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 • University of Texas, Austin, TX 78712, USA International Workshop on Computational Methods for Nanoscale Systems, Hong Kong December 12, 2006
C. Bekas, Y. Saad, Y. Zhou, M. Alemany, X. Huang Collaborators S. Ogut G. Dalpian M. Tiago L. Kronik
The Nano Regime • Motivation • Algorithms: Recent Developments • Applications: Role of Quantum Confinement on Optical, Magnetic and Doping Properties • Semiconductor nanocrystals and nanowires • Magnetic clusters (iron) • Excited state properties • Summary Nanocrystals Nanowires
Nano-science: Changing optical properties with size Bawendi Group: Colloidal CdSe quantum dots dispersed in hexane. Size 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.
R Quantum Confinement Electronic energies scale as ~ 1/R2 In the nano regime, the “size” of an electronic excitation is often determined by the physical size of the nanocrystal or quantum dot. Electron confined to spherical well of radius R
Nano Regime: Length Scale • Crude estimate of the “size” of dopant, e.g., P in Si, or exciton: • A quantum dot of 12 nm in diameter contains roughly 50,000 atoms. For dots much smaller than this, we expect quantum confinement to play an important role in determining the electronic properties.
Methods capable of describing matter at the nanoscale have been developed, but they are difficult to implement: Numerous degrees of freedom (both electronic and nuclear). Would like to solve systems with at least 10,000 atoms...or more. Low symmetry Goals: 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 implement Predicting Properties at the Nanoscale
Pseudopotential theory Focus on chemically active electronic (valence) states Capture the physical content of the periodic table Pseudopotential model: “Standard Model” Physical Basis • Density functional theory • Map all electron problem to one electron problem • Solve Kohn-Sham equation: Solution to the Kohn-Sham theory yields all relevant ground state properties.
Algorithms: Solving the Kohn-Sham problem (The prelude to applications...)
System of interest (quantum dot) 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.
Nonlocal Pseudopotentials 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)
Most of the time is spent on the diagonalization part. One can use ARPACK, variant of the Lanczos process (Implicitly Restarted Lanczos) “Traditional Approach to the Kohn-Sham Problem”
are a better estimate of the true eigenstates Damped 6th degree polynomial Window for Filtering Eigenvalues Chebyshev Subspace Iteration • Main ingredient: Chebyshev filtering. Given a set of basis vectors , filter according to • Pk is a Chebyshev polynomial of low degree: • Polynomials damped by applying an affine mapping to window region of interest.
New Approach:Chebyshev Filtering Most of the time is now spent on filtering! Much faster and requires fewer orthogonalization operations.
Chebyshev Filtering • Chebyshev filtering enhances the desired eigen-components. This step is not used to compute the eigenvectors accurately as a traditional diagonalization. • Convergence remains good and robust. • Need estimates for windowing, e.g., get bounds from initial diagonalization. • Scales as O(N3) because of orthogonalization, but fewer orthogonalizations performed! Reference: Y. Zhou, Y. Saad, M.L. Tiago, and J.R. Chelikowsky, J. Comp. Phys. 218, 172 (2006) and Phys. Rev. E (in press.)
This is what counts! Benchmarks 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).
Bond Length 1.21 A (expt) 1.19A (Calc.) Energy (eV) Oxygen Molecule Orbital Matlab “Standard” -32.53 -32.48 -19.49 -19.51 -13.57 -13.38 R(a.u.) -13.02 -13.06 -6.04 -6.06 PARSEC This software is free..... including a MatLab version! http://www.ices.utexas.edu/parsec/
Density of States (eV-1) Energy (eV) Density of States for Si Quantum Dot Si9041H1860 diameter = 7 nm Critical point structure clearly evolved by a length scale of 7 nm
Electron Ionization: I Final Initial Electron Affinity: A Final Initial Quasiparticle energy gap: Eqp= I -A Ionization and Affinities Energies in Quantum Dots Dangling bonds are passivated by attaching hydrogen atoms.
Electron Affinity and Ionization Potential 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)
Doping Si with P atoms: Crystalline Limit Conduction band Ed Valence band Donor ionization energy, Ed, is roughly 50 meV, which is comparable to kT≈25 meV. Ed= I(extrinsic) - A(intrinsic) What happens for nanocrystals?