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Jahn–Teller Distortion in Clusters and Lithiated Manganese Oxides
R. PrasadPhysics Department, IIT Kanpur
Outline1. Jahn–Teller Effect2. Clusters3. Lithiated Mn–Oxides4. Conclusions
Collaborators:1. D. Balamurugan, M. K. Harbola, Phys. Rev. A (2004).2. R. Benedek, M. M. Thackeray, Argonne National Lab, Phys. Rev. B 68, 012101 (2003).
Jahn–Teller Theorem : Any complex occupying an energy level with electronic degeneracy is unstable against a distortionthat removes the degeneracy in first order.
Mn3+ JT Ion
Mn4+ No distortionMn3+O
Mn3+ 3d4
Q = distanceU(Q) = ½ k Q2
Phenomenology
Ee (Q) = complicated function of Q - A QE(Q) = ½ k Q2 – A Q
k
Α 0Qοr
Α Qk 0dQ
dE(Q)
Static Jahn–Teller effectDyanamic Jahn–Teller effectCooperative Jahn–Teller effect
Density Functional Theory
• Hohenberg and Kohn, 1964
1. The ground state energy E of an inhomogeneous
interacting electron gas is a unique functional of the
electron density .
2. The total energy E{} takes on its minimum value for the
true electron density.
Exc= exchange-correlation energy
T0 = Kinetic energy of a system with
density without electron-electron interaction
xc
ext
xcH
Erdrdrr
rr
rdrrvT
EEE
'|'|
)'()(
2
1
)()(}{
}{}{}{
0
)rρ(
Kohn-Sham Equation
Minimize E subject to the condition
Local density approximation (LDA)
= contribution of exchange and correlation to the
total energy per particle in a homogeneous but
interacting electron gas of density ρ
)('
|'|
)'()()(
|)(|)(
)()()(2
)(
2
22
r
Erd
rr
rrvrV
rr
rrrVm
Nrdr
xcext
i
occ
i
iii
))(()(}{ rrrdE xcxc
))(( rxc
Extension to spin-polarised systems
•Von Barth and Hedin 1972
•Rajagopal and Callaway 1973
for uniform spin directions (σ = or )
niσ = Occupation no.
• Local spin density approximation (LSDA)
εxc= exchange correlation energy per particle of a
homogeneous, spin-polarized electron gas with density ρ, ρ.
)()()](
|'|
)'(')(
2[ 22
2
rrrv
rr
rrderv
m
iiixc
ext
2|)(| rn ii
i
)](),([)( rrrrdE xcxc
xc
xcE
v
Beyond the LSDABeyond the LSDAFor higher accuracy, need to go beyond the LSDAGradient expansion approximation (GEA)Kohn and Sham 1965, Herman 1969For slowly varying densities, the energy functional can be expanded as a Tylor series in terms of gradient of the density
For real system GEA often is worse than LSDA
Generalized Gradient Approximation (GGA)Generalized Gradient Approximation (GGA)Ma and Bruckner ; Langreth, Perdew, Wang
where f is chosen by some set of criteria.Many function have been proposed :Perdew - Wang 1986 (PW86)Becke 1988 (B88)Perdew and Wang 1991 (PW91)
3/2
'
'3/2
''
LSDA
XC
GEA
XC
),(
],[ ],[
C
EE
),,,f(r d ],[ 3GGA
XCE
Pseudopotential MethodPseudopotential MethodHistorically pseudopotential were introduced to justify nearly free electron model.The core electrons are removed and the potential is replaced by an effective potential which reproduces the same energy levels and the same valence wavefunctions beyond the cut-off radius.Vanderbilt’s ultra-soft potential 1990
Advantages of the methodAdvantages of the method1. FFT’s can be used to speed up the method.2. Calculations of energy and forces are very simple.3. There are no Pulay forces on the nuclei.
'2
22
2'k',
k
XC2
2
r).Gki(
GGk
|Gk|2m
'Gk|2m
- |Gk
0 | |
v w 2m
-
e C )(
GG
GGKS
GkGk
kkKS
KS
k
HH
H
r
SiH4 SiH4+
H H
HH
H
H
H
HSi
Si
Symmetry Breaking
Symmetry Breaking
(CH4) (CH4+) (CH4
-)
(SiH4) (SiH4+) (SiH4
-)
(GeH4) (GeH4+) (GeH4
-)
(SnH4)
(PbH4)
(SnH4+)
(PbH4+)
(SnH4-)
(PbH4-)
MO construction
Effect of different occupations
Symmetric and Asymmetric charge density
Mechanism of structural distortion
1. Ionize SiH4
2. Look at the structure as it evolves
Bond Lengths SiH4+
SiH4+
Energy lowering comes from electrostatic interaction and not from level splitting.
Is it really Jahn-Teller effect?
Mechanism of structural distortion
1. When the cluster is charged, in general, charge asymmetry is created.
2. Electrostatic repulsion between charged atoms creates structural distortion.
Consider Tetrahedral SiH4
H is more electronegative than SiH has small –ve charge -0.12eSi has small +ve charge +0.52e
Tetrahedral SiH4+
H has -0.12e -0.02e , +0.06e +0.30eSi has 0.77e
Consider CH4
C is more electronegative than H
C has small –ve charge
H has small +ve charge
Consider CH4+
H has +0.20e, +0.23e +0.42e , +0.55eC has -0.40e
Negatively charged clusters
SiH4
-
CH4
-
H has charge -0.14e, -0.15e, -0.16, -0.37eSi has charge -0.18e
Nearly 0.0e charge on H atomC has ~ -1.0e
Jahn – Teller distortion occurs because of creation of charge asymmetry
LiMnOLiMnO22
• Potential material for rechargeable batteries
• Electron correlations are expected to play important role.
• How far one can push LSDA and GGA?
• Layered oxides, similarity with High Tc oxides
• Mixed valent systems
• Charge ordered systems
• Magnetism plays an important role.
• Jahn-Teller distortion
Rhombohedral LiMnORhombohedral LiMnO22
Monoclinic LiMnOMonoclinic LiMnO22
D. Singh, Phys. Rev. B55, 309 (1997)
Magnetism plays an important role in phase stabilityMagnetism plays an important role in phase stability
Total energies of LiMnO2 at experimental lattice constants.
1. Non spin-polarised calculation does not give the correct structure.2. LSDA gives monoclinic AF3 structure to be of lower energy, in agreement with experiment.3. GGA also gives monoclinic AF3 structure to be of lower energy.
Structure Non-spin polarised(eV)
Ferro (eV) AF3 (eV)
Monoclinic -118.812 -120.580 -124.127 LSDA Rhombohedral -121.264 -123.851 -123.584 Monoclinic -108.363 -114.667 -115.204 GGA Rhombohedral -110.269 -113.440 -113.663
Effect of Co doping
1. About 10% Co doping suppresses Jahn–Teller distortion in favor of rhombohedral structure
2. We have calculated total energies of various structures E (m, AF, x) = Total energy of Monoclinic AF structure
with x concentration of Co E = E (m, AF, 0) - E (r, F, 0) = -359 meV/unit cell After 25% Co doping
E = -111 meV/unit cell E will be zero at x = xc = 0.32 The system will become ferromagnetic rhombohedral at About 32% Co doping.
Questions
1. Why does theory predict large xc?
2. Why does Co suppress Jahn–Teller effect?
3. Why is transformed rhombohedral phase ferromagnetic?
Monoclinic AF3
Charge transfer from Mn to Co
Mn3+ Mn4+
Co3+ Co2+
How does it explain the suppresion of JT distortion
Mn3+ is JT ion
Mn4+ is not
Why is the transformed rhombohedral phase ferromagnetic?
Double exchange interaction
Experimental support for charge transfer
Co2+
Co3+
Divalent – dopant Criterion
•We have studied other dopants like Ni, Fe, Al, Zn, Mg, Cr, Cu etc.
•We find that dopants which are most effective in suppressing JT distortion are those which adopt divalent state in both JT distorted and the transformed structure.
Conclusions1. Jahn – Teller distortion results from creation of charge asymmetry.
2. Charge transfer can play an important role in creating / suppressing Jahn-Teller distortion in clusters as well bulk materials.
3. We find an unusual bonding between two hydrogen atoms in SiH4+.
The structural distortion is caused by electrostatic repulsion.
4. Our calculations explain the suppression of JT distortion in Co doped LiMnO2 in terms of charge transfer from Mn to Co, which has been verified by the XAS experiment.
5. Charge transfer also explains the transition of monoclinic AF3 structure to rhombohedral ferromagnetic structure.
6. We propose a divalent–dopant criterion for the suppression of JT distortion in Mn–Oxides.