Transcript Slide 1

Nearly Free Electron Approximation
Has had success in accounting for many properties of metals:
Electrical conductivity
Thermal conductivity
Light absorption of metals
Wiedemann-Franz law (universal proportionality)
However, until quantum theory was incorporated there were
problems with predicting specific heat.
Solids rarely have heat capacities greater than 3R, and if the
electrons are freely moving, they would contribute 3/2 R to
the heat capcity. This is a problem since a great deal of
the heat capacity is accounted for by lattice vibrations
Model also fails, without Quantum Mechanics, to
account for existence of band gaps and insulators
Quantum Treatment
Assume that the electrons in a lattice act as if they were particles in a box
of length L. This is just the particle in a box model used to model
translational motion in elementary quantum mechanics
The solution to Hy = Ey for this model gives both the energy and the
wavefunction where y2 describes the probability that the electron is in some
region.
For the 1 dimensional model, the solution is
yn = A sin (npx/L) or A sin(kx)
k = np/L
Using Euler’s relation this becomes yn = A/2 [exp (ikx) – exp (-ikx)
The wavefunction exp (ikx) is the wavefunction of a free
(unbound particle), that is one moving without a potential
present, but k is not quantized as it is in the particle in a
box
The edge of the crystal at x=0 and x=L quantizes k. L is very
large compared to the lattice spacing and this means that
there will be many many quantum states that are very
closely spaced in energy
En = n2 h 2 / (8mL2) (Examples)
There are two parts of the wavefunction in
yn = A/2 [exp (ikx) – exp (-ikx)]
exp(ikx) relates to electrons traveling in a positive
direction in the lattice
exp(-ikx) relates to electrons traveling in a negative
direction in the lattice
This comes largely from the fact that y is an
eigenfunction of the momentum operator as will with
momentum
p = kh/2p and E = k2 h2/ 8p2me
(for a free electron)
The model can be extended to three dimensions by:
y (x,y,z) = exp (ik r)
There are many many electrons in a macroscopic crystal, but there is a
vastly greater number of k states.
At 0K the electrons will occupy the lowest states available, but as they
are fermions, they will obey the Pauli exclusion principle meaning
So they will fill up the states to some energy level, the Fermi Energy (EF)
the firmi energy. At T>0 the electrons can populate higher states,
higher states will be thermally populated according to the Fermi Dirac
Law.
This picture allows resolution of the heat capacity trouble mentioned
before. As seen in figure, only those electrons near the Fermi
surface can be promoted to a higher energy.
SO only a small number of the electrons contribute to the heat capacity.
What about the band gaps?
This is largely brought on by the lattice of positively charged nuclei that can
interact with the electrons, so essentially the electrons are not free ie
(nearly free approximation), but periodically disrupted by an attractive
potential.
The wavefunction for an electron in a periodic potential is given by:
y (x) = exp(ikx) U(x)
U(x) is a function that has the periodicity of the lattice. For an electron interacting
with a potential, its wave-like properties become crucial.
What tells us about the wave-like properties? DeBroglies relation:
l = h/p and from before we found the momentum to be p = kh/2p
So l = 2p/k
Effectively, the periodic potential scatters the electrons whose wavelengths meet
the Bragg condition.
Need to first look at the Bragg Equation
William and Lawrence Bragg developed a method for
treating X-ray diffraction treating scattering as reflections
from parallel planes of the crystal.
Constructive interference occurs when the path difference
between waves reflected from the planes is equal to an
integral multiple of the wavelength.
2d sin q = n l
where d = plane spacing
Waves not adding constructively will cancel.
So for the present case with lattice spacing “a”
2a sin q = n l = 2pn/k
So at 90o the Bragg reflections occur when k = n p / a
Again, electrons of a wavelength that satisfies the Bragg condition will be scattered
and cannot pass through the lattice. That means that the energies corresponding
to the values of k where k = n p / a will not be allowed.
There will effectively be discontinuities in the energy. Remember the energy for a
free electron depends on k
E = k2 h2/ 8p2me
A plot of E vs k would be in the shape of a parabola
But near the Bragg reflection condition, the parabola is modified,
producing band gaps
These gaps, are in fact zones in three dimensional k space. They are
called the Brillouin Zones, zones where the electron is scattered
off of the positive nuclear potential of the atoms making up the
lattice.
Thus in the Nearly Free Electron Approximation, the band gaps
appear due to scattering of the electron waves by the atoms that
make up the lattice
So lets review how this Model allows us to account for the
different types of materials
With the model, we can explain insulators as materials for
which the Fermi level coincides with the beginning of
the band gap
Effectively to a k-value which is equal to np/a
Why can’t the electrons below the Fermi-level conduct
electricity?
Remember that the k values specify something
beyond the energy
According to this model, the k values also specify both the
momentum and the direction of the electrons as well
If all k states in a given energy range are occupied, what
does that mean regarding the direction vectors for the
electrons?
If all occupied ten electrons travel uniformly in all direction,
and so effectively the electrons must be stationary as a
whole.
To move, a force must be impressed to change the
momentum direction of the electrons.
Can this happen?
Can a force be impressed on the electrons to cause them
to drift in a particular direction?
Below the Fermi level all states are occupied, so there
cannot be a net change into any particular direction or
particular set of them.
So then the material cannot conduct electricity, (no net
momentum in a particular direction)
In a metal however, even at 0K, there is not a problem
because in the case of the alkali metals and noble
metals there are always states available directly above
the Fermi level. Why?
If the gap between the states is much more that the thermal
energy, k T, then these states are not available.
What about the case of the alkaline earth metals? These
have an even number of valence electrons.
Here one has to consider whether or not the energy bands
overlap with one another in energy.
Expect that the alkaline earth metals might be insulators,
but looking at the energy diagram in k space, we find
that the bands overlap
Thus there are two ways to get a metal (or metal like
conductivities
a. because of electron concentration
b. because of band overlap
Semiconductors
Electrical Resistivity
Good Conductors 10-6
ohm-cm
Semiconductors
10-2 – 10-9
Insulators
1014 – 1022
Define as an insulator in which in thermal equilibrium some
charge carriers are mobile.
Characteristic semiconducting properties are usually
brought about by thermal agitation, impurities, lattice
defects, or a lack of stoichiometry (departure from
nominal chemical composition)
Two types of conductivity associated with semiconductors,
intrinsic conductivity and impurity conductivity
Intrinsic Conductivity
Intrinsic temperature range – range in which the electrical
properties of a semiconductor are not essentially
modified by impurities within the crystal.
Based on the idea of a vacant conduction band separated
by energy gap from a filled valence band (the old song
and dance)
Both the electrons in the conduction band and the vacant
states or holes left behind in the valence band will
contribute
At temps below the intrinsic range impurities become the
dominant mechanism, at sufficiently high temps. Intrinsic
takes over because there are more electrons excited
than there are on the impurities.
The value of the intrinsic conductivity is controlled by
The value of Eg relative to kT.
Table of energy gap in some semiconductors.
Values obtained two ways, by optically and by analysis of the
dependence of conductivity on temp.
Besides thermal excitation, photon absorption may also occur
and the threshold to continuous optical absorption is at the
frequency vg Eg = hvg
The photon can be absorbed in two ways either in a direct
process or in an indirect process (Direct Semiconductor or
Indirect Semiconductor)
Direct process
Lowest point in the conduction band is at the same k value as the high
point in the conduction band.
Here the threshold frequency vg determines the bandgap directly.
Indirect process
Involves both a photon and a phonon as the band edges of the
conduction and valence bands are widely separated in k
(momentum) space. Essentially the lowest point of the conduction
band does not coincide with the highest point in valence band
k(photon) = kc + K
hvg = Eg + hW
where
hW is the
phonon energy, and kc is wavevector of separation, K is the phonon
wavevector
Photon wavevectors are negligible in magnitude at the
energy range of 1eV
In effect the phonon becomes an inexpensive source of
crystal momentum.
Energies of phonons are small 0.1 to 0.3 eV at room temp
so with the photon energy this can occur.
Direct absorption of phonons is also possible if the
necessary band gap energy is present in the phonon.
The band gap determination from the temp dependence of
the conductivity or the carrier concentration in the
intrinsic range can be obtained from an experiment
known as the Hall Effect sometimes with additional
coductivity measurements