Basics of periodic systems calculations

(Electronic structure theory for materials)

Sergey V. Levchenko

FHI Theory department

Extended (periodic) systems

There are 1020 electrons per 1 mm3 of bulk Cu

x

y

z

a1

a2

a3

Position of every atom in the crystal (Bravais lattice):

332211321 )0,0,0(),,( aaarr nnnnnn

lattice vector:

,2,1,0,,

),,(

321

332211321

nnn

nnnnnn aaaR

Example: two-dimensional Bravais lattice

1a

2a

primitive unit cells12 23 aa

The form of the primitive unit cell is not unique

From molecules to solids

Electronic bands as limit of bonding and anti-bonding combinations ofatomic orbitals:

electronic band

bandgap

Adapted from: Roald Hoffmann, Angew. Chem. Int. Ed. Engl. 26, 846 (1987)

Bloch’s theorem

Rr

rR

0

)()exp()( rkRRr i

In an infinite periodic solid, the solutions of the one-particle Schrödinger equations must behave like

)()( rRr UU Periodic potential(translational symmetry)

Consequently:

)()(),()exp()( rRrrkrr uuui

332211 aaaR nnn

Index k is a vector in reciprocal space

332211 gggk xxx ijji 2ag

1g

2g

nm

laa

g 2 – reciprocal lattice vectors

The meaning of k

chain of hydrogen atoms

j

sjk ajikx )()exp( 1

Adapted from: Roald Hoffmann, Angew. Chem. Int. Ed. Engl. 26, 846 (1987)

k shows the phase with which the orbitals are combined:

k = 0: )2()()()0exp( 1110 aaaj ssj

s

k = : )3()2()()()exp( 11110 aaaajji sssj

s πa

a a a a

k is a symmetry label and a node counter, and also represents electron momentum

Bloch’s theorem: consequences

kGkGk krGrkr nn uiiui ~)exp()]exp()[exp(

a Bloch state at k+G with index n

a Bloch state at k with a different index n’

kkk nnnh ˆ

In a periodic system, the solutions of the Schrödinger equations are

characterized by an integer number n (called band index) and a vector k:

For any reciprocal lattice vector

332211 gggG nnn

a lattice-periodic function u~

1g

2g

21 ggG

Can choose to consider only k within single primitive unit cell in reciprocal space

)()(),()exp()( rRrrkrr kkkk nnnn uuui

Brillouin zones

A conventional choice for the reciprocal lattice unit cell

For a square lattice

For a hexagonal lattice

Wigner-Seitz cell

Wigner-Seitz cell

In three dimensions:

Face-centered cubic (fcc) lattice

Body-centered cubic (bcc) lattice

Time-reversal symmetry

For Hermitian , can be chosen to be realh kn

)()(ˆ)()(ˆ ** rrrr kkkkkk nnnnnn hh

From Bloch’s theorem:

)()exp()()()exp()( ** rkRRrrkRRr kkkk nnnn ii

)(*

kk nn kk nn )(

Electronic states at k and –k are at least doubly degenerate

(in the absence of magnetic field)

Electronic band structure

kn

k0 π/a-π/a

For a periodic (infinite) crystal, there is an infinite number

of states for each band index n, differing by the value of k

Band structure represents dependence of on k

)(k

Electronic band structure in three dimensions

z

ΓΛ

LU

XWK

ΔΣ yx

Brillouin zone of the fcc lattice

By convention, are measured (angular-resolved photoemission spectroscopy, ARPES) and calculated along lines in k-space connecting points of high symmetry

kn

ε n(k

), e

V

Al band structure (DFT-PBE)

Finite k-point mesh

kk nn , – smooth functions of k can use a finite mesh, and then interpolate and/or use perturbation theory to calculate integrals

Charge densities and other quantities are represented by Brillouin zone integrals:

occ

BZ

32

BZ

)()(j

j

kdn rr k 8x4 Monkhorst-Pack grid

occ

1

2kpt

)()(j

N

mjm m

wn rr k

xk

yk1b

2b

H.J. Monkhorst and J.D. Pack, Phys. Rev.B 13, 5188 (1976); Phys. Rev. B 16, 1748 (1977)

Band gap and band width (dispersion)

a a a …0.8 Å – hydrogen molecule chain(DFT-PBE)

a = 10.0 Å

ε(k

)

a = 5.0 Å

ε(k

)

0a

a

0

a

a

a = 2.0 Å

ε(k

)

a = 1.6 Å

ε(k

)

Overlap between interacting orbitals determines band gap and band width

Band structure – test example

Adapted from: Roald Hoffmann, Angew. Chem. Int. Ed. Engl. 26, 846 (1987)

Orbital energies are smooth functions of k

Example: chain of Pt-L4 complexes (K2[Pt(CN)4])

a

z

xy

xz

z2k

k = π/ay

xz

π/a0

z2

xy

xz,yz

x2-y2

ε(k

)

z

yz

x

Insulators, semiconductors, and metals

Eg>>kBT

Insulators (MgO, NaCl,ZnO,…)

Eg~kBT

Semiconductors (Si, Ge,…)

Eg=0

Metals (Cu, Al, Fe,…)

kk

εF

In a metal, some (at least one) energy bands are only partially occupied

The Fermi energy εF separates the highest occupied states from lowest unoccupied

Plotting the relation

The grid used in k-space must be sufficiently fine to accurately sample the Fermi surface

Fermi surface

in reciprocal space for different n yields different parts of the Fermi surface

F)( kn

For free electrons, Fermi surface is a sphereF

22

2

em

k

K Cu Al

Periodic table of Fermi surfaces: http://www.phys.ufl.edu/fermisurface/

Density Of States (DOS)

n

N

jjnj

nn wkdg

kpt

BZ 1

3 ))(())(()( kk

Number of states in energy interval dε per unit volume,

d

d1

1

Atom-Projected Density Of States (APDOS)

Decomposition of DOS into contributions from different atomic functions :i

nnnii kdrdg

BZ

323 ))(()()()( krr k

Recovery of the chemical interpretation in terms of orbitals

Qualitative analysis tool; ambiguities must be resolved by truncating

the r-integral or by Löwdin orthogonalization of i

O(2s)

O(2p)

Mg(3s)Mg(3p)Mg(3d)

O(3d)

Potential of an array of point charges

To

tal p

ote

nti

al at

r=(0

.05,0

.05,0

.05)

Å,

eV

Number of point charges

…

0,)(11

N

ii

N

i i

i qq

VR Rrr

r

Convergence of the potential with number of charges is extremely slow

+ –

Ewald summation

+ ≡

screening gaussian charge distribution

R Rrr

rN

i i

iqV1

)(

(Poisson's equation)

R Rrr

Rrrr

,1

/erfc)(

i i

iiqV

Decays fast with R

)(4)(2 rr V

22 /)(exp1

irr

0G

rrGG

Gr

,

22

22 )(4

exp4

)(i

ii iqV

Decays fast with G

Diverges at , but divergence

is cancelled for

0G

0i

iq

There is no universal potential energy reference (like vacuum level) for periodic systems – important when comparing different systems

Modeling surfaces, interfaces, and point defects –the supercell approach

The supercell approach

Can we benefit from periodic modeling of non-periodic systems?

Yes, for interfaces (surfaces) and wires (also with adsorbates), and defects (especially for concentration or coverage dependences)

Supercell approach to surfaces (slab model)

supercell

• Approach accounts for the lateral periodicity

• Sufficiently broad vacuum region to decouple the slabs

• Sufficient slab thickness to mimic semi-infinitecrystal

• Semiconductors: saturate dangling bonds on theback surface

• Non-equivalent surfaces: use dipole correction

• Alternative: cluster models (for defects and adsorbates)

Surface band structureExample: fcc crystal, (111) surface

ΓΓ K

M

M

surface Brillouin zone

kz kzkz

Surface band structure of Cu(111)

-9

-8

-7

-6

-5

-4

-3

-2

-1

εF

1

2

En

erg

y, e

V

Cu(111)

Shockley surface state

Tamm surface state

Γ MM

Shockley surface states

For near-free electrons:

]exp[~ z ])(exp[~ ziki

z0

)exp(~)( rkrk i

matching condition

potential

Decaying states can be treated as Bloch states with complex k (W. Kohn, Phys. Rev., 115, 809 (1959))

Complex band structures can give useful information about conductance through interfaces and molecular junctions

Tamm surface states

In the tight-binding (localized orbital) picture, surface states may appear due to ‘dangling orbitals’ split off from the band edge

Surface reconstruction and band structure

Dimerization at (001)-surface of group IV-elements

[001]

side view

bulk-terminated atomic structure

top view

[110]

[110]

side view

Surface reconstruction and band structure

Buckling of dimers at Si (100) surface

π-bond re-hybridization and charge transfer (from down to up)

see, e.g., J. Dabrowski and M. Scheffler,Appl. Surf. Sci. 56-58, 15 (1992)

Surface reconstruction and band structure

symmetric dimer model (SDM)

asymmetric dimer model (ADM)

Experimental results from angular-resolved photo-emission spectroscopy

total density contour plot

contour plot of electron density difference with respect to free Si atoms (dashed = decrease)

P. Krüger & J. Pollmann, Phys. Rev. Lett. 74, 1155 (1995)

Electron-repulsion integrals in periodic systems

Standard DFT and the self-interaction error

][)()(

2

1

2

1)(][ XC

33

1 1

3

1tot nErrdd

nnZZrd

nZnTE

M

I

M

J JI

JIM

I I

I

rr

rr

RRRr

r

)(rn -- electron density

LDA, GGA, meta-GGA: ][][][ locC

locXXC nEnEnE

Standard DFT: (Semi)local XC operator low computational cost

(includes self-interaction)

exchange-correlation (XC) energy

Removing self-interaction + preserving fundamental properties (e.g., invariance with respect to subspace rotations) is non-trivial residual self-interaction (error) in standard DFT

Consequences of self-interaction (no cancellation of errors): localization/delocalization errors, incorrect level alignment (charge transfer, reactivity, etc.)

Hybrid DFT

][][)α1(}][{α}][{ locC

locX

HFXXC nEnEEE

-- easy in Kohn-Sham formalism ( )n

nnfn2

Perdew, Ernzerhof, Burke (J. Chem. Phys. 105, 9982 (1996)): α = 1/N

MP4 N = 4, but “An ideal hybrid would be sophisticated enough to optimize N for each system and property.”

PBEC

PBEX

HFX

PBE0XC )α1(α EEEE

Hartree-Fock exchange – the problem

rrddDDE

ljki

lkji

jkil33

,,,

HFX

)()()()(

2

1

rr

rrrr

)(ri)(rk

)'(rj

)'(rl

rr

1

electron repulsion integrals

Lots of integrals, naïve implementation N4 scaling (storage impractical for N > 500 basis functions)

• need fast evaluation• need efficient use of sparsity (screening)

“Resolution of identity” (RI) (density fitting)

rrddklij

lkji 33)()()()()|(

rr

rrrr

μ

μμ )()()( rrr PCijji

μ

μμ )()()( rrr PCkllk

independent auxiliary basis

Basis-pair space is overcomplete, since approaches completeness size of ~4-5 times size of

)}({ ri)}({ μ rP )}({ ri

“Resolution of identity” (RI) (density fitting)

rrddklij

lkji 33)()()()()|(

rr

rrrr

μ

μμ )()()( rrr PCijji

μ

μμ )()()( rrr PCkllk

independent auxiliary basis

Whitten, J. Chem. Phys. 58, 4496 (1973); Dunlap, Rösch, and Trickey, Mol. Phys. 108, 3167 (2010); Dunlap, J. Mol. Structure: THEOCHEM 501-502, 221 (2000)

Finding the RI coefficients: RI-SVS

νμ,

νμν

μ,

μ

μμ )|()()()( klijklijijji CVCIklijPC rrr

rdPC jiij

C

3

2

μ

μμ )()()(min rrr Minimize the error norm:

rdPOOSVSOI jiijklijklij3

νν

ν

μ1μσσσ

1νσ

ν, )()()(,

rrr

Number of molecule in the S22 setErro

r in

to

tal e

ner

gy (

meV

)

0

60RI-SVS

Finding better RI coefficients

rdklij

klijjiij

3

||

)(ρ)(ρ)|(),()()(ρ

rr

rrrrr

Let us calculate the error in (ij|ij) – the largest integrals:

)δρ(||

)()(δρ

2)|(δ 233ν

νν

ij

ijij

rrdd

PC

ijij

rr

rr

first-order correction does not vanish in RI-SVS!

Is it possible to do better without increasing the size of the auxiliary basis?

Finding better RI coefficients: RI-V

Set the first-order correction to zero, minimize second-order:

rrddP

QQVQIji

ijklijklij

33νν

ν

μ1-νμ

ν,

||

)()()(,

rr

rrr

Number of molecule in the S22 set

Erro

r in

to

tal e

ner

gy (

meV

)

0

60

0)δρ(min)δρ( 2 ijij

RI-SVS

RI-V

Finding better RI coefficients: RI-V

rrddP

Qji

ij

33νν

||

)()()(

rr

rrr rdPO jiij3

νν )()()( rrr

Distant aux. functions contribute to the integrals a lot of memoryand operations scaling N3

Localized RI-V (RI-LVL, Jürgen Wieferink)

0)δρ(min)δρ( 2 ijij

)atom(or )atom(μ,)()()(μ

μμ jiPCijji rrr

νσ1

νσ

σσ μ

μμσσσ

ν

νσν

, )(,

VLQLVLQI klijklij

local inverse Coulomb matrix

Number of molecule in the S22 set

Erro

r in

to

tal e

ner

gy (

meV

)

0

60RI-SVS

RI-V RI-LVL

Hartree-Fock exchange in extended systems

',,

33*

''*

HFX

),(),(),(),(

2

1

kk

kkkk

rr

rrrr

nm

mnnm ddrrddE

rrdd

DDE

ljki

lkji

jkil

33

,,,,,

HFX

)()()()(

)()(2

1

rr

RRrRrRrr

RRRR

RRR

R

kRkk Rrr

,

e)()(),(i

iimim cs

Electron repulsion integrals in periodic systems

)(ri

)( Rr j

Q Q

QRQQR

QRRRRR

μ ν

ν,)(μν

μ,, lkjilkij CVCI

RI-LVL: Sparse, easier to calculate, linear scaling is possible!

RI-V is impractical for extended systems

)(μ Qr P

0μ, QRjiC

Hybrid functionals -- scaling with system size

Periodic GaAs, HSE06 hybrid functional

Hybrid functionals -- CPU scaling

Periodic GaAs, HSE06 hybrid functional

Concluding remarks

1) Periodic models can be efficiently used to study concentration/coverage dependence, including infinitely dilute limit (low-dimensional systems, defects, etc.)

2) A lot of useful and experimentally testable information on material’s properties can be obtained from the analysis of its electronic structure (band structure, DOS, APDOS, etc.)

3) A lot of development (in both computational methods and code efficiency) is still necessary to go beyond standard DFT for periodic systems, and to approach accuracy that can be achieved nowadays for molecules

Recommended literature

1) “How Chemistry and Physics Meet in the Solid State”, Angew. Chem. Int. Ed. Engl. 26, 846-878 (1987)

Neil W. Ashcroft and N. David Mermin, “Solid state physics”

Roald Hoffmann (1981 Noble Prize in Chemistry (shared with Kenichi Fukui)):

2) “A chemical and theoretical way to look at bonding on surfaces”, Reviews of modern physics, 60, 601-628 (1988)

Axel Groß, “Theoretical surface science: A microscopic perspective”

Surface modeling: important issues

1) Finite slab thickness (surface-surface interaction)

2) Finite vacuum layer thickness (image-image interactions)

3) Long-range interactions (charge, dipole moment)

4) Surface polarity

- +- +

- +- +

- +- +

- +- +

…

φperiodic boundary conditions

artificial electric field

- +- +

- +- +

- +- +

- +- +

…

φ

Li (+1)

Nb (+5)

O (-2)+q -q +q -q+q -q +q -q +q -q +q -q

Δφ

From molecules to solids

R. Hoffmann, Solids and Surfaces - A chemist’s view of bonding in extended structures, VCH Publishers, 1998

Electronic bands as limit of bonding and anti-bonding combinations ofatomic orbitals:

bands band gap

Shockley surface states

For nearly-free electrons:

)( k

k0

a

GV2

gap

i

]exp[~ z ])(exp[~ ziki

z0