Theory of semiconductor nanostructures

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Theory of semiconductor nanostructures Funded by
BES LAB 03-17
Alex Zunger Alberto Franceschetti Gabriel
Bester Joonhee An
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• Theory of is generally available in two size
  regimes
• Tiny dots (lt100 atoms)
• First-principles LDAGW calculations are
  feasible
• 2. Huge dots (gt107 atoms)
• Empirical effective mass models are OK
• Theory is not available for 103 -106atom quantum
  dots
• (most interesting size range)

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What we offerAtomistic theory for 103-106 atom
nanostructures
Step 1 Relax atomic positions
Step 2 Solve single-particle problem
Step 2 Solve many-particle problem
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Summary of present methodAtomistic theory for
103-106 atom nanostructures
Step 1 Relax atomic positions
Valence-force field (VFF)
Step 2 Solve single-particle problem
Step 2 Solve many-particle problem
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Step 1 Relax atomic positions
Example InAs/GaAs dot molecule
Hydrostatic Strain Field
Valence Force Field model (VFF)
The strains on the two dots are different, even
when the dots are identical
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Summary of present methodAtomistic theory for
103-106 atom nanostructures
Step 1 Relax atomic positions
Semi-empirical pseudopotential method (EPM)
Step 2 Solve single-particle problem
Step 2 Solve many-particle problem
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Step 2 Solve the single-particle Schroedinger
equation

• Input
• Atomic positions Ri
• Screened atomic potentials vi (fitted to bulk
  electronic structure)
• Output
• Single-particle energies ei and wave functions
  yi

• Pseudopotential method includes
• Inter-band coupling
• Inter-valley coupling
• Spin-orbit coupling
• Anisotropy of effective masses
• Realistic potential barrier

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Folded Spectrum Method
?dot?(r) ?i ci ji (r)
Hi, j lt ?i H ?j gt
ji (r)
plane waves
Ai, j lt ?i (H Eref )2 ?j gt
bulk wave functions
(E?- Eref )2
E
Conduction band
Valence band
Eref
The lowest eigenvalue of A is the eigenvalue of H
closest in energy to Eref
Up to 1,000,000 atoms !
Memory N CPU time N
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Summary of present methodAtomistic theory for
103-106 atom nanostructures
Step 1 Relax atomic positions
Step 2 Solve single-particle problem
Configuration interaction method (CI)
Step 2 Solve many-particle problem
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Step 3 Solve the many-body problem
Example Biexciton
e-e interaction
e-h interaction
h-h interaction
Configuration interaction
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Configuration-interaction Hamiltonian
Coulomb and exchange integrals
W screened Coulomb potential
is diagonalized in a basis set of electronic
configurations
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Modern theory of nanostructures Current
capabilities
Materials Si, Ge, GaAs, InAs, InP, GaP, CdSe,
PbSe Dimensionality Films (2D), Wires (1D),
Dots (0D) Physical properties/effects Optical
absorption Emission Excitons,
bi-excitons, multi-excitons Auger
cooling rates Radiative recombination
rates Impact ionization rates
Charging energies Excitonic fine
structure
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ExampleCarrier multiplication in PbSe
nanocrystals
Ellingson et al., Nano Letters (2005)
Schaller and Klimov, PRL (2004), APL (2005)
Carrier multiplication One photon in ? several
electron-hole pairs out
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Mechanism of carrier multiplication in quantum
dots
(a)
II
II
AR
AR
(c)
(b)
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II and AR transition rates
W screened Coulomb interaction
Open questions1. Is II faster than AR? 2. What
is the energy threshold for II?
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1. II vs AR transition rates
PbSe dot R15.3 Å
Transition rate (arb. units)
II becomes faster than AR

• For

II is 500 times faster than AR

• For

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2. II threshold
Impact ionization threshold
A. Franceschetti, J.M. An, A. Zunger, Nano
Letters (in press)
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Modern theory of nanostructures Current
capabilities
Materials Si, Ge, GaAs, InAs, InP, GaP, CdSe,
PbSe Dimensionality Films (2D), Wires (1D),
Dots (0D) Physical properties/effects Optical
absorption Emission Excitons,
bi-excitons, multi-excitons Auger
cooling rates Radiative recombination
rates Impact ionization rates
Charging energies Excitonic fine
structure