Magnetic Properties of Fe3+ based

honeycomb lattice

A project report submitted in partial fulfillment for the award of the degree of
Master of science by
Sayak Das

Roll no: PH21C040

Under the guidance of

Dr. Panchanana Khuntia

DEPARTMENT OF PHYSICS

INDIAN INSTITUTE OF TECHNOLOGY MADRAS

May 24, 2023

1
 CERTIFICATE
This is to certify that the project titled Magnetic Properties of Fe3+ based honey-
comb lattice is a bona fide record of work done by Sayak Das towards the partial
fulfillment of the requirements of the Master of Science degree in Physics at the
Indian Institute of Technology, Madras, Chennai 600036, India.

(Dr. Panchanana Khuntia, Prjoect supervisor)

2
 ACKNOWLEDGEMENT

I want to start by sincerely thanking my supervisor, Dr. Panchanana

Khuntia, for including me in this study and introducing me to this
fascinating topic about which I was completely unaware of.

Also, I would like to thank my seniors Ms. Maneesha Barik , Mr.

Umashankar Jena without whom the work might not have been com-
plete so far. Their continuous care, insightful comments and guidance
helped throughout of my project work and writing of this thesis.

3
 ABSTRACT

The interplay between dimensionality, spin correlation, and charge transport assumes
an important role in the realization of unconventional phenomena, including quan-
tum spin liquid, spin glass, superconductivity, anomalous Hall effect, and fractional
Hall effect, within the realm of condensed matter physics. In the case of two di-
mensional (2D) spin structure, the dimensional confinement, large surface-to-volume
ratio, and enhanced quantum fluctuation (Mermin-Wagner theorem) give rise to var-
ious emergent phenomena. Insulators(Mott-insulator) with triangular, kagome, hon-
eycomb, and square lattices are the current focus of research to study various spin
dynamics in the low energy limit. We have successfully synthesized a polycrystalline
honeycomb lattice with the chemical formula SrFeSn(PO4 )3 . The high temperature
synthesis crystallizes it into a trigonal lattice with R 3̄ space group. The magnetic
study of the material has revealed the presence of ZFC-FC bifurcation around the
temperature 14.8 K in the field of 100 Oe. Owing to the Curie-Weiss fitting of in-
verse susceptibility with temperature, the Curie-Weiss temperature (θCW ) is found
to be -32 K, which suggests the presence of antiferromagnetic interaction between
Fe3+ moments arrayed in the honeycomb lattice. Further, the five-quadrant M H
curve rules out the possibility of ferromagnetic ordering and the bifurcation of ZFC-
FC DC susceptibility can be due to the appearance of spin glass in the low tempera-
ture region. We carried out the ac susceptibility experiment at various frequencies to
comprehend the ground state behavior of the system. The shifting of peaks in the ac
susceptibility may require higher frequencies as there is absence of any such shifts
in the measured frequency window.

4
Contents

1 Introduction to Magnetism 6
1.1 Origin of Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Orbital Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 Spin Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 An atom in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Magnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Classification of magnetic materials . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.1 Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.2 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.3 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.4 Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Phase transition and critical phenomena 17

2.0.1 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.0.2 Landau theory of Symmetry breaking . . . . . . . . . . . . . . . . . . . . . 19
2.0.3 Critical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Unconventional ground states of magnetism 22

3.1 Frustrated Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Spin Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Quantum Spin Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Experimental techniques 27
4.1 X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.1 Production of X-Ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.2 Principle of XRD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.3 X-ray Diffraction Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Principle of VSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Interpreting Magnetic susceptibility data . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Magnetic studies on SrFeSn(PO4 )3 35

5.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 XRD refinement and structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.4 Magnetic behaviour of SrFeSn(PO4 )3 . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.6 Future Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5
Chapter 1

Introduction to Magnetism

Magnetism refers to the fundamental property of material associated with the interaction of moments
of constituent entities. Research in magnetism is motivated by the significant influence it has on a
variety of facets of daily life, including data storage and electrical motors. Apart from the real world
applications, the underlying physics of magnetism has fascinated a generation of physicists which
is now at the heart of modern condensed matter physics. Starting from the conventional symmetry
breaking mechanism to the development of topological behavior, this field constitutes a rich variety of
physics. The collective behavior of moments brings about exotic states to emerge and the underlying
mechanisms are important for the development of quantum technologies. Recent scientific progress
in this area has focused on the synthesis of transition metal and rare-earth based compounds where
the elongated d and f-bands play vital roles in the magnetic behavior of such materials. Here we will
briefly address the origin of conventional magnetism from the atomic level.

1.1 Origin of Magnetism

We’ll look at the characteristics of isolated magnetic moments in this chapter. At this stage, in-
teractions between magnetic moments on different atoms, or between magnetic moments and their
immediate environments, are ignored. Electrons are fermions obeying Pauli’s exclusion principle and
follow Fermi-Dirac statistics. There are two possible origins for the magnetic properties of atoms.
The orbital motion of electrons gives rise to an orbital magnetic moment and because electrons are
fermions, they have a half-integer spin (1/2 for an electron), which is connected to a spin magnetic
moment.

1.1.1 Orbital Magnetic Moment

Consider an electron of mass m and charge −e moving with the velocity of magnitude v in a circular
Bohr orbit of radius r as shown in figure 1.1. The charge circulating in a loop constitutes a current of
magnitude
e e
I=− =−
T 2πr
where T is the orbital period of the electron. A magnetic field produced by such a current loop is
equivalent to a fictional magnetic dipole moment that produces a similar field from the loop. For a
current I in a loop of area A, the magnitude of the orbital magnetic dipole moment is
µL = IA

6
Because of the negative charge, the magnetic dipole moment is anti-parallel to its orbital angular
momentum L = mvr. Thus
ev 2 evr e e
µL = IA = − r =− =− mvr = − L
2πr 2 2m 2m
eℏ
This can be written in term of Bohr magneton µB = 2m and
is often regarded as the elementary unit of magnetic moment.
Its value is 9.27 × 10−24 .
eℏ |L| p
|µL | = − = µB l(l + 1)
2m ℏ
The z-component of the orbital angular momentum of the elec-
tron is
e
µz = − Lz = γLz (1.1)
2m
e
where γ = − 2m . The constant γ is called the gyro-magnetic
ratio of the electron. In particular, its z-component is quantized
and restricted to the values

µLz = γml ℏ ; ml = l, l − 1, ....., −l Figure 1.1: Orbital magnetic mo-

ment
eℏ
Where, the Bohr Magneton is also defined as µB = −γml ℏ = 2m . In term of Bohr Magneton, the
z-component of orbital magnetic moment is

µLz = −µB ml (1.2)

1.1.2 Spin Magnetic Moment

Now we consider the magnetic moment that arises from the spin of the electron.The relation between
the spin and its magnetic moment can be derived from the relativistic Dirac equation,
which gives µs = 2γS. The experimental value of the magnetic
moment can be determined by observing the effect of a mag-
netic field on the motion of an electron beam, and it is found
that
µs = gγS where g = 2.0023
The factor g is called the g-factor of the electron. The magnetic
moment due to electron spin can be written as
e
µs = − S (1.3)
2m
p
The magnitude of S is equal to s(s + 1)ℏ, here s is called spin
Figure 1.2: Spin magnetic moment quantum number. As for the orbital magnetic moment, the spin
magnetic moment has quantized components on the z-axis (µSz ), and we write
e 1
µSz = g Sz = −gµB ms ; ms = ± (1.4)
2m 2
where Sz is the spin moment along z direction.

7
1.2 An atom in a magnetic field
Let us consider that there are Z electrons in an atom, consider ith electron let its position be ri and
momentum be pi . Let us now consider an atom with Hamiltonian Ĥ0 [2] given by
Z  2 
X pi
Ĥ0 = + Vi (1.5)
i=1
2m e

2
p i
where 2m e
is the kinetic energy for the ith electron and Vi is the potential energy of ith electron. Let
us assume that Ĥ0 has known eigen states and eigenvalues. We now add Magnetic field B given by

B =∇×A (1.6)

where A is the magnetic vector potential. We choose a gauge such that

B×r
A= (1.7)
2
We know that in the presence of magnetic field effective momentum (pef f ) becomes

pef f = p + eA(r) (1.8)

(p+eA(r))2
So kinetic energy becomes 2me
and hence the perturbed Hamiltonian must now be written

Z 
(p + eA(r))2
X 
Ĥ = + Vi + gµB B · S
i=1
2me
X  p2 
e2 X
i
= + Vi + µB (L + gS) · B + (B × r i )2 (1.9)
i
2m e 8m e i
2 X
e
= Ĥ0 + µB (L + gS) · B + (B × r i )2
8me i

Hence, in equation 1.9, the first term represents the original Hamiltonian (the one in the absence of
a magnetic field), which is normally the dominant term, the second term represents the paramagnetic
term, and the third term represents the diamagnetic term. We might refer to the last two terms as the
perturbation terms. The second term is more prominent than the third term.[9]

1.3 Magnetic Susceptibility

Now let us introduce a term called magnetic susceptibility, it is defined as χ = M /H where M
is magnetization which is defined as number of magnetic moments per unit volume and H is the
magnetic field and χ is known as magnetic susceptibility. We assume that these two quantities are
linearly related, which is typically most valid at high temperatures and low fields [8]. Magnetic
susceptibility measurements, in general, tell how the material responds to an applied magnetic field,
which can be used to reveal that material’s magnetic properties.

8
1.4 Classification of magnetic materials
When a magnetic substance is subjected to an external magnetic field, the magnetic moments of the
material behave differently. Materials are divided into different categories based on the response. The
main categories are Diamagnetic , Paramagnetic , Ferromagnetic , Antiferromagnetic .

1.4.1 Diamagnetism
All materials show some degree of weak, negative magnetic susceptibility. This is known as Diamag-
netism. For a diamagnetic substance, a magnetic field induces a magnetic moment which opposes the
applied magnetic field that caused it. This effect is frequently described from a classical standpoint as
the action of a magnetic field on an electron’s orbital motion creates a back e.m.f, which, according to
Lenz’s equation, opposes the magnetic field that causes it. So the magnetic moment is always oppo-
site to the direction of external magnetic field. Although diamagnetism can be explained classically,
it is a quantum phenomenon that should be studied using quantum mechanics.
Consider the case of an atom with no unfilled electronic shells so that the paramagnetic term in
eqn 1.9 can be ignored. If B is parallel to the z axis, then we have B = B ẑ. From the third term in
eqn 1.9 we have
(B × r i )2 = B 2 (xi 2 + yi 2 ) (1.10)
so that the first-order shift in the ground state energy due to the diamagnetic term is
Z
e2 B 2 X
∆E0 = ⟨0| (xi 2 + yi 2 ) |0⟩ (1.11)
8me i=1

where ⟨0| is the ground state wave function. If we assume a spherically symmetric atom (This is a
good assumption if the total angular momentum J is zero) , ⟨xi 2 ⟩ = ⟨yi 2 ⟩ = ⟨zi 2 ⟩ = 31 ⟨ri 2 ⟩ then we
have
Z
e2 B 2 X
∆E0 = ⟨0| ri2 |0⟩ (1.12)
12me i=1
Consider a solid composed of N ions (each with Z electrons of mass m) in volume V with all shells
filled . Now magnetization M can be written as:
Z
∂F N ∂∆E0 N e2 B X 2
M =− =− =− ⟨ri ⟩ (1.13)
∂B V ∂B 6me i=1

where F is the Helmholtz function. Hence we can extract the diamagnetic susceptibility χ = M/H ≈
µ0 M/B (assuming that χ ≪ 1). Following this procedure, we have the result that
Z
N e2 µ0 X 2
χ=− ⟨ri ⟩ (1.14)
V 6me i=1

This expression has assumed first-order perturbation theory. Diamagnetic susceptibilities are usually
PZef f 2
largely temperature independent. Now we can crudely approximate that i=1 ⟨ri ⟩ ≈ Zef f r2 , where
Zef f is the number of electrons in the outer shell of an ion. Diamagnetism is present in all materials,
but it is a weak effect that can either be ignored or is a small correction to a larger effect[2].

9
1.4.2 Paramagnetism
In paramagnetic material, the magnetic moments tend to align in the same direction as the applied
magnetic field, resulting in a positive susceptibility. Even in the absence of any magnetic field, the
atoms have some magnetic moment due to the presence of unpaired electrons. The interaction be-
tween neighboring magnetic moments is assumed to be weak, so they are considered independent.
When there is no applied magnetic field, the magnetic moments are randomly oriented, resulting in a
net magnetization of zero. However, when a magnetic field is applied, the magnetic moments tend to
align with the field, which produces a nonzero magnetization.
The magnetic moment of an atom is dependent on the total angular momentum J which is the
sum of orbital angular momentum L and spin angular momentum S given by:

J =L+S (1.15)

Now J can take any integer or half integer values. Consider a system that has n number of magnetic
moments per unit volume and is subjected to an external magnetic field B ẑ. Now the magnetic
moment for an atom is given as:
gJ e
µ=− J (1.16)
2m
Where gJ is the Lande-g-value given by:
J(J + 1) + S(S + 1) − L(L + 1)
gJ = 1 + (1.17)
2J(J + 1)
Now the average magnetic moment is given by:
gJ e
⟨µ⟩ = − ⟨J ⟩ (1.18)
2m
Since the magnetic field is pointing in the z direction we have ⟨J ⟩ = ⟨J z ⟩ẑ = ⟨mJ ⟩ℏẑ . So we have

⟨µ⟩ = ⟨µz ⟩ = gJ µB ⟨mJ ⟩ (1.19)

Now the partition function for the system can be written as :

J
X 
mJ gJ µB B

Z= e kB T
(1.20)
mJ =−J

From this partition function we find ⟨mJ ⟩ which is given as:

J
X mJ emJ y
⟨mJ ⟩ = (1.21)
m =−J
emJ y
J

gJ µB B
Where y = kB T
. Now magnetization is given by :

∂lnZ
M = nkB T (1.22)
∂B
Now taking x = yJ with substitution we find that

M = Ms BJ (x) (1.23)

10
WhereMs is called Saturation magnetization and its value is given by Ms = ngJ µB J. BJ (x) is the
Brillouin Function defined as:
 
2J + 1 2J + 1 1 x
BJ (x) = coth x − coth (1.24)
2J 2J 2J 2J

For small magnetic field we can approximate BJ (x) = (J+1)

3J
+ O(x3 ). Hence susceptibility is given
by:
M µ0 M nµ0 µ2 ef f
χ= = = (1.25)
H B 3kB T
p
Where µef f = gJ µB J(J + 1). We see that χ is inversely related to temperature, this suggests that
susceptibility has a Curie’s law dependence where χ = CT .

Figure 1.3: Curie’s law states that χ ∝ T1 as shown in (a). Thus a straight line in the graph is obtained
by plotting χ1 against T as shown in (b) [2]

1.4.3 Ferromagnetism
Ferromagnetism emerges at a finite temperature where all moments point along an easy axis. The
spin-exchange strength J is positive and the excited state is characterized by the presence of bosonic
excitation such as magnon. At high temperature regime, the thermal energy randomizes the moment
hence resulting in paramagnetic behavior with zero average magnetization in the absence of any
external field. When a ferromagnetic substance is in an unmagnetized state, the magnetization vector
in different domains is oriented in different domains such that the resultant magnetization is zero.
When an external magnetic field is applied the domain rotates and the individual magnetic dipole
aligns its magnetic moment along the field direction hence results in a finite moment which is depicted
in the fig. 1.4.
When a ferromagnet is placed in an applied magnetic field B, the appropriate Hamiltonian to solve
is [2]: X X
Ĥ = − Ji,j S i · S j + gµB Sj · B (1.26)
i,j j

11
 Figure 1.4: (a) In absence of magnetic field (b) In presence of magnetic field

The first term in equation 1.26 is Heisenberg exchange energy and the second term is Zeeman
energy. For a ferromagnetic model the exchange interaction constant Ji,j > 0. One of the simple
models to understand ferromagnetism is the Weiss model. From equation 1.26 we can say that the ith
spin experiences an internal effective molecular field is given by:
2 X
B mf = − Ji,j S j (1.27)
gµB j

In Weiss model we assume that: (1) all the magnetic ions experience the same internal molecular field
Bmf given by equation 1.27. (2) the molecular field is assumed to be of the form Bmf = λM . Let us
now focus on ith ion. The interaction energy for an ith ion is given as:
X X
Ei = −2 Ji,j S i · S j + gµB Sj · B
j j (1.28)
= gµB S i · B mf
Now the total Hamiltonian becomes :
X X
H = gµB S i · (B mf + B) = gµB S i · (λM + B) (1.29)
i i

Here we can interpret that all the atoms experience a magnetic field of B + λM . So we can treat this
model as a simple paramagnetic material that is experiencing a B + λM field. At low temperatures,
the internal molecular field can align the moments even in the absence of an applied field. It is
noteworthy that the alignment of these magnetic moments creates the internal molecular field that
causes the alignment initially. Now the magnetization M can be expressed as
M = ngJ µB JBJ (x) (1.30)
g µ JB gJ µB J(B+λM )
where BJ (x) = 2J+1 coth 2J+1 1 x



2J 2J
x − 2J coth 2J and x = J kBB T ef f = kB T
. In case of
spontaneous magnetization , B = 0, so
gJ µB JλM
x= (1.31)
kB T
xkB T
and M (T ) = (1.32)
λgJ µB J

12
 as T −→ 0 or x −→ ∞ , BJ (x) −→ 1 : the magnetic moments align themselves parallel to the
field and the magnetization M becomes the saturation magnetization Ms (0). Then we can write

Ms (0) = ngJ µB J (1.33)

Dividing equation 1.32 by 1.33 we get

M (T ) xkB T
= (1.34)
Ms (0) λngJ 2 µB 2 J 2

also dividing equation 1.32 by 1.30 , we get

M (T )
= BJ (x) (1.35)
Ms (0)

Now the magnetization M (T ) at a given temperature can be obtained by solving equation 1.34 and
1.35 simultaneously.

Figure 1.5: The graphical solutions for equation 1.34 and 1.35 [2]

In the above graph, the three straight lines passing through the origin represent the RHS of equation
1.34, at different values of temperature T . The curve represents equation 1.35 i.e. the BJ (x) function.
Now the only non-zero solution ( where these two curves intersect) exists for T < Tc . This indicates
a non-zero value of M even for zero external magnetic fields and hence corresponds to spontaneous
magnetization.
Above graph also shows that spontaneous magnetization decreases with an increase in temperature
and vanishes beyond the temperature Tc , which is known as ferromagnetic Curie temperature. For
T > Tc , spontaneous magnetization is zero. So external field will have to be applied to produce some
magnetization. But the field should be small enough to avoid the saturation state.

13
 Now as we know M = ngJ µB JBJ (x) and BJ (x) ≈ J+1


3J
x (for x << 1) . Also we know
gJ JµB (B+λM )
x= kB T
. Thus :
ng 2 J µ2 B J(J + 1)
M= (B + λM ) (1.36)
3kB T
Now susceptibility χ is given by

M ng 2 J µ2 B J(J + 1)B ng 2 J µ2 B J(J + 1)λM

χ= = +
H 3kB T H 3kB T H
 2 2
 2 2 (1.37)
ng J µ B J(J + 1)λ µ0 ng j µ B J(J + 1)
χ 1− =
3kB T 3kB T

Now we can define

ng 2 J µ2 B J(J + 1)λ
Tc = (1.38)
3kB T
So equation becomes
 
Tc µ0 Tc /λ
χ 1− =
T T
(1.39)
µ0 Tc /λ C
χ= Tc

=
T 1− T T − Tc

Figure 1.6: This plot depicts the variation of χ wrt T of ferromagnetic system [6]

Where C = µ0λTc and the equation 1.39 is known as Curie-Weiss law. It agrees well with the
temperature dependence of χ in the paramagnetic region provided T > Tc .

14
1.4.4 Antiferromagnetism
Antiferromagnetism occurs when the exchange interaction is negative (J < 0) and the molecular field
aligns in a way that benefits the nearest neighbor magnetic moments to be anti-parallel to each other.
This phenomenon commonly happens in systems that can be viewed as two overlapping sublattices
(refer to Figure 1.7 ), where one sublattice has magnetic moments pointing upwards and the other has
them pointing downwards. In Figure 1.7, the magnetic moments’ closest neighbors are entirely on
the other sublattice. To begin with, we will assume that the magnetic field acting on one sublattice is
directly proportional to the magnetization of the other sublattice and that there is no external magnetic
field applied.

Figure 1.7: An antiferromagnet can be decomposed into two interpenetrating sublattices [2]

The Weiss model is another simple model for explaining antiferromagnetism. We assume that a
sublattice’s molecular field is proportional to the magnetization of another sublattice. Therefore we
label ’+’ for the ’up’ sublattice and use the symbol ’−’ for the down. Hence the molecular field
experienced by the up sublattice is B+ and the molecular field experienced by the down sublattice is
B− . So far, we have:
B+ = −|λ|M− (1.40)
B− = −|λ|M+ (1.41)
 
gJ µB JλM−
M+ = BJ (1.42)
kB T
 
gJ µB JλM+
M− = BJ (1.43)
kB T
Equations 1.42 and 1.43 came from equation 1.23. In these sublattices everything is equivalent apart
from direction of the moments , such that |M− | = |M+ | = M . So that :
 
gJ µB JλM
M = BJ (1.44)
kB T
This form of the equation is identical to the corresponding equation of a ferromagnet. So we will
have a similar plot to that of Figure 1.5 and have have transition temperature also known as Neel
temperature TN above which there is no more antiferromagnetic ordering, below TN we can find
antiferromagnetic ordering. We can follow the same approach as done for ferromagnet to find sus-
ceptibility and Neel temperature for an antiferromagnet they are given as:
gJ µB (J + 1)|λ|Ms
TN = (1.45)
3kB
and
1
χ∝ (1.46)
T + TN

15
 Figure 1.8: This plot shows variation of χ with temperature on antiferromagnetic systems [2]

Figure 1.8 illustrates the susceptibility of an antiferromagnetic material under different magnetic
field orientations. The susceptibility represented by χ⊥ occurs when a magnetic field is applied per-
pendicular to the magnetic moments of the material, while χ∥ occurs when the field is applied parallel
to the moments. At temperatures below TN , χ⊥ remains constant while χ∥ decreases as temperature
decreases, reaching zero at T = 0. However, above TN , the system behaves as a paramagnet and
follows the equation 1.46 .

16
Chapter 2

Phase transition and critical phenomena

The diagram in figure 2.1 illustrates the phase transition behavior of an Ising ferromagnet. The graph
displays the average magnetization as a function of magnetic field (H) and temperature (T). If we
follow path 1, where the temperature remains constant and below the critical temperature (Tc ), we
observe that the magnetization suddenly switches from positive to negative as the magnetic field (H)
changes, reaching zero at H = 0. This indicates a change in the direction of the spins from up
to down. On the other hand, if we follow path 2, where the magnetic field is held at zero and the
temperature is gradually lowered from Tc , the magnetization continuously increases from zero.

Figure 2.1: This figure represents the Phase diagram of Ising ferromagnet [12]

The phase diagram reveals that a ferromagnetic substance can undergo a sudden change in its
overall behavior when subjected to altered external conditions such as temperature or magnetic field.
The points at which this abrupt shift in behavior occurs are known as critical points, and they repre-
sent the phase transition from one state to another. In Figure 2.1, Path 1 shows a sudden change in
magnetization, which involves a discontinuous change in thermodynamic properties and is referred
to as a first-order phase transition. On the other hand, Path 2 demonstrates that the magnetiza-
tion gradually decreases to zero as the system transitions from ferromagnet to paramagnet, and such
transitions are called second-order phase transitions.
If we modify the magnetic field and temperature of the sample, there may be some fluctuation
in microscopic properties (such as magnetization) across the sample. The parameter which charac-

17
terizes this fluctuation is called Correlation length. During a first-order phase transition, the Cor-
relation length has a finite value. However, during a second-order transition, the Correlation length
becomes infinite, causing fluctuations to be correlated across the entire sample and resulting in the
system being in a distinct critical phase. Therefore, as the correlation length increases indefinitely,
the magnetization gradually decreases to zero.

2.0.1 Phase Transitions

Let us consider more carefully the classic example of a phase transition involving the condensation
of a gas into liquid. The phase diagram represented in figure 2.2 exhibits several important features
and generic features of phase transitions:

• In the (P, T ) plane, the phase transition occurs at a line that terminates at a critical point
(Pc , Tc ).

• In the (P, v) plane, the transition appears as a coexistence interval corresponding to a mixture
of gas and liquid of densities ρg = 1/vg and ρl = 1/vl at a temperature T < Tc .

• Due to the termination of the coexistence line it is possible to go from gas phase to liquid phase
continuously by going around the critical point. Thus there are no fundamental differences
between liquid and gas phases

• The difference between the densities of the coexisting liquid and gas phases vanishes on ap-
proaching Tc , i.e. ρl −→ ρg as T −→ Tc −

• The pressure versus volume isotherms become progressively more flat on approaching T c from
the high temperature side. This implies that the isothermal compressibility, the rate of change
of density with pressure, κT = −(1/V )(∂V /∂P )T diverges as T −→ Tc

Figure 2.2: Phase diagram of (a) Liquid-Gas transition (b) ferromagnetic transition. Here P ←→
H, 1/v ←→ m . Isotherms above, below and at Tc are sketched [9]

18
The isotherm displayed in figure 2.2 reveals a discontinuity in magnetization when the magnetic field
H, passes through zero. Similar to the condensation problem, the magnetization isotherm shares
several characteristics. Both scenarios involve a line of abrupt transition that ends at a critical point,
where the isotherm displays unique behavior. The symmetry of H −→ −H simplifies the phase
diagram’s appearance, guaranteeing that the critical point happens at Hc = Mc = 0.

2.0.2 Landau theory of Symmetry breaking

The Russian physicist Lev Landau proposed a model that produces a phase transition in a straightfor-
ward way, based on some fundamental concepts. In this model, the free energy of a ferromagnet with
magnetization M is expressed as a series of powers of M . Since there is no energy preference for
either the ”up” or ”down” direction, this series only contains even powers of M , and we can represent
the free energy as F (M ) using this formula [2]

F (M ) = F0 + a(T )M 2 + bM 4 (2.1)

where F0 and b are constants (we assume b > 0) and a(T ) is temperature dependent. By allowing
the temperature dependent function a(T ) to change its sign at the transition temperature TC , we can
demonstrate that the system undergoes a suitable phase transition. Thus near the transition, we write
a(T ) = a0 (T − TC ) where a0 is a positive constant. To find the ground state of the system, it is
necessary to minimize the free energy so we look for solutions of ∂F/∂M = 0. This condition
implies
2M [a0 (T − TC ) + 2bM 2 ] = 0 (2.2)
The left-hand side of this equation is a product of two terms, so either of them could be zero. This
means  1/2
a0 (T − TC )
M = 0 or M = ± (2.3)
2b

Figure 2.3: [2]

The second condition is only valid when T < TC , otherwise one is trying to take the square root
of a negative number. The first condition, on the other hand, is valid for temperatures above or below

19
TC , but if the temperature is below TC , it only results in an unstable equilibrium position (which is
determined by evaluating ∂ 2 F/∂M 2 ). Thus the magnetization follows the curve shown in Fig. 6.6(b);
it is zero for temperatures T ≥ TC and is non-zero and proportional to (T c − T )1/2 for T < TC .
Landau’s approach to studying phase transitions is called a mean-field theory which means that
it assumes that all spins ’feel’ an identical average exchange field produced by all their neighbors.
This field is proportional to the magnetization.

2.0.3 Critical Behaviour

The set of critical exponents describes the unique behavior that occurs near the critical point. This
behavior is associated with the non-analytic nature of different thermodynamic exponents. An impor-
tant parameter that can be used to distinguish between different phases or to track changes during a
phase transition is known as the order parameter. In the case of a magnet, the order parameter is
represented by the magnetization M (H). For condensation, the order parameter is determined by the
density. Now m is defined by:
M (H, T )
m(H, T ) = (2.4)
N
In the zero field, m vanishes for a paramagnet and is non-zero for a ferromagnet. So near Tc we have:
(
0 T > Tc
m(T, H → 0+ ) ∝ β (2.5)
|t| T < Tc

Here t = (T − Tc )/Tc denotes the reduced temperature. The singular behaviour of m along the
coexistence line is therefore characterised by a critical exponent β.
The singular behaviour of m along the critical isotherm is governed by another exponent, δ given by:

m(T = Tc ) ∝ H 1/δ (2.6)

Critical systems are extremely responsive to external perturbations. For instance, at the liquid-gas
critical point, the compressibility given by κT = −(1/V )(∂V /∂P )T becomes infinite. The parameter
corresponding in a ferromagnetic system is magnetic susceptibility, the divergence of susceptibility is
characterized by critical exponent γ, so for a ferromagnetic system we have :
 
∂m
+
χ± (T, H −→ 0 ) = ∝ |t|γ± (2.7)
∂H H=0

Where in principle two exponents γ+ and γ− are required to specify the divergence on both sides of
the phase transition. Actually, in almost all cases we have γ+ = γ− .
Similarly, we have Heat capacity which also diverges at critical point and its singularity at zero field
are described by the exponent α.
∂E ±
C± = ∝ |t|α (2.8)
∂T
E here refers to the internal energy and again the critical exponents usually coincide i.e α+ = α− .

20
 Figure 2.4: Critical points of the response functions [12]

The parameter such as magnetic susceptibility (χ), specific heat (C), compressibility (κT ), mag-
netization (m) etc are known as response function. Some typical response function are plotted at
critical points as shown in 2.3.
The plot of m2 vs r provides information on the system’s fluctuations, while ξ represents the correla-
tion length. Figure 2.3 illustrates that susceptibility diverges at T = Tc , and the specific heat (C) and
correlation length (ξ) follow a similar pattern. Moreover, magnetization goes to zero at T = Tc , as
observed in figure 2.3.

21
Chapter 3

Unconventional ground states of magnetism

We know that when a magnetic material is exposed to an external magnetic field, the material’s
magnetic moments respond differently. Based on their response materials are classified into various
categories, such as paramagnet, ferromagnet, and antiferromagnet. All of these materials exhibit a
symmetry-breaking phase transition below a certain transition temperature, which is why they are
referred to as conventional magnets. However, there are some systems where no symmetry-breaking
phase transition is observed, and they are called unconventional magnets. Frustration-induced quan-
tum spin liquid, spin glass, and spin ice are some of the unconventional phenomena.

3.1 Frustrated Magnet

The term ”frustration” in the field of physics describes a situa-
tion where opposing forces cannot be satisfied simultaneously.
This idea has been broadly applied in areas such as magnetism.
The study of frustration originated with antiferromagnets, where
geometric frustration is typically the cause. This type of frus-
tration arises in systems where spins are present on lattices that
include triangular patterns. The nearest-neighbor interactions
encourage anti-aligned spins, but on a triangle, all three spins
cannot be antiparallel at the same time. Depending on the spe-
cific circumstances, the spins may fluctuate down to low tem-
perature or the spin correlation brings long range or short range
ordering.
The most basic example of frustration is a triangle of Ising
spins that interact in an antiferromagnetic manner and can only
point up or down. It is impossible for all three spins to be with
each others, resulting in six ground states instead of the two Figure 3.1: A triangle of antiferro-
mandated by Ising symmetry. This degeneracy can persist on magnetically interacting Ising spins
2D and 3D lattices, which enhances fluctuations and suppresses [1]
ordering. Ramirez introduced an empirical measure of frustra-
tion based on this fact, which has become widely used. This measurement is obtained by comparing
the Curie-Weiss temperature, ΘCW , which provides a estimate for the strength of magnetic interac-
tions (θCW < 0 for an antiferromagnet), with the temperature at which order freezes, Tc . The ratio
these temperatures gives the frustration parameter f (f = |θCWTc
|
). Typically, f > 5–10 [1] indicates a

22
strong suppression of ordering, as a result of frustration.

Figure 3.2: Frustrated lattices [7] [1]

3.2 Spin Glass

Spin glasses are a type of disordered material that can be described in very simple terms. They consist
of magnetic moments, or ”spins,” that interact with one another in a random manner, both in terms of
the sign and magnitude of their interactions [14]. In contrast to ferromagnets, which have positively
interacting moments that all align to produce a macroscopic magnetization, and antiferromagnets,
which have negatively interacting moments that form two interlocking sublattices oriented in opposite
directions, spin glasses are a relatively simple archetype of disordered materials.

Figure 3.3: Plaquette of the square AF lattice with one bond replaced by a FM bond, illustrating
frustration induced by site disorder common to most spin glasses [10]

The case of spin glasses can be simply described as a mixture of both ferro- and antiferromagnetic
situations. The magnetic moments, or spins, interact randomly with one another, such that each

23
moment experiences conflicting constraints from its neighbors, which can be either ferromagnetic or
antiferromagnetic. This situation of conflicting influences is referred to as ”frustration”. Unlike in
simple symmetric configurations, there is no equilibrium state with a clear minimum of energy in
spin glasses. Rather, there are numerous possible arrangements of the spins with comparable energy,
leading to a large number of metastable states. It is very difficult to find the absolute minimum energy
configuration, and in practice, spin glasses are almost always out of equilibrium.
The initial spin glass materials that were discovered were made up of non-magnetic metals (Au,
Ag, Pt. . . ). These metals had a small percentage of magnetic atoms (Fe, Mn. . . ) scattered randomly
within them. For instance, in Cu:Mn 3%, there are three manganese magnetic atoms that are posi-
tioned at random distances from each other. The RKKY interaction between these atoms is oscillating
in nature, which causes their coupling constants to have random signs[14].
The dynamic behavior of spin-glass is usually characterized by AC susceptibility[5]. When mag-
netic spins are in a spin-glass material, they interact randomly with other magnetic spins. This creates
a state that is highly irreversible residing in a metastable. The spin-glass state is realized below the
freezing temperature, and the system is paramagnetic above this temperature.

Figure 3.4: AC susceptibility of CuMn (1 at% Mn) showing the cusp at the freezing temperature. The
inset shows the frequency dependence of the cusp from 2.6 Hz (triangles) to 1.33 kHz (squares)[5].
′
To determine the freezing temperature of a system, a curve of χ vs temperature is taken into con-
sideration. This curve shows a sharp bend or cusp around the freezing temperature. The measurement
of AC susceptibility is vital for spin-glass systems because the freezing temperature cannot be de-
duced from the specific heat. Moreover, the position of the cusp is influenced by the frequency of the
AC susceptibility measurement, a characteristic of the change in the relaxation time with the change
of frequency and it validates the existence of the spin-glass phase. These two aspects are visible in
the AC susceptibility data for Cu1−x M nx , as illustrated in Figure 3.4.

24
3.3 Quantum Spin Liquid
Quantum spin liquid refers to a dynamic state of a collection of spins (collective behavior) persisting
down to zero temperature. Typically, quantum spin liquids are identified by their long-range quantum
entanglement, lack of conventional magnetic order, and fractional excitations[1]. The concept of
quantum spin liquids was initially introduced by physicist Phil Anderson in 1973. Anderson proposed
that a system of spins arranged in a triangular lattice, interacting antiferromagnetically with their
nearest neighbors, would result in a ground state characterized by a quantum spin liquid.

Figure 3.5: Valence-bond states of frustrated antiferromagnets[1].

The valence bond is a natural building block for non-magnetic states, consisting of a pair of spins
that form a spin singlet state due to an antiferromagnetic interaction. Valence bonds are quantum
phenomena with maximal entanglement and non-classical behavior. When all the spins in a system
are part of valence bonds, the resulting ground state is non-magnetic and has a spin of 0. One way
to achieve this is by partitioning all the spins into specific valence bonds that are static and localized.
In this case, the ground state can be well approximated as a product of the valence bonds, with each
spin highly entangled with only one other spin, its valence-bond partner. This state is known as a
valence-bond solid (VBS) state and occurs in various materials.
A VBS state is not, however, a true QSL, because it typically breaks lattice symmetries (because
the arrangement of valence bonds is not unique) and lacks long-range entanglement. To create a QSL,
the valence bonds must be allowed to undergo quantum mechanical fluctuations. The resulting ground
state is a superposition of different partitionings of spins into valence bonds, forming a valence-bond
’liquid’ (Fig. 3.5 b, c). This type of wavefunction is known as a resonating valence bond (RVB) state,
where there is no preference for any specific valence bond if the distribution of partitionings is broad.
RVB states became the focus of intense theoretical interest when Anderson proposed in 1987 that

25
there might be the underlying physics of high-temperature superconductivity. However, it was only
recently that RVB wavefunctions were demonstrated to be ground states of several specific model
Hamiltonians.
There is no direct method to observe the QSL state however the thermodynamic and microscopic
measurements indirectly can indicate the presence of such a dynamic phase. One good indication is
a large frustration parameter, f > 100 − 1000 [1]( may be limited by material complications, such
as defects or weak symmetry-breaking interactions). An even more rigorous test for a quantum spin
liquid is the absence of any static moments, even if they are disordered, at low temperatures. This
feature can be investigated through experiments using nuclear magnetic resonance (NMR) and muon
spin resonance. To gain further insight into the nature of a putative quantum spin liquid, specific-heat
measurements can provide information on its low-energy density of states, which can then be com-
pared to theoretical models. Thermal transport experiments can determine whether these excitations
are itinerant or localized. The use of elastic and inelastic neutron-scattering experiments, especially
on single crystals, can provide critical information on the nature of correlations and excitations, which
may uncover spinons. An important difference between spin glass and quantum spin liquid is that un-
like spin glass there is no ZFC-FC splitting in QSL below the transition temperature.

26
Chapter 4

Experimental techniques

4.1 X-Ray Diffraction

X-ray diffraction is a technique used to study the structure of materials at the atomic level. It involves
shining a beam of X-rays onto a sample and measuring the diffraction pattern that results from the
interaction of the X-rays with the atoms in the sample. The X-rays are diffracted by the regular
arrangement of atoms in the sample, resulting in a pattern of spots on a detector. The positions and
intensities of these spots can be used to determine the arrangement of atoms in the sample, as well as
other structural properties such as the spacing between atoms and the crystal symmetry.
The wavelength of X-ray is approximately equal to the distance between atoms in a crystal. There-
fore, we use X-ray scattering to study atomic arrangements.

4.1.1 Production of X-Ray

The production of X-rays takes place in a vacuum chamber
in an X-ray tube with two metal electrodes. The cathode,
made of tungsten filament, is heated to produce electrons.
These electrons are negatively charged particles and accel-
erated towards the anode, which is typically at ground po-
tential. Upon reaching the water-cooled anode, the elec-
trons collide with it at a high velocity, resulting in a loss of
energy that is exhibited as X-rays. However, only a small
portion (less than 1%) of the electron beam is converted
into X-rays[13]. The majority of the beam dissipates as
heat in the water-cooled metal anode. A cross-sectional
view of the X-ray tube is shown in Figure 4.1.
If an electron loses all its energy in a single collision
with a target atom, an x-ray photon with the maximum en-
ergy or the shortest wavelength is produced. This wave-
length is known as the short-wavelength limit. When an Figure 4.1: Schematic showing the es-
incoming electron has enough energy to displace an elec- sential components of a modem x-ray
tron in the inner shell of an atom, the atom is left in an ex- tube [13]
cited state with an empty space in the electron shell. This
phenomenon is demonstrated in Figure 4.2(b). When this empty space is filled by an electron from

27
an outer shell, an X-ray photon is emitted with an energy level equal to the difference in energy be-
tween the two electron shells. The energy level of the X-ray photon is specific to the metal target. As
shown in Figure 4.2(a), these characteristic lines are clearly visible as sharp peaks superimposed on
a continuous spectrum. In X-ray diffraction, these characteristic lines are particularly valuable[13].

Figure 4.2: [13]

4.1.2 Principle of XRD

The three basic components of an X-ray diffractometer are the X-ray source, Specimen, and X-ray
detector and they all lie on the circumference of a circle, which is known as the focusing circle.
The angle between the plane of the specimen and the x-ray source is θ, the Bragg angle. The angle
between the projection of the X-ray source and the detector is 2θ. For this reason, the x-ray diffraction
patterns produced with this geometry are often known as θ − 2θ (theta- two theta) scans.

Figure 4.3: Geometry of an x-ray diffractometer[13].

28
 In the θ − 2θ geometry, the X-ray source remains stationary while the detector moves across a
range of angles. The radius of the focusing circle varies and gets larger as the angle 2θ decreases,
as illustrated in Figure 42. Typically, the 2θ measurement range extends from 0◦ to approximately
170◦ [13]. However, during an experiment, it is not necessary to scan the entire range of detector
angles. A 2θ range of 30◦ to 140◦ is a common example of a scan range. The selection of the range
depends on the crystal structure of the material (if it is known) and the amount of time one wants to
spend obtaining the diffraction pattern. In the case of an unknown specimen, a broad range of angles
is often used because the positions of the reflections are unknown.

4.1.3 X-ray Diffraction Pattern

In a polycrystalline material or powder, the grains tend to be oriented randomly. However, some grains
will always be favorably oriented with respect to the X-ray beam, allowing diffraction to occur from
a particular set of lattice planes. Each set of lattice planes in the crystal having spacings d1 , d2 , d3 , . . .
will diffract at different angles θ1 , θ2 , θ3 , . . . where θ increases as d decreases in such a way as to
satisfy Bragg’s law (2dsinθ = nλ)[13]. The intensity of the diffracted beam at each of these different
angles is detected, and this is what forms the x-ray diffraction pattern.

Figure 4.4: Visualization of Bragg’s law[13]

Fig. 4.5 displays an example of a typical X-ray diffraction pattern, which shows the general
features of the pattern regardless of the specimen being analyzed. The pattern is consists of a series
of peaks, with the y-axis representing peak intensity and the measured diffraction angle 2θ, along the
x-axis.
Each reflection or peak in the diffraction pattern corresponds to X-rays diffracted from a specific
set of planes in the specimen. These peaks have different intensities, which are proportional to the
number of X-ray photons of a particular energy detected by the detector for each angle. The intensity
is typically measured in arbitrary units because it is challenging to measure the absolute intensity
accurately.
If the specimen is textured (ie., there is a preferred grain orientation) then some of the reflections in
the x-ray diffraction pattern may be anomalously intense or some may be absent compared to a pattern
obtained from a randomly oriented powder specimen. By comparing the intensities of the reflections
from the textured specimen with those from the powder pattern, we can determine the extent of the
preferred orientation. The reflections that are missing in a diffraction pattern obtained from a textured
material are not forbidden by the structure factor but are absent because the grains are not aligned in

29
 Figure 4.5: X-ray diffraction pattern [13].

the correct way to allow diffraction from those planes. For example, in a cubic material where all
grains have the [100] direction oriented perpendicular to the specimen’s surface, diffraction occurs
only from the (100) planes and not from the (001) or (010) planes. Therefore, the intensity of the 100
reflections in the textured specimen is expected to be weaker than in the randomly oriented one.
The positions of the peaks in an X-ray diffraction pattern depend on the crystal structure (more
specifically, the shape and size of the unit cell) of the material, and this is what enables us to determine
the structure and lattice parameter of the material.

Figure 4.6: Comparison of x· ray diffraction patterns from materials with different cubic crystal
structures[13].

30
 Materials with lower symmetry crystal structures have diffraction patterns with more peaks than
those with higher symmetry structures. This can be observed easily in cubic materials, which have
fewer peaks compared to those of non-cubic materials. Fig. 4.6 displays the calculated diffraction
patterns of the four cubic structures (simple cubic, bcc, fcc, and diamond cubic). The diffraction
pattern of the fcc structure displays peaks appearing alternately as a pair and a single peak, while in
the diamond cubic structure, the peaks are alternatively more widely and less widely spaced. These
differences occur due to reflections that are forbidden by the structure factor.
Modern instruments have the ability to perform peak searches on a computer. The computer
calculates the 2θ angles, FWHM, and integrated intensities for each peak. In addition, the peak shape
can be defined, and overlapping peaks deconvoluted. Most commercial software allows the user to
compare standard patterns (from the JCPDS-ICDD data base) with experimentally observed patterns,
allowing rapid matching of patterns and material identification. Certain software packages also allow
us to

1. Determine lattice strain

2. Calculate crystallite size

3. Refine calculation of lattice parameter(s) (Rietveld method)

4. Calculate diffraction patterns, and other operations

4.2 Principle of VSM

The Quantum Design Physical Property Measurement System (PPMS) represents a unique concept in
laboratory equipment. It is an open architecture, variable temperature-field system that is optimized
to perform a variety of automated measurements.
The Quantum Design Vibrating Sample Magnetometer (VSM) option for the Physical Property
Measurement System (PPMS) family of instruments is a fast and sensitive DC magnetometer. Its
fundamental measurement technique involves oscillating the sample close to a detection coil and
detecting the induced voltage in sync. The system employs a compact gradiometer pickup coil ar-
rangement and achieves a relatively high oscillation amplitude (1-3 mm peak) and a frequency of 40
Hz. This setup enables the instrument to detect magnetization changes of less than 10−6 emu at a 1
Hz data rate[4].

Theory of Operation
The basic principle of operation for a vibrating sample magnetometer is that a changing magnetic flux
will induce a voltage in a pickup coil. The time-dependent induced voltage is given by the following
equation:
dΦ
Vcoil =
dt    (4.1)
dΦ dz
=
dz dt

31
 In equation (4.1), Φ is the magnetic flux enclosed by the pickup coil, z is the vertical position of
the sample with respect to the coil, and t is time. For a sinusoidally oscillating sample position, the
voltage is based on the following equation:

Vcoil = 2πf CmAsin(2πf t) (4.2)

In equation (4.2), C is a coupling constant, m is the DC magnetic moment of the sample, A

is the amplitude of oscillation, and f is the frequency of oscillation. The acquisition of magnetic
moment measurements involves measuring the coefficient of the sinusoidal voltage response from the
detection coil. Figure 1-1 illustrates how this is done with the VSM option.

Figure 4.7: [4]

The sample is attached to one end of a rod that undergoes sinusoidal motion. The gradiometer
pickup coil is located at the vertical center of the oscillation. The VSM motor module regulates the
precise position and amplitude of oscillation by utilizing the optical linear encoder signal from the
VSM linear motor transport. The voltage produced in the pickup coil is magnified and identified
through lock-in detection in the VSM detection module. The VSM detection module employs the
position encoder signal as a reference for synchronous detection, which it obtains from the VSM
motor module interpreting the encoder signals from the VSM linear motor transport. The in-phase and
quadrature-phase signals from the encoder and amplified voltage from the pickup coil are perceived
by the VSM detection module, which averages them and relays them over the CAN bus to the VSM
application running on the PC[4].

32
4.3 Interpreting Magnetic susceptibility data
Magnetic materials are of tremendous importance in both fundamental science and in applications-
driven research. For newly synthesized materials, there is one indispensable characterization tech-
nique that is as old as the field of magnetism itself: magnetic susceptibility, χ,

M = χH (4.3)

which is the quantity that relates a material’s magnetization, M , to the strength of an applied magnetic
field, H. We will proceed with the conventional usage, which assumes that these two quantities are
linearly related, which is typically most valid at high temperatures and low fields[8]. In simple words,
magnetic susceptibility measurements show how a material reacts to a magnetic field and can help
reveal its magnetic properties. Susceptibility measurements can be conducted in either a direct current
(DC) field to study static magnetic properties or an alternating current (AC) field to examine dynamic
properties.

Curie–Weiss law
The Curie–Weiss law, which is derived as an extension of Curie’s law by incorporating the concept of
Weiss’s molecular field is
C
χ = χ0 + (4.4)
T − θCW
where C is known as the Curie constant (with units cm3 K mol−1 in CGS) and θCW (with units
K) is often referred to as the Curie–Weiss temperature. And χ0 is the temperature independent con-
tribution to susceptibility. This temperature independent term can have many origins including: core
diamagnetism (both from the sample or from the sample holder), Pauli paramagnetism, or Van Vleck
paramagnetism. The equation expresses how paramagnetic moments tend to align themselves with
an external field, resulting in increased susceptibility at lower temperatures as thermal fluctuations
decrease (as shown in Figure 4.8(a)). The number of unpaired electrons directly relates to the Curie
constant C, which can be calculated and used to determine the magnetic moment per ion in units of
Bohr magnetons, µB , √
µef f = 8C µB (4.5)
This effective moment can be directly compared to the calculated value for the ion in question,
given by[8] p
µcal = gJ J(J + 1)µB (4.6)
which depends only on its total angular momentum J and its g-tensor gJ . The Curie-Weiss tem-
perature, θCW , is proportional to the intensity of the molecular field, which can be taken as an ap-
proximate indicator of the strength of the magnetic correlations between ions. Positive θCW values
indicate that the molecular field aligns with the external field, implying ferromagnetic interactions,
while negative θCW values suggest antiferromagnetic interactions[8] (as illustrated in Figure 4.8(b)).

33
Figure 4.8: Appearance of Curie–Weiss behavior in direct and inverse susceptibility[8].

34
Chapter 5

Magnetic studies on SrFeSn(PO4)3

Here we discuss the magnetic properties of SrFeSn( PO4 )3 . These include the XRD data of SrFeSn(
PO4 )3 , magnetization data measured in different fields and M-H isotherms. With the help of these
data, we are going to determine the crystal structure of SrFeSn( PO4 )3 and its magnetic behavior.

5.1 Sample preparation

The polycrystalline sample of SrFeSn(PO4 )3 was prepared by conventional solid-state reaction route
using high purity starting materials. At first, the stoichiometric mixture of SrCO3 (Puratronic®,
99.994% (metals basis), preheated in oven at 100◦ C for 12 hrs to remove moisture) , Fe2 O3 (Puratronic®,
99.998% (metal basis)) , SnO2 ,(99.95% Alfa) (NH4 )H2 PO4 (Puratronic®, 99.995% (metal basis))
was prepared. The chemical equation is given below

4SrCO3 + 2Fe2 O3 + 4SnO2 + 12(NH4 )H2 PO4 = 4SrFeSn(PO4 )3 + 4CO2 + 12NH3 + 18H2 O (5.1)

Then this mixture was grounded for an hour and pressed into a pellet and placed in an alumina
crucible and heated via the following profile[3]. The sample was first placed in a box furnace and
heated up to 300◦ C by 60◦ C / hr. Then it was kept at 300◦ C for 6 hrs. And then it was cooled down
to room temperature by 60◦ C / hr. Here slow heating and cooling is necessary for good homogeneity.
The sample was then heated up to 600◦ C by 60◦ C / hr. Then it was kept at 300◦ C for 12 hrs. And
then it was cooled down to room temperature by 60◦ C / hr. The sample was then heated up to 900◦ C
by 60◦ C / hr. Then it was kept at 900◦ C for 24 hrs. And then it was cooled down to room temperature
by 60◦ C / hr. The colour of the sample was white in this step. In next step the sample was heated up
to 1100◦ C by 60◦ C / hr. Then it was kept at 1100◦ C for 24 hrs. And then it was cooled down to room
temperature by 60◦ C / hr. The final colour of the sample was reddish.

5.2 XRD refinement and structure

The crystal structure of SrFeSn(PO4 )3 was analyzed using Powder X-ray diffraction. The diffraction
pattern was obtained using CuKα radiation with a wavelength of 1.5406 Å, and the sample was
scanned from 2θ = 10.005◦ to 69.988◦ with a step size of 0.01087◦ . For a better fit, a pseudo-voigt
profile shape is chosen which is a mixture of contributions from Lorentzian and Gaussian shapes.
The refined atomic coordinates were obtained using the Rietveld refinement method[11] through the

35
FullProf software. The refined atomic positions and the results of the refinement are shown in table
5.1 and fig 5.1, respectively.

Figure 5.1: Rietveld refinement of XRD data of SrFeSn( PO4 )3 . The orange open circle is experimen-
tal data (Iobs ), green line is the theoretically calculated one (Icalc ) , blue line is the difference between
experimental and theoretical (Iobs − Icalc ) and purple vertical lines are corresponding Bragg peaks

From Rietveld refinement as shown in fig 5.1, we get R-factor (Rp ) = 20.6%, weighted profile
R-factor (Rwp ) = 16.6%, expected R-factor (Rexp ) = 5.06% and goodness of the fit χ2 = 10.08. Here
χ2 value is comparatively higher because the sample has few percentage of unreacted SnO2 impurity.
From atomic positions in table 5.1 we see that occupancy is of all the atoms is 1, so we conclude that
there is a distinct ordering of all atoms.
Table 5.1: The crystallographic data for SrFeSn( PO4 )3 after Rietveld refinement.
Atoms Label x y z Occupancy
Sr Sr1 0.0000 0.0000 0.0000 1
Sr Sr2 0.0000 0.0000 0.5000 1
Fe Fe1 0.0000 0.0000 0.1485 1
Sn Sn1 0.0000 0.0000 0.6505 1
P P1 0.2880 -0.0070 0.2452 1
O O1 0.1470 -0.0560 0.1900 1
O O2 0.03810 0.8320 0.7000 1
O O3 0.1870 0.1180 0.1010 1
O O4 0.8130 0.7900 0.5880 1

36
5.3 Crystal Structure

SrFeSn(PO4 )3 crystallizes into a trigonal structure with space group : R 3̄ . The unit cell is shown
in the fig 5.2a. The lattice parameters are given by a = 8.3980 Å, b = 8.3980Å, c = 22.8739
Å, α = β = 90◦ , γ = 120◦ . F e3+ form honeycomb layers as shown in the fig 5.2b . The
antiferromagnetic exchange interaction among the spins placed on the corners are satisfied with the
nearest neighbours . However, the competing interaction as a result of the next nearest neighbour
interactions give rise to novel ground state in such honeycomb system.

Figure 5.2: Structure of the SrFeSn(PO4 )3

5.4 Magnetic behaviour of SrFeSn(PO4)3

From magnetization data we plot χ vs T , in field-cooled (FC) and zero field cooled (ZFC) mode as
shown in the fig 5.3(a). The ZFC-FC bifurcation below T = 14.8 K exists in an applied field of 100
Oe shown in fig 5.3(a). This temperature is known as spin freezing temperature (Tg ).
This may be associated with the spin glass nature of SrFeSn( PO4 )3 . From susceptibility data we
plot 1/χ vs T and from this data plot we fit the plot with Curie-Weiss law given by:

1 T − θCW
= (5.2)
χ χ0 (T − θCW ) + C

Here θCW is Curie-Weiss temperature. The corresponding Curie-Weiss fit is shown in figure
5.3(b) . The presence of Curie temperature θCW is due to the magnetization caused by the F e3+ ions.
There is presence of some transition around 14.8 K the origin of which,we will discuss below. From
Curie-Weiss
√ fit we obtained θCW = −32.4 K, C = 4.87 cm3 K/mol . The effective magnetic moment
8CµB for magnetic ion is 6.24 µB which is close to expected value of 5.92 µB for F e3+ ion.

37
 (a) ZFC-FC data showing bifurication at 14.8 K.

(b) 1/χ vs T plot fitted with Curie-Weiss law in an applied field of H=1000 Oe.
From the fitting parameter we get χ0 = 7.99 × 10−4 ,C = 4.87, and θCW =
−32.37K

Figure 5.3

38
 Figure 5.4: ZFC-FC spliting at 100 Oe, 1T , 3T

In fig 5.4 the suppression of ordering at 14.8 K denotes that it’s not due to long range ordering
which will require around 15 T to get suppressed. It can be spin glass transition.
Fig 5.5 represent the five-quadrant M - H curve. This rules out the possibility of ferromagnetic
ordering and hence supporting the conclusion from the susceptibility.
To understand the ground state of the system , we performed the ac-susceptibility experiment (fig
5.5 b) in different frequencies. The frequency independence nature of the susceptibility may either be
associated with the unconventional spin glass nature or the requirement of higher frequency to cause
the shift in the susceptibility.

39
 (a) M - H plot at 5 K and 10 K

(b) AC susceptibility plot at different frequencies

Figure 5.5

40
5.5 Conclusion
We have successfully synthesized the sample SrFeSn(PO4 )3 . It crystallizes into a trigonal R 3̄ space
group. The DC susceptibility infers the antiferromagnetic exchange interaction between F e3+ mo-
ments. The appearance of ZFC-FC bifurcation around 14.8 K indicates the spin glass nature of this
honeycomb lattice. AC-susceptibility shows no anomaly of frequency shifting response of spin glass
which can be attributed to the fact that the glass dynamic has different window of frequency range.

5.6 Future Plan

To understand the nature of magnetic ground state precisely, further magnetic measurements in higher
frequencies and thermodynamic measurements are to be performed.

41
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