An Introduction to Band An Introduction to Band Theory, A Molecular Theory, A Molecular Orbital ApproachOrbital Approach

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

Lectures #17-18Lectures #17-18

References – Band TheoryReferences – Band TheoryThe material for this lecture comes primarily from the work The material for this lecture comes primarily from the work of Roald Hoffmann and those who came through his group of Roald Hoffmann and those who came through his group (Mike Whangbo, Jeremy Burdett, Thomas Albright, Tim (Mike Whangbo, Jeremy Burdett, Thomas Albright, Tim Hughbanks, etc.). His treatment (and many of the figures Hughbanks, etc.). His treatment (and many of the figures used in this lecture) are nicely summarized in a small book used in this lecture) are nicely summarized in a small book he wrote some years ago:he wrote some years ago:

““Solids and Surfaces: A chemists view of Solids and Surfaces: A chemists view of bonding in extended structures”bonding in extended structures”

Roald Hoffmann, VCH Publishers, (1988)Roald 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).

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 For an extended solid the band structure plays

the same role that an MO diagram does for a the same role that an MO diagram does for a molecule.molecule.

• Why it is important to be able to comprehend Why it is important to be able to comprehend the electronic band structure?the electronic band structure?– The band structure is the link between crystal The band structure is the link between crystal

structure, bonding and properties. In structure, bonding and properties. In particular details of the band structure are particular details of the band structure are closely linked to the following properties:closely linked to the 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-

OctahedronOctahedronIn the last lecture we discussed In the last lecture we discussed

the MO diagram of an octahedron the MO diagram of an octahedron (shown to the left). How does (shown to the left). How does this diagram change when we this diagram change when we have an extended structure?have an extended structure?

Band StructureBand Structure

(aka Spaghetti diagram): MO (aka Spaghetti diagram): MO diagram with translational diagram with translational

symmetry taken into account.symmetry taken into account.

Density of States (DOS)Density of States (DOS)

Integration of the band Integration of the band structure. Shows the # of structure. Shows the # of available levels between E available levels between E

and E+dE as dE and E+dE as dE 0. 0.

Band Structure ReOBand Structure ReO33

Constructing a DOS Diagram: Constructing a DOS Diagram: TiOTiO22

It’s possible to construct a It’s possible to construct a reasonable approximation of the reasonable approximation of the

DOS diagram from the MO diagram DOS diagram from the MO diagram of the building block.of the building block.

The energy levels of each block of The energy levels of each block of bands or “BOB” comes from the MO bands or “BOB” comes from the MO diagram (based on electronegativity diagram (based on electronegativity

and bonding interactions)and bonding interactions)

The area of each BOB is The area of each BOB is proportional to the number of MO’s proportional to the number of MO’s

at that approximate energy.at that approximate energy.

The width of each BOB is derived The width of each BOB is derived from the overlap between building from the overlap between building

blocksblocks

Constructing a DOS Diagram: Constructing a DOS Diagram: TiOTiO22

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

and MgTiOand MgTiO33 (ilmenite) are all going to be qualitatively similar, because the (ilmenite) are all going to be qualitatively similar, because the electronic 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 minor contributions near the Fermi energy. The main difference will be the width of contributions near the Fermi energy. 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 **

Ti eTi egg

Ti-O Ti-O **

EEff

Chain of 5 H Chain of 5 H atomsatoms

HH22 MoleculeMolecule

EE

Bonding

Antibonding

Nonbonding

00

11

22

33

44

# of Nodes

Infinite 1D Chain of H atomsInfinite 1D Chain of H atomsIf there are N atoms in the chain If there are N atoms in the chain there will be N energy levels and N there will be N energy levels and N electronic states ("MOs"). The electronic states ("MOs"). The wavefunction for each electronic wavefunction for each electronic state is:state is:

kk = = e eiknaiknann

Where:Where:

•aa is the lattice constant (spacing is the lattice constant (spacing between H atoms), between H atoms),

•nn identifies the individual atoms identifies the individual atoms within the chain, within the chain,

• nn represents the atomic orbitalsrepresents the atomic orbitals

•kk is a quantum # that identifies is a quantum # that identifies the wavefunction and tells us the the wavefunction and tells us the phase of the orbitals.phase of the orbitals.

k=0

k=/a

k=/2a

a

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

/a = = 00+(exp{i+(exp{i})})1 1 +(exp{i2+(exp{i2})})2 2 ++

(exp{i3(exp{i3})})33+(exp{i4+(exp{i4})})44+…+…

/2a = = 0 0 - - 1 1 + + 2 2 - - 3 3 + + 4 4 +…+…

k = /2a

/2a = = 00+(exp{i+(exp{i/2})/2})1 1 +(exp{i+(exp{i})})2 2

+(exp{i3+(exp{i3/2})/2})33+(exp{i2+(exp{i2})})44+…+…

/2a = = 0 0 + 0 - + 0 - 2 2 + 0 + + 0 + 4 4 +…+…

k = 0 k = 0

00 = = 00++1 1 ++2 2 ++3 3 ++4 4 +…+…k=0

k=/a

k=/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=/a /a orbital phase reverses when we translate byorbital phase reverses when we translate by a a

Band Band Structure Structure Linear H Linear H

ChainChain

•The Fermi energy separates the filled states (E < EThe Fermi energy separates the filled states (E < EFF at T = 0 at T = 0 K) from the empty states (E > EK) from the empty states (E > EFF at T = 0 K). Here it splits the at T = 0 K). Here it splits the band (band (each band can hold 2 electronseach band can hold 2 electrons))•A 1D chain of H atoms is predicted to be metallic because the A 1D chain of H atoms is predicted to be metallic because the Fermi level cuts a band (there is no gap so it takes only an Fermi level cuts a band (there is no gap so it takes only an infinitesimal 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 //a) because the in phase (at a) because the in phase (at k=0) combination of orbitals is bonding and the out of phase k=0) combination of orbitals is bonding and the out of phase (at k=(at k=/a) is antibonding. /a) is antibonding.

00 /a/akk

E(k)E(k)

EEFF

Effect of Orbital OverlapEffect of Orbital OverlapIf we reduce the lattice If we reduce the lattice parameter parameter aa it has the it has the

following effects:following effects:•The spatial overlap of the The spatial overlap of the orbitals increasesorbitals increases•The band becomes more The band becomes more bonding at k=0bonding at k=0•The band becomes more The band becomes more antibonding k=antibonding k=/a. /a. •The increased antibonding is The increased antibonding is larger than the increased larger than the increased bonding.bonding.•The bandwidth increases.The bandwidth increases.•The electron mobility The electron mobility increases.increases.

Wide bands Wide bands Good orbital Good orbital overlap overlap High carrier High carrier

mobilitymobility

Band Structure: Linear Chain of Band Structure: Linear Chain of FF

00 /a/akk

E(k

)E

(k)

EEFF

00 /a/akk

EEFF

00 /a/akk

EEFF

00 /a/akk

EEFF

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

FF FF FF FF FF FF FF FF FF FF(a(a))

(b(b))

(c(c))

(d(d))

Band Structure: Linear Chain of Band Structure: Linear Chain of FF

00 /a/akk

EEFF

Doubly degenerateDoubly degenerate

Doubly degenerateDoubly degenerate

Antibonding 2pAntibonding 2pzz

Bonding 2pBonding 2pz z

Bonding 2s Bonding 2s

Antibonding 2s Antibonding 2s Bonding 2pBonding 2pxx/2p/2pyy

Antibonding 2pAntibonding 2pxx/2p/2pyy

Band Structure: Linear Chain of Band Structure: Linear Chain of FF

00 /a/akk

EEFF

Doubly degenerateDoubly degenerate

A more accurate A more accurate treatment of the band treatment of the band structure would show structure would show an avoided crossing an avoided crossing

between the 2pbetween the 2pzz and and 2s 2s * interactions at * interactions at k=k=/a. There would /a. There would be mixing between be mixing between

these two bands these two bands (creating sp-hybrid (creating sp-hybrid

like states) similar to like states) similar to the 2s/2p the 2s/2p & & * *

interactions seen in interactions seen in the MO diagram of Nthe MO diagram of N22..

Antibonding 2pAntibonding 2pzz

Bonding 2s Bonding 2s

Bonding 2pBonding 2pxx/2p/2pyy

Lessons from Linear F ChainLessons from Linear F Chain

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

• The fact that the wavefunction corresponding to a The fact that the wavefunction corresponding to a p-orbital changes sign at the nucleus causes the p-orbital changes sign at the nucleus causes the 2p 2p band to run downhill (opposite of the 2s band to run downhill (opposite of the 2s band).band).

• The reduced spatial overlap of the The reduced spatial overlap of the interaction interaction causes the causes the bands to be narrower than the bands to be narrower than the bands. bands.

• The 2p orbitals start out at a higher energy than The 2p orbitals start out at a higher energy than the 2s orbitals (from atomic quantum theory)the 2s orbitals (from atomic quantum theory)

More than 1 atom in the Unit More than 1 atom in the Unit CellCell

The Pierls DistortionThe Pierls DistortionWhat happens when there is more than one atom in the What happens when there is more than one atom in the

unit cell? To illustrate the consequences consider a unit cell? To illustrate the consequences consider a distortion of the 1D chain of H atoms which causes the distortion of the 1D chain of H atoms which causes the

atoms to dimerize. The consequences will be as follows:atoms to dimerize. The consequences will be as follows:

•There are now 2 orbitals in each unit cell, so there will There are now 2 orbitals in each unit cell, so there will be 2 bands in the band structure.be 2 bands in the band structure.

•Let’s construct 1 band to be bonding within the unit cell Let’s construct 1 band to be bonding within the unit cell (HOMO - valence band), and the other to be antibonding (HOMO - valence band), and the other to be antibonding within the unit cell (LUMO - conduction band).within the unit cell (LUMO - conduction band).

•Translational symmetry and the value of k will take care Translational symmetry and the value of k will take care of the rest.of the rest.

Pierls Distortion (HPierls Distortion (H2 2 Chain)Chain)

k=0

k=/a

k=/2a

k=0

k=/a

k=/2a

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

Valence Band (HOMO Valence Band (HOMO ))

IntraIntra Inter Inter

**

** non non

** **

non non

IntraIntra Inter Inter

* *

Band runs Band runs downhilldownhill

**

Band runs Band runs uphilluphill

Band Structure HBand Structure H22

00 /a/akk

E(k

)E

(k)

EEFF

•There are two electrons per unit cell so that the valence band is full There are two electrons per unit cell so that the valence band is full and the conduction band empty. Thus Eand the conduction band empty. Thus EFF occupies a position midway occupies a position midway

between the two bands and the system is now a between the two bands and the system is now a semiconductor/insulator. semiconductor/insulator.

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

•The net effect of this distortion is to lower the energy of the filled The net effect of this distortion is to lower the energy of the filled states (which originate from bonding MO’s) while raising the energy of states (which originate from bonding MO’s) while raising the energy of empty states (antibonding MO’s). In 1D such a distortion will always empty states (antibonding MO’s). In 1D such a distortion will always be stable when you have a half filled band (as we did in the H band be stable when you have a half filled band (as we did in the H band structure). Physicists call this a Peierls distortion, it is the solid state structure). Physicists call this a Peierls distortion, it is the solid state equivalent of a 1equivalent of a 1stst order Jahn-Teller distortion. order Jahn-Teller distortion.

Lessons – Pierls Distortion Lessons – Pierls Distortion (H(H22))

PtHPtH44 a Molecular Chain a Molecular Chain CompoundCompound

Now let’s consider a compound that contains different types of Now 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 +1

cation, such as Kcation, such as K++, and X –1 anion, such as Cl, and X –1 anion, such as Cl--, CN, CN-- or H or H--) forms ) forms linear chains and is a good example to consider. Let’s construct linear chains and is a good example to consider. Let’s construct the band structure of PtHthe band structure of PtH44

2-2- (the K (the K++ are very electropositive and are very electropositive and serve primarily as donors of electrons, we will ignore them except serve primarily as donors of electrons, we will ignore them except when it comes to counting electrons). The process consists of 3 when it comes to counting electrons). The process consists of 3

steps.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 PtH44 Square 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 in Consider the orbital overlap between units (molecules in this 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 and Consider the degree of orbital overlap (spatial and energetic) 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 the Consider the MO Diagram of the Building unit (here a PtHBuilding unit (here a PtH44 Square Planar Square Planar

Unit). This gives an idea of the band energy Unit). This gives an idea of the band energy levels.levels.

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

downhill.downhill.

Linear PtHLinear PtH44 Chain Chain

Linear PtHLinear PtH44 Chain Chain

Step 3: Step 3: Consider the degree of orbital Consider the degree of orbital overlap (spatial and energetic) overlap (spatial and energetic)

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

Calculated Band Calculated Band StructureStructure

Geometry and PropertiesGeometry and Properties

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

optimal geometry and properties.optimal geometry and properties.

Which configuration would be Which configuration would be more stable a more stable a staggeredstaggered (PtH (PtH44

units rotated by 45º to each other) units rotated by 45º to each other) or an or an eclipsedeclipsed geometry (PtH geometry (PtH44

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

Would you expect this material to Would you expect this material to be a semiconductor or a metal?be a semiconductor or a metal?

How would the Pt-Pt distance react How would the Pt-Pt distance react if the material were oxidized? if the material were oxidized?

KK22Pt(CN)Pt(CN)44ClClxx

KK22Pt(CN)Pt(CN)44

• PtPt2+2+ (d (d88) - d) - dz2z2 band is band is completely filledcompletely filled

• Pt-Pt distance = 3.3 APt-Pt distance = 3.3 A

• Eclipsed geometry (the Eclipsed geometry (the Pt-Pt distance is too Pt-Pt distance is too

long for Pt-Pt long for Pt-Pt interactions to matter).interactions to matter).

• Semiconductor (Fermi Semiconductor (Fermi level 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) - ) - ddz2z2 band is partially band is partially

filledfilled

• Pt-Pt distance = 2.7 A Pt-Pt distance = 2.7 A (electrons have been (electrons have been removed from Pt-Pt removed from Pt-Pt antibonding statesantibonding states

• Staggered geometry Staggered geometry (minimize d(minimize dxzxz-d-dxzxz and and ddyzyz-d-dyzyz interactions) interactions)

• Conductor (Fermi Conductor (Fermi Level cuts dLevel cuts dz2z2 band) band)

Two dimensionsTwo dimensions

= (kx=0, ky=0, kz=0) X = (/a, 0, 0)

M = (/a, /a, 0) Y = (0, /a, 0) R = (/a, /a, /a) Z = (0, 0, /a)

Band Structure of BaBand Structure of Ba22SnOSnO44

To 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 the band structure for one of these sheets.

xx

yy zz

Constructing the Band Constructing the Band StructureStructure

Step 1: Step 1: Consider the MO Diagram of the building unit. In Consider the MO Diagram of the building unit. In this case the building unit is a SnOthis case the building unit is a SnO66 octahedron. The 4d octahedron. The 4d orbitals of Sn are essentially core orbitals and can be orbitals of Sn are essentially core orbitals and can be neglected. The 2s orbitals of oxygen play a role, but for neglected. The 2s orbitals of oxygen play a role, but for the sake of simplicity they will be ignored.the sake of simplicity they will be ignored.

Step 2: Step 2: Consider the orbital overlap between units. In this Consider the orbital overlap between units. In this case the Sn 5pcase the Sn 5pxx-O 2p-O 2pxx, the Sn 5p, the Sn 5pyy-O 2p-O 2pyy, and the Sn 5s-O , and the Sn 5s-O 2p2pxx/O 2p/O 2pyy interactions need to be considered. The Sn interactions need to be considered. The Sn 5p5pzz-O 2p-O 2pzz interactions are essentially isolated on interactions are essentially isolated on individual building units and will give rise to flat bands.individual building 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 Sn 5s-O 2p and Sn 5p-O 2p interactions have a Sn 5s-O 2p and Sn 5p-O 2p interactions have a favorable spatial overlap. The Sn 5s-O 2p interaction favorable spatial overlap. The Sn 5s-O 2p interaction has a better energetic overlap though.has a better energetic overlap though.

*In total there are 3*In total there are 36-4 = 14 non-bonding O 2p orbitals 6-4 = 14 non-bonding O 2p orbitals (neglecting (neglecting -bonding and Sn 4d overlap). Only two are shown -bonding and Sn 4d overlap). Only two are shown

here.here.

Sn 5sSn 5s

Sn 5pSn 5p

eegg

tt1u1u

aa1g1g

O 2pO 2p**

1a1a1g1g

2a2a1g1g

1t1t1u1u

2t2t1u1u

eegg

SnOSnO66 MO Diagram MO Diagram

Sn 5s-O 2p Sn 5s-O 2p Valence BandValence Band

Majority O 2p characterMajority O 2p character

XX

MM

xx

yy

O 2p NonbondingO 2p Nonbonding

Sn 5s-O 2p Sn 5s-O 2p BondingBonding

Sn 5s-O 2p Sn 5s-O 2p Conduction BandConduction Band

Majority Sn 5s Majority Sn 5s charactercharacter

Sn 5s Sn 5s NonbondingNonbonding

Sn 5s-O 2p Sn 5s-O 2p AntibondingAntibonding

XX

MM

xx

yy

Sn 5pSn 5pxx-O 2p-O 2pxx Valence BandValence Band

Majority O 2p characterMajority O 2p character

XX

MM

xx

yy

O 2p O 2p NonbondingNonbonding

Sn 5p-O 2p Sn 5p-O 2p BondingBonding

Sn 5pSn 5pxx-O 2p-O 2pxx

Conduction BandConduction BandMajority Sn 5p Majority Sn 5p

charactercharacter

Sn 5p Sn 5p NonbondingNonbonding

Sn 5p-O 2p Sn 5p-O 2p AntibondingAntibonding

XX

MM

xx

yy

Sn 5pSn 5pyy-O 2p-O 2pyy Valence BandValence Band

Majority O 2p characterMajority O 2p character

XX

MM

xx

yy

O 2p O 2p NonbondingNonbonding

Sn 5p-O 2p Sn 5p-O 2p BondingBonding

Sn 5pSn 5pyy-O 2p-O 2pyy

Conduction BandConduction BandMajority Sn 5p Majority Sn 5p

charactercharacter

Sn 5p Sn 5p NonbondingNonbonding

Sn 5p-O 2p Sn 5p-O 2p AntibondingAntibonding

XX

MM

xx

yy

EEFF

XX MM

Sn 5pSn 5pZZ-O 2p-O 2pZZ

Sn 5pSn 5pZZ-O 2p-O 2pZZ

Sn 5pSn 5pxx-O 2p-O 2pxx

Sn 5pSn 5pxx-O 2p-O 2pxx

Sn 5pSn 5pyy-O 2p-O 2pyy

Sn 5pSn 5pyy-O 2p-O 2pyy

Sn 5s-O 2p Sn 5s-O 2p

Sn 5s-O 2p Sn 5s-O 2p

O 2p O 2p nonbondingnonbonding

Band GapBand Gap

BaBa22SnOSnO44: Band Structure: Band Structure

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

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

1.1. Because of the 3-dimensionality all of the Re based bands will be disperse Because 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).

2.2. The most important interactions are the Re 5d-orbitals, tThe most important interactions are the Re 5d-orbitals, t2g2g ( (//*) & e*) & egg ( (//*), *), rather than the Sn 5s & 5p.rather than the Sn 5s & 5p.

3.3. The Fermi level cuts the The Fermi level cuts the * bands creating a metallic conductor rather than * bands creating a metallic conductor rather than a semiconductor.a semiconductor.

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

W(tW(t2g2g//**) ~ 5 eV) ~ 5 eV

W(eW(eg g //**) ~ 7 eV) ~ 7 eV

ReRe

ReReReRe

ReReOO

OO

OO

OO

ReRe

ReReReRe

ReReOO

OO

OO

OO

**

pointnon-bonding

Overlap at the M-pointOverlap at the M-pointkkxx=k=kyy==/a, k/a, kzz=0=0

Overlap at the Overlap at the -point-pointkkxx=k=kyy=k=kzz=0=0

ReOReO33 BandstructureBandstructure

**

**

The points circled in blue are the The points circled in blue are the bonding counterparts to the bonding counterparts to the

antibonding interactions shown to the antibonding interactions shown to the right.right.

The eThe egg//* band is wider than t* band is wider than t2g2g//* due to larger orbital overlap (spatial).* due to larger orbital overlap (spatial).

O 2p O 2p nonbondinnonbondin

gg

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

that gives the phase of the AO’s as well as the wavelength of the that gives the phase of the AO’s as well as the wavelength of the electron wavefunction (crystal momentum).electron wavefunction (crystal momentum).

• How many lines are there in a band structure diagram?How many lines are there in a band structure diagram? As As many as there are orbitals in the unit cell.many as there are orbitals in the unit cell.

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

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

• How do we distinguish metals from semiconductors and How do we distinguish metals from semiconductors and insulators?insulators? The Fermi level cuts a band in a metal, whereas there The Fermi level cuts a band in a metal, whereas there is a gap between the filled and empty states in a semiconductor.is a gap between 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 This depends on the degree of orbital overlap between building units.the degree of orbital overlap between building units.

Wide bands Wide bands Large intermolecular overlap Large intermolecular overlap delocalized e delocalized e--

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