HUMBOLDT-UNIVERSITÄT ZU BERLIN

A chemist‘s view on the electronicstructure of solids

Joachim SauerInstitut für Chemie

International Max Planck Research School, "Surfaces" Oct 2004

Literature(1) R. Hoffmann, "Solids and surfaces", VCH, Weinheim, 1988.

ISBN 3-527-26905-3 VCH Verlagsgesellschaft(2) A. P. Sutton,"The electronic structure of materials, Oxford University

Press, 1993.German edition: VCH, 1966 ISBN 3-527-29395-7

(3) R. A. van Santen, "Theoretical Heterogeneous Catalysis", WorldScientific, Singapore 1991.

What we would like to understand

Band structure Density of states

Brillouin zone

N-membered Polyene Chain

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Ek = α + 2β ⋅ cos2π kN k = 0, ± 1, ± 2,KN

2n even

k = 0, ± 1, ± 2,K ± N − 12

n uneven

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ψ± k = c jj=1

N∑ ⋅ χ j = exp ±2πi k

N ⋅ j

j=1

N∑ ⋅ χj

Model for 1-dim periodic systemwith one orbital and one electron per cell

N-membered Polyene Chain - Real orbitals

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ψ 0 =1N

1⋅j=1

N∑ χj

ψ k ,1 = 2N

j=1

N∑ cos 2π k

N ⋅ j

⋅ χ j

ψ k ,2 = 2N

j=1

N∑ sin 2π k

N ⋅ j

⋅ χj

ψN/ 2 =1N

( -1)j ⋅j=1

N∑ χj

HMO solution for cyclobutadiene (N=4)

2β

-2β

HMO solution for benzene (N=6)

Orbital energies for cyclic systems as a function of N

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Ek = α + 2β ⋅ cos2π kN k = 0, ± 1, ± 2,KN

2 n even

Orbital energy as a function of parameter k - band structure

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2π kN

k

one orbital per unit cell

Illustration of cj along infinite structure for orbitals with different k

lowest energyk=0

k=±N/2

k= N/12

highest energy

Density of states

How do the bands run ?

For chains of p orbitals along the periodic direction the crystalorbital for k=0 has the maximum number of nodal planes

is therefore highest in energyforms the top of the band

The orbital for k=N/2 has no nodal planes across the bonds (onlywithin the atom)

forms therefore the bottom of the band

p and d bands

Note:Increase or decrease with increasing k is determined by nodal behaviourBand width is determined by strength of interaction (magnitude of β)

Band dispersion (band width)

Difference between the highest point and the lowest point of a bandFor our model: 4 βMagnitude of β depends on strength of interaction

Reciprocal spacelattice vector (cell size)

k is vector of reciprocal spacek-space

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a =2πN

shifts by one cell

aj =2πN

⋅ j = a ⋅ j shifts by j cells

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ψk = e− i⋅2π k

Njχ jj=1

N∑ = e− i⋅a j ⋅kχ jj=1

N∑

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a ⋅b = 2π (ai ⋅bj = 2π ⋅δij)

defines reciprocal cellBRILLOIN zone

includes permitted k values0 ... N; - N/2 ....+ N/2

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b = 2π /a =N

Two dimensions

crystal lattice(direct space)

reciprocal lattice(k space)

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−πa1/2

≤ kx/y ≤ +πa1/2

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a1 ⋅b1 = 2π; a2 ⋅b2 = 2π

band structurecorresponds todifferent planesthrough k space

E(k)=α + 2β (cos kxa+cosky)€

−πa≤ kx,ky ≤ +

πa

4β

2β

-4β

Different nodal behaviour depending onminimum/maximum k value in one, the other, two directions

FERMI surface

includes all k values in the Brillouin zone whichbelong to occupied states

E(k)=α + 2β (cos kxa+coskya)

only a few states occupied(lowest energy for k=0)

half of the states occupiedEFermi(k)= αcos kxa+coskya=0

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kx,ky = ±πa

Double unit cell - two solutions per k valueonly half of the k values - "folding" of band structure

We use the double cell to study the effect of distortion

bonding interaction becomesstronger - stabilization

antibonding interaction becomesstronger - destabilization

Alternation of long (weak - small b)and short (strong - large b) bondsleads to a splitting for large k values

Peierls Distortion

2 of the 4 electrons are indegenerate orbitals. Distortionof symmetry lifts degeneracy.The 2 electrons occupy thestabilized orbital while thedestabilized orbital remainsempty - this is the drivingforce for the distortion.

Cf. Jahn-Teller Effect

Orbitalschema -CO

3σ

4σ

5σ

6σ

Dewar-Chatt-Duncanson-Modell*(Oberflächen: Blyholder-Modell)

bes. 5σ-Orbital: CO → M donationunbes. 2π*-Orb. CO ← M back don.

*Chatt-Duncanson, JCS 1953, 2929:C2H4-PtCl2

Festkörper: Bänder von Orbitalenergien ersetzen die diskreten Orbitalenergien von Molekülend-Bänder sind schmal - geringe Überlappung

Orbitalmischung

5σ - CO

2π* - CO

M - dz2

M - dxz dyz

R. Hoffmann, Solids & Surfaces, VCH 1988.

Mixing: dxz, dyz,and 2π*

Mixing: dz2 and 5σ

Rh6CO16

Diamond structure Si and C

• Tetrahedral coordination• localized bonds built from sp3 hybrid orbitals• model: one-dimensional chain of sp-hybrid orbitals

One-dimensional chain of sp-hybrid orbitals

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(2β cosL +αs −E)cs(L) − i2β 'sinL cp(L) = 0i2β 'sinL cs(L) (−2β"cosL +αp −E)cp(L) = 0L = 2kπ /N

Hückel equation for one s and one p orbital per cell€

β = βss α s

β '= βsp

β"= β pp α p

No interaction between s and p at neighboring sites:2 separate bands, s and p, as we know already

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(2β cosL +αs −E)cs(L) = 0(−2β"cosL +αs −E)cp(L) = 0

General solution for s-p-interaction

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E1/2 = 12 (αs + αp)+ 2(β −β")cosL

± 12 αs -αp + 2(β + β")cosL[ ]

2+16β '2 sin2L

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β = β '= β" αs =αp€

E1/2 =α ± 2β

Ideal hybridisation

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4β