Transcript Solid State Physics

Solid State Physics
2
Section 10-3,4
1
Topics

Recap

Free Electron Gas in Metals

Quantum Theory of Conduction

Heat Capacity of Electron Gas

Summary
2
Recap
The number of particles (here electrons) with
energy in the small interval E to E+dE is given by
n( E )dE  f ( E ) g ( E)dE
where,
V 4 p 2 dp
g ( E )dE  W
h3
E+dE
E
is the number of states
in that interval and f (E) is the probability that
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these states are occupied
Recap
The electron has two spin states, so W = 2.
Assuming the electron’s speed << c, we can
take its energy to be
p2
E
2m
in which case the number of states in the
interval E to E+dE is given by
  8m 
3/ 2
g ( E )dE  V
 2
2 h 
1/ 2
E dE
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Free Electron Gas in Metals
The number of electrons in the interval
E to E+dE is therefore
n( E)dE 
1
( E  EF ) / kT
e
  8m 
3/ 2
1
V
 2
2 h 
E1/ 2 dE
The first term is the Fermi-Dirac distribution
and the second is the number of states g(E)dE
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Free Electron Gas in Metals
From n(E) dE we can calculate many global
characteristics of the electron gas. Here are a
few
 The Fermi energy – the maximum energy
level occupied by the free electrons at
absolute zero


The average energy
The equation of state; that is, the
equivalent of PV = nRT for the electron gas
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Free Electron Gas in Metals
The total number of electrons N is given by

N   n( E )dE  V
0
  8m 
3/ 2
 2
2 h 


0
1/ 2
E dE
( E  EF ) / kT
e
1
The average energy of a free electron is given by
1
E 
N


0
V   8m 
En( E )dE 
 2
N 2 h 
3/ 2


0
3/ 2
E dE
( E  EF ) / kT
e
1
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Free Electron Gas in Metals
At T = 0, the integrals are easy to do.
For example, the total number of electrons is
N V
V
  8m 
3/ 2
 2 
2 h 
  8m 
 2 
2 h 

EF
0
3/ 2
E1/ 2 dE
2 3/ 2
EF
3
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Free Electron Gas in Metals
The average energy of an electron is
V   8m 
E 
 2 
N 2 h 
3/ 2
V   8m 

 2 
N 2 h 
This implies

EF
0
3/ 2
E 3/ 2 dE
2 5/ 2
EF
5
EF = k TF defines
3
E  EF the Fermi
5
temperature
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Free Electron Gas in Metals
  8m 
3/ 2
n( E)  V
 2
2 h 
E1/ 2
e( E  EF ) / kT  1
At T > 0, only
the electrons
near the
Fermi energy
can be excited
to higher
energy states
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Free Electron Gas in Metals
  8m 
3/ 2
n( E)  V
 2
2 h 
E1/ 2
e( E  EF ) / kT  1
Their energy
increases by
about kT,
which at
T = 300 K is
only about
1/40 eV, to
be compared
with
EF ~ 8 eV
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Quantum Theory of Conduction
In a very large (strictly infinite)
perfect crystal, calculations
show that electrons suffer
no scattering.
That is, their
mean free path is infinite.
In real crystals, electrons scatter
off imperfections and the thermal
vibrations of the lattice ions
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Quantum Theory of Conduction
Mean Free Path – This is the average distance
traveled between collisions. Consider a box of
length L and cross-sectional area A that contains
n particles per unit volume. Suppose each particle
presents a cross sectional area a.
What is the probability of a collision
between a single incident particle, of
L
negligible size, and a particle
within the box?
A
a (nAL)
Pr 
 naL
A
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Quantum Theory of Conduction
Mean Free Path – A collision is guaranteed to
occur when Pr = 1. This occurs for a particular
value of L = l, called the mean free path,
where
1
l
 v 
na
L
A
and <v>, the average speed of
the incident particle, defines
the relaxation time 
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Quantum Theory of Conduction
Resistance – For a wire of length L and cross
sectional area A, the electrical resistance can
be written as
L
R
A
where , the resistivity, is
inversely proportional to the
mean free path  = C / l.
L
A
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Quantum Theory of Conduction
Classically, lattice ions are modeled
as spheres of cross-sectional area
r2. In the quantum theory, we
model ions as points vibrating in
three dimensions and that present
a cross section of
a r
2
where r is the oscillation amplitude
of the ions
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Quantum Theory of Conduction
For a simple harmonic oscillator,
the potential energy is
1 2 1
2 2
U  Kr  M  r
2
2
Its average value is kT, assuming
that the equipartition theorem
holds, at least approximately,
that is,
1
2 2
U  M  r  kT
2
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Quantum Theory of Conduction
The mean free path is therefore
M 1
l

2
n r 2 nk T
2
1
So, in this simple model, quantum
theory predicts that the
resistivity is proportional to the
temperature
 T
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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
20
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:
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before classes end
Heat Capacity of Electron Gas
At T= 0, the total energy of the electron gas is
3
U  NEF
5
At T > 0, the number of electrons that can be
excited to higher energy states is roughly
and
each
is
kT
N raised in energy
EF
by roughly kT
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Heat Capacity of Electron Gas
Therefore, the total energy can be written as
 kT 
3
U  NEF    N  kT
5
 EF 
where  = 2/4, as first shown by Sommerfeld
The heat capacity of the
electron gas is predicted to
be
2
dU 
T
CV 

R
dT
2 TF
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Summary



The energy distribution of an electron gas
does not vary much until the temperature is
near and above the Fermi temperature
The free electron gas model predicts a
resistivity that is proportional to the
temperature
The heat capacity of the electron gas is small
compared with that of the ions
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