Theory of Orbital-Ordering in LaGa1-xMnxO3 Jason Farrell

Supervisor: Professor Gillian Gehring

1. Introduction

•LaGaxMn1-xO3 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.

• La1-xCaxMnO3 (Mn3+ and Mn4+) 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: LaGaxMn1-xO3 (Mn3+ only; no CMR) has not been extensively studied.

2. General Physics of Manganites

• Ion of interest is Mn3+.

• Neutral Mn: [Ar]3d7 electronic configuration.→ Mn3+ has valence configuration of 3d4.• Free ion: 5 (= 2l +1; l = 2) d levels are wholly degenerate.• Ion is spherical.

Place ion into cubic crystal environment with six Oxygen O2- 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: eg and t2g.

• t2g are localised; the eg orbitals are important in bonding.• On-site Hund exchange, JH, 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 O2- 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 LaMnO3

•Based upon the perovskite crystal structure:

•Jahn-Teller effect associated with each Mn3+

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 TN ~ 140 K.

• Long-range orbital order is more robust: destroyed above TJT ~ 750 K.

→ Structural transition to cubic phase.

• On-site Coulomb repulsion U (4 eV) is greater than electron bandwidth W (1 eV) →LaMnO3 is a Mott-Hubbard insulator.

orbitals spins

6. Gallium Doping• Randomly replace some of the Mn3+ with Ga3+ to give LaMn1-xGaxO3.• Ga3+ has a full d shell (10 electrons):→ Ion is diamagnetic (no magnetic moment)→ Not a Jahn-Teller ion; GaO6 octahedra, unlike MnO6, 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 Mn3+ magnetic moment = 4 µB; CF-quenching of orbital moment.

Begin with LaGaO3 and dope with Mn3+:

• 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, xc, a ‘supercluster’ will extend over the entire lattice.

→ Determine the magnetisation per Mn3+ as a function of doping:

• Excellent agreement at small x: evidence for magnetic percolation.

• As x → xc (= 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 Ga3+ ion, neighbouring x and y Mn3+ 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.

t2g e g

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Orthorhombic Strain in

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Experimental Data: Vertruyen B. et al., Cryst. Eng., 5 (2002) 299

20 x 20 x 20 Simulation

2(b-

a)/(

b+a)

Magnetisation of LaGa1-xMnxO3

@ T = 5 K; applied B = 5 Tesla

xPolycrystalline experimental data: Vertruyen B. et al., Cryst. Eng., 5 (2002) 29920 x 20 x 20 percolation simulation

M (

µB/M

n)

Mn O Mn

Mn O Mn

Mn O Mn

(a) (b) (c)

LaMn1-xGaxO3 @ T = 5 K