Transcript Solid State Physics

Solid State Physics
3
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
R / R0   / 0  r / r
2
2
0
As can be seen from the
graph, the prediction
 T
U
1
M  2 r 2  kT
2
fails at very low
temperatures. This is
due, in part, to the
failure of the
equipartition theorem
at low temperatures.
Challenge: create a
3
better model!
Special Extra Credit
Derive the temperature
R / R0   / 0  r / r
dependence of R/R0 by
computing the average potential energy <E>
of a lattice ion
assuming that the energy
1
level of the nth
2 2
U M
 r  E
1
U
vibrational state is
2 2 M  r  kT
2
2
2
0
2
En  (n  12 )
rather than En = n as
Einstein had assumed
Due:
4
before classes end
Heat Capacity of Electron Gas
By definition, the heat capacity (at constant
volume) of the electron gas is given by
dU
CV 
dT
where U is the total energy of the gas. For a gas
of N electrons, each with average energy <E>,
the total energy is given by
UN E
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Heat Capacity of Electron Gas
Total energy

U  N E   E n( E )dE
0
V
  8m 
 2 
2 h 
3/ 2


0
E 3/ 2 dE
( E  EF ) / kT
e
1
In general, this integral must be done
numerically. However, for T << 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
3 
U  N E  N  EF 
5 
For 0 < T << 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
 kT 
3
U  NEF     NkT
5
 EF 
where  = 2/4, as first shown by Sommerfeld
The heat capacity of the
electron gas is predicted to
be
dU  2
T
CV 

Nk
dT
2
TF
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Heat Capacity of Electron Gas
Consider 1 mole of copper. In this case Nk = R

2
T
CV 
R
2 TF
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
1
2
3
d 2 ( x)

 U ( x) ( x)  E ( x)
2
2m dx
2
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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
ikx and the wavefunction must
 ( x)  uk ( x)e reflect the periodicity of
the lattice:
1
2
3
 ( x  n(a  b))
  ( x)e
ikn ( a b )
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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
ka   n
1
2
3
The energy gaps
are basically energy
levels that cannot
occur in the solid
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Band Theory of Solids
Completely free
electron
2
2
2
p
k
E

2m 2m
electron in a
lattice
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Band Theory of Solids
When, ka   n the wavefunctions become
standing waves. One wave peaks at the lattice
sites, and another peaks between them. Ψ2, has
lower energy
than Ψ1. 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: -/a < k < /a; zone 2: -2/a < - /a;
zone 3: /a < k < 2/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
T=300K
Therefore, only a tiny fraction of electrons are
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in the conduction band
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
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