Electronic structure of solids

Sergey V. Levchenko

Fritz Haber Institute of the Max Planck Society, Berlin, Germany

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 cells 12 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 of

atomic orbitals:

electronic

band

band

gap

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

Bloch’s theorem

Rr

r

R

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 ss

j

s

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

j

s π a

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 real h 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

Γ Λ

L U

X W K

Δ Σ y

x

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

εn(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

m

jm mwn rr k

xk

yk 1b

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

)

0 a

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

Insulators, semiconductors, and metals

Eg>>kBT

Insulators (MgO, NaCl,

ZnO,…)

Eg~kBT

Semiconductors (Si,

Ge,…)

Eg=0

Metals (Cu, Al, Fe,…)

k k

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

Density Of States (DOS)

n

N

j

jnj

n

n wkdgkpt

BZ1

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

n

nnii kdrdgBZ

32

3 ))(()()()( 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)

Periodic and cluster models of defects

Embedded cluster model Periodic model

+ Higher-level ab initio methods

can be applied

+/- Defects in dilute limit

- Effect of embedding on the

electronic structure and Fermi

level – ?

+ Robust boundary conditions

+ Higher defect concentrations

+/- Higher defect concentrations

- Artificial defect-defect

interactions

DFT and wave-function methods for solids

})({extiV r

Ground-state electronic structure problem

}{},{ ii r – many-body wave function, depends on spatial ( )

and spin ( ) coordinates of particles (also on nuclear

coordinates ( ) and )

iri

JR

}{},{

}{},{})({1

2

1 ext

2

2

ii

ii

i i J ji

i

jiJi

J

i

E

VZ

r

rrrrRrr

• already include approximations (Born-Oppenheimer,

non-relativistic, no spin-orbit coupling, no magnetic field)

• wave function depends on 4N variables (spatial + spin)

• electrons interact via Coulomb forces

Non-interacting fermions

i i J ji

i

jiJi

J

i

VZ

H })({1

2

1ˆ ext

2

2

rrrRrr

i

ii vV )(})({extrr

),()()(2

12

2

rrrRrr

nnn

J J

Jv

Z

),(,),,(det!

1,, 11111 NNNNN

N rrrr

one-particle states

Non-interacting fermions – periodic system

)()()(2

12

2

rrrRRrr

kkk

R

nnn

J J

Jv

Z

n – band index,

k – k-point

41 ,,,1 kkk n

41 ,,,2 kkk n

41 ,,,3 kkk n

Born-von Karman periodic boundary conditions

finite number of k-points, infinite (macroscopic) system as physical

limit

)()( Rrr kk nn

),(,),,(det!

1,, 11111 NNNNN

N rrrr

Interacting fermions (electrons)

i i J ji jiJi

J

i

ZH

rrRrr

1

2

1ˆ2

2

0ˆ

ˆmin

i

HH

i

Introduce basis set: ,

variational principle

j

jiji cs )()(),( rr 0ˆ

ijc

H

Approximations to the electronic problem: Basis set

Widely used basis sets:

gaussians (localized, analytic integrals)

plane waves (delocalized, analytic integrals)

Slater-type (localized, nuclear cusp)

grid-based (localized, analytic integrals)

)exp( 2rzyx kji

)exp( rk i

)exp( rzyx kji

)( irr

Core electrons are often treated separately

(pseudopotentials, plane-wave + localized basis)

Introduce basis set: , j

jiji cs )()(),( rr 0ˆ

ijc

H

k

kkkR

k

R rr )(e1

)( )(m

m

nmi

n UN

w

From reciprocal to real space

Wannier functions: not unique, can

be chosen more or less localized An arbitrary unitary

transformation within

occupied and virtual

subspaces

)()(det!

111 11 NNww

Nrr RR

Wannier functions in each

unit cell

The Hartree-Fock (HF) approximation

),(,),,(det!

1,, 11111 NNNNN

N rrrr

rrddnnZZ

hEM

I

M

J JI

JIN

n

nn

33

1 11

tot

)()(

2

1

2

1ˆrr

rr

RR

N

nm

mnnmddrrdd

1,

33** ),(),(),(),(

2

1

rr

rrrr

HF (exact) exchange energy

• No self-interaction

• Coulomb mean-field no dynamic correlation, single determinant

no static correlation

The Hartree-Fock (HF) approximation

Dynamic correlation:

versus

Coulomb

potential

Non-dynamic (static) correlation:

versus +

HF approximation ≥90% of total energy, overestimates ionicity

smaller e density, large r /

(quasi)degenerate HOMO-LUMO)

Density functional theory

Density functional theory: Hohenberg-Kohn theorem

rn

H NN rr ,,11

totE

– many-body Hamiltonian

– many-body wave function

– total energy

][)()(

2

1

2

1)(][ XC

33

1 1

3

1

tot nErrddnnZZ

rdn

ZnTEM

I

M

J JI

JIM

I I

I

rr

rr

RRRr

r

Approximations to : Local density approximation (LDA),

generalized gradient approximation (GGA), meta-GGA

][XC nE

Standard DFT and the self-interaction error

][)()(

2

1

2

1)(][ XC

33

1 1

3

1

tot nErrddnnZZ

rdn

ZnTEM

I

M

J JI

JIM

I I

I

rr

rr

RRRr

r

)(rn -- electron density

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

C

loc

XXC 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}][{ loc

C

loc

X

HF

XXC 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)

Hartree-Fock exchange in extended systems

',,

33*

''*

HFX

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

2

1

kk

kkkk

rr

rrrr

nm

mnnmddrrddE

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

Q

RQQRQRRRRR

μ ν

ν

,)(μνμ,, lkjilkij CVCI

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

RI-V is impractical for extended systems

)(μ Qr P

0μ, Q

RjiC

Q

QR QrRrr

μ

μμ, )()()( PC jiji

Hybrid functionals -- scaling with system size

Periodic GaAs, HSE06 hybrid functional

Hybrid functionals -- CPU scaling

Periodic GaAs, HSE06 hybrid functional

Beyond mean-field for extended systems

The concept of mixing excitations

iii EH 00ˆ – non-interacting effective particles (HF, DFT, etc.)

}{ i – a basis set for N-electron wave functions

j

j

ijij

j

ijj

j

iji cEcHHHc )]ˆˆ(ˆ[, 00

ikki

j

jkij cEEHc )(ˆ 0

Project onto equations for : k ijc

configuration interaction

,,, 210

Configuration interaction – matrix diagonalization

abij

abij

abij

ai

ai

ai cc

,,

0

},{},{,0abij

ai DS

00ˆ H0

0ˆ HSS

D

T

Q

0

0ˆ HD

0

0

S D T Q

SH0

SHS ˆ

SHD ˆ

SHT ˆ

0

DH0

DHS ˆ

DHD ˆ

DHT ˆ

DHQ ˆ

0 0

THS ˆ

THD ˆ

THT ˆ

THQ ˆ

0

QHD ˆ

QHT ˆ

QHQ ˆ

!)!(

!

nnM

M

n-tuple

excitations

Configuration interaction in periodic systems

n – band index,

k – k-point

41 ,,,1 kkk n

41 ,,,2 kkk n

41 ,,,3 kkk n

0

rqpk

rqpkrpqk

rqpk

qk

qkqk

qk

baji

bjai

baji

ai

ai

ai cc ,

,

,,

,,

,

0

excitations can change not only n but also k-point

momentum conservation

rqpk

qk

baji

ai H ,

,ˆ

RI is a must!

Hierarchies of GS wave-function methods

Møller-Plesset

perturbation theory

(MP2, MP3, MP4,…)

Coupled-cluster

(CCD,CCSD,CCSDT,…)

0)0()0(

0

2

0)2(

0

ˆ

i i

i

EE

HE

MPm: ~nNm+2

0

ˆ}{

0 e mi

iTm

95

100

CID

CISD

CISDT

CISDTQ

MP2 MP3

MP4 MP5 MP6

CCD

CCSD

CCSD(T) CCSDT

CCSDTQ

R.J. Bartlett, private communication

Truncated CI

(CISD, CISDT,…)

0}{

0ˆ

mi

im T

CI{m}: ~nmNm+2

90

85

80

75

70

CC{m}: ~nmNm+2

% c

orr

ela

tion e

nerg

y

Ripples in the charged Fermi liquid

0000ˆ EH VHH λˆˆ

00

λ0

1

tadiabatic switching

Long-range correlation (screening) contributions can be

summed up to infinite order (EX+cRPA, rPT2, GW, BSE)

- - - - - - - - -

- - - - -

+ + + + + + + + +

+ + + +

fluctuating charge density

polarization (screening)

i i

i

EE

VE

)0()0(0

2

0)2(0

λ

Random-phase approximation

Second-order energy correction (MP2, HF orbitals):

rr

rrrr )()()()()|(

**bjai

jbia

abij bajii i

i jaibjbiajbia

EE

VE

,)0()0(

0

2

0)2(0

εεεε

))|()|)((|(2

Random-phase approximation

correlation exchange

Second-order energy correction (MP2, HF orbitals):

rr

rrrr )()()()()|(

**bjai

jbia

abij bajii i

i jaibjbiajbia

EE

VE

,)0()0(

0

2

0)2(0

εεεε

))|()|)((|(2

Random-phase approximation

correlation exchange

Second-order energy correction (MP2, HF orbitals):

abij bajii i

i jaibjbiajbia

EE

VE

,)0()0(

0

2

0)2(0

εεεε

))|()|)((|(2

rr

rrrr )()()()()|(

**bjai

jbia

- - - - - - - - -

- - - - -

+ + + + + + + + +

+ + + + cE

kcjbia cakibaji

iakckcjbjbia

,, )εεεε)(εεε(ε

)|)(|)(|(

Ren, Rinke, Joas, and Scheffler, Invited Review, Mater. Sci. 47, 21 (2012)

Random-phase approximation and beyond

RPAcE

0

μν0μν0RPAc ω)}ω)(χ({Tr))}ω)(χ{1ln(det( dVVE

.c.c)εε(ω

)(ψ)(ψ)(ψ)(ψω),,(χ

occ unocc **

0

i a ia

iaai

i

rrrrrr

rPT2cE

Kohn-Sham polarizability:

Renormalized second-order perturbation theory (rPT2):

single excitations second-order screened exchange RPA

Conclusions

• Periodic models can be efficiently used to study

concentration/coverage dependence, including infinitely dilute limit

(low-dimensional systems, defects, etc.)

• Practically all beyond-DFT methods involve calculation of the

electron repulsion integrals (ERIs)

• ERIs can be evaluated efficiently by using resolution of identity

(density fitting) and taking into account sparsity

• Wave function methods slowly but steadily enter the arena of

condensed matter physics

• Exploration of the connection between wave-function methods

(coupled-cluster) and many-body perturbation theory (beyond RPA)

continues

Acknowledgements

Volker Blum

Patrick Rinke

Rainer Johanni

Jürgen Wieferink

Xinguo Ren

Igor Ying Zhang

Matthias Scheffler