Chap 6 Free Electron :Metal

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Chap 6 Free Electron Metal
6.1 Drude Model of simple metals
Conduction electron move freely inside the metal
(Free electron gas) neglect
week attractive potential between the ion-cores
and the conduction electron
e
Electronic properties depend only on the
number of conduction electron and statistics
Why the electrons move freely? 1) A conduction
electron is not deflected by the ion cores
arranged on a periodic lattice. 2) A conduction
electron scatter only infrequently by other
conduction electrons and crystal
imperfections (Pauli Exclusion Principle)
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6.2 A free electron in 3-Dimension
Schrodinger Equation
Boundary Condition (Born-Von Karman)
periodic boundary condition
L
Assumption Boundary condition cannot change
bulk properties significantly (neglect surface
effect)
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Solution
momentum eigenstate
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6.2 N-electron system Many body problem

• N independent electron approximation
• Pauli Exclustion Principle identical fermions,
  spin 1/2
• (No two identical fermions can occupy the same
  states)

• Ground state T0
• Electrons fill N lowest energy engenstates

kz
kF
ky
kx
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Fermi Energy
EF Highest energy of electrons in the ground
state
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• Density of States

of states in e e d e
ede
e
EF
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Density of state at Fermi Energy
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6.3 Excited State at T
kz
kF
Excited electronic state
ky
kx
Empty state (hole)

Fermi -Dirac Statistics
f(e) Occupation probability of a state with
energy e m Chemical Potential
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m
Chemical potential is the energy where the
occupation becomes 1/2
N (total number of electrons)
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6.3 Heat Capacity of Metal electrons
Classical thermodynamics Cv 3/2
kBT Experimental values less than 1 of
classical value
f(e)
eF
1) Rough argument
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2) Quantitatively
0
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Note
C/T
T
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6.5 Electrical Conductivity and Ohms Law
Motions of electrons in the presence of
electromagnetic fields
E0
E?0
Collision time t
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Electric Conductivity
Main Source of Collision - 1. Impurity atoms
(defects) - 2. Phonon (thermal lattice vibrations
T)
Matthiessens Rule - The collisions are
independent of each other and contribute
independently to r.
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Electron-Phonon Scattering
zone boundary
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At very low temperature, normal scattering is
dominant
Blochs theory
Motion in magnetic field
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B
centrifugal
Magnetic force
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Thermal conductivity of Metals (electronic
thermal transport)
t collision time l mean free path
TDT
T-DT
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Both electric conductivity thermal
conductivity are proportional to the electron
density and the collision time
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6.6 Hall Effects
Generation of electrical field in response to
applied magnetic field
Bz

Ey
Jx
Ex
- - - - - - - - - - - - -
-

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In the steady state, the force due to Ey is
balanced by the magnetic force
For free electron model
i) Give carrier concentration n ii) Sign
negative
Experimentally some positive materials Al, Be, Zn
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6.7 Pauli Paramagnetism
Conduction electron has spin Spin interacts
with magnetic field
Magnetic energy
f(e)
eF
e
e
eF
k
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Curie-law without F.D