1. Phase separation in strongly correlated electron systems with Jahn-Teller ions K.I.Kugel, A.L. Rakhmanov, and A.O. Sboychakov Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Izhorskaya Str. 13/19, Moscow, 125412 Russia

2. OUTLINE  Phase separation and the doping level  Interplay between the localization and metallicity: a minimal model  Properties of homogeneous states: FM metallic and AFM insulating  Inhomogeneous states: FM-AFM phase separation etc.  Phase diagram in the temperature-doping plane  Effects of applied magnetic field  Conclusions

3. Phase separation and the doping level The phase separation phenomena: a key issue in the physics of strongly correlated electron systems, especially in manganites and related compounds The simplest type: formation of nanoscale inhomogeneities such as ferromagnetic metallic droplets (magnetic polarons or ferrons) located in an insulating antiferromagnetic matrix self-trapping of charge carriers What charge carries are self-trapped? If all, it leads to unphysical results The doping level and the number of self-trapped carriers can differ drastically Possible cause: The competition between the localization induced by lattice distortions due to the Jahn-Teller effect (orbital ordering) and the gain in kinetic energy due to intersite hopping of charge carriers The aim: To analyze a simplified model of such competition

4. Mn 3+ (Jahn-Teller ion)Mn 4+ d x 2 -y 2 – stretching of the octahedron in xy plane d 3z 2 -r 2 - stretching of the octahedron along z axis As a result, the Jahn-Teller gap E JT arises Electron structure of Mn ions

5. Electron Hamiltonian: general form

6. Effective Hamiltonian: localized and itinerant charge carriers is the canting angle Two groups of electrons: localized, l, at JT distortions an itinerant (band), b  JT is the JT energy gain for l electrons counted off the bottom of b-electron band  is the chemical potential Similar model: T.V. Ramakrishnan et al., PRL 92, 157203 (2004)

7. Analysis of the effective Hamiltonian Mean field approach Gives n b and n l Hubbard I type decoupling at fixed magnetic structure Green function for b electrons (U/t>>1): Band width depends on n l polaronic band narrowing

8. Densities of localized and itinerant electrons  JT 0 x 2 <x<1 x1<x<x2x1<x<x2 0<x<x10<x<x1 n l =0 n b =0 n b ≠0, n l ≠0

9. Comparison of energies at T=0 Ferromagnetic (FM): Antiferromagnetic (AF):

10. Energies of ferro- and antiferromagnetic states

11. Homogeneous states at T  0 2. Ferromagnetic (FM): 1. Antiferromagnetic (AF): 3. Canted: favorable near x=1 4. Paramagnetic (PM): (a) (b) with the growth of T transforms to PM with

12. Free energies for homogeneous states F= min (F FM,F AF,F PM, F Cant )

13. Phase diagram for homogeneous states

14. Inhomogeneous states (FM-AF phase separation) AF insulator FM metal Spatial separation of localized and itinerant electrons is favorable in energy What determines the size of inhomogeneities?

15. Energy of inhomogeneous state p is the content of FM phase E Coul is the Coulomb energy related to inhomogeneous charge distribution E surf is surface energy of the droplets E FM is the minimum energy of ferromagnetic phase (at x=x 2, n l =0) E AF is the minimum energy of antiferromagnetic phase (at x=0, n b =0)

16. Coulomb energy x f and x a are the densities of charge carriers in FM and AF phases, respectively. Here x f =x 2, x a =0. d is lattice constant,  is average permittivity R is the droplet radius Spherical model: Wigner-Seitz approximation Each droplet is surrounded by spherical layer of the opposite charge.

17. Surface energy is surface tension of metallic droplet calculated using bulk density of states with size-effect corrections Minimization of the sum E Coul ~R 2 and E surf ~1/R gives size R of the droplet

18. Radius of a droplet

19. Energy of inhomogeneous state Dashed lines correspond to the energies of phase-separated state at different values of parameter V 0 /w 0 : 1 - 0, 2 – 1/4, 3 – 1, 4 – 3/2, 5 – 2.

20. Phase diagram including inhomogeneous states 1.– homogeneous AF state (n b =0) 2.– PM(n b  0) – PM(n b =0) phase-separated state

21. Effect of applied magnetic field The most interesting situation – near the transition from PS to a homogeneous state. At the transition, the density of b electrons undergoes a jump  n b 0 The magnetic field shifts the transition point: T PS =T PS (H) => a jump in the relative change of electron density

22. SUMMARY A “minimal” model dealing with the competition between the localization and metallicity in manganites was formulated It is demonstrated that the number of itinerant charge carriers can be significantly lower than that implied by the doping level. A strong tendency to the phase separation was revealed for a wide doping range