Sinclair- Band Theory and Hopping

Question 1

General requirement for high conductivity in TMOs

Answer 1

Need mixed metal valence TMs. Like Cu2+/Cu3+

Question 2

Methods of getting mixed valency in TMOs

Answer 2

Process in an oxidising or reducing atmosphere. Chemical doping

Question 3

Oxidation method for mixed valency

Answer 3

Sample gains oxygen to become non-stoichiometric. Example NiO becomes NiO1+δ when processed at 1500C in air. Get Ni2+ (d8) and Ni3+ (d7). Goes from pale green insulator to black semiconductor. Right of d block tend to be oxidised

Question 4

Reduction method for mixed valency

Answer 4

Sample loses oxygen to become non-stoichiometric. Example TiO2 goes to TiO2-δ when processed at 1500C in Ar (inert atmosphere). Get Ti3+ (d1) and Ti4+ (d0). Goes from white insulator to blue semiconductor. Left of d block tend to be resuced

Question 5

Normal difference in radii allowed for doping

Answer 5

About +/-15%

Question 6

What does aliovalent doping mean?

Answer 6

Dopant has different charge to host ion. Need for doping method for mixed valency

Question 7

Acceptor doping for mixed valency

Answer 7

Dopant ion of lower charge. E.g Li+ doping of NiO turns it into a black semiconductor. Get mixed Ni2+ and Ni3+. Hole doping, p-type conductor

Question 8

Donor doping for mixed valency

Answer 8

Dopant ion of higher charge. E.g Nb5+ doping of Ti site in StTiO3 can induce semiconductivity. Get mixed Ti3+ and Ti4+. Electron doping, n-type conductor

Question 9

What is a hopping semiconductor?

Answer 9

One in which electrons or holes can hop between ions of the same type but different valence.

Question 10

How is conductivity of hopping semiconductor influenced by temperature?

Answer 10

σ=nqμ still holds. Hopping is thermally activated so mobility of charge carrier is temperature dependent (contrast to band model where μ of charge carriers has little temperature dependence).

Question 11

What happens when NiO heated in air?

Answer 11

Normally rocksalt structure. It picks up O2 molecules which adsorb on the surface, dissociate and pick up 2e- to form O2-. The electrons come from oxidation of Ni2+ to make Ni3+ (d7). To partially balance extra layer of O2- ions at surface Ni2+ ions migrate leaving cation vacancies throughout the crystal. Each O2- generated creates one Ni vacancy somewhere and two Ni3+ ions.
Ni(1-x)O becomes Ni2+(1-3x)Ni3+(2x)O

Question 12

How does hopping conduction work in NiO after heating in air?

Answer 12

Electrons hop from Ni2+ to an adjacent Ni3+ (or holes hop in opposite direction. Electrons localised on Ni2+ so concept of conduction and valence bands doesn’t apply. Main process
Ni2+ -> Ni3+ + e-. Activation energy needed to cause hopping. Number of charge carriers, n, is [Ni3+]

Question 13

NiO heated in air graph of lnσ vs 1/T

Answer 13

Straight line negative gradient. Slope -E/k

Question 14

Why is it difficult to control degree of oxidation for NiO heated in air?

Answer 14

Depends on oxygen partial pressure, temperature and solid/gas equilibria/kinetics. Means n varies and therefore so does σ

Question 15

How to control oxidation of NiO

Answer 15

Dope with Li+ to obtain Ni(1-x)LixO.
2Ni2+ -> Ni3+ + Li+. Compound becomes
Ni2+(1-2x)Ni3+(x)Li+(x)O
Means [Ni3+] and σ controlled by x so Li+. It forms a solid solution with rocksalt structure and Ni3+, Ni2+ and Li+ distributed at random over crystal octahedral sites. Fixed x at fixed field E means n constant. μ is lowish and Ea 0.2 to 1eV

Question 16

Why can Li+ doped NiO be used for thermistors?

Answer 16

σ=f(T). Can use its lnσ vs 1/T graph. Measure conductivity and read off temperature. σ reproducible over 1000s of cycles

Question 17

Polaron model

Answer 17

Polaroid Ar due to interaction of mobile electron (or hole) with lattice ions which cause localised lattice distortions. When distortion sufficiently strong, electron or hole may be trapped at a particular lattice site via polarisation and can only move by thermally activated hopping through the solid.

Question 18

Polarons in Fe3O4

Answer 18

Fe2+ (d6) next to Fe3+ (d5). Fe2+ is larger ion so has longer bond lengths with O. Electron can move from Fe2+ to Fe3+ much quicker than cation and anion movement. The hopping e- can polarise the surrounding lattice. As electron moves through the solid the distortion follows it. This lower mobility of the e- and increases its effective mass m*. Is called a polaron and can be considered as a quasi-particle

Question 19

The two extreme cases for polarons

Answer 19

e- localised on either metal ion
e- delocalised between metal ions
An oxygen separates the adjacent metal ions

Question 20

Energy vs distortion coordinate, D, graph

Answer 20

Two y=x^2 curves. One has minimum on right and the other on the left but both same level above x axis. The curves cross at the y axis. Activation energy height difference between minimum and point of intersection. Means Ea is proportional to D^2 and small changes in ionic radii have dramatic effect on Ea for hopping conduction

Question 21

What is distortion coordinate equal to?

Answer 21

Difference in ionic radius between the two metal ions

Question 22

Normal and inverse spinel structures

Answer 22

Normal is [A][B2]O4 where A is 2+ and in tet sites, B is 3+ and in oct sites.
Inverse is [B][A,B]O4 where A is 2+ and in half of oct sites, B is 3+ with half occupying all tet sites and half occupying half of oct sites.
Example inverse is [Fe3+][Fe2+,Fe3+]O4
Octahedral sites always half full and tetrahedral sites 1/8 full

Question 23

What happens if the localised state for a polaron is more stable?

Answer 23

The e- is effectively trapped by the local distortion it creates around a given lattice site. This is the small polaron model for an e- in a metal orbital in an oxide. The extra electron changes the oxidation state of the atom in the solid. This changes the ionic radius which produces a local distortion that can trap the charge l

Question 24

How to identify distinct M2+ and M3+ ions

Answer 24

Use x-ray absorption spectroscopy (XAS) to examine the crystallography.

Question 25

What happens if the e- or h+ polaron can become delocalised?

Answer 25

All ions have some fraction of oxidation state. For Fe3O4 all ions on oct sites appear identical as Fe2.5+ instead of distinct Fe2+ and Fe3+

Question 26

Phase transition in Fe3O4

Answer 26

Structural phase transitions often induce changes in electrical properties. Changes from an insulator to a semiconductor at 120K and conductivity increases by 3 orders of magnitude. At higher temperatures all Fe-O bonds in oct sites are the same Fe2.5+-O and there is easy e- hopping process and is semiconductor. At lower T cooperative lattice distortions create distinct oct sites so distinct Fe2+-O and Fe3+-O bonds, e- hopping difficult, insulator

Question 27

Why is inverse spinel Fe3O4 a semiconductor/metal and normal spinel Mn3O4 a poor semiconductor/insulator?

Answer 27

In inverse Fe3O4, there are short distances between the oct sites which have Fe2+ or Fe3+ so the hopping
Fe2+=Fe3+ + e- is easier
In normal Mn3O4the distance between tet and oct sites is long so the hopping
Mn2+ = Mn3+ + e- is harder