Chap 7 Energy Band

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Chap 7 Energy Band
7.1 Bloch Theorem
Failure of free electron model

• 1. Conductor / Insulator / Semiconductor
• 2. Hall Effect RH -1/nec magnitude, sign
• Semiconductor Crystal with small band gap
• Insulator Crystal with large band gap
• Conductor Crystal with overlapping band gap

E
k
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Blochs Theorem
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Proof Any function obeying Born-von Karman
B.C. can be expanded as
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Equivalent to Schrodinger equation written in
momentum space.
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• Significance of k-label

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7.2 Explicit write out of the central equation
Schrodinger Equation Central Equation
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Note 1) To have a non-trivial solution, the
determinent should vanish. Give
eigenvalue Ek.n n k the wave vector that
labels Ck n the index for the order of energy
(band index) 2) If we had started with kg
instead of k, we would have obtained the same
set of eigenvalues. Hence we can restrict k
values in the first Brillouin Zone
Reduced Zone Scheme
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Quantum perturbation theory
Most important
-G/2
G/2
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• Empty Lattice (V0), Free electron

Example ) Low-lying free electron bands of a
simple cubic lattice along the 100 direction
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7.3 Band Calculations
Approximate solution at a zone boundary under a
weak periodic potential

• At the zone boundary, a bandgap is created
• Band gap Forbidden energy range
• electrons in a crystal cannot have energy value
  in this range

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Eigenstate (Blochs wave function)
Free electron Traveling wave
Crystal electron Standing wave
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Physical origin of the band gap
Electron density
V is generally negative Bragg Diffraction
standing wave eigenstate different spatical
charge distribution gt difference in potential
energy gt band gap
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• Solution of Blochs equation near a zone boundary

When k is near the zone boundary, then the
central equation becomes,
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Hence the band energy varies as quadratic
in k-tilda as we move away from the zone boundary
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7.4 Equation of Motion
1). Electron velocity ltvgt
Velocity of a wave packet Group velocity
lt Quantum mechanics gt
Blochs theorem
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2). Equation of Motion
Response of electrons under external force

• Time evolution of a quantity in quantum mechanics

• Fext External Force
• Hamiltonian HH0-Fx
• H, T H -Fx, T -Fx, T -FxTTFx
• TFxF(xa)TFxTFaT
• H,TFaT

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• Hand waving argument

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• Physical Force

• In DC electric field,

Fext -eE
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E, v
velocity
p/a
Energy

• In Magnetic field,

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7.5 Effective Mass

• Mass ratio of force to acceleration

F total force m electron mass a electron
acceleration
F ma

• Effective mass ratio of external force to
  acceleration

1/m determines curvature of E(k) . Free
electron
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E, v
Negative mass
Positive mass
p/a
Force
acceleration
Light electron
Heavy electron
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Nearly Free electron band
E
In a typical semiconductor, l 5eV, V
0.11eV m/m 0.10.01
p/a