1.1) Band theory

How do we know if an element conducts or not

atomic config + crystal structure + chemical bonding (overlap of atomic orbitals)

Simple rules for molecular orbital theory

no AO = no MO

only consider valence AOs

consider relative energy of valence AOs on the atoms and their symmetry and overlap

greater variety of MOs in polyatomic molecules

lager molecules = more MOs and closer spacing

MOs generally extended over all constituent atoms

crystal orbitals extend throughout the molecule/solid are a property of the molecule/solid not individual atoms

What affects the width of an energy band

interatomic spacing which changes the degree of overlap; large overlap = wide bands

Bands in metals

valence s and p orbitals give large overlaps (6-8eV), contracted core orbitals give very narrow bands and do not contribute to bonding; partially filled bands can conduct electrons

What is the Fermi level

where probability of electron occupation = 1/2

What are the two types of band structures in metals

partly full bands eg Cu; overlapping bands where both are partly occupied eg Mg where 3s and 3p overlap (3s is technically full with 2 e but the combined 3s3p band is not full)

Insulators

many s and p block metal oxides are white insulating solids as the VB is full and CB is empty with a large energy gap between them

Why is Si semiC while Al is metal

Al has one fewer e in its outer shell (3 vs 4) so doesn’t have enough to form covalent bonds and a band structure that doesn’t overlap

Semiconductors

full VB, empty CB and E_g<3eV

Points about band theory

ignores e-e repulsion effects within bands (works well for wide bands but not for narrow ones); band width is related to relative energy of orbitals involved and degree of overlap; actual E_f depends on relative DoS in VB and CB but often approx 1/2Eg

Equation for conductivity and it varies in metals and SC

σ = nqμ; Metals: n is large, μ decreases slightly with temp; intrinsic SC (and insulators): n is small but increases expo with T, μ decreases slightly with T

Arrhenius plot

σ = A exp(-E/kT), plot ln(σ) vs 1/T: metal straight line w small positive gradient (higher σ at lower T); SC: straight line w large negative gradient; Insulator: larger negative gradient

Why does free electron theory break down for transition metals

free e theory breaks down; d orbitals are more contracted so do not overlap strongly; d band has high DoS due to 10 e/atom within narrow energy range; e-e repulsion becomes important

Describe the bonding behaviour of d band electrons

bottom of the band: bonding, e here stabilise the bonding; middle: non-bonding, little effect on cohesion; top: antibonding, adding electrons destabilises the metal

How does bond strength change across the table

d band becomes narrower across 3d metals as larger nuclear charge causes d orbital to contract so they overlap less with each other; bond strength increases towards the middle then decreases after as e have antibonding effect

Why do 3d and 4d show local minima at d5

strong magnetic effects: at half full you have one e in each orbital maximising exchange energy which stabilises the local atomic config and makes e resist delocalising into the band weaking the bonds; this doesn’t occur in 5d metals as spin orbit coupling is stronger (makes spin not a good quantum number reducing effect of exchange energy)

Band vs bond model

band: delocalisation of e in solid; bond: localisation of e on atoms (requires hopping)

Transition metal oxides

wide variety of oxidation states, variety of electrical properties (insulators→semiconductors); most have rocksalt structure (except CuO and ZnO); number of d electrons largely affects properties

Spatial orientation of d orbitals in TMOs

rocksalt MO: M2+ occupies octahedral sites with CN=6 to O2- ions, all AO increase in energy; eg set: dz2 and dx2-y2 are highest in energy as they point directly along the bond axes; t2g set: other orbitals are lower in energy than eg

Crystal field splitting

energy splitting between eg and t2g sets; depends on charge and type of both ions

Why is TiO a conductor but NiO an insulator

TiO has 2 d e so dxy is partially full and these orbitals overlap between Ti ions and so its a conducto; the t2g band is 1/3 full;

NiO has 8 d e so dxy, dyz, dxz are all full and dx2-y2 and dz2 are partially full but they are aligned with O ions so don’t overlap between Ni ions, so it’s an insulator; eg band is partially full; unpaired e in these orbitals can couple with adjacent O p orbital e in the super exchange mechanism (makes NiO an antiferromagnetic insulator)