Sinclair- Band Theory and Jahn Teller Effect and Insulators

Question 1

Jahn-Teller theorem

Answer 1

Electron configurations with unequal occupancy of degenerate orbitals are not stable

Question 2

Jahn-Teller effect

Answer 2

A geometric distortion of a non-linear molecular system that reduces it’s symmetry and energy. The distortion is most commonly observed among octahedral complexes where the two axial bonds can be shorter or longer than the equatorial bonds. Effects depends on electronic state of system (for TMOs corresponds to number of d electrons on TM cation).

Question 3

JT elongation

Answer 3

Axial bonds are longer than equatorial. Stabilisation of d orbitals with a z component means all orbitals with a z component have lower energy than the other on in t2g and eg. Other orbitals are destabilised (higher energy). Lowers electrostatic repulsion between electron pair on non-metal ligand and any electrons in orbitals with z component thus lowering energy of complex.

Question 4

JT compression

Answer 4

Equatorial bonds longer than axial. De-stabilisation of d orbitals with z component (higher energy). Orbitals without z component in t2g and eg stabilise (lower energy)

Question 5

What are elongation and compression effects dictated by?

Answer 5

The amount of overlap between the metal and ligand orbitals

Question 6

How does distortion from JT effect vary?

Answer 6

Varies greatly depending on type of metal and ligands. In general the stronger the metal-ligand orbital interactions are, the greater the chance for a JT effect to be observed

Question 7

Structure of La2CuO4

Answer 7

Hybrid of rocksalt (AX) and perovskite (ABX3) so is (LaO)(LaCuO3). The Cu is 2+ with d9. Orthorhombic cell. Corner sharing CuO6 octahedra which are elongated

Question 8

What does band theory predict about La2CuO4?

Answer 8

Predicts it to be a metal. Is d9 cation. Orbitals with z component stabilise. Full band for t2g orbitals. Full narrow band for d(z^2) orbital. Half full narrow band for d(x^2-y^2) orbital. This is wrong as it is actually an antiferromagnetic insulator. In narrow bands electron-electron repulsion effects become significant.

Question 9

Features of Mott-Hubbard model

Answer 9

Electron-electron repulsion effects cause narrow band to split into 2 sub bands. Lower band is lower Hubbard band (LHB) and upper band is upper Hubbard band (UHB). The total narrow band has width W. The peaks of the sub bands are separated by energy gap U. Ef in middle of U

Question 10

When is something a metal or insulator for Mott-Hubbard model?

Answer 10

If W>U there will be finite overlap between sub bands and metallic ground state will result.
If W is less than U the LHB and UHB will not overlap and the material will be insulating

Question 11

How to modify W in TMOs

Answer 11

Chemical doping (size and/or charge that alter bond lengths/angles and/or [e/h]). Can switch metal to insulator or reverse

Question 12

Mott-Hubbard model for La2CuO4

Answer 12

W is less than U and the e-e repulsion effects are strong and two, one electron bands colour rather than the formation of a half filled d(x2-y2) band. Means the single d(x2-y2) electron is strongly localised on the Cu ions and that a large activation energy U inhibits any electron migration (and therefore conduction) via the Cu ions

Question 13

Why is La2CuO4 AFMI?

Answer 13

Spin-spin correlations that arise through PEP serve to render the insulating ground state antiferromagnetic. Superexchange mechanism between Cu2+ 3d(x^2-y^2) ions via O2- 2px and 2py ions.

Question 14

What does the covalency between Cu and O ions mean?

Answer 14

Mixing of O 2p-band and Cu 3d(x^2-y^2) LHB in the final band picture.

Question 15

Overall band picture for La2CuO4

Answer 15

LHB for Cu 3d9 in middle and sticks out of O 2p band which is wider. UHB above and empty with no overlap. Energy gap Eg between top of O 2p band and bottom of UHB less than U so is known as a charge transfer insulator (CTI) rather than a Mott-Hubbard insulator

Question 16

Charge transfer energy

Answer 16

Introduced by Zaanen, Sawatsky and Allen (ZSA). Symbol Δ. Is the cost of exciting an electron from a filled band associated with the highest occupied ligand states to the lowest unoccupied metal state usually UHB.

Question 17

Two types of insulating ground state in ZSA scheme

Answer 17

Charge-transfer insulator where U>Δ>W
Original Mott-Hubbard insulator where Δ>U>W
U and Δ measured between peaks of bands

Question 18

When are CTI and MHI descriptions appropriate generally?

Answer 18

For transition metal compounds, TM near end of series CTI is appropriate. TM near beginning of series MHI holds. This is in general

Question 19

Why are TM oxides of interest?

Answer 19

For the oxide ligand the ZSA parameters Δ, U and W are close in energy and relative magnitudes can be manipulated through changes in materials chemistry. Hole doping can be used in MHI and CTI to create metallic conduction

Question 20

Doping of NdTiO3

Answer 20

Perovskite. Is MHI but A site alkaline earth metal doping oxidises some Ti3+ to Ti4+ (d0) for charge neutrality which creates holes in LHB and gives rise to metallic conduction.
Nd3+ + Ti3+ = Sr2+ + Ti4+
Ef now where LHB is now filled up to (not fully)
Full O 2p band below mostly full LHB below UHB

Question 21

Doping of La2CuO4

Answer 21

Dope with Sr2+ to oxidise some Cu2+ to Cu3+ (d8) for charge neutrality creating holes in the hybrid Cu 3d / O 2p band. Transforms from AFM CTIB into metal.
La3+ + Cu2+ = Sr2+ + Cu3+
When doped by sufficient amount (0.03) metallic behaviour switches to superconducting at 30K.
Holes could also be due to reduction of O2- to O-.
Ef now near top of hybrid band (where it is filled up to)

Question 22

Another way of defining what is an insulator

Answer 22

How it’s resistivity varies with temperature. If resistivity decreases as T increases then is insulator. If increases then is metal

Question 23

RETiO3 compounds

Answer 23

Distorted perovskite structures at 300K. Contains rare earth metal of A site. The series has U about 4eV (essentially fixed) but W can be changed.
La, CE, Pr, Nd, SM, Gd

Question 24

What happens when RE ion size decreases in RETiO3?

Answer 24

Decrease in ionic radii across lanthanide series. Leads to increased internal stress in the crystal structure that is accommodated by decrease of the Ti-O-Ti bond angle which results in reduction of the width of the lower Hubbard band, W. Means Mott-Hubbard gaps increase. Resistivity increases by 3 orders of magnitude. Atom on A site gets too small and its CN reduces from 12 to about 8.