1. Defects and semiconductors
Learning Outcomes:
By the end of this section you should:
know about vacancies, interstitials and Frenkel defects
be able to calculate the energy of vacancy formation from quenching data
be able to describe the different types of line defect and use the Burgers’ vector
know the difference between intrinsic and extrinsic conduction, p- and n-type silicon and donor and acceptor doping
be able to calculate number densities of holes and electrons in doped materials

2. Defects
Up to now we have considered perfect crystals, i.e. crystals with perfect periodic arrangements.
Most “good” crystals show very little departure from this idea, e.g. silicon single crystals can be grown without defects over a range of several mm
This sounds small but is about 10 million unit cells!
However, defects are very important in processing and for optical and electrical properties.

3. 1. Vacancies A vacancy is the absence of an atom in the lattice.
In ionic crystals (e.g NaCl) vacancies occur in pairs (Na + Cl) so that charge balance is maintained.
Also called a Schottky Defect.
Vacancies allow diffusion through the crystal:
Vacancy : point defect - associated with a point in the crystal

4. Vacancies
Vacancies are not energetically favourable - the number of vacancies increases with temperature (i.e. putting energy into the system)
Mathematically, for a crystal containing N atoms, there is an equilibrium number of vacancies, n, at temperature T (in K) given by:
where EV is the energy of vacancy formation and kB is Boltzmann’s constant. Applies to pairs also.

5. Diffusion Similary, the diffusion coefficient, D, is given by:
where ED is the energy of diffusion and DO is a diffusion constant specific to the element.
Strictly this applies only to self-diffusion, that is diffusion in an elemental substance.

6. Quenching
Non-equilibrium concentrations of vacancies may be obtained by rapidly cooling (quenching) metals from high temperatures.
These defects can cause additional resistivity proportional to the number of defects:
where C is a proportionality constant.
R is the relative increase in resistance at low temperature after quenching from the temperature T.

7. Uses
so:
y = c mx
EV can be obtained from a graph of lnR against (1/T)

8. Example - Gold

9. 2. Interstitials
Previously we discussed small tetrahedral and octahedral interstitial atoms within the close packed structure.
If the interstitial atom is the same size as the close packed atoms, then considerable disruption to the structure occurs.
Again, this is a point defect and requires much energy

10. 3. Frenkel Defects
Often a vacancy and interstitial occur together - an ion is displaces from its site into an interstitial position.
This is a Frenkel Defect (common in e.g. AgCl) and charge balance is maintained.
Frenkel defects can be induced by irradiation of a sample

11. 4. Line Defects - 1. Stacking Faults
We discussed h.c.p which has sequence ABABABA and c.c.p. which has sequence ABCABCA.
A stacking fault occurs when the sequence goes wrong, e.g. ABCBCABCABC (A missing) or ABCABACABC (extra A)
Often these do not extend right across the plane, e.g.
This is also known as a partial dislocation

12. Line Defects - 2. Edge dislocations
Originally proposed to account for mechanical strength in crystals.
Consists of an extra plane of atoms which terminates within the crystal. This distorts the local environment.

13. Burgers Vector
If the dislocation was not present, then atom at A would be at A’
We define a vector B which shows the displacement of A due to the dislocation.
B is known as the Burgers’ Vector.
For an edge dislocation, the Burgers’ vector is perpendicular to the dislocation

14. Slipping
Such defects are produced by part of the crystal “slipping” with respect to the rest.
Consider a close packed structure:
For the top layer to slip to the right, to another close packed position, it must pass through a non-equilibrium position

15. Line Defects - 3. Screw dislocations
Here there are no extra planes - the defect appears as though part of the crystal has been cut in two, then shifted down on one side of the cut.

16. Burgers’ Vector
In this case, A would have been at A’ had the dislocation not occurred.
The Burgers’ Vector B is hence parallel to the direction of the screw dislocation.
Screw dislocations are important in the growth of single crystals since they provide nucleation sites for the growth of a new layer

17. Line Defects - 4. Twinning
Crystals are often grown with a fault in which one region of the crystal is a mirror image of the other:
In c.p. structures, twins are produced by stacking faults
ABCABCBACBA
Here C is the twin plane
Polymorphic compounds (i.e. ones with more than one crystal structure) are prone to twinning, e.g. YBa2Cu3Od

18. 4. “Impurities”
Preparing pure crystals is extremely difficult - often foreign atoms enter the structure and substitute for “native” atoms - often by contamination from container
This can have a large effect (either detrimental or beneficial) on the properties of the crystal. We can also add impurities (or dopants) deliberately.
An important example is that of silicon.

19. Silicon
Silicon is a group IV element and, like carbon, bonds to four nearest neighbours: T
At elevated temperatures bonds are broken to produce a (positive) gap - known as a hole - and a conduction electron.
This is known as the intrinsic effect in semiconductors

20. Doped Silicon
If we take a group V element (e.g. As) and substitute (at low levels) for Si there is a spare electron for conduction and no positive hole:
This process is known as “doping”. Arsenic acts as an electron donor to Si, making it easier to conduct electricity.
Si doped with As is an extrinsic semiconductor and because the electron is negative this is an n-type semiconductor

21. Doped Silicon
If we take a group III element (e.g. B) and substitute (at low levels) for Si there is a positive hole and no conduction electron
Boron acts as an electron acceptor to Si.
Holes can move by diffusion - “hopping” into the hole leaves behind a new hole.
Again this is an extrinsic semiconductor and because the hole is positive this is a p-type semiconductor

22. The band picture
Red = filled energy states, light blue = empty, white = forbidden (energy gap)
Green dotted = donor states, blue dotted = acceptor states
Bottom band = valence band, top band = conduction band.
Points to note: Energy gap is big (>3eV) in an insulator, ~1eV in a semiconductor

23. (where ni = intrinsic no. of electrons)
The numbers!
In the intrinsic material, electrons and holes are created as pairs
If n=no. of electrons, p=no. of holes, then
n=p=ni
(where ni = intrinsic no. of electrons)
We can state that
pn=ni2
As we add donors, the number of holes will decrease. Thus this condition is generally true.

24. Charge neutrality
We define ND and NA as the number of donor/ acceptor atoms. At 300K, these are fully ionised
All materials, doped or intrinsic, have the condition of charge neutrality
p + ND = n + NA
Combine the 2 equations and solve as quadratic, e.g.
n = p + ND - NA = ni2/p
 p2 + p(ND - NA) - ni2 = 0

25. Solution and similar for n. Use with care! Example
Find the electron and hole densities in a semiconductor if ni = 1016 m-3, ND = 1020 m-3 and NA = 0.
(a) calc n using quadratic
(b) calc p using quadratic
(c) calc p from pn = ni2

26. Long Solution n = ½ (1020 + (1040 + 41032) ) = 1020 m-3
p = ni2/n = 1012 m-3.
But if we use
p = ½ ( ( 1032) )
= ½ (  1020) (!!)
Could miss this

27. Useful simplifications
We define “effective doping” in cases where one type dominates:
1) if NA > ND then we define
NA’ = NA - ND (effective acceptor doping)
2) if ND > NA then we define
ND’ = ND - NA (effective donor doping)

28. Useful simplifications
Then:
1) If ND = NA dopants cancel n = p = ni
2) If NA’ >> ni p = NA’ n = ni2/NA’
p-type semiconductor
3) If ND’ >> ni n = ND’ p = ni2/ND’
n-type semiconductor

29. Example
Example
Find the electron and hole densities in a semiconductor if ni = 1016 m-3, ND = 1020 m-3 and NA = 1018 m-3.
ND’ = = 9.9  1019 m-3 = n
p = ni2/n = 1.01  1012 m-3

30. Summary Most crystals contain defects
Extra vacancies can be produced by quenching; this can produce an increase in resistivity which can be calculated.
Line defect formation can be described using the Burgers’ Vector, B
Defects can be used to advantage, e.g. doped silicon
We can calculate numbers of holes and electrons in doped materials  4th year