MSE 5310: Modeling Materials

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MSE 5310 Modeling Materials

• Instructor Prof. Rampi Ramprasad
• Class Wednesday, 500 pm 800 pm, Gentry 119E
• Grade Homework (50), Midterm (25), Final term
  presentation/paper (25)
• Course Objectives
• This course is intended to provide an overview
  of the theory and practices underlying modern
  electronic structure materials computations,
  primarily density functional theory (DFT).
  Students involved primarily/partially in
  materials computations, as well as those focused
  on experimental materials research wishing to
  learn about DFT techniques will benefit from this
  course. Several case studies will be presented.
  The course will culminate in a term project that
  will provide the students with an opportunity to
  address a problem close to their research using
  an appropriate computational method. Prior
  programming experience is not essential. This
  course is not about curve fitting, data analysis
  or data visualization!
• Primary Text
• Density Functional Theory A Practical
  Introduction, D. Sholl J. Steckel, Wiley
  (2009).
• Supplementary Texts
• Methods of Electronic-Structure Calculations
  From Molecules to Solids, M. Springborg, Wiley
  (2000).
• Electronic Structure Calculations for Solids and
  Molecules, Jorge Kohanoff, Cambridge University
  Press (2006).
• Electronic Structure Basic Theory and
  Practical Methods, R. Martin, Cambridge
  University Press (2004).

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Planned Topics

• Introduction the course in a nutshell (Chap. 1)
• Theory From Quantum Mechanics to Density
  Functional Theory (Chap. 1)
• Density Functional Theory (DFT) nuts bolts
  (Chap. 3)
• Reciprocal space total energy formalism
• Approximations k-point sampling,
  pseudopotentials, exchange-correlation
• Simple molecules solids
• Structure (Chap. 2)
• Geometry optimization (Chap. 3)
• Vibrations (Chap. 5)
• Electronic structure (Chap. 8)
• Surface science
• Periodic boundary conditions, relaxation,
  reconstruction, surface energy (Chap. 4)
• Chemisorption and reaction on surfaces (Chap. 4,
  5, 6)
• Chemical processes transition state theory
  (Chap. 6)
• Non-zero temperatures Thermodynamics
• Phase diagrams (Chap. 7)
• Thermal properties specific heat, thermal
  expansion
• Response functions elastic, dielectric,
  piezoelectric constants
• Beyond standard DFT

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Lecture Topics

1. Introductory comments
2. Overview of theory Total energy methods and
   density functional theory (DFT)
3. Predictions of known properties using DFT (i.e.,
   validation)
4. Practical value of DFT calculations Insights,
   and design of new materials (i.e., success
   stories)

4
The need for computational science

• Anytime the future has to be predicted or
  forecasted, simulation is used, generally based
  on well understood scientific notions/principles
• Anytime experimental analysis is too expensive or
  too impractical, simulation becomes necessary
• Simulations complement experiments and could
  provide insights
• Examples from everyday contemporary experience
• Weather modeling involves solution of Newtons
  equations of motion and fluid dynamics
• Astrophysical predictions (eclipses, comets)
  involve solution of Newtons equations of motion
  and/or general relativity
• Other notable examples economic (stock market)
  modeling, drug design, mechanical properties
  (auto industry), electromagnetic simulations
  (microelectronics industry)
• Challenges
• Models by themselves may not be representative of
  the real situation
• Practical treatment of model (or numerical
  solution) is time intensive
• Sometimes the physical principles (or theory)
  involved are not well known
• Unknown extraneous factors, e.g., stock market
• Major numerical problems non-linear systems,
  e,g, chaotic pendulum, weather
• Fortunately, in Computational Materials Science
  (CMS), we need to worry mainly about the first
  two challenges, and the others are listed in
  decreasing order of importance

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Theory, Models, Simulation Experiments

• Theory experiment go hand in hand.
• A set of results may come out of experiment, but
  one needs a theory to put it all in a framework
  of understanding. A theory cannot be formulated
  in the absence of experimental data
• Goal of science construct theory based on
  available experimental data, make predictions
  using theory outside the regime of experimental
  input, and modify theory if predictions are not
  satisfactory
• A model is a representation of physical reality,
  along with a set of assumed equations that govern
  that reality
• Simulation is the process of using the model
  using numerical techniques
• CMS involves theory, modeling and simulation,
  with the terminologies generally used
  interchangeably!

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Theory, Models, Simulation Experiments Example

• Let us consider an example tensile testing
• In the elastic region, we know that there is a
  linear relationship between stress and strain,
  which is at the heart of elasticity theory
  merely having experimental data on stress versus
  strain for a few materials does not constitute
  true understanding the experimental data
  together with the realization that stress
  constantstrain constitutes understanding
• Now, one can do two types of computations (1)
  use the constant obtained from experimental data
  to look at complicated geometries (FEM, used
  widely in the auto industry), or (2) we can use a
  more fundamental theory to determine the constant
  from first principles the second approach
  results in a even more fundamental understanding
  of the origin of the constant, namely, in terms
  of atomic level bond stretching
• CMS sometimes complements experimental studies,
  and sometimes provides insights, and increasingly
  is being used to design materials

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CMS at different scales
Time
Engineering Design
Years
Hours
Finite element Analysis (Continuum/classical)
Minutes
Seconds
Mesoscale Modeling (Semi-classical)
Microseconds
Molecular Mechanics
Nanoseconds
Quantum Mechanics
Picoseconds
Femtoseconds
mm
Å
nm
mm
m
Distance
e.g., density functional theory (DFT)

• If more than one box is involved in a
  computation ? multi-scale modeling

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Overview of CMS Course contd.

• Central themes
• Our system is represented as a collection of
  atoms, or a collection of electrons and ions
• We can determine the total potential energy of
  this collection of particles
• Equilibrium (stable, unstable and metastable)
  situations correspond to features (minima,
  maxima, saddlepoint) in the total potential
  energy function
• Analogous approaches in other types of
  computations
• Electromagnetic simulations involve minimization
  of electromagnetic energy density
• Mechanical simulations involve minimization of
  strain energy
• Let us for a moment assume that we do have a
  prescription for computing the total energy of a
  group of atoms, given their spatial positions
• What kind of properties can we compute? And how?

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Diatomic molecule, A-B

• Only one degree of freedom ? RAB
• If we new E(RAB), then we can determine potential
  energy surface (PES)

Thus, IF E(RAB) is known, then we can trivially
determine equilibrium bond length, bond energy
and vibrational frequency!
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Triatomic system, A-B-C

• Consider the reaction A-B C ? A B-C
• Two degrees of freedom in 1-dimensional A-B-C
  system ? RAB, RBC
• If we new E(RAB, RBC), then we can determine
  potential energy surface (PES)

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Bulk cubic material

• Only one degree of freedom ? lattice parameter a,
  or Volume V ( a3)
• If we new E(V), then we can determine potential
  energy surface (PES)

Curvature B/V0 B bulk modulus Note slope
stress
Cohesive energy
V0
Thus, IF E(V) is known equation of state, then
we can trivially determine equilibrium lattice
parameter, cohesive energy and bulk modulus!
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Lecture Topics

1. Introductory comments
2. Overview of theory Total energy methods and
   density functional theory (DFT)
3. Predictions of known properties using DFT (i.e.,
   validation)
4. Practical value of DFT calculations Insights,
   and design of new materials (i.e., success
   stories)

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Prescriptions for computing energy

• Until 1950s, no reliable prescription was
  available to practically compute the energy
• breakthrough quantum mechanics, 1920s Wigner
  Seitz, 1930s more later
• Hence energy as a function of geometry was
  parameterized using experimental data then, and
  even to-date!
• Lennard-Jones, Morse, etc. (physicists),
  embedded-atom method, etc. (materials
  scientists), force fields (chemists)
• Referred to as empirical or semi-empirical
  methods (as experimental data was used entirely
  or partially)
• Today, reliable parameter-free methods are
  available to compute energy, which come with a
  (rapidly diminishing) price tag of large
  computational time
• Density functional theory (DFT), and higher level
  quantum mechanics based methods

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Empirical approach example

• Suppose that our system contains M atoms, and
  that atoms interact pairwise

summation over i and j run over the number of
atoms M

• Lennard-Jones
• Morse
• By fitting A and B (Lennard-Jones), or V0, d and
  r0 (Morse) to experimental data so that
  equilibrium bulk properties are reproduced, we
  can in principle have a scheme to compute E
• We could make the scheme more sophisticated by
  defining E in terms of 3- or many-atom
  interactions (e.g., embedded atom method) or
  angular (e.g., Stillinger-Weber)

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The quantum mechanical prescription

• Building blocks are N electrons and M nuclei,
  rather than M atoms
• The N-electron, M-nuclei Schrodinger (eigenvalue)
  equation

The N-electron, M-nuclei wave function
The total energy that we seek
The N-electron, M-nuclei Hamiltonian
Nuclear kinetic energy
Electronic kinetic energy
Nuclear-nuclear repulsion
Electron-electron repulsion
Electron-nuclear attraction

• The problem is completely parameter-free, but
  formidable! Why?
• Cannot be solved analytically when NM gt 3
  (really?!?)
• Too many variables (for a 100 atom Pt cluster,
  the wave function is a function of 23,000
  variables!!!)

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Formidable ? Manageable!
Density Functional Theory (DFT) W. Kohn,
Chemistry Nobel Prize, 1999
1-electron wave function (function of 3
variables!)
1-electron energy (band structure energy)
The average potential seen by electron i
Energy can be obtained from r(r), or from ?i and
ei (i labels electrons)

• Still parameter-free, but has a few acceptable
  approximations
• DFT is versatile in principle, it can be used to
  study any atom, molecule, liquid, or solid
  (metals, semiconductors, insulators, polymers,
  etc.), at any level of dimensionality (0-d, 1-d,
  2-d and 3-d)

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Lecture Topics

1. Introductory comments
2. Overview of theory Total energy methods and
   density functional theory (DFT)
3. Predictions of known properties using DFT (i.e.,
   validation)
4. Practical value of DFT calculations Insights,
   and design of new materials (i.e., success
   stories)

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The first convincing DFT calculationYin and
Cohen, PRB 26, 5668 (1982)
Slope is transition pressure

• The correct equilibrium phase (diamond cubic) is
  predicted
• The lattice parameter of the equilibrium phase,
  and the pressure for the diamond cubic ? beta-tin
  phase transition (common tangent) are also
  predicted to a good level of accuracy

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Predictions of geometry

• Structural details predicted typically to within
  1 of experiments for a wide variety of solids
  and molecules
• Results from various sources collected in
  Ramprasad, Shi and Tang, in Physics and Chemistry
  of Nanodielectrics, Dielectric Polymer
  Nanocomposites (Springer)

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Predictions of other basic properties
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Band offsets
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Polarization in Periodic SystemsThe Fundamental
Difficulty

• Textbook definition
• Polarization dipole moment per unit cell volume
• inadequate depends on how unit cell is defined

unless we are in the Clausius-Mossotti limit
Dipole per unit cell well defined
Each unit cell will give a different net dipole
Resolution provided by Resta and King-Smith
Vanderbilt, in terms of phases of the
wavefunctions (Berrys phase)
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Phase transformations involving solids
Experiments
Tin
Boron Nitride
Kern et al, PRB 59, 8551 (1999)
Pavone et al, PRB 57, 10421 (1998)
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Phase transformations involving melting
expt.
expt.
DFT
DFT
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Most-cited papers in APS journals

• Six out of the top eleven most-cited papers are
  DFT-foundational papers!

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Lecture Topics

1. Introductory comments
2. Overview of theory Total energy methods and
   density functional theory (DFT)
3. Predictions of known properties using DFT (i.e.,
   validation)
4. Practical value of DFT calculations Insights,
   and design of new materials (i.e., success
   stories)

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Extreme pressures

• Extreme geophysical pressures may be difficult to
  create in the lab, but can be simulated easily

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Extreme pressures contd.
Liquid
Solid
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Extreme pressures contd.
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Thermal expansion

• Can a material contract when heated?

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Si reconstruction

• When heated to high temperatures in ultra high
  vacuum the surface atoms of the Si (111) surface
  rearrange to form the 7x7 reconstructed surface

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Bi makes Cu-Cu bonds softer (hence, brittleness
NOT due to electronic effects)
Grain boundary decohesion due to larger size of
Bi atoms
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• High activity of transition metals in oxidation
  catalysis is due to the presence of surface
  oxides under catalytic conditions

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What can DFT (not) do?

• Systems that can be represented in terms of up to
  a few 100 atoms per repeating unit cell are okay
• Geometric details to within 1 of experiments
• Other properties to within 5 of experiments
• Challenges
• Investigations requiring a large number of atoms
• Systems in which periodicity is absent
• Band gaps and excited state energies
• Non-zero and high temperatures
• Highly correlated-electron systems

40
Lecture Topics

1. Introductory comments
2. Overview of theory Total energy methods and
   density functional theory (DFT)
3. Predictions of known properties using DFT (i.e.,
   validation)
4. Practical value of DFT calculations Insights,
   and design of new materials (i.e., success
   stories)

41
Planned Topics

• Introduction the course in a nutshell (Chap. 1)
• Theory From Quantum Mechanics to Density
  Functional Theory (Chap. 1)
• Density Functional Theory (DFT) nuts bolts
  (Chap. 3)
• Reciprocal space total energy formalism
• Approximations k-point sampling,
  pseudopotentials, exchange-correlation
• Simple molecules solids
• Structure (Chap. 2)
• Geometry optimization (Chap. 3)
• Vibrations (Chap. 5)
• Electronic structure (Chap. 8)
• Surface science
• Periodic boundary conditions, relaxation,
  reconstruction, surface energy (Chap. 4)
• Chemisorption and reaction on surfaces (Chap. 4,
  5, 6)
• Chemical processes transition state theory
  (Chap. 6)
• Non-zero temperatures Thermodynamics
• Phase diagrams (Chap. 7)
• Thermal properties specific heat, thermal
  expansion
• Response functions elastic, dielectric,
  piezoelectric constants
• Beyond standard DFT