Density-Functional Theory in Materials Science
Transcript of 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~~
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 3
Structure of Matter
(I) Electronic structure
calculations
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 4
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 5
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
i j
nucleus
electrons
E = Enn + Ene + Eee + Te
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 6
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
i j
Vijnn =
|Ri - Rj|ZiZje2
Vijee =
|ri - rj|e2
Vijne =
|Ri - rj|-Zie2
Te = 1/2 mv2E = Enn + Ene + Eee + Te
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 7
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 8
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 9
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 10
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]
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 11
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]
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 12
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 13
Structure of Matter
(II) Electronic structure
of the solid state
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 14
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 15
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)
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 16
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 17
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 18
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 19
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-
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 20
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 21
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)
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 22
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 23
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)
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 24
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]
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 25
Structure of Matter
(III) Applications
2D structures – domain boundaries1D structures – wires
0D structures – clusters on surfaces
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 26
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~~
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 27
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 28
Structure of Matter
Piezo ||
109°
180°
conductivity
+
+
+
71°
109°
180°
P|| ⟨⟨⟨⟨1
11⟩⟩⟩⟩
71°
109°
180°
Domains in BiFeO3
2D structures
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 29
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 30
Structure of Matter
MoS2 – based nanowires: S-deficient Mo6S6
(Nano Lett. 8 (2008) 3928-3931)
1D structures
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 31
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 32
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 33
Structure of Matter
Mo6S6 : Electromechanic switch
(Nano Lett. 8 (2008) 4093-4097)
Mo S
tors
ion
[°/n
m]
energy [eV]
DFPT: Transmission
1D structures
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 34
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 35
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 36
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~~
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 37
Structure of Matter
Scale-bridging: Growth modes at vicinal surfaces
layer-by-layer
islandsroughening
phase-field (PF)
(kinetic) Monte-Carlo (KMC)
0D structures
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 38
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 39
Structure of Matter
Particle-based Monte-Carlo approach
J12
J13
J12'
H
Lateral interactions:
Vertical interactions:
J13'
step
H-HsH+Hs
0D structures
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 40
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 41
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 42
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 43
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)
∂tc = D ∇2c – c/τ + F – ½ ∂tΨ
diffusion desorption flux coupling term
(I)
(II)
0D structures
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 44
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)
∂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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 45
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 46
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
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 47
Structure of Matter
Thank you!
Institute of Ion Beam Physics and Materials Research � FZ Dresden-Rossendorf � Sibylle Gemming � www.fzd.de � Slide 48
Structure of Matter