1. VII. Semiconducting Materials & Devices
Band Structure and Terminology
Intrinsic Behavior
Optical Absorption by Semiconductors
Impurity Conductivity
Extrinsic Behavior
Hall Effect and Hall Mobility
The Diode: A Simple p-n Junction

2. A. Band Structure and Terminology
Semiconductors and insulators have qualitatively similar band structures, with the quantitative distinction that the band gap Eg > 3.0 eV in insulators.
conduction band
valence band
Energy band diagram in k-space
“Flat-band” diagram in real space
Ec = conduction band edge
Ev = valence band edge

3. Fermi-Dirac Distribution Function
The important external parameter that determines the properties of a semiconductor is the temperature T, which controls the excitation of electrons across the band gap in a pure (intrinsic) semiconductor.
Fermi-Dirac distribution function
The probability for an electron to be in an energy level  at temperature T
 = chemical potential  EF for T << TF
For nearly all T of interest:
This is the Maxwell-Boltzmann (classical) limit of Fermi-Dirac statistics
So we can approximate:

4. B. Intrinsic Behavior
“Intrinsic” means without impurities. Electrical conductivity is zero at T = 0, but for T > 0 some electrons are excited into the conduction band, also creating holes (H+) in the valence band. In general the conductivity can be written (using the nearly FEG model):
The conductivity is controlled by the magnitude of n and p, which rise exponentially as T increases. The relaxation times are only dependent on 1/T, and this dependence is often ignored because the exponential behavior dominates.
  as T  due to increased scattering ( )
high-T (intrinsic) region
low-T (extrinsic) region
Experimentally we find that for a pure semiconductor:

5. Intrinsic Carrier Statistics
It is relatively simple to calculate n(T) and p(T) for the intrinsic region, where the conductivity is caused by excitation of e- across the energy gap:
e- in the conduction band:
Ec = Eg 
h+ in the valence band:
Ev = 0
For parabolic bands the density of states are:
Think: why are we justified in assuming parabolic E(k) here?

6. Intrinsic Carrier Statistics, cont’d.
We can write the density of states per unit volume:
And now calculate the carrier concentration n(T):
Now rearrange cleverly and pull out a factor of (kT)1/2:
Which allows:

7. Intrinsic Carrier Statistics, cont’d.
Now make a variable substitution:
The integral becomes:
So finally:
Whew! And next we can do the same calculation for holes to get p(T)!

8. Intrinsic Hole Carrier Statistics
Now for holes in the valence band:
And since
we have
Just as before, calculate the carrier concentration p(T):
Now rearrange :
Replace –E with E and flip limits due to minus sign:
Pull out a factor of (kT)1/2:

9. Intrinsic Hole Carrier Statistics, cont’d.
Now make the variable substitution:
The integral becomes:
again!
So finally:
So far these relations for n(T) and p(T) are true for any semiconductor, with or without impurities. It is very convenient to calculate the product np:

10. Intrinsic Carrier Statistics: Results
Now for an intrinsic semiconductor (or in the intrinsic region of a doped semiconductor) ni = pi, so:
And equating the earlier expressions for n and p:
This gives an expression for (T):
So the chemical potential, or Fermi level, has some dependence on T, but if mh and me are similar, then this is very small.

11. Carrier Mobility
The total conductivity, including both the electron and hole contributions, is:
It is common to define a quantity that expresses the size of the drift speed for each type of carrier in an electric field E:
(Note: the carrier mobility is directly related to the switching speed of a solid-state electronic device)
Definition of carrier mobility:
Earlier FEG result:
Now we can rewrite the total conductivity as:
Experiment shows that  has a power-law temperature dependence:
Thus the exponential temperature dependence of n and p dominates, and we can approximate the intrinsic conductivity
So a plot of vs. 1/T gives a straight line with slope –Eg/2k. Conductivity measurements allow us to determine Eg!

12. C. Optical Absorption by Semiconductors
Examine the following calculated 3-D band structures for semiconductors Si and GaAs. What difference(s) do you see? Si
GaAs Ec Ec Ev Ev
Indirect band gap
(kgap  0)
Direct band gap
(kgap = 0)

13. Optical Absorption and Conservation Laws
Absorption of a photon by a semiconductor can promote an electron from the valence to the conduction band, but both energy and momentum must be conserved:
For semiconductors Eg  1 eV so the photon wavevector can be estimated:
But this is utterly tiny compared to a typical BZ dimension:
So essentially we have:
A direct-gap (vertical) transition

14. Direct vs. Indirect Gap Semiconductors
But for indirect gap semiconductors it is clear that:
So for an indirect gap transition momentum can only be conserved by absorption or emission of a phonon (lattice vibration)
To estimate a typical phonon energy, we know:

15. Optical Absorption: Experimental Results
Experimental absorption coefficients () are measured to be:
While for an indirect gap material with a direct transition at a slightly higher energy:

16. D. Impurity Conductivity in Semiconductors
Consider two types of substitutional impurities in Si:
pure Si:
(each line represents an e-)
Si:P
weakly bound extra electron
donor energy level
(n-type material)
Si:B
missing electron
acceptor energy level
(p-type material)

17. Donor Impurities in Semiconductors
We can estimate the ionization energy of a pentavalent donor impurity using the Bohr model:
Bohr model for H: +e
-e, m r Ed
For an electron orbiting a positive ion inside a semiconductor, what changes must we make in the Bohr model equations?
periodic potential (effective mass):
dielectric medium:
So in a semiconductor:

18. Donor Ionization Energy
Assuming that the electron is initially in its lowest energy level, the donor ionization energy Ed is:
For Si we can use representative values of the effective mass and dielectric constant to obtain: P As Sb Si 45 49 39
All within a factor of two of our rough estimate!
Experimental data reveal ionization energies (in meV):
The orbital radius is predicted to be:
So this electron moves through a region that includes hundreds of atoms, which supports the use of the dielectric constant of the bulk semiconductor.

19. Acceptor Impurities in Semiconductors
What happens when the impurity atom is trivalent?
At 0 K the acceptor level is empty, so a “hole” is bound to the impurity atom.
However, the energy Ea is so small (50 meV in Si) that at room T electrons in the valence band bound to other Si atoms can be excited into the acceptor level, leaving behind a mobile hole in the valence band. Ea
Summary: Both donor (P, As) and acceptor (B, Ga) impurities provide an easy way to increase either n or p even at low T. When such impurity-related carriers dominate the electrical properties, the semiconductor material displays extrinsic behavior.
Note: If impurity concentration is very large, the Bohr orbits (wavefunctions) of the donor electrons can overlap and form an “impurity band” that extends throughout the material. This leads to a so-called insulator-metal transition and causes an abrupt increase in the conductivity (see problem 10.5).

20. E. Extrinsic Behavior and Statistics
Let’s consider donor impurities in a semiconductor (n-type):
Nd = concentration of donor atoms
Nd+ = concentration of ionized donor atoms
Nd0 = concentration of neutral donor atoms
Now in the presence of a large donor e- concentration, then n >> ni so p must decrease in order to keep the product np = constant. What physical process causes p to decrease?
Essentially the large number of e- in the conduction band will be sufficient to fill most available holes in the valence band, so that:
and
Ev = 0 Eg
Eg-Ed

21. Extrinsic Carrier Statistics
Now solve for the electron concentration n:
Now from our earlier treatment of intrinsic behavior:
Equating the expressions for n:
You can always use this exact master eqn. to solve for  and thus n, but you have to do it numerically.

22. Limits of Low and High Impurity Concentrations
This discussion is relevant to several HW problems in Myers (see 10.4, 10.8, 10.9). It provides simple approximations for n and  corresponding to very small Nd and very large Nd.
We will develop approximations to simplify the solution of this eqn.
Now consider two extreme limiting cases:
Ev = 0 Eg 
Nd << n In Si at 300K (high T limit) 
Since Eg >> 2kT at room temp, this means
And solving for :
Thus,

23. Limits of Low and High Impurity Concentrations
Ev = 0 Eg
Eg-Ed 
2. Nd >> n (low T limit) 
We can neglect the “1” in the denominator here:
And now solve for : 
Does the low T limit make sense?
Substituting into the above eqn. for n:
Thus,

24. Summary of Impurity Semiconductor Behavior
Now our schematic plot of ln  vs. 1/T is even easier to understand:

25. Extrinsic Behavior for p-Type Semiconductors
Let’s consider acceptor impurities in a semiconductor:
Na = concentration of acceptor atoms
Na- = concentration of occupied acceptor levels
Na0 = concentration of neutral (unoccupied) acceptor levels
Now in the presence of a large hole concentration, then p >> pi so n must decrease in order to keep the product np = constant. What physical process causes n to decrease?
Essentially most of e- in the conduction band will fall down to fill up holes in the valence band, so that:
and
Ev = 0 Eg Ea

26. Extrinsic p-Type Carrier Statistics
Now from our earlier treatment of intrinsic behavior:
Equating the expressions for p:
Again, you can always use this exact master eqn. to solve for  and thus p, but you have to do it numerically.

27. Limits of Low and High Impurity Concentrations
Here we provide simple approximations for p and  corresponding to very small Na and very large Na.
We will develop approximations to simplify the solution of this eqn.
Now consider two extreme limiting cases:
Ev = 0 Eg 
Na << p0 (high T limit)
As before, we argue that  is near Eg/2 as in the intrinsic case: 
Since Eg >> 2kT at room temp, this means
Thus,
And solving for :

28. Limits of Low and High Impurity Concentrations
Ev = 0 Eg Ea 
2. Na >> p (low T limit) 
We can neglect the “1” in the denominator here:
And now solve for : 
Does the low T limit make sense?
Substituting into the above eqn. for p:
Thus,

29. F. Hall Effect and Mobility
The Hall effect is easier to measure in semiconductors than in metals, since the carrier concentration is smaller:
When one carrier dominates, we have a Hall coefficient:
where
Hall measurements can tell us whether a semiconductor is n-type or p-type from the polarity of the Hall voltage: B I w + - - +
n-type
p-type
When one carrier dominates, we can write the conductivity:
Measuring RH and  will thus give: sign, concentration, and mobility of carrier,
So the mobility can be written:

30. General Form of Hall Coefficient
For a semiconductor with significant concentrations of both types of carriers:
So if holes predominate (ph > ne ), RH > 0 and the material is said to be p-type, while if RH < 0 (as for simple metals), the material is said to be n-type.

31. G. The Diode: A Simple p-n Junction
When p- and n-type materials are fabricated and brought together to form a junction, we can easily analyze its electronic properties. p n Na Nd
Near the junction the free electrons and holes “diffuse” across the junction due to the concentration gradients there. As this happens, a contact potential  develops. p n - +
The field E due to the contact potential inhibits further flow of electrons and holes toward the junction, and equilibrium is established at finite .
depletion region E

32. Physics of a Simple p-n Junction
We can also describe the situation in terms of flat band diagrams.
Ecp
Evp
Ecn
Evn
Initially (before equilibrium)
Ecn
Evn
Ecp
Evp e
Finally (equilibrium established)
Here we see “band bending” in equilibrium. This reflects the potential at the junction and the equalization of the chemical potential throughout the system.

33. Physics of a Simple p-n Junction
In this dynamic equilibrium two types of carrier fluxes are equal and opposite:
1. recombination flux: electrons in the n-type region and holes in the p-type region “climb” the barrier, cross the junction, and recombine with h+/e- on the other side.
2. generation flux: thermally-generated electrons in the p-type region and holes in the n-type region are “swept” across the junction by the built-in electric field there.
We can picture the carrier fluxes (currents):
In equilibrium at V=0:
Jng
Jnr
Jpr
Jpg
p-type
n-type

34. A Simple p-n Junction With Applied Voltage
Now when an external voltage V is applied to the junction, there are two cases:
1. Forward bias: electrons in the n-type region are shifted upward in energy
Generation currents are not affected since they depend on excitation across band gap: e eV
Recombination currents are increased by a Boltzmann factor, since they depend on carriers climbing the potential energy step at the junction (and Maxwell-Boltzmann statistics applies):

35. Current-Voltage Relation for A Simple p-n Junction
We can now calculate the net current density from both holes and electrons:
I > 0 so current flows from p  n
Is = “saturation current”
2. Reverse bias: electrons in the n-type region are shifted downward in energy e eV
Here the only difference is that recombination currents are decreased by a Boltzmann factor, which just changes the sign of V in the exponential terms. So the resulting current is:
I < 0 so current flows from n  p (leakage current)
We can express both cases in one “ideal diode equation” if we define forward bias to be V > 0 and reverse bias to be V < 0:

36. Current-Voltage Relation for A Real Diode
What about in the real world? I V
-Is
Vbr
The ideal diode equation is approximately correct, but we have made some assumptions that are not rigorously true, and have neglected other effects, so in a real diode we see behavior like this:
Mechanisms for breakdown in reverse bias include:
1. Zener breakdown: a large reverse bias allows tunneling of electrons from valence band of p-type region to conduction band of n-type region, where they can carry current!
2. Avalanche breakdown: electrons generated in p-type region and swept across the junction acquire enough kinetic energy to generate other electrons, which in turn generate more, etc.