Chapter 3
Energy Band and Charge
Carriers in Semiconductors
OBJECTIVES
1. Understand conduction and valence energy bands, and how
band gaps are formed
2. Appreciates the idea of doping in semiconductors
3. Use the density of states and Fermi Dirac statistics to
calculates carrier concentrations
4. Calculates drift currents in an electric field in terms of carrier
mobility, and how mobility is affected by scattering
5. Discuss the idea of “effective” masses
1. Energy Bands, E-k relation,
Effective mass
• Bonding Forces in Solid
- The interaction of electrons in neighboring atoms of
a solid serves the very important function of holding
the crystal together
ex) ionic bonding, metallic bonding, covalent bonding
Electro-negativity(0.78~3.90), symbol  (the Greek
letter chi), is a chemical property that describes the
tendency of an atom or a function group to attract
electrons towards itself and thus the tendency to form
negative ions
Occurs between + (low ionization energy) and – (high Electronegativity) ions.
 Requires electron transfer.
 Large difference in electro-negativity required
 Good insulator

Na11=[Ne]3S1
 = 
Cl17=[Ne]3S23p5
 = 
3D
2D
NaCl Ionic bonding
 Arises from a sea of donated valence electrons (1, 2,
or 3 from each atom)
 High electrical conductivity
Requires shared electrons (next page)
 After bonding, two electrons are indistinguishable

- Energy band diagram for two noninteracting hydrogen nuclei.
no interaction of
electron wave
functions between
them → identical
electronic structure
- As the nuclei are brought together, the upper energy levels merge and
electrons in those levels are shared between the atoms.
electron wave functions
begin to overlap →
exclusion principle
dictates that no two
electrons in a given
interacting system may
have the same
quantum state

No free electron at 0K
As in the case of ionic crystal, no free electrons are available
to the lattice in the covalent diamond structure.
 Ge & Si should also be insulators at 0 K.
 However an electron can be thermally or optically exited out
of a covalent bond and thereby become free to participate in
conduction.


Formation of Energy bands
3S2 & 3P2
2
4SP3
2
2
6
2
6
2
2
2
2
+14
+14
As isolated atoms are brought together to form a solid,
various interactions occur between neighboring atoms.
 In the process of interactions, important changes occur in the
electron energy level configurations, and these changes
result in the varied electrical properties of solids.

 Formation of Energy bands
*
<Probability density function of hydrogen atom>
“splitting of energy level”
By Pauli exclusion principle.
- Suppose the Hydrogen atom that are initially very far apart, and
let’s begin pushing the atoms together. Then, the initial quantized
energy level will split into a band of discrete energy levels.
- The parameter r0 represents the equilibrium interatomic distance
(spacing) in the crystal.
EIS (ro or ao ): equilibrium is lowest energy state, attractive and repulsive forces are balanced
Interaction among a large number of
electrons will create an almost continuous,
closely spaced energy levels, which are
called an energy band.
due to Pauli Exclusion principle.
Avogadro’s number : number of units
in one mole of any substance (defined
as its molecular weight in grams),
equal to 6.02214076 × 1023.

Outer orbital

Inner orbital
ao
a < ao

SP3 hybridization & Bonding / Antibonding orbitals
LCAO : linear combinations of the individual atomic orbitals

Conduction band & Valence band
energy difference!
Atomic Size and Coulombic Interaction
 Band Structure Differences
 Bonding and Atomic Interactions

3S2 3P2
4S2 4P2
core
valence
antibonding energy
level(anti-symmetric
LCAO)
bonding energy level
(symmetric LCAO)
equilibrium interatomic distance, a0
At absolute 0K, electrons are in the lowest energy state, so that all states in
the lower band(the valence band) will be full & all states in the upper band(the
conduction band) will be empty.
 The band gap energy Eg between the top of the valence band & the bottom of
the conduction band is the width of the forbidden energy.

antibonding energy
level(anti-symmetric
LCAO)
Solve wave functions
using Schrö dinger
equation
bonding energy level
(symmetric LCAO)
EE, Kwang-Seok Yun

Valence band (Filled band)
◦ completely filled with electrons at 0K
◦ no empty states into which electrons
can move

Conduction band (Empty band)
◦ unoccupied states
◦ allowed energy states which are not
already occupied by electrons
Every solid has its own characteristic
energy band structure → variation in band
structure is responsible for the wide range
of electrical characteristics
(refer Neamen’s book)

The concept of allowed and forbidden energy bands can be
developed by considering quantum mechanics and
Schrodinger’s wave equation.
- The potential function of a single, non-interacting, one-electron atom is shown
Figure (a)
- Figure(b) shows the same type of potential function for the case when several atoms
are in close proximity arranged in a on dimensional array.
(a) Potential function of a
single isolated atom
Atom
(b) Overlapping potential
functions of adjacent atoms
Atom
Atom
Atom
Atom
(refer Neamen’s book)
- The potential functions of adjacent atoms overlap, and the net potential
function for this case is shown Figure(c)
-
Figure (d) is the one-dimensional Kronig-Penney model of the periodic
potential function. → solve Schrodinger’s wave equation in each region.
(c) Net potential function of a onedimensional single crystal
Atom Atom Atom Atom
(d) One-dimensional periodic potential
function of the Kronig-Penney model
Figure (d)
- Solve the wave equation in given potential model
Assume that the wave function of electron is a plane wave
ψ(x) = u(x)eikx
Bloch theorem : if it is
assumed to travel
through a perfectly
periodic lattice, all
single-electron wave
function must be of the
form, u(x)eikx
Where the function u(x) modulates the wave function according to the periodicity of
the lattice(a+b) and k is wave vector (propagation constant, motion constant)
1. Solve the wave equation in region I (V = 0)
ψ1(x) = u1(x)eikx
, where  2 =
2. Solve the wave equation in region II (V = V0)

2mE
2
ψ2(x) = u2(x)eikx
3. Apply boundary conditions
→ This equation relates the parameter k to the total energy, E (through the parameter )
and the potential function V0 (through the parameter )
► Let the potential barrier width b → 0 and the barrier height V0 → 
Then, the product bV0 remains finite
- Then, finally, we have the relation
P’
this equation is not a solution of Schrodinger’s wave equation but gives the conditions for which
Schrodinger’s wave equation will have a solution.
 E-k relation in free space (V0=0 in Kronig-Penney
ere  2 =
model)
cos a = cos ka
 =k
2mE
2
free particle
the total energy E is equal to the kinetic energy
1
2m( mv 2 )
2mE
p
2
=
=
=
=k
2
2
(p : particle momentum)



→ p = k
The constant of the motion parameter, k is related to the particle
momentum (p) for the free electron.
- Change of k means change of p
-
➢ The wave vector (denoted by the symbol k) is a mathematical vector
that represents the spatial and directional properties of a wave.
➢ It provides essential information about the wave's propagation
direction, magnitude, wavelength, and phase.
➢ The wave vector is typically measured in meters-1 (m-1).
➢ The magnitude of the wave vector is called the wave number and is
given by
2
k=

[1/m]
Where λ is the wavelength in the medium
 E-k relation in free space (V0=0 in Kronig-Penney
model)
- The relation of energy and momentum
p2 2 2
E=
=
k
2m 2m
n
kn =
L
E vs. k for free particle
 E-k relation in single crystal lattice
-As the parameter P’ increases, the
particle becomes more tightly bound to
the potential well or atom.
P’
-Let define f(a) = P’
. Figure (a) is a plot of the 1st term of above eq.
. Figure (b) is a plot of the 2nd term of above eq.
. Figure (c) is a the sum of the two terms.
f(a)
P’
- We also have that f(a) = cos ka
. For above eq. to be valid, the allowed values of the f(a) function must be
bounded between +1 & -1.
. Fig.(c) shows the allowed values of f(a) and the allowed values of a in the
shaded areas.
. The values of ka are also shown on the figure from the right side of equation
which correspond to the allowed values of f(a).
The certain range of energy (allowed energy
values), separated by ranges of energy for which
the value of the function lies outside the range
of +1 to -1 (forbidden energy values)
 → discrete
E vs. k in solid (1-D, K-P model)
-The parameter  is related to the
total energy E of the particle
through Eq.
2 =
2mE
2
- Figure shows the concept of
allowed energy bands for the
particle propagating in the crystal
lattice.
A plot of the energy E of the particle as a
function of the wave number k can be
generated from Figure (c)
E-k diagram
+
e
h
+
T = 0K
T > 0K
Hole energy increases oppositely to electron energy, because the two
carriers have opposite charge
 Hole energy increases downward in valence band, seeking lowest
energy state available → move upward

◦ Electron energy increases upward: seeking lowest energy level → move
downward
(c)an almost full band separated
by a small bandgap(~1eV) from
an almost empty band
(b)
(a)
(d)a full band and an empty
band separated by a large
bandgap (~10eV)
(a)a half filled band & (b)two
overlapping bands,
a) This situation occurs in materials consisting of atoms, which contain
only one valence electron per atom. Most highly conducting metals
metal including copper, gold and silver satisfy this condition.
b) Materials consisting of atoms that contain two valence electrons can
still be highly conducting if the resulting filled band overlaps with an
empty band.
c) It depicts the situation in a semiconductor. The completely filled band
is now close enough to the next higher empty band that electrons can
make it into the next higher band. This yields an almost full band
below an almost empty band. We will call the almost full band the
valence band since it is occupied by valence electrons. The almost
empty band will be called the conduction band, as electrons are free
to move in this band and contribute to the conduction of the material.
d) No conduction is expected for where a completely filled band is
separated from the next higher empty band by a larger energy gap (>
~10 eV). Such materials behave as insulators.
EE, Kwang-Seok Yun

E vs. k for GaAs and Si (in real periodic structure)
k is dependent on
crystal direction
The band structure of GaAs has a minimum in the conduction
band and a maximum in the valence band for the same k
value(k=0). i.e. Energy vs. crystal momentum for a
semiconductor with a direct band gap, showing that an electron
can shift from the lowest-energy state in the conduction band to
the highest-energy state in the valence band without a change
in k value. An electron can directly emit a photon.
Si has its valence band maximum at a different value of k than its
conduction band minimum. Energy vs. crystal momentum for a
semiconductor with an indirect band gap, showing that an electron
cannot shift from the lowest-energy state in the conduction band to
the highest-energy state in the valence band with a change in
momentum. A photon cannot be emitted because the electron must
pass through an intermediate state and transfer momentum to the
crystal lattice.
There are 2 classes of semiconductor energy bands : direct & indirect
➢ In semiconductor, the band gap of a semiconductor is always one of two
types, a direct band gap or an indirect band gap.
• The band gap is called "direct" if the momentum of electrons and holes is
the same in both the conduction band and the valence band; an electron
can directly emit a photon.
• In an "indirect" gap, a photon cannot be emitted because the electron must
pass through an intermediate state and transfer momentum to the crystal
lattice.
•The minimal-energy state in the conduction band, and the maximal-energy
state in the valence band, are each characterized by a certain crystal
momentum (k-vector). If the k-vectors are the same, it is called a "direct
gap". If they are different, it is called an "indirect gap".
• Crystal momentum is a momentum-like vector associated with electrons
in a crystal lattice. It is defined by the associated wave vectors k of this
lattice, according to pcrystal = k
At absolute zero (0 kelvins), every electron is in the lowest possible energy state. In
a perfect semiconductor, every state in the valence band is occupied by an electron.
Every state in the conduction band is empty. → No conduction
At room temperature(by thermal excitations), electrons are being excited up to the
conduction band. In addition, after electrons are exited to the conduction band,
the empty states left in the valence band can contribute to the conduction process.
→ Conduction
If an electron in the valence band acquires
enough energy to reach the conduction band,
it can flow freely among the nearly empty
conduction band energy states.
For convenience, an empty state in the
valence band is referred to as a hole.
→ They are called Electron-Hole Pair (EHP)
**In valence band(filled band):
- Electron: negative charge, negative effective mass
- Hole: positive charge, positive effective mass
Si at 0K (no free electron)
Bond model
Energy Band model
After EHP generation, we have two charge carriers
When an electron is moving inside a solid material, the force between
other atoms will affect its movement and it will not be described by
Newton's law.
 When force is applied on electron in solid crystal,
Ftotal = Fext + Fint =ma
Collection of internal periodic crystal force
Instead of caring internal effect, we use
Fext =m*a, where m* is effective mass
The electron is represented by a wave packet. So the velocity of
the electron is the group velocity (Vg) of the wave packet which is
defined by
d 1 dE
vg =
=
dk  dk
From angular frequency,  = 2π  energy  = h
→  = E/ℏ
1 d dE
1 dk d dE
1 dp d 2 E
a=
=
( )=
( )= 2
dt
 dt dk
 dt dk dk
 dt dk 2
dv g
1
m =
1 d 2E
 2 dk 2
*
From p= m* vg, dp/dt = m* a

negative m*:
d 2E
The curvature of
dk 2
is negative* at the valence
band maxima.

Thus, the electron near the top of the valence band
have negative effective mass from eq.
1
*
k
m =
a
2
positive m* The curvature of d E
dk 2
is positive at the
1 d 2E
 2 dk 2
conduction band minima.

Eg
k
d2E
→ small → " heavy hole"
dk 2
d 2E
→ large → " light hole"
dk 2
* Physically speaking the electrons in these regions are accelerated in a direction opposite to the direction of the applied field.
This is called the negative mass behaviour of the electrons. / The electrons with the negative effective mass is considered as a
new entity having the same positive mass of that of an electron but with positive charge. The new entity is given the name
"hole".