Download Antiferromagnetic resonance in frustrated system Ni5(TeO3)4Br2

Antiferromagnetic resonance in
frustrated system Ni5(TeO3)4Br2
Matej Pregelj
Mentor: doc. dr. Denis Arčon
Contents
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Introduction
Magnetic materials
Frustration
Measurements of magnetic order
Antiferromagnetic resonance
Experimental results on Ni5(TeO3)4Br2
Analysis
Conclusion
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.
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
e
 Electrostatic repulsion favors unpaired electrons
 Pauli exclusion principle
 Spin configuration in Ni2+ - 8 electrons in d-orbitals
S=1
t
 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.
Magnetic materials
Paramagnetism
 Partition function:
Thermal fluctuations:
 mJ g J  B B 

z   exp 
mJ   J
 k BT 
J
 Magnetization
M  M S BJ ( y)
y  B kBT
Brillouin function:
BJ ( y ) 
2J 1
 2J 1  1
 y 
coth 
y 
coth 

2J
 2J
 2J
 2J 
 In the limit of classical spins - Curie law:
M CCurie


H
T

T
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:
 
E  J Si  S j
 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.
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):
 
 
H   J ij Si  S j  g B  S j  B0 ;
ij
J >0
j
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.
paramagnetic behavior
domain 'rotation'
' irreversible growth ' of magnetic domains
'reversible growth' of magnetic domains
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 < HSF
H≠0
H > HSF
H >HC1
H > HC2
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.
Frustration geometries
 The simplest example – three spins coupled
antiferromagnetically:
J
?
J
J
?
Frustration geometries
 The simplest example – three spins coupled
antiferromagnetically:
J
J
J
?
Once two spins orient in the opposite directions the
third one cannot be antiparallel to both of them.
Frustration geometries
 The simplest example – three spins coupled
antiferromagnetically:
J
J
J
All the pairwise interactions can not be
simultaneously satisfied!
Frustration geometries
 The simplest example – three spins coupled
antiferromagnetically:
J
J
J
All the pairwise interactions can not be
simultaneously satisfied!
 Triangular lattice – What will be the ground state?
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.
Frustration geometries
2D
triangular
Kagomé
3D
fcc cubic
pyrochlore
spinel
In all frustration geometries triangle is a basic building block!
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
Electron Spin Resonance
Classical picture:
 Magnetic moment M in the magnetic field B0
exhibit precession described by Bloch equation:

 
dM
  M  B0
dt




applied B0
S = 1/2
 If we apply radio-frequency, which matches the
precession frequency the absorption occurs.

E  E ( ,  )

S = 1/2

E  E ( ,  )
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:
E (mS ) 
g B
B0 mS
h
 Transition can be induced by radio-frequency EM field:
  E  E (1 / 2)  E (1 / 2) 
g B
B0
h

B0
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:


  
 

 
 

H  2 J ij Si  S j   2 De j (ee j  S j ) 2  Din j (ein j  S j ) 2   d ij  (Si  S j )  g B  S j  B0
ij
j
ij
j
 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.
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
Antiferromagnetic resonance – MFA
 Applying the mean field approximation to the
Hamiltonian, we formulate the free energy, F:

i j


 
 2
 2


 



ij M i  M j   Be j (ee j  M j )  Bin j (ein j  M j )   Cij  ( M i  M j )  E  M j  B0
F   A
j
i j
j
 Mi (i = 1, … , n) is the magnetization of the i-th sublattice,
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:
Aij 
4 Z J ij
N ( g B ) 2
exchange interaction
Be j 
4 Z De j
N ( g B ) 2
Bin j 
4 Z Din j
N ( g B ) 2
crystal field anisotropy
Cij 
which
Z ... number of neighbors
4 Z d ij
N ( g B ) 2
Dzyaloshinsky-Moriya
interaction
E = g / g0
Zeeman interaction
Antiferromagnetic resonance – MFA
 The mean field Hi acting on the sublattice
magnetization Mi is derived as:

F
Hi   
M i
 Time dependence of the magnetization Mi
is described with the equations of motion:



dM i
  M i  Hi
dt

Hi
Mi

 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.
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:



M i  M 0 i  mi (t )

M 0 i ... the equilibrium orientation


 it
mi (t )  mi e ... the oscillating part, perpendicular to M 0 i .
 Similarly we can write the mean field acting on the i-th sublattice
magnetization:   


H i  H 0 i ( M 0 i )  dH i (mi (t ))
Antiferromagnetic resonance – MFA
 The equation of motion:




dmi

  M 0 i  mi  H 0 i  dH i  
dt





 

  M 0 i  H 0 i   M 0 i  dH i   mi  H 0 i   mi  dH i 




 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.
Crystal structure of Ni5(TeO3)4Br2
Monoclinic unit cell: C2/c
a*||bc
c
b
Spin network in Ni5(TeO3)4Br2
c
b
Ni2
J2 Ni1 J Ni2
2
J1
J3
Ni3
J1
J3
Ni3
Three different Ni-sites
Octahedra:
NiO6 (yellow) and NiO5Br (purple)
Spin network in Ni5(TeO3)4Br2
c
b
Ni2
Ni3
J2 Ni1 J Ni2
2
J6
J1
J6
Ni2
J3
J3
Ni3
J4
Ni2
J1
Ni3
J5
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.
Experiments performed on Ni5(TeO3)4Br2
Neutron scattering
Magnetization measurements
Electron spin resonance
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 °
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
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.
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.
a*
c
Ni3
Ni1
M1
Ni2
A2
a)
Ni2
Ni1
Ni3
b)
M3 A6
A3
A1
M2
M5
A2
A1
A6
M6
A3
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.
Analysis
 We were able to:
15000
12500
M (a. u.)
 satisfy the orientation of the
magnetic moments measured by
neutron scattering
 explain magnetization curve
 frequency dependence.
17500
 The obtained parameters imply:
 the magnetic moments are
coupled between the Ni – O layers
 the exchange interactions are
anisotropic
7500
5000
 large Dzyaloshinsky-Moriya
contribution
 very strong crystal field
anisotropy.
2500
2
4
6
8
10
12
14
H (T)
1000
800
ν (GHz)
 Still there is a big chance, that the
obtained set of parameters is not
the only one.
 Other possible contributions:
10000
600
400
200
2
4
6
8
H (T)
10
12
14
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