Density-Functional Theory in Materials Science

Institute of Ion-Beam Physics and Materials Research � PD Dr. Sibylle Gemming � www.fzd.de � Member of the Leibniz Association

Structure of Matter

Density-Functional Theoryin Materials Science

S. GemmingInstitute of Ion Beam Physics and Materials Research

FZ Dresden-Rossendorf

Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 2

Structure of Matter

Time and length scales in materials modelling

electronicstructure

atomisticmodelling

discrete particles, processesfinite-element

continuummodels

piko nano micro macro

Time/length scale of modelled phenomenon

Sys

tem

siz

elo

g N

(at)

0

3

6

20~~


Structure of Matter

(I) Electronic structure

calculations


Structure of Matter

I Electronic structure calculations –Quantum mechanics

• Explicit calculation of electron distribution and energies

• Separation of nucleus- and electron dynamics(= Born-Oppenheimer approximation)and stepwise optimisation of both systems

• Energy conservation in Hamilton formalism


Structure of Matter





i j

nucleus

electrons

E = Enn + Ene + Eee + Te


Structure of Matter





i j

Vijnn =

|Ri - Rj|ZiZje2

Vijee =

|ri - rj|e2

Vijne =

|Ri - rj|-Zie2

Te = 1/2 mv2E = Enn + Ene + Eee + Te


Structure of Matter

Standard Electronic Structure Methods

• Hartree-Fock (HF) formalism: wave function

scaling: N4

and spectroscopy with correlation correctionsscaling: N4 - N7

• Density Functional Theory (DFT): electron density

and atom arrangement via Car-Parrinello dynamicsscaling: N3

• Tight-Binding (TB) and semiempirical methodsscaling: N, NlogN, N2, N3

N = number of atoms


Structure of Matter

Hartree-Fock Method

• Idea: calculate everything, up to ∫∫∫∫∫∫∫ (7-fold integration !!) of interactions

• Virtue:inclusion of further interactions straightforward

strong electronic correlations = more than two electron cross-talkelectric and magnetic fields, relativistic corrections (spin-orbit)

• Limitation: system sizes up to 10-30 atoms

• Accuracy: spectroscopic accuracy (meV, pm)

• Acronyms: MPn = Møller-Plesset perturbation theory

CAS = Complete Active Space, CI = Configuration InteractionCC = Coupled Cluster

Very high predictive power, but computationally expensive


Structure of Matter

Density Functional Theory I

• Idea: Calculate up to 3-fold integrals, include others in "mean-field" potential

• Virtue:Inclusion of further interactions for larger systems

electric and magnetic fields, relativistic corrections (spin-orbit)

• Limitation: up to 103 atomsband gap too small, correction terms complicatedground state, no easy description of excited states

• Accuracy: atom distances ~ pm, lattice constants ± 3% occupied levels ~ meV

Very good predictive power for ground state (!) systems


Structure of Matter

Density Functional Theory II

• Basis: Hohenberg-Kohn Theorems (Hohenberg/Kohn/Sham, 1964/65)

Ø Theorem I:all observables = function(al)s of the ground state electron density n0.

Ø Theorem II:variational principle → ground state electron density n0

(total energy reaches is minimal, if n0 is inserted in the functional)

E[n0] = Enn[n0] + Ene[n0] + Eee [n0] + Exc[n0] + Te[n0]


Structure of Matter

Density Functional Theory II

• Basis: Hohenberg-Kohn Theorems (Hohenberg/Kohn/Sham, 1964/65)

Ø Theorem I:all observables = function(al)s of the ground state electron density n0.

Ø Theorem II:variational principle → ground state electron density n0

(total energy reaches is minimal, if n0 is inserted in the functional)

• Acronyms: LDA = Local Density ApproximationGGA = Generalized Gradient ApproximationVWN, PZ, PBE, BP, BLYP, B3LYP, … = different forms of Exc[n]

E[n0] = Enn[n0] + Ene[n0] + Eee [n0] + Exc[n0] + Te[n0]


Structure of Matter

Tight-Binding Method

• Idea: calculate only double integrals, parameters for other onesinner electrons + nucleus → effective core potential

separation into: on-site term (Coulomb)+ hopping term for electron interactions

between (directly) neighbouring atoms

• Easy incorporation of other interactions, including excitation

• Limitation: system sizes up to several 104 atoms

• Accuracy: mostly explanatory, limited predictive power


Structure of Matter

(II) Electronic structure

of the solid state


Structure of Matter

Modelling electronic structure of crystals

• Idea: exploit long-range periodicity

• Realisation:calculate smallest unit cell explicitlyapply Periodic Boundary Conditions

PBCa1a2

a3

basis vectors


Structure of Matter

Electrons in the lattice

• Basis set representation of electron distribution:ψk(r) = ΣG ck+G ei(k+G)r

• Typical basis sets:

plane waves

local orbitals

LAPW (linearised augmented plane waves =local orbitals for core, plane waves for rest)

s dp

FPLO (full potential local orbitals)LMTO (linearised muffin-tin orbitals)


Structure of Matter

The trick: reciprocal space

• not all plane waves eikr match latticeonly discrete subset G witheiG(r+R) = eiGr

= reciprocal lattice

• reciprocal space unit cell = Brillouin Zone


Structure of Matter

The trick: reciprocal space

• not all plane waves eikr match latticeonly discrete subset G witheiG(r+R) = eiGr

= reciprocal lattice

• reciprocal space unit cell = Brillouin Zone


Structure of Matter

Pseudopotentials

• Lattice periodic potential comprisesVnuc-nuc, Vnuc-el, Vel-el (Coulomb, XC)

• Problem:strong spatial modulation of core e-

requires many plane waves or local functions


Structure of Matter

Pseudopotentials

• Lattice periodic potential comprisesVnuc-nuc, Vnuc-el, Vel-el (Coulomb, XC)

• Problem:strong spatial modulation of core e-

requires many plane waves or local functions

• Solution: treat only valence e- explicitlyscreen Vnuc by potential of core e-


Structure of Matter

Pseudopotentials

• General form of ion-electron termVion-el = Vloc + Vnl

• Norm-conserving PP

∫ nPP dr = ∫ nAE dr in core region

• Ultrasoft PP

specially smooth nPP, few plane waves

long-rangelocal part

short-rangel-dependent part


Structure of Matter

Basic properties

• Total energy E and energy levels Enk

band structure (occupied) = levels vs. k,E

density of states (DOS) = levels vs. E

• Derivatives of E

Hellman-Feynman forces, geometry = ∂E / ∂Ri

stress tensor components = ∂2E / ∂Ri ∂Rj

phonons = ∂E / ∂Ri vs. k

external electric field (long-wave)


Structure of Matter

k0aπ

aπ-

Band structure

• Energy Enk depends on band index n (Pauli principle)wave vector k dispersion

• band structure – density of states (DOS)

k0aπ

2aπ

aπ

2aπ- -

E3

E2

E1

EFermi

DOS

occupied

virtual


Structure of Matter

Spectroscopy – ELNES, XANES

S. Köstlmeier, C. Elsässer, Phys. Rev. B 60 (1999) 14025.S. Köstlmeier, C.Elsässer, B. Meyer, Ultramicroscopy 80 (1999) 145.S. Köstlmeier, Ultramicroscopy 86 (2001) 319.

dens

ityof

sta

tes

DO

S

core statesEELS

valence statesVEELS

empty states Fermi'sGolden Rule:

ELNES ~ M(E) x DOS(E)


Structure of Matter

-10 0 10 20 -10 0 10 20 -10 0 10 20

exp.exp.

theo. theo.theo.

exp.

C

C

B

AA'

A

BA'

A'

A

B C

A' A

C

A'

A

B CA'

A

C

O-K Edges in Oxides - ELNES

MgO MgAl2O4 α-Al2O3

Energy [eV]


Structure of Matter

(III) Applications

2D structures – domain boundaries1D structures – wires

0D structures – clusters on surfaces


Structure of Matter


electronicstructure

atomisticmodelling


continuummodels



Sys

tem

siz

elo

g N

(at)

0

3

6

20~~


Structure of Matter

Piezo ||

109°

180°

conductivity

71°

109°

180° BiFeO

pseudo-cubicBi arrangement

tilted,distortedFeO6

Oktaeder

Seidel et al., Nature Mater. 8 (09) 229.

⟨111⟩ polarisation Tc = 650 Kantiferromagnet TN = 1103 K

Domains in BiFeO3

2D structures


Structure of Matter

Piezo ||

109°

180°

conductivity

+

+

+

71°

109°

180°

P|| ⟨⟨⟨⟨1

11⟩⟩⟩⟩

71°

109°

180°

Domains in BiFeO3

2D structures


Structure of Matter

Piezo ||

109°

180°

conductivity

+

+

+

71°

109°

180°

P|| ⟨⟨⟨⟨1

11⟩⟩⟩⟩

71°

109°

180°0.36 J/m2

0.21 J/m2

0.83 J/m2

0.02 eV

0.18 eV

0.15 eV

Domain boundaries in BiFeO3

2D structures


Structure of Matter

MoS2 – based nanowires: S-deficient Mo6S6

(Nano Lett. 8 (2008) 3928-3931)

1D structures


Structure of Matter

Ø maxima at SØ distances 4.4 Å, 10.2 ÅØ wire height 9.4(±0.1)Å

Connolly sphere onDFT density contour4 – 10 Å

experiment simulation

Mo6S6 nanowires: STM - structure

Electromechanics


Structure of Matter

Mo d+

S p

states

Mo dstates

=conductionband edge

Ø metallic conductance through Mo part, S insulates

Besenbacher(Aarhus)

energy [eV]

Mo6S6 nanowires: STS - conductivity

1D structures


Structure of Matter

Mo6S6 : Electromechanic switch

(Nano Lett. 8 (2008) 4093-4097)

Mo S

tors

ion

[°/n

m]

energy [eV]

DFPT: Transmission

1D structures


Structure of Matter

Mo6S6 : Electromechanic switch

(Nano Lett. 8 (2008) 4093-4097)

Mo S

tors

ion

[°/n

m]

energy [eV]

DFPT: Transmission

gap

1D structures


Structure of Matter

ideal: C3v

alloweda1-a2-crossing

distorted: C3

forbiddena-a-crossing

Ene

rgie

[eV

]E

nerg

ie [e

V]

Γ k X

a1a2

e

aa

e

EF

EF

Mo6S6 : Structure-induced metal-insulator transition!

1D structures


Structure of Matter


electronicstructure

atomisticmodelling


continuummodels



Sys

tem

siz

elo

g N

(at)

0

3

6

20~~


Structure of Matter

Scale-bridging: Growth modes at vicinal surfaces

layer-by-layer

islandsroughening

phase-field (PF)

(kinetic) Monte-Carlo (KMC)

0D structures


Structure of Matter

Motivation – Growth modes at vicinal surfaces

layer-by-layer

islandsroughening

phase-field (PF)

(kinetic) Monte-Carlo (KMC)

PF-KMC hybrid model

islands+step-flow

meandering

0D structures


Structure of Matter

Particle-based Monte-Carlo approach

J12

J13

J12'

H

Lateral interactions:

Vertical interactions:

J13'

step

H-HsH+Hs

0D structures


Structure of Matter

Particle-based Monte-Carlo approach

J12

J13

J12'

H

Total energy:

Etot

= Σij

J12

sis

j+ Σ

ijJ

13s

is

j+

+ Σij

J12‘

sis

j+ Σ

ijJ

13‘s

is

j

+ ΣiH s

i+

+ Σi' (H±Hs) si

NNNNN

adsorption strength

Schwöbel barrier

Monte-Carlo simulationMetropolis algorithm

si= 1: occupied; s

i= 0: empty

Lateral interactions:

Vertical interactions:

(Loppacher, Gemming, et al., Nanotechnology, 17 (06) 1568)(Kunze, Gemming, Numaza, Schreiber, CPC, accepted)

J13'

step

H-HsH+Hs

0D structures


Structure of Matter

Burton-Cabrera-Frank model for surface growth

desorptionflux

F

diffusionD

τadatoms

adatom concentration c(r,t) surface topology= phase field

0D structures


Structure of Matter

Continuum-theoretical phase-field description

Coupled differential equations

Evolution of the adatom concentration field c(r;t)

Evolution of the surface topology = phase field Ψ(r;t)

(I)

(II)

0D structures


Structure of Matter





∂tc = D ∇2c – c/τ + F – ½ ∂tΨ

diffusion desorption flux coupling term

(I)

(II)

0D structures


Structure of Matter





∂tc = D ∇2c – c/τ + F – ½ ∂tΨ

τΨ∂tΨ = W2 ∇2Ψ – sin(πΨ ) – λc[1+cos(πΨ )]

diffusion desorption flux coupling term

coupling termtopologysteps

interface motiondiffuse width W

(I)

(II)

0D structures


Structure of Matter

Calculated properties

adatom density c(r,t), si

(T = 400 K, D = 3.2 x 105 a2/s, F = 3 ML/ms, τ = 104 s)

interface length Q(t)

nucleation rate τi(t)

topology Ψ(r,t), Hi

island density ni(r,t)

step density ns(r,t)

roughness R(t)

growth modes

(Radke, Kundin, Emmerich, Gemming, Physica B, 2009)

0D structures


Structure of Matter

Scale-CouplingScale-Hopping

MicromechanicsBCF-Phase-field with concentrations

Ising/HeisenbergMonte-Carlo (MC) with molecules/spins

(Semi-)Empirical Theorymolecular dynamics (MD) with atoms

First-principles Theoryquantum mechanics with electrons

Pa

ram

ete

r-T

ran

sfe

r

QM/MM

QM/QM‘TUD

PP/MCRWTHTUD

L [m] t [s]

10-9 10-9

10-6 10-6

10-3 10-3

100 100

10-12 10-12

Scale-bridging approaches

Goal – method cooperation


Structure of Matter

Thank you!


Structure of Matter

Density-Functional Theory in Materials Science

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