Energy Band Theory

Energy Band Formation

In an isolated atom, the electrons
have discrete energy levels.
But in solids a large number of E1 E1
E1
atoms remain closely spaced. Fig. (a) Fig. (b) E2 E2
E3

 According to Pauli’s exclusion principle, no two electrons in a crystal can have the
same quantum state.
 Thus the energy levels of one atom cannot superimpose on the levels of the other,
instead, they remain very close and get modified.
 This modification is not appreciable in the case of energy levels of electrons in the
inner shells (completely filled).
 But in the outermost shells, modification is appreciable because the electrons are
shared by many neighbouring atoms.
 Due to influence of high electric field between the core of the atoms and the shared
electrons, energy levels are split-up or spread out forming energy bands.
Formation of Energy Bands in Si

Energy

Conduction Band
• • 3p2
Forbidden Energy Gap • • 3s2
Valence Band

•••••• 2p6 Ion

• • 2s2 core
state
• • 1s2

O a b c d Inter atomic spacing (r)

Energy Band Formation
Consider a single crystal of silicon having N atoms. Each atom is associated with the same lattice.
Electronic configuration of Si is 1s2, 2s2, 2p6,3s2, 3p2. (Atomic No. is 14)
Formation of Energy Bands in Si
 r = Od (>> Oa): The outer two sub-shells (3s and 3p of n = 3 shell) of silicon atom contain two s electrons
and two p electrons. So, there are 2N electrons completely filling 2N possible s levels, all of which are at the
same energy. Of the 6N possible p levels, only 2N are filled and all the filled p levels have the same energy.
 Oc < r < Od: There is no visible splitting of energy levels but there develops a tendency for the splitting of
energy levels.
 r = Oc: The interaction between the outermost shell electrons of neighbouring Silicon atoms becomes
appreciable and the splitting of the energy levels commences.
 Ob < r < Oc: The energy corresponding to the s and p levels of each atom gets slightly changed.
Corresponding to a single s level of an isolated atom, we get 2N levels. Similarly, there are 6N levels for a
single p level of an isolated atom.
 r = Ob: The energy gap disappears completely. 8N levels are distributed continuously. We can only say that
4N levels are filled and 4N levels are empty.
 r = Oa: The band of 4N filled energy levels is separated from the band of 4N unfilled energy levels by an
energy gap called forbidden gap or energy gap or band gap.
The lower completely filled band (with valence electrons) is called the valence band and the
upper unfilled band is called the conduction band.
Crystalline Solids

Crystalline
Solids

Metal
Insulator Semiconductor
(Conductor)
Insulator

 The resistivity of an insulator is very large.

 Figure shows a simplified energy-band diagram of an insulator. The bandgap energy
Eg of an insulator is usually of the order of 3.5 to 6 eV or larger, so that at room
temperature, there are essentially no electrons in the conduction band and the
valence band remains completely full.
Semiconductor

 The above figure shows the simplified energy-band diagram for

semiconductor for T > 0 K.
 The bandgap energy may be of the order of 1 eV.
 The resistivity of a semiconductor can be controlled and varied over many
orders of magnitude by doping.
Semiconductor
 At absolute zero temperature, no electron has energy to jump from valence
band to conduction band and hence the semiconductor acts as an insulator.

 At room temperature, some valence electrons gain energy more than the
energy gap and move to conduction band to conduct even under the
influence of a weak electric field.

 As an electron leaves the valence band, it leaves some energy level in the
valence band as unfilled.

 Such unfilled states are termed as ‘holes’ in the valence band. They are
positive charge carriers.

 Any movement of this empty state is referred to the movement of a

positive charge carrier (hole).
Metal

 Figure shows possible energy-band diagram of a metal. It shows a

case in which conduction and valence bands overlap at the
equilibrium interatomic distance.
 Metals have very low resistivity of the order of 10-8 Ωm.
Insulator, Semiconductor & Metal (Conductor)
Distinction between Insulator, Semiconductor and Metal
At T > 0K few valence electrons
move to the conduction band
Fermi-Dirac Distribution and the Fermi-level
 Electrons in solids obey Fermi-Dirac statistics.
 Density of states tells us how many states can exist at a given energy E.
 The Fermi function f(E) specifies how many of the existing states at the energy E
will be occupied by the electrons.
 The function f(E) specifies, under thermal equilibrium condition, the probability
that an available state at an energy E will be occupied by an electron. It is a
probability distribution function given by:

 EF = Fermi energy or Fermi level

 k = Boltzmann constant = 1.38  10−23 J/K = 8.6  10−5 eV/K
 T = absolute temperature in K
 Fermi-Dirac distribution (Considering T = 0K)
1
For E > EF : f ( E  EF ) = = 0
1 + exp (+)

1
For E < EF : f ( E  EF ) = =1
1 + exp (−)

At T = 0K all the states are occupied (probability one) up to EF0 and all the states above EF0 are empty
(occupation probability zero).

As the temperature increases, we might expect more and more electrons to be in states above EF (E > EF)
Fermi-Dirac distribution (Considering T > 0K)
1 1
For E = EF : 𝑓 𝐸 = 𝐸𝐹 =
1 + exp 0
=
2

1
For E >> EF : f ( E  EF ) = = 0
1 + exp (+)

1
For E << EF : f ( E  EF ) = =1
1 + exp (−)
Energy Band Representations
Energy Band Structure Representation…

Band
Gap
Eg
 E-k Diagram

 An E-k diagram shows the relationship between the energy and

momentum of available quantum mechanical states for electrons in
the material. They arise from the solution of Schrodinger equation in
periodic lattice.
 The x-axis can be either momentum, p, or wavenumber, k, since
p = ℏk, where ℏ = h/2π = 1.054 × 10-34 J-s is called modified
Plank's constant. h = 6.625× 10-34 J-s is the Plank's constant
 The band gap (EG) is the difference in energy between the top of the
valence band and the bottom of the conduction band.
 The effective mass of electrons and holes in the material can be given
by the curvature of each of the bands.
E-k Diagram of Si & GaAs

1.43 eV

1.11 eV
 Why Silicon is Preffered?

 Due to lower band-gap, Ge has higher leakage current (reverse saturation

current) as compared to Si.

 Si-SiO2 interface has the least defect density among all semiconductors.

 Si is the second most abundant element on earth.

Semiconductor Classification

Semiconductor

Presence
of Dopant

Intrinsic Extrinsic

N-type P-type
Intrinsic Semiconductor
Valence electrons
Covalent Bond
Ge Ge Ge Ge
Broken Covalent Bond

Free electron ( - )
Ge Ge Ge Ge Hole ( + )

Ge Ge Ge Ge C.B
+

Eg 0.66 eV
Ge Ge Ge Ge V.B
+ +

Heat Energy
Intrinsic Semiconductor

 Intrinsic Semiconductor is a pure semiconductor.

 The energy gap in Si is 1.1 eV and in Ge is 0.66 eV.
 Si: 1s2, 2s2, 2p6,3s2, 3p2. (Atomic No. is 14)
 Ge: 1s2, 2s2, 2p6,3s2, 3p6, 3d10, 4s2, 4p2. (Atomic No. is 32)
 In intrinsic semiconductor, the number of thermally generated electrons
always equals the number of holes.
If ni and pi are the concentration of electrons and holes respectively, then
ni = pi
The quantity ni or pi is referred to as the ‘intrinsic carrier concentration’.
Energy Band Diagram of Intrinsic Semiconductor
Doping of a Semiconductor
 Doping is the process of deliberate addition of a very small amount of impurity into an intrinsic
semiconductor.
 The impurity atoms are called ‘dopants’.
 The semiconductor containing impurity is known as ‘impure or extrinsic semiconductor’.

 Methods of Doping:
i) Thermal Diffusion
ii) Ion Implantation
In both cases, impurities are brought in contact with the substrate surface and are given sufficient energy to make
them enter below the substrate surface.
In case of diffusion, the required energy is supplied thermally, whereas for ion implantation, an electric field is
applied to energize the dopants.

 Criteria of Doping:
i) Size of Dopant (must be of the same order)
ii) Valency of Dopant (must be ±1)
iii) Position of Dopant (should be substitutional not interstitial)
Extrinsic or Impure Semiconductor:
N - Type Semiconductors:

Ge Ge Ge
C.B

-
Eg = 0.66 eV
Ge As Ge
+ V.B

Ge Ge Ge Donor level
+

 When a semiconductor of Group IV (tetra valent) such as Si or Ge is doped with a penta

valent impurity (Group V elements such as P, As or Sb), N – type semiconductor is
formed.
 When germanium (Ge) is doped with arsenic (As), the four valence electrons of As form
covalent bonds with four Ge atoms and the fifth electron of As atom is loosely bound.
N - Type Semiconductors
 A small amount of energy provided due to thermal agitation is sufficient to detach
fifth loosely bound electron and it is ready to conduct current.
 Such electrons from impurity atoms will have energies slightly less than the energies
of the electrons in the conduction band.
 Therefore, the energy state corresponding to the fifth electron is in the forbidden gap
and slightly below the lowest level of the conduction band.
 This energy level is called ‘donor level’.
 The impurity atom is called ‘donor’.
 N – type semiconductor is called ‘donor – type semiconductor’.
Energy Band Diagram of N-type Semiconductor
P - Type Semiconductors

Ge Ge Ge
C.B

Ge In Ge
Eg = 0.66 eV
+
V.B

Ge Ge Ge Acceptor level
+

 When a semiconductor of Group IV (tetra valent) such as Si or Ge is doped with a tri

valent impurity (Group III elements such as In, B or Ga), P – type semiconductor is
formed.
 When germanium (Ge) is doped with indium (In), the three valence electrons of In
form three covalent bonds with three Ge atoms. The vacancy that exists with the
fourth covalent bond of the particular Ge atom constitutes a hole.
P-Type Semiconductors
 The hole which is deliberately created may be filled with an electron from neighbouring
atom, creating a hole in that position from where the electron ejected.
 Therefore, the tri-valent impurity atom is called ‘acceptor’.
 Since the hole is associated with a positive charge, therefore, this type of semiconductor
is called P – type semiconductor.
 The acceptor impurity produces an energy level just above the valence band.
 This energy level is called ‘acceptor level’.
 The energy difference between the acceptor energy level and the top of the valence band
is much smaller than the band gap.
 Electrons from the valence band can, therefore, easily move into the acceptor level by
being thermally agitated.
 P – type semiconductor is called ‘acceptor – type semiconductor’.
 In a P – type semiconductor, holes are the majority charge carriers and the electrons are
the minority charge carriers.
Energy Band Diagram of P-type Semiconductor
Energy Band Diagrams of Intrinsic & Extrinsic Semiconductor

Donor
Level
Acceptor
Level
Carrier Transport Phenomena
Carrier Transport Phenomena
 The net flow of the electrons and holes in a semiconductor will
generate currents.
 The process by which these charged particles move is called transport.
 Here we consider two basic transport mechanisms in a semiconductor
crystal:
 drift—the movement of charge due to electric fields, and
 diffusion—the flow of charge due to density gradients.
 The carrier transport phenomena are the foundation for finally
determining the current-voltage characteristics of semiconductor
devices.
Drift
An electric field applied to a semiconductor will produce a force on electrons and
holes so that they will experience a net acceleration and net movement, provided there
are available energy states in the conduction and valence bands. This net movement of
charge due to an electric field is called drift. The net drift of charge gives rise to a
drift current.

Electric Field = 0
Drift Velocity
 The carriers in extrinsic semiconductors move randomly like gas molecules.
 Under the influence of a constant electric field (E), the carriers will be accelerated to a certain direction.
 However, the carriers will be also retarded due to mutual collision.
 The steady velocity attained by the carriers under such an influence of electric field is known as drift
velocity.

Electric Field = 30 V/m

Drift Velocity
 The drift velocity (vd) is proportional to the applied electric field:

where μ is a proportionality constant, known as mobility (m2V-1s-1) of the carrier.

 If a voltage V be applied across the semiconductor of length L and cross-sectional are A,
the field is E = V/L.
 Let there be N number of carriers within the crystal, each having charge q. If a carrier takes
time T to cross length L, then T = L/vd.
 Therefore, the current (total charge per second in ampere) passing through any cross-
section is

 The current density (J) i.e. the current per unit area (A. m-2) of the conducting medium is
given by

where n = N/LA is the carrier concentration per unit volume.

Drift Velocity
 The current density can also be expressed as:

where σ = nqμ is called the conductivity [(Ω.m)-1] of the semiconductor. The above is a
general expression.
 For n-type semiconductor with electron concentration n and electron mobility μn, the
current density is

 For p-type semiconductor with hole concentration p and electron mobility μp, the current
density is

 As a semiconductor contains both electrons and holes, the total drift current density is
given by

 Thus the total conductivity is:

 Diffusion
 In addition to drift, there is a second mechanism that can induce a current in a semiconductor.
 We may consider a classic physics example in which a container, as shown in figure, is divided
into two parts by a broken membrane. The lower side contains gas molecules at a higher
concentration and the upper side is initially almost empty. The gas molecules are in continual
random thermal motion so that, when the membrane is broken, the gas molecules flow into the
upper side of the container.
 Diffusion is the process whereby particles flow from a region of high concentration toward a
region of low concentration. If the gas molecules were electrically charged, the net flow of
charge would result in a diffusion current.
Diffusion

(a) Diffusion of electrons due to a density gradient. (b) Diffusion of holes due to a density gradient.

 Let the concentration of carriers varies with distance x in a semiconductor. Now, if dn/dx
is the concentration gradient in the semiconductor, then the diffusion current density is
given by:

negative
Total Current Density