Phy 2009 Lecture 17
Transcript of Phy 2009 Lecture 17
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Failure of the classical gas model
heat capacity
The free-electron Fermi gas
confinement within crystal
density of states function
Paul exclusion principle
Fermi energy
Fermi distribution function
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Free electron gas model: application to a metal
Fermi energy
Heat capacity
Electrical conductivity
Successes and failures
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Fermi energy estimation
The total number of states from E = 0 to E = EF must be equal tothe total number of electrons in the system:
For a 3D metal, there is typically one free electron for every
atom, or in other words one electron for every (31010)3 m3 of
volume (typical volume of a primitive unit cell)
Ne =V
322mE F
2
3 2
EF =
2
2m32
Ne
V
2 3
Ne
V
1
31010( )
3= 4 10
28m3
EF = 4 eV
Ne =A
2
2mE
2
EF=
2Ne
Am
3D:
2D:
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Fermi energy estimation
Comments:
over 100 times kBT at room temperature
EF = 4 eV
EF = kBTF TF 50,000 K
often useful to talk about the Fermi temperature
Electrons at EF have velocities ~ 106 to 107 ms1
step-like T=0 behaviour of Fermi function a
good approximation
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When is a thin film 2 or 3 dimensional?Lx
Ly
Lz
E=2kx
2+ ky
2+ kz
2( )2m
kx =nx
Lx,ky =
ny
Ly,kz =
nz
Lz
The energy levels associated with different nx
and ny
values are more
closely spaced than those associated with different nz values
Electrons will fill levels with increasing nx
and ny
values while nz
= 1
D(E)
E
nz
= 2
nz
= 3
The nz
= 2 level begins to be
occupied when either the
number of electrons or the
temperature is increased
sufficiently
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Heat capacity
In the classical gas every particle acquires extra thermal
energy as the temperature increased, so every particle
contributes to the heat capacity:
C =3
2NekB
(T/TF~1/100 at room
temperature)C =
3
2NekB
T
TF
In the Fermi gas, only electrons within kBT of EF acquire
extra thermal energy
That is roughly a fraction T/TF of the total number
therefore expect
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Electrical conductivity
In equilibrium we now have a
picture of the electrons in a
solid filling up allowed states in
k space up to an energy EF:
ky
kx
Apply an electric field:
Equation of motion is
or
since
dp
dt= eE
dkdt = eEp = k
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Electrical conductivity
In equilibrium we now have a
picture of the electrons in a
solid filling up allowed states in
k space up to an energy EF:
ky
kx
Apply an electric field:
After a time t, an electron with
wavevector k will acquire an extra
k from the field: k = eEt
ky
kx
E
k
Electric field causes all electrons
to transfer from k-state to k-state,
in unison
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Electrical conductivity
If the electrons were in a vacuum, or an infinite, perfect
crystal at absolute zero the Fermi sphere would continue to
shift in this way forever as the electrons accelerated
indefinitely
ky
kx
E
In a real solid it is assumed that
the electrons scatter, on
average, after a time scattering
k = eE
... so each electron acquires, on
average
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Electrical conductivity
Now electrical conductivity is defined by the equation
j=E
Here, vd is the drift velocity the extra velocity due to the
field, which is related to k by:
vd =p
m
=
k
m
=
eE
m hence j=
nee2E
m
=nee
2
m
j= neqvdand (note: ne = Ne/V)
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Electrical conductivity
This is exactly the same formula as for the classical gas
model
BUT the mean free path will be different:
=nee
2
m
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Electrical conductivity
In the classical model, the mean free path was calculated
from
mfp = vthermal electrons were assumed to move at the
thermal velocity
=nee
2
m
1
2mvthermal
2=
3
2k BT
vthermal =3kBT
m
1 2
105 ms1
Remember this gave a mean free path ~ a lattice constantusing a typical room temperature value of
Note: the drift velocity caused by the electric field is very
small compared with this a few mm s1
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Electrical conductivity
In the free-electron fermi-gas model, the velocity to use is
the Fermi velocity
mfp = vF where
=nee
2
m
Consequently, the value of obtained from measurement of
electrical conductivity gives a mean free path ~ 10 100
lattice constants.
1
2mvF
2= EF
vF =2E F
m
1 2
106 ms1
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Furthermore, if we measure the low-temperature
conductivity in very pure crystals, we get mfp ~ size of
crystal scattering is from edges of crystal!
Electrical conductivity
In the free-electron fermi-gas model, the velocity to use is
the Fermi velocity
mfp = vF where
=nee
2
m
1
2mvF
2= EF
vF =2E F
m
1 2
106 ms1
WHAT HAPPENED TO SCATTERING FROM THE LATTICE?
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Consider the effect of applying a magnetic field B to theelectron gas. The electron has magnetic moment B and
so the electrons gain additional energy BB depending
upon whether their spin lies parallel or anti-parallel to B.
EF
D E( )
D E( )
EF
D E( )
D E( )
B
EF
D E( )
D E( )
B
2BB
D EF( )2
m = 2B BB( )D EF( )
2
= m /B = B
2D EF( )
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Successes and failures
explains the small heat capacity with linear T dependence
as observed
predicts small contribution to paramagnetic susceptibility
("Pauli paramagnetism") in metals
gives same (successful) formula for electrical conductivity
as did the classical theory
BUT unexpectedly long mean free path (turns out to be
correct though!)
no explanation of insulators:
insulators have no free electrons BUT
WHY?
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Free electron gas model: application to a metal
Fermi energy
Heat capacity
Electrical conductivity
Successes and failures