1. Jahn–Teller Distortion in Clusters and Lithiated Manganese Oxides R. Prasad Physics Department, IIT Kanpur Outline 1.Jahn–Teller Effect 2.Clusters 3.Lithiated Mn–Oxides 4.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).

2. Jahn–Teller Theorem : Any complex occupying an energy level with electronic degeneracy is unstable against a distortion that removes the degeneracy in first order. Mn 3+ JT Ion Mn 4+ No distortion Mn 3+ O Mn 3+ 3d 4

3. Q = distance U(Q) = ½ k Q 2 Phenomenology E e (Q) = complicated function of Q  - A Q E(Q) = ½ k Q 2 – A Q Static Jahn–Teller effect Dyanamic Jahn–Teller effect Cooperative Jahn–Teller effect

4. 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. E xc = exchange-correlation energy T 0 = Kinetic energy of a system with density  without electron-electron interaction

5. 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

6. Extension to spin-polarised systems Von Barth and Hedin 1972 Rajagopal and Callaway 1973 for uniform spin directions (σ =  or  ) n iσ = Occupation no. Local spin density approximation (LSDA) ε xc = exchange correlation energy per particle of a homogeneous, spin-polarized electron gas with density ρ , ρ .     xc E v 

7. Beyond the LSDA Generalized Gradient Approximation (GGA) Beyond the LSDA For higher accuracy, need to go beyond the LSDA Gradient expansion approximation (GEA) Kohn and Sham 1965, Herman 1969 For 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) 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)

8. Pseudopotential Method Advantages of the method Pseudopotential Method Historically 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 method 1. 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.

9. SiH 4 SiH 4 + H H H H H H H H Si Symmetry Breaking

10. (CH 4 ) (CH 4 + )(CH 4 - ) (SiH 4 ) (SiH 4 + )(SiH 4 - )

11. (GeH 4 )(GeH 4 + )(GeH 4 - ) (SnH 4 ) (PbH 4 ) (SnH 4 + ) (PbH 4 + ) (SnH 4 - ) (PbH 4 - )

13. MO construction

14. Effect of different occupations

15. Symmetric and Asymmetric charge density

16. Mechanism of structural distortion 1.Ionize SiH 4 2.Look at the structure as it evolves

17. Bond Lengths SiH 4 +

18. SiH 4 +

19. Energy lowering comes from electrostatic interaction and not from level splitting. Is it really Jahn-Teller effect?

20. 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 SiH 4 H is more electronegative than Si H has small –ve charge -0.12e Si has small +ve charge +0.52e Tetrahedral SiH 4 + H has -0.12e -0.02e, +0.06e +0.30e Si has 0.77e

21. Consider CH 4 C is more electronegative than H C has small –ve charge H has small +ve charge Consider CH 4 + H has +0.20e, +0.23e +0.42e, +0.55e C has -0.40e

22. Negatively charged clusters SiH 4 - CH 4 - H has charge -0.14e, -0.15e, -0.16, -0.37e Si has charge -0.18e Nearly 0.0e charge on H atom C has ~ -1.0e Jahn – Teller distortion occurs because of creation of charge asymmetry

24. Rhombohedral LiMnO 2

25. Monoclinic LiMnO 2

26. D. Singh, Phys. Rev. B55, 309 (1997)

27. Magnetism plays an important role in phase stability Magnetism plays an important role in phase stability Total energies of LiMnO 2 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

28. 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 = x c = 0.32  The system will become ferromagnetic rhombohedral at  About 32% Co doping.

29. Questions 1. Why does theory predict large x c ? 2. Why does Co suppress Jahn–Teller effect? 3. Why is transformed rhombohedral phase ferromagnetic?

30. Monoclinic AF3

33. Charge transfer from Mn to Co Mn 3+ Mn 4+ Co 3+ Co 2+ How does it explain the suppresion of JT distortion Mn 3+ is JT ion Mn 4+ is not

34. Why is the transformed rhombohedral phase ferromagnetic? Double exchange interaction

35. Experimental support for charge transfer Co 2+ Co 3+

36. 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.

40. Conclusions 1. 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 SiH 4 +. The structural distortion is caused by electrostatic repulsion. 4. Our calculations explain the suppression of JT distortion in Co doped LiMnO 2 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.