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First Principle Simulations in Nano-science

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First Principle Simulations in Nano-science

Tianshu Li

University of California, Berkeley

University of California, Davis

- Introduction to the First Principle method
- Overview of Density Functional Theory
- Density Functional Theory calculation in Nano science
- Limitations in current theory
- Summary remarks

- Nano scope where the focus is electron, atom, or molecule.

- The fundamental law in atomic world is Quantum Mechanics

- Bloch theorem-1928
- Wilson-Implication of band theory-Insulators/metal-1931
- Wigner-Seitz-Quantitative calculation for Na-1935
- Slater-Bands of Na-1934
- Bardeen-Fermi surface of a metal-1935
- Invention of the Transistor-1940
- Bardeen & Shockley

- BCS theory for superconductivity-1957
- Bardeen, Cooper, & Schrieffer

- Kohn-Density Functional Theory-1965

- “First principle” means things that cannot be deduced from any other
- Ab initio : “From the beginning”
- Most of physical properties are predictable based on the quantum mechanics laws.
- Unlike many other simulation methods, the only input information in “First Principle” calculation is just the atomic number!
- The most accurate simulation technique.

- The simplest problem: single s electron in H atom
- A little more difficult problem: two s electrons in H2 molecule (2x2 matrix)
- A more tougher one: four s and eight p electrons in O2 molecule (12x12 matrix)
- An overwhelming case: 1023 electrons (s, p, d, f,…) in real materials? (1023x1023 matrix)

“The difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble”

--Dirac

- Hylleras-Numerically exact solution for H2-1929
- Slater-Augmented Plane Waves-1937
- Herring-Orthogonalized Plane Waves-1940
- Boys-Gaussian basis functions-1950
- Phillips, Kleinman, Antoncik-Pseudopotentials-1950
- Kohn-Density Functional Theory-1964
- Anderson-Linearized Muffin Tin Orbitals-1975

- Density Functional Theory
- Most popular and widely adopted technique

- Quantum Chemistry
- Finite systems like molecules

- Quantum Monte Carlo
- Explicit many-body method, yet computationally demanding

- Tight-binding Method
- Fast, but parameters are adjustable. Empirical method in its nature

- Physics Today, June 2005

- What is Density Functional Theory (DFT)?
- Walter Kohn, 1998 chemistry Nobel prize
- Hohenberg-Kohn theorems
- Kohn-Sham

- What can it do?
- Mapping any interacting many-body system exactly to a much easier-to-solve non-interacting problem

- Why is it important?
- Numerous applications in both science and engineering. Open a new field.

Tel: Kinetic energy of electrons

Vel-ion: Electron-ion interactions

Vel-el: Electron-electron interactions

Vion-ion: Ion-ion interactions

A nasty problem!

- HK theorem:
- All properties of many-body system are determined by the ground state density n0(r)
- The ground state total energy E is a functional of n0(r)

Aside: Functional vs. Function

- Function maps a variable to a result
- For example: g(x)->y

- Functional maps a function to a result
- For example: f[n(r)]->y

- The functional is universal
- Independent on the external potential

- Exact theory, no approximation
- Proof is rather simple, by contradiction

- A new idea of solving ground state problem
- The ground state total energy should only depend on the ground state electron density n(r).

- A non-interacting system should have the same ground state as interacting system
- Only the ground state density and energy are required to be the same as in the original many-body system

Original system

Interacting

Kohn-Sham system

Non-interacting

KS

HK

HK

Initial guess

n(r)

Solve Kohn-Sham equation

EKSФKS=εKSФKS

Calculate electron density

ni(r)

No

Self consistent?

Yes

Output

- Solving an interacting many-body electrons system is equivalent to minimizing the Kohn-Sham functional with respect to electron density.

- The Kohn-Sham wavefunctions do not have explicit physical interpretations
- Without further approximation, DFT remains “useless” in practice.

Exc[n]: contains everything that we don’t know. Unknown functional!

- The simplest and easiest approximation
- Assume Exc[n(r)] is a sum of contributions from each point depending only on the density at each point, i.e.,
- εxc(n) can be computed exactly from Quantum Monte Carlo method

- In principal, only supposed to work in a uniform electronic system

- In practice, LDA works surprisingly well for many systems.
- One of the most successful approximations
- Still the most frequently used approximation nowadays, especially in materials science and physics.

- LDA underestimates the Ex by 10% while overestimates the Ec by 200~300%. Usually Ex~10Ec, so net Exc (=Ex+Ec) is typically underestimated by ~7%.

- Include the gradient of the density in functional, so that the exchange-correlation functional is non-local.
- Improve performance in finite systems, like molecules.
- Widely adopted in chemistry and biology
- Why Kohn got a Nobel prize in Chemistry rather than Physics

- Phase transitions of Si under Pressure:
- Si was predicted to be metal under very high pressure (>110GPa), which was then verified by experiments.

* M.Y. Tin and M.L. Cohen, Physical Review B 26, 5668(1982)

* Experimental data of phonon frequencies are extracted from M. Mostoller et al., Physical Review B 40, 2856(1989)

* NiAl: {100} being unfavorable

* FeAl: Preference of {100} type of cleavage

* K.-M. Chang, R. Darolia, and H.A. Lipsitt,, Acta. Metall. Mater. 40, 2727 (1992)

- Tianshu Li, J.W. Morris, Jr., D.C. Chrzan, Phys. Rev. B 70, 054107 (2004)
- Tianshu Li, J.W. Morris, Jr., D. C. Chrzan, Phys. Rev. B 73, 024105 (2006)

Tianshu Li, J.W. Morris, Jr., D.C. Chrzan, to be submitted

- Nano-materials containing 100~1000 atoms are the perfect match to the first principle (quantum) simulation.
- Experimental technique alone is not adequate to probe all the features in nano structure
- For example, surface structure

- First Principle method can separate different physical effects and assess their relevance in determining various properties.

- Optical properties of nano-materials depend on the size (Quantum Dots)

CdSe: Size tunable energy gap provides size dependent emission

- Visible light carries the photon energies 1.7eV~3eV.

Size of nano particles

A.J. Williamson J. Grossman, R.Q. Hood, A. Puzder and G. Galli, Phys. Rev. Lett, 89, 196803 (2002).

Reboredo FA, Galli G, Phys. Chem. B 109, 1072 (2005).

Density Functional Theory and Quantum Monte Carlo calculations

5

Experiments

“Perfect” Si Q-Dots

4

Optical Absorption Gap (eV)

3

2

1

0

0

2

4

6

8

Q-Dot Diameter (nm)

- Consider core, surface and solvent effects, one at a time:
- Gaps reveal quantum confinement.
- Key role of surfacechemistry (e.g. oxygen) and surface reconstruction.

A.Puzder et al. J Am Chem Soc 2003.;A Puzder et al, Phys Rev Lett 2003.;E Draeger et al, Phys Rev Lett 2003; F.Reboredo et al., J.Am Chem Soc 2003

D.Prendergast et al., JACS 2004; F.Reboredo et al. Nanolett. 2004 and JPC-B 2005

J.-Y. Raty, G. Galli, C. Bostedt, T. W. van Buuren, and L. J. Terminello, Phys. Rev. Lett. 90, 037401 (2003)

Nano diamonds have stable size distribution between 2~5nm

J.-Y. Raty and G. Galli, Nature Materials 2, 792 (2003)

A. Puzder, A.J. Williamson, F. Gygi, and G. Galli, Phy. Rev. Lett. 92, 217401 (2004)

- Size restrictions. Max ~ 1000 atoms
- Excited states properties, e.g., optical band gap.
- Strong or intermediate correlated systems, e.g., transition-metal oxides
- Soft bond between molecules and layers, e.g., Van de Waals interaction

- Excitations are not well described by LDA or GGA within DFT. The Kohn-Sham orbitals are only exact for the ground states.
- Famous “band gap” problem.
- Promising solutions:
- GW calculation
- QMC

- Electrons are strongly localized. Sparse system
- Wrong ground state
- Promising solutions
- LDA+U (semi-ab initio)

- Soft bonding. Sparse system
- Wrong ground state
- Promising solution:
- Van der Waals Density Functional

H. Rydberg et al., Phy. Rev. Lett. 91, 126402 (2003)