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Bloch Oscillations
Alan Wu
April 14, 2009
Physics 138
Outline
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Phenomenon Description
Semi-Classical Derivation
Wannier-Stark States
Implications and Applications
– Terahertz Oscillations
– Bloch Oscillation Transistors
Bloch Oscillation Phenomenon
• Described by Bloch (1928)
• Imagine a particle in a periodic potential
acted on by a constant force.
• Example: electrons in crystal lattice
exposed to constant electric field
• Classically, we expect Ohmic behavior
Bloch Oscillation Phenomenon
• But quantum mechanics predicts that the
particle will undergo an oscillation
• The periodicity causes the group velocity
of the wavefunction to oscillate
• Ohmic behavior results from scattering
Bloch Oscillation Frequency
• Use invariance: shift 1 period d and shift
energy ΔE
• Phase shift now
• Corresponding frequency is
Semi-Classical Derivation
• Schrodinger’s Equation can be
transformed into the form:
• Known as the Acceleration Theorem, since
it describes change in momentum
• Like classical relation between momentum
and force
K-Space in a Lattice
• Potential periodicity in real space =>
periodicity in k-space
• Also known as a reciprocal lattice
The Brillouin Zone
• Brillouin Zone: a basic cell in the reciprocal
lattice
• The dispersion relation gives an oscillating
k within this zone
Kronig Penney Model
• The Kronig Penney model for a lattice can
be used to find the potential in k-space.
Source: http://fermi.la.asu.edu/schmidt/applets/kp/plugkp.html
Dispersion in lattice
Wannier-Stark Resonance States
• At each well, a series of energies are
available, much like that of a harmonic
resonator.
• These states form what is known as a
Wannier-Stark energy ladder.
Tight-binding model
• Consider just interactions between
neighboring wells (known as Wannier
representation)
• Also have energy difference from constant
force
Experimental Confirmations
• Bloch oscillations have been observed in
semiconductor lattices
• Shining a laser will excite the Wannier
Stark states, which then oscillate.
• These oscillations can be measured
Terahertz Radiation
• Changing the electric field allows for a
tunable radiation source.
• Can get frequencies in the terahertz
Bloch Oscillation Transistors
• Bloch oscillations can control Josephson
Junctions
• Act much like bipolar transistors
Conclusion
• Bloch oscillations are just another strange
quantum phenomenon
• They can be used for frequencies in the
terahertz range
• Bloch oscillator transistors are an
interesting way of amplifying signals