The Quantized Free Electron Theory

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The Quantized Free Electron Theory
Electron sees effective smeared potential
Jellium model
Energy E
electrons shield potential to a large extent
Nucleus with localized core electrons








Spatial coordinate x
2
Electron in a box
In three dimensions
In one dimension
where
where
3
x
0
L
Periodic boundary conditions
Fixed boundary conditions
free electron parabola
Remember the concept of
density of states
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1. approach
use the technique already applied for phonon
density of states
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1/ Volume occupied by a state in k-space
kx
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Free electron gas
2
Each k-state can be occupied with 2 electrons of
spin up/down
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2. approach
calculate the volume in k-space enclosed by the
spheres
and
of states between spheres with k and kdk
2
with
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D(E)dE
of states in dE / Volume
E
EdE
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Statistics of the electrons (fermions)
Fermions are indistinguishable particles which
obey the Pauli exclusion principle
Let us distribute 4 electrons spin
T0
4 electrons spin
En6
En5
EF
En4
En3
En2
En1
1
Probability that a qm state is occupied
f(E,T0)
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Fermi Dirac distribution function at Tgt0
With accuracy sufficient for many estimations
f(E,T) linearized at EF
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More detailed approach to Fermi statistics
The grand canonical ensemble
Particle reservoir
Average energy
Average particle
Normalized probabilities
Heat Reservoir R
Tconst.
Now we consider independent particles
occupation ni0,1 of single particle state i
with energy ?i
where
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average occupation of state j is given by
Chemical potential
For details see
additional info see
means
where the summation
Repeat this step
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The Fermi gas at T0
EF0
Fermi energy depends on T
of states in E,EdE/volume
Probability that state is occupied
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Energy of the electron gas _at_ T0
there is an average energy of
per electron without thermal stimulation
with electron density
we obtain
Click for a table of Fermi energies, Fermi
temperatures and Fermi velocities
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only a few electrons in the vicinity of EF can be
scattered by thermal energy into free states
Specific heat much smaller than expected from
classical consideration
Specific Heat of a Degenerate Electron Gas
here strong deviation from classical value
2kBT
These
of electrons
increase energy from
to
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Significant contributions only in the vicinity of
EF
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EF
with
and
18
with
and
in comparison with
for relevant temperatures
Heat capacity of a metal
electronic contribution
lattice contribution _at_ Tltlt?D
W.H. Lien and N.E. Phillips, Phys. Rev. 133,
A1370 (1964)
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Selected phenomena which dont require detailed
knowledge of the band structure
Temperature dependence of the electrical
resistance
Impurities temperature independent imperfection
scattering
phonon scattering
Matthiessens rule
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scattering rate
Remember Drude expression
scattering cross section
scattering cross section
of scattering centers/volume
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Note T5 low temperature dependence not
described by this simple approach
average atomic spacing
Lindemann melting temperature TM
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Thermionic Emission
work function
generalized
Current density for k-dependent velocity
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Again
Spin degeneracy
Fermi distribution approximated by Maxwell
Boltzmann distribution
approximated
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Let us investigate the integral
Remember integrals of the type
Richardson-Dushman
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Vacuum tube
Tungsten
A (1-r)0.72 X 106 A/m2 K
Universal constant A1.2 X 106 A/m2K
Reflection at the potential step
Nobel prize  in 1928 "for his work on the
thermionic phenomenon and especially for the
discovery of the law named after him".
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Field-Aided Emission
Image potential
tunneling through thin barrier
cold emission