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 Semiconductors: have electrical resistivities in the range of .01 to
10**9 ohmcm at room temperature

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 The conduction band is vacant at absolute zero
 and is separated by an energy gap Eg from the filled valence band
 The lowest point in the conduction band is called the conduction band
edge; the high point in the valence band is called the valence band
edge. The band gap is the
difference between these two band edges.

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 Thermal energy excites electrons from the valence band to the conduction
band.
 The thermally excited electrons in the conduction band then contribute
to the electrical conductivity of the semiconductor.
 The vacant orbitals or holes left behind in the valence band also
contribute to the electrical conductivity!

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 These are largely controlled by the ratio of the band gap to the thermal
energy. If this ratio is large,
there will be few electrons excited.
Therefore the concentration of the carriers will be low, which
means the conductivity will also be small.

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 The best way to measure the value of the band gap for a semiconductor is
via optical absorption.
 The threshold of continuous optical absorption at a frequency determines the band gap via

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 In a direct absorption process the crystal absorbs a photon with the creation
of an electron and hole.

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 In this case the minimum energy gap involves electrons and holes that
are separated by a nonnegligible wavevector
 Hence one cannot have a direct photon transition at the minimum gap, as
one cannot safisfy the requirement of conservation of “momentum” (or
wavevector).

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 Consider a wave packet made of up wave functions near a particular
wavevector k

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 Consider the work done on the electron by an electric field E in a small
time interval

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 Equating these two equations for energy:

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 Note that an electron in a crystal is subject to forces from the crystal
(e.g. interactions with the ions), in addition to the external
electromagnetic force F.

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 Consider an electron of group velocity v in a constant magnetic field B

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 The gradient of the energy is normal to the energy surface. The rate of change of k is in a
direction normal to this gradient and to the magnetic field.
 Therefore, an electron moves in k space under the influence of a
constant B field
 on a surface of constant energy.

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 Consider the Bloch energy eigenfunction for an electron in three
dimensions with wavevector k

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 Suppose we now apply an external force to the electron, changing its
state from k to k + k
 We will assume the crystal is an insulator and
 Electrostatically neutral except for a single electron in the state k
of a single band.
 The total impulse J given to the
crystal system by a force F acting during a time interval t
is J=F t

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 The impulse will equal the total change in momentum of the system,
namely the change of momentum of the lattice plus electron

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 Next, we need to calculate what the change in the lattice momentum is
that results from the change in the state of the electron, due to the
external force F
 We first note that an electron that is reflected by the lattice
transfers momentum to the lattice, as we have seen earlier.

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 Namely, if an incident electron with a plane wave component of momentum
hk is reflected with momentum h(k+G), then the lattice gains the
momentum –hG.
 If we extend this argument to a wavepacket, then there will be a
contribution to the lattice momentum from each individual component of
the wavepacket.

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 Therefore the total change in momentum of the system is

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 The semiclassical model describes the response of the electrons to
external E and B fields that vary slowly over the dimensions of the wave
packet describing the electron (and therefore slowly over a few
primitive cells)
 Such E and B fields give rise to ordinary classical forces in the
equations describing the position and wave vector of the wave packet
 However, the periodic potential of the lattice varies over dimensions
that are small compared with the spread of the wave packet and cannot be
treated classically.

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 The semiclassical model predicts how (in the absence of collisions) the
position r and the wave vector k of each electron evolve in the presence
of externally applied electric and magnetic fields.
 It assumes a knowledge of the band structure of the material, i.e. of
energy eigenvalues as a function of k.
We often label each band of the energy by a subscript integer n.

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 The band index n is a constant of the motion. The semiclassical model ignores
transitions between bands.
 In thermal equilibrium the average occupation number of the electrons is
given by the FermiDirac distribution function

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 3. Given an energy band with band
index n the equations of motion are

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 A small phase space element dk about k contributes a term –e v(k) dk to
the electrical current, where v(k) is the group velocity. Thus the total contribution over the
band is

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 But one can prove that the integral over any primitive cell of the
gradient of a periodic function must vanish. Therefore for a completely filled band
the integral for both the electric and energy currents must vanish, as
the energy eigenvalue is periodic
in the reciprocal lattice.

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 Since electrons in a volume element dk about k contribute a term
proportional to
 e v(k)dk to the current density, the contribution of all electrons in a
given band is

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 The unoccupied levels in a band evolve in time under the influence of
external applied fields precisely as if they were occupied by real
electrons of charge –e.
 The reason is that given the values of k and r at t=0, the semiclassical
equations of motion (six firstorder differential eqns in 6 variables)
uniquely determine their subsequent orbit in the presence of E and
B. We can therefore separate the
orbits into unoccupied or occupied orbits, depending on their states at
t=0. (No two distinct orbits can
intersect.)

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