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Year 2 - Chem > Solid State > Flashcards

Solid State Flashcards

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1
Q

In a short chain system, what occurs when the chain increases?

A

As chain increases the number of nodes increase

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2
Q

What occurs as a short chain of 1s orbitals increase to infinity?

A

In order of low to high E:

Bonding/conduction band - e- delocalied in band of crystal orbitals

Ef - fermi level where chemical potential is mean of HOMO and LUMO

Antibonding band

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3
Q

What does poorer overlap look like in band structure?

A

Poor overlap means narrower band - larger density of states

Usually observed in TM

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4
Q

What does good overlap look like in band structure?

A

Broader band

Higher anti / lower bonding band

Comnmon in main group

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5
Q

What is the density of states in band structure?

A

This is the x-axis, with energy on the y-axis

Measure of # of states in a given energy interval

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6
Q

What is the fermi level equivalent to?

A

Mean of HOMO and LUMO

Equates to chemical potential

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7
Q

In solids how does the band gap change down a group of the periodic table?

A

Down the group, the band gap diminishes as there is decreased bonding-antibonding separation

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8
Q

What is a semimetal in terms of band structure?

A

No band gap

But 0 density at the EF

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9
Q

Why are metal d bands narrow?

A

Due to covalency

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10
Q

What is the band structure of ReO3?

A

Metal conductor - partially filled band & finite density of states at the Fermi level

Good covalency of Re as high oxn states

O 2p band is low conduction band - broad and strongly covalent

Re 5d band is partially filled

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11
Q

What is the band structure of metals?

A
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12
Q

What is the band structure of insulators?

A
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13
Q

What is the band structure of semiconductors?

A
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14
Q

What is the band structure of a semimetal?

A
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15
Q

What occurs to meal d compared to s/p bands across a row of TM?

A

d bands decrease in E faster than s/p bands

Causes insulator when d high at start then metal in middle as partially filled is at Ef , back to insulator when filled lower than Ef

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16
Q

What is a mott-hubbard insulator?

A

When eg band is at fermi level and partially filled as strong e- e- repulsion causes it to split and populate localised only states

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17
Q

How is LCAO adapted for periodic systems?

A

Crystal orbitals are delocalised over the whole solid

e- density distribution is same in each unit cell, ψ*(x+a)ψ(x+a) = ψ*(x)ψ(x)

Therefore where μ is a phase factor, ψ(x+a) = μψ(x) and μ*μ = 1

As periodic are infinite include boundary, ψ(x+Na) = ψ(x) where N is v large

ψ(x+Na) = μNψ(x) and μN = 1

So μ = exp(2πip/N) where p is a quantum number

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18
Q

What definitions are dervied from μ = exp(2π​ip/N)?

A

k = 2π​p/Na

ψk(x+a) = exp(ika)ψk(x)

k is a wavevector in units of inverse length, allows following of Bloch’s theorem, and for defining first brillouin zone

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19
Q

What is Bloch’s theorem?

A

For a wavefn that obeys SE, there is a vector k such that translation by lattice vector a is equivalent to multiplication by a phase factor exp(ika)

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20
Q

What is the first brillouin zone?

A

“unit cell” for band structure

Solutions for crystal orbitals in range -π/a < k < +π/a

Plotted these energies against k (spatial coordinate) which gives band structure for a solid

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21
Q

What is the value of k and λ for anti, non, and bonding orbitals?

A

Antibonding: λ = 2a, and k = +/- (π/a), where adj orbitals are out of phase

Non-bonding: λ = 4a, and k = +/- (π/2a), adj out of phase but further apart so insig overlap

Bonding: λ = ∞, and k = 2π/λ and =0 as no nodes

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22
Q

What is value of the wavevector k at the centre of the 1st brillouin zone?

A

k = 0

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23
Q

How is the band dispersion related to density of states?

A

E of the different wavevectors (k) from the band dispersion is on the y of both the band dispersion and density of states

Then sum the number of states from the band dispersion to give number of states, n(E), on the x-axis

Gives the full density of states graph

24
Q

What is β wrt the band dispersion and density of states?

A

β is a resonance integral

E of orbitals ranges from α-2β (highest E) to α+2β (lowest E) in band dispersion graphs

Therefore the range of energy is 4β

25
Q

Why do metals conduct wrt the first brillouin zone?

A

Applied electric field causes a distortion to distribution of the 1st brilluin zone, which causes net electronic momentum

26
Q

What does k=0, and k=+/- π/a look like in an infinite s-pσ chain?

A

Ionic s-pσ chain in A+X-

27
Q

What is the 1st brillouin zone and band density in the ionic A-X chain with s-pσ bonding?

A
28
Q

How do you consider density of states of crystals?

A

Assume crystal orbitals filled 2e- per orbital in order of increasing E

Consider: Basis orbital energies (anion orbitals less E than metal) & how strongly orbitals overlap

Band widths: σ>π>δ, 4s/4p>3d, and 5d>>4f

29
Q

How is the band gap in semiconductors related to χ?

A

If more than one element (in isoelectronic series)

Larger the Δχ the bandgap is increased

30
Q

What is the energy in the free electron model?

A

Free e- travelling waves: ψk = exp(ikx), where k=2π/λ

electronic momentum is p = h/λ = hk/2π = ℏ*k, so k is a momentum vector

Electronic KE: T = p2/2m = ℏ2k2/2m

Total E, Ek = (ℏ2k2/2m) + V0

Gives parabolic curve for E and k, predicts a single band which is never filled so all solids would be metallic (as more states at higher k)

31
Q

What does Fermi-Dirac stats predict for density of states for a free e-?

A

Sharp cutoff in occupancy with energy

EF is chemical potential at T=0

Not actually sharp cutoff

32
Q

What is the equation for conductivity, σ?

A

σ = 1/ρ = neμ

Where: ρ is resitivity - density of electrons in free electron gas

n - number of carriers

e - charge of carriers

μ - carrier mobility

33
Q

How does conductivity change with temperature?

A

As temperature increases the μ (carrier mobility) decreases and conductivity therefore decreases

34
Q

What occurs to band structures in a magnetic field?

A

Magnetisation proportional to n(EF)

Narrow bands have larger density of states at fermi level so bigger effect

35
Q

What is the effect of periodic potential in a 1D system?

A

With periodic potential of lattice two solutions at k = π/a

Energy gap opens up - one potenital in phase the other is not

Gap depends on strength of potential - in higher dimensions a v strong potential required to open the gap

36
Q

How does the free e- model relate to tight-binding LCAO model in the 1st brillioun zone?

A
37
Q

How does the bond order and band filling change across the TM periods?

A

Increased bond order to middle, thne decreased as e- in antibonding increases

As more e- are inserted into d-orbitals, the d-band is filled

38
Q

What occurs in ferromagnetic metals?

A

Spin polarisation of narrow d bands is spontaneous at ground state and is larger than pauli paramagnet

Only occurs when narrow d band such as Fe, Co, Ni

39
Q

What is the stoner criterion for ferromagnetism?

A

P x n(EF) > 1

Where P is the pairing energy, & n(EF) is density of states at fermi level

When narrow band can occupy just spin up to decrease repulsion, fills higher in band

40
Q

What is the spin of ferromagnetic metals?

A

Spin-polarised so high spin metal due to narrow band width, gives a net magnetic polarisation

Band ferromagnet: Fe, Co, Ni, CrO2

41
Q

How does the splitting energy Δ (oct or tet) relate to band width?

A

Δ is analogue of band width

small Δ is required for ferromagnetic

42
Q

What is the band structure of group 10 metals?

A
43
Q

What is the Peierls distortion?

A

Conjugated C system behaves as 1-D s chain

Gap at Ef opens, which causes electronic energy to decrease

2 bands and therefore 2 orbitals per cell, so doubling of cell and halving of the 1st BZ

44
Q

Why can a 1D metal not exist?

A

Always a distortion possible to open a gap at the Fermi level

Peierls

45
Q

What d states exhibit a JT distortion?

A

Must be degeneracy in either the t2g or eg orbitals

d1, d2, low spin d4, low spin d5, low spin d7, and d9

high spin d4, high spin d6, high spin d7

46
Q

How can the peierls distortion be suppressed in a 1d chain?

A

Fill the bands: Se, Si2- (in BaSi)

Increase dimensionality or increase interaction between chains e.g. polyacene distorts but graphite doesnt (as not 1d)

47
Q

What is a molecular metal / charge transfer salt?

A

No metal in compound

Co-crystallise donor and acceptor molecules, which stack in mostly 1D systems and e- transfered to give 2 partially filled bonds in 1D

48
Q

What are the structures of layered TM sulfides?

A

ZrS2 - CdI2 type, Oh (triangular antiprismatic)

NbS2 (d1), MoS2 (d2)- triangular prismatic

Layered is favouured when sig polarisation effects, layers are edge-shared triagular prisms instead of Oh so different ligand field compared to ZrS2

49
Q

What are the trends in 2nd row sulfides?

A
50
Q

Why does RuS2 form with Ru2+?

A

Cannot stabilise Ru4+

Instead forms [S-S]2-

51
Q

Why is NbS2 metallic and MoS2 insulating?

A

Nb: d1 partially fills the band at EF

Mo: d2 raises EF and fills band to make it an insulator

52
Q

How does rutile VO2 structure change with T?

A

At standard/high T - significant 1D character as Oh along one direction

At low temperature - a bit like peierls distortion opens up gap at fermi level, and decreases energy of states containing e- (through V-V bonds)

53
Q

Which rutile structures are distorted?

A

Distorted - VO2 (low T), NbO2 which are insulators, MoO2, TcO2 which are mettalic

Undistorted - TiO2 which is a band gap insulator, VO2, CrO2, and RuO2 which are metallic, and MnO2 which is a mott-hubbard insulator

54
Q

Why is NbO2 a distorted rutile structure?

A

Better Nb-Nb bonds as 4d is more radially extended

Distorts at a lower temperature

55
Q

Why do electrons localise and therefore deviate from metallic bands?

A

Band instabilities e.g. Peierls distortion - gap at EF which localises e- in a localised bond or CDW

e- e- interactions make a state with localised e- more favourable then delocalise, e.g. Mott-Hubbard insulators

Ferromagnetic metals further deviation - but they remain metallic

56
Q
A

Year 2 - Chem (23 decks)

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