1. Mechanisms for effects of EDM in solids. Oleg P. Sushkov University of New South Wales, Sydney

2. Acknowledgments T.Mukhamedjunov S. Kuenzi S. Buhmann J. Ludlow V. Dzuba S. Cadogan Private communications K. Rushchanskii N. Spaldin

3. Electric Dipole Moment (EDM) of electron d e Present upper limit d e < 1.6 10 -27 e cm comes from experiment with atomic Tl beam. Regan, Commins, Schmidt, DeMille (2002). Solid state ideas Ancient history (< 2000): Experiments with solids were suggested by F. L. Shapiro in 1968 Nickel Zinc ferrite measurement, Vasil’ev and Kolycheva in 1978 Modern history ( >2000): Gadolinium Gallium Garnet S. Lamoreaux, 2002 Gadolinium Iron Garnet L. Hunter, 2002 Present: Eu 05 Ba 05 TiO 3 A. Sushkov, S. Eckel, S. Lamoreaux, 2008

4. Gadolinium Gallium Garnet Gd 3 Ga 5 O 12 Gadolinium Iron Garnet Gd 3 Fe 5 O 12 Europium Barium Titanate Eu 0.5 Ba 0.5 TiO 3 These compounds are ionic Mott insulators Ga 3+, Fe 3+, Ba 2+, Ti 4+ and O 2- are spinless while Gd 3+ and Eu 2+ have spin 7/2 (half filled 4f-shell) Conducting band Valence band Band insulatorNormal metal gap

5. “Conducting band” “Valence band is half filled” Mott insulator Spin and charge degrees of freedom are separated. Spin excitations are gapless while charge excitations have gap about 4 eV.

6. LDA +U band structure calculation for Eu 0.5 Ba 0.5 TiO 3 K. Rushchanskii, private communication

7. Idea of the experiment is pretty simple Energy shift due to EDM: Induced magnetization: Naively E eff ~ E external in a paraelectric E eff ~ 10 7 - 10 8 V/cm in a ferroelectric Question to theory: how to calculate E eff ?

8. The estimates have been performed for GGG Gd 3+ in the cage of O 2- ions The spin-dependent energy shift reads S=7/2 is spin of the Rare Earth ion. x is displacement of the ion from center of the cage x replaces E eff We need to calculate the dimensionless constant A

9. Mechanism (2) is dominating : 2p-electrons of O 2- ions penetrate to the Rare Earth nucleus. There are two mechanisms for the coefficiant A 1)Chemical hybridization: direct mixing of Eu 4f and 5d orbitals due to ferroelectric lattice distortion. 2) Correlations: ferroelectric distortion mixes 6s and 6p of Eu and the mixing is transferred on 4f and 5d via noncentral Coulomb interaction

10. EDM Hamiltonian Account for the Schiff theorem Diagramatic notation for V d

11. Jelly model: spread oxygen ions of the cage over a spherical shell Rare Earth ion inside spherical oxygen cage. x is displacement with respect to the cage center. The displacement is due to external electric filed in a paraelectric. The displacement is due ferroelectricity in a ferroelectroic The displacement mixes s- and p-orbitals centered at the Rare Earth ion Diagramatic notation for influence of the displacement on external orbitals of Rare Earth ion

12. There is also a residual (noncentral) Coulomb intreraction between electrons

13. A= -0.094 -0.159 +0.080 +0.133 +0.141 -0.295 -0.257 -0.040 +0.610 -0.295 -0.009 -0.001 +5.055 -4.615 -0.159 =0.095 A= -0.008 -0.016 -0.006 -0.010 +0.186 -0.596 -0.614 -0.049 +0.010 +0.671 -0.189 -0.019 -0.656 +1.396 -0.005 =0.095 NPP

14. The calculations is somewhat messy because we use atomic physics methods to approach the solid state problem. Result: A ~ 0.1 An experimental estimate for x in Eu 0.5 Ba 0.5 TiO 3 is x ~ 0.01 Å Together with the calculated value of A this gives the effective electric field E eff ~ 10MV/cm

15. Conclusions E eff ~ 10MV/cm This is an estimate. A calculation with accuracy 20% ~ 30% is feasible