2. Semiconductor Conductivity
It is well-known that in semiconductors, there are
Two charge carriers!
Electrons  e- & Holes  e+
What is a hole?
We’ll use a qualitative definition here!
A quantitative definition will come in
Physics 5335, Fall, 2018!
Holes are often treated as “positively charged electrons”.
How is this possible?
Are holes really particles?
Physics 5335 answers both of these questions.

3. Are holes really particles?
Holes are often treated as if they were
“positively charged electrons”.
How is this possible?
Are holes really particles?
We’ll eventually answer both of
these questions as the course
proceeds.

4. A Qualitative Picture of Holes
Consider an idealized, 2 dimensional, “diamond” lattice for e- & e+ conduction

5. “Thought Experiment”# 1
Add an extra e- (“conduction electron”) & apply an electric field E (the material is n-type: negative charge carriers) e-
E Field Direction 
 e- Motion Direction
(“almost free”)

6. “Thought Experiment”# 2
Remove an e- leaving, a “hole” e+ & apply an electric field E. (the material is p-type: positive charge carriers) e+
E Field Direction 
 e- Motion Direction
e+ Motion Direction 

7. A Crude Analogy: CO2 Bubbles in Beer!
Glass
Beer
Bubbles
g (gravity)
Bubble
Motion
We could develop a formal theory of bubble motion in the earth’s gravitational field. Since the bubbles move vertically upward, in this theory, the
Bubbles would need “negative mass”!

8. Thermal Pair Generation & Annihilation
Now: A classical Treatment. A simple, classical, statistical analysis. Later: Quantum Treatment
Define: Eg  Binding energy of a valence electron.
(In the Band Picture: This is the band gap energy).
Apply an energy Eg to an atom (from thermal or other excitation).
An e- is promoted out of a valence level (band) into a conduction level (band). Leaves a hole (e+) behind.
Later: e- - e+ pair recombine, releasing energy Eg
(in terms of heat, lattice vibrations, …)

9. Conductivity σ increases with increasing T.
Schematically:
e e+  Eg
This chemical “reaction” can go both ways.
As the temperature T increases, more e- - e+ pairs are generated & the electrical conductivity increases & the
Conductivity σ increases with increasing T.
 e-, e+ Pair Generation
Recombination 

10. T Dependences of e- & e+ Concentrations
Define: n  concentration (cm-3) of e-
p  concentration (cm-3) of e+
Can show: np = CT3 exp[- Eg /(kBT)]
(C = material dependent constant)
From the “Law of mass action” from statistical physics
In a pure material: n = p  ni (np = ni2)
ni  “Intrinsic carrier concentration”
ni = C1/2T3/2exp[- Eg /(2kBT)]
At T = 300K
Si : Eg= 1.2 eV, ni =~ 1.5 x 1010 cm-3
Ge : Eg = 0.67 eV, ni =~ 3.0 x 1013 cm-3

11. Also: Band Gaps are (slightly) T dependent!
It can be shown that:
Eg(T) = Eg(0) - αT
Si : α = 2.8 x 10-4 eV/K
Ge : α = 3.9 x 10-4 eV/K
But this doesn’t affect the T dependence of ni!
ni2 = CT3exp[- Eg(T)/(kBT)]
= Cexp(α/kB)T3exp[- Eg(0)/(kBT)]
= BT3exp[- Eg(0)/(kBT)]
where B = Cexp(α/kB) is a new constant prefactor

12. Intrinsic Concentration vs. T Measurements/Predictions
Note the different scales on the right & left figures!

13. Consider an idealized carbon (diamond) lattice
Doped Materials: Materials with Impurities! These are more interesting & useful!
Consider an idealized carbon (diamond) lattice
(we could do the following for any Group IV material).
C : (Group IV) valence = 4
Replace one C with a phosphorous.
P : (Group V) valence = 5
4 e-  go to the 4 bonds
5th e- ~ is “almost free” to move in the lattice
(goes to the conduction band; is weakly bound).
P donates 1 e- to the material
 P is a DONOR (D) impurity

14. But, it works both directions
Doped Materials
The 5th e- is really not free, but is loosely bound with energy
ΔED << Eg
The 5th e- moves when an E field is applied!
It becomes a conduction e-
Let: D  any donor, DX  neutral donor
D+ ionized donor (e- to the conduction band)
Consider the chemical “reaction”:
e- + D+  DX + ΔED
As T increases, this “reaction” goes to the left.
But, it works both directions
We’ll show how
to calculate this!

15. Consider very high T  All donors are ionized
 n = ND  concentration of donor atoms
(constant, independent of T)
It is still true that
np = ni2 = CT3 exp[- Eg /(kBT)]
 p = (CT3/ND)exp[- Eg /(kBT)]
 “Minority carrier concentration”
All donors are ionized
 The minority carrier concentration is T dependent.
At still higher T, n >>> ND, n ~ ni
The range of T where n = ND
 the “Extrinsic” Conduction region.

16. n vs. 1/T   High T Low T   Almost no ionized donors & no
intrinsic carriers
lllll
  High T Low T  

17. n vs. T

18.  B is an ACCEPTOR (A) impurity
Again, consider an idealized C (diamond) lattice.
(or any Group IV material).
C : (Group IV) valence = 4
Replace one C with a boron.
B : (Group III) valence = 3
B needs one e- to bond to 4 neighbors.
B can capture e- from a C
 e+ moves to C (a mobile hole is created)
B accepts 1 e- from the material
 B is an ACCEPTOR (A) impurity

19.  e+ moves when an E field is applied! NA  Acceptor Concentration
The hole e+ is really not free. It is loosely bound by energy
ΔEA << Eg
Δ EA = Energy released when B captures e-
 e+ moves when an E field is applied!
NA  Acceptor Concentration
Let A  any acceptor, AX  neutral acceptor
A-  ionized acceptor (e+ in the valence band)
Chemical “reaction”: e++A-  AX + ΔEA
As T increases, this “reaction” goes to the left.
But, it works both directions
Just switch n & p in the previous discussion!

20. “Compensated Material”
Terminology
“Compensated Material”
 ND = NA
“n-Type Material”
 ND > NA
(n dominates p: n > p)
“p-Type Material”
 NA > ND
(p dominates n: p > n)

21. Doping in Compound Semiconductors
This is MUCH more complicated!
Semiconductor compound constituents can act as donors and / or acceptors!
Example: CdS, with a S vacancy
(One S-2 “ion” is missing)
The excess Cd+2 “ion” will be neutralized by 2
conduction e-. So, Cd+2 acts as a double
acceptor, even though it is not an impurity!
 CdS with S vacancies is a p-type material, even with no doping with impurities!