Electrons on surfacesDensity functional theory, surface states and electron correlations

O.PankratovTheoretische Festkörperphysik

1

What is special about a surface?

2

Solid state “dogmas”

• Adiabatic approximationseparation of electrons and phonons

• Electron correlations are weakmean field description: “nearly freeelectrons”

Surface electrons

• Strong electron-phonon couplingrelaxation & reconstruction,surface polarons

• Strong correlation effectssurface Hubbard insulatorssurface excitonssurface magnetismetc

The outline

• Surface electron states• Density functional theory

Background: homogeneous and inhomogeneous electron gasFundamentals: interacting vs. non-interacting particlesWeak electron correlations: LDAStrong correlations: Mott-Hubbard insulator

• Electron-phonon interaction on GaAs (110) and Si (100)• Mott-Hubbard correlations on SiC (1000)

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Surface states

Tamm (1932) and Shockley (1939) states

4

Surface states are very model-sensitive!

Surface states

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Bulk states

Bulk states

Ener

gy

2D electron gas on Cu (111)

6

fcc Cu (111) surface

R.Courths et al m*=0.46m

4H-SiC (0001) Si-terminated surface

Si

C

7

Electron correlations on SiC(0001)

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Strong on-site correlation: surface insulator

Rohlfing et al : U ≈2eV

Weak correlations: metallic surface

Anisimov et al, Bechstedt et al…

Strong electron-lattice coupling on GaAs(110)

GaAs

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The theory should be able to…

• be material-specific (metals, semiconductors etc)• describe surface relaxation & reconstruction• allow calculations for a large collections of atoms• describe electron-lattice coupling• correctly describe electron states and excitations• include effects of electron-electron correlations

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Density Functional Theory

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The 1998 Nobel Prize in Chemistry in the area of quantum chemistry to Walter Kohn, University of California at Santa Barbara, USA and John A. Pople, Northwestern University, Evanston, Illinois, USA.

Citation: "to Walter Kohn for his development of the density-functional theory and to John Pople for his development of computational methods in quantum chemistry."

Electron density n(r)

Electron density and electrostatic potential in the amino acidcystein molecule.

The time arrow: from H atom to DFT

1926 Schrödinger equation: wave function

1927 Thomas - 1928 Fermi approximation: electron density

1928 Hartree - 1930 Fock approximation: self-consistent field

Configuration Interaction (CI) methods

1930 Dirac: local approximation for exchange energy

1964 Hohenberg - Kohn theorems: foundation of Density

Functional Theory (DFT)

1965 Kohn - Sham Equations of DFT

1998 Kohn, Pople: Nobel Prize

Density Functional Theory: background

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Gar Manches rechnet Erwin schonMit seiner Wellenfunktion.Nur wissen möch’t man gerne wohlWas man sich dabei vorstell’n soll.

2322 |),...,,,(|...)( NNddNn rrrrrrr Ψ= ∫

Density Functional Theory: background

Homogeneous electron gas

ε(k)=ħ2k2/2m

εF

kN particlesElectron density n=const

on a positive background

14

Fermi energy 3/222

)3(2

nmF πε h

= Total (kinetic) energy NE Fε53

=

Electron’s pressure VEP /32

= Typically andeVF 61−≈ε MbarP 1≈

Density functional theory: backgroundHomogeneous electron gas : Coulomb interaction

Exchange-correlation hole Correlation function

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⇒−

= ∑≠ ji ji

CeE

||1

2

2

rr0)(

2

2

<= ∫ rdr

rnNeE xcxc

Exchange-correlation energy

Density functional theory: background

Homogeneous electron gas: Coulomb interaction

16is a distance between electrons:

nar Bs

1)(34 3 =πBsar

Total energy density VEE xckin /)( +

25.3=Lisr

.104uaE−⋅

Exchange-correlation energy NExc /

Density Functional Theory: background

Inhomogeneous electron gas: kinetic energy

+

17

+

External potential )(rextv : quantization Inhomogeneous density

)()(53 rrr ndE Fkin ε∫≈

Density Functional Theory: background

Inhomogeneous electron gas: Coulomb energy

+

18

+

Exact Coulomb energy Hartree energy

rrrrrr ′′−′

= ∫ ddnneEH ||)()(

2

2

∑≠ −

=ji ji

CeE

||1

2

2

rr

xcHC EEE +=

Density Functional Theory: backgroundThomas-Fermi approximation

)()()(||)()(

2)()(

53)]([

2

rrrrrrrrrrrrrrr nvddddnnendnE extxcF ∫∫∫∫ ++′′−′

+= εε

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)()()(][ rr

r SF vn

nE+== ε

δδµextxcHartreekin EEEEnE +++=)]([ r

const=µ

)(

)(

rv

r

S

Fε)()()()( rrrr extxcHS vvvv ++=

)(rnv xc

xc δδε

= Exchange-correlation potential

Density Functional Theory: background

µε =++ )())(())(( rrr extHF vnvn

Thomas-Fermi equation Thomas-Fermi screening

)/exp()( 0 TFLxnxn −=)]()([4)( 2 rrr extF nne +=∆ πε

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FTF

neL ε

π 2261

=⎟⎟⎠

⎞⎜⎜⎝

⎛

const=µ

• No quantization• No sharp size of atoms• No molecules (e.g. H2)

Problems with Thomas-Fermi theory)(rFε

)(rvtot

Density Functional Theory: fundamentals

Hohenberg-Kohn theorems

I. determines the quantum state:

II. results from

,...),()()( 2rrrr 1Ψ⇒⇒ extvn

)()( 0 rr nn =

)(rn

)]([)]([min 0)(rr nEnE

rn=

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What is the energy functional )]([ rnE ?

extxcHartreekin EEEEnE +++=)]([ r

Density Functional Theory: fundamentals

Kohn-Sham energy functional

)()()(||)()(

2|

2|)]([

2

1

22rrrrrrr

rrrrr xcexti

N

ii dnvdddnne

mnE εϕϕ ∫∫∫∑ ++′

′−′

+⟩∇−

⟨==

h

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)(riϕAuxiliary one-particle Kohn-Sham wave functions

∑=

=N

iin

1

2|)(|)( rr ϕ and giveallow to represent any density

a much better approximation to kinetic energy than TF-method

Density Functional Theory: fundamentals

Variation of the Kohn-Sham functional ii

nE εδϕδ

=)(][

r

leads to Kohn-Sham equations for )(riϕ and parameters iε

Nivm iiiS ÷==⎥

⎦

⎤⎢⎣

⎡+

∇− 1,)()()(2

22

rrr ϕεϕh

not a many-body problem anymore!

)()()()( rrrr xcextHS vvvv ++=

)()(

rnv xc

xc δδε

=r

where the Kohn-Sham potential is

the exchange-correlation potential is the only unknown

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Weak correlations: LDALocal density approximation for xc energy and potential

))(()]([ rrr ndnE hxcxc ε∫= dn

dn

vhxc

hxc

xcε

δδε

==)(

)(r

rand

Correlations are assumed the same as in a homogeneous electron gas

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xc-hole is assumed to be spherical)(rxcn

rrrr

rrr ′′−

′−= ∫∫ ddnneE xc

xc |||)(|)(

2

2

LDA is not perfect in inhomogeneous system

The xc-hole is not spherical: ),(|)(| rrrr ′⇒′− xcxc nn

Correlations are not the same as in a homogeneous electron gas!

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rrrr

rrr ′′−

′= ∫∫ ddnneE xc

xc ||),()(

2

2

LDA: a success story (simple example)

GaAs lattice constant

a

Ga

As

26

Exp

LDA: a success story (protein’s structure)

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What if electron correlations are strong? Homogeneous electron gas:

Coulomb interaction versus kinetic energy

is a distance between electrons:n

d 134 3=πdar Bs =

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SBF

rad

nme

mnnede

=≈≈≈ 3/12

2

3/22

3/122

//

hhεCoulomb interaction energy

kinetic energy:

Wigner crystal

3D: rs~ 672D: rs~ 371D: rs~ 1-5

Metal or insulator?

Allowed states

Allowed states

Energy gap

εF

Allowed states

Allowed states

Energy gap

Ener

gy

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Correlations in a localized state

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Correlation gap:on-site repulsion U

0ε

U+0ε

1

2

ε

ε

“Band structure” gap:no e-e interaction

Mott-Hubbard insulator

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t

Mott-Hubbard parameter U/tU/t >>1 Insulator

U/t <<1 Metal

Lower Hubbard band

Upper Hubbard band

U

0ε

LDA does not reproduce Hubbard bands

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)1(21)]([ −= nUnnE LDA r

0

0

ε

ε U+

LDAε

Average occupation number: n

Fε LDA potential

)21(][

−== nUdn

ndEvLDA

LDA energy level

)21(0 −+= nULDA εε

How to repair LDA: LDA +U

occupation numbers of two (e.g. spin) states: nnn =+ 21

U jji

iLDAULDA nnUnUnnEnnE ∑

≠

+ +−−=21)1(

21][],[ 21

LDAε

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LDA+U potential

21,0,)

21( UvvnnUv

dndEv LDA

iiiLDA

i

ULDA

i ±=⇒=−+==+

LDA+U energy levels:2ULDA

i ±= εε Upper and lower Hubbard states!