Semiconductor Conductivity
Ch. 1, S
• 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 for now!
A quantitative definition will come later!
• Holes are often treated as “positively charged electrons”.
How is this possible?
Are holes really particles?
We’ll eventually answer both of these questions
as the course proceeds.
A Qualitative Picture of Holes
(from Seeger’s book)
An idealized, 2 dimensional, “diamond”
lattice for e- & e+ conduction
“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”)
“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 
Crude Analogy: CO2 Bubbles in Beer!
Beer
Glass
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”!
Thermal Pair Generation & Annihilation
• Now: A classical Treatment. 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, …)
• Schematically:
e-
+
e+
 Eg
 e-, e+ Pair Generation
Recombination 
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.
T Dependences of e- & e+ Concentrations
• Define: n  concentration (cm-3) of ep  concentration (cm-3) of e+
• Can show (& we will): 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
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
Intrinsic Concentration vs. T
Measurements/Predictions
Note the different scales on the right & left figures!
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
Doped Materials
• The 5th e- is really not free, but is loosely bound with energy
We’ll show later how
to calculate this!
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
• 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.
n vs. 1/T
Almost no ionized
donors & no
intrinsic carriers
lllll
  High T
Low T  
n vs. T
• 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
• 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!
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)
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!