First Principle Simulations in Nano-science Tianshu Li University of California, Berkeley University of California, Davis

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

It’s all about quantum mechanics! Nano scope where the focus is electron, atom, or molecule. The fundamental law in atomic world is Quantum Mechanics

Contribution of Quantum Mechanics to the Technology Bloch theorem-1928 Wilson-Implication of band theory-Insulators/metal 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

What is the “First Principle” method? “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.

Real difficulty The simplest problem: single s electron in H atom A little more difficult problem: two s electrons in H 2 molecule (2x2 matrix) A more tougher one: four s and eight p electrons in O 2 molecule (12x12 matrix) An overwhelming case: electrons (s, p, d, f,…) in real materials? (10 23 x10 23 matrix) “The difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble” --Dirac

Basic Methods of Electronic Structure Calculations Hylleras-Numerically exact solution for H Slater-Augmented Plane Waves-1937 Herring-Orthogonalized Plane Waves-1940 Boys-Gaussian basis functions-1950 Phillips, Kleinman, Antoncik-Pseudopotentials Kohn-Density Functional Theory-1964 Anderson-Linearized Muffin Tin Orbitals-1975

Most frequently used methods in simulating nano-scale systems 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

One of the most active areas in science Physics Today, June 2005

Overview of DFT 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.

Total Hamiltonian for the interacting electron-ion system T el : Kinetic energy of electrons V el-ion : Electron-ion interactions V el-el : Electron-electron interactions V ion-ion : Ion-ion interactions A nasty problem!

Hohenberg-Kohn (HK) theorem HK theorem: All properties of many-body system are determined by the ground state density n 0 (r) The ground state total energy E is a functional of n 0 (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

A diagram of HK theorem HK

Comments on HK theorem Exact theory, no approximation Proof is rather simple, by contradiction The functional is universal Independent on the external potential A new idea of solving ground state problem The ground state total energy should only depend on the ground state electron density n(r).

Kohn-Sham (KS) equation Interacting->Non interacting 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

KS Diagram Ф0(r) E0[n0(r) ] n0(r) Фks(r) E0[n0(r) ] n0(r) HK KS Original system Interacting Kohn-Sham system Non-interacting

Solving KS self-consistently Initial guess n(r) Solve Kohn-Sham equation E KS Ф KS =ε KS Ф KS Calculate electron density n i (r) Self consistent? Output No Yes Solving an interacting many-body electrons system is equivalent to minimizing the Kohn- Sham functional with respect to electron density.

Comments of DFT The Kohn-Sham wavefunctions do not have explicit physical interpretations Without further approximation, DFT remains “useless” in practice. E xc [n]: contains everything that we don’t know. Unknown functional!

Approximation in Solving KS equation Local Density Approximation (LDA) The simplest and easiest approximation Assume E xc [n(r)] is a sum of contributions from each point depending only on the density at each point, i.e., ε xc (n) can be computed exactly from Quantum Monte Carlo method In principal, only supposed to work in a uniform electronic system

Approximation in Solving KS equation Local Density Approximation (LDA) 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 E x by 10% while overestimates the E c by 200~300%. Usually E x ~10E c, so net E xc (=E x +E c ) is typically underestimated by ~7%.

Approximation in Solving KS equation Generalized Gradient Approximation (GGA) 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

Comparisons between predictions based on DFT and experiments Lattice ConstantBulk Modulus DFTExp.DFTExp. C (diamond) Si Au W Mo Ta

Prediction of new phase of Si based on First Principle calculations 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)

Comparison of the calculated and measured phonon band structures of NiAl * Experimental data of phonon frequencies are extracted from M. Mostoller et al., Physical Review B 40, 2856(1989)

Cleavage anisotropy in Transition-metal Aluminides * FeAl: Preference of {100} type of cleavage * NiAl: {100} being unfavorable * 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, (2004) Tianshu Li, J.W. Morris, Jr., D. C. Chrzan, Phys. Rev. B 73, (2006)

Elastic moduli predicted by First principle method in Ti-V alloys Tianshu Li, J.W. Morris, Jr., D.C. Chrzan, to be submitted

Application in Nano Science 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.

Schematic representations of Nano-structures

Quantum Confinement CdSe: Size tunable energy gap provides size dependent emission Optical properties of nano-materials depend on the size (Quantum Dots) Visible light carries the photon energies 1.7eV~3eV. Size of nano particles

Si nano clusters Surface compensation A.J. Williamson J. Grossman, R.Q. Hood, A. Puzder and G. Galli, Phys. Rev. Lett, 89, (2002). Reboredo FA, Galli G, Phys. Chem. B 109, 1072 (2005).

Si Quantum Dots Q-Dot Diameter (nm) Optical Absorption Gap (eV) “Perfect” Si Q-Dots Experiments Density Functional Theory and Quantum Monte Carlo calculations Consider core, surface and solvent effects, one at a time: Gaps reveal quantum confinement. Key role of surface chemistry (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 and JPC-B 2005

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

Ultradispersity of nano- diamonds Nano diamonds have stable size distribution between 2~5nm J.-Y. Raty and G. Galli, Nature Materials 2, 792 (2003)

CdSe nano-particles Surface reconstruction A. Puzder, A.J. Williamson, F. Gygi, and G. Galli, Phy. Rev. Lett. 92, (2004)

Limitations of current techniques 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

The band gap problem 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 DFT-LDA (eV) Exp (eV) Silicon Diamond MgO DFT-LDAGWExp Silicon Diamond MgO

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

Weak non-local bonding Soft bonding. Sparse system Wrong ground state Promising solution: Van der Waals Density Functional H. Rydberg et al., Phy. Rev. Lett. 91, (2003)

Summary Remarks