Bilayer heterostructures

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Bi-layer heterostructures
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Carrier density dependence on gate voltage
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Carrier density dependence on gate voltage
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2DEG Fermi surface
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Effect of a parallel B field on k states
Landau Gauge
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Magneto-tunnelling spectroscopy

• Conservation of energy and momentum
• Tunnelling only at the Fermi surface

d2I (f(e) f(eeV)).d(k- k)
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Magneto-tunnelling spectroscopy J. P. Eisenstein
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Magneto-tunnelling spectroscopy
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Weak tunnelling
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Weak tunnelling
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Phys.Rev.Lett 91 226804-1 (2003).
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Quantum Electronics in Semiconductors
Dr C. H. W. Barnes
Lecture2 Classical Electron transport
Mott Seminar Room Monday 12.00pm Friday
9.00am E-mail chwb101_at_cam.ac.uk
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Semi-classical transport theory
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Equilibrium
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The Lorentz force
F -e ( E v B )
F m dv/dt
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Velocity in k-space
- Average No. of electrons in volume d2k around k
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Acceleration from Lorentz force
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Scattering with impurity centers
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The Non-Linear Boltzmann Equation
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The Linear Boltzmann Equation
Non-Linear Boltzmann Equation
Zero-term
Non-linear term
Linear Boltzmann Equation
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Solution in Zero B- field
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Solution in Zero B- field
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Mobility
Mobilities In a GaAs-AlGaAs heterostructure m
50m2/Vs -gt 1000m2/Vs t 19 ps -gt 380ps vd
0.6 ms-1 -gt 12.5 ms-1 for 1mV across an 80mm
sample
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Boltzmann Conductivity
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Drude Conductivity
Constant relaxation approximation
x
x
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Drude Conductivity
Zero temperature
Change of variables
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Drude Conductivity
Mobilities In a GaAs-AlGaAs heterostructure m
50m2/Vs -gt 1000m2/Vs t 19 ps -gt 380ps vd
0.6 ms-1 -gt 12.5 ms-1 for 1mV across an 80mm
sample
Mean free path In a GaAs-AlGaAs
heterostructure EF 10meV lF vF t 5.3mm -gt
106 mm
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Impurity scattering
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Impurity scattering simplifying assumptions
- Probability per unit time k -gt small volume
d2k around k
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Impurity scattering
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Impurity scattering
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Impurity scattering
Small angle scattering absent from mobility
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Particle-like motion of electrons
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Two-dimensional electron systems in GaAs-AlGaAs
heterostructures
AlGaAs
GaAs-AlGaAs heterostructure
GaAs
CB edge
VB edge
Conduction band edge after equilibration
CB edge
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Ohmic Contacts
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Electrostatic surface-gates
Vg 0V
Vg -0.4V
Vg
V(x,y) - 2D subband energy
2D Schroedinger equation in the effective mass
approximation
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The electrostatic aperture Quantum Point Contact
x
y
V(x,y)
SEM image of a typical surface-gate pattern for a
1D system
x
y
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Cyclotron Orbits
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Cyclotron Orbits (Molenkamp)
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Classical reflection from a smooth boundary
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Collimation I
Narrow
Wide
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Collimation II
High Potential
Low Potential
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Cyclotron Orbits and Obstacles (Ford)
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Skipping Orbits (Specter)
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Imaging Collimation (Westervelt)
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Imaging Cyclotron Orbits (Crook)
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Chaotic Motion (Marcus)
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Chaotic Motion (Marcus)
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Imaging Chaotic Orbits (Crook)
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(No Transcript)
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Refraction in Optics

n gt 1
q
d
Refractive index gradient
d
q
n 1
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Classical Refraction
Vg 0

Graded Potential
Vg 0
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Classical Refraction (Specter)
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Classical Lens (Specter)