Solid State Physics 3

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Solid State Physics3

• Section 10-4,6

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Topics

• Heat Capacity of Electron Gas
• Band Theory of Solids
• Conductors, Insulators and Semiconductors
• Summary

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Special Extra Credit
As can be seen from the graph, the prediction
fails at very low temperatures. This is due, in
part, to the failure of the equipartition
theorem at low temperatures. Challenge create
a better model!
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Special Extra Credit
Derive the temperature dependence of R/R0
by computing the average potential energy ltEgt of
a lattice ion
assuming that the energy level of the
nth vibrational state is
rather than En ne as Einstein had assumed
Due before classes end
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Heat Capacity of Electron Gas
By definition, the heat capacity (at
constant volume) of the electron gas is given by
where U is the total energy of the gas. For a
gas of N electrons, each with average energy
ltEgt, the total energy is given by
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Heat Capacity of Electron Gas
Total energy
In general, this integral must be done
numerically. However, for T ltlt TF, we can use a
reasonable approximation.
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Heat Capacity of Electron Gas
At T 0, the total energy of the electron gas is
For 0 lt T ltlt TF, only a small fraction kT/EF of
the electrons can be excited to higher energy
states
Moreover, the energy of each is increased
by roughly kT
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Heat Capacity of Electron Gas
Therefore, the total energy can be written as
where a p2/4, as first shown by Sommerfeld
The heat capacity of the electron gas is
predicted to be
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Heat Capacity of Electron Gas
Consider 1 mole of copper. In this case Nk R
For copper, TF 89,000 K. Therefore, even
at room temperature, T 300 K, the
contribution of the electron gas to the
heat capacity of copper is
small CV 0.018 R
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Band Theory of Solids

• So far we have neglected the lattice of
  positively charged ions
• Moreover, we have ignored the Coulomb repulsion
  between the electrons and the attraction between
  the lattice and the electrons
• The band theory of solids takes into account the
  interaction between the electrons and the lattice
  ions

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Band Theory of Solids
Consider the potential energy of a 1-dimensional
solid
which we approximate by the Kronig-Penney Model
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Band Theory of Solids
The task is to compute the quantum states
and associated energy levels of this simplified
model by solving the Schrödinger equation
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2
3
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Band Theory of Solids
For periodic potentials, Felix Bloch showed
that the solution of the Schrödinger equation
must be of the form
and the wavefunction must reflect the periodicity
of the lattice
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2
3
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Band Theory of Solids
By requiring the wavefunction and its
derivative to be continuous everywhere, one finds
energy levels that are grouped into bands
separated by energy gaps. The gaps occur at
The energy gaps are basically energy levels that
cannot occur in the solid
1
2
3
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Band Theory of Solids
Completely free electron
electron in a lattice
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Band Theory of Solids
When, the wavefunctions become standing
waves. One wave peaks at the lattice sites, and
another peaks between them. ?2, has lower energy

than ?1. Moreover, there is a jump in energy
between these states, hence the energy gap
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Band Theory of Solids
The allowed ranges of the wave vector k
are called Brillouin zones. zone 1 -p/a lt k lt
p/a zone 2 -2p/a lt - p/a zone 3 p/a lt k lt
2p/a etc.
The theory can explain why some substances are
conductors, some insulators and others
semi conductors
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Conductors, Insulators, Semiconductors
Sodium (Na) has one electron in the 3s state,
so the 3s energy level is half-filled.
Consequently, the
3s band, the valence band, of solid sodium is
also half-filled. Moreover, the 3p band, which
for Na is the conduction band, overlaps with the
3s band. So valence electrons can easily be
raised to higher energy states. Therefore,
sodium is a good conductor
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Conductors, Insulators, Semiconductors
NaCl is an insulator, with a band gap of 2
eV, which is much larger than the thermal energy
at T300K
Therefore, only a tiny fraction of electrons
are in the conduction band
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Conductors, Insulators, Semiconductors
Silicon and germanium have band gaps of 1 eV
and 0.7 eV, respectively. At room temperature, a
small
fraction of the electrons are in the conduction
band. Si and Ge are intrinsic semiconductors
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Summary

• The heat capacity of the electron gas is small
  compared with that of the ions
• Energy gaps arise in solids because they contain
  standing wave states
• The size of the energy gap between the valence
  and conduction bands determines whether a
  substance is a conductor, an insulator or a
  semiconductor