in an artificial way to minimize the effect of collisions and make it possible to observe the Bloch oscillations of electrons (or atoms!).

36. Eg & m* versus T
Strong dependence of Eg on T Weak dependence of m* on T

37. Absorption/emission of photon - Vertical transition
Absorption/emission of phonon - horizontal transition
The onset of indirect transition should include both Eg-Ephonon, Eg+Ephonon,
In any transition, K must be conserved as well as E.
A direct gap semiconductor; on the left is the E-K diagram, and on the right the conventional energy band diagram.
An indirect gap material (so called because conduction band minimum and the valence band maximum do not occur at the same value of K).

38. Past Homework Hint: How do constants affect the effective mass?
Is there a way to get around dealing with M-1?
Any way to simplify matrix given H direction?
The formula for an ellipse (the orbit) may help. Why an ellipse?