Texas Advanced Computing Center. National Energy Research Scientific Computing Center. Algorithms for Spanning the Nano-Regime. Jim Chelikowsky Center for Computational Materials Institute for Computational Engineering and Sciences Departments of Physics and Chemical Engineering
National Energy Research Scientific Computing CenterAlgorithms for Spanning the Nano-Regime
International Workshop on Computational Methods for Nanoscale Systems, Hong Kong December 12, 2006
optical properties with size
Bawendi Group: Colloidal CdSe quantum dots dispersed in hexane.
The optical properties of semiconductors like CdSe can be tuned to span the optical region of the spectrum by varying the size of the dot. Intensive property becomes “extensive.” However, not like thermodynamic varaibles in that “extensive properties” need not scale linearly with the system size.
Numerous degrees of freedom (both electronic and nuclear). Would like to solve systems with at least 10,000 atoms...or more.
Avoid transferring parameters from bulk to nanophase (Scaling is often not clear.)
Unified theory from atoms to crystals
Numerical methods targeted at high performance platforms, i.e., multiprocessor machines
Easy to implementPredicting Properties at the Nanoscale
Focus on chemically active electronic (valence) states
Capture the physical content of the periodic table
“Standard Model”Physical Basis
Solution to the Kohn-Sham theory yields all relevant ground state properties.
(The prelude to applications...)
Wave function vanishes outside the domain
No supercells: One can readily handle charged systems.No plane waves: Avoid Fourier transforms of the vacuum.
FLEXIBLE BOUNDARY CONDITIONS.
Express ionic potential as nonlocal operator
ulm are pseudo atomic states
Vlocal chosen for convenience.
L. Kleiman and D. Bylander, Phys. Rev. Lett. 48, 1425 (1982)
One can use ARPACK, variant of the Lanczos process (Implicitly Restarted Lanczos)
“Traditional Approach to the Kohn-Sham Problem”
of the true eigenstates
Damped 6th degree polynomial
Window for Filtering
EigenvaluesChebyshev Subspace Iteration
Most of the time is now spent on filtering! Much faster and requires fewer orthogonalization
Reference: Y. Zhou, Y. Saad, M.L. Tiago, and J.R. Chelikowsky,
J. Comp. Phys. 218, 172 (2006) and Phys. Rev. E (in press.)
Largest cluster to date: Si9041H1860 a total of 10,901 atoms and matrix size of 2,992,832 requiring 19,015 eigenpairs. Took roughly 1 week using 48 processors (IBM Power4).
Si525H276 leads to a matrix size of 290,000, requiring a solution of 1,194 eigenpairs. In 1997, this took roughly 20 hours of CPU time on the Cray T3E, using 48 processors.
TODAY: 2 hours on one SGI Madison processor (1.3GHz) (Could be done on a good workstation!)
Undiminished accuracy from “traditional” algorithm.
Used a machine that is NOT on the “Top 500.’’
This method yields a superb TIME TO SOLUTION (TTS).
1.21 A (expt)
This software is free.....
including a MatLab version!
Density of States for Si Quantum Dot
diameter = 7 nm
Critical point structure clearly evolved by a length scale of 7 nm
Electron Affinity: A
Quasiparticle energy gap:
Eqp= I -A
Ionization and Affinities Energies in Quantum Dots
Dangling bonds are passivated by attaching hydrogen atoms.
for Si Quantum Dots
IP = IP0 + A/Dα
EA = EA0 + B/Dβ
IP0 = 4.5 eV EA0 = 3.9 eV
α = 1.1 β = 1.08
The LDA quasi-particle gap (IPo-EAo) as D→∞ approaches the HOMO-LUMO gap from the Kohn-Sham eigenvalues (0.6 eV).
Itoh, Toyoshima & Onuki, J. Chem. Phys. 85, 4867 (1986)
Donor ionization energy, Ed, is roughly 50 meV, which is
comparable to kT≈25 meV.
Ed= I(extrinsic) - A(intrinsic)
What happens for nanocrystals?