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2) Phase space combines position and momentum space, specifying the complete state of a system. For classical particles, the Maxwell-Boltzmann distribution describes average particle numbers. Quantum statistics include Bose-Einstein and Fermi-Dirac distributions.
3) A photon gas in an enclosure reaches thermal equilibrium where the Bose-Einstein distribution applies. The number of photon energy states is calculated from phase space considerations.
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The chapter gives brief knowledge about formation of bands in solids. What are free electrons how they contribute for conductivity in conductors, but can be extended to semiconductors also.
Derive the thermal-equilibrium concentrations of electrons and holes in a semiconductor as a function of the Fermi energy level.
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Determine the thermal-equilibrium concentrations of electrons and holes in a semiconductor as a function of the concentration of dopant atoms added to the semiconductor.
Determine the position of the Fermi energy level as a function of the concentrations of dopant atoms added to the semiconductor.
explain the k-space diagrams of si and GaAs and the difference between the direct and the indirect bands, Additional Effective Mass Concepts, Mathematical Derivation for the density of state function with extension ti to semiconductors
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The chapter gives brief knowledge about formation of bands in solids. What are free electrons how they contribute for conductivity in conductors, but can be extended to semiconductors also.
Derive the thermal-equilibrium concentrations of electrons and holes in a semiconductor as a function of the Fermi energy level.
Discuss the process by which the properties of a semiconductor material can be favorably altered by adding specific impurity atoms to the semiconductor.
Determine the thermal-equilibrium concentrations of electrons and holes in a semiconductor as a function of the concentration of dopant atoms added to the semiconductor.
Determine the position of the Fermi energy level as a function of the concentrations of dopant atoms added to the semiconductor.
explain the k-space diagrams of si and GaAs and the difference between the direct and the indirect bands, Additional Effective Mass Concepts, Mathematical Derivation for the density of state function with extension ti to semiconductors
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1. ELEMENTS OF STATISTICAL MECHANICS
Definition: The Subject which deals with the relationship between the overall behavior of the
system and the properties of the particles is called statistical mechanics.
Macro state: Each compartment distribution of a system of particles is known as a macrostate.
Ex: Distribution of x, y, z particles in 2 compartments.
Compartment Possible distributions of particles
1 0 1 2 3
2 3 2 1 0
Micro State: Each distinct arrangement is known as the micro state of the system.
Ex: Distinct arrangement of particles x, y, z in 2 compartments.
Macro
State
Compartment
1
Compartment
2
No of
Micro states
0, 3 0 X Y Z 1
1, 2
X
Y
Z
Y Z
X Z
X Y
3
2, 1
X Y
Y Z
X Z
Z
X
Y
3
3, 0 X Y Z 0 1
Position space: The three dimensional space in which the location of a particle is completely
specified by the three position coordinates is known as position space.
A small volume element in position space dv is given by dv = dxdydz
Momentum Space: The three dimensional space in which the momentum of particle is
completely specified by the three momentum coordinates px, py and pz is known as momentum
space.
A small volume element in momentum space dฯ = dpxdpydpz
Phase space: A combination of the position space and momentum space is known as phase
space.
๏ท A point phase space is completely specified by six coordinates x, y, z,px, py,pz
๏ท If there are N particles. 6N coordinates provide complete information regarding the
position and momentum of all N particles in the phase space in a dynamic system.
๏ท A small volume in phase space dT is given by
2. dT = dxdydzdpxdpydpz
dT = dvdฯ
Thus a volume element dT in phase space is the product of a volume element dv in position
space and Volume element dฯ in momentum space.
โด ๐โ๐ ๐๐๐ก๐๐ phase space volume T = โซ๐๐ = โซ๐๐ฃ โซ ๐๐
T =ัด ฯ
๏ท The small volume dT in phase space is called a cell.
๏ท
๐๐๐ก๐๐ ๐๐ฃ๐๐๐๐๐๐๐ ๐๐๐๐ข๐๐ ๐๐ ๐โ๐๐ ๐ ๐ ๐๐๐๐ (ัด ฯ)
๐๐๐๐ข๐๐ ๐๐ ๐๐๐ ๐๐๐๐ ๐๐
= ๐โ๐ ๐ก๐๐ก๐๐ ๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐ ๐๐ ๐โ๐๐ ๐ ๐ ๐๐๐๐
๏ท Each spherical shell is called an energy compartment.
๏ท A cell is a sub-compartment.
๏ท Each compartment is divided into a very large number of cells in such a way that each
cell is of the same size. Hence all the cells have the same a priori probability of a
particle going into a cell.
๏ท In classical mechanics the volume of a cell in phase space can even approach zero(dT =
dvdฯโ 0).
๏ท In quantum mechanics the volume of the cell in phase space cannot be less than h3.
๏ท Three kinds of distributions are possible corresponding to 3 different kinds of particles.
Classical Statistics โ Maxwell Boltzmann Distribution:
๏ท The main assumptions of M B Statistics.
1. The particles are identical and distinguishable.
2. The volume of each phase space cell chosen is extremely small and hence chosen
volume has very large number of cells.
3. Since cell are extremely small, each cell can have either one particle or no particle
though there is no limit on the number of particles which can occupy a phase space cell.
4. The systemis isolated which means that both the total number of particles of the
system and their total energy remain constant.
5. The state of each particle is specified either by its cell number in phase space or
instantaneous position and momentum co-ordinates.
6. Energy levels are continuous.
7. ๐๐(๐ธ๐) = ๐โโ
๐
โ
๐ธ๐
๐พ๐
= ๐๐๐ต(๐ธ๐)
is the Maxwell โ Boltzmann distribution function. It gives the
average number of particles โnโ with energy. E in equilibrium at a given temperature for
a system of classical particles. Since โnโ is the average number of particles, it need not
be an integer.
๏ท ๐คโ๐๐๐ โ =
๐ถ๐๐๐ ๐ก๐๐๐ก ๐๐๐ ๐๐๐๐๐๐๐ ๐๐ ๐กโ๐ ๐๐๐๐๐๐๐ก๐ฆ ๐๐ ๐กโ๐ ๐ ๐ฆ๐ ๐ก๐๐ ๐๐๐ ๐๐๐๐๐๐๐๐ก๐ข๐๐ ๐
๏ท Ei = ith energy state
๏ท ni = number of particles at ith energy state.
3. ๏ท K = Boltzmann constant = 1.38 x 10-23j/k.
Limitation of classical Statistics.
1. The observed energy distribution of electrons in metals.
2. The observed energy distribution of photons inside an enclosure.
3. The behavior of helium at low temperatures.
Quantum Statistics:
Bose EinsteinStatistics: Assumptions:
1. In Bose Einstein particles are known as Bosons.
2. The Bosons of the system are identical and indistinguishable.
3. The Bosons have integral spin angular momentum in units of h/2ฯ.
4. Bosons obey uncertainty principle.
5. Any number of Bosons can occupy a single cell in phase space.
6. Bosons do not obey the exclusion principle.
7. The number of phase space cells is comparable with the number of Bosons.
8. Wave functions representing the bosons are symmetric i.e. ๐(1,2) = ๐(2,1).
9. The wave functions of bosons do overlap slightly i.e. weak interaction exists.
10. Energy states are discrete.
11. The probability f(E) that a boson occupies a state of energy E is given by
๐ ๐๐(๐) =
๐
๐
(๐+
๐
๐๐
)
โ ๐
This is called Bose โ Einstein distribution function. The quantity ฮฑ is a constant and
depends on the property of the system and temperature T.
If โnโ is the number of Bosons with total energy E when in equilibrium at temperature T, the
most probable distribution of bosons among the various energy levels is given by
๐ง๐ข =
๐ ๐ข
๐
(๐+
๐
๐๐
)
โ ๐
Where gi is the number of quantum states (phase space cells) available for the same
energy Ei.
๏ผ The quantity gi is called degeneracy.
Examples of Bosons are photons, He4, ฯ and K โ mesons.
Fermi Dirac Statistics: Assumptions:
1. Half integral spin particles are known as fermions.
2. Fermions are identical and indistinguishable.
3. They obey Pauliโs exclusion principle i.e. there cannot be more than one particle in a single
cell in phase space.
4. Wave function representing Fermions are antisymmetric. i.e. ๐(1,2) = โ๐(2,1).
5. Weak interaction exists between the particels.
6. Uncertainty principle is applicable.
7. Energy states are discrete.
8. The probability f(E) that a Fermion occupies a state of energy E is given by
4. ๐ ๐ ๐(๐) =
๐
๐
(๐+
๐
๐๐
)
+ ๐
This is called Fermi โ Dirac distribution function. The quantity ฮฑ is a constant and depends on
the property of the system and Temperature โTโ.
๏ผ Let the total energy of the electrons be E, this energy is distributed among all the
โnโ electrons. According to the Fermi โ Dirac distribution law.
๐ง๐ =
๐๐
๐
(๐+
๐
๐๐
)
+ ๐
Where gi is the number of phase space cells with energy Ei.
๏ผ Examples of Fermions: Free electrons in conductor, protons, neutrons and He3
atoms.
Fermi energy: Fermi โ Dirac Distribution function.
๐ง(๐ฌ) =
๐(๐ฌ)
๐
(๐+
๐
๐๐
)
+ ๐
But ฮฑ=
โยต
KT
๐ง(๐ฌ) =
๐(๐ฌ)
๐
(
โยต
KT
+
๐
๐๐
)
+ ๐
=
๐(๐ฌ)
๐
(
Eโยต
KT
)
+ ๐
Since electrons have spin 1/2, there are two states with spin projection +1/2 and -1/2
with the same energy.
โด ๐(๐ธ) = 2 ๐๐๐ ๐๐๐ ๐ธ
Then the average number of particles in a state with energy โEโ or the probability of
occupation of a state of energy E is given by
๐(๐ฌ) =
๐
๐
(
Eโ๐ธ๐
KT
)
+ ๐
๏ผ This is known as Fermi function. We have replaced the Chemical potential ยต by
๐ธ๐ which is known as Fermi Energy.
๏ผ The meaning of Fermi Energy becomes clear from the following considerations.
If T=0, then
E โ Ef
KT
= โโ, if E โบ Ef and
E โ Ef
KT
= โ, if E > Ef
โด ๐
Eโ๐ธ๐
KT + 1 = {
1 ๐๐ E < Ef
0 ๐๐ E > Ef
๐๐ ๐ ๐๐๐ ๐ข๐๐ก
๐(๐ธ) = {
1 ๐๐ E < Ef
0 ๐๐ E > Ef
The probability of occupation of a state of energy E is one upto E=EF and also is zero for E > Ef.
That means all states with energy upto EF are occupied and all states with energy higher than EF
are empty at T=0. This is the meaning of EF at T=0.
5. If T > 0
๏ผ For T > 0 the exponential function ๐
Eโ๐ธ๐
KT is well behaved for all E. As a result 0<f(E)<1 for
all E.
๏ผ That means the probability of occupation is not one even if E < Ef and the probability of
occupation of no state is zero even if E > Ef.
๏ผ Further if E = Eff(E) = ยฝ. Therefore we interpret Ef as the energy at which the
probability of occupation is ยฝ for T > 0.
PHOTON GAS:
๏ถ Radiation enclosed in a container which is at thermal equilibrium with the walls of the
container is called Black Body Radiation.
๏ถ In thermal equilibrium with the walls of the container means there is a constant
exchange of energy between the radiation and the walls of the container so that the
temperature of the walls is the same as the temperature of radiation.
๏ถ This radiation contains electromagnetic waves of all wavelengths (frequencies) from
zero to infinity. Such a radiation can be considered to be a collection of photons of all
wavelengths from zero to infinity at thermal equilibrium.
๏ถ The spin of these particles (photons) is known to be 1. Because of this they have
polarization index +1 or -1. Therefore there can be two photons of the same frequency
one with polarization index +1 and another with -1.
๏ถ Within the container the number of photons of a given frequency ฮฝ is not constant. A
photon of certain energy can split itself into two or more photons of lower energy or
two or more photons can fuse together to form a photon higher energy.
๏ถ This process goes on all time. As a result the number of photons of a given frequency is
not a constant in time.
๏ถ i.e. without spending or gaining energy one can increase or decrease the number of
photons of a particular energy.
โด โ ๐๐ = ๐ ๐๐ ๐๐๐๐๐๐๐๐๐๐ ๐
๐
This we take into account by making ฮฑ = 0 Thus the chemical potential of a photon gas is
zero.
Black body radiationand the Planck radiationlaw
๏ท The momentum of photons = p =
โ๐
๐
๏ท The Volume of each allowed cell in the phase space can be dะ = dxdydzdpxdpydpz = h3
๏ท Now the phase space is split into the position space and momentum space. If the
range of position coordinates is continuously increased till they embrace the whole
volume โVโ of the enclosure then the particles can be found anywhere in the
enclosure.
๏ท So according to Heisenbergโs uncertainty relation an element of volume in the
momentum space can be written as = ๐๐ =
โ3
๐
6. ๏ท ๐ด๐ ๐๐Denotes the size of an elementary cell in the momentum space only a single
value of momentum can be recognized within a cell.
๏ท At any instant all photons having their momenta between p and p+dp will be within a
spherical shell described in momentum space with radii p and p+dp.
๏ท Therefore the volume of this shell is 4ฯp2dp
Therefore the total number of energy states between momenta p and p+dp is given by
g(p)dp = [
4ฯp2
๐
โ3 ]๐๐ ------๏ 1
๏ท For a photon p =
โ๐
๐
๐๐ ๐p =
โ๐๐
๐
Substituting these values in above equation the total number of energy states
between frequencies ฮฝ and ฮฝ+dฮฝ is given by
g(ฮฝ)dฮฝ =[
4ฯV๐2
๐3 ]๐๐ ----๏ 2
Taking into account the doubling of the states due to polarization of photons the total
number of Eigen states (Energy States) available for the photons in the frequencies
range ฮฝ and ฮฝ+dฮฝ
g(ฮฝ)dฮฝ =[
8ฯV๐2
๐3 ]๐๐ ----๏ 3
Introducing this result in Bose Einsteinโs distribution law
๐๐ =
g(ฮฝ)dฮฝ
๐๐ผ +๐ฝ๐ธโ1
-------๏ 4
In this case ฮฑ = 0, ฮฒ=1/KT, E=hฮฝ
โด ๐๐ =
g(ฮฝ)dฮฝ
๐๐ผ +๐ฝ๐ธโ1
=
8ฯV๐2
๐3
1
๐
hฮฝ
๐พ๐โ1
๐๐ -----๏ 5
๐๐
๐
=
8ฯ๐2
๐3
1
๐
hฮฝ
๐พ๐ โ1
๐๐ ------๏ 6
This equation represents the number of photons per unit volume lying in the frequency range ฮฝ
and ฮฝ+dฮฝ.
The energy density of radiation of frequencies between ฮฝ and ฮฝ+dฮฝ can now be found by
multiplying above equation by the energy of the photon hฮฝ.
Therefore if Eฮฝdฮฝ represents the energy density of radiation within the specified frequency
range then the energy distribution can be written as
Eฮฝdฮฝ =
๐๐
๐
hฮฝ =
8ฯ๐2
๐3
hฮฝ
๐
hฮฝ
๐พ๐ โ 1
๐๐
๐๐๐๐ =
๐๐๐ก๐๐
๐๐
๐
๐
๐ก๐
๐ฒ๐ปโ๐
๐ ๐-----๏ 7
This is well known Plankโs law of radiation in terms of frequency.
V=nฮป ---๏ c= ฮฝฮป -๏ ฮฝ = c/ฮป and dฮฝ =
๐
๐2 ๐๐
7. E(ฮป)dฮป =
8ฯh๐3
๐3๐3
1
๐
hc
๐๐พ๐ โ 1
๐
๐2
๐๐
โด ๐(๐)๐๐ =
๐๐๐ก๐
๐๐
๐
๐
๐ก๐
๐๐ฒ๐ปโ๐
๐ ๐-----๏ 8
This is Plankโs law of radiation in terms of wavelength โฮปโ
๏ท Wienโs Distribution Law for shorter wavelength range.
In the shorter wavelength range e
hc
ฮปKT โซ 1 hence we neglect 1 in the denominator of the
equation
โด ๐(๐)๐๐ =
๐๐๐ก๐
๐๐
๐
๐
๐ก๐
๐๐ฒ๐ป
๐ ๐ ------๏ 9
This equation is known as Wienโs Distribution law.
๏ท Rayleigh โ Jeans law for Larger Wavelength range.
In the longer wavelengths range e
hc
ฮปKT โช 1 ๏ โ 1 +
hc
ฮปKT
โ 1 =
hc
ฮปKT
โด E(ฮป)dฮป =
8ฯhc
๐5
๐๐
hc
ฮปKT
โด ๐(๐)๐๐ =
๐๐๐๐
๐๐
๐ ๐ ------๏ 10
This equation is called Rayleigh โ Jeans law.