ENE 311
Lecture V
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, and Feynman model.
Let us concentrate on Kronig-Penney model.
Kronig-Penney Model
Band theory uses V  0. The potential is periodic in space due to the presence
of immobile lattice ions.
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 1-dimension (x-only), we have
d 2 ( x) 2m
 2 E  V ( x) ( x)  0
dx 2
(1)
For this equation to have solution, the following must be satisfied
cos(ka) 
P sin( a)
 cos( a)
a
(2)
where
P

maV0 w
2
1
2mE
(3)
(4)
2
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. From
equation (2), we have
1. If P increases, allowed bands get narrower and the forbidden bands get
wider.
2. If P decreases, allowed bands get wider and forbidden bands get narrower.
3. If P = 0, then cos(a)=cos(ka)
2  k2 
E
2mE
2
or
2
k2
(like a case of free electron of V = 0)
2m
4. 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, …
How to plot E-k diagram
1. Choose values between -1 to +1, then find argument of right-hand side (a)
which satisfies chosen values.
2. Likewise to left-hand side (ka).
3. a = (any number in radian)
3
(#) 2
  2 substitute to (4), then
a
(#) 2 2
E 2
a 2m
2
4. ka = (any number in radian)
5. Plot E-k diagram
4
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


0
0
n   n( E )dE   N ( E ) F ( E )dE
(5)
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”.
The density of states per unit volume in three dimensions can be expressed as
 2m 
N ( E )  4  2 
h 
3/ 2
E1/ 2
The probability of occupancy is given by the Fermi-Dirac- distribution as
where EF = Fermi energy level (the energy at F(E) = 0.5)
k = Boltzmann’s constant
T = absolute temperature (K)
For T = 0 K: If E > EF, F(E) = 0  F(E) = 1/(e + 1) = 0
E < EF, F(E) = 1  F(E) = 1/( e- + 1) = 1
For T > 0, F(EF) = 0.5
5
From equation (5),

EF
0
0
n   N ( E ) F ( E )dE 
 N ( E) F ( E)dE
For T = 0 and E < EF
n
EF
 N ( E )(1)dE
0

1  2m 


2 2  2 
EF (T  0)   3 2 n 
2/ 3
3/ 2
 2 3/ 2 
 EF 
3

 2 


 2m 
For T > 0
  2  kT 2


EF (T )  EF (0) 1  

...

 12  EF (0) 

Li
4.72 eV
Na
3.12 eV
K
2.14 eV
Cu
7.04 eV
Ag
5.51 eV
Al
11.70 eV
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”.
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 zero.
6
Intrinsic carrier concentration
Free charge carrier density or the number of electrons per unit volume

n   N ( E ) F ( E )dE
0
 2m 
N ( E )  4  2 e 
 h 
3/ 2
E1/ 2
For electrons: E1/2 = (E - EC)1/2 and m  me
*
For holes: E1/2 = (EV - E)1/2 and m  mh
*
At room temperature, kT = 0.0259 eV and (E – EF) >> kT, so Fermi function
can be reduced to Maxwell-Boltzmann distribution.
F ( E )  e ( E  EF ) / kT

3/ 2

3/ 2
 2me* 
n   4  2  .e  ( E  EF ) / kT .( E  EC )1/ 2 dE
 h 
0
E  EC
Let
 xC ,
kT
dx
1
then C 
or dE  kTdxC
dE kT
so e  ( E  EF ) / kT  e  ( kTxC  EC  EF ) / kT  e  xC .e  ( EC  EF ) / kT
Then
 2me* 
n   4  2 
 h 
0
 2me* 
 4  2 
 h 
3/ 2
.e  xC .e  ( EC  EF ) / kT .(kTxC )1/ 2 kTdxC
 kT 
3/ 2
e
 ( EC  EF ) / kT

x
1/ 2  xC
C
e
dxC
0
Therefore, the electron density in the conduction band at room temperature
can be expressed by
n  NC exp  ( EC  EF ) / kT 

where N C  2 2 me kT / h
in the conduction band.
*

2 3/ 2
(6)
= effective density of states
7
Similarly, we can obtain the hole density p in the valence band as
(7)
where
NV  2  2 mh*kT / h 2 
3/ 2
= effective density of states
in the valence band
(a) Schematic band diagram. (b) Density of states. (c) Fermi distribution function.
(d) 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  ni2
where ni = intrinsic carrier density
From (8);
N C exp  ( EC  EF ) / kT .NV exp  ( EF  EV ) / kT   ni2
N C .NV .exp  ( EC  EV ) / kT   ni2
EC  EV  Eg
ni2  N C .NV .exp   Eg / kT 
ni  N C NV exp   Eg / 2kT 
The Fermi level of an intrinsic semiconductor can be found by equating (6) = (7) as
EF  Ei   EC  EV  / 2   kT / 2  ln  NV / NC  .
(8)
8
Ex. Calculate effective density of states NC and NV for GaAs at room temperature if
GaAs has me  0.067m0 and mh  0.65m0 .
*
*
9
Ex. From previous example, calculate intrinsic carrier density ni for GaAs at room
temperature where energy gap of GaAs is 1.4 eV.
Soln
We may have a conclusion that
1. As EF  EC, then n increases.
2. As EF  EV, then p increases.
3. As T = 0 K, then EF is at Eg/2
4. 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. The figure shows schematic bond
pictures for n-type and p-type.
(a) n-type Si with donor (arsenic) and (b) p-type Si with acceptor (boron).
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
10
called “a donor”. The donor ionization energy is EC – ED where ED is donor level
energy.
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.
The ionization energy or binding energy, producing a free charge carrier in
semiconductor, can be approximately expressed by
E
where
m*e4
8  0 r  h2
2
m*  me* for donor atoms
m*  mh* for acceptor atoms
Ex. Calculate approximate binding energy for donors in Ge, given that r = 16 and
me* = 0.12m0.
Soln
11
(a) donor ions and (b) acceptor ions.
Consider an n-type semiconductor, if ND is the number of donor electrons at

the energy level ED, then we define N D to be the number of free electron carrier

(number of ND that have gone for conduction). N D or ionized donor atom density
can be written as
N D  N D 1  F  ED 

For a p-type, the argument is similar. Therefore, N A or free-hole density or
ionized acceptor atom density is written as
N A  N A F  E A 
We can obtain the Fermi level dependence on temperature for three cases:
very low temperature, intermediate temperature, and very high temperature. Let us
examine them by using n-type semiconductor.
1. Very low temperature
12
2. Intermediate temperature
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. From (6), we have
n  N C exp  ( EC  EF ) / kT 
 NC exp  ( EC  Ei ) / kT  exp ( EF  Ei ) / kT 
n  ni exp  ( EF  Ei ) / kT 
Similarly to p-type, we have
where n. p  ni =mass action law.
2
This n. p  ni is valid for both intrinsic and extrinsic semiconductors under thermal
2
equilibrium.
n-Type semiconductor. (a) Schematic band diagram.
(b) Density of states. (c) Fermi distribution function (d) Carrier concentration. Note
that np = ni2.