Post on 04-Apr-2018
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1
Free Electron Model for Metals
Metals are very good at conducting both heat and electricity. Metals were described as behaving like a set of nuclei forming a
lattice with a sea of electrons shared between all nuclei/lattice(moving freely between them):
This is referred to as the free electron model for metals.
This model explains many of the properties of metals: Electrical Conductivity: The mobile electrons carry current.
Thermal Conductivity: The mobile electrons can also carry heat.
Malleability and Ductility: Deforming the metal still leaves eachcation surrounded by a sea of electrons, so little energy isrequired to either stretch or flatten the metal.
Opacity and Reflectance (Shininess): The electrons will have awide range of energies, so can absorb and re-emit many differentwavelengths of light.
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The Free Electron Gas Model
Plot U(x) for a 1-D
crystal lattice:
Simple and
crude finite-
square-well
model:
Can we justify this model? How can one replace the entire lattice by a
constant (zero) potential?
U
U = 0
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Free-Electron Model
m
k
m
p
22
222
classical description
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Em
02
22
zkykxkL
xyz zyx sinsinsin8
3
In a 3D slab of metal, es are free to move
but must remain on the inside
Solutions are of the form:
L
nz
2222
2
8
zyx nnn
mL
h
With energy:
Quantum Mechanical Viewpoint
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At T = 0, all states are filled
up to the Fermi energy
max
222
2
2
8zyxFo
nnnmL
h
A useful way to keep track of the states that are filled is:
max2222
nnnn zyx
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Electron follow Fermi-Dirac Distribution
1
1)(
/
kTfEEeEf
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total number of states up to an energy fermi:
3
3max
4
8
1
8
122
n
sphereof
volume
N
3/2
219
3/22
.10646.33
8
V
NmeVx
V
N
m
hFo
max2
2
2
8n
mL
hFo
# states/volume ~ # free es / volume
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Example: Numerical Values for Copper slab
V
N
= 8.96 gm/cm3
1/63.6 amu 6e23 = 8.5e22 #/cm3
= 8.5e28 #/m3
nmax = 4.3 e 7
so we can easily pretend that theres a smooth distrib of nxnynz-states
eV
V
NmeVx
V
N
m
hFo
0.7.10646.33
8
3/2
219
3/22
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Density of StatesHow many combinations of are there
within an energy interval to + d ?3/2
23
8
V
N
m
hFo
2/3
2
8
3
h
mEVN
dEhm
hmEVdN
2
2/1
288
23
3
2/12/3
2/12/13
3
2
32
8)( E
E
NEm
h
V
dE
dNEg
Fo
dgfKETot
0
Huge number of states, it can be treated in
continuous energy
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At T 0 the electrons will be spread out among the allowed states
How many electrons are contained in a particular energy range?
occuringenergythis
ofyprobabilit
energyparticulara
havetowaysofnumber
1
12
8)(
/)(
2/12/13
3
kTfEeEm
h
VEn
)()()( EfEgEn
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this assumes there are no other issues
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Electrical transport (relaxation time) in conductor
eEv
mdt
dvm
dd j nev
ne
md
2E
Electron Conductivity
F
Fr
v
l
m
en
v
l
v
l
m
e
n
EjLawOhm
*
2
*
2
V
Nn
tyconductivi
velocityFermiv
densityelectronn
pathfreemeanelectronl
timecollisiontimerelaxation
F
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Simple Kinetic Theory of Heat Conduction
tyconductivithermalCv
3
qx
x
Hot Cold
xvx
vxxvxxx xx
nEv2
1nEv
2
1q
2
( )-
( )-
xx x
x
d nEvq v
dxd nE
vdx
2- xdu dT
vdT dx
/C du dT
,
2 2 / 3xv v
2
-3
x
v dTq C
dx
2
3
Cvk
Taylor Expansion:
local thermodynamics equilibrium: u=u(T)
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Assigments
Proof that at T=0 the total energy of
electron
Calculate specific heat capacity using FEM
What fraction of free electron in Cu have a
kinetic energy between 3.95 eV and 4.05 eV
at room temperature
FoNEU 5
3
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Heat Capacity of the Quantum-Mechanical FEM
1
1)(
/
kTEe
Ef
where = chemical potential EF for kT
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Heat Capacity of Metals: Theory vs. Expt. at low T
Very low temperature
measurements reveal:
Meta
lexpt FEG
expt/FEG =
m*/m
Li 1.63 0.749 2.18
Na 1.38 1.094 1.26
K 2.08 1.668 1.25
Cu 0.695 0.505 1.38
Ag 0.646 0.645 1.00
Au 0.729 0.642 1.14
Al 1.35 0.912 1.48
Results for simple
metals (in units
mJ/mol K) show
that the FEG values
are in reasonable
agreement with
experiment, but are
always too high:
The discrepancy is
accounted for by
defining an effectiveelectron mass m* that is
due to the neglected
electron-ion interactions
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Problems with Free Electron Model
* * * * * * * * * * * * * * * * * * * * * * * * * * * *
1) Bragg reflection
2) .
3) .
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Other Problems with the Free Electron Model
graphite is conductor, diamond is insulator
variation in colors of x-A elements
temperature dependance of resistivity
resistivity can depend on orientation of crystal & current I direction frequency dependance of conductivity
variations in Hall effect parameters
resistance of wires effected by applied B-fields
.
.
.
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Nearly-Free Electron Model
version 1SP221
2/2
2/k
a
2/k
2/2
2/
k
a
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Nearly-Free Electron Model
version 2SP324
Bloch Theorem
Special Phase Conditions, k = +/- m /a
the Special Phase Condition k = +/- /a
This treatment assumes that when
a reflection occurs, it is 100%.
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(x) ~ u e i(kx-wt)
(x) ~ u(x) e i(kx-wt)
~~~~~~~~~~amplitude
In reality, lower energy waves are sensitive to the lattice:
Amplitude varies with location
u(x) = u(x+a) = u(x+2a) = .
BlochsTheorem
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u(x+a) = u(x)
(x+a) e -i(kx+ka-wt) (x) e -i(kx-wt)
(x) ~ u(x) e i(kx-wt)
(x+a) e ika (x)
Something special happens with the phase when
e ika = 1
ka = +/ m m = 0 not a surprise
m = 1, 2, 3,
...,2,aa
k
What it is ?
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ak
Consider a set of waves with +/ k-pairs, e.g.
k = + /a moves
k = /a moves
This defines a pair of waves moving right & left
Two trivial ways to superpose these waves are:
+ ~ e ikx + e ikx ~ e ikx e ikx
+ ~ 2 cos kx ~ 2i sin kx
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+ ~ 2 cos kx ~ 2i sin kx
Kittel
|+|2 ~ 4 cos2 kx ||2 ~ 4 sin2 kx
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Free-electron Nearly Free-electron
Kittel
Discontinuities occur because the lattice is impacting the movement of electrons.
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Effective Mass m*
A method to force the free electron
model to work in the situations where
there are complications
ER Ch 13 p461 starting w/ eqn (13-19b)
*2
22
m
k
free electron KE functional form
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Effective Mass m*-- describing the balance between applied ext-E and lattice site reflections
2
2
2
1
*
1
km
m* a = S Fext
q Eext
2)
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No distinction between m & m*,
m = m*, free electron, lattice structure does
not apply additional restrictions on motion.
m = m*
greater curvature, 1/m* > 1/m > 0, m* < m
net effect of ext-E and lattice interaction
provides additional acceleration of electrons
greater |curvature| but negative,net effect of ext-E and lattice interaction
de-accelerates electrons
At inflection pt
1)
2)
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*
2222
22 m
k
m
k latticefromonperturbatiapply
Another way to look at the discontinuities
Shift up implies effective mass has decreased, m* < m,
allowing electrons to increase their speed and join
faster electrons in the band.
The enhanced e-lattice interaction speeds up the electron.
Shift down implies effective mass has increased, m* > m,
prohibiting electrons from increasing their speed and making
them become similar to other electrons in the band.
The enhanced e-lattice interaction slows down the electron
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From earlier: Even when above barrier,
reflection and transmission coefficients can
increase and decrease depending upon the energy.
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change in motion
due to reflections
is more significantthan change in motion
due to applied field
change in motion
due to applied fieldenhanced by change in reflection coefficients
Nearly Free Electron Model
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Nearly-Free Electron Model
version 3 la Ashcroft & Mermin, Solid State Physics
This treatment recognizes
that the reflections of electronwaves off lattice sites can
be more complicated.
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A reminder:
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Waves from the left behave like:
iKxiKx
leftthe
from ere
iKx
left
thefrom et
m
K
2
22
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Waves from the right behave like:
iKxiKx
rightthefrom
ere iKx
right
thefrom et
m
K
2
22
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rightleftsum BA
Blochs Theorem defines periodicity of the wavefunctions:
xeax sumika
sum
xeax sumika
sum
unknown weights
Related to
Lattice spacing
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xeax sumika
sum
xeax sumika
sum
Applying the matching conditions at x a/2
A + B
leftright
A + B
left right
A + B
left right
A + B
left right
iKaiKae
te
t
rtka
2
1
2cos
22
m
K
2
22
And eliminating the unknown constants A & B leaves:
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For convenience (or tradition) set:
221 rt
iett ierir
kat
Ka
cos
cos
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ka
t
Kacos
cos
Related to
possible
Lattice spacings
Related to
Energy
m
K
2
22
allowed solution regions
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allowedsolu
tionregions