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Solid State Devices-I
Dr. Vaishali V. Deshmukh
Dept. of Physics
Shri Shivaji Science College,
Amravati
Fermi Level and Fermi Energy
Lecture - 3
Fermi Level in Intrinsic and Extrinsic Semiconductors
Fermi Level
01
Expression for electrical conductivity
Mobility and Conductivity
03
Theory of Hall Effect
Hall Effect
04
Outline
Drift Velocity
Drift current in Semiconductors
02
Fermi Level and Fermi Energy
Fermi level is the highest energy state occupied by electrons in a material at absolute zero
temperature. Fermi level is also defined as the work done to add an electron to the system.
The value of the Fermi level at absolute zero temperature (−273.15 °C) is known as the Fermi energy.
It is also the maximum kinetic energy an electron can attain at T=0K. Fermi energy is constant for each
solid. The Fermi energy is a concept in quantum mechanics usually referring to the energy difference
between the highest and lowest occupied single-particle states in a quantum system of non-interacting
fermions at absolute zero temperature.
Enrico Fermi, the physicist who first proposed Fermi level. It is important in determining the
electrical and thermal properties of solids.
The term “Fermi level” comes from Fermi-Dirac statistics, which describes a distribution of particles
over energy states in systems consisting of fermions (electrons) that obey the Pauli exclusion principle.
In semiconductors the position of the Fermi level is within the band gap, approximately in the middle of
the band gap.
Fermi level of intrinsic semiconductor
For an intrinsic semiconductor, every time an electron moves from the valence band to the conduction band, it leaves a
hole behind in the valence band. The density of electrons in the conduction band equals the density of holes in the
valence band. At temperature T(K), the electron density ‘n’ is equal to hole density ‘p’ in an intrinsic semiconductor.
Here Nc is the effective density of states in the
conduction band, Nv is the effective density of
states in the valence band, EF is the Fermi
energy, Ec is the lower energy level of conduction
band, Ev is the lower energy level of valence
band, KB is Boltzmann's constant, and T is the
temperature in K.
Conduction Band
Valence Band
EF
Fermi level
Band
Energy
Band Gap
The Fermi energy is in the middle of the band gap
(Ec + Ev)/2 plus a small correction that depends
linearly on the temperature. The correction term is
small at room temperature since Eg ~ 1 eV
while kBT ~ 0.025 eV. For Si and Ge, Nc > Nv and
the correction term is negative while for
GaAs Nc < Nv and the correction term is positive.
EV
EC
Eg
Fermi level of extrinsic semiconductor
In extrinsic semiconductor, the number of electrons in the conduction band and the number of holes in the valence
band are not equal. Hence, the probability of occupation of energy levels in conduction band and valence band are not
equal. Therefore, the Fermi level for the extrinsic semiconductor lies close to the conduction or valence band.
Fermi level in n-type semiconductor
In n-type semiconductor pentavalent impurity is added. Each pentavalent impurity donates
a free electron.The addition of pentavalent impurity creates large number of free electrons
in the conduction band. At room temperature, the number of electrons in the conduction
band is greater than the number of holes in the valence band. Hence, the probability of
occupation of energy levels by the electrons in the conduction band is greater than the
probability of occupation of energy levels by the holes in the valence band. This
probability of occupation of energy levels is represented in terms of Fermi level.
Therefore, the Fermi level in the n-type semiconductor lies close to the conduction band.
The Fermi level for n-type semiconductor is given as,
Conduction Band
Valence Band
EF
Fermi level
Band
Energy
Band Gap
EV
EC
Fig: Fermi level in n-type semiconductor
Where, EF is the fermi level, EC is the conduction band, KB is the Boltzmann constant, T is the absolute
temperature, NC is the effective density of states in the conduction band, ND the concentration of donar atoms.
Fermi level of extrinsic semiconductor
In p-type semiconductor trivalent impurity is added. Each trivalent impurity creates a hole in the valence band and
ready to accept an electron. The addition of trivalent impurity creates large number of holes in the valence band.
Fermi level in p-type semiconductor
At room temperature, the number of holes in the valence band is greater than the number
of electrons in the conduction band. Hence, the probability of occupation of energy levels
by the holes in the valence band is greater than the probability of occupation of energy
levels by the electrons in the conduction band. This probability of occupation of energy
levels is represented in terms of Fermi level. Therefore, the Fermi level in the p-type
semiconductor lies close to the valence band. The Fermi level for p-type semiconductor is
given as
Conduction Band
Valence Band
EF
Fermi level
Band
Energy
Band Gap
EV
EC
Fig: Fermi level in p-type semiconductor Where , NV is the effective density of states in the valence band, NA is the concentration of acceptor atoms.
Current density in Semiconductors
Drift current in Semiconductors
The flow of charge (ie) current through a semiconductor material are of two types namely drift & diffusion.
The net current that flows through a (PN junction diode) semiconductor material has two components
(i) Drift current
(ii) Diffusion current
When an electric field is applied across the semiconductor material, the charge carriers attain a certain drift velocity
Vd , which is equal to the product of the mobility of the charge carriers and the applied Electric Field intensity E ;
Drift velocity Vd = mobility of the charge carriers X Applied Electric field intensity.
Holes move towards the negative terminal of the battery and electrons move towards the positive terminal of the
battery. This combined effect of movement of the charge carriers constitutes a current known as the drift current.
Thus the drift current is defined as the flow of electric current due to the motion of the charge carriers under the
influence of an external electric field.
Drift current due to the charge carriers such as free electrons and holes are the current passing through a square
centimeter perpendicular to the direction of flow.
It is possible for an electric current to flow in a semiconductor even in the absence of the applied voltage provided a
concentration gradient exists in the material.
In a semiconductor material the charge carriers have the tendency to move from the region of higher concentration
to that of lower concentration of the same type of charge carriers. Thus the movement of charge carriers takes place
resulting in a current called diffusion current.
Drift current in Semiconductors
When an electric field is applied to a conductor at room temp, electron move towards the +Ve terminal of the applied volt but they
continuously collide with atoms along the way. Each time the electron collides with an atom, it rebounds in a random fashion. At each
collision, the electron loses some kinetic energy, then accelerates again, gains certain component of velocity in the direction of –E and
loses its energy at the next collision. Obiviously, presence of the electric field does not stop collisions and random motion but it does
cause the electrons to drift towards the +Ve terminal of the applied voltage V. Consequently, the electrons gain an average directed drift
velocity v which is directly praportional to E.
V = µe E Where, µe is called electron mobility.
V (meter/second)
E(volt/meter)
µe =
The resulting flow of electrons carrying negative charge at drift velocity V constitutes electric
current called drift current.
Let n = number of electrons per unit volume of the conductor
A = conductor cross-section (m2)
I = conductor length (m)
E= V/l = applied electric field (V/m)
I = n × (v × A)
Toal no. of electrons which cross any plane P of cross-section A in one second
Charge carried by them per second is = e n v A
Drift current in Semiconductors
It represents the drift current I = v e n A
Substituting the value of v, we get, I= n e A µe E = n e A µe
V
l
Now, R=
1
e
V l
l A neµ
 
  
 
Þ =
l
A
Resistivity þ
1
neue

Conductivity = neµe siemens/m
Mobility and Conductivity
The ability of an electron to move through a semiconductor, in the presence of
applied electric field is known as electron mobility.
Vn= μnE, Therefore, Mobility of electron μn= Vn/E (SI unit of mobility m2/(V.s)
The ability of an hole to move through a semiconductor, due to applied electric field
is known as hole mobility.
Vp= μpE, Therefore, Mobility of electron μp= Vp/E (SI unit of mobility m2/(V.s)
In intrinsic semiconductors, the number of electrons is equal to the number of holes.
ne = np
σ = ne e μn + np e μp = ni e (μn + np)
HALL EFFECT
When a magnetic field is applied to a current carrying conductor
in a direction perpendicular to that of the flow of current, a
potential difference or transverse electric field is created across a
conductor. This phenomenon is known as Hall Effect.
Hall Effect was discovered by Edwin Hall in 1879.
The voltage or electric field produced due to the application of
magnetic field is also referred to as Hall voltage or Hall field.
When a voltage is applied to a conductor or semiconductor,
electric current starts flowing through it. In conductors, the
electric current is conducted by free electrons whereas in
semiconductors, electric current is conducted by both free
electrons and holes.
The free electrons in a semiconductor or conductor always try to flow in a straight path. However, because of the
continuous collisions with the atoms, free electrons slightly change their direction. But if the applied voltage is strong
enough, the free electrons forcefully follow the straight path. This happens only if no other forces are applied to it in other
direction. If we apply the force in other direction by using the magnetic field, the free electrons in the conductor or
semiconductor change their direction.
HALL EFFECT
Consider a material, either a semiconductor or conductor as shown in the below
figure. When a voltage is applied, electric current starts flowing in the positive
x-direction (from left to right). If a magnetic field is applied to this current
carrying conductor or semiconductor in a direction perpendicular to that of the
flow of current (that is z-direction), an electric field is produced in it that exerts
force in the negative y direction (downwards).
Mathematical expression for the Hall voltage is given by
VH =
IB
qnd
Where,
VH = Hall voltage
I = current flowing through the material
B = magnetic field strength
q = charge
n = number of mobile charge carriers per unit volume
d = thickness of the material
Hall Voltage is directly proportional to the
electric current and applied magnetic field.
Derivation of Hall Coefficient
e EH = e Bv
EH = Bv
JX = nev V =
Thus , EH =
EH = RH JX B
RH = =
RH =
JX
ne
B JX
ne
EH
JX B
1
np
If v is the velocity of electrons at right angles to the magnetic field, there is a
downward force acts on each electron of magnitude Bev. This results in the electron
current to be deflected in a downward direction and causes a negative charge to
gather on the bottom face of the specimen. A potential difference is therefore
generated from top to bottom of the specimen. This potential difference causes a
field EH in the negative y-direction, and thus a force of eEH acting in the upward
direction on the electron. Hence at equilibrium condition, the force downwards due
to magnetic field will be equal to the upward electric force.
The Hall effect is described by means of the Hall coefficient RH defined in terms
of current density
Negative sign indicates that the electric field developed is in the negative y-
direction.
In case of p-type specimen, when current is completely due to holes,
-1
ne
Applications of hall effect
Hall Effect is used to find whether a semiconductor is N-type or P-type.
Hall Effect is used to find carrier concentration.
Hall Effect is used to calculate the mobility of charge carriers (free
electrons and holes).
Hall Effect is used to measure conductivity.
Hall Effect is used to measure a.c. power and the strength of magnetic
field.
Hall Effect is used in an instrument called Hall Effect multiplier which
gives the output proportional to the product of two input signals.
Charge carriers in semiconductors are negatively charged electrons and positively
charged holes. Holes are essentially vacancies in an otherwise filled band so that when an
electron moves to such a vacancy, the effect is equivalent to movement of a hole in reverse
direction.
In the absence of an electric field, charge carriers move randomly so that their average
velocity is zero. When an electric field is applied, the positve charge carriers move in the
direction of the field and the negative charge carriers move against the field direction.
This directed motion is superimposed over the random direction and is called drift.
Drift velocity is proportional to the direction of the field, the constant of proportionality
is the mobility of the carrier.
If a magnetic field is applied on a flat strip of semiconductor in a direction perpendicular
to the direction of current flow, a voltage develops in a direction which is perpendicular to
both the direction of the field and the magnetic field. This is known as Hall voltage.
Hall voltage arises due to Lorentz force that acts on charge carriers and provides a direct
means of verifying existence of holes.
Without semiconductors, the
world would be a very
different place; we would
have no electronic hand
calculators, microwave
ovens, digital alarm clocks,
cellphones, personal
computers, electronically
controlled transmissions, or
washing machines.
Anyone who
has never
made a
mistake has
never tried
anything
new.
Albert Einstein
THANK YOU

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Lecture3: Fermi Level and Fermi Energy.pdf

  • 1. Solid State Devices-I Dr. Vaishali V. Deshmukh Dept. of Physics Shri Shivaji Science College, Amravati
  • 2. Fermi Level and Fermi Energy Lecture - 3
  • 3. Fermi Level in Intrinsic and Extrinsic Semiconductors Fermi Level 01 Expression for electrical conductivity Mobility and Conductivity 03 Theory of Hall Effect Hall Effect 04 Outline Drift Velocity Drift current in Semiconductors 02
  • 4. Fermi Level and Fermi Energy Fermi level is the highest energy state occupied by electrons in a material at absolute zero temperature. Fermi level is also defined as the work done to add an electron to the system. The value of the Fermi level at absolute zero temperature (−273.15 °C) is known as the Fermi energy. It is also the maximum kinetic energy an electron can attain at T=0K. Fermi energy is constant for each solid. The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. Enrico Fermi, the physicist who first proposed Fermi level. It is important in determining the electrical and thermal properties of solids. The term “Fermi level” comes from Fermi-Dirac statistics, which describes a distribution of particles over energy states in systems consisting of fermions (electrons) that obey the Pauli exclusion principle. In semiconductors the position of the Fermi level is within the band gap, approximately in the middle of the band gap.
  • 5. Fermi level of intrinsic semiconductor For an intrinsic semiconductor, every time an electron moves from the valence band to the conduction band, it leaves a hole behind in the valence band. The density of electrons in the conduction band equals the density of holes in the valence band. At temperature T(K), the electron density ‘n’ is equal to hole density ‘p’ in an intrinsic semiconductor. Here Nc is the effective density of states in the conduction band, Nv is the effective density of states in the valence band, EF is the Fermi energy, Ec is the lower energy level of conduction band, Ev is the lower energy level of valence band, KB is Boltzmann's constant, and T is the temperature in K. Conduction Band Valence Band EF Fermi level Band Energy Band Gap The Fermi energy is in the middle of the band gap (Ec + Ev)/2 plus a small correction that depends linearly on the temperature. The correction term is small at room temperature since Eg ~ 1 eV while kBT ~ 0.025 eV. For Si and Ge, Nc > Nv and the correction term is negative while for GaAs Nc < Nv and the correction term is positive. EV EC Eg
  • 6. Fermi level of extrinsic semiconductor In extrinsic semiconductor, the number of electrons in the conduction band and the number of holes in the valence band are not equal. Hence, the probability of occupation of energy levels in conduction band and valence band are not equal. Therefore, the Fermi level for the extrinsic semiconductor lies close to the conduction or valence band. Fermi level in n-type semiconductor In n-type semiconductor pentavalent impurity is added. Each pentavalent impurity donates a free electron.The addition of pentavalent impurity creates large number of free electrons in the conduction band. At room temperature, the number of electrons in the conduction band is greater than the number of holes in the valence band. Hence, the probability of occupation of energy levels by the electrons in the conduction band is greater than the probability of occupation of energy levels by the holes in the valence band. This probability of occupation of energy levels is represented in terms of Fermi level. Therefore, the Fermi level in the n-type semiconductor lies close to the conduction band. The Fermi level for n-type semiconductor is given as, Conduction Band Valence Band EF Fermi level Band Energy Band Gap EV EC Fig: Fermi level in n-type semiconductor Where, EF is the fermi level, EC is the conduction band, KB is the Boltzmann constant, T is the absolute temperature, NC is the effective density of states in the conduction band, ND the concentration of donar atoms.
  • 7. Fermi level of extrinsic semiconductor In p-type semiconductor trivalent impurity is added. Each trivalent impurity creates a hole in the valence band and ready to accept an electron. The addition of trivalent impurity creates large number of holes in the valence band. Fermi level in p-type semiconductor At room temperature, the number of holes in the valence band is greater than the number of electrons in the conduction band. Hence, the probability of occupation of energy levels by the holes in the valence band is greater than the probability of occupation of energy levels by the electrons in the conduction band. This probability of occupation of energy levels is represented in terms of Fermi level. Therefore, the Fermi level in the p-type semiconductor lies close to the valence band. The Fermi level for p-type semiconductor is given as Conduction Band Valence Band EF Fermi level Band Energy Band Gap EV EC Fig: Fermi level in p-type semiconductor Where , NV is the effective density of states in the valence band, NA is the concentration of acceptor atoms.
  • 8. Current density in Semiconductors
  • 9. Drift current in Semiconductors The flow of charge (ie) current through a semiconductor material are of two types namely drift & diffusion. The net current that flows through a (PN junction diode) semiconductor material has two components (i) Drift current (ii) Diffusion current When an electric field is applied across the semiconductor material, the charge carriers attain a certain drift velocity Vd , which is equal to the product of the mobility of the charge carriers and the applied Electric Field intensity E ; Drift velocity Vd = mobility of the charge carriers X Applied Electric field intensity. Holes move towards the negative terminal of the battery and electrons move towards the positive terminal of the battery. This combined effect of movement of the charge carriers constitutes a current known as the drift current. Thus the drift current is defined as the flow of electric current due to the motion of the charge carriers under the influence of an external electric field. Drift current due to the charge carriers such as free electrons and holes are the current passing through a square centimeter perpendicular to the direction of flow. It is possible for an electric current to flow in a semiconductor even in the absence of the applied voltage provided a concentration gradient exists in the material. In a semiconductor material the charge carriers have the tendency to move from the region of higher concentration to that of lower concentration of the same type of charge carriers. Thus the movement of charge carriers takes place resulting in a current called diffusion current.
  • 10. Drift current in Semiconductors When an electric field is applied to a conductor at room temp, electron move towards the +Ve terminal of the applied volt but they continuously collide with atoms along the way. Each time the electron collides with an atom, it rebounds in a random fashion. At each collision, the electron loses some kinetic energy, then accelerates again, gains certain component of velocity in the direction of –E and loses its energy at the next collision. Obiviously, presence of the electric field does not stop collisions and random motion but it does cause the electrons to drift towards the +Ve terminal of the applied voltage V. Consequently, the electrons gain an average directed drift velocity v which is directly praportional to E. V = µe E Where, µe is called electron mobility. V (meter/second) E(volt/meter) µe = The resulting flow of electrons carrying negative charge at drift velocity V constitutes electric current called drift current. Let n = number of electrons per unit volume of the conductor A = conductor cross-section (m2) I = conductor length (m) E= V/l = applied electric field (V/m) I = n × (v × A) Toal no. of electrons which cross any plane P of cross-section A in one second Charge carried by them per second is = e n v A
  • 11. Drift current in Semiconductors It represents the drift current I = v e n A Substituting the value of v, we get, I= n e A µe E = n e A µe V l Now, R= 1 e V l l A neµ        Þ = l A Resistivity þ 1 neue  Conductivity = neµe siemens/m
  • 12. Mobility and Conductivity The ability of an electron to move through a semiconductor, in the presence of applied electric field is known as electron mobility. Vn= μnE, Therefore, Mobility of electron μn= Vn/E (SI unit of mobility m2/(V.s) The ability of an hole to move through a semiconductor, due to applied electric field is known as hole mobility. Vp= μpE, Therefore, Mobility of electron μp= Vp/E (SI unit of mobility m2/(V.s) In intrinsic semiconductors, the number of electrons is equal to the number of holes. ne = np σ = ne e μn + np e μp = ni e (μn + np)
  • 13. HALL EFFECT When a magnetic field is applied to a current carrying conductor in a direction perpendicular to that of the flow of current, a potential difference or transverse electric field is created across a conductor. This phenomenon is known as Hall Effect. Hall Effect was discovered by Edwin Hall in 1879. The voltage or electric field produced due to the application of magnetic field is also referred to as Hall voltage or Hall field. When a voltage is applied to a conductor or semiconductor, electric current starts flowing through it. In conductors, the electric current is conducted by free electrons whereas in semiconductors, electric current is conducted by both free electrons and holes. The free electrons in a semiconductor or conductor always try to flow in a straight path. However, because of the continuous collisions with the atoms, free electrons slightly change their direction. But if the applied voltage is strong enough, the free electrons forcefully follow the straight path. This happens only if no other forces are applied to it in other direction. If we apply the force in other direction by using the magnetic field, the free electrons in the conductor or semiconductor change their direction.
  • 14. HALL EFFECT Consider a material, either a semiconductor or conductor as shown in the below figure. When a voltage is applied, electric current starts flowing in the positive x-direction (from left to right). If a magnetic field is applied to this current carrying conductor or semiconductor in a direction perpendicular to that of the flow of current (that is z-direction), an electric field is produced in it that exerts force in the negative y direction (downwards). Mathematical expression for the Hall voltage is given by VH = IB qnd Where, VH = Hall voltage I = current flowing through the material B = magnetic field strength q = charge n = number of mobile charge carriers per unit volume d = thickness of the material Hall Voltage is directly proportional to the electric current and applied magnetic field.
  • 15. Derivation of Hall Coefficient e EH = e Bv EH = Bv JX = nev V = Thus , EH = EH = RH JX B RH = = RH = JX ne B JX ne EH JX B 1 np If v is the velocity of electrons at right angles to the magnetic field, there is a downward force acts on each electron of magnitude Bev. This results in the electron current to be deflected in a downward direction and causes a negative charge to gather on the bottom face of the specimen. A potential difference is therefore generated from top to bottom of the specimen. This potential difference causes a field EH in the negative y-direction, and thus a force of eEH acting in the upward direction on the electron. Hence at equilibrium condition, the force downwards due to magnetic field will be equal to the upward electric force. The Hall effect is described by means of the Hall coefficient RH defined in terms of current density Negative sign indicates that the electric field developed is in the negative y- direction. In case of p-type specimen, when current is completely due to holes, -1 ne
  • 16. Applications of hall effect Hall Effect is used to find whether a semiconductor is N-type or P-type. Hall Effect is used to find carrier concentration. Hall Effect is used to calculate the mobility of charge carriers (free electrons and holes). Hall Effect is used to measure conductivity. Hall Effect is used to measure a.c. power and the strength of magnetic field. Hall Effect is used in an instrument called Hall Effect multiplier which gives the output proportional to the product of two input signals.
  • 17. Charge carriers in semiconductors are negatively charged electrons and positively charged holes. Holes are essentially vacancies in an otherwise filled band so that when an electron moves to such a vacancy, the effect is equivalent to movement of a hole in reverse direction. In the absence of an electric field, charge carriers move randomly so that their average velocity is zero. When an electric field is applied, the positve charge carriers move in the direction of the field and the negative charge carriers move against the field direction. This directed motion is superimposed over the random direction and is called drift. Drift velocity is proportional to the direction of the field, the constant of proportionality is the mobility of the carrier. If a magnetic field is applied on a flat strip of semiconductor in a direction perpendicular to the direction of current flow, a voltage develops in a direction which is perpendicular to both the direction of the field and the magnetic field. This is known as Hall voltage. Hall voltage arises due to Lorentz force that acts on charge carriers and provides a direct means of verifying existence of holes.
  • 18. Without semiconductors, the world would be a very different place; we would have no electronic hand calculators, microwave ovens, digital alarm clocks, cellphones, personal computers, electronically controlled transmissions, or washing machines.
  • 19. Anyone who has never made a mistake has never tried anything new. Albert Einstein