Solid State Flashcards
In a short chain system, what occurs when the chain increases?
As chain increases the number of nodes increase
What occurs as a short chain of 1s orbitals increase to infinity?
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
What does poorer overlap look like in band structure?
Poor overlap means narrower band - larger density of states
Usually observed in TM
What does good overlap look like in band structure?
Broader band
Higher anti / lower bonding band
Comnmon in main group
What is the density of states in band structure?
This is the x-axis, with energy on the y-axis
Measure of # of states in a given energy interval
What is the fermi level equivalent to?
Mean of HOMO and LUMO
Equates to chemical potential
In solids how does the band gap change down a group of the periodic table?
Down the group, the band gap diminishes as there is decreased bonding-antibonding separation
What is a semimetal in terms of band structure?
No band gap
But 0 density at the EF
Why are metal d bands narrow?
Due to covalency
What is the band structure of ReO3?
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
What is the band structure of metals?
What is the band structure of insulators?
What is the band structure of semiconductors?
What is the band structure of a semimetal?
What occurs to meal d compared to s/p bands across a row of TM?
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
What is a mott-hubbard insulator?
When eg band is at fermi level and partially filled as strong e- e- repulsion causes it to split and populate localised only states
How is LCAO adapted for periodic systems?
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
What definitions are dervied from μ = exp(2πip/N)?
k = 2πp/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
What is Bloch’s theorem?
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)
What is the first brillouin zone?
“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
What is the value of k and λ for anti, non, and bonding orbitals?
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
What is value of the wavevector k at the centre of the 1st brillouin zone?
k = 0
How is the band dispersion related to density of states?
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
What is β wrt the band dispersion and density of states?
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
Why do metals conduct wrt the first brillouin zone?
Applied electric field causes a distortion to distribution of the 1st brilluin zone, which causes net electronic momentum
What does k=0, and k=+/- π/a look like in an infinite s-pσ chain?
Ionic s-pσ chain in A+X-
What is the 1st brillouin zone and band density in the ionic A-X chain with s-pσ bonding?
How do you consider density of states of crystals?
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
How is the band gap in semiconductors related to χ?
If more than one element (in isoelectronic series)
Larger the Δχ the bandgap is increased
What is the energy in the free electron model?
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)
What does Fermi-Dirac stats predict for density of states for a free e-?
Sharp cutoff in occupancy with energy
EF is chemical potential at T=0
Not actually sharp cutoff
What is the equation for conductivity, σ?
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
How does conductivity change with temperature?
As temperature increases the μ (carrier mobility) decreases and conductivity therefore decreases
What occurs to band structures in a magnetic field?
Magnetisation proportional to n(EF)
Narrow bands have larger density of states at fermi level so bigger effect
What is the effect of periodic potential in a 1D system?
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
How does the free e- model relate to tight-binding LCAO model in the 1st brillioun zone?
How does the bond order and band filling change across the TM periods?
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
What occurs in ferromagnetic metals?
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
What is the stoner criterion for ferromagnetism?
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
What is the spin of ferromagnetic metals?
Spin-polarised so high spin metal due to narrow band width, gives a net magnetic polarisation
Band ferromagnet: Fe, Co, Ni, CrO2
How does the splitting energy Δ (oct or tet) relate to band width?
Δ is analogue of band width
small Δ is required for ferromagnetic
What is the band structure of group 10 metals?
What is the Peierls distortion?
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
Why can a 1D metal not exist?
Always a distortion possible to open a gap at the Fermi level
Peierls
What d states exhibit a JT distortion?
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
How can the peierls distortion be suppressed in a 1d chain?
Fill the bands: Se, Si2- (in BaSi)
Increase dimensionality or increase interaction between chains e.g. polyacene distorts but graphite doesnt (as not 1d)
What is a molecular metal / charge transfer salt?
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
What are the structures of layered TM sulfides?
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
What are the trends in 2nd row sulfides?
Why does RuS2 form with Ru2+?
Cannot stabilise Ru4+
Instead forms [S-S]2-
Why is NbS2 metallic and MoS2 insulating?
Nb: d1 partially fills the band at EF
Mo: d2 raises EF and fills band to make it an insulator
How does rutile VO2 structure change with T?
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)
Which rutile structures are distorted?
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
Why is NbO2 a distorted rutile structure?
Better Nb-Nb bonds as 4d is more radially extended
Distorts at a lower temperature
Why do electrons localise and therefore deviate from metallic bands?
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
