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.