LectureStructure
8: The effects
electron repulsion
Electronic
andofChemistry
of Solids
A course given as part of the joint Masters programme
Condensed Matter Science
of the Universiteit van Amsterdam
and the Vrije Universiteit, Amsterdam.
Mark Golden (UvA
(UvA)) & Bernard Dam (VU)
Chapter 5 (lecture 8): The Effects of Electron Repulsion
Mark S. Golden, Van der Waals-Zeeman Institute, Universiteit van Amsterdam,
Room 2.09, Valckenierstraat 65, 1018 XE Amsterdam, Tel.: 020 525 6363.
[email protected]
http://www.science.uva.nl/research/wzi/cmp
Lecture
Lecture
7: finishing
8: The off
effects
bandoftheory
electron
(from
repulsion
last week)
Crystal orbitals in one dimension
Lecture 6
Bloch functions
LCAO theory for monoatomic chain
The binary chain
NearlyNearly-free electron model
Effective mass
Electronic conductivity
Band structure and spectroscopy
& lecture
7
Two dimensions
LCAO square lattice
Nearly free electrons in 2D
The Hall effect
Graphite
Band structure from ARPES
Three dimensions
BS of simple solids
Metals and alloys
Orbital and overlap populations
http://www.science.uva.nl/research/cmp/Golden/Teaching
Lecture 8: The effects of electron repulsion
Fermi surfaces and HumeHume-Rothery rules
weak periodic potential not enough to form band gap in 3D
it does modify N(E)
Lecture 8: The effects of electron repulsion
E.g. HumeHume-Rothery rules: f.c.c.
f.c.c. vs. b.c.c.
b.c.c.
Aim of the game:
crystal structure will tune itself so that the band filling
exploits the N(E) peaks (and dips) to lower total E
when FS touches BZ face, E(k
E(k) flattens out
flatter E(k
E(k) → peak in the DOS, followed by a drop
f.c.c.
f.c.c.
b.c.c.
b.c.c.
for n.f.e.
n.f.e. metals, positions of DOS peaks and dips should be
linked to the structure (real lattice ↔ reciprocal lattice)
….simple justification of the HumeHume-Rothery rules:
crystal structure of an alloy ∝ average no. of valence e
Lecture 8: The effects of electron repulsion
Structure vs. N(E)
e/atom
3
4
5
6
7
8
9
10
11
predicted
hcp
hcp
bcc
bcc
hcp
hcp / fcc
fcc
fcc
fcc
alloy with 1.4 e/atom 'fits' f.c.c.
f.c.c.
well
add more e (i.e. up to 1.5
e/atom), then e have to go into
the dipping N(E) → quickly fills
higher E states → b.c.c.
b.c.c. better
Lecture 8: The effects of electron repulsion
The ideal chemical electronic structure machine…….
However, in the transition metals, the DOS (not just n.f.e.
n.f.e.like spsp-bands but including d bands) can help explain the
crystal structure……
hcp
bcc
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
kn.f.e.
n.f.e. touches
BZ boundary
f.c.c.:
f.c.c.:
n.f.e.
n.f.e. Fermi surface touches
BZ edge for 1.36 e/atom
b.c.c.:
b.c.c.: 1.48 e/atom
chemical
composition
(structure)
real
hcp
hcp
Mn
bcc
bcc
cubic (a mess)
bcc (fcc for T>T
T>TCurie)
fcc
fcc
fcc
fcc
bonding, gaps, frontier orbitals/levels,
orbitals/levels, e, m & o properties…….
1
Lecture 8: The effects of electron repulsion
How to get from spaghetti…… to chemistry
Orbital (and overlap) populations from band structure
e.g. ReO3
Lecture 1: Introduction
Solids
Chemical classification:
Lecture 8: The effects of electron repulsion
bonding
molecular
ionic
→ atomic & bonding character of bands
…..not much
chemical insight
...that's
better !
covalent
metallic
© Mark S.
Golden 2002
C M S M a s t e r s : Electronic Structure and Chemistry of Solids.
Lecture 8: The effects of electron repulsion
Orbital (and overlap) populations from band structure
consider again the 1D -A-B-A-B- chain….
ψ k = ∑n=1 exp(ikna)[ak χ ( A) n + bk χ ( B) n ]
N
Lecture 8: The effects of electron repulsion
Orbital (and overlap) populations from band structure
….still better is to show the degree of contribtion from each
AO to the band
vs. k
summed over k in the N(E)
amplitudes of the AO of atom A and B in the crystal orbital
in Hückel theory (non(non-orthogonality neglected)
2
(ak) = electron density on atom A
or DOS
partial densities of
states: pDOS
(bk)2 = electron density on atom B
sum over all occupied orbitals of each type:
population of each AO
Lecture 8: The effects of electron repulsion
bonding
antibonding
Lecture 8: The effects of electron repulsion
Modern approach:
sophisticated codes based on the Local Density
Approximation (LDA
(LDA)) to Density Functional Theory (DFT
(DFT))
self consistent picture of total electron density
project this result back onto the basis set used to construct
the potential (e.g. LMTO, LCAO)
Last week's question
atomic character of the bands vs. k and
partial densities of states: pDOS
very useful approach for:
sorting out the interesting strands of spaghetti from the rest
i.e. those that control the electronic properties
comparison with site and orbital selective spectroscopies (XAS,
(XAS, XES)
2
Lecture 8: The effects of electron repulsion
Lecture
The effects
electron
This8:week's
q uof e
s t irepulsion
o n
For Thursday: carbon nanotubes
Explain why an
isolated 'armchair'
SWCNT is expected to
be metallic, and some
zigzig-zag tubes are not
look it up on www
fullerenefullerene-like
structures made
purely of carbon
conceptually, a CNT
is a 'rolled'rolled-up’
graphene sheet with
endend-caps
Cees Dekker
tip:
it involves the
graphene
bandstructure and
finitefinite-size effects
(k(k-quantisation)
from
Cees Dekker
Cees Dekker
© Mark S. Golden
© Mark S. Golden
Lecture 8: The effects of electron repulsion
Graphite
Lecture 8: The effects of electron repulsion
finite size → k-quantisation parallel to ΓM for
armchair tube:
Graphite
'armchair' direction is ΓM
K
M
Γ
M: k=(2π
=(2π/√3a, 2π
2π/3a)
K
k⊥ . 2π
2πr = n2π
n2π
k⊥ = n/r
1/r
always an allowed kk-value (0) crossing the K point
→ metallic band structure
Lecture 8: The effects of electron repulsion
Graphite
finite size → k-quantisation parallel to ΓK
for zigzig-zag tube:
Graphite
finite size → k-quantisation parallel to ΓK
for zigzig-zag tube:
M K
M K
Γ
Lecture 8: The effects of electron repulsion
M
K
for this rNT, no allowed kk-value crosses the K point
→ semiconducting band structure
Γ
M
K
for this rNT, there are allowed kk-values crossing the
K point
→ in this case, metallic band structure
3
Lecture 8: The effects of electron repulsion
The Hubbard Model
Lanthanides
Lecture 8: The effects of electron repulsion
Now,
lecture 8
4f orbitals and the Hubbard U
Divalent compounds of the lanthanides
Electron repulsion
Transition metal compounds
Monoxides
General trends
Properties of localised electrons: ligand field effects
Magnetic properties of localised electrons
Other theories of electron repulsion
Wigner crystallisation
The polarisation catastrophe
MetalMetal-insulator transition at high and low densities
Lecture 8: The effects of electron repulsion
What's so interesting about electron repulsion ? I
Understanding how electrons behave in (real) solids
is one of the outsanding challenges facing physics……
…….it represents the many body problem at a huge
scale, coupled to extreme complexity
Lecture 8: The effects of electron repulsion
What's so interesting about electron repulsion ? II
In fact a number of effects are only aproximately (or not at
all) treated in band theory:
Electron repulsion
Coupling of the ee- system to the lattice
Defects and doping
strenuous and highhigh-level intellectual challenge
at the cutting edge of theory and experiment
potential spinspin-offs for society are enormous
Surfaces
Cox
Ch.
Ch. 5, 6, 7
Excited state properties (excitons
(excitons etc.)
In some cases, these effects hold the key to the interesting
electronic, magnetic and optical properties of novel materials
(think of HTSC, CMR, polaronic systems…..)
Lecture 8: The effects of electron repulsion
In general….
……a
……a theory is not much practical use if one doesn't have a
feel for it's limitations: where is doesn't work.
Lecture 8: The effects of electron repulsion
When does electron repulsion matter ?
to a certain degree, electron repulsion is 'built into' band theory,
theory, as
the potential felt by 'the' electron includes the average repulsion
repulsion from
the other electrons
LCAO: hidden in atomic orbital energies
n.f.e. hidden in the periodic potential
Band theory (and in particular DFT) is
such a cenral tool in condensed matter
science, it's important to also discuss
the cases where it struggles or fails,
rather than to only concentrate on the
myriad cases for which it works.
repulsion needs to be more specifically taken into
account when the bands are narrow
an average potential is not sufficient to capture the
essential physics
transition metal and lanthanide compounds
plus: molecular, and low dimensional systems
4
Lecture 8: The effects of electron repulsion
Hubbard model
Lecture
. 5): effects
The Effects
of Electron
Repulsion
Lecture8 (Ch
8:
The
of electron
repulsion
(Ch.
nothing to do do with Ron L.
full treatment of repulsion still around the corner…….
……..approximate approaches are used
Hubbard model
Now consider SMALL overlap between the orbitals
low β, narrow band, high m*
very widely applied approach is Hubbard model:
important electron repulsion effects only felt between
electrons (holes) on the same atom
ground state: one electron localised on each atom:
….this is an approximation, but it is the intraatomic
intraatomic
repulsion that kills band theory in many cases
as ever, start simple:
1D chain of atoms with an ss-orbital with one e per site
band theory says: metal with halfhalf-filled band ]
[
Lecture
. 5): effects
The Effects
of Electron
Repulsion
Lecture8 (Ch
8:
The
of electron
repulsion
(Ch.
WHY ?
Lecture 8: The effects of electron repulsion
Ionic bonding
Hubbard model
Remember ? From Ch.
Ch. 3……..
electron repulsion leads to localisation:
localisation:
U
Electron
affinity
Ionisation
energy
Na + 5 eV--> Na++e
Cl+e --> Cl- + 3.5 eV
E required to remove an electron from an orbital = I,
to add it to another site = A
U=I-V
Lecture 8: The effects of electron repulsion
Hubbard model
Lecture 8: The effects of electron repulsion
Hubbard model
U = I - A = repulsion between two ee- in same atom
e.g. single band Hubbard model
e.g.
H
H
H
H
H
if our atoms in the chain were hydrogen…….
what would U be ?
U = II-A = 13.6 eV - 0.8 eV = 12.8 eV
transfer integral
large = strong intersite
overlap, large W
Hubbard U
punishes double occupancy
repulsion drives a halfhalf-filled band insulating, when the
intersite interaction (and thus W ) is small
5
Lecture 8: The effects of electron repulsion
Hubbard model
Lecture 8: The effects of electron repulsion
competition:
Hubbard model
no DOS anymore!
electron removal or electron addition spectral weight
EF
U = E0(N(N-1)
E=E=-A a single eeadded to give one
doubly occupied
orbital
E
U wants to localise
W rewards delocalisation
U
-A
-I
+ E0(N+1)
- 2E0(N)
EF
for W>U:
delocalisation wins
out → no gap
EF
U shows up when
U
IPES
XAS
PES
altering the
occupation
(e.g. transport,
spectroscopy)
Lecture 8: The effects of electron repulsion
Hubbard model
E=E=-I: all orbitals
singly occupied
W=U
W
this MottMott-Hubbard gap is NOT the
same thing a LCAO bandband-gap, but
is due to e-e repulsion
Lecture 8: The effects of electron repulsion
Hubbard model
in practice, polarisation screening of U takes place in the
solid
Upper levels: upper subsub-band, upper
Hubbard band (UHB)
Ueff < UI-A
E
in addition, this polarisation screening ↑, as W ↑
-A
-I
EF
one must pay the MottMott-Hubbard
gap to excite an electron into
another orbital:
W=U
W
→ direct relevance to transport
Lower levels: lower subsub-band,
lower Hubbard band (LHB)
Lecture 8: The effects of electron repulsion
modify diagram of E vs.
vs. W
as W increases, U decreases
MottMott-Hubbard gap closes earlier than in 'atomic'
version
Lecture 8: The effects of electron repulsion
without effects of
polarisation
Lanthanides
including polarisation
screening:
E
unoccupied levels ↓E
occupied levels ↑E
U reduced
6
Lecture 8: The effects of electron repulsion
Lanthanides and rare earth compounds
Lecture 8: The effects of electron repulsion
4f orbitals
radial components of
atomic wavefunctions:
wavefunctions:
Tm4f
lanthanides:
lanthanides:
what are they ?
what's special ?
Tm5dTm6s
lanthanides have partiallypartially-filled 4f orbitals
4f orbitals are very compact
in the solid:
4f's behave more like corecore-levels, or atomic levels
Pic.: Jensen and Mackintosh: Rare Earth
Magnetism: Structures and Excitations, OUP, 1991
Lecture 8: The effects of electron repulsion
The lanthanides and their compounds
Lecture 8: The effects of electron repulsion
The lanthanides and their compounds
only small (if any) overlap between 4f and the 'outside world'
very small band widths, W4f
crystal field splittings very small: ca. 10 meV or 100cm-1
remarkable chemical similarity:
4f
narrow 4f bands make RE systems the
clearest examples of MottMott-Hubbard physics
U
in spectroscopy: often strong intraintra-atomic
multiplet structures visible
remember ?
Chapter 2: Spectroscopic Methods
Using final state multiplets to determine valencies
changing no. of 4f ee- doesn't affect reactivity
Tm@C82
4f configuration does affect the 'valence', or oxidation state:
Tm is
purely
divalent
RE generally trivalent:
4f13
initial state
this does not necessarily mean Ln3+
three valence (i.e. nonnon-4f) electrons in 5d & 6s, others in 4f shell
air stable
PES
Tm4d→
Tm4d→4f excitations
Phys. Rev. Lett.,
Lett., 79,
79, 3026 (1997)
Lecture 8: The effects of electron repulsion
The lanthanides and their compounds
Ln:
Ln: Uff values in the elemental metals
4f levels have high cross section for electron removal or
addition involving high energy photons
PES
IPES
Uff
Lecture 8: The effects of electron repulsion
all Uff > 5 eV
extra
Hund's rule
4f7 penalty !
Wf < 0.1 eV
example for Gd metal
Uff in solid ~12 eV
cf.
cf. 25 eV in atom
repulsion depends
upon 4f filling
fine structure
metallic nature is from 5d and 6s overlap
7
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Uff values vs.
vs. I 's and A 's for the 4f levels
'Oxidation states' of lanthanides in solids
Three contributions :
4f n → 4f n+1
↑nuclear charge:
↑I and ↑A
break at halfhalf-filling
greater for 2nd half of the
series
in the solid, the vast majority of RE systems are trivalent with
4fn'6s25d1
can participate in bonding
three valence electrons
halfhalf-filling (4f7) is specially
stable, thus to go from 4f7
to 4f6 or 4f8 costs 'extra' E
f-f angular momentum
coupling
→curvature
in the gas phase, many RE are stable in a 4fn6s2 configuration
beware: chemists and physicists can hang different meanings
on terms like valence & oxidation state
physicists define valence in RE system w.r.t. 4f count
4f n → 4f n-1
metals
Tb gas phase
in solid
TbS,
TbS, although for Cox 'formal oxidation state +2' has 8 4f
electrons and thus a physicist would call it trivalent
Lecture 8: The effects of electron repulsion
'Oxidation states' of lanthanides in solids
4f9 6s2
4f8 6s2 5d1
Lecture 8: The effects of electron repulsion
4fn=trivalent ion
'Divalent' compounds of lanthanides
in the solid, the vast majority of RE systems are trivalent
4fn6s25d1
4fn→4fn+1
Exceptions:
4fn→4fn+1
Ce,
Ce, Pr: 4f0, 4f1 = tetravalent
Eu:
4f7 = divalent
Eu:
(in order to please Hund)
Hund)
Tm:
4f13 = divalent, partially
Yb:
4f14 = divalent
Yb:
(in order to fill shell)
4fn→4fn+1
La
Eu,
Eu, Yb
Tm
no RE valence e outside
4f → nonnon-metal
metallic from RE 5d band
Lecture 8: The effects of electron repulsion
Mixed valence
mixed valent / mixed
configuration
Lecture 8: The effects of electron repulsion
TK=
35K
Kondo effect in heavy fermions
e.g. CeSi2
e.g. TmSe
all Tm are structurally identical
spectroscopy says:
it contains both 4f125d1 and
the 4f135d0 configuration
conduction electrons: Ce6s5d
trivalent Ce 4f electron: 4f1
as lower T, local 4f moment
becomes screened by conduction
electrons forming a many body
singlet state = Kondo resonance
characteristic T = Kondo temp, TK
electron hops (slowly) in and
out of the 4f level
f a s t probe such as PES
4f125d1
or
4f135d0
theory: Anderson impurity model
high resolution PES (∆
(∆E=5meV)
can directly measure the Kondo
resonance
Garnier et al, PRL78, 4127 (1997)
IPES
f0
Uff
f1
f2
E
8
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Transition metals
TM: what are
they ?
Transition metals
Lecture 8: The effects of electron repulsion
Transition metals and ee-repulsion
what's special ?
transition metals have partiallypartially-filled
d orbitals
eg
σ-bonding to ligands
t2g
π-bonding to ligands
Lecture 8: The effects of electron repulsion
Transition metals and ee-repulsion
less clearclear-cut (more interesting):
extension of the 'compact' orbitals:
orbitals:
4f < 5f < 3d < 4d < 5d
band
formation
crystal
field
overlap
smaller
Hubbard
U
?
orbital degrees
of freedom
Gd
exchange
(Hund)
Hund)
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Transition metals and ee-repulsion
transition metals: W > Ueff → band theory can be used
but repulsion does lead to large exchange energies and so
to magnetism
transition metal compounds:
compounds: W < Ueff
→ splitting into Hubbard bands
depending on system, can be quite close to border line
U~W
thus many interesting (and useful) transitions can
occur on changing T, P, filling, structure etc.
9
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Why transition metal oxides ?
fundamental importance
the outstanding challenge to solid state
physics and chemistry:
understand and master strongly correlated
electronic systems
technological potential
superconductors
superionic conductors
spintronic materials
catalysts (making money already)
What is the intellectual challenge ?
TM oxides are real, complex solids
high Tc
superconductors
electron-electron interaction important
electron-lattice interaction important
defect chemistry / complex structural properties
CMR systems
myriad quantum
ground states:
FM
AFM
PM
insulator
semiconductor
metal
superconductor
MITs
QPTs
CO / OO
subtle interplay between:
charge
spin and
orbital degrees of freedom
many of the usual theoretical simplifications are not applicable
hard
Lecture 8: The effects of electron repulsion
2pz
3dx2-y2
images: http://buffer.bu.edu/acrosby/orbitals
2py
3dxz
Energy scheme for a TMTM-O
Transition metal compounds
M-H gap
closure
General factors:
↑W
larger d orbitals:
orbitals:
right to left in period
3d → 4d → 5d
↑W
narrow: or
MHI
3dyz
3dxy
t2g
Lecture 8: The effects of electron repulsion
Eg
occ.
occ. ↑
unocc.
unocc.↓
3dz2-r2
eg
Lecture 8: The effects of electron repulsion
Transition metal compounds
interesting
3d orbitals
images: http://buffer.bu.edu/acrosby/orbitals
2px
but
Lecture 8: The effects of electron repulsion
1s and 2p orbitals
1s
1023 particles
per cm3 !!
wide bands:
metal
↑W
↓U
oxidation state:
early TM: low oxidation state good for MM-M overlap
late TM: high oxidation state good for MM-O-M
covalency
anion partner:
if W from covalency:
covalency: low electronegativity anions
better
(halides, O, S, Se, Te,
Te, phosphides....)
phosphides....)
10
Lecture 8: The effects of electron repulsion
Transition metal compounds
General factors contd:
contd:
electron configuration
Mn 2+ has 3d5 - halfhalf-filled shell ..... U↑
U↑
↑U
other cations
e.g. perovskites:
perovskites: ABO3 (B= TM)
Lecture 8: The effects of electron repulsion
TM oxides: LEGO on atomic dimensions
Just like all crystalline solids, TMTM-oxides are built up
of regular repeat units such as:
as:
TM-O6 octahedron
TM-O5 square pyramid
TM-O4 plaquette
? what’s a perovskite ?
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Perovskites
Variations on perovskites
Very large class of transition metal (and other) oxides
'classic' cubic ABO3
transition metal
(B)
lanthanide or
group II metal
(A)
oxygen
pic: Matt Rosseinsky
Lecture 8: The effects of electron repulsion
Transition metal compounds
Brain teaser......
General factors contd:
contd:
electron configuration
Mn 2+ has 3d5 - halfhalf-filled shell ..... U↑
U↑
↑U
↑W
Lecture 8: The effects of electron repulsion
Y2Ru2O7 vs.
vs.
Bi2Ru2O7
how many valence electrons does Ru have?
other cations
e.g. perovskites:
perovskites: ABO3 (B= TM)
if A has empty levels way above TMd no effect
if A is, e.g., a postpost-TM metal such as Bi:
Bi:
filled s orbitals at ~E cf.
cf. d levels of TM
broaden dd-band
e.g.
Y2Ru2O7
vs.
vs.
Bi2Ru2O7
11
Lecture
1: Introduction
Lecture 8: The
effects
of electron repulsion
Lecture 8: The effects of electron repulsion
Brain teaser......
Y3+
O2Ru:
Ru: 8 valence e
Y2Ru2O7 vs.
vs.
Bi2Ru2O7
how many valence electrons does Ru have?
8
how many Ru e are left (where?) on oxide formation ?
7 x O2-
1414-
2 x Y3+
6+
2 x Ru =
8+ formally Ru4+ = 4d4
metallic ? insulating ?
for Y, U wins out: localised 4d4
www.webelements
.com
www.webelements.com
for Bi,
Bi, the Bi6s mixes in the fight (Bi3+ has 6s2), W↑
W↑
and the system becomes metallic
Bi3+ or Bi5+
Lecture 8: The effects of electron repulsion
E.g. U/W tuning for d1 compounds
all compounds are d1 oxides
tuning U/W e.g.
3,4,5: M-O-M distortion from
180° reduces W
1 - metal
2 - metal
3 - metal
4 - barely metallic AFM (low T=FM)
5 - FM insulator
still see 'memory' of lower
Hubbard band as incoherent
emission in correlated metals
Lecture 8: The effects of electron repulsion
What else effects U ?
1
up to now we discussed halfhalf-filled bands
the dd-band can contain 10 e
2
what about away from halfhalf-filling ?
for integer filling, we can
still sink the occupied and
raise unoccupied states by
opening a MM-H gap
3
4
4/10 filling
Q:
what about
nonnon-integer
filling ?
5
Fujimori et al., PRL69, 1796 (1992)
Lecture 8: The effects of electron repulsion
Doping
Lecture 8: The effects of electron repulsion
Doping
unlike in athletics, doping is to be encouraged in TM compounds
add one additional electron into the UHB........
unlike in athletics, doping is to be encouraged in TM compounds
or remove one electron from LHB (add one hole)
U
........free to move without paying U !!
U
........free to move without paying U !!
12
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Doping
nonnon-stoichiometric systems or systems with nonnon-integral
valency should be metallic even if U>W
BUT
Feedback:
the system still has narrow bands and so the extra carriers
are susceptible to being trapped (e.g. as polarons)
polarons)
structure + electrons, spins
Lecture 8: The effects of electron repulsion
Effects
of structure on the electronic levels
Lecture 8: The effects of electron repulsion
Interplay:
Hund's rules and crystal field
Coulomb potentials in oxides lead to lifting of the
orbital degeneracy of the 3d levels:
E
b1g
eg
d
2
a1g
b2g
5
3
t2g
free ion
eg
cubic
Hund's rules
1) maximise spin multiplicity, S
x2 − y 2
e- in different orbitals,
orbitals, with parallel spin
z2 − r2
xy
xz , yz
2) maximise total orbital angular momentum, L
tetragonal
3) maximise total angular momentum, J
crystal
field
splitting
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Ligand field effects
Ligand field effects
competition between ligand field and exchange interaction
(Hund's 1st rule)
eg
d
5
free ion
2
gap
10Dq
t2g
For localised electron configurations in TM oxides and
halides:
3
cubic
d6
eg
sometimes 10Dq wins out
→
low spin
more often, exchange wins
→
high spin
2
Mn2+
eg
Fe2+
Co3+
2
low spin
.....could promote electrons to the eg states in order to
maximise no. of parallel spins
t2g
3
t2g
3
13
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
8
Ni 3d optical transitions: solution vs.
vs. NiO
once the dd-levels are no longer degenerate, we can have
electronic transitions between them
e.g. Ni 3d8 in solution or in NiO
eg
d
3 triplet final states
spanning (t
(t2g)5(eg)3
and (t
(t2g)4(eg)4
5
t2g
free ion
3 peaks
cubic
Lecture 8: The effects of electron repulsion
solution and solid spectrum similar !
→ localised levels also in NiO
in solution: width is from vibrational
excitations
in NiO WNi~1eV in NiO,
NiO, but spectra
equally narrow
Egap
NiO
Absorption
Ligand field effects & spectroscopy
Frenkel exciton,
exciton,
like in molecular
solids
Ni(H
Ni(H2O)62+
1
2 3 4
Photon energy (eV
(eV))
Lecture 8: The effects of electron repulsion
Ligand field effects contd.
contd.
Symmetry adapted linear combinations: 2p (O
(Oh) + 3d
ligand field effects partly control
occupancy of different d orbitals
average interatomic
distance
bonding effect changes as
d occupancy does:
ligand field effects have
influence on structure, too
here:
(i) TM contraction
(ii) ↑ ionic radius d3-d5 and d8-d10
all compounds are high spin
(i) radius ↑ when eg (anti(antibonding) becomes occupied
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
JahnJahn-Teller effect
b1g
4
eg
d
2
a1g
eg
t2g
eg
JT ions are when:
t2g has
1, 2, 4, 5 electrons
eg has
1, 2 electrons
JT effects generally only significant in eg levels (σ
(σ bonding!)
cubic → tetragonal
b1g
9
d
2
b2g
3
Oh
JahnJahn-Teller effect contd.
contd.
last slide showed average structural parameter…..
consider each group of bonds MM-ligand separately:
uneven
Recipe for a JahnJahn-Teller distortion:
forces
take a degenerate set of orbitals….
orbitals….
occupy them unevenly with electrons:
distortion
d4
a1g
b2g
t2g
3
d9
eg
orbitals in the distorted field
14
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
JahnJahn-Teller effect in Rb2CrCl4
JahnJahn-Teller effect contd.
contd.
most common examples for d4 : Cr2+ and Mn3+
in nonnon-metallic compounds they show JT distortion
coco-operative JahnJahn-Teller distortion
e.g. Rb2CrCl4
K2NiF4 based structure: CrCl4 squares with apical Rb's.
Rb's.
for d9 : Cu2+
in the CrCl2 plane:
formed the initial inspiration to look for supersuperconductivity in squaresquare-planar cuprates
two short CrCr-Cl
two long CrCr-Cl
per CrCl4 unit
when JT distortion 'fits' the lattice, one talks of a:
coco-operative JahnJahn-Teller distortion
e.g. Rb2CrCl4
K2NiF4 based structure: CrCl4 squares with apical Rb's.
Rb's.
Lecture 8: The effects of electron repulsion
Magnetic properties of localised electrons
in a normal metal, the electrons are unpaired
χ = 2µ0µB2 N(EF )
Pauli susceptibility for f.e. metal
(Ch.
Ch. 3)
unpaired localised electrons have very different
magnetic properties
in an isolated complex, susceptibility follows Curie law:
χ = C /T
Curie const:
const:
similar effects seen in K2CuF4 and the manganite perovskites
Lecture 8: The effects of electron repulsion
Magnetic properties of localised electrons
in transition metal complexes, the orbital moment is
quenched by ligand field effects
use spinspin-only expression for µ:
µ = g {S ( S + 1)}½ µ B
S is the spin quantum no.
g factor is close to 2 for free electrons
(deviations can occur due to SS-O coupling)
C = Nµ0 µ 2 /(3k B )
magnetic moment of the electrons
in a real solid, things are less simple, as the ions always
interact with each other to some extent……
in 4f systems the electrons have both spin moment and
orbital moment
Lecture 8: The effects of electron repulsion
Magnetic properties of localised electrons
paramagetism
ferromagetism
TC
Lecture 8: The effects of electron repulsion
Magnetic properties of localised electrons
above any ordering temperatures, the magnetic susceptibility
follows CurieCurie-Weiss formula:
χ = C /(T − θ )
antiferromagetism
Weiss constant, θ :
TN
positive for FM
negative for AFM
ferrimagnetism also possible (antiparallel
(antiparallel alignment of
spins, but without cancellation)
for an overview of magnetic data for FM and AFM
compounds, see Cox Tables 5.1 and 5.2 (pp
(pp 156156-157)
15
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Magnetic properties of some TM compounds
Q:
Exchange interaction
WHY are some TM compounds FM with a Curie
temp of 6K and others AFM with a Neel temp of
2000K ?
magnetic ordering → high entropy state
'the bath' wants to randomise the spins
to get a high transition T one needs strong
interaction between TM ions
exchange interaction, J , can come from direct overlap
between the TM orbitals holding the unpaired spins
however, just as in bonding, the indirect route involving
overlap with intervening ligand orbital(s) is very important:
this is called
superexchange
direct TMTM-TM overlap is not the important factor …….
…….what
…….what is ?
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
pdσ
pdσ = AFM
Superexchange - I
Superexchange - I
pdσ
pdσ = AFM
3d3d-2p2p-3d superexchange leads to AFM exchange interaction
covalent mixing
there is an O2p sitting
between each Mn site in
MnO:
MnO:
Pauli exclusion principle enforces antiparallel arrangement
3d3d-2p2p-3d superexchange leads to AFM exchange
interaction
'planar' late TM systems such nickelates and
cuprates (including HTSC) Hubbard model:
J = 4t 2 / U
Lecture 8: The effects of electron repulsion
Superexchange - II
Covalency as driving
force:
TN (K)
MnO
122
FeO
198
CoO
292
NiO
530
magnetic structure from neutron scattering
Lecture 8: The effects of electron repulsion
Superexchange - III
FM
high spin Fe
Fe2+ (d6)
Pauli exclusion principle enforces
antiparallel arrangement
Hund's rule encourages parallel
arrangement (different orbitals)
orbitals)
here superexchange combined with coco-operative
JT distortion (Rb2CrCl4) leads to a FM interaction
Fe3+ (d5)
double exchange:
hopping of minority spin e favourable as it leaves maximum
spin multiplicity behind (Hund
#1)
(Hund#1)
but only if the majority spins are FM can the minority spin
hop to the next site
delocalisation lowers E, drives FM
16
Lecture 8: The effects of electron repulsion
o
rb
it
i
n
ro
Lecture 8: The effects of electron repulsion
Playing
with the knobs of transition metal oxides
cs
charge
coupled dynamics
spin
orbital
complex (quantum) electronic matter:
liquid-like phases
crystal-like phases
liquid crystal-like phases
orbital degeneracy: a new degree of freedom (like the spin)
electronic phase separation, pattern formation
Lecture 8: The effects of electron repulsion
Lecture 8: The effects of electron repulsion
Localisation plays a crucial role
Ordering patterns: example 1: LaVO3
correlation (Mott
(Mott)) physics means that the electrons
are localised at the atomic sites
La = 3+, O3 = 66-, thus V = 3+
this means we have clear occupation of orbitals,
orbitals, and
a clear local spin orientation (e.g. via
superexchange)
superexchange)
V3+ means 3d2: a system with partial t2g occupation
orbital-spin pattern:
spin z: FM chains
spin xy: AFM
opens the door for quantum electronic texture.....
.....with versatile ordering patterns
orbital: dyz and dxz
alternate in x, y and z
dxy left out
Lecture 8: The effects of electron repulsion
Ordering patterns: example 2: LaMnO3
La = 3+, O3 = 66-, thus Mn = 3+
anisotropic despite
cubic structure !
Lecture 8: The effects of electron repulsion
Questions for next time
what are the crystal field levels for La2CuO4 (CuO6
octahedra,
octahedra, tetragonally distorted) ?
which levels have how many electrons ?
Mn3+ means 3d4: a JahnJahn-Teller system.
what do you expect the magnetic and electronic
ground states to be ? why ?
orbital degeneracy lifted by
JT:
favours occupation of either
3dz2-r2
or
3dx2-y2
linear combination
alternating:
3dx2-r2
and
what will happen when we replace some La with Sr ?
- to the formal Cu d count ?
- which orbitals really change their electron count ? why ?
- what then happens to the magnetic properties ? why ?
what is the microscopic mechanism for the formation
of superconducting Cooper pairs in the holehole-doped
cuprates ?
3dy2-r2
17
Lecture 8: The effects of electron repulsion
The End
18