Antiferromagnetic resonance in frustrated system Ni 5 (TeO 3 ) 4 Br 2

1. Antiferromagnetic resonance in frustrated system Ni5(TeO3)4Br2 Matej Pregelj Mentor: doc. dr. Denis Arčon

2. Contents • Introduction • Magnetic materials • Frustration • Measurements of magnetic order • Antiferromagnetic resonance • Experimental results on Ni5(TeO3)4Br2 • Analysis • Conclusion

3. Introduction • Magnetic materials are present all around us! • Compass, magnets, ... • Memory devices: hard drives, memory cards, ... • Electric generators, transformers, motors, ... • Their magnetic properties become pronounced in the vicinity of other magnetic materials or in the presence of the external fields. • We distinguish materials with: • permanent magnetic moments • induced magnetic moments • Ordering of magnetic moments depends on the crystal structure (arrangement of the magnetic ions) • Frustration geometries – crystal geometries, which prevent magnetic moments to satisfy all the inter-spin interactions at the same time. • Such system is also Ni5(TeO3)4Br2, where spins lie on a triangular lattice.

4. Magnetic materials Magnetic moments of transition metal ions Permanent magnetic moment is a consequence of unpaired electrons in the atomic orbitals. • Filling of the atomic orbitals - Hund’s rules • Electrostatic repulsion favors unpaired electrons • Pauli exclusion principle • Spin configuration in Ni2+ - 8 electrons ind-orbitals • In the applied magnetic field, magnetic moments tend to line up parallel with the field – paramagnetic behavior. • This phenomenon is opposed by thermal vibrations of the moments. e S = 1 t

5. Magnetic materials Paramagnetism • Partition function: Thermal fluctuations: • Magnetization • In the limit of classical spins - Curie law: Brillouin function:

6. Magnetic materials Order in magnetic materials • If the magnetic ions are close together they start to interact(internal fields). • Overlapping of the atomic orbitals in association with Pavli principle manifests in the so called exchange interaction: • Therefore below certain temperature some magnetic materials exhibit magnetically ordered state. • Most representative types of magnetically ordered materials are: • Ferromagnets - magnetic moments are in parallel alignment • Antiferromagnets - magnetic moments lie in the antiparallel alignment.

7. paramagnetic behavior domain 'rotation' ' irreversible growth 'of magnetic domains 'reversible growth' of magnetic domains ; J > 0 Ferromagnets • All the magnetic moments lie along the unique direction. • They exhibit spontaneous magnetization in the absence of an applied field. • This effect is generally due to the exchange interactions. • Hamiltonian for a ferromagnet in the applied magnetic field (system with no orbital angular momentum): Here μB is Bohr magneton and g is gyromagnetic ratio, and B0 is the magnetic field • Real systems usually exhibit a hysteresis loop, due to domain structure.

8. H = 0 H ≠ 0 H >HC1 H < HSF H > HSF H > HC2 Antiferromagnets • antiparalel ordering – at least two interpenetrating sublattices • no magnetization in the absence of the external field • the exchange constantJ is negative • easy direction - along which the magnetic moments are aligned • Response to the magnetic field: • B0 _|_ easy: the magnetic moments are being turned in the direction of the field; beyond certain field all the moments point in the direction of the applied field. • B0 || easy: the magnetic moments do not turn until the applied field exceeds critical value HSF. At that point magnetic moments snap into different configuration - spin flop. Beyond this point the magnetic moment act as the filed was perpendicularto them.

9. Frustration • In frustrated antiferromagnetic materials ordinary two sublattice antiferromagnetic ordering is being altered. • The long-range order of strongly interacting spins is frustrated by their geometric arrangement in the crystal lattice – all interactions among the spin pairs cannot have simultaneously their optimal values. • Typical for these systems is that they remain magnetically disordered even when cooled well below the ordering temperature, expected from the strength of pairwise interaction.

10. J J J Frustration geometries • The simplest example – three spins coupled antiferromagnetically: ? ?

11. J J ? J Frustration geometries • The simplest example – three spins coupled antiferromagnetically: Once two spins orient in the opposite directions the third one cannot be antiparallel to both of them.

12. J J J Frustration geometries • The simplest example – three spins coupled antiferromagnetically: All the pairwise interactions can not be simultaneously satisfied!

13. J J J Frustration geometries • The simplest example – three spins coupled antiferromagnetically: All the pairwise interactions can not be simultaneously satisfied! • Triangular lattice – What will be the ground state?

14. Frustration geometries • Triangular lattice - spin arrangement is not defined • A variety of different spin orientations, with minimal energy. • Non of them satisfy all the pairwise interactions simultaneously. • No long-range order. • Other frustrated systems: • other 2D and 3D regular lattices, where spins are coupled through the uniform exchange interactions. • spin glasses – magnetic moments are randomly distributed through the whole crystal matrix.

15. Frustration geometries 2D triangular Kagomé 3D fcc cubic pyrochlore spinel In all frustration geometries triangle is a basic building block!

16. Measurements of magnetic order • Magnetization and magnetic susceptibility • superconducting quantum interference device (SQUID) • torque magnetometer • extraction magnetometer • Neutron scattering • non-zero magnetic moment • no electromagnetic charge • neutrons scatter from: • nuclei via the strong nuclear force • variations in the magnetic fieldwithin a crystal via the electromagnetic interaction • When the sample becomes magnetic, new peaks can appear in the neuron diffraction pattern. • Electron Spin Resonance • electrons act as a local probes

17. S = 1/2 S = 1/2 Electron Spin Resonance Classical picture: • Magnetic moment M in the magnetic field B0 exhibit precession described by Bloch equation: • If we apply radio-frequency, which matches the precession frequency the absorption occurs. Quantum picture • For spins is energetically favorable to orient in the direction opposite to the applied magnetic field. • Energy gap occurs between two possible orientations of spin: • Transition can be induced by radio-frequency EM field: applied

18. Antiferromagnetic resonance • Ordered spin system - collective response • In ESR experiment we observe excitation between different collective spin energy states, called the magnons. • The Hamiltonian for such spin system: • The first term corresponds to the exchange interaction between the neighboring spins J. • The second term is due to the crystal field anisotropy, determining the easy De, eej and intermediate Din, einj axis according to the energetically favorable orientation. • The third term stands for antisymmetric Dzyaloshinsky-Moriya exchange interactiond. • The last term is due to the Zeeman interaction.

19. Antiferromagnetic resonance – MFA • If we know the magnetic crystal structure, the mean field approximation can be applied. • The goal is to describe large number of individual spins in the crystal with a few sublattice magnetizations, coupled with each other. • Each sublattice magnetization presents mean field of N spins, lying in the same spot in the primitive cell of the magnetic lattice of the crystal. M1 M2

20. Antiferromagnetic resonance – MFA • Applying the mean field approximation to the Hamiltonian, we formulate the free energy, F: • Mi (i = 1, … , n) is the magnetization of the i-th sublattice, which is given by Mi = – N gμB <Si> • N is the number of magnetic ions in i-th sublattice • < > represent the thermal average. • The parameters in equation are: Z ... number of neighbors E = g / g0 exchange interaction crystal field anisotropy Dzyaloshinsky-Moriya interaction Zeeman interaction

21. Hi Mi Antiferromagnetic resonance – MFA • The mean field Hi acting on the sublattice magnetization Mi is derived as: • Time dependence of the magnetization Mi is described with the equations of motion: • The magnetizations oscillates, with angular frequency ω, therefore we ascribe them time dependence eiωt. • The resonant frequencies are consequently eigenvalues of a matrix with 3n × 3n elements Since it is impossible to exactly solve such system, we are forced to make some approximations.

22. Antiferromagnetic resonance – MFA • This is done in the following steps: • First we calculate the equilibrium orientations of each sublattice magnetization by minimizing the free energy, F. • Approximation:The deviations of each magnetization are small. Hence, we take in to consideration only deviations, perpendicular to the equilibrium orientation. • We can write the each sublattice magnetization as: ... the equilibrium orientation ... the oscillating part, perpendicular to . • Similarly we can write the mean field acting on the i-th sublattice magnetization:

23. Antiferromagnetic resonance – MFA • The equation of motion: • The first term on the right is equal to zero, since the equilibrium orientation of i-th magnetization is parallel to mean field acting on it. • Approximation: In sense of the mean field theory we neglect the last term, as we expect it to be small compared to the other contributions. • What we achieved is: • The oscillating part of each sublattice magnetization is linearly dependent on the oscillating parts of the remaining sublattice magnetizations. • The oscillating part of each sublattice magnetization is perpendicular to its equilibrium orientation – we can describe it with two components instead of three. • We are able to reduce 3n × 3n nonlinear matrix to a 2n × 2n linear matrix, which can be numerically solved for a reasonable number of sublettice magnetizations.

24. Crystal structure of Ni5(TeO3)4Br2 Monoclinic unit cell: C2/c a*||bc b c

25. J2 J2 J1 J1 J3 J3 Spin network in Ni5(TeO3)4Br2 Ni2 b Ni1 Ni2 c Ni3 Ni3 Three different Ni-sites Octahedra: NiO6 (yellow) and NiO5Br(purple)

26. Spin network in Ni5(TeO3)4Br2 Ni2 b Ni3 Ni1 Ni2 c J2 J2 J6 J1 J1 J6 J3 J3 Ni2 Ni3 Ni3 J4 J5 Ni2 Ni1 • Ni-Ni distances d1 = 2.82 Å, d2 = 2.98 Å, d3 = 3.29 Å, d4 = 3.40 Å, d5 = 3.57 Å,d6 = 3.58 Å Six different exchange pathways We can not distinguish between J1, J4 and J3, J5: J1’ = J1 + J4 J3’ = J3 + J5.

27. Experiments performed on Ni5(TeO3)4Br2 • Neutron scattering • Magnetization measurements • Electron spin resonance

28. Neutron scattering • Two spectra were measured, first at 4 K, well bellow the transition temperature TN, and second one above TN. • From the difference in the diffraction spectra the orietation of the magnetic moments was determined: The angles between the magnetic moments and a* are: Ni1 site φ = 1 ° Ni2site φ = 46 ° Ni3 site φ = 33 °

29. Magnetization measurements • The change of the slop around 11 T implies the spin flop transition, which is more obvious if we draw the field dependence of dM/dT • Angular dependence around all three axes: • That magnetization is the smallest, when the applied field is in the a*c plane ~ 25 ° twisted from a* towards c. – easy axis. • The magnetization is the greatest in the b direction – intermediate axis, • The hard axis is in the a*c plane ~ 25 ° twisted from c towards -a*. easy hard intermediate

30. Electron spin resonance • Wide frequency range: from 50 GHz up to 550 GHz in fields up to 15 T. • Detected antiferromagnetic resonance corresponds to the antifferomagnetic ordering expected from neutron diffraction. • Angular dependence was performed at 240 GHz in the range from 5 T up to 12 T.

31. c a* M3 A3 A6 M5 Ni3 M1 Ni2 Ni1 A2 A1 A1 A2 A6 M4 M2 Ni1 A3 M6 a) Ni2 b) Ni3 Analysis • Essential terms in the spin Hamiltonian: • Symmetric exchange interaction • Crystal field anisotropy, since there are three different Ni-sites. • Antisymmetric Dzyaloshinsky-Moriya exchange interaction (DM) • We will attempt to described the system as a combination of: • six different sublattice magnetizationsMi • coupled via four different exchange interactions Ai • and DM interaction between M2 - M1,M3- M1,M5 - M4, and M6- M4. • The only think we have to keep in mind is that M2, M3, M5 and M6 are now twice as big as measured, since every Ni1 has two Ni2 and Ni3 neighbors.

32. Analysis • We were able to: • satisfy the orientation of the magnetic moments measured by neutron scattering • explain magnetization curve • frequency dependence. • The obtained parameters imply: • large Dzyaloshinsky-Moriya contribution • very strong crystal field anisotropy. • Still there isa big chance, that the obtained set of parameters is not the only one. • Other possible contributions: • the magnetic moments are coupled between the Ni – O layers • the exchange interactions are anisotropic M (a. u.) H (T) ν (GHz) H (T)

33. Conclusion • Ordering of the magnetic moments depends on the crystal structure. • Frustration in antiferromagnetic materials is a consequence of a crystal lattice geometry • The mean field method introduced in this seminar is quite a powerful tool to resolve magnetic properties of antiferromagnetic materials. Consequently we were able to determine the dominant terms in spin Hamiltonian of the Ni5(TeO3)4Br2 system. Surprisingly large contribution of Dzyaloshinsky-Moriya interaction Strong crystal field anisotropy The obtained set of parameters is still not completely optimized. • The frustration in the Ni5(TeO3)4Br2 system is obviously suppressed due to the strong interactions – it does not play a significant role at temperatures around 4 K. • Further studies • Explain angular dependences of the magnetization, and AFMR • Consider other contributions