ENE 311
Lecture 5
The band theory of solids
• Band theories help explain the properties of
materials.
• There are three popular models for band theory:
- Kronig-Penney model
- Ziman model
- Feynman model
Kronig-Penney Model
• Band theory uses V  0. The potential is periodic
in space due to the presence of immobile lattice
ions.
Kronig-Penney model
w
Ideal
Kronig-Penney Model
• Ions are located at x = 0, a, 2a, and so on. The potential
wells are separated from each other by barriers of
height U0 and width w.
• From time-independent Schrödinger equation in 1dimension (x-only), we have
(1)
Kronig-Penney Model
• For this equation to have solution, the following must
be satisfied
(2)
(3)
(4)
Kronig-Penney Model
Allowed band
Forbidden band
Kronig-Penney Model
• We plot the right-hand side of (2) as a function
of a and since the left-hand side of the same
equation is always between -1 and +1, a solution
exists only for the shaded region and no solution
outside the shaded region.
• These regions are called “allowed and forbidden
bands of energy” due to the relation between 
and E.
Kronig-Penney Model
From equation (2), we have
• If P increases, allowed bands get narrower and
the forbidden bands get wider.
• If P decreases, allowed bands get wider and
forbidden bands get narrower.
• If P = 0, then cos(a) = cos(ka)
Kronig-Penney Model
• If P  , then sin(a) = 0
• At the boundary of an allowed band cos(ka) =
1, this implies k = n/a for n = 1, 2, 3, …
Kronig-Penney Model
How to plot E-k diagram
• Choose values between -1 to +1, then find
argument of right-hand side (a) which
satisfies chosen values.
• Likewise to left-hand side (ka).
• a = (any number in radian)
Kronig-Penney Model
• ka = (any number in radian)
• Plot E-k diagram
Brillouin Zone
Reduced Brillouin Zone
Number of electrons per unit volume
• The total number of electrons per unit volume
in the range dE (between E and E + dE) is given
by
where N(E) = density of states (number of energy
levels per energy range per unit volume)
F(E) = a distribution function that specifies
expectancy of occupation of state or called
“probability of occupation”.
Number of electrons per unit volume
• The density of states per unit volume in three
dimensions can be expressed as
Number of electrons per unit volume
• The probability of occupancy is given by the Fermi-Diracdistribution as
0.5)
where EF = Fermi energy level (the energy at F(E) =
k = Boltzmann’s constant
T = absolute temperature (K)
Number of electrons per unit volume
• For T = 0 K:
If E > EF, F(E) = 0 
F(E) = 1/(e +1) = 0
If E < EF, F(E) = 1 
F(E) = 1/( e- + 1) = 1
• For T > 0, F(EF) = 0.5
Number of electrons per unit volume
• From equation (5),
Number of electrons per unit volume
• For T > 0
Fermi levels of various materials
Li
4.72 eV
Na
3.12 eV
K
2.14 eV
Cu
7.04 eV
Ag
5.51 eV
Al
11.70 eV
Number of electrons per unit volume
Characteristics of F(E)
1. F(E), at E = EF, equals to 0.5.
2. For (E – EF) > 3kT
This is called “Maxwell – Boltzmann distribution”.
Number of electrons per unit volume
Characteristics of F(E)
3. For (E – EF) < 3kT
4. F(E) may be distinguished into 3 regions for T >
0 as
− E = 0 to (E = EF – 2.2kT): F(E) is close to
unity.
− (E = EF – 2.2kT) to (E = EF + 2.2kT): F(E)
changes from nearly 1 to nearly 0.
− (E = EF + 2.2kT) to E = : F(E) is close to
Intrinsic carrier concentration
• Free charge carrier density or the number of
electrons per unit volume
• For electrons: E1/2= (E - EC)1/2 and
• For holes: E1/2 = (EV - E)1/2 and
Intrinsic carrier concentration
• At room temperature, kT = 0.0259 eV and
(E – EF) >> kT, so Fermi function can be
reduced to Maxwell-Boltzmann
distribution.
Intrinsic carrier concentration
Intrinsic carrier concentration
Intrinsic carrier concentration
• Therefore, the electron density in the
conduction band at room temperature can be
expressed by
(6)
= effective density of states in the
conduction band.
Intrinsic carrier concentration
• Similarly, we can obtain the hole density p in
the valence band as
(7)
= effective density of states in the
valence band
Intrinsic carrier concentration
(a) Schematic band diagram.
(b) Density of states.
(c)Fermi distribution function.
(d) Carrier concentration
Intrinsic carrier concentration
• For intrinsic semiconductors, the number of
electrons per unit volume in the conduction
band equals to the number of holes per unite
volume in the valence band.
n  p  ni
n. p  n
2
i
where ni = intrinsic carrier density
(8)
Intrinsic carrier concentration
• From (8);
Intrinsic carrier concentration
• The Fermi level of an intrinsic semiconductor
can be found by equating (6) = (7) as
Intrinsic carrier concentration
Intrinsic carrier concentration
Ex. Calculate effective density of states NC and NV
for GaAs at room temperature if GaAs has and .
Intrinsic carrier concentration
Soln
We clearly see that the only difference between
NC and NV is the values of effective electron and
hole mass.
Intrinsic carrier concentration
Soln
Intrinsic carrier concentration
Ex. From previous example, calculate intrinsic
carrier density ni for GaAs at room temperature
where energy gap of GaAs is 1.4 eV.
Intrinsic carrier concentration
Soln
Intrinsic carrier concentration
We may have a conclusion that
• As EF  EC, then n increases.
• As EF  EV, then p increases.
• As T = 0 K, then EF is at Eg/2
• If EF > EC or EF < EV, then that semiconductor is
said to be “degenerate”.
Donors and Acceptors
• When a semiconductor is doped with some
impurities, it becomes an extrinsic
semiconductor.
• Also, its energy levels are changed.
Donors and Acceptors
The figure shows schematic bond pictures
for n-type and p-type.
Donors and Acceptors
• For n-type, atoms from group V impurity release
electron for conduction as free charge carrier.
• An electron belonging to the impurity atom
clearly needs far less energy to become
available for conduction (or to be ionized).
• The impurity atom is called “a donor”.
• The donor ionization energy is EC – ED where ED
is donor level energy.
Donors and Acceptors
• For p-type, atoms from group III capture
electron from semiconductor valence band and
produce hole as free charge carrier.
• EA is called “acceptor level” and EA – EV is
called “acceptor ionization level energy”.
• This acceptor ionization level energy is small
since an acceptor impurity can readily accept an
electron.
Donors and Acceptors
• The ionization energy or binding energy,
producing a free charge carrier in
semiconductor, can be approximately expressed
by
Donors and Acceptors
Donors and Acceptors
Ex. Calculate approximate binding energy for
donors in Ge, given that r = 16 and = 0.12m0.
Donors and Acceptors
Soln
Donors and Acceptors
(a) donor ions and (b) acceptor ions.
Donors and Acceptors
• Consider an n-type semiconductor, if ND is the
number of donor electrons at the energy level
ED, then we define to be the number of free
electron carrier (number of ND that have gone
for conduction). or ionized donor atom density
can be written as
Donors and Acceptors
• For a p-type, the argument is similar.
Therefore, NA- or free-hole density or ionized
acceptor atom density is written as
Donors and Acceptors
We can obtain the Fermi level dependence on
temperature for three cases:
• very low temperature
• intermediate temperature
• very high temperature.
Donors and Acceptors
1. Very low temperature
Donors and Acceptors
Donors and Acceptors
2. Intermediate temperature
Donors and Acceptors
Donors and Acceptors
3. Very high temperature
• In this case, all donors are ionized and
electrons are excited from valence band to
conduction band.
• This is acting like an intrinsic semiconductor or
EF = Ei.
• It may be useful to express electron and hole
densities in terms of intrinsic concentration ni
and the intrinsic Fermi level Ei.
Donors and Acceptors
• From (6), we have
Donors and Acceptors
• Similarly to p-type, we have
• This
is valid for both intrinsic and
extrinsic semiconductors under thermal
equilibrium.
Donors and Acceptors
n-Type semiconductor.
(a) Schematic band diagram.
(b) Density of states. (c) Fermi distribution
function (d) Carrier concentration. Note that
np = ni2.