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
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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• 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

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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
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Density of state at Fermi Energy
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6.3 Excited State at T
Fermi -Dirac Statistics
f(e) Occupation probability of a state
with energy e m Chemical Potential
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Chemical potential is the energy where the
occupation becomes 1/2
m
N (total number of electrons)
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6.4 Heat Capacity of Metal electrons
Classical thermodynamics Cv 3/2
kBT Experimental values less than 1 of
classical value
f(e)
eF
1) Rough estimate
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2) Quantitatively
0
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Note
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6.5 Electrical Conductivity and Ohms Law
Motions of electrons in the presence of
electromagnetic fields
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
Single Phonon absorption Single Phonon emission
Phonons energy is much smaller than
electrons energy -gt electrons move in constant
energy surface
Normal
Umklapp
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At high temperature, the number of phonons in any
normal mode is,
At low temperature (below Debye temperature),
only phonons with energy Comparable to or less
than kT can be involved.
1.
2. Electron-phonon coupling decline with q
scattering rate
3.
Effective scattering rate
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At very low temperature, normal scattering is
dominant
Blochs theory
Umklapp Process
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Thermal conductivity of Metals (electronic
thermal transport)
t collision time l mean free path
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Both electric conductivity thermal conductivity
are proportional to the electron density and the
collision time
Independent of materials Wiedemann-Franz law
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6.6 Motion in Magnetic field and Hall Effects
Magnetic force centrifugal
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Hall Effects
Generation of electrical field in response to
applied magnetic field
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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 materials Al, Be, Zn have
positive Hall coefficient
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6.7 Pauli Paramagnetism
Conduction electron has spin Spin interacts
with magnetic field
Magnetic energy
e
eF
k
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Magnetization
Magnetic susceptibility
Curie-law without F.D
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6.8 Landau Diamagnetism
Free electron under external magnetic field Bz
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B
B0
EF
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N-electrons upto s-level completely
filled s1 level partially filled
Total energy of electrons
Oscillatory with in inverse magnetic field
de Hass van Alphen Effect
There are oscillations in M/H
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If s-level is completely filled, and s1 is empty,
S Extremal area of the Fermi sphere in
reciprocal space