An Introduction to Band Theory, A Molecular Orbital …woodward/ch...1 An Introduction to Band...

1

An Introduction to BandAn Introduction to BandTheory, A MolecularTheory, A MolecularOrbital ApproachOrbital Approach

Chemistry 754Chemistry 754Solid State ChemistrySolid State ChemistryDr. Patrick WoodwardDr. Patrick Woodward

Lectures #17-18Lectures #17-18

References � Band TheoryReferences � Band TheoryThe material for this lecture comes primarily from the work ofThe material for this lecture comes primarily from the work ofRoaldRoald Hoffmann and those who came through his group (Mike Hoffmann and those who came through his group (MikeWhangboWhangbo, Jeremy Burdett, Thomas Albright, Tim, Jeremy Burdett, Thomas Albright, Tim Hughbanks Hughbanks,,etc.). His treatment (and many of the figures used in thisetc.). His treatment (and many of the figures used in thislecture) are nicely summarized in a small book he wrote somelecture) are nicely summarized in a small book he wrote someyears ago:years ago:

�Solids and Surfaces: A chemists view of bonding�Solids and Surfaces: A chemists view of bondingin extended structures�in extended structures�

RoaldRoald Hoffmann, VCH Publishers, (1988) Hoffmann, VCH Publishers, (1988)

See alsoSee also�The Electronic Structure and Chemistry of Solids��The Electronic Structure and Chemistry of Solids�P.A. Cox, Oxford University Press, Oxford (1987).P.A. Cox, Oxford University Press, Oxford (1987).

�Chemical Bonding in Solids��Chemical Bonding in Solids�Jeremy K. Burdett, Oxford University Press, Oxford (1995).Jeremy K. Burdett, Oxford University Press, Oxford (1995).

2

Relevance of Band StructureRelevance of Band Structure

�� What is the electronic band structure?What is the electronic band structure?�� For an extended solid the band structure plays theFor an extended solid the band structure plays the

same role that an MO diagram does for a molecule.same role that an MO diagram does for a molecule.�� Why it is important to be able to comprehend theWhy it is important to be able to comprehend the

electronic band structure?electronic band structure?�� The band structure is the link between crystalThe band structure is the link between crystal

structure, bonding and properties. In particularstructure, bonding and properties. In particulardetails of the band structure are closely linked to thedetails of the band structure are closely linked to thefollowing properties:following properties:

�� Electronic conductivityElectronic conductivity�� Optical properties, including colorOptical properties, including color�� Electronically driven structural distortionsElectronically driven structural distortions�� Mechanical and Magnetic PropertiesMechanical and Magnetic Properties�� Catalytic ActivityCatalytic Activity

MO Diagram ReOMO Diagram ReO666-6- Octahedron Octahedron

In the last lecture we discussed theIn the last lecture we discussed theMO diagram of an octahedronMO diagram of an octahedron

(shown to the left). How does this(shown to the left). How does thisdiagram change when we have andiagram change when we have an

extended structure?extended structure?

3

Band StructureBand Structure((aka aka Spaghetti diagram): MOSpaghetti diagram): MO

diagram with diagram with translationaltranslationalsymmetry taken into account.symmetry taken into account.

Density of States (DOS)Density of States (DOS)Integration of the bandIntegration of the band

structure. Shows the # ofstructure. Shows the # ofavailable levels between E andavailable levels between E and

E+E+dE dE as as dE dE →→ 0. 0.

Band Structure ReOBand Structure ReO33

Constructing a DOS Diagram: TiOConstructing a DOS Diagram: TiO22

It�s possible to construct a reasonableIt�s possible to construct a reasonableapproximation of the DOS diagramapproximation of the DOS diagram

from the MO diagram of the buildingfrom the MO diagram of the buildingblock.block.

The energy levels of each block ofThe energy levels of each block ofbands or �BOB� comes from the MObands or �BOB� comes from the MOdiagram (based on diagram (based on electronegativityelectronegativity

and bonding interactions)and bonding interactions)

The area of each BOB is proportionalThe area of each BOB is proportionalto the number of MO�s at thatto the number of MO�s at that

approximate energy.approximate energy.

The width of each BOB is derived fromThe width of each BOB is derived fromthe overlap between building blocksthe overlap between building blocks

4

Constructing a DOS Diagram: TiOConstructing a DOS Diagram: TiO22

The DOS plots for TiOThe DOS plots for TiO22 ((rutilerutile), SrTiO), SrTiO3 3 (perovskite), CaTiO(perovskite), CaTiO33 (cubic perovskite) (cubic perovskite)

and MgTiOand MgTiO33 ( (ilmeniteilmenite) are all going to be qualitatively similar, because the) are all going to be qualitatively similar, because theelectronic building block is the same in each case, the TiOelectronic building block is the same in each case, the TiO66 octahedron. The octahedron. The

electropositive cations: such as Srelectropositive cations: such as Sr2+2+, Ca, Ca2+2+ and Mg and Mg2+2+ only make minor only make minorcontributions near the contributions near the Fermi Fermi energy. The main difference will be the width ofenergy. The main difference will be the width of

the bands (as the octahedra are connected differently in each structure).the bands (as the octahedra are connected differently in each structure).

O 2sO 2s

O 2p,O 2p,Ti-O bondingTi-O bonding

Ti tTi t2g2g

Ti-O Ti-O ππππππππ**

Ti Ti eegg

Ti-O Ti-O σσσσσσσσ**

EEff

Chain of 5 H atomsChain of 5 H atoms

χχχχ0000 χχχχ1111 χχχχ2222 χχχχ3333 χχχχ4444

HH22 Molecule Molecule

χχχχ1111 χχχχ2222

EE

Bonding

Antibonding

Nonbonding

00

11

22

33

44

# of Nodes

5

Infinite 1D Chain of H atomsInfinite 1D Chain of H atomsIf there are N atoms in the chainIf there are N atoms in the chainthere will be N energy levels andthere will be N energy levels andN electronic states ("MOs"). TheN electronic states ("MOs"). Thewavefunctionwavefunction for each electronic for each electronicstate is:state is:

ΨΨkk = = ΣΣ eeiknaiknaχχnn

Where:Where:��aa is the lattice constant (spacing is the lattice constant (spacingbetween H atoms),between H atoms),��nn identifies the individual atoms identifies the individual atomswithin the chain,within the chain,�� χχχχχχχχnn represents the atomic represents the atomic orbitalsorbitals��kk is a quantum # that identifies is a quantum # that identifiesthe the wavefunctionwavefunction and tells us the and tells us thephase of the phase of the orbitalsorbitals..

k=0


k=π/a

k=π/2a

a

Infinite 1D Chain of H atomsInfinite 1D Chain of H atomsk = ππππ/a

ΨΨπ/a = = χχ00+(exp{+(exp{iiππ})})χχ1 1 +(exp{i2+(exp{i2ππ})})χχ22

+(exp{i3+(exp{i3ππ})})χχ33+(exp{i4+(exp{i4ππ})})χχ44+�+�

ΨΨπ/2a = = χχ0 0 - - χχ1 1 + + χχ2 2 - - χχ3 3 + + χχ4 4 +�+�

k = ππππ/2aΨΨπ/2a = = χχ00+(exp{+(exp{iiππ/2})/2})χχ1 1 +(exp{+(exp{iiππ})})χχ22

+(exp{i3+(exp{i3ππ/2})/2})χχ33+(exp{i2+(exp{i2ππ})})χχ44+�+�

ΨΨπ/2a = = χχ0 0 + 0 - + 0 - χχ2 2 + 0 + + 0 + χχ4 4 +�+�

k = 0k = 0ΨΨ00 = = χχ00++χχ1 1 ++χχ2 2 ++χχ3 3 ++χχ4 4 +�+�

k=0


k=π/a

k=π/2a

a

k=0 k=0 →→→→→→→→ orbital phase does not change when we translate byorbital phase does not change when we translate by a ak=k=ππππππππ/a /a →→→→→→→→ orbital phase reverses when we translate byorbital phase reverses when we translate by a a

6

BandBandStructureStructure

Linear H ChainLinear H Chain

��The The FermiFermi energy separates the filled states (E < E energy separates the filled states (E < EFF at T = 0 K) at T = 0 K)from the empty states (E > Efrom the empty states (E > EFF at T = 0 K). Here it splits the band at T = 0 K). Here it splits the band((each band can hold 2 electronseach band can hold 2 electrons))��A 1D chain of H atoms is predicted to be metallic because theA 1D chain of H atoms is predicted to be metallic because theFermi Fermi level cuts a band (there is no gap so it takes only anlevel cuts a band (there is no gap so it takes only aninfinitesimal energy to excite an electron into an empty state).infinitesimal energy to excite an electron into an empty state).��The band runs "uphill" (from 0 toThe band runs "uphill" (from 0 to ππ//a) because the in phase (at k=0)a) because the in phase (at k=0)combination ofcombination of orbitals orbitals is bonding and the out of phase (at k= is bonding and the out of phase (at k=ππ/a) is/a) isantibondingantibonding..

00 ππππππππ/a/akk

E(k)E(k)

EEFF

Effect of Orbital OverlapEffect of Orbital OverlapIf we reduce the latticeIf we reduce the lattice

parameter parameter aa it has the following it has the followingeffects:effects:

��The spatial overlap of theThe spatial overlap of theorbitals orbitals increasesincreases��The band becomes moreThe band becomes morebonding at k=0bonding at k=0��The band becomes moreThe band becomes moreantibonding antibonding k=k=ππ/a./a.��The increased The increased antibondingantibonding is islarger than the increasedlarger than the increasedbonding.bonding.��The bandwidth increases.The bandwidth increases.��The electron mobility increases.The electron mobility increases.

Wide bands Wide bands →→→→→→→→ Good orbital Good orbitaloverlap overlap →→→→→→→→ High carrier mobility High carrier mobility

7

Band Structure: Linear Chain of FBand Structure: Linear Chain of F

00 ππ/a/akk

E(k

)E

(k)

EEFF

00 ππ/a/akk

EEFF

00 ππ/a/akk

EEFF

00 ππ/a/akk

EEFF

Which of the following is the correct band structure forWhich of the following is the correct band structure fora linear chain of F atoms?a linear chain of F atoms?

FF FF FF FF FF FF FF FF FF FF(a)(a) (b)(b) (c)(c) (d)(d)


00 ππ/a/akk

EEFF

Doubly degenerateDoubly degenerate


AntibondingAntibonding 2p 2pzz σ∗σ∗σ∗σ∗σ∗σ∗σ∗σ∗

Bonding 2pBonding 2pz z σσσσσσσσ

Bonding 2s Bonding 2s σσσσσσσσ

AntibondingAntibonding 2s 2s σ∗σ∗σ∗σ∗σ∗σ∗σ∗σ∗Bonding 2pBonding 2pxx/2p/2pyy ππππππππ

AntibondingAntibonding 2p 2pxx/2p/2pyy π∗π∗π∗π∗π∗π∗π∗π∗

8


00 ππ/a/akk

EEFF


A more accurateA more accuratetreatment of the bandtreatment of the band

structure would show anstructure would show anavoided crossingavoided crossing

between the 2pbetween the 2pzz σσσσσσσσ and and2s 2s σσσσσσσσ * interactions at * interactions at

k=k=ππππππππ/a. There would be/a. There would bemixing between thesemixing between these

two bands (creating sp-two bands (creating sp-hybrid like states)hybrid like states)

similar to the 2s/2p similar to the 2s/2p σσσσσσσσ & &σσσσσσσσ* interactions seen in* interactions seen inthe MO diagram of Nthe MO diagram of N22..

AntibondingAntibonding 2p 2pzz σ∗σ∗σ∗σ∗σ∗σ∗σ∗σ∗

Bonding 2s Bonding 2s σσσσσσσσ

Bonding 2pBonding 2pxx/2p/2pyy ππππππππ

Lessons from Linear F ChainLessons from Linear F Chain

�� There are now 4 There are now 4 orbitals orbitals in the unit cell (a single Fin the unit cell (a single Fatom with 1 atom with 1 2s2s + 3 + 3 2p2p orbitalsorbitals) giving rise to 4) giving rise to 4bands in the band structure.bands in the band structure.

�� The fact that the The fact that the wavefunctionwavefunction corresponding to a corresponding to ap-orbital changes sign at the nucleus causes the 2pp-orbital changes sign at the nucleus causes the 2pσσσσσσσσ band to run downhill (opposite of the 2s band to run downhill (opposite of the 2s σσσσσσσσ band). band).

�� The reduced spatial overlap of the The reduced spatial overlap of the ππππππππ interaction interactioncauses the causes the ππππππππ bands to be narrower than the bands to be narrower than the σσσσσσσσbands.bands.

�� The 2p The 2p orbitals orbitals start out at a higher energy thanstart out at a higher energy thanthe 2s the 2s orbitalsorbitals (from atomic quantum theory) (from atomic quantum theory)

9

More than 1 atom in the Unit CellMore than 1 atom in the Unit CellThe The Pierls Pierls DistortionDistortion

What happens when there is more than one atom in theWhat happens when there is more than one atom in theunit cell? To illustrate the consequences consider aunit cell? To illustrate the consequences consider a

distortion of the 1D chain of H atoms which causes thedistortion of the 1D chain of H atoms which causes theatoms to atoms to dimerizedimerize. The consequences will be as follows:. The consequences will be as follows:

��There are now 2 There are now 2 orbitals orbitals in each unit cell, so there willin each unit cell, so there willbe 2 bands in the band structure.be 2 bands in the band structure.��Let�s construct 1 band to be bonding within the unit cellLet�s construct 1 band to be bonding within the unit cell(HOMO - valence band), and the other to be (HOMO - valence band), and the other to be antibondingantibondingwithin the unit cell (LUMO - conduction band).within the unit cell (LUMO - conduction band).��Translational Translational symmetry and the value of k will take caresymmetry and the value of k will take careof the rest.of the rest.

Pierls Pierls Distortion (HDistortion (H2 2 Chain)Chain)

χχχχ0000 χχχχ1111 χχχχ2222 χχχχ3333 χχχχ4444 χχχχ5555

k=0

k=π/a

k=π/2a

χχχχ0000 χχχχ1111 χχχχ2222 χχχχ3333 χχχχ4444 χχχχ5555

k=0

k=π/a

k=π/2a

Conduction Band (LUMO Conduction Band (LUMO σσσσσσσσ**))

Valence Band (HOMO Valence Band (HOMO σσσσσσσσ))

IntraIntra Inter Inter

σσσσσσσσ** σσσσσσσσ

σσσσσσσσ** non non

σσσσσσσσ** σ σ σ σ σ σ σ σ **

σσσσσσσσ σσσσσσσσ∗∗∗∗∗∗∗∗

σσσσσσσσ non non

σσσσσσσσ σσσσσσσσ

IntraIntra Inter Inter

σσσσσσσσ* * →→ σσσσσσσσ

Band runsBand runsdownhilldownhill

σσσσσσσσ →→ σ σ σ σ σ σ σ σ **

Band runsBand runsuphilluphill

10

Band Structure HBand Structure H22

00 ππ/a/akk

E(k

)E

(k)

EEFF

��There are two electrons per unit cell so that the valence band is full andThere are two electrons per unit cell so that the valence band is full andthe conduction band empty. Thus Ethe conduction band empty. Thus EFF occupies a position midway between occupies a position midway betweenthe two bands and the system is now a semiconductor/insulator.the two bands and the system is now a semiconductor/insulator.

��The minimum energy gap between bands occurs at k = The minimum energy gap between bands occurs at k = ππ/a. Since the/a. Since themaximum in the VB and the minimum in the CB occur at the same value of k,maximum in the VB and the minimum in the CB occur at the same value of k,the compound is said to be a direct gap semiconductor.the compound is said to be a direct gap semiconductor. ��Compared with the band structure of a linear chain of hydrogen atoms theCompared with the band structure of a linear chain of hydrogen atoms thebands in this system will be narrower. This is a consequence of the reducedbands in this system will be narrower. This is a consequence of the reducedoverlap between molecules.overlap between molecules.

��The net effect of this distortion is to lower the energy of the filledThe net effect of this distortion is to lower the energy of the filledstates (which originate from bonding MOstates (which originate from bonding MO��s) while raising the energy ofs) while raising the energy ofempty states (empty states (antibondingantibonding MO MO��s). In 1D such a distortion will always bes). In 1D such a distortion will always bestable when you have a half filled band (as we did in the H band structure).stable when you have a half filled band (as we did in the H band structure).Physicists call this aPhysicists call this a Peierls Peierls distortion, it is the solid state equivalent of a distortion, it is the solid state equivalent of a11stst order Jahn-Teller distortion. order Jahn-Teller distortion.

Lessons � Lessons � Pierls Pierls Distortion (HDistortion (H22))

11

PtHPtH44 a Molecular Chain Compound a Molecular Chain CompoundNow let�s consider a compound that contains different types ofNow let�s consider a compound that contains different types of

atoms within the unit cell. The compound Aatoms within the unit cell. The compound A22PtXPtX44 (where A is a +1 (where A is a +1cation, such as Kcation, such as K++, and X �1 anion, such as , and X �1 anion, such as ClCl--, CN, CN-- or H or H--) forms linear) forms linearchains and is a good example to consider. Let�s construct the bandchains and is a good example to consider. Let�s construct the band

structure of PtHstructure of PtH442-2- (the K (the K++ are very electropositive and serve are very electropositive and serve

primarily as donors of electrons, we will ignore them except when itprimarily as donors of electrons, we will ignore them except when itcomes to counting electrons). The process consists of 3 steps.comes to counting electrons). The process consists of 3 steps.

Step 1: Step 1: Consider the MO Diagram of the Building unit (here a PtHConsider the MO Diagram of the Building unit (here a PtH44Square Planar Unit). This gives an idea of the band energy levels.Square Planar Unit). This gives an idea of the band energy levels.

Step 2: Step 2: Consider the orbital overlap between units (molecules inConsider the orbital overlap between units (molecules inthis case). This tells us if the bands run uphill or downhill.this case). This tells us if the bands run uphill or downhill.

Step 3: Step 3: Consider the degree of orbital overlap (spatial andConsider the degree of orbital overlap (spatial andenergetic) between units. This gives the band widths.energetic) between units. This gives the band widths.

Linear PtHLinear PtH44 Chain Chain

PtPtHH

HH

HH

HH

PtPtHH

HH

HH

HH

PtPtHH

HH

HH

HH

PtPtHH

HH

HH

HHStep 1: Step 1: Consider the MO Diagram of theConsider the MO Diagram of the

Building unit (here a PtHBuilding unit (here a PtH44 Square Planar Unit). Square Planar Unit).This gives an idea of the band energy levels.This gives an idea of the band energy levels.

12

Step 2: Step 2: Consider the orbital overlapConsider the orbital overlapbetween units (molecules in this case). Thisbetween units (molecules in this case). Thistells us if the bands run uphill or downhill.tells us if the bands run uphill or downhill.



Step 3: Step 3: Consider the degree of orbitalConsider the degree of orbitaloverlap (spatial and energetic) betweenoverlap (spatial and energetic) between

units. This gives the band widths.units. This gives the band widths.

Calculated BandCalculated BandStructureStructure

13

Geometry and PropertiesGeometry and Properties

Consider the salt KConsider the salt K22PtHPtH44 (or K (or K22PtClPtCl44))and predict what would be it�sand predict what would be it�s

optimal geometry and properties.optimal geometry and properties.

Which configuration would be moreWhich configuration would be morestable a stable a staggeredstaggered (PtH (PtH44 units units

rotated by 45º to each other) or anrotated by 45º to each other) or aneclipsedeclipsed geometry (PtH geometry (PtH44 units units

related by a mirror plane)?related by a mirror plane)?

Would you expect this material to beWould you expect this material to bea semiconductor or a metal?a semiconductor or a metal?

How would the Pt-Pt distance reactHow would the Pt-Pt distance reactif the material were oxidized?if the material were oxidized?

KK22Pt(CN)Pt(CN)44ClClxxKK22Pt(CN)Pt(CN)44

�� PtPt2+2+ (d (d88) - d) - dz2z2 band is band iscompletely filledcompletely filled

�� Pt-Pt distance = 3.3 APt-Pt distance = 3.3 A

�� Eclipsed geometry (theEclipsed geometry (thePt-Pt distance is too longPt-Pt distance is too longfor Pt-Pt for Pt-Pt ππππππππ interactions interactions

to matter).to matter).

�� Semiconductor (Semiconductor (FermiFermilevel falls between dlevel falls between dz2z2

band and the dband and the dx2-y2x2-y2 band) band)

KK22Pt(CN)Pt(CN)44ClCl0.30.3

�� PtPt2.3+2.3+ (mixed d (mixed d77/d/d88) - d) - dz2z2band is partially filledband is partially filled

�� Pt-Pt distance = 2.7 APt-Pt distance = 2.7 A(electrons have been(electrons have beenremoved from Pt-Ptremoved from Pt-Ptantibonding antibonding statesstates

�� Staggered geometryStaggered geometry(minimize (minimize ddxzxz--ddxzxz and and ddyzyz--

ddyzyz interactions)interactions)

�� Conductor (Conductor (Fermi Fermi LevelLevelcuts dcuts dz2z2 band) band)

14

Two dimensionsTwo dimensions

Γ = (kx=0, ky=0, kz=0) X = (π/a, 0, 0)M = (π/a, π/a, 0) Y = (0, π/a, 0) R = (π/a, π/a, π/a) Z = (0, 0, π/a)

Band Structure of BaBand Structure of Ba22SnOSnO44To give a real example of a band structure which is

predominantly two dimensional consider the Ruddlesden-Popper compound Ba2SnO4. This compound contains

sheets of corner sharing octahedra. Let�s derive theband structure for one of these sheets.

xx

yy zz

15

Constructing the Band StructureConstructing the Band StructureStep 1: Step 1: Consider the MO Diagram of the building unit. In thisConsider the MO Diagram of the building unit. In this

case the building unit is a SnOcase the building unit is a SnO66 octahedron. The 4d octahedron. The 4d orbitals orbitalsofof Sn Sn are essentially core are essentially core orbitals orbitals and can be neglected. and can be neglected.The 2sThe 2s orbitals orbitals of oxygen play a role, but for the sake of of oxygen play a role, but for the sake ofsimplicity they will be ignored.simplicity they will be ignored.

Step 2: Step 2: Consider the orbital overlap between units. In thisConsider the orbital overlap between units. In thiscase the case the SnSn 5p 5pxx-O 2p-O 2pxx, the, the Sn Sn 5p 5pyy-O 2p-O 2pyy, and the, and the Sn Sn 5s-O 5s-O2p2pxx/O 2p/O 2pyy interactions need to be considered. The interactions need to be considered. The Sn Sn 5p 5pzz-O-O2p2pzz interactions are essentially isolated on individual building interactions are essentially isolated on individual buildingunits and will give rise to flat bands.units and will give rise to flat bands.

Step 3: Step 3: Consider the degree of orbital overlap. Both the Consider the degree of orbital overlap. Both the SnSn5s-O 2p and 5s-O 2p and SnSn 5p-O 2p interactions have a favorable 5p-O 2p interactions have a favorablespatial overlap. The spatial overlap. The Sn Sn 5s-O 2p interaction has a better5s-O 2p interaction has a betterenergetic overlap though.energetic overlap though.

*In total there are 3*In total there are 3××6-4 = 14 non-bonding O 2p 6-4 = 14 non-bonding O 2p orbitals orbitals (neglecting(neglectingππππππππ-bonding and -bonding and Sn Sn 4d overlap). Only two are shown here.4d overlap). Only two are shown here.

Sn Sn 5s5s

Sn Sn 5p5p

eeggtt1u1u

aa1g1g

O 2pO 2p**

1a1a1g1g

2a2a1g1g

1t1t1u1u

2t2t1u1u

eegg

SnOSnO66 MO Diagram MO Diagram

16

Sn Sn 5s-O 2p5s-O 2pValence BandValence Band

Majority O 2p characterMajority O 2p character

ΓΓΓΓΓΓΓΓ

XX

MM

xx

yy

O 2p NonbondingO 2p Nonbonding

SnSn 5s-O 2p 5s-O 2pBondingBonding

Sn Sn 5s-O 2p5s-O 2pConduction BandConduction Band

Majority Majority Sn Sn 5s character5s character

Sn Sn 5s Nonbonding5s Nonbonding

SnSn 5s-O 2p 5s-O 2pAntibondingAntibonding

ΓΓΓΓΓΓΓΓ

XX

MM

xx

yy

Sn Sn 5p5pxx-O 2p-O 2pxxValence BandValence Band


ΓΓΓΓΓΓΓΓ

XX

MM

xx

yy


SnSn 5p-O 2p 5p-O 2pBondingBonding

SnSn 5p 5pxx-O 2p-O 2pxxConduction BandConduction Band

Majority Majority Sn Sn 5p character5p character

Sn Sn 5p Nonbonding5p Nonbonding

SnSn 5p-O 2p 5p-O 2pAntibondingAntibondingΓΓΓΓΓΓΓΓ

XX

MM

xx

yy

17

Sn Sn 5p5pyy-O 2p-O 2pyyValence BandValence Band


ΓΓΓΓΓΓΓΓ

XX

MM

xx

yy


SnSn 5p-O 2p 5p-O 2pBondingBonding

SnSn 5p 5pyy-O 2p-O 2pyyConduction BandConduction Band

Majority Majority Sn Sn 5p character5p character

Sn Sn 5p Nonbonding5p Nonbonding

SnSn 5p-O 2p 5p-O 2pAntibondingAntibondingΓΓΓΓΓΓΓΓ

XX

MM

xx

yy

ΓΓΓΓΓΓΓΓ

EEFF

XX MM ΓΓΓΓΓΓΓΓ

SnSn 5p 5pZZ-O 2p-O 2pZZ σσσσσσσσ

SnSn 5p 5pZZ-O 2p-O 2pZZ σ∗σ∗σ∗σ∗σ∗σ∗σ∗σ∗

SnSn 5p 5pxx-O 2p-O 2pxx σ∗σ∗σ∗σ∗σ∗σ∗σ∗σ∗

SnSn 5p 5pxx-O 2p-O 2pxx σσσσσσσσ

SnSn 5p 5pyy-O 2p-O 2pyy σ∗σ∗σ∗σ∗σ∗σ∗σ∗σ∗

SnSn 5p 5pyy-O 2p-O 2pyy σσσσσσσσ

SnSn 5s-O 2p 5s-O 2p σ∗σ∗σ∗σ∗σ∗σ∗σ∗σ∗

SnSn 5s-O 2p 5s-O 2p σσσσσσσσ

O 2p nonbondingO 2p nonbondingBand GapBand Gap

BaBa22SnOSnO44: Band Structure: Band Structure

18

We can now go full circle and return to the ReOWe can now go full circle and return to the ReO33 spaghetti diagram that spaghetti diagram thatstarted our discussion. The band structure of this compound is similar instarted our discussion. The band structure of this compound is similar in

many ways to the Bamany ways to the Ba22SnOSnO44 band structure. The main differences are: band structure. The main differences are:

�� Because of the 3-dimensionality all of the Re based bands will be disperseBecause of the 3-dimensionality all of the Re based bands will be disperse(only O 2p non-bonding states will give rise to flat bands).(only O 2p non-bonding states will give rise to flat bands).

�� The most important interactions are the Re 5d-The most important interactions are the Re 5d-orbitalsorbitals, t, t2g2g ( (ππ//ππ*) & *) & eegg((σσ//σσ*), rather than the *), rather than the Sn Sn 5s & 5p.5s & 5p.

�� The The Fermi Fermi level cuts the level cuts the ππ* bands creating a metallic conductor rather* bands creating a metallic conductor ratherthan a semiconductor.than a semiconductor.

3D: The Band Structure ReO3D: The Band Structure ReO33

DOS

0 3

k

En

erg

y (v

s. O

2s)

10

15

20

25

30

X R M Γ R

10

15

20

25

30

DOS

0 6

O 2pM t2g

M eg

X R M Γ R

k

EF

W(tW(t2g2g//ππππππππ**) ~ 5 ) ~ 5 eVeVW(W(eegg //σσσσσσσσ**) ~ 7 ) ~ 7 eVeV

ReRe

ReReReRe

ReReOO

OO

OO

OO

ReRe

ReReReRe

ReReOO

OO

OO

OO

ππππππππ**

ππππππππ

ΓΓΓΓ pointnon-bonding

Overlap at the M-pointOverlap at the M-pointkkxx==kkyy==ππππππππ/a, /a, kkzz=0=0

Overlap at the Overlap at the ΓΓΓΓΓΓΓΓ-point-pointkkxx==kkyy==kkzz=0=0

ReOReO33 BandstructureBandstructure

ππππππππ**

ππππππππ**

The points circled in blue are the bondingThe points circled in blue are the bondingcounterparts to the counterparts to the antibondingantibondinginteractions shown to the right.interactions shown to the right.

The The eegg//σσσσσσσσ* band is wider than t* band is wider than t2g2g//ππππππππ* due to larger orbital overlap (spatial).* due to larger orbital overlap (spatial).

O 2pO 2pnonbondingnonbonding

19

Summary Band StructuresSummary Band Structures�� What is being plotted?What is being plotted? Energy vs. k, where k is the Energy vs. k, where k is the wavevector wavevector thatthat

gives the phase of the AO�s as well as the wavelength of the electrongives the phase of the AO�s as well as the wavelength of the electronwavefunction wavefunction (crystal momentum).(crystal momentum).

�� How many lines are there in a band structure diagram?How many lines are there in a band structure diagram? As many asAs many asthere are there are orbitals orbitals in the unit cell.in the unit cell.

�� How is the center of gravity energy level of each band determined?How is the center of gravity energy level of each band determined?Usually follows from the MO diagram.Usually follows from the MO diagram.

�� How do we determine whether a band runs uphill or downhill?How do we determine whether a band runs uphill or downhill? ByBycomparing the orbital overlap at k=0 and k=comparing the orbital overlap at k=0 and k=ππ/a./a.

�� How do we distinguish metals from semiconductors and insulators?How do we distinguish metals from semiconductors and insulators?The The FermiFermi level cuts a band in a metal, whereas there is a gap between level cuts a band in a metal, whereas there is a gap betweenthe filled and empty states in a semiconductor.the filled and empty states in a semiconductor.

�� Why are some bands flat and others steep?Why are some bands flat and others steep? This depends on theThis depends on thedegree of orbital overlap between building units.degree of orbital overlap between building units.Wide bands Wide bands → → → → → → → → Large intermolecular overlap Large intermolecular overlap →→→→→→→→ delocalized delocalized e e--

Narrow bands Narrow bands → → → → → → → → Weak intermolecular overlap Weak intermolecular overlap →→→→→→→→ localized e localized e--

An Introduction to Band Theory, A Molecular Orbital …woodward/ch...1 An Introduction to Band...

Documents

Transcript of An Introduction to Band Theory, A Molecular Orbital …woodward/ch...1 An Introduction to Band...