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
Intrudaction: basic properties of mag mater, where we encounter magnetic materials, stress the importance of crystal structure for the mag. behavior
Mag materials: origin of the magnetic moment in atoms, response to the external fields, ordering of the moments,
Frustration: In case of a special crystal structure the magnetic oreder is supresed
Measuing tehniques – stress o the ESR
AFMR – basic properties, mean field approach
Experimens performed
Analysis – present our calculations, and how they match with the experimental results
Conclusion – summarize, stress main congnitions

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 Ni electrons in d-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. 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.
'reversible growth' of magnetic domains
' irreversible growth ' of magnetic domains
domain 'rotation'
paramagnetic behavior
; J > 0

8. Antiferromagnets
antiparalel ordering – at least two interpenetrating sublattices
no magnetization in the absence of the external field
the exchange constant J 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 perpendicular to them.
H = 0
H ≠ 0
H >HC1
H < HSF
H > HSF
H > HC2

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

11. 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. ? J

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

13. 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? J

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 geometries2D
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 field within 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. 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:
S = 1/2
S = 1/2
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 interaction d.
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 g μB <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. 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. Mi Hi

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. Spin network in Ni5(TeO3)4Br2c
Ni2 J2 J1 J3
Ni3
Ni3
Three different Ni-sites
Octahedra:
NiO6 (yellow) and NiO5Br (purple)

26. Spin network in Ni5(TeO3)4Br2c
Ni2 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 °
Ni2 site 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. 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 magnetizations Mi
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. M1 M3 M2 M5 M6 M4 A3 A2 A1 A6
Ni1
Ni3
Ni2 a) c b) a*

32. Analysis We were able to: The obtained parameters imply:
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 is a 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