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## Algorithms for Spanning the Nano-Regime

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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

- 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

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.

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 NanoscalePseudopotential 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.

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”

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 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.)

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).

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/

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

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?

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