Quantum Electronics in Semiconductors

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Quantum Electronics in Semiconductors
Dr C. H. W. Barnes
Mott Seminar Room Monday 12.00pm Friday
9.00am E-mail chwb101_at_cam.ac.uk
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Phys.Rev.Lett 91 226804-1 (2003).
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GaAs Crystal Structure
Face-centred cubic lattice.
GaAs -- Zincblend lattice structure a
5.6Angstrom
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Si qbit
E. G. Emiroglu, D. G. Hasko and D. A. Williams.
Phys. Rev. Lett
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Si Crystal Structure
Face-centred cubic lattice.
Si -- diamond lattice structure a 5.4Angstrom
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Hamiltonian - Blochs Theorem
The crystals symmetry causes its Hamiltonian to
be invariant under real-space transformations
H(r )H( r R(i,j,k))
This in turn results in the invariance of the
wavefunctions and eigenenergies under Reciprocal
lattice transformations k -gt kG
Fn,k(r) Fn,kG (r)
En,k En,k G(k,l,m)
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Reciprocal lattice of a face-centred cubic lattice
Brillouin boundaries for a face-centred cubic
lattice.
Body-centred cubic lattice.
Fn,k(r) Fn,kG (r)
En,k En,k G(k,l,m)
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Band structure of Si and GaAs
Fn,k(r) Fn,kG (r)
En,k En,k G(k,l,m)
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Fermi surfaces of conduction electrons in Si and
GaAs
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Free-electron-like behaviour
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Dopants in Si and GaAs
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Band engineering
1. Modification of the chemical potential and
carrier density of a semiconductor. 2. Band
bending with unbalanced charge.
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Modification of carrier density and chemical
potential
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Modification of carrier density and chemical
potential
Charge neutrality
Occupied impurity sites
Occupied conduction and valance-band states
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Modification of carrier density and chemical
potential
p - type
n - type
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Band bending with unbalanced charge
Poissons eqn.
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Band bending with unbalanced charge
EC -ef
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Example the n p junction
The n-p junction
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Example the n p junction
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The general case
Poissons equation
Semiconductor properties
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The Si metal-oxide-semiconductor junction
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The Si metal-oxide-semiconductor junction
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The Si metal-oxide-semiconductor junction
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The Si metal-oxide-semiconductor junction
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The Si metal-oxide-semiconductor junction
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GaAs AlGaAs heterostructure
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Modulation doping
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Equilibration
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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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The two-dimensional electron system
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The capacitor model
d
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The capacitor model
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The capacitor model
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The capacitor model
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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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Magneto-tunnelling spectroscopy
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Magneto-tunnelling spectroscopy
Landau Gauge
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Magneto-tunnelling spectroscopy
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Magneto-tunnelling spectroscopy
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Magneto-tunnelling spectroscopy
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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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Syllabus
The free-electron gas Si and GaAs crystal
properties and band structure. Effective-mass
theory. Doping. Band engineering The
creation of three-dimensional, two-dimensional
and multi-layer electron gases in semiconductor
materials. Semi-classical electron transport
A semi-classical approach to understanding the
effect of Fermi statistics and impurity
scattering on the transport properties of a
two-dimensional electron gas.
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Particle-like motion of electrons The
observation of particle-like motion of electrons
in clean two-dimensional electron systems. The
Ohmic contact, the point source, cyclotron
motion, collimation, skipping orbits, chaotic
motion, classical refraction, the ballistic
lens. The quantum Hall and Shubnikov de Haas
effects The predictions of semi-classical
transport theory for finite magnetic field and
their breakdown at high field. Eigenstates in
a magnetic field, disorder effects, screening
effects, and the quantum Hall and Shubnikov de
Haas effects.
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Quantum transport in one dimension the
one-dimensional electron gas.
Magnetotunnelling as a means to observe the
eigenstates of these systems. The effective
potential, density of states, quantum capacitance
and conductance of one-dimensional systems.
The Landauer quantum transport
formalism. General quantum transport theory
one-dimensional electron systems in finite
perpendicular magnetic field. The multi-probe
Landauer-Büttiker formalism, edge states and
disorder, the quantum Hall effect and edge-state
networks.
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Quasi-particles Landaus theory of
quasi-particles. Experimental proof of the
existence of quasi-particles in strongly
interacting systems. The formation of novel
quasi-particles at high magnetic field, the
composite-fermion picture of the fractional
quantum Hall effect, the composite-fermion
edge-state model. The Luttinger
liquid. Quantum dots and artificial atoms
Zero-dimensional electron systems. Eigenenergie
s and eigenstates. Single-particle conductance,
classical Coulomb blockade, quantum Coulomb
blockade, quantum dots as artificial atoms, the
Kondo effect, the Aharonov-Bohm effect.
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Semiconductor Quantum Computation Electrons and
quasi-electrons in quantum dots as qubits, The
DiVincenzo criteria. State preparation,
manipulation, entanglement and measurement.
Experiments that show Rabi oscillations and
single-spin detection in quantum dots.
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The Free-Electron Gas
Si and GaAs crystal properties. Effective-mass
theory. Doping. Band engineering The Si
metal-oxide-semicondutor junction. The
GaAs-AlGaAs heterostructure. Bi-layer
heterostructures.
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Blochs Theorem
The crystal symmetry in a lattice
r r R(i,j,k)
causes Schrodingers equation to have solutions
of the form
F(r) U(r) .exp(i k.r)
where U is a periodic function in r and k is a
wave vector with the symmetry of the reciprocal
lattice
k k G(k,l,m)
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GaAs