basics for ab initio calculations - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/2-basics.pdf ·...

P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Basics for ab initio Calculations

Basics for ab initio Calculations

http://folk.uio.no/ravi/CMT2015/

Prof.P. Ravindran, Department of Physics, Central University of Tamil

Nadu, India

Approximation #1: Separate the electrons into 2 types:

Core Electrons & Valence Electrons

The Core Electrons: Those in the filled, inner shells of the

atoms. They play NO role in determining the electronic properties

of the solid.

Example: The Si free atom electronic configuration:

1s22s22p63s23p2

Core Shell Electrons = 1s22s22p6 (filled shells)

These are localized around the nuclei & play NO role in the

bonding.

Lump the core shells together with the Nuclei Ions

(in ∑i , include only the valence electrons)

Core Shells + Nucleus Ion Core

He-n He-i , Hn Hi

The Valence Electrons

Those in the unfilled, outer shells of the free atoms. These determine the electronic properties of the solid and take part in the bonding.

Example:

The Si free atom electron configuration:

1s22s22p63s23p2

The Valence Electrons = 3s23p2 (unfilled shell) In the solid, these hybridize with the electrons on neighbor

atoms. This forms strong covalent bonds with the 4 Si nearest-neighbors in the Si lattice

Methods for Efficient Computation

K-points (k)

– Discrete points specified in Brillouin Zone used to perform numerical integration during calculation.

Energy Cut-off Value (ecut)

– Energy value for maximum energy state included in a summation over electron states.

Pseudopotentials (pp)

– Offers specific exchange-correlation functional form which represents frozen “core electrons.” They are based largely on empirical data.

LDA vs. GGA Approximations

“Local-density approximations (LDA) are a class of

approximations to the exchange-correlation (XC) energy

functional in DFT that depend solely upon the value of the

electronic density at each point in space (and not, for example,

derivatives of the density or the Kohn-Sham orbitals).” This is

more of a first-order approximation.

“Generalized gradient approximations (GGA) are still local but

also take into account the gradient of the density at the same

coordinate.”

Ensuring k and Ecut Lead to a Converged Energy

The most important skill in performing DFT calculations is the

ability to get converged energies. Since the appropriate choice of

k-points and Ecut vary wildly among different geometries (and

even different required accuracies), it is important to be able to

form the following graphs every time you perform ‘scf’

calculations on new geometries.

Converged values of Ecut and k should be reported any time you

publish DFT results, so that someone else may reproduce your

calculation and agree on the same numerical error.

Energy Cut-off Convergence Plot

Not only should the graph look converged, but the difference in energy between

the last two consecutive points should be smaller or equal to your required accuracy!

K-point Convergence Plot

As we can see from the

convergence plots, the

presence of smearing does

little to ensure convergence

with fewer k-points.

Note: In an automatic

distribution of k-points, the value

of k specifies how many discrete

points there are equally-spaced

along each lattice vector to

populate the Brillouin Zone.

K-point Convergence Plot (cont.)

When unit cells do not have equal-length lattice vectors, it is sometimes computationally rewarding to “geometrically-optimize” your automatic k-point distribution.

For example, if one had a unit cell that was four times taller in one direction than its other two directions, one should specify only a quarter as many k-points along the taller direction.

– This makes sense, because in reciprocal space, the taller distance will only be a quarter as long as the other two distances.

Comparing the Relaxed Structure to Literature

Comparing the Relaxed Structure to Literature (cont.)

On top of comparing the ‘a’ and ‘c/a’ lattice parameters to literature and to experiment, it is also useful to compare a quantity called “cohesive energy”

– The difference between the average energy of the atoms of a crystal and that of the free atoms.

It is important to compare cohesive energies and not specifically the self-consistent energies, because only differences in energies are physically meaningful.

– The energy datum for an ‘scf’ calculation is specified by the choice of pseudopotential.

Unit Cell: the smallest group of atoms possessing the

system of the crystal which, when repeated in all

directions, will develop the crystal lattice.

Primitive Cell: is a type of minimum-volume cell that

will fill all space by the repetition of suitable translation

operation.

Primitive Cell

Unit Cell

Bravais Lattice and Crystal structures

The Fourteen Space (Bravais) Lattice : well known

P - primitive cell;

I - body-centered cell;

F - face-centered cell,

C - Centered: additional point in the center of each end

R - Rhombohedral: Hexagonal class only

Monoclinic and

Triclinic Cells

Orthorhombic Cells

Hexagonal and

Triagonal Cells

Tetragonal Cells

Cubic Cells

Rhombohedral

Rotation

Reflection

Inversion

32 point groups

Translation

14 Space Lattices

230 Space Groups

▲Miller Indices:

Indices for Direction: [x,y,z] represents a specific

crystallographic direction; <x,y,z> represents all

of the directions of the same form

Indices for Planes: (x,y,z) represents a specific plane;

{x,y,z} represents all of the planes of the same

form [x,y,z] is the directional normal of plane

(x,y,z)

Miller Indices for Hexagonal Crystals: 4-digit system

(hklm); h+k=-l => reduced to 3-digit system

c hkl: reciprocals of the intercepts with

a1, a2, and a3, m reciprocals of the

intercepts with c.

Miller Indices (Example)

Reciprocal Lattices and the First Brillouin Zone

Reciprocal Lattice

– Crystal is a periodic array of lattices

Performing a spatial Fourier transform

– Reciprocal Lattice

Expression of crystal lattice in fourier space

Reciprocal lattice

FT for time

FT for space

Reciprocal Lattice

b-c plane

Next lattice plane

* b ca

* 1| |a

Defined as a Wigner-Seitz unit cell in the reciprocal lattice

The waves whose wavevector starts from origin, and ends at this

plane satisfies diffraction condition

Brillouin construction exhibits all the

wavevectors which can be Bragg

reflected by the crystal

Wigner-Seitz unit cell

22k G G

Brillouin zone

First Brillouin Zone

The smallest of a Wigner-Seitz cell in the reciprocal lattice

The reciprocal lattices

(dots) and corresponding

first Brillouin zones of

(a) square lattice and

(b) hexagonal lattice.

The first brillouin zone is the

smallest volume entirely enclosed

by planes that are the

perpendicular bisection of the

reciprocal lattice vectors drawn

from the origin

The Brillouin zone (BZ)

Irreducible BZ (IBZ)

– The irreducible wedge

– Region, from which the whole BZ can be obtained by applying all symmetry operations

Bilbao Crystallographic Server:

– www.cryst.ehu.es/cryst/

– The IBZ of all space groups can be obtained from this server

Wigner-Seitz Cell

Form connection to all neighbors and span a plane normalto the connecting line at half distance

Wigner-Seitz Cell & Brillouin Zone

1st Brillouin Zone is the Wigner-Seitz Cell of the reciprocal lattice

BCC WS cell BCC BZ

FCC WS cell FCC BZ

Real Space Reciprocal Space

29The first Brillouin zone of a FCC structure

Face-centered cubic

KMiddle of an edge joining two hexagonal faces

L Center of a hexagonal face

UMiddle of an edge joining a hexagonal and a square face

W Corner point

X Center of a square face

Concepts when solving Schrödingers-equation

non relativistic

semi-relativistic

fully-relativistic

“Muffin-tin” MT

atomic sphere approximation (ASA)

Full potential : FP

pseudopotential (PP)

Hartree-Fock (+correlations)

Density functional theory (DFT)

Local density approximation (LDA)

Generalized gradient approximation (GGA)

Beyond LDA: e.g. LDA+U

Non-spinpolarized

Spin polarized

(with certain magnetic order)

non periodic

(cluster)

periodic

(unit cell)

plane waves : PW

augmented plane waves : APW

atomic orbitals. e.g. Slater (STO), Gaussians (GTO),

LMTO, numerical basis

Basis functions

Treatment of

Representation

of solid

Form of

potential

exchange and correlation potential

Relativistic treatment

of the electrons

Schrödinger - equation

Muffin-Tin Potential

K-point sampling

where the wave vector k is located in the first Brillouin zone (BZ).

Bloch’s theorem states that the wave function in a periodic

crystal can be described as:

It is therefore necessary to sample the wave

function at multiple k-points in BZ to get a

correct description of the electron density

and effective potential.

Using symmetry lowers the number of

necessary k-point to the ones in the

Irreducible Brillouin zone (IBZ).

1. In a periodic solid:

Number of atoms

Number and electrons

Number of wave functions ??

Bloch theorem will rescue us!!

2. Wave function will be extended over the entire solid

Periodic systems are idealizations of real systemsComputational problems

A periodic potential commensurate with the lattice. The Bloch theorem

Bloch Theorem: The eigenstates of the one-electron Hamiltonian in a

periodic potential can be chosen to have the form of a plane wave times a

function with the periodicity of the Bravais lattice.

Periodicity in reciprocal space

Reciprocal lattice vector

The wave vector k and the band index n allow us to label each electron (good quantum numbers)

The Bloch theorem changes the problem

Instead of computing an infinite

number of electronic wave functionsFinite number of wave functions at an

infinite number of k-points in the 1BZ

Many magnitudes require integration of Bloch functions over Brillouin zone (1BZ)

Charge density

In practice: electronic wave functions at k-points that are very close

together will be almost identical

It is possible to represent electronic wave functions over a region of

k-space by the wave function at a single k-point.

Band structure energy

In principle: we should know the eigenvalues and/or eigenvectors at

all the k-points in the first BZ

Recipes to compute sets of spetial k-points for thedifferent symmetries to accelarate the convergence of BZ integrations

Baldereschi (1973)

Chadi and Cohen (1973)

Monkhorst-Pack (1976)

The magnitude of the error introduced by sampling the Brillouin zone with a

finite number of k-points can always be reduced by using a denser set of points

11 papers published in APS journals since 1893 with >1000 citations in

APS journals (~5 times as many references in all science journals)

From Physics Today, June, 2005

The number of citations allow us to gauge the importance of the works on DFT

k-samplingMany magnitudes require integration of Bloch

functions over Brillouin zone (BZ)

r dk n k BZ

In practice: integral sum over a finite uniform grid

sampling in k is essential Essential for:

Small systems

Real space Reciprocal space

Metals Magnetic systems

Good description of the Bloch

states at the Fermi level

Sampling of the k-points

K-point sampling in the BZ

Linear Interpolation of eigen values to improve the BZ

integration

Brillouin Zone Integration – Tetrahedrons method

Temperature Effects

“Occupancy” of state =

PE,T = Probability that state at energy E will be occupied at

temperature T

= f(E) (“Fermi function)

k=Boltzmann constant = 8.62 × 10-5 eV/K

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

T=300K

T=600K

Fermi Function vs. T

Ef=1 eV

kT=26 meV (300K)

kT=52 meV (600K)

Density of states (DOS)

Method:

Calculate the Kohn-Sham eigenvalues with a very

dense k-point mesh.

Use a Gaussian or Lorentzian broadening function

for the delta function.

Perform the summation of the states over the

Brillouin zone.

Projected density of states (PDOS)

Method:

Calculate the Kohn-Sham eigenvalues i and wave

functions i.

Calculate the overlap between the Kohn-Sham wave

functions i and atomic wave functions fal

Use a Gaussian or Lorentzian broadening function

for the delta function.

Band Structure & DOS for Graphene

K G M K G -F [eV]

k[Å-1] van Hove singularities

Brillouin Zone

PDOS for Graphene

-F [eV]

Brillouin Zone

-F [eV]

DOS of real materials: Silicon, Aluminum, Silver

Measuring DOS: Photoemission spectroscopy

1( ) ( ) ( ) ( ) ( )

( )kin bin bin kin kin

I D f D f f f

Fermi Golden Rule: Probability per

unit time of an electron being ejected

is proportional to the DOS of occupied

electronic states times the probability

(Fermi function) that the state is

occupied:

Measuring DOS: Photoemission spectroscopy

Once the background is subtracted

off, the subtracted data is proportional

to electronic density of states

convolved with a Fermi functions.

We can also learn about DOS above the Fermi

surface using Inverse Photoemission where electron

beam is focused on the surface and the outgoing flux

of photons is measured.

Band Filling Concept

Electron bands determined by lattice and ion core potentials

Bands are filled by available conduction and valence electrons

Pauli principle → only one electron of each spin in each (2π)3 volume in “k-space”

Bands filled up to “Fermi energy”

– Fermi energy in band → metal

– Fermi energy in gap → insulator/semiconductor

For T≠0, electron thermal energy distribution = “Fermi function”

Energy bands

The inter-atomic spacing varies with the direction you are

moving in the crystal {111}planes would have a smaller d

than {100} planes, thus the energy gaps are different in

each direction.

This gives rise to different bands, but usually effect due to

this can be averaged.

1D case

3D case

- Band structure Indirect gap

Direct gap

“Density of States”: N(E)

N(E) dE = number of electron states between E and E+dE

n = number of electrons per unit volume

Isotropic Parabolic Band

Density of States:

Isotropic, Parabolic Bands

NT(E) = number of states/unit volume with energy<E

= “k-space” volume/(2π)3 (per spin direction)

Electron Density of States: Free Electrons

1 1 2( )

dN mD E E

3 23 3 3

/k L L mE

Electron Density of States: Free Electrons

Density of States (DOS)

81Bandstructure and DOS

Electronic Structure and Chemical BondingJ.K. Burdett, Chemical Bonding in Solids

L G X W L K G

Electronic Structure of Si:

Fermi Level

Electronic Band Structure Electronic Density of States

Real Band Effects

Thermal Eg ≠ Optical Eg

Electron effective mass ≠ Hole effective mass

More than one electron/hole band

– Multiple “pockets”

– Overlapping bands

Anisotropic electron/hole pockets

Non-parabolic bands

Real Band Effects

85LDA vs. LSDA

What we can calculate

Murnaghan Equation of State is usually used to extract Bulk

modulus from Total energy curve

What About the Kohn-Sham Eigenvalues?

Attempts on Improving LDA

Slab structure of Al(110)

Surface Relaxation and Reconstruction

Surface Relaxation

Relaxation of Pt (210)

Surface Reconstruction: Missing Row Reconstruction

Semiconductor Surfaces: Si(100)

basics for ab initio calculations - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/2-basics.pdf ·...

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