Lecture+14+MAK +Free+Electron+Gas1
Transcript of Lecture+14+MAK +Free+Electron+Gas1
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PHY 3201 FIZIK KEADAAN PEPEJAL
Free Electron Fermi Gas
The properties of materials depends to a large extent
on the type of bonding and the energy distribution of
the bonding electrons. The bonding electrons in the
solid find themselves in the potential of their parent
nuclei, other nuclei and other electrons. Because ofthe intricate nature of the potential, the theory of
solids is quite complex.
However, it may be simplified by assuming simple
models for the potential appropriate to different typesof solids
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Many solids conduct electricity.
There are electrons that are not bound to atoms but are able tomove through the whole crystal.
Conducting solids fall into two main classes; metals and
semiconductors.
Metal - Resistivity increases by the addition of small amounts
of impurity. The resistivity normally decreases monotonically
with decreasing temperature.
Semiconductor Resistivity can be reduced by the addition ofsmall amounts of impurity.
Semiconductors tend to become insulators at low T.
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Why mobile electrons appear in some
solids and not others?
According to free electron model (FEM), the valance
electrons are responsible for the conduction of
electricity, and for this reason these electrons are
termed conduction electrons.
Na11 1s2 2s2 2p6 3s1
This valance electron, which occupies the third atomic
shell, is the electron which is responsible chemical
properties of Na.
Valance electron (loosely bound)
Core electrons
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The removal of the valance electrons leaves a
positively charged ion. The charge density associated the positive ion
cores is spread uniformly throughout the metal sothat the electrons move in a constant electrostaticpotential. All the details of the crystal structure islost when this assunption is made.
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A valance electron really belongs to the whole crystal,since it can move readily from one ion to its neighbour,and then the neighbours neighbour, and so on.
This mobile electron becomes a conduction electron in asolid.
According to Free Electron Fermi Gas this potential istaken as zero and the repulsive force betweenconduction electrons are also ignored.
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Free Electron Fermi Gas
The main assumptions for metals is as follows:
The atomic nuclei are fixed on the crystal lattice atregular positions.
The electrons in the inner shells of each atom are
tightly bound to the atom; the atomic nucleus ownsthem.
Electrons in the outer shells are not bound to anyparticular atom. They are fairly free to move around inthe crystal lattice, but they are still bound to the solid.
All atoms together provide a collective potential inwhich all the outer electrons are moving. The electronsin turn, by floating around from one atom to the other,provide bonds that keep all the atoms together.
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A free electron model is the simplest way to represent theelectronic structure of metals. According to this model, thevalence electrons of the constituent atoms of the crystalbecome conduction electrons and travel freely throughout
the crystal. The only restriction being that they cannotescape from the metal. Within this model we also neglectthe interaction of conduction electrons with ions of thelattice and the interaction between the conductionelectrons. Hence the conduction electrons in a metal are
modeled by a free electron gas.
This theory is called the free electron theory of metals
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The Time-IndependentSchrodinger Equation
The Schrodinger Equation is a mathematical description
of an electron as a wave as suggested by Schrodinger in
1926. The time independent Schrodinger Equation is
used when the properties of atomic systems have to be
calculated in stationary conditions, i.e. when the propertyof the surroundings of the electron does not change with
time. If the potential energy V, depends only on location
(and not, in addition, on time), :
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The time independent Schrodinger equation has the
following form:
nnH
H is the Hamiltonian = Kinetic Energy +
Potential Energy
nis the energy of the electron in orbital n
n is the wavefunction for the orbital n
If we use the assumptions above, the Schrodinger equation for a
metal is simply that of a particle in a box and since the potential is
constant we might as well set it to zero. The Hamiltonian will then
consist of only the kinetic term
2
222
2))((
2
1
2 dx
d
mdx
di
dx
di
mm
pH
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nnnndx
d
mH
2
22
2
Thus, the Schrodinger eqn. reduces to
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Free electron
For one dimension the Schrodinger equation becomes0
222
2
nn
n m
x
This is the differential equation for an undampedvibration and have the following solution
kxAx sin)(
2
2 2
mk
2
2
2k
m
where
Let
02
2
2
nn
n kx
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Since no boundary condition had to be considered for
the calculation of the free electrons, all values of theenergy are allowed i.e. one obtains an energy
continuum.
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Electon in a potential
well (bound electron)
Consider a dimensional potential box in which the
electron can move freely between two infinitely high
potential barriers. The potential barriers do not allow
the electron to escape from the potential well, which
means that = 0 forx0 andxL The potential energyinside the well is zero. Therefore
02
22
2
nn
n m
x
V
xL
xAn
n
2sin
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The boundary conditions are= 0 atx = 0
= 0 atx = L
Atx = L,
0
2
sin
LA nn
Which can only be satisfied if
Thus,
nL
n
2
or2
nnL
x
L
nAn
sin
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These solutions correspond to standing waves with a
different number of nodes within the potential well asis shown in Fig.1.
n
nmm
kmp
2
222
2222
2222
2/2
2
2
L
n
mnLmn
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Now we need to accommodate Nvalence electrons in
these quantum states. According to the Pauli exclusion
principle no two electrons can have their quantum
number identical.
The electronic state in a 1D solid is characterized bytwo quantum numbers that are n and ms, where n
describes the orbital n and msdescribes the magnetic
quantum number. Therefore, each orbital labeled by the
quantum numbern can accommodate two electrons,
one with spin up and one with spin down orientation.
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Let nFdenote the highest filled energy level, where we
start filling the levels from the bottom (n = 1) andcontinue filling higher levels with electrons until all N
electrons are accommodated. It is convenient to
suppose that Nis an even number.
The condition 2nF= Ndetermines nF, the value ofn forthe uppermost filled level.
The energy of the highest occupied level is called the
Fermi energy, F. For the one dimensional system ofN
electrons we find2222
222
L
N
mL
n
m
FF
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The ground state of the Nelectron system is illustrated
in Fig.2a: All the electronic levels are filled up to theFermi energy. All the levels above are empty.
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The distribution of electrons among the levels is usually
described by the distribution function, f(), which isdefined as the probability that the level is occupied by
an electron. Thus if the level is certainly empty, then, f()
= 0, while if it is certainly full, then f() = 1. In general, f()
has a value between zero and unity.It follows from the preceding discussion that the
distribution function for electrons at T = 0K has the form
F0
1
Ff )(
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The distribution function at non-zero temperature is
given by the Fermi distribution function. The Fermidistribution function determines the probability that an
orbital of energy is occupied at thermal equilibrium.
The quantityis called the chemical potential.
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Tkf
B/exp)(