1. Why does glass absorb IR but not visible?UV
Visible
Onset due to band gap, which varies with type of glass
Much of the infrared absorption of standard glass is due in part to residual –OH groups, which largely act like water in terms of their vibrational absorption spectra. “An ongoing challenge with these glasses is removal of impurities such as hydroxyl ions (OH−), which are characterized by absorption in the wavelength region of interest.” (
Asked by Pedram. (attach to last class)
Not a huge deal for your car, but think about laser systems!

2. Learning Objectives for Today: Adding math to our discussion of semiconductors
After today’s class you should be able to:
Apply density of states equations to semiconductors
Find the chemical potential with temperature
Understand n and p doping in semiconductors and their approximate energy levels
Determine the number of electron and hole carriers in a doped semiconductor
Gaps may happen in the next lecture

3. The most important property of semiconductors: Conductivity
What do we need to know in order to predict the conductivity for semiconductors?
For metals?
How did we find n, the density of conduction electrons, precisely for metals?
 = n e2  / m*

4. Electron and Hole Densities
To accurately determine the number of carriers in each band, we will modify what we derived for free electrons in metals
Electron Fermi-Dirac distribution function means what?
Actually, we normally approximated it from valence electrons, but we could do this to be more accurate.

5. Chemical potential  or Fermi level
At a temperature T the probability that an available electron state is occupied is given by the Fermi-Dirac function
The chemical potential, m , is the energy for which f = ½
(50% likely to be occupied).
Fermi energy: all energy states are occupied below EF at T = 0 (metal).
If intrinsic, m is within the energy gap. In the conduction band
Therefore
Typically called chemical potential to separate from the Fermi level in metals which is defined a little differently
Note that if the density of states is not exactly symmetric about the centre of the band gap, then the chemical potential does not have to be exactly in the centre of the band gap. However, under such circumstances, it will still be extremely close to the centre of the band gap whatever the temperature, and for all practical purposes can be considered to be in the centre of the band gap.
Then, how should f(E) change with temperature in a semiconductor?

6. Number of electrons in conduction band
Silicon
@ 300K
n ~ 2x1016 m-3
log plot
Note Units are cm-3
Electron density increases ~ exponentially with temperature
(exponential would be straight line in log plot)

7. Electron and Hole Densities
To determine the number of carriers in each band, we will modify what we derived for free electrons
Fermi-Dirac distribution function for conduction (c) and valence (v) bands
What dependence on energy should the DOS have?
See notes just before making approximation that just the exponential
Note that g is density of states. I have some picture files in here so I haven’t changed all of them yet. Sorry.

8. Density of States
Assume: bottom of conduction band and the top of valence band parabolic
conduction band E = Ec + 2k2/2me*
valence band E = Ev - 2k2/2mh*
Conduction
band
Valence
Conduction Band
Ec = Eg
Ev = 0
Valence Band
Valence Band
Conduction Band Eg E
D(E) E
Note that if the density of states is not exactly symmetric about the centre of the band gap, then the chemical potential does not have to be exactly in the centre of the band gap. However, under such circumstances, it will still be extremely close to the centre of the band gap whatever the temperature, and for all practical purposes can be considered to be in the centre of the band gap.
DOS 1

9. Electron density in conduction band
Density of states
Distribution function
To find the electron density of occupied states:

10. Number of holes and electrons
Exactly same argument for holes in the valence band gives
Distribution function
Total density of hole states
Group: Find the product NeNh.
As Ne=Nh(intrinsic) we can say:
This result is important; known as law of mass action; find carrier density without µ.
True for both intrinsic & extrinsic semiconductors.

11. Chemical Potential
For intrinsic SCs, determine an approach that will find  (the energy for which f = ½). When is  = Eg/2? Results of recent discussion:

12. Intrinsic semiconductors Ne=Nh
In pure “intrinsic” semiconductors the electrons and holes arise only from excitation across the energy gap.
Therefore Ne = Nh
(Do algebra to derive this equation on the board)
Note that if the density of states is not exactly symmetric about the center of the band gap, then the chemical potential does not have to be exactly in the center of the band gap. However, under such circumstances, it will still be extremely close to the center of the band gap whatever the temperature, and for all practical purposes can be considered to be in the center of the band gap.
Silicon Eg=1.15eV
m*e = 0.2me & m*h = 300K. Gives
Chemical potential near middle of gap

13. Doping  Extrinsic Semiconductors
Semiconductors can be easily doped
Doping is the incorporation of impurities (different atoms) into a semiconductor in a controlled manner
In a doped material, Ne  Nh !
Impurities change the electrical properties (e.g. conductivity) of the material, which can be useful for devices
In fact, it can be hard to manufacture pure semiconductors. We’ve only gotten good at Si due to huge amounts of investment

14. Impurties have a big effect on semiconductors Why?
~ cm-3
Close enough to form bands when wavefunction overlap
Impurities add states within gap, excitation more likely
Impurities can get close enough to form their own bands
Causes the material to become a metal (Mott transition)

15. Donors Let’s use Silicon (Si) as an example
Substitute one Si (Group IV) atom with a Group V atom (e.g. As or P)
Si atoms have four valence electrons that participate in covalent bonding
When a Group V atom replaces a Si atom, it will use four of its electrons to form the covalent bonding
What happens with the remaining electron?

16. Donors
The remaining electron will not be very tightly bound, and can be easily ionized at T > 0K
Ionized electron is free to conduct
In terms of the band structure, this electron is now in the conduction band
Such Group V impurities are called Donors, since they “donate” electrons into the Conduction Band
Semiconductors doped by donors are called n-type semiconductors (extra electrons with negative charge)

17. This crystal has been doped with a pentavalent impurity.
The free electrons in n type silicon support the flow of current.

18. Donors: Energy Levels
Such impurities create “shallow” levels - levels within the band gap, close to the conduction band
A donor is similar to a hydrogen atom (called hydrogenic donors)
A positive charge with a single electron within its potential
the small ionization energy means a sizable fraction of donor atoms will be ionized at room temperature 
Because most of the electrons are busy in bonds

19. Another View: Hydrogenic Donors
An electron added to an intrinsic semiconductor at T=0 would go into the lowest empty state i.e. at the bottom of the conduction band.
When one adds a donor atom at T=0
the extra electron is bound to the positive
charge on the donor atom.
The electron bound to the positive
ion is in an energy state ED = Eg- DE
where DE is the binding energy.
How much is E?
Conduction Band
Ec = Eg DE
+ve ion -e ED
Valence Band
Ev = 0

20. Treat Similar to Hydrogen
Employ the solution for hydrogen atom
Consider the energy of the bottom of the conduction band to be zero – “free” electron
Substitute the effective mass for the electron mass
Charge shielded in a solid so modify the Coulomb interaction by the dielectric constant of the solid (dielectric constant for free space, κ = 1)

21. Magnitude of binding energy
Similar to a hydrogen atom. Ground state wavefunction is
The Bohr radius, a0 = 4ro2/mee2 determines the spatial extent of the wavefunction. Hydrogen atom (= 1 ) a0 = 0.53 Å.
+ve ion -e
Binding energy of an electron in the ground state of a hydrogen atom is  
Typical Semiconductor:
  ~ 50 Å
me*  ~0.15 me and  ~15.
  ~10 meV.

22. Examples Ge: me* = 0.04m0; κ = 16  ED = -2.1 meV
GaAs: me*= 0.067m0; κ = 13  ED = -5.4 meV
Si: me* = 0.26m0; κ = 12  ED = -25 meV
ZnSe: me* = 0.21m0; κ = 9  ED = -35 meV

23. Chemical Potential in n-type
At low temp, the chemical potential must lie somewhere between the donor levels and the conduction band.
At higher temp, when the donor level is depleted of electrons, the contribution from intrinsic electrons to the electrical conductivity becomes more substantial and the chemical potential tends towards ~Eg/2.

24. Acceptors Again use example: silicon (Si)
-ve ion +e
Acceptors
Again use example: silicon (Si)
Substitute one Group III atom (e.g. Al or In) with a Si (Group IV) atom
Si atoms have 4 electrons for covalent bonding
When a Group III atom replaces a Si atom, it cannot complete a tetravalent bond scheme
A hole is formed.
If the hole leaves the impurity, the core would be negatively charged, so the hole created is then attracted to the negative core
At T = 0 K this hole “stays” with atom – localized hole

25. Acceptors
At T > 0 K, electron from the neighboring Si atom can jump into this hole – the hole starts to migrate, contributing to the current
We can say that this impurity atom accepted an electron, so we call them Acceptors
Acceptors accept electrons, or they “donate holes”
Such semiconductors are called p-type semiconductors since they contribute positive charge carriers

26. This crystal has been doped with a trivalent impurity.
The holes in p type silicon contribute to the current.
Note that the hole current direction is opposite to electron current
so the electrical current is in the same direction

27. Acceptor: Energy Levels
Such impurities create “shallow” levels - levels that are very close to the valence band
Energy to ionize the atom is still small
They are similar to “negative” hydrogen atoms
Such impurities are called hydrogenic acceptors

28. Acceptor energy levels are bigger. Why?
Examples
Since holes are generally heavier than electrons, the acceptor levels are deeper (larger) than donor levels Why the range?
The valence band has a complex structure and this formula is too simplistic to give accurate values for acceptor energy levels
Acceptor energy levels are bigger. Why?
Ge: 10 meV
Si: 45 – 160 meV
GaAs: 25 – 30 meV
ZnSe: 80 – 114 meV
GaN: 200 – 400 meV
Acceptor and donor impurity levels are often called ionization energies or activation energies
Why acceptor energies bigger?

29. Impurity Bands
Have considered the impurities as isolated atoms. Reasonable as doping level normally ~ one donor per 106 semiconductor atoms.
At very high donor concentrations, one has substantial overlap between the donor or acceptor wavefunctions.
Above a critical doping level one has an impurity energy band with a finite conductivity.
Electron density at which this “metal insulator transition” occurs?
aB ~ 50 Å & lattice constant, a ~ 2.5 Å.
Need b ~ aB = 20a . i.e. one donor per 203 = 8000 semiconductor atoms
f(r) aB + + b

30. Mott Transition
~ cm-3
Impurities get close enough to form their own bands when wavefunctions overlap
Causes the material to become a metal
Also happens for high density Excitons  forms plasma

31. Carrier Concentrations in Extrinsic Semiconductors
The carrier densities in extrinsic semiconductors can be very high
Depends on doping levels and ionization energy of the dopants
Often both types of impurities are present
If the total concentration of donors (ND) is larger than the total concentration of acceptors (NA) have an n-type semiconductor
In the opposite case have a p-type semiconductor

32. How can we measure whether our material is more n or p type?

33. Standard Hall Effect Experiment
In a current carrying wire when in a perpendicular magnetic field, the current should be drawn to one side of the wire. As a result, the resistance will increase and a transverse voltage develops.
Current from the applied E-field
e- v
e+ v
Lorentz force from the magnetic field on a moving electron or hole
Top view
e- leaves + & – charge on
the back & front surfaces– Hall Voltage
The sign is reversed for holes
Top view—electrons drift from back to front
Standard Hall Effect Experiment

34. Charge Neutrality Equation
For an intrinsic semiconductor, nc = pv
Remind of what it is for intrinsic semiconductor pv=nc. Now we add a bunch of levels.
From poisson’s equation (the divergence of the electric field is proportional to the free charge density, which are charges from outside)
Here, I’m just adding up the charge.

35. Simplify: Consider n-type semiconductor with small NA~0
Conduction Band
Ec = Eg ED
What is nd at T=0?
At T>>0,
Valence Band
Ev = 0

36. Compensated semiconductor
ND donors per m3 and NA acceptors per m3
For ND > NA have an n-type S.C.
with n ~ ND - NA for T ~ 300K
For NA > ND have an p-type S.C.
(homework)
Conduction Band
Ec = Eg ED
NA electrons fall into acceptor states EA
Explain first what happens at T=0 and then discuss how changes as increase temperature.
Valence Band
Ev = 0
Don’t overthink final homework. Just wants you to explain where carriers go in terms of formula. Start by discussing where the levels are.

37. Extrinsic Semiconductors
Electrical Properties of Semiconductors can be altered drastically by adding very small amounts of suitable impurities to the pure crystals
Impurities: Atoms of the elements different from those forming solid
Substitutional: “foreign” atoms occupying the sites of host atoms
Interstitial: “foreign” atoms “squeezed” between regular sites crystal sites