DFT Density Functional theory and the description of Nature Edison Z. da Silva Instituto de Física...

DFT Density Functional theory

and the description of Nature

Edison Z. da SilvaInstituto de Física "Gleb Wataghin", UNICAMP, Campinas - SP, Brazil

[email protected]

Outline•DFT, What it is, how it came about, what it is used for.....

•Solids and DFT

•Probing the Earth´s Inner Core

• Gold Nanowires

We have a well defined systemWe have a well defined system

Nuclei+

ElectronsAtomsAtoms

Molecules

Solids

Liquids

Description of known materials, prediction of new materials, explanation of many questions

Description of known materials, prediction of new materials, explanation of many questions

DFT, theoretical frameworkDFT, theoretical framework

Electrons are the “glue” which holds matter togetherElectrons are the “glue” which holds matter together

Fundamental laws of electronic and nuclear

motions well known

Electromagnetic Forces+

Quantum Mechanics

Electromagnetic Forces+

Quantum Mechanics

H ( {ri , r} ) = ETOT ( {ri , r} ) H ( {ri , r} ) = ETOT ( {ri , r} ) ^

H = ( -½2i ) + ( -½ 2

) + ( - ) + + Zri

___i<j

Ne

i=1

Ne^ 1___rij

=1

Nn

i=1

Ne

=1

Nn

<

Ne ZZ_____r

M

Te^ Tn

^ Ve-n^ Ve-e

^ Vn-n^

“There is an oral tradition that, shortly after Schrödinger’s equation for the electronic wavefunction had been put forward and spectacularly validated for small systems like He and H2, P. A. M. Dirac declared that chemistry had come to an end - its content was entirely contained in that powerful equation. Too bad, he is said to have added, that in almost all cases, this equation was far too complex to allow solution.”

W. Kohn, Reviews of Modern Physics 71, 1253 (1999).

•Therefore, we have a well posed problem, but too complex and difficult to solve

•Impossible to have exact solution except in very few cases

•Need of approximations

Born-Oppenheimer (1927) Approximation (Qualitatively)

•Based on the fact that nuclear masses are much larger than the electronic ones

•Therefore, electrons move on a different time-scale than the nuclei (nuclei move much more slowly)

•Thus, electrons follow the nuclei “instantaneously”

•This allows us to consider the nuclei fixed when solving for the electronic motion

Born-Oppenheimer Approximation

Helecaelec({ri};{r}) = Ea

elec({r}) aelec({ri};{r}) Heleca

elec({ri};{r}) = Eaelec({r}) a

elec({ri};{r}) ^

aelec({ri};{r}) depends explicitly on the electronic

coordinates {ri} but only parametrically on the nuclear coordinates {r}, as does Ea

elec({r})

Helec = ( -½2i ) + ( - ) + + Z

ri

___i<j

Ne

i=1

Ne^ 1___rij

i=1

Ne

=1

Nn ZZ_____r<

Ne

Te^ Ve-n

^ Ve-e^ Vn-n

^


• As the {aelec(r;Q)}, for a fixed Q, form a complete

orthonormalized set, one can write:

(r , Q) = a(Q) aelec(r;Q) (r , Q) = a(Q) aelec(r;Q)

a

[Tn + Ebelec(Q)] b(Q) + ba(Q) a(Q) = Etot b(Q)

^a

• Roughly speaking, if the kinetic energy of the nuclei is small when compared to |Eb

elec(Q) - Eaelec(Q)|, the

non-diagonal terms ba can be neglected


[Tn + Ebelec(Q)] b(Q) = Etot b(Q)

^

• The total electronic energy Ebelec(Q) acts as a

potential energy for the nuclear motion, which occurs in a single electronic state b

elec .

• The nuclear motion simply deforms the electronic distribution, and does not cause transitions between different electronic states.

(r , Q) = b(Q) belec(r;Q) (r , Q) = b(Q) b

elec(r;Q)


• Important concepts:– Molecular geometry



– Crystal structure



– Crystal structure

Geometricalcoordinate

PES Ex: Carbon

Amorphous

DiamondGraphite

– Potential Energy Surface (PES)

Two issues

• How to calculate (as accurately as possible) the PES– electronic degrees of freedom

• How to sample the PES (or how to move the atoms)– nuclear degrees of freedom

Traditional way of thinking would be to describe the wavefunction as well as possible

However, WF for a many electron system is a very complicated object!

How to solveHelecelec = Eelecelec

Electronic density e(r) as the fundamental object(and not wavefunction elec)

Electronic density e(r) as the fundamental object(and not wavefunction elec)

YESYES

Is there another way to solve the problem (ab initio)?Is there another way to solve the problem (ab initio)?

Density Functional Theory (DFT)

• Formally well founded in the works of W. Kohn

– P. Hohenberg and W. Kohn, Phys. Rev 136, 864B (1964)

– W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965)

Chemistry Nobel prize - 1998

• Roots in the works of Thomas (1926), Fermi (1928), Dirac (1930), Slater, etc.

Proc. Cambridge Philos. Soc. 26, 376 (1930)

DFT - Two basic theoremsP. Hohenberg and W. Kohn, Phys. Rev 136, 864B

(1964)

• Let’s consider a system of N interacting electrons in the ground-state associated with an external potential v(r)

• 1) The ground state density 0e(r) uniquely

determines the potential v(r)– Since with v(r) the full electronic Hamiltonian is known,

0e(r) completely determines all properties of the system

• 2) There exists a functional E[e(r)] which has its minimum for the ground-state density 0

e(r)

– Can use this variational principle to calculate 0e(r)

DFT - How to calculate 0e

W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965)

E[e] = v(r) e(r) d3r + T[e] + U[e]

• I don’t know how to solve this problem, but I know how to solve the following problem:– If I had a system of non-interacting electrons, in an

external potential veff(r), then:

Es[e] = veff(r) e(r) d3r + Ts[e]

Ts[e] = kinetic energy of a non-interacting system with density e



• The ground-state wavefunction in this case is a Slater determinant of N orbitals i (i=1,N), which satisfy the equations:

[-½ 2 + veff(r)] i(r) = i i(r)

• And the ground-state density can be written as:

0e(r) = |i(r)|2lowest N

i



• However, from the Hohenberg-Kohn theorem, a completely equivalent way of calculating 0

e(r) would be through the minimization of E[e(r)]:

e(r) Es[e] - e(r’)

d3r’ = 0

+ veff(r) - = 0Ts[e]

e(r) Gives same density as0

e(r) = |i(r)|2i



• Application of the Hohenberg-Kohn theorem to E[e] gives

e(r) E[e] - e(r’) d3r’ = 0

Ts[e] + v(r’) e(r’) d3r’ + UH[e] + EXC[e] - e(r’) d3r’

e(r) = 0

+ v(r) + d3r’e(r’)/|r-r’| +Ts[e]

e(r)EXC[e]

e(r) - = 0



• Comparison with similar equation for non-interacting electrons gives:

+ v(r) + d3r’e(r’)/|r-r’| +Ts[e]

e(r)EXC[e]

e(r) - = 0

Interacting electrons

+ veff(r) - = 0Ts[e]

e(r)

Non-interacting electrons

v(r) + d3r’e(r’)/|r-r’| +EXC[e]

e(r) veff(r) =



• We can, therefore, calculate exactly (in principle) the problem of the interacting electrons through the solution of the following equations:

v(r) + d3r’e(r’)/|r-r’| +EXC[e]

e(r) veff(r) =

[-½ 2 + veff(r)] i(r) = i i(r)


i



Problem of many-electrons mapped exactly (in principle) into the problem of non-interacting electrons subjected to an

effective potential!

v(r) + d3r’e(r’)/|r-r’| +EXC[e]

e(r) veff(r) =

• However, we don’t know how to calculate this effective potential:

• We need approximations for EXC[e] and vXC[e] =EXC[e]

e(r)

DFT - Local Density Approximation (LDA)

• Let’s consider the exchange-correlation energy, per particle, for a homogeneous electron gas, eh

XC

• This quantity can be calculated very accurately (Quantum Monte Carlo)• Let’s imagine the real system divided in small cells:

i

Ni Vi i = Ni / Vi


• If we consider that the density is approximately homogeneous inside cell i, then we can write for the XC energy in cell i

i

Ni Vi i = Ni / Vi

EiXC eh

XC (i) Ni = ehXC (i) Ni (Vi / Vi) =

= ehXC (i) (Ni / Vi) Vi = eh

XC (i) i Vi


• Summing over all the cells i will give for the total XC energy

i

Ni Vi i = Ni / Vi

EXC ehXC(i) i Vi

i

EXC ehXC((r)) (r) d3r = ELDA

XC


• And from this we can get the XC potential vLDAXC

i

Ni Vi i = Ni / Vi

ELDAXC

e

vLDAXC = = eh

XC((r)) (r) d3r e

= ehXC()

d

d e

DFT - Self-consistency cycle

• Still a problem: veff depends one ande depends on veff

• Solution: self-consistency cycle

1. Choose an initial density 0e and determine the effective

potential veff (0e)

2. Solve the Kohn-Sham equations [-½ 2 + veff(r)] i(r) = i i(r)


i

3. Calculate the density

4. Check if 1e = 0

e. If yes, end of cycle. If not, go back to 2. (with 0

e = 1e)

Crystal - Bloch functions• veff has the periodicity of the crystal

• According to Bloch’s theorem, the i will have the following form:

(n,k)(r) = exp(i k•r) u(n,k)(r)

• The function u(n,k) also has the periodicity of the crystal

• Therefore, it can be expanded in a discrete set of plane waves whose wave vectors are reciprocal lattice vectors of the crystal

u(r) = cGexp(i G•r)G

Crystal - Bloch functions• Therefore

• For a crystal, the density will be given by

(n,k)(r) = cnk+G exp[i (k+G)•r]

G

e(r) = |i(r)|2 = 2 |(n,k)(r)|2 lowest N

i

occ.

bands

n k

BZ

Crystal - Bloch functions• Two approximations:

– 1) Maximum value of G=GC number of basis function (plane wave)

e(r) = 2 |(n,k)(r)|2

occ.

bands

n k

SP


G

Gc

– To determine GC one gives the kinetic energy associated with the GC

– Cutoff energy EC = GC2/2m

– 2) Number of Brillouin-zone points - set of “special points”

All electronvalence WF

Crystal - Bloch functions

Strong oscillations in the core region

• These strong oscillations are bad for plane-wave expansion (large value of GC to describe small features in real space)

• Either use a different basis function in the core region, or change the function in the core region

Pseudopotentials

rc

r

V,

• Valence electrons are the chemically relevant ones• Removes the core electrons and replaces them and the strong

potential by an effective, weaker potential that acts on the valence electrons

All electron

Pseudo

Si, The credibility of DFT

How to solve the KS equations

• After all these approximations, I have to find the cnk+G through

the self-consistent solution of the KS equations


G

Gc

HKS i(r) = i i(r)

Traditional method - diagonalization

• Write the matrix for the KS Hamiltonian in the plane wave basis set and diagonalize it

HKSk+G,k+G’ cn

k+G’ = nk+G cn

k+GG’

HKS c = c

Perfect crystal

Crystal with a defect

Surfaces

Applications

• Understanding the Earth´s inner core

• Breaking of Gold nanowires

Elasticity of Iron at the temperature of the Earth´s inner

core

G. Stainle-Neumann, L. Stixrude, R. Cohen & O. Gulseren

Nature, 413, 57 (2001)

General picture

•Seismological Studies:Compressional waves travel faster along polar paths than in equatorial plane.

•Earth´s center, 2,440 km across is nearly pure Iron

•At high temperatures and pressures iron goes to anhcp phase called -phase

Iron at Temperatures and pressures of the Earth´s core.

•Crystal structure distorts unexpectedly

•Inner core believed to be made up of Iron crystals in hcp structure

•Observed seismic anisotropy must arise from differencesin elastic response of hcp Iron

•Densities for iron 7.87 Mg m-3 atmospheric pressure13.04 Mg m-3 at Earth´s core pressures

Iron at Temperatures and pressures of the Earth´s core.

• DFT calculations of elastic constants

• Thermodynamics given by the Helmholtz free energy

F(V,T) = E0(V) + Eel(V,T) -T Sel(V,T) + Fvib(V,T)

E0(V) static total energyEel(V,T) due to thermal excitationsSel(V,T) electronic entropyFvib(V,T) phonon contribution (particle in a cell )

Equilibrium structure of hcp iron over a range of densities and temperatures of the Earth´s core.Computing F for different c/a and finding the minima

(Mg m-3)12.52

13.0413.62

Elastic constants

Elasticity of hcp iron for a density 13.04 Mgm-3

Velocity

Acoustic properties of iron for the Earth´s core. Adiabatic bulk modulus (KS)and shear modulus ()as function of temperature compared with inner core atthe same density 13.04 Mgm-3

Travel time anomalySolid line , model with 1/3 of crystals with basal planesaligned with rotation axis

Conclusions

Seismological Studies:Compressional waves travel faster along polar paths than in equatorial plane.

At high temperatures and pressures iron goes to anhcp phase called -phase. With c/a = 1.7 seismic waves travel 12 % faster in the ab plane.This anisotropy is too strong, if crystals are aligned it accounts for the experimental findings.

Breaking of Gold Nanowires: Breaking of Gold Nanowires: A Computer Simulation studyA Computer Simulation study

E. Z. da Silva

Instituto de Física "Gleb Wataghin", UNICAMP, Campinas - SP, Brazil

[email protected]

Participants, motivation...

Theory

Antonio José R. da Silva(IF-USP)Adalberto Fazzio (IF-USP)Frederico D. Novaes (PhD-IF-USP)

Experimental motivation

Daniel Ugarte (LNLS)Varlei Rodrigues

When we get to the very, very smallworld - say circuits of seven atoms -we have a lot of new things that wouldhappen that represent completelly new opportunities for design. Atoms on a small scale behave like nothing on the large scale, for they satisfy the laws of quantum mechanics.

Richard Feynman Plenty of Room at the Bottom(1959)

Program

• Why study Au nanowires?

• Experiments

• Theory

• TBMD simulations

• Impurities

• Conclusions

Why study Au nanowires?

Gold nanowires are produced in the lab!Understanding of its properties is fundamental.Gold is the electrical contact in nanotechnology.

When Gold nanowires become very thin,interesting surprises are found:• Conductivity is quantized.• Possibility of helical structures.• One-atom thick wires are produced.

Gold in devicesGold in devices

J Reichert, et al., Phys. Rev. Lett., 88, 176804 (2002)

U.Landman, W.D.Luedtke, B.E.Salisbury, and R.L.Whetten, Phys. Rev. Lett. 77, 1362, (1996)

Experiments, How it all started!Pin-plate experiment:• nanowire formed• conductance during elongation and contraction

O. Gulseren, F Ercolessi, and E. Tosatti, Phys. Rev. Lett. 80, 3775, (1998)

Theoretical Predictions

Computer simulations with a “glue” type empiricalmany-body potential (for Al and Pb) predicted: As the diameter decrease, thin metal wires develop exotic stable noncrystalline structures:

Existence of critical diameter Formation of helical, spiral-structured wires

O. Gulseren, F Ercolessi, and E. Tosatti, Phys. Rev. Lett. 80, 3775, (1998)

Theory Weird Nanowires

How to make a wire

Theorist´s view: production and imaging process!

Stable Au Nanowires from surface structures

ProductionAu thin films (3nm) in UHVTEMelectron beam irradiation (100 A/cm2)produced many holes nanowires

ImagingStraight nanowiresimages produced with electronbeam irradiation (5 A/cm2)

Y. Kondo and K. Takayanagi, Phys. Rev. Lett. 79, 3455, (1997)

1-7

4-11

6-13

1-7-14

1-8-15

Interpretation of theExperiments

Cylindrical folding of a

triangular lattice for

an (m, n) tube, with

views of several coaxial

tube nanowires. Each atom

is pictured as a sphere of

atomic radius. The (7, 3) gold

nanowire (note its chirality)

was reported to be magic in (3).

Folding of a (111) plane

V. Rodrigues and D. Ugarte, Phys. Rev. B 63, 073405-1, (2001)

Real time imaging of one-atom-thick Au nanowires

The film irradiation techniquefrom Kondo and Takayanagiproduced stable one-atom-thick Au- nanowires.

Time sequence of formation,elongation and fracture of achain of gold atoms.

Is it possible to simulate the evolution of a gold nanowire under stress with a reliable electronic structure method?

Dynamical Simulations of Gold nanowires

• Wire under tension.• It forms, starts to thin ( atoms going to the wire edges)neck is produced (one atom constriction). • Atoms from the wire,( near the tips) are draw back and are incorporated into the one -atom thick neck.• This thin neck grows to 4-5 atoms• The wire finally breaks

AC

CU

RA

CY

Nu

mb

er o

f at

oms

MD with Classical PotentialsEmbedded-atom, Effective medium,Finnis-Sinclair, Second Moment(TB)

(No electrons!)

Ab Initio DFT Methods(costly in terms of computer time!)

NRL-TBMDkeeps the electrons

(three orders of magnitude faster)

The Tight -Binding Method

Slater and Koster proposed in 1954 a modified combination of atomic orbitals (LCAO) as a interpolation scheme to determine energy bands.

In 1986 Papaconstantopoulos published a handbook of SK parameters for most elements in the periodic table.

As DFT ab initio methods improve, so does the Papaconstantopoulos TB fits with updates the of SK parameters for most elements in the periodic table.

M.J.Mehl and D. A.Papacontantopoulos, Phys. Rev. B 54, 4519 (1996)

The NRL tight-binding method

Procedure, The method fits TB parameters to reproduce ab initio total energies (DFT) calculated with full-potential linear aumented plane wave method.

Slater-Koster parameters are fitted to LAPW bands and total energies as function of volume for 10 fcc structures, 6 bcc structures and 5 sc structures. Perdue-Wang parametrization to LDA

Tight-binding and Molecular Dynamics

• TB basis set

• The Slater-Koster (SK) parameters

• Tight-binding Fit of the ab initio Potential energy surface

• Ab initio molecular dynamics

TB basis set

bN

1ii c

2r2z2y2xzxyzxyzyx dddddppps ,,,,,,,,

Minimum basis set of atomic like orbitals, for transition metals;

Number of basis functions per atom N0. Number of atoms Na,dimension space spanned by this basis set Nb = N0 Na.

States are expressed as linear combinations of these orbitals:

The Slater-Koster parameters

Overlap terms S

HHHopping terms

ppppspss

ppppspss

H,H,H,HH

S,S,S,SS

Matrix elements of the Hamiltonian and overlap gives:

SK-parameters

Tight-Binding Fit of the ab initio Potential energy surface

matrixoverlapS,nhamiltoniaH

0SHdet

ofeigenvalue

Tk1;zexp1zf

functionFermiwith

frnE

i

B

1

íi

í

ijji

ij2iji

2ii

34ii

32iiii

10

rh

rFRexp

"density"theanddorp,swith

dcbah

:termssiteonDiagonal

l/rrexp1rF

rFerferP

SandHofelementsMatrix2

Formalism Parametrization

Complicated Transition Metal-Mn

Manganese crystallizes in the -Mn structure with 29 atoms

Above 710 0C, Mn transforms to the -Mnwith 20 atoms

Calculated TB total energy

F.Kirchhhoff et al. Phys. Rev. B, 63 195101 (2001)

Fitting Procedure for Au, using the static TB code

Equation of state

Phonon dispersion, T = 0

How good is the TBparametrization forfcc bulk Au?

Data basis:10 fcc structures, 6 bcc structures, 5 sc structures.

Fcc structure, lowest energy

Ab initio molecular dynamics steps of the simulation

iiii RERH

Input, Ri, positions of the atoms H the Hamiltonian The electronic ground state of the system is calculated.

bN

1ii cWhere states are expanded

using the TB orbitals

I

Ii

IIII dR

Rc,RdEFRM

Eigenvectors

Forces on the atoms. Forces (EOM) integrated

and new positions found.

F.Kirchhhoff et al. Phys. Rev. B, 63 195101 (2001)

Electronic DOS for Au, using TBMD codeSolid, T dependence

Liquid Au, T = 1773 K

MD simulations:Microcanonical ensemble,EM with Verlet algo,t = 2 fs, point sampling.

Au-nanowires, simulationsE. Z. da Silva. A J. R.da Silva and A. Fazzio,

Phys. Rev. Lett., 87 , (2001)

Starting structures:nanowire along thealong the(111) direction- 10 atom planes - 7atoms in each plane.

Top view

Front view

Simulation Protocol

• EM integrated using Verlet algo, t =1 fs

• The system is heated to T = 600 K, 7000 MD steps, 7ps

• Elongation of the wire by L = 0.5 Å

• The system is heated to T = 400 K, 4000 MD steps, 4ps

• Previous two steps are repeated

• Procedure done for 24.0 Å < LW < 44.0 Å

Au nanowire, 6 (111) planes of 7 Au atoms,

relaxed structure

unrelaxed structure

Similarities for the breaking the one-atom necklace

Experiment

2.99 2.98

2.89

2.74

2.63 2.91

2.61 2.89

Simulation

Evolution and breaking

Atomic configurations(in Å)(a) 25.5, (b) 33.0, (c) 37.0

(a) 38.0, (b) 40.5, (d) 41.0

Defect structure, one atom neck

(i) Defect structure (iii) 4-5 atoms planes

(iv) 4-2-3 atoms planes (v) one atom neck

Forces

Saw-toothbehavior

Before breaking the one-atom necklace

The nanowire develops aone-atom thick necklacewith five atoms in its finalstructure.

Just before breaking, the apex-apex distance gets toda-a ~ 11.33 Å.

Animation of the simulation

• See this animation at• www.ifi.unicamp.br/~zacarias/nano

Pending problems

• Are TBMD results reproducible using ab initio DFT.

• Large Au-Au distances seen in the experiments

HRTEM image of atom chainwith three gold atoms. Atomicpositions are the dark spots

Large Au-Au distances!!!!

Schematic representation



V. Rodrigues and D. Ugarte, Phys. Rev. B 63, 073405, (2001)

Before breaking: one-atom necklaceBefore breaking: one-atom necklace

Simulations give shorter Au-Au distances

Ab initio DFT (Siesta)

• Fully self-consistent DFT LCAO– Extremely fast simulation using minimal

basis set

• Kohn-Sham equations– Exchange-correlation (GGA)– Norm conserving pseudopotentials

(Troullier-Martins)

• Total energy• Forces

Ab initio evolution

Starting structure

Before breaking

After breaking

Au-Au distance 3.0 Å

Large distances, Impurities?

The ImagesAu-Au distances 3.6±0.2

The statistics

Legoas,Galvão, Rodrigues , Ugarte , PRL, 88, 106105, (2002)

(Au-X-Au)Max 3.6 Å

X = ?X = ?

(Au-Au)Max 3.1 Å

Reasoning for impurities

2.9

2.8

3.6

3.1

TEM simulationStructure simulation

Carbon in Au nanowire

Carbon intermediate structure,

3.73

Carbon, initial structure Au-nanowire

Carbon, structure right before breaking

Quebra

Impurities and the Au-Au Bond Length

Frederico D. Novaes, A.J.R. da Silva, E.Z. da Silva and A.Fazzio, Phys. Rev.

Lett. 90,036101 (2003)22 january

• Use ab initio DFT to study contamination of the pure Au nanowire by impurities.

• Considered the effect of the impurities, H, B, C, N, O and S

• Experiment gives Au-?-Au = 3.6 Å

• Au-??-Au = 4.8 Å

Contamination by H and S

H Impurities?

Experiments:Legoas,Galvão, Rodrigues , Ugarte , PRL, 88, 106105, (2002)

3.5

4.9 3.5

Our simulation

General Conclusions• Importance of Gold nanowires.• Simulations can help understand the dynamical evolutionand finally the breaking of metal nanowires.• TBMD is reliable and fast.• Can be used to study dynamical evolution of metal nanowires.

Future:• Study of other directions for wire formation.• Investigation of nanowires of other metals (Ag, Pt etc..)• Use nanowires and their tips to design devices.

Conclusions•DFT, A effective way of studying and projecting interesting materials.

•Solids and DFT

•Probing the Earth´s Inner Core

• Gold Nanowires

DFT Density Functional theory and the description of Nature Edison Z. da Silva Instituto de Física...

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Transcript of DFT Density Functional theory and the description of Nature Edison Z. da Silva Instituto de Física...