Download Empty Lattice Approximation – used for calculating Energy Band Gaps

November 4, 2010
Class 17
Hall Effect
If a magnetic field is applied perpendicular to a current, the electrons will suffer a force
perpendicular to both, the drift velocity and B, of a magnitude
F=qvxB
This force produces an accumulation of charge and an electric field that generates an
electrical force that opposes to the magnetic force. In equilibrium
-eEH=evB or EH=-vB
By defining the Hall coefficient RH as
EH=-RH JB
vB= RH JB or RH=v/J
(notice that RH can be measured by measuring the electric field perpendicular to the current and
it is a property of the material).
Using J=nqv RH=1/nq (where q is negative for electrons)
For Li, monovalent, the measured Hall coefficient at room temperature is -1.7x10-10 m3/C and
the calculated value -1.4x10-10 m3/C. For Al (trivalent) both values are ~ -0.3x10-10 m3/C. But
for Zinc, the calculated value is -0.5x10-10 m3/C
while the measured one is +0.3x10-10 m3/C. The
reason is that the carriers are positive. Although
electrons are always the ones moving, some
material, especially divalent, behave as if the
carriers were positive. We’ll talk about that later.
Nearly Free-electron Theory
Free electron (FE) theory produces very good results for many things but there are fundamental
issues that are not accounted for within that simple theory. Band structure will not come out
from FE, thus there is no way to understand or define conductors, insulators, semiconductors, etc.
Also FE, as indicated above, does not account for a positive Hall coefficient, in other words,
from the FE point of view only free electrons can conduct.
Real life shows us that the above is a way too simplified picture, electrons states do NOT
form a continuous from 0 to infinity but the energy breaks in bands with gaps of forbidden
energy. Also experiments show that some materials behave as if the particles being dragged by
the electric field were actually positive particles. In addition, it is observed that the mass of the
electrons being conducted (or that of the positive particles) is NOT necessarily equal to the free
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electron mass. All of this can be explained with a “simply correction” to the FE, we know this as
Nearly-FE (NFE) theory.
Empty Lattice Approximation – used for calculating Energy Band Gaps
Figures
taken
http://people.deas.harvard.edu/~jones/ap216/lectures/ls_2/ls2_u7/ls2_unit_7.html
from
One way of approximating the band structure is to use the free electron approximation within the
brillouin zones,  
 2k 2
. Using that k  k' G , where k’ is within the first brillouin zone we
2m
can bring all the bands into the first zone for drawing. If G is a reciprocal translation vector-- the
 2 ( k' G )2
solution within the free electron theory then becomes  
. Supporting info.
2m
Bands (nearly free electrons)
For  >> a (small k), the waves do not see structures and electron behave as free electrons, then
E=ħ2k2/2m (this is the low energy regime, electrons near the bottom of the band).
Let’s suppose that instead /2=a. Now there is diffraction (the Bragg condition is met) since
k=/a. This condition means standing waves and electrons are not traveling back and forth. The
solutions to the Schrödinger equation cannot be a single complex exponential, since a complex
exponential is never zero and standing waves must have fixed nodes. However, a linear
combination of complex exponentials is also a solution, and the more appropriate solutions are
sine’s and cosines (which can be obtained as a linear combination of two plane waves, one
moving to the right and the other moving to the left).
(+)=2A cos(x/a)Aeiπ/a+A e-iπ/a
(-)=2Bi sin(x/a)= Beiπ/a-Be-iπ/a
A and B do not need to be real numbers since what is physically observable is the density which
is the square of the wave functions, thus
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(+)=|(+)|2cos2(x/a)
(-)=|(-)|2sin2(x/a)
For x=0,a,2a,..,na (+) is maximum (cos=1) and it is zero in between (a/2, 3a/2,…) etc thus the
electron density is maximum in the position of the nucleus and the nodes are in between. The
opposite happens for (-) thus the first wavefunction represent electrons piled up at the position
of the nucleus while the second wave function represents electrons piled up in between atoms.
If the wave’s maximum is on the nuclei position, there is maximum (negative) interaction with
the lattice and electrons have lower energy than free electrons. If instead the wave’s maximum
is in between nuclei positions, there is maximum (positive) interaction with the lattice and
electrons have higher energy than free electrons, this is the origin of the band gap.
When NFE is considered, near the band edge, the E(k) differs
from a parabola determining two values at the border separated
by a gap. By taking these values to within the first Brillouin
zone (for visual purposes) we get a clear picture of the gaps. You
will find energy band plots of the form shown in the figure.
http://www.mtmi.vu.lt/pfk/funkc_dariniai/quant_mech/bands.htm
Bloch Functions
“When I started to think about it, I felt that the main problem was to explain how the electrons
could sneak by all the ions in a metal… By straight Fourier analysis I found to my delight that
the wave differed from the plane wave of free electrons only by a periodic modulation.”
F. Bloch
Bloch proved that the solution of the Schrödinger equation in a periodic potential must be of
the form
 k ( r )  uk r e ik .r
Where u(r) has the periodicity of the lattice, thus u(r+T)= u(r) where T is a translation
vector of the lattice.
A proof restricted to the case where the states are non-degenerated is as follows:
Consider a ring with N identical lattice points at a distance a from each other. By symmetry,
let’s look at solutions of the form
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 x  a   C x  where C is a complex number, thus  x  Na  C N x 
Since x+Na=x in the circle,  x  Na   x , then CN=1 and C is one of the N roots of the
unity
Ce
i
2 s
N
Thus
 x   ux e
i
2sx
Na
where u(x) must satisfy u(x+a)= u(x) such that  x  a   C x 
This is a simply proof to the block theorem for the case where non-degenerate solutions
exist. Block functions are the electron wave functions within the NFE
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