1. Theory of Orbital-Ordering in LaGa 1-x Mn x O 3 Jason Farrell Supervisor: Professor Gillian Gehring 1. Introduction LaGa x Mn 1-x O 3 is an example of a manganese oxide known as a manganite. The electronic properties of manganites are not adequately described by simple semiconductor theory or the free electron model. Manganites are strongly correlated systems: Electron-electron interactions are important. Electron-phonon coupling is also crucial. → Magnetisation is influenced by electronic and lattice effects. La 1-x Ca x MnO 3 (Mn 3+ and Mn 4+ ) and similar mixed-valence manganites are extensively researched. These may exhibit colossal magnetoresistance (CMR). → Very large change in resistance as a magnetic field is applied. → Possible use in magnetic devices; technological importance. BUT: LaGa x Mn 1-x O 3 (Mn 3+ only; no CMR) has not been extensively studied. 2. General Physics of Manganites Ion of interest is Mn 3+. Neutral Mn: [Ar]3d 7 electronic configuration. → Mn 3+ has valence configuration of 3d 4. Free ion: 5 (= 2l +1; l = 2) d levels are wholly degenerate. Ion is spherical. Place ion into cubic crystal environment with six Oxygen O 2- neighbours: Electrostatic field due to the neighbours; the crystal field. Stark Effect: electric-field acting on ion. Some of the 5-fold degeneracy is lifted. Cubic crystal: less symmetric than a spherical ion. → d orbitals split into two bands: e g and t 2g. t 2g are localised; the e g orbitals are important in bonding. On-site Hund exchange, J H, dominates over the crystal field splitting ∆ CF. → 4 spins are always parallel; a “high-spin” ion. 3. The Jahn-Teller Effect Despite crystal field splitting, some degeneracy remains. Fundamental Q.M. theory: the Jahn-Teller effect. Lift as much of the ground state degeneracy as possible → Further splitting of the d orbitals Orbitals with lower energy: preferential occupation → JTE introduces orbital ordering. Lift degeneracy ↔ reduce symmetry. Strong electron-lattice coupling. → Jahn-Teller effect distorts the ideal cubic lattice. 4. Interplay of Spin- and Orbital-Ordering Coupling between spins in neighbouring Mn orbitals is determined by the amount of orbital overlap → Pauli Exclusion Principle. Large orbital overlap: antiferromagnetic ↑↓ spin coupling. Less orbital overlap: ferromagnetic ↑↑ spin coupling. Also have to consider the intermediate O 2- neighbours. Extended treatment considers virtual interorbital electron hopping: the Goodenough-Kanamori-Anderson (GKA) rules. Gives the same result; also gives each exchange constant. 5. Physics of LaMnO 3 Based upon the perovskite crystal structure: Jahn-Teller effect associated with each Mn 3+ act coherently throughout the entire crystal. This cooperative, static, Jahn-Teller effect is responsible for the long-range orbital ordering. Long and short Mn-O bonds in the basal plane → a pseudo-cubic crystal. The spin-ordering is a consequence of the orbital ordering (Section 4). → A-type spin ordering: spins coupled ferromagnetically in the xy plane; antiferromagnetic coupling along z. Long-range magnetic order is (thermally) destroyed above T N ~ 140 K. Long-range orbital order is more robust: destroyed above T JT ~ 750 K. → Structural transition to cubic phase. On-site Coulomb repulsion U (4 eV) is greater than electron bandwidth W (1 eV) →LaMnO 3 is a Mott-Hubbard insulator. orbitals spins 6. Gallium Doping Randomly replace some of the Mn 3+ with Ga 3+ to give LaMn 1-x Ga x O 3. Ga 3+ has a full d shell (10 electrons): → Ion is diamagnetic (no magnetic moment) → Not a Jahn-Teller ion; GaO 6 octahedra, unlike MnO 6, are not JT-distorted. How does such Gallium-doping affect the orbital ordering and hence the magnetic and structural properties of the material? 7. Theoretical Approach Finite cubic lattice (of Mn and Ga) with periodic boundary conditions. Spin-only Mn 3+ magnetic moment = 4 µ B ; CF-quenching of orbital moment. Begin with LaGaO 3 and dope with Mn 3+ : Theory: ferromagnetic spin exchange along the Mn-O-Mn axes. Period of rotation of these axes is faster than spin relaxation time. → Isotropic ferromagnetic coupling between nearest-neighbour Mn spins. Try a percolation approach: As Mn content increases, ferromagnetic Mn clusters will form. At higher Mn content, larger clusters will form. At a critical Mn fraction, the percolation threshold, x c, a ‘supercluster’ will extend over the entire lattice. → Determine the magnetisation per Mn 3+ as a function of doping: Excellent agreement at small x: evidence for magnetic percolation. As x → x c (= 0.311 for a simple cubic lattice) simple approach fails. This is expected: percolation is a critical phenomenon. Change in orbital-ordering also leads to change in the crystal dimensions: Hypothesis: upon introducing a Ga 3+ ion, neighbouring x and y Mn 3+ orbitals in the above/below planes flip into z direction. Good qualitative agreement: the orbital-flipping hypothesis is correct. → Crystal c-axis evolution (not shown) is also predicted correctly. → True understanding of how Ga-doping perturbs the long-range JT order. Future Work: investigate the behaviour of the high-x (Mn-rich) magnetisation. t 2g e g hello Orthorhombic Strain in x Experimental Data: Vertruyen B. et al., Cryst. Eng., 5 (2002) 299 20 x 20 x 20 Simulation 2( b-a )/( b+a ) Magnetisation of LaGa 1-x Mn x O 3 @ T = 5 K; applied B = 5 Tesla x Polycrystalline experimental data: Vertruyen B. et al., Cryst. Eng., 5 (2002) 299 20 x 20 x 20 percolation simulation M (µ B /Mn) Mn O Mn (a) (b) (c) LaMn 1-x Ga x O 3 @ T = 5 K