metals i: free electron model
DESCRIPTION
Metals I: Free Electron Model. Physics 355. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. Free Electron Model. Schematic model of metallic crystal, such as Na, Li, K, etc. - PowerPoint PPT PresentationTRANSCRIPT
Metals I: Free Electron Model
Physics 355
Free Electron Model
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Schematic model of metallic crystal, such as Na, Li, K, etc.
The equilibrium positions of the atomic cores are positioned on the crystal lattice and surrounded by a sea of conduction electrons.
For Na, the conduction electrons are from the 3s valence electrons of the free atoms. The atomic cores contain 10 electrons in the configuration: 1s22s2p6.
Free Electrons?
How do we know there are free electrons?
You apply an electric field across a metal piece and you can measure a current – a number of electrons passing through a unit area in unit time.
But not all metals have the same current for a given electric potential. Why not?
Paul Drude
(1863-1906)
• resistivity ranges from 108 m (Ag) to 1020 m (polystyrene)
• Drude (circa 1900) was asking why? He was working prior to the development of quantum mechanics, so he began with a classical model:• positive ion cores within an electron
gas that follows Maxwell-Boltzmann statistics
• following the kinetic theory of gases- the electrons in the gas move in straight lines and make collisions only with the ion cores – no electron-electron interactions.
Paul Drude
(1863-1906)
• He envisioned instantaneous collisions in which electrons lose any energy gained from the electric field.
• The mean free path was approximately the inter-ionic core spacing.
• Model successfully determined the form of Ohm’s law in terms of free electrons and a relation between electrical and thermal conduction, but failed to explain electron heat capacity and the magnetic susceptibility of conduction electrons.
Ohm’s Law
Experimental observation: IV
LjA
IL
IA
L
RIV
jE
jL
V
Ej or
E
Ohm’s Law: Free Electron Model
dnevj
nee volume
number
Conventional current
The electric field accelerates each electron for an average time before it collides with an ion core.
Ohm’s Law: Free Electron Model
m
eEv
vm
maeEF
d
d
Em
nenevj d
2
m
ne 2
Ohm’s Law: Free Electron Model
If electrons behave like a gas…
m
Tkv B
8
The mean free time is related to this average speed…
v
a
v
Then,
Tk
m
m
ane
B8
1 2
typical valueAbout 1014 s
Ohm’s Law: Free Electron Model
0 20 40 60 80 100 120
0
2
4
6
8
10
12
Res
istiv
ity
Temperature
B
Predictedbehavior
The mean free path is actually many times the lattice spacing – due to the wave properties of electrons.
Low T: Resistivity limited by lattice defects.
High T: Resistivity limited by lattice thermal motion.
Wiedemann-Franz Law (1853)
Conductivities
Electrical Thermal
vm
ne 2
Bkvn 21
wherem
Tkv B
8
2
24
e
TkB
Lorentz number (Incorrect!!)
Wiedemann-Franz Law (1853)
(Ludwig) Lorenz Number(derived via quantum mechanical treatment)
28
2
2B
2
K
W 1045.2
3
1
e
k
TL
• Assume N electrons (1 for each ion) in a cubic solid with sides of length L – particle in a box problem.
• These electrons are free to move about without any influence of the ion cores, except when a collision occurs.
• These electrons do not interact with one another.
• What would the possible energies of these electrons be?
• We’ll do the one-dimensional case first.
Free Electron Model: QM Treatment
0 L
Free Electron Model: QM Treatment
At x = 0 and at L, the wavefunction must be zero, since the electron is confined to the box.
One solution is:
Free Electron Model: QM Treatment
Free Electron Model: QM Treatment
Free Electron Model: QM Treatment
If an electron is added, it goes into the next available energy level, which is at the Fermi
energy. It has little temperature dependence.
1
11
1)(
/)(
/)(
TBkF
TBk
e
ef
Fermi-Dirac Distribution
For lower energies,f goes to 1.
For higher energies,f goes to 0.
Free Electron Model: QM Treatment
From thermodynamics,the chemical potential, and thus the Fermi Energy, is related to the Helmholz Free Energy:
whereVT
NFNF,
)()1(
TSUF
Free Electron Model: QM Treatment
where nx, ny, and nz are integers
Free Electron Model: QM Treatment
and similarly for y and z, as well
... ,4
,2
,0
... ,4
,2
,0
... ,4
,2
,0
LLk
LLk
LLk
z
y
x
rkik e
Free Electron Model: QM Treatment
m
k
m
pv
m
k
2
2F
2
F
Free Electron Model: QM Treatment
• Each value of k exists within a volume3
2
LV
• The number of states inside the sphere of radius kF is
3/12
F32
3F3
4 3
2
V
Nk
k
V
VN
L
s
3/22
22F
2
F 322
V
N
mm
k
• This successfully relates the Fermi energy to the electron density.
Free Electron Model: QM Treatment
B
FF k
T
1/ 32
F F3 N
v km m V
Free Electron Model: QM Treatment
3/22
22F
2
F 322
V
N
mm
k
2/3
F22
2
3
mVN
2/12/3
22
2
2 )(
mV
d
dNg
N
d
dNg
N
mVN
2
3
then
constantlnln
2
3
23
2/3
F22
Free Electron Model: QM Treatment
The number of orbitals per unit energy range at the Fermi energy is approximately the total number of conduction electrons divided by the Fermi energy.
Free Electron Model: QM Treatment
As the temperature increases above T = 0 K, electrons from region 1 are excited into region 2.
This represents how many energies are occupied as a function of energy in the 3Dk-sphere.