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SEMICONDUCTOR PHYSICS by Dibyendu Chowdhury ECE HIT Haldia DC
SEMICONDUCTOR PHYSICS
by
Dibyendu Chowdhury
ECE HIT Haldia DC
Semiconductors
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The materials whose electrical conductivity lies between those of
conductors and insulators, are known as semiconductors.
Silicon 1.1 eV
Germanium 0.7 eV
Cadmium Sulphide 2.4 eV
Silicon is the most widely used semiconductor.
Semiconductors have negative temperature coefficients of resistance,
i.e. as temperature increases resistivity deceases
Energy Band Diagram
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Conduction
electrons
Energy Band Diagram
Forbidden energy band is small for
semiconductors.
Less energy is required for
electron to move from valence to
conduction band.
A vacancy (hole) remains when an
electron leaves the valence band.
Hole acts as a positive charge
carrier.
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Intrinsic Semiconductor
Both silicon and germanium are
tetravalent, i.e. each has four
electrons (valence electrons) in
their outermost shell.
Each atom shares its four
valence electrons with its four
immediate neighbours, so that
each atom is involved in four
covalent bonds.
A semiconductor, which is in its extremely pure form, is known as an
intrinsic semiconductor. Silicon and germanium are the most widely
used intrinsic semiconductors.
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Intrinsic Semiconductor
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When the temperature of an intrinsic semiconductor is increased,beyond room temperature a large number of electron-hole pairs aregenerated.
Since the electron and holes are generated in pairs so,
Free electron concentration (n) = concentration of holes (p)
= Intrinsic carrier concentration (ni)
Extrinsic Semiconductor
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Pure semiconductors have negligible conductivity at roomtemperature. To increase the conductivity of intrinsicsemiconductor, some impurity is added. The resulting semiconductoris called impure or extrinsic semiconductor.
Impurities are added at the rate of ~ one atom per 106 to 1010
semiconductor atoms. The purpose of adding impurity is to increase
either the number of free electrons or holes in a semiconductor.
Extrinsic Semiconductor
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Two types of impurity atoms are added to the semiconductor
Atoms containing 5
valance electrons
(Pentavalent impurity atoms)
Atoms containing 3
valance electrons
(Trivalent impurity atoms)
e.g. P, As, Sb, Bi e.g. Al, Ga, B, In
N-type semiconductor P-type semiconductor
N-type Semiconductor
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The semiconductors which are obtained by introducing pentavalentimpurity atoms are known as N-type semiconductors.
Examples are P, Sb, As and Bi. These elements have 5 electrons intheir valance shell. Out of which 4 electrons will form covalent bondswith the neighbouring atoms and the 5th electron will be available asa current carrier. The impurity atom is thus known as donor
atom.
In N-type semiconductor current flows due to the movement ofelectrons and holes but majority of through electrons. Thus electronsin a N-type semiconductor are known as majority charge carrierswhile holes as minority charge carriers.
P-type Semiconductor
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The semiconductors which are obtained by introducing trivalentimpurity atoms are known as P-type semiconductors.
Examples are Ga, In, Al and B. These elements have 3 electrons intheir valance shell which will form covalent bonds with theneighbouring atoms.
In P-type semiconductor current flows due to the movement ofelectrons and holes but majority of through holes. Thus holes in a P-type semiconductor are known as majority charge carriers whileelectrons as minority charge carriers.
The fourth covalent bond will remain incomplete. A vacancy,
which exists in the incomplete covalent bond constitute a hole.
The impurity atom is thus known as acceptor atom.
Fermi Energy
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The Fermi energy is a quantum mechanical concept and it usuallyrefers to the energy of the highest occupied quantum state in asystem of fermions at absolute zero temperature..
Fermi Level
The Fermi level (EF) is the maximum energy, which can be occupiedby an electron at absolute zero (0 K).
Fermi Energy Diagram for Intrinsic Semiconductors
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Fermi
Level (EF)
Forbidden
Energy
Gap
The Fermi level (EF) lies at the middle of the forbidden energy gap.
Fermi Energy Diagram for N-type Semiconductors
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Donor
LevelEnergy (eV)
The Fermi level (EF) shifts upwards towards the bottom of theconduction band.
Fermi
Level (EF)
Fermi
Level (EF)
Fermi Energy Diagram for P-type Semiconductors
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Acceptor
LevelEnergy (eV)
The Fermi level (EF) shifts downwards towards the top of the valanceband.
Fermi
Level (EF)
Fermi
Level (EF)
Addition of n-type impurities decreases the number of holes below a
level. Similarly, the addition of p-type impurities decreases the number
of electrons below a level.
It has been experimentally found that
“Under thermal equilibrium for any semiconductor, the
product of no. of holes and the no. of electrons is constant and
independent of amount of doping. This relation is known as mass
action law”
where n = electron concentration, p = hole concentration
and ni = intrinsic concentration
Mass Action Law
2. inpn
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The free electron and hole concentrations are related by the Law of
Electrical Neutrality i.e.
Total positive charge density is equal to the total negative charge
density
Let ND = Concentration of donor atoms = no. of positive charges/m3
contributed by donor ions
p = hole concentration
NA=Concentration of acceptor atoms
n = electron concentration
By the law of electrical neutrality
ND + p = NA + n
Charge carrier concentration in N-type and
P-type Semiconductors
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For N-Type semiconductor
NA = 0 i.e. Concentration of acceptor atoms
And n>>p, then
ND + 0 = 0 + n
ND = n
i.e. in N-type, concentration of donor atoms is equal to the
concentration of free electrons.
According to Mass Action Law
2. inpn
Dii Nnnnp // 22
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For P-Type semiconductor
ND = 0 i.e. Concentration of donor atoms
And p>>n, then
NA + 0 = 0 + p
NA = p
i.e. in P-type, concentration of acceptor atoms is equal to the
concentration of holes.
According to Mass Action Law
2. inpn
Aii Nnpnn // 22
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The Concept of Effective Mass :
• If the same magnitude of electric field is applied toboth electrons in vacuum and inside the crystal, theelectrons will accelerate at a different rate from eachother due to the existence of different potentialsinside the crystal.
• The electron inside the crystal has to try to make itsown way.
• So the electrons inside the crystal will have a differentmass than that of the electron in vacuum.
• This altered mass is called as an effective-mass.
Comparing
Free e- in vacuum
An e- in a crystal
In an electric field
mo =9.1 x 10-31
Free electron mass
In an electric field
In a crystal
m = ?
m*effective mass
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What is the expression for m*
• Particles of electrons and holes behave as a wave under certain conditions.So one has to consider the de Broglie wavelength to link particalbehaviour with wave behaviour.
• Partical such as electrons and waves can be diffracted from the crystal justas X-rays .
• Certain electron momentum is not allowed by the crystal lattice. This is theorigin of the energy band gaps.
sin2dn n = the order of the diffraction
λ = the wavelength of the X-ray
d = the distance between planes
θ = the incident angle of the X-ray beam
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The energy of the free e-
is related to the k
free e- mass , m0
is the propogation constant
dn 2=
k
2=
kP =
The waves are standing waves
The momentum is
(1)
(2)
By means of equations (1) and (2)certain e- momenta are not allowedby the crystal. The velocity of theelectron at these momentum valuesis zero.
The energy of the free electron
can be related to its momentum
mE
P
2
2
=
hP =
2
1
2 2 (2 )
22 2
2 2
2 2E
m
kh hEm m
k
2
h=
momentum
k
Energy
E versus k diagram is a parabola.
Energy is continuous with k, i,e, all
energy (momentum) values are allowed.
E versus k diagram
or
Energy versus momentum diagrams
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To find effective mass , m*
2
2 2
2
2 2
2*
dE k
dk m
d E
mdk
m
d E dk
We will take the derivative of energy with respect to k ;
Change m* instead of m
This formula is the effective mass of
an electron inside the crystal.
- m* is determined by the curvature of the E-k curve
- m* is inversely proportional to the curvature
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Direct an indirect-band gap materials :
• For a direct-band gap material, the minimumof the conduction band and maximum of thevalance band lies at the same momentum, k,values.
• When an electron sitting at the bottom of theCB recombines with a hole sitting at the topof the VB, there will be no change inmomentum values.
• Energy is conserved by means of emitting aphoton, such transitions are called asradiative transitions.
Direct-band gap s/c’s (e.g. GaAs, InP, AlGaAs)
+
e-
VB
CBE
k
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• For an indirect-band gap material; theminimum of the CB and maximum of theVB lie at different k-values.
• When an e- and hole recombine in anindirect-band gap s/c, phonons must beinvolved to conserve momentum.
Indirect-band gap s/c’s (e.g. Si and Ge)
+
VB
CB
E
k
e-
Phonon
Atoms vibrate about their mean
position at a finite temperature.These
vibrations produce vibrational waves
inside the crystal.
Phonons are the quanta of these
vibrational waves. Phonons travel with
a velocity of sound .
Their wavelength is determined by the
crystal lattice constant. Phonons can
only exist inside the crystal.
Eg
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• The transition that involves phonons without producing photons are callednonradiative (radiationless) transitions.
• These transitions are observed in an indirect band gap s/c and result ininefficient photon producing.
• So in order to have efficient LED’s and LASER’s, one should choosematerials having direct band gaps such as compound s/c’s of GaAs,AlGaAs, etc…
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• For GaAs, calculate a typical (band gap) photon energy and momentum , and compare this with a typical phonon energy and momentum that might be expected with this material.
.:: CALCULATION
photon phonon
E(photon) = Eg(GaAs) = 1.43 ev
E(photon) = h = hc / λ
c= 3x108 m/sec
P = h / λ h=6.63x10-34 J-sec
λ (photon)= 1.24 / 1.43 = 0.88 μm
P(photon) = h / λ = 7.53 x 10-28 kg-m/sec
E(phonon) = h = hvs / λ
= hvs / a0
λ (phonon) ~a0 = lattice constant =5.65x10-10 m
Vs= 5x103 m/sec ( velocity of sound)
E(phonon) = hvs / a0 =0.037 eV
P(phonon)= h / λ = h / a0 = 1.17x10-24 kg-m/sec
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• Photon energy = 1.43 eV
• Phonon energy = 37 meV
• Photon momentum = 7.53 x 10-28 kg-m/sec
• Phonon momentum = 1.17 x 10-24 kg-m/sec
Photons carry large energies but negligible amount of momentum.
On the other hand, phonons carry very little energy but significant
amount of momentum.
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Positive and negative effective mass
• The sign of the effective mass is determined directlyfrom the sign of the curvature of the E-k curve.
• The curvature of a graph at a minimum point is apositive quantity and the curvature of a graph at amaximum point is a negative quantity.
• Particles(electrons) sitting near the minimum have apositive effective mass.
• Particles(holes) sitting near the valence bandmaximum have a negative effective mass.
• A negative effective mass implies that a particle willgo ‘the wrong way’ when an extrernal force isapplied.
2 2
2*m
d E dk
Direct-band gap s/c’s (e.g. GaAs, InP, AlGaAs)
+
e-
VB
CBE
k
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-1
-2
0
2
3
1
4GaAs
Conduction
band
Valance
band
0
ΔE=0.31
Eg
[111] [100] k
En
erg
y (
eV
)
-1
-2
0
2
3
1
4Si
Conduction
band
Valance
band
0
Eg
[111] [100] k
En
erg
y (
eV
)
Energy band structures of GaAs and SiECE HIT Haldia DC
-1
-2
0
2
3
1
4GaAs
Conduction
band
Valance
band
0
ΔE=0.31
Eg
[111] [100] k
En
erg
y (
eV
)
Energy band structure of GaAs
Band gap is the smallest energy
separation between the valence and
conduction band edges.
The smallest energy difference occurs at
the same momentum value
Direct band gap semiconductor
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-1
-2
0
2
3
1
4Si
Conduction
band
Valance
band
0
Eg
[111] [100] k
En
erg
y (
eV
)
Energy band structure of Si
The smallest energy gap is between the
top of the VB at k=0 and one of the CB
minima away from k=0
Indirect band gap semiconductor
•Band structure of AlGaAs?
•Effective masses of CB satellites?
•Heavy- and light-hole masses in VB?
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Egk
EEE
direct
transition
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Eg
k
EE
direct
transition
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Eg
k
E
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Eg indirect
transition k
E
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Eg indirect
transition k
E
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