1. 1 The investigation of charge ordering in colossal magnetoresistance Shih-Jye Sun Department of Applied Physics National University of Kaohsiung 2005/9/30 in NCKU

2. 2 Colossal Magnetoresistance La 1-x (Ca,Sr…) x MnO 3

3. 3 Urushibara et al (1995)Cheong and Hwang (1999) Phase diagram of CMR

4. 4 2p （A）（A） (1) egeg t 2g Mn 3+ (2) O 2- 2p (3) egeg t 2g Mn 4+ （B）（B） O 2- 2p (3) (2) egeg t 2g Mn 3+ (1) egeg t 2g Mn 3+ （C）（C） (2) egeg t 2g Mn 3+ (1) O 2- (3) egeg t 2g Mn 4+ Double exchange mechanism

5. 5 John Teller distortion

6. 6 The motivation para-insulator(PI) CO AFM FI CO PI x Temp x~0.2 0.5<x<0.85 La 1-x Ca x MnO 3 TCTC T CO TNTN TCTC χ T C (T CO or T N ) T Susceptibility instability I II III From region I to II and II to III

7. 7 Theoretical formulas derivation Hamiltonian: (kinetic energy) (inter-Coulomb repulsion) (on-site Coulomb repulsion) Local spin Itinerant spin

8. 8 Hamiltonian in momentum representation

9. 9 Greens function for susceptibilities Charge-charge susceptibility

10. 10 Spin-spin susceptibility

11. 11 Equation of motion method (1) (2)(3) (1)

12. 12 (2)

13. 13 Fermi-Dirac distribution Wick’s theorem Random Phase Approximation

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17. 17 Spin dependent in PI state

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19. 19 PI to CO transition Similarly, for spin-spin susceptibility

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23. 23 (spin dependent in PI) PI to AFM In CO state Mn +4 Mn +3

24. 24 CO to AFM x TC TN 0.55 222 156 0.60 260 143 0.65 265 130 0.70 250 125 0.75 215 113 0.80 180 106 0.85 130 102 Substituting to Experimental data To determine interaction relations Cheong and Hwang (1999)

25. 25 Results and discussion Reflection different transitions

26. 26 Consistent with John Teller distortion non-symmetrysymmetry More distortion

27. 27 Charge gaps are depressed by U

28. 28 Charge gap fluctuation The competition between H V and H U

29. 29 Conclusions Substituting experimental critical transition temperatures of T CO s and T N s to charge-charge and spin-spin susceptibility functions offer the determination of the inter-Coulomb repulsions and charge gaps for x > 0.5, respectively. These Inter-Coulomb repulsions increase with x increasing but not in linear. In small on-site repulsion U the phase transitions only occur pare-insulator to charge-ordering transitions and in large U only occur para-insulator to antiferromagnetic transitions. The consequential phase transitions for para- insulator to charge-ordering following charge-ordering to antiferromagnetic transitions occur in a moderate U. In charge ordering states the charge gaps are opened and are depressed by U. The scale of the charge gap increases linearly with x increasing excluding a small range of deviation. This deviation comes from the charge gap fluctuation according to the competition between inter-Coulomb and on-site Coulomb interactions.

30. 30 Thanks for your attendance!!