
Master Dissertation

Investigation of molecular conductors, the

magneto-caloric effect and the binary alloy

FeSe1−x

Keywords: Molecular Conductors, Magneto-caloric Effect, Superconductivity

M. Sc. Candidate: Lucas Cesar Gomes Squillante
Prof. Dr. Valdeci Pereira Mariano de Souza (Advisor)

Departamento de F́ısica do Instituto de Geociências e Ciências
Exatas da Universidade Estadual Paulista “Júlio de Mesquita Filho”

Câmpus de Rio Claro


Lucas Cesar Gomes Squillante

Investigation of molecular conductors, the

magneto-caloric e�ect and the binary alloy

FeSe1−x

Orientador: Prof. Dr. Mariano de Souza

Dissertação de Mestrado apresentada ao Instituto

de Geociências e Ciências Exatas da Universidade

Estadual Paulista “ Júlio de Mesquita Filho” -
Campus de Rio Claro, para obtenção do título

de Mestre em Física.

Rio Claro

2018


Squillante, Lucas Cesar Gomes 
      Investigation of molecular conductors, the
magneto-caloric effect / Lucas Cesar Gomes Squillante. - Rio
Claro, 2018
      76 f. : il., figs., gráfs., tabs., fots.

      Dissertação (mestrado) - Universidade Estadual Paulista,
Instituto de Geociências e Ciências Exatas
      Orientador: Valdeci Pereira Mariano de Souza

      1. Eletrônica. 2. Correlação eletrônica. 3. Molecular
conductors. 4. Magneto-caloric effect. 5. Superconductivy. I.
Título.

 
537.5
S773i

	 Ficha Catalográfica elaborada pela STATI - Biblioteca da UNESP
Campus de Rio Claro/SP - Adriana Ap. Puerta Buzzá / CRB  8/7987 


Lucas Cesar Gomes Squillante

Investigation of molecular conductors, the

magneto-caloric e�ect and the binary alloy

FeSe1−x

Dissertação de Mestrado apresentada ao Instituto
de Geociências e Ciências Exatas da Universidade
Estadual Paulista “ Júlio de Mesquita Filho” -
Campus de Rio Claro, para obtenção do título
de Mestre em Física.

Comissão Examinadora

Prof. Dr. Mariano de Souza (orientador)
IGCE/UNESP - Rio Claro, SP

Prof. Dr. Ricardo Paupitz B. dos Santos
IGCE/UNESP - Rio Claro, SP

Prof. Dr. Lúcio Campos Costa
UFABC - Santo André, SP

APROVADO

Rio Claro, SP, 24 de Novembro de 2017.


3

Abstract

The phenomenon of superconductivity is currently one of the most relevant
topics in Solid State Physics, making strongly correlated systems a very high-
attractive topic due to the possibility of studying the fundamental aspects of
the electron-electron interaction that are the core of superconductivity. Thus,
the class of molecular conductors (TMTTF)2X (where TMTTF is tetram-
ethyltetrathiafuvalene and X is a counter-anion) plays a systematic and fun-
damental role to study such correlation aspects. In this Master Thesis, the
materials of interest were the (TMTTF)2PF6-H12 and (TMTTF)2PF6-D12,
where a different dielectric anomaly at the Mott-Hubbard ferroelectric tran-
sition was observed for the two salts and the relaxor behavior of the hydro-
genated variant was analysed based on the mean-field theory. A review of
classical and quantum phase transitions was performed aiming to study the
so-called magneto-caloric effect (the magnetic Grüneisen parameter) for the
Brillouin paramagnet model, which is a powerful and unique physical quan-
tity to experimentally detect a quantum phase transition induced by magnetic
field in a real system. Also, a comparative study between the δ (hexagonal)
and δ′ (tetragonal) phases of the binary alloy FeSe1−x was performed and
single-crystals were synthesized employing the solid-state reaction method in
order to achieve the δ phase.


4

Resumo

O fenômeno da supercondutividade é atualmente um dos mais relevantes
tópicos na F́ısica da Matéria Condensada, tornando os sistemas fortemente
correlacionados um tópico de grande interesse devido à possibilidade de estu-
dar os aspectos fundamentas da interação elétron-elétron, que são o âmago
da supercondutividade. Desta forma, a classe de condutores moleculares
(TMTTF)2X (onde TMTTF é tetrametiltetratiafuvaleno e X é um contra-
ânion) desempenha um papel sistemático e fundamental no estudo de tais as-
pectos de correlação. Nesta tese de mestrado, os materiais de interesse foram o
(TMTTF)2PF6-H12 e o (TMTTF)2PF6-D12, onde uma anomalia na constante
dielétrica diferente para os dois sais foi observada na transição ferroelétrica
de Mott-Hubbard através de medidas de constante dielétrica quasi-estática
no eixo c∗ (contribuição iônica) e o comportamento tipo relaxor da variante
hidrogenada foi analisado com base na teoria de campo médio. Uma revisão de
transições de fase clássicas e quânticas também foi realizada com o objetivo
de estudar o chamado efeito magneto-calórico para o modelo do paramag-
neto de Brillouin (o parâmetro de Grüneisen magnético), que é uma grandeza
F́ısica única e poderosa para detectar experimentalmente uma transição de
fase quântica induzida por campo magnético em um sistema real. Ainda,
um estudo comparativo entre as fases δ (hexagonal) e δ′ (tetragonal) da liga
binária FeSe1−x foi realizado e monocristais foram sintetizados utilizando o
método de śıntese de estado sólido visando atingir a fase δ.


Contents

1 Introduction and motivation 7

2 Introduction to magnetism, phase transitions, and the magnetic Grüneisen
parameter 9
2.1 Fundamental aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Paramagnetism in insulators − the Brillouin model . . . . . . . . . . . . . 10
2.3 Paramagnetism in Metals − Pauli paramagnetism . . . . . . . . . . . . . 14
2.4 The 1D−Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.1 Introduction to classical phase transitions . . . . . . . . . . . . . . 20
2.5.2 Classical first-order phase transitions . . . . . . . . . . . . . . . . . 21
2.5.3 Classical second-order phase transitions . . . . . . . . . . . . . . . 22
2.5.4 Quantum phase transitions . . . . . . . . . . . . . . . . . . . . . . 23

2.6 The Grüneisen parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6.1 The magneto-caloric effect (the magnetic Grüneisen parameter) . . 25
2.6.2 Results and discussion (Brillouin paramagnet) . . . . . . . . . . . 26

3 Superconductivity 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Materials of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 The binary alloy FeSe1−x . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 The synthesis of FeSe1−x . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Correlation phenomena in the (TMTTF)2X (X = PF6-H12, PF6-D12)
Fabre-salts 36
4.1 Relevant fundamental literature and theoretical aspects . . . . . . . . . . 36
4.2 The Extended Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 The charge-ordered phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4 The dielectric constant and the Clausius-Mossoti relation . . . . . . . . . 39
4.5 The Solid State Physics laboratory in Rio Claro, SP − Brazil . . . . . . . 41
4.6 Experiments performed in the frame of this Master Thesis . . . . . . . . . 42

4.6.1 (TMTTF)2PF6 (H12) . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.6.2 (TMTTF)2PF6 (D12) . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6.3 The mean-field theory . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Summary and Conclusions 52

6 Perspectives and Outlook 54

Appendix 1: ICAM-I2CAM Junior Exchange Program - National High
Magnetic Field Laboratory 58

Participation in scientific events 64

5


6 CONTENTS

Awards, Grants & Honours 65

Teaching activities 66

Publications 67

Fundamental literature 68

Acknowledgements 69


Chapter 1

Introduction and motivation

In order to explore fundamental aspects of matter, it is worth to make a systematic
comparison between the fundamental literature taught in the undergraduate course of
Physics and the most recent investigated topics of Physics as well.

Since the discovery of the electron followed by the foundation of quantum mechanics,
several years went through and the importance of the electron is only increasing regarding
the so-called border science and its interaction gives rise to several exotic phenomena of
matter, e.g. superconductivity.

In this regard, systems where the interaction between electrons is significant (strongly
correlated electron systems) plays a fundamental role in order to unlock how such exotic
phases behave. Understanding its fundamental aspects will give our scientific community
the tools to manipulate it and create great scientific advances, e.g. room-temperature
superconductivity.

This Master Thesis is divided into three main chapters:

• Chapter 1: A review of fundamental models of magnetism is performed regarding the
paramagnetism in both metals (Pauli) and insulators (Brillouin), as well as the exact
solution for the one-dimensional (1D) Ising-model. A review of classical phase tran-
sitions is discussed based on the Landau theory and also the fundamental aspects
of quantum phase transitions is presented. Yet, the so-called Grüneisen parame-
ter is presented with the focus on the magneto-caloric effect (magnetic Grüneisen
parameter), which was analytically calculated for the Brillouin model.

• Chapter 2: A historical discussion regarding superconductivity is presented in de-
tails followed by a systematic description of a particular synthesis method (solid
state physics reaction) employed to synthesize several superconducting alloys. Em-
phasis was given to the binary alloy FeSe1−x, since its critical temperature (Tc) can
be increased through hydrostatic external pressure. The focus of this chapter was to
discuss the synthesis of single-crystals of the δ phase of FeSe1−x, which is predicted
to be insulating, and compare directly with its superconducting δ′ phase in order
to understand the suppression of superconductivity in the δ phase.

• Chapter 3: A relevant fundamental literature review is performed regarding the
molecular conductors of the family (TMTTF)2X, the extended Hubbard model, the
dielectric constant and the Clausius-Mossoti relation and the charge-ordered phase.

7


8

The main focus of this Master Thesis was to compare the physical properties of the
fully hydrogenated and the 97.5% deuterated variant of the system (TMTTF)2PF6.
Electrical resistivity ρ(T ) and ionic dielectric constant ε′(T,B) measurements were
carried out in such systems exclusively in the Solid State Physics Laboratory in Rio
Claro, SP − Brazil, and the data were analyzed regarding the insertion of defects
into the system and the mean-field theory.

The motivation to investigate such topics is not only to characterize or describe the
physical systems of interest in this Master Thesis, but to make important contributions
to the scientific community in order to advance Science a little further.


Chapter 2

Introduction to magnetism, phase
transitions, and the magnetic Grüneisen
parameter

The discussion presented below is based on Blundell’s, Nolting’s and Reif’s books [1–3],
in which a comprehensive and detailed description of the involved quantities is reported.
The physical quantities discussed in the following are of relevance for the discussion of
the Physics explored in this work.

2.1 Fundamental aspects

Regarding statistical mechanics, the aim is always to analyze the average properties of
a certain system by mathematically predicting the physical macroscopic behavior through
analyzing the microscopic ones. A good example is the liquid-to-gas phase transition of
water. Supposing that the forces between the water molecules are very well known so it
can be calculated how much energy it would be necessary to break the bonding between
the atoms of hydrogen (H) and oxygen (O). When such physical phenomenon occurs, the
water goes from liquid to steam (observable discontinuity of its density at the transition
temperature) characterizing a phase transition.

On the other hand, a proper analogy with such discontinuity on the water’s density
can be done with the magnetization, in a system which presents the so-called spontaneous
magnetization. Suppose a one-dimensional chain of spins without applied magnetic field
so they are all pointing in different directions. Increasing the magnetic field will result in
starting to orientate all the spins in the direction of the applied field until a determined
value is achieved that all spins are pointing in the same direction, so the system is mag-
netized (see Figure 2.1). Depending on the system, ferromagnetic materials for example
can remain magnetized even after the field decreases to zero, this is called spontaneous
magnetization.

9


10 2.2. PARAMAGNETISM IN INSULATORS − THE BRILLOUIN MODEL

B = 0

B ≠ 0

a)

b)

Figure 2.1: a) One dimensional spin chain without applied magnetic field so they are all non-aligned;
b) after applying a magnetic field all spins align in the same direction, there are finite magnetic moment
so the system is magnetized [4].

2.2 Paramagnetism in insulators − the Brillouin model

Consider a system of N atoms with no interaction between them at temperature T
with the local field B acting on the atom along the ~z direction. The magnetic energy E
of a single atom is given by:

E = −~µ · ~B, (2.1)

where ~µ is the magnetic moment vector of one single atom and it is related to the total
angular momentum by the expression:

~µ = gµB ~J, (2.2)

where ~J is the total angular momentum ~J = ~L+ ~S, ~L is the orbital angular momentum,
~S is the spin angular momentum, µB is the Bohr magneton and g is the Landé factor
described as:

g = 1 +
J(J + 1) + S(S + 1)− L(L+ 1)

2J(J + 1)
.

The Bohr magneton µB is given by:

µB =
eh̄

2me
,

where e is the elementary charge of the electron [5], h̄ is the reduced Planck’s constant,
namely h/2π, and me is the electron rest mass. Considering e = 1.60 × 10−19 C, h̄ =
1.05× 10−34 J.s and me = 9.11× 10−31 kg:

µB =
(1.60× 10−19 C).(1.05× 10−34 J.s)

2 · (9.11× 10−31 kg)
= 9.22× 10−24 J.T−1.

Regarding the magnetic field in Equation 2.1 it is relevant to mention that the field
B is pointing in the z direction and is not quite the same as an external one since it also
includes the magnetic moment contributions from all the other atoms. In the case of only
a few atoms, this contribution is usually not relevant since it is very small. Replacing
Equation 2.2 into 2.1:

E = −gµB ~J · ~B = −gµBBJz, (2.3)

where Jz is:
Jz = m = −J,−J + 1,−J + 2, ..., J − 1, J, (2.4)

where m is all the possible values between −J and J in integer steps.
Furthermore, m has 2J+1 possible values which correspond to all the projections of

the angular momentum vector along the z-axis.
Considering Equation 2.3, the magnetic energy of the atoms is given by:


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER11

Em = −gµBBm. (2.5)

The probability to encounter an atom at a state m is:

Pm ∝ e−βEm = eβgµBBm, (2.6)

where:

β =
1

kBT
, (2.7)

and kB is the Boltzmann constant.
The z component of the magnetic moment in this state is:

µz = gµBm. (2.8)

In order to calculate the mean z component of the magnetic moment µ̄z, the Maxwell-
Boltzmann statistics is employed:

µ̄z =

J∑
m=−J

µze
−βEm

Z
. (2.9)

Considering the partition function Z as:

Z =

J∑
m=−J

eβgµBBm, (2.10)

the mean z component of the magnetic moment is given by:

µ̄z =

J∑
m=−J

eβgµBBm(gµBm)

J∑
m=−J

eβgµBBm
. (2.11)

Conveniently writing:

J∑
m=−J

eβgµBBm(gµBm) =
1

β

∂Z

∂B
. (2.12)

Inserting Equation 2.12 into 2.11:

µ̄z =
1

β

1

Z

∂Z

∂B
=

1

β

∂lnZ

∂B
. (2.13)

Now consider:

η = βgµBB =
gµBB

kBT
, (2.14)

as a dimensionless parameter to analyze the ratio of the magnetic energy gµBB to the
thermal energy kBT . Thus, Equation 2.10 becomes:

Z =

J∑
m=−J

eηm = e−ηJ + e−η(J−1) + ...+ eηJ . (2.15)

Multiplying Equation 2.15 by eη:

Zeη = eη · e−ηJ + eη · e−η(J−1) + ...+ eη · eηJ ,


12 2.2. PARAMAGNETISM IN INSULATORS − THE BRILLOUIN MODEL

Zeη = e−η(J−1) + e−η(J−2) + ...+ e−η(J+1). (2.16)

By subtracting Equations 2.15 and 2.16, the only terms left are the first one from
Equation 2.15 (e−ηJ) and the last from Equation 2.16 (eη(J+1)). Thus:

Z − Z · eη = e−ηJ − eη(J+1),

Z(1− eη) = e−ηJ − eη(J+1).

Therefore:

Z =
e−ηJ − eη(J+1)

1− eη . (2.17)

Multiplying the numerator and the denominator of Equation 2.17 by e−η/2:

Z =
e−ηJ−η/2 − e[η(J+1)−η/2]

e−η/2 − eη−η/2 =
e−η(J+1/2) − eη(J+1/2)

e−η/2 − eη/2 . (2.18)

Considering that:

sinh(y) =
ey − e−y

2
,

Equation 2.18 becomes:

Z =
��−2 · sinh [η (J + 1/2)]

��−2 · sinh
(η

2

) . (2.19)

Thus,

Z =
sinh [η (J + 1/2)]

sinh
(η

2

) . (2.20)

Replacing Equation 2.20 into Equations 2.13 and 2.14:

µ̄z =
1

β

∂lnZ

∂B
=

1

β

∂lnZ

∂η

gµBβ︷︸︸︷
∂η

∂B
=

1

��β

∂lnZ

∂η
gµB��β = gµB

∂lnZ

∂η
, (2.21)

µ̄z = gµB

[
(J + 1/2)cosh[η(J + 1/2)]

sinh(J + 1/2)η
− (1/2)cosh(η/2)

sinh(η/2)

]
, (2.22)

µ̄z = gµBJBJ(η), (2.23)

where BJ(η) is the so-called Brillouin function, given by the form:

BJ(η) =
1

J

[(
J +

1

2

)
coth

(
J +

1

2

)
η − 1

2
coth

(η
2

)]
. (2.24)

The hyperbolic cotangent can also be written by the form:

cosh(y) ≡ cosh(y)

sinh(y)
=
ey + e−y

ey − e−y . (2.25)

In the case where y � 1, the trivial result is coth(y) = 1. Considering the case where
y � 1, it is possible to expand Equation 2.25 in a power series by the form:

coth(y) =
1 + 1

2y
2 + ...

y + 1
6y

2 + ...
' 1

y

(
1 +

1

2
y2

)(
1 +

1

6
y2

)−1

. (2.26)

Considering the binomial expansion:


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER13

(1 + x)n = 1 + nx+
n(n− 1)x2

2!
+ ..., (2.27)

it is possible to rewrite (1 + 1
6y

2)−1 from Equation 2.26 by the form:

(
1 +

1

6
y2

)−1

' 1 + (−1)
y2

6
+

(−1)[(−1)− 1]

2!

(
y2

6

)2

= 1− y2

6
+

�
�
�y4

36
+ ... '

(
1− y2

6

)
.

(2.28)
Since it is being considered that y � 1, the term in the expansion y4/36 is close to

zero and can be neglected, as well as the higher order terms of the expansion.
Thus, replacing Equation 2.28 into 2.26:

coth(y) =
1

y

(
1 +

1

2
y2

)(
1− 1

6
y2

)
=

1

y

(
1− y2

6
+
y2

2
−

�
�
�y4

12

)
'
(

1

y
+
y

3

)
. (2.29)

Analogously from Equation 2.28, the term −y4/12 can be neglected since it is close
to zero (y � 1).

Replacing Equation 2.29 into 2.24:

BJ(η) =
1

J

{
(J + 1/2)

[
1

(J + 1/2)η
+

1

3
(J + 1/2)η

]
− 1

2

[
2

η
+
η

6

]}
, (2.30)

BJ(η) =
1

J

 �
�
�1

η

1︷ ︸︸ ︷
(J + 1/2)

(J + 1/2)
+

1

3
(J + 1/2)2η −

�
�
�1

η
− η

12

 , (2.31)

BJ(η) =
1

J

{η
3

(J2 + J + 1/4)− η

12

}
, (2.32)

BJ(η) =
1

J

{
η

3

[
J2 + J +

�
��
1

4
−

�
��
1

4

]}
, (2.33)

BJ(η) =
η

3J
(J2 + J) =

η�J(J + 1)

3�J
=
η

3
(J + 1). (2.34)

Considering N atoms per unit of volume, the mean magnetization M̄z is given by:

M̄z = Nµ̄z = NgµBJBJ(η). (2.35)

So, Equation 2.35 is:

M̄z = NgµBJ
η

3
(J + 1) =

Ng2µB
2J(J + 1)

3kB︸ ︷︷ ︸
C

B

T
, (2.36)

where C is the so-called Curie’s constant.
The magnetic susceptibility χ can be defined as:

χ =
M̄z

B
. (2.37)

Thus, Equation 2.36 becomes:

χ =
C

T
. (2.38)

Equation 2.38 represents the so-called Curie’s Law, describing a linear behavior of 1/χ
as a function of temperature for an insulating paramagnet.


14 2.3. PARAMAGNETISM IN METALS − PAULI PARAMAGNETISM

2.3 Paramagnetism in Metals − Pauli paramagnetism

To describe the paramagnetism in a metal [4], i.e. itinerant electrons, consider that
each electron will contribute with −µB/V (g = 2) to the magnetization density if the spin
is parallel to a field B, and µB/V if antiparallel, where V is the volume per conduction
electron.

Considering the total numbers of electrons per unit of volume n± (where n+ represents
the number of electrons with parallel spins and n− with antiparallel), the magnetization
can be written:

M = −µB(n+ − n−). (2.39)

If only the magnetic moments of the electrons are affected, it means there will be
an energy shift in each electronic level of ±µBB accordingly if the spin is parallel or
antiparallel to B. Such interaction can be expressed in terms of density levels.

Consider g±(ε)dε the number of spins between the values of energy ε and ε + dε, so
it can be expressed:

g±(ε) =
1

2
g(ε) [B = 0], (2.40)

where g(ε) is the ordinary density levels. Since the energy levels with a specified spin
parallel to B is shifted up from its zero-field value by µBB, the number of levels with
energy ε in the presence of B is the same as the number with energy (ε − µBB) in the
absence of B:

g+(ε) =
1

2
g(ε− µBB). (2.41)

g−(ε) =
1

2
g(ε+ µBB). (2.42)

The number of electrons per unit of volume of each spin configuration is:

n± =

∫
dεg ± (ε)f(ε), (2.43)

where,

f(ε) =
1

eβ(ε−µ) + 1
. (2.44)

The chemical potential µ can be determined from the fact that the total electronic
density is:

n = n+ + n−. (2.45)

Thus, it is possible to use Equations 2.39 and 2.40 to determine the magnetization
density as a function of n, by employing a Taylor series:

g±(ε) =
1

2
g(ε± µBB) =

1

2
g(ε)± 1

2
µBBg

′(ε). (2.46)

Considering Equation 2.40:

n± =
1

2

∫
g(ε)f(ε)dε∓ 1

2
µBB

∫
dεg′(ε)f(ε). (2.47)

Thus, from Equation 2.45:

n =

∫
g(ε)f(ε)dε. (2.48)

Such equation is exactly the electronic density of states in the absence of a magnetic
field B, so the chemical potential can be encountered in the form:


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER15

µ = εF −
π2(kBT )2

6

g′(εF )

g(εF )
, (2.49)

where εF is the Fermi energy.

Since g(ε) varies as ε1/2 by the expression:

g(ε) =
3

2

n

εF

(
ε

εF

)1/2

, (2.50)

µ = εF

[
1− 1

3

(
πkBT

2εF

)2
]
, (2.51)

µ = εF

[
1 +O

(
kBT

εF

)2
]
. (2.52)

Together with Equations 2.39 and 2.47 gives a magnetization density:

M = µB
2B

∫ εF

0
g′(ε)f(ε)dε. (2.53)

In order to properly rewrite Equation 2.53, a few mathematical steps are required.
Thus, it is possible to write:

∂

∂ε
(f · g) = g

∂f

∂ε
+ f

∂g

∂ε
,

∫ εF

0

∂

∂ε
(f · g)dε =

∫ εF

0
g
∂f

∂ε
dε+

∫ εF

0
f
∂g

∂ε
dε,

∫ εF

0
f
∂g

∂ε
dε = f · g

∣∣∣∣∣
εF

0

−
∫ εF

0
g
∂f

∂ε
dε, (2.54)

f · g
∣∣∣∣∣
εF

0

=
g(ε)

e(ε−µ)/kBT + 1

∣∣∣∣∣
εF

0

.

Replacing g(ε) from Equation 2.50:

f · g
∣∣∣∣∣
εF

0

=
g(ε)

e(ε−µ)/kBT + 1

∣∣∣∣∣
εF

0

=

(
3n

2ε
3/2
F

)
ε1/2

e(ε−µ)/kBT + 1

∣∣∣∣∣
εF

0

,

f · g
∣∣∣∣∣
εF

0

=

(
3n

2ε
3/2
F

)
0, since T = 0 K at εF︷ ︸︸ ︷

ε
1/2
F

e(εF−µ)/kBT + 1
− 0

e−µ/kBT + 1

 = 0.

Thus, Equation 2.54 becomes:∫ εF

0
f
∂g

∂ε
dε =

∫ εF

0
g(ε)

(
−∂f
∂ε
dε

)
.

So, Equation 2.53 becomes:

M = µ2
BB

∫ εF

0
g(ε)

(
−∂f
∂ε

)
dε. (2.55)

Expanding the function −∂f/∂ε in a Taylor series until the first-order term so that
T 2 terms can be despised, and evaluating the result at T = 0 K, it is possible to conclude
that:


16 2.3. PARAMAGNETISM IN METALS − PAULI PARAMAGNETISM

−∂f
∂ε

= δ(ε− εF ) (2.56)

So Equation 2.55 can be rewritten by the form:

M = µ2
BB

g(εF )︷ ︸︸ ︷∫ εF

0
g(ε)δ(ε− εF )dε . (2.57)

Thus,

M = µ2
BBg(εF ). (2.58)

Since the T 6= 0 corrections to −∂f/∂ε are of orders (kBT/εF )2, then Equation 2.58
is also valid at high temperatures. Since we are dealing with the magnetism associated
with the conduction electrons, it is natural to consider that the energies associated with
those entities are in the range of the Fermi ones [4].

εF =
h̄2kF

2

2me
, (2.59)

where me is the electron mass. Considering that:

a0 =
h̄2

mee2
, (2.60)

where e is the electron charge. It is possible to write Equation 2.59 in terms of the Bohr
radius a0 by the expression:

εF =

(
e2

2a0

)
(kFa0)2, (2.61)

where (e2/2a0) is known as the Rydberg (Ry) = 13.6 eV.
The expression for kF can be given as a function of the Bohr radius a0 and the radius

of the free electron sphere rs, by the form:

kF =
3.63

rs/a0
. (2.62)

Thus, Equation 2.61 is given by:

εF =
50.1 eV

(rs/a0)2
=

8.026909× 10−18 J

(rs/a0)2
. (2.63)

From Equation 2.63 it is possible to calculate the Fermi temperature TF by the expression:

εF = kBTF → TF =
εF
kB
, (2.64)

TF =
8.026909× 10−18

1.38064852× 10−23
· 1

(rs/a0)2
. (2.65)

So, the Fermi temperature is given by:

TF =
58.2

(rs/a0)2
× 104 K. (2.66)

From 2.58 it is possible to write the magnetic susceptibility:

χ =

(
∂M

∂B

)
= µ2

Bg(εF ). (2.67)


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER17

This is the so-called Pauli paramagnetic susceptibility. Differing from the paramagnetic
behavior of the Curie’s Law, the susceptibility of the conducting electrons does not depend
on the temperature. Regarding the free electrons case, the density of levels is given by:

g(εF ) =
mkF

h̄2π2
. (2.68)

Replacing 2.68 into 2.67:

χPauli =
( α

2h̄

)2
(a0kF ), (2.69)

where:

α =
e2

hc
=

1

137
, (2.70)

is the so-called fine structure constant.
Also, the Pauli susceptibility can be written as:

χPauli =
2.59

rs/a0
× 10−6 . (2.71)

Equation 2.71 express the fact that Pauli susceptibility is extremely low because the
exclusion principle is much more effective than the thermal disorder in suppressing the
tendency of the spin magnetic moments to align with B.

Also, Pauli paramagnetism can be compared with Curie’s Law and it can be verified
that Pauli susceptibility is hundreds of times smaller than the Curie one and is non-
temperature dependent.

2.4 The 1D−Ising model

There are many existing phenomena that can be looked upon a one-dimensional sys-
tem concomitant with nearest-neighbor interaction, see e.g. [6]. Therefore, one model
of particular interest is the so-called Ising model. In this section is discussed its exact
solution [7].

Considering a chain of N spins interacting only with the two nearest ones, so the
one-dimensional open chain is replaced by a curved one so that the N th spin interacts
with the first one, not altering the thermodynamic properties of the chain.

1 2
3

4

5

6

N - 3

N - 2

N - 1

N

J

Figure 2.2: Ising closed infinite chain where each small circle represent a spin and J represents the
coupling constant between the spins [7].

With such configuration the Hamiltonian of the system can be written:


18 2.4. THE 1D−ISING MODEL

HN{σi} = −J
N∑
i=1

σiσj − µBB
N∑
i=1

σi, (2.72)

where J is the coupling term between the spins, σi and σj are the interaction between
two nearest-neighbor spins. Since they are neighbors it is possible to write σj = σi+1, so
the Hamiltonian:

HN{σi} = −J
N∑
i=1

σiσi+1 −
1

2
µBB

N∑
i=1

(σi + σi+1). (2.73)

From Equation 2.73 it is possible to calculate the partition function ZN (T,B):

ZN (T,B) =

N∑
i=1

e−βHN . (2.74)

Inserting Equation 2.73 into Equation 2.74:

ZN (T,B) =
∑
σ1=±1

...
∑

σN=±1

exp

{
β

N∑
i=1

[Jσiσi+1 + 1/2µBB(σi + σi+1)]

}
. (2.75)

It is possible to rewrite Equation 2.75 by a matrix elements operator P by the form:

〈σi|P |σi+1〉 = exp {β [Jσiσi+1 + 1/2µBB (σi + σi+1)]} . (2.76)

So Equation 2.75 becomes:

ZN (T,B) =
∑
σ1=±1

...
∑
σN±1

〈σ1|P|σ2〉〈σ2|xP|σ3〉...〈σN−1|P|σN 〉〈σN |P|σ1〉. (2.77)

Regarding Equation 2.76, the matrix P is given by:

(P) =

(
〈+1|P |+1〉 〈+1|P |−1〉
〈−1|P |+1〉 〈−1|P |−1〉

)
. (2.78)

〈+1|P |+1〉 = eβ[J ·(+1)·(+1)+1/2µBB(1+1)] = eβ[J+µBB],

〈+1|P |−1〉 = eβ[J ·(+1)·(−1)+1/2µBB(1−1)] = e−βJ ,

〈−1|P |+1〉 = eβ[J ·(−1)·(+1)+1/2µBB(1−1)] = e−βJ ,

〈−1|P |−1〉 = eβ[J ·(−1)·(−1)+1/2µBB(−1−1)] = eβ[J−µBB].

So the matrix P is:

(P) =

(
eβ[J+µBB] e−βJ

e−βJ eβ[J−µBB]

)
. (2.79)

So, Equation 2.77 results in:

ZN (T,B) =
∑
σ1±1

〈σ1|PN |σ1〉 = Tr (PN ) = λ+
N + λ−

N , (2.80)

where λ1 and λ2 are the eigenvalues of P (that is, the eigenenergies of HN ) and can be
found making:


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER19

∣∣∣∣eβ(J+µBB) − λ e−βJ

e−βJ eβ(J−µBB) − λ

∣∣∣∣ = 0

eβ(J+µBB) · eβ(J−µBB) − λeβ(J+µBB) − λeβ(J−µB) + λ2 − e−2βJ = 0.

λ2 +

2sinh(2βJ)︷ ︸︸ ︷
e2βJ − e−2βJ −λ(eβ(J−µBB) + e−β(−J+µBB)︸ ︷︷ ︸

eβJ (eµBB + e−µBB)︸ ︷︷ ︸
2cosh(βµBB)

) = 0.

So,

λ2 − λ · 2eβJcosh(βµBB) + 2sinh(2βJ) = 0. (2.81)

Solving Equation 2.81 for λ:

λ± =
2eβJcosh(βµBB)± {4e2βJcosh2(βµBB)− 8sinh(2βJ)}1/2

2
. (2.82)

λ± =
�2eβJcosh(βµBB)± �2[e2βJ

1+sinh2(βµBB)︷ ︸︸ ︷
cosh2(βµBB)−

(e2βJ−e−2βJ )︷ ︸︸ ︷
2sinh(2βJ)]1/2

�2
.

λ± = eβJcosh(βµBB)± {e2βJ [1 + sinh2(βµBB)]− [e2βJ − e−2βJ ]}.

λ± = eβJcosh(βµBB)± {���e2βJ + e2βJsinh2(βµBB)−�
��e2βJ + e−2βJ ]}.

λ± = eβJcosh(βµBB)± {e2βJsinh2(βµBB) + e−2βJ} .

Considering that λ+ > λ−, when N → ∞ it is considered only the λ+ eigenvalue due
to the fact that λ− is tiny when compared with λ+ in the thermodynamic limit. Thus,
the partition function from Equation 2.80 becomes:

ZN (T,B) = λ+
N . (2.83)

ln[ZN (T,B)] = ln(λ+)N .

ln[ZN (T,B)] = N ln(λ+). (2.84)

Inserting Equation 2.84 into the Helmholtz free energy:

F (T,B) = −kBT ln[ZN (T,B)]. (2.85)

F (T,B) = −kBTN ln[eβJcosh(βµBB) + {e2βJsinh2(βµBB) + e−2βJ}1/2].

F (T,B) = −kBTN ln[eβJcosh(βµBB) + {e2βJ [sinh2(βµBB) + e−4βJ ]}1/2].

F (T,B) = −kBTN ln[eβJcosh(βµBB) + eβJ · {sinh2(βµBB) + e−4βJ}1/2].


20 2.5. PHASE TRANSITIONS

F (T,B) = −kBTN ln
[
eβJ ·

(
cosh(βµBB) + {sinh2(βµBB) + e−4βJ}1/2

)]
.

F (T,B) = −kBTN{lneβJ︸ ︷︷ ︸
βJ

+ln[cosh(βµBB) + {sinh2(βµBB) + e−4βJ}1/2]}.

F (T,B) = −N kBTβ︸ ︷︷ ︸
1

J −NkBT ln[cosh(βµBB) + {e−4βJ + sinh2(βµBB)}1/2].

Thus, the free energy F (T,B) is given by:

F (T,B) = −NJ −NkBT ln[cosh(βµBB) + {e−4βJ + sinh2(βµBB)}1/2]. (2.86)

Thus, free energy per spin:

f(T,B) =
F (T,B)

N
, (2.87)

is given by:

f(T,B) = −J − kBT ln[cosh(βµBB) + {e−4βJ + sinh2(βµBB)}1/2]. (2.88)

From Equation 2.88 it is possible to derive all the thermodynamic physical quantities,
such as:

• Magnetization:

M(T,B) =

(
∂f(T,B)

∂B

)
T

; (2.89)

• Entropy:

S(T,B) = −
(
∂f(T,B)

∂T

)
B

; (2.90)

• Specific heat:

C(T,B) = −T
(
∂2f(T,B)

∂T 2

)
B

. (2.91)

2.5 Phase transitions

In this section is presented the mathematical detailed discussion regarding classic
phase transitions of first- and second-order, employing the Landau theory [8]. A brief
discussion of quantum phase transitions is presented as well.

2.5.1 Introduction to classical phase transitions

Regarding the first-order phase transition [8], an example is a transition between a
ferroelectric and a paraelectric state, where a discontinuous change of the saturation polar-
ization can be observed at a transition temperature Tc. A transition between the normal
and the superconducting state of a physical system is also a classic phase transition, but
a second-order one. One theory that describes the classic phase transition of both first
and second order is the Landau theory of phase transitions.

The so-called free energy is the available energy in a solid (for example) and in order
to transition between two different phases, there must have free energy in the system.
Considering the so-called Landau equation of the free energy density F̂ it is possible


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER21

to mathematically describe such free energy employing the polarization P (it is worth
mentioning that the Landau’s theory can be applied for any order parameter, not only for
the polarization P ), temperature T and the electric field E as one mathematical example
of it, given by:

F̂ (P, T,E) = −EP + g0 +
1

2
g2P

2 +
1

4
g4P

4 +
1

6
g6P

6 + ..., (2.92)

where the gn coefficients are temperature dependent.
The reason that odd coefficients are not present is due to the unpolarized crystal

taking in consideration has a center of inversion symmetry, but crystals can also behave
anisotropically regarding the polarization P so in such cases the odd coefficients take
place in Equation 2.92. It is possible to find the minimal point of Equation 2.92 by the
form:

∂F̂

∂P
= 0 = −E + g2P + g4P

3 + g6P
5 + ... (2.93)

Aiming to achieve such example of a ferroelectric phase, the coefficient in P 2 (g2) can
be defined with the condition that it passes through zero at a temperature T0, given by
the form:

g2 = γ(T − T0), (2.94)

where γ is a positive constant and T0 is a temperature near the transition temperature
Tc.

2.5.2 Classical first-order phase transitions

Regarding a first order transition, the free energy is expandable until the coefficient
g6, so the free energy is:

F̂ = −EP + g0 +
1

2
g2P

2 +
1

4
g4P

4 +
1

6
g6P

6. (2.95)

To find the minimum point of F̂ :

∂F̂

∂P
= 0. (2.96)

So:

−E + g2P + g4P
3 + g6P

5 = 0. (2.97)

Dividing Equation 2.97 for P :

−E/P + g2 + g4P
2 + g6P

4 = 0. (2.98)

Considering no applied electrical field (E = 0) and P 2 = y, Equation 2.98 is:

g6y
2 + g4y + g2 = 0. (2.99)

Now solving for y2 employing Bhaskara’s formula:

y =
−g4 ±

√
g4

2 − 4g6g2

2g6
. (2.100)

So, P is given by:

P =

(
−g4 ±

√
g4

2 − 4g6g2

2g6

) 1
2

. (2.101)

Replacing the g2 coefficient from Equation 2.94 into Equation 2.101:


22 2.5. PHASE TRANSITIONS

P =

(
−g4 ±

√
g4

2 − 4g6γ(T − T0)

2g6

) 1
2

. (2.102)

Considering T = T0:

P =

(
−g4 ±

√
g4

2

2g6

) 1
2

. (2.103)

The two possible solutions are:

P1 =

(
−g4 +

√
g4

2

2g6

) 1
2

=

(−g4 + g4

2g6

) 1
2

=

(
0

2g6

) 1
2

= 0, (2.104)

or

P2 =

(
−g4 −

√
g4

2

2g6

) 1
2

=

(−g4 − g4

2g6

) 1
2

=

(−�2g4

�2g6

) 1
2

=

(
−g4

g6

) 1
2

. (2.105)

In order to obtain a finite real valor, it is assumed that the coefficient g4 is negative,
so the solution is real.

Thus, Equation 2.97 is given by:

γ(T − T0)P − |g4|P 3 + g6P
5 = 0. (2.106)

Considering T0 = Tc:

g2︷ ︸︸ ︷
γ(T − Tc)P − |g4|P 3 + g6P

5 = 0, (2.107)

where P is the so-called order parameter in this equation, which is finite below the tran-
sition temperature Tc and goes to zero at T = Tc for first-order phase transitions.

2.5.3 Classical second-order phase transitions

In the case of a second order phase transition, the coefficient g4 is positive so g6 can
be neglected and the expansion goes only until g4:

∂F̂

∂P
= 0 = −E + g2P + g4P

3. (2.108)

Replacing g2 from Equation 2.94 and E = 0:

γ(T − T0)P + g4P
3 = 0. (2.109)

So:

P (γ(T − T0) + g4P
2) = 0, (2.110)

either P = 0 or:

γ(T − T0) + g4P
2 = 0. (2.111)

P 2 = −γ(T − T0)

g4
. (2.112)

P =

√
−γ(T − T0)

g4
=

(
γ

g4

) 1
2

(T0 − T )
1
2 . (2.113)


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER23

Thus, T0 in this particular case is the Curie temperature for this phase transition.
The transition is considered second-order because the polarization goes continuously to
zero at the transition temperature.

Yet, it is possible to relate Equation 2.113 with Equation 4.10 from the mean-field
theory, since when the temperature T is exactly the transition temperature (T0 or Tco)
both the polarization P and 1/ε′ goes to zero (in fact, 1/ε′ never goes exactly to zero
since ε′ has finite values).

2.5.4 Quantum phase transitions

In the last years, quantum phase transitions have regained a wide attention especially
by the Solid State Physics community regarding systems with strong correlation effects [9],
e.g. the metal-insulator transition in charge-transfer salts [10–12]. The physical properties
regarding non-interacting systems formed the primary basis upon which condensed matter
Physics developed the view of the behavior of such systems near a quantum critical
point. On the other hand, systems with strong electronic interaction play a fundamental
role in the fundamental understanding of their behaviors near a quantum critical phase
transition.

0
QCP Control parameter r

T
e
m

p
e
ra

tu
re

 T

ordered 

phase

Figure 2.3: Schematic phase diagram temperature T versus control parameter r indicating a quan-
tum critical point (red) at zero temperature, an ordered phase at low temperatures and the crossover
temperatures represented by the dashed lines. Figure adapted from Ref. [13].

The quantum phase transitions (QPT) are non temperature-driven phase transitions
and occurs at T = 0 K [9], different from the classical one driven by the temperature T [14].
The so-called quantum critical point (QCP) is where the quantum phase transition takes
place. The quantum phase transition is driven by the so-called control parameter r, such
as pressure, external magnetic field or doping.

One way to experimentally detect a quantum phase transition is associated with the
divergence of the Grüneisen parameter, namely the ratio between the thermal expansion
and the specific heat [13, 15]. In a magnetic-field-induced quantum phase transition,
namely the magneto-caloric effect, the magnetic Grüneisen parameter diverges, associated
with a sign change of the magnetic Grüneisen parameter near a quantum critical point [16].

The fundamental aspects of the Grüneisen parameter and the magneto-caloric effect
are discussed in the next sections.


24 2.6. THE GRÜNEISEN PARAMETER

2.6 The Grüneisen parameter

Regarding the volumetric thermal expansion β [17]:

β =
1

V0

(
∂V

∂T

)
P

, (2.114)

where V0 is the initial volume and V the volume at a certain temperature T . The volu-
metric thermal expansion is the sum of the linear thermal expansion from the a, b and
c-axes, determined by the expression:

β = αa + αb + αc, (2.115)

where αa, αb and αc are the linear thermal expansion coefficient of the a, b and c-axes
respectively. For the sake of completeness, the linear thermal expansion coefficient ex-
pression is recalled, given by:

α =
1

l0

(
∂l

∂T

)
P

. (2.116)

It is possible to relate the isothermal compressibility κT with β by the form:

β = − 1

V

(
∂V

∂P

)
T︸ ︷︷ ︸

κT

·
(
∂P

∂T

)
V

(2.117)

It is possible to write the expression for β by the alternative form:

β = κT

 ∂

∂V

(
−∂F
∂T

)
V︸ ︷︷ ︸

S


T

= κT

(
∂S

∂V

)
T

. (2.118)

Also, a relation between β and the specific heat as well. The specific heat expression
is given by:

CV = −T
(
∂2F

∂T 2

)
V

= −T

 ∂

∂T

(
∂F

∂T

)
V︸ ︷︷ ︸

−S


V

= T

(
∂S

∂T

)
V

. (2.119)

Inserting Equation 2.119 into Equation 2.118:

β = −κT ·
(
∂S

∂T

)
V︸ ︷︷ ︸

CV /T

·
(
∂T

∂V

)
S

, (2.120)

β = −κT · CV
T

·
(
∂T

∂V

)
S

. (2.121)

Multiplying Equation 2.121 by V /V :

β = −κT · CV
T

·
(
∂T

∂V

)
S

V

V
= −κT · CV

1

V

V

T

(
∂T

∂V

)
S︸ ︷︷ ︸

∂lnT

∂lnV

. (2.122)

So:

β = −
(
∂lnT

∂lnV

)
S︸ ︷︷ ︸

Γ

1

V
· κT · CV =

κT · CV
V

Γ (2.123)


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER25

where Γ is the so-called Grüneisen parameter. It can also be expressed by:

Γ =
α

cB
, (2.124)

where cB is the molar specific heat.
The volumetric thermal expansion microscopic contributions are the phononic, elec-

tronic and magnetic. So, it is possible to write:

β = βphononic + βelectronic + βmagnetic, (2.125)

where βphononic, βelectronic and βmagnetic are respectively the phononic, electronic and mag-
netic contributions to the volumetric thermal expansion. Comparing Equations 2.125 and
2.123 it is possible to write:

β =
κT
V

(ΓphononicCphononic + ΓelectronicCelectronic + ΓmagneticCmagnetic), (2.126)

where Γphononic, Γelectronic and Γmagnetic are the respectively Grüneisen parameters.
In this particular Master Thesis, emphasis will be given to the magnetic Grüneisen

parameter Γmagnetic or the so-called magneto-caloric effect.

2.6.1 The magneto-caloric effect (the magnetic Grüneisen parameter)

The so-called magneto-caloric effect Γmag (or the magnetic Grüneisen parameter) is
defined if the control parameter of a quantum phase transition is not the pressure but an
external magnetic field B. In this case, the Grüneisen ratio is given by [13]:

Γmag = −(∂M/∂T )B
cB

, (2.127)

where M is the magnetization per mole and cB is the molar specific heat. The molar
specific heat is given by the form:

c =
∆Q

∆T
; dQ = TdS,

c = T

(
∂S

∂T

)
B

. (2.128)

Also, the magnetization can be given by the expression:

M = −
(
∂F

∂B

)
T

.

Deriving the magnetization over the temperature:(
∂M

∂T

)
=

∂

∂T

(
−∂F
∂B

)
=

∂

∂B

(
−∂F
∂T

)
︸ ︷︷ ︸

S

=

(
∂S

∂B

)
T

. (2.129)

Replacing Equations 2.128 and 2.129 into Equation 2.127:

Γmag = − (∂S/∂B)T
T · (∂S/∂T )B

= − 1

T

(∂S/∂B)T
(∂S/∂T )B

. (2.130)

Equation 2.130 relates the variation of the entropy S in respect to the temperature
T and the magnetic field B. This equation was employed in the next section in order
to calculate the magneto-caloric effect for the Brillouin paramagnet model. Also, it is
possible to rewrite Equation 2.130 by the form:


26 2.6. THE GRÜNEISEN PARAMETER

Γmag = − 1

T

(∂S/∂B)T
(∂S/∂T )B

= − 1

T
·
(
∂S

∂B

)
T

·
(
∂T

∂S

)
B

=
1

T

(
∂T

∂B

)
S

. (2.131)

From Equation 2.131 it is clear that the magneto-caloric effect Γmag determines a
change in the temperature T in response to an adiabatic (constant entropy S) change of
the magnetic field B.

2.6.2 Results and discussion (Brillouin paramagnet)

Considering the Brillouin paramagnet and J = 1/2, it is possible to calculate the
magnetocaloric effect (Γmag) by the expression [13]:

Γmag = − 1

T
.
(∂S/∂B)T
(∂S/∂T )B

, (2.132)

where the entropy S is given by [1]:

S(T,B) = nkB

{
ln

[
2 cosh

(
µBB

kBT

)]
− µBB

kBT
tanh

(
µBB

kBT

)}
.

Deriving the entropy in respect to B:(
∂S

∂B

)
T

= nkB

 1

2 cosh
(
µBB
kBT

) [2 sinh

(
µBB

kBT

)
µB
kBT

]

−

 µB
kBT

tanh

(
µBB

kBT

)
+
µBB

kBT

1

cosh2
(
µBB
kBT

) µB
kBT


(
∂S

∂B

)
T

= nkB

����������
tanh

(
µBB

kBT

)
µB
kBT�����������

− tanh

(
µBB

kBT

)
µB
kBT

− µB
2B

kB
2T 2cosh2

(
µBB
kBT

)
 .

Therefore, (
∂S

∂B

)
T

= − nBµ2
B

kBT 2cosh2
(
µBB
kBT

) . (2.133)

Analogously,(
∂S

∂T

)
B

= nkB

 1

2 cosh
(
µBB
kBT

) [2 sinh

(
µBB

kBT

)(
− µBB
kBT 2

)]

−

(− µBB
kBT 2

)
tanh

(
µBB

kBT

)
+

(
µBB

kBT

)
1

cosh2
(
µBB
kBT

) (− µBB
kBT 2

) .

(
∂S

∂T

)
B

= nkB


�������������

− tanh

(
µBB

kBT

)(
µBB

kBT 2

)
�������������

+ tanh

(
µBB

kBT

)(
µBB

kBT 2

)
+

µB
2B2

kB
2T 3cosh2

(
µBB
kBT

)
 .

Therefore, (
∂S

∂T

)
B

=
nµB

2B2

kBT 3cosh2
(
µBB
kBT

) . (2.134)


CHAPTER 2. INTRODUCTION TO MAGNETISM, PHASE TRANSITIONS, AND THE MAGNETIC GRÜNEISEN PARAMETER27

Replacing Equations (2.133) and (2.134) into (2.132):

Γmag = −
− nBµB

2

kBT 2cosh2
(
µBB

kBT

)
nµB2B2

kBT 2cosh2
(
µBB

kBT

) .

Resulting:

Γmag =
1

B
. (2.135)

From Equation 2.135 one can directly conclude that the Brillouin paramagnet can be
considered intrinsically quantum critical to B → 0. In other words, Γmag diverges as B
→ 0, a fingerprint of a quantum phase transition [13,15,18].

Also, it is possible to calculate the magneto-caloric effect for general values of J . The
Helmholtz free energy for n spins per unit of volume is given by:

F = −nkBT ln[ZJ(y)], (2.136)

where ZJ(y) is the partition function, given by:

ZJ(y) =
sinh[(2J + 1) y

2J ]

sinh[ y2J ]
, (2.137)

where y = gJµBJB/kBT .

Thus, the Helmholtz free energy is:

F = −nkBT ln

sinh
[

(2J+1)gJµBB
2kBT

]
sinh

[
gJµBB
2kBT

]
 . (2.138)

From Equation 2.138, the entropy S can be calculated by the form:

S = −
(
∂F

∂T

)
B

. (2.139)

(2.140)S = −

−nkBln

ZJ (y)︷ ︸︸ ︷sinh
[

(2J+1)gJµBB
2kBT

]
sinh

[
gJµBB
2kBT

]

− [Θ],

where Θ is:

Θ = −nkBT
sinh

[
gJµBB
2kBT

]
sinh

[
(2J+1)gJµBB

2kBT

]
·

sinh
[
gJµBB
2kBT

]
cosh

[
(2J+1)gJµBB

2kBT

] (
− (2J+1)gJµBB

2kBT 2

)
− sinh

[
(2J+1)gJµBB

2kBT

]
cosh

[
gJµBB
2kBT

] (
−gJµBB

2kBT 2

)
sinh2

[
gJµBB
2kBT

]
 .

Θ = −nkBT ·

 sinh
[
gJµBB
2kBT

]
sinh

[
(2J+1)gJµBB

2kBT

] · cosh
[

(2J+1)gJµBB
2kBT

]
sinh

[
gJµBB
2kBT

] ·
(
−(2J + 1)gJµBB

2kBT 2

)
−

−
sinh

[
gJµBB
2kBT

]
sinh

[
(2J+1)gJµBB

2kBT

] · sinh
[

(2J+1)gJµBB
2kBT

]
· cosh

[
gJµBB
2kBT

]
·
(
−gJµBB

2kBT 2

)
sinh2

[
gJµBB
2kBT

]
 .


28 2.6. THE GRÜNEISEN PARAMETER

Θ =

−y·BJ (y)︷ ︸︸ ︷
−nkBT

{
coth

[
(2J + 1)gJµBB

2kBT

]
·
(
−(2J + 1)gJµBB

2kBT 2

)
−
(
−gJµBB

2kBT 2

)
· coth

[
gJµBB

2kBT

]}
.

Θ = −nkB · y

BJ (y)︷ ︸︸ ︷[
(2J + 1)

2J
coth

[
(2J + 1)

2J
· y
]
− 1

2J
·
[ y

2J

]]
= −nkBTBJ(y) · y.

Replacing Θ into 2.140:

S(y) = nkB · [lnZJ(y)− yBJ(y)] . (2.141)

Now, it is possible to calculate:

(
∂S

∂B

)
T

=


[

(2J+1)gJµB
2kBT

]2
· nkBB

sinh2
[

(2J+1)gJµBB
2kBT

] − [gJµB2T

]2 · nkBB
sinh2

[
gJµB
2kBT

]
 . (2.142)

and

(
∂S

∂T

)
B

=


[
gJµBB
2kBT

]2
· nkBT

sinh2
[
gJµBB
2kBT

] −
[

(2J+1)gJµBB
2kBT

]2
· nkBT

sinh2
[

(2J+1)gJµBB
2kBT

]
 . (2.143)

Replacing Equations 2.142 and 2.143 into 2.132:

Γmag =

−
{[

(2J+1)gJµB
2kBT

]2
·nkBB

sinh2
[
(2J+1)gJµBB

2kBT

] − [ gJµB2T ]
2·nkBB

sinh2
[
gJµB
2kBT

]
}

T ·
{[

gJµBB

2kBT

]2
·nkB
T

sinh2
[
gJµBB

2kBT

] −
[
(2J+1)gJµBB

2kBT

]2
·nkB
T

sinh2
[
(2J+1)gJµBB

2kBT

]
} . (2.144)

Γmag =

�
��nkB ·B

��������������������
[
gJµB
2kBT

]2

sinh2
[
gJµBB
2kBT

] −
[

(2J+1)gJµB
2kBT

]2

sinh2
[

(2J+1)gJµBB
2kBT

]


���nkB ·B2

��������������������
[
gJµB
2kBT

]2

sinh2
[
gJµBB
2kBT

] −
[

(2J+1)gJµB
2kBT

]2

sinh2
[

(2J+1)gJµBB
2kBT

]


=
B

B2
. (2.145)

Thus:

Γmag =
1

B
. (2.146)

From Equation 2.146 it can be concluded that Γmag diverges as B → 0 T, a fingerprint
of a quantum phase transition [13]. Thus, although the Brillouin model is purely classical,
there is an intrinsic quantum critical-like behavior at T = 0 K for the particular case
of B = 0 T. For both positive and negative infinitesimal values of B, Γmag diverges.
The sign change in the magneto-caloric effect is also a fingerprint of a quantum phase
transition [13,15,18].


Chapter 3

Superconductivity

In this chapter, current and fundamental aspects of superconductivity are presented.
An analysis of the high-temperature superconductors are performed and the binary alloy
FeSe1−x is introduced in this context.

3.1 Introduction

One of the main goals of exploring low-temperature Physics is the fact that exotic
phenomena emerge when a sufficiently low temperature is achieved. One of the most
investigated properties of matter in low temperatures is superconductivity, where the
resistance of a determined system goes to zero at a critical temperature Tc. In other
words, there is a current flow without energy dissipation by the Joule effect (P = Ri2),
where P is the power dissipated, R is the resistance and i the electrical current.

After several years since Kamerlingh Onnes [19] first observed superconductivity in
a mercury (Hg) sample with Tc ∼ 4 K, Meissner and Ochsenfeld [20] demonstrated ex-
perimentally the Meissner-Ochsenfeld effect, where a magnetic field is expelled from a
superconductor in the so-called magnetic levitation. Such effect is crucial in defining su-
perconductivity as a new state of matter. This phenomenon is so counter-intuitive that
more than half-century passed since the BCS theory [21] was developed, which proposed
the electron-electron coupling by the phonon of the lattice in the so-called Cooper pairs.

There is a technological interest into synthesizing superconducting materials with
higher critical temperatures since it would boost such industry with non-dissipative ma-
terials. Most metallic elements were discovered to superconduct below 10 K and even
a group of metal alloys have shown superconductivity. However, until the 80’s tempera-
tures above 30 K had not been observed, as shown in Fig 3.1. In 1986 Bednorz and Müller
observed for the first time superconductivity in the so-called cuprates (alloys containing
copper and oxygen) above the 30 K range [22]. Less than one year after, YBa2Cu3O7−x
(YBCO) presented a critical temperature of 90 K being above the liquid N2 range. In
the last decade, superconductivity was discovered in iron-based structures, the so-called
iron pnictides or iron-based superconductor (IBSC) [23]. Such discovery was unexpected
for two main reasons: the magnetic ions would destroy superconductivity and the non-
superconducting phases of this family are metallic, totally different from the cuprate
families which were insulating.

29


30 3.2. MATERIALS OF INTEREST

1 9 0 0 1 9 2 0 1 9 4 0 1 9 6 0 1 9 8 0 2 0 0 0 2 0 2 0
0

2 0
4 0
6 0
8 0

1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
2 0 0

 c o n v e n t i o n a l
 p o s s i b l y  u n c o n v e n t i o n a l
 u n c o n v e n t i o n a l

 
T c(K)

Y e a r

H g N b N b N i n s u l a t o r s
C 6 C a

m e t a l s

N B S 2

C h e v r e l  p h a s e s

b i s m u t h a t e s

o r g a n i c
h e a v y  f .

A 3 C 6 0
M g B 2

e - d o p e d  c u p r .
B 4 C

S r R u
P u C o G a 5

C o H 2 OU G e 2
B i S 2

C 6 H 6

F e S e
F e A s

( S N ) x

  I n t e r m e t a l l i c
  C o m p o u n d s
( A 1 5  s t r u c t u r e )

h o l e - d o p e d  c u p r a t e s

H 2 S

Figure 3.1: Increasing of the critical temperature Tc over the years of conventional, possibly unconven-
tional and unconventional superconductors [24].

The idea is still in development but years of collective effort revealed that the Physics
of Fe-based superconductors is way richer than foreseen and they feature a few unique
properties. The discovery of IBSC has also added more degrees of freedom for theoretical
work since the non-conventional superconductivity (not explained by the BCS theory)
was observed just in systems with strong electrical correlation such as cuprates, heavy
fermions [25] and organic conductors [26].

The BCS theory of conventional superconductors reveals how to achieve high Tc with-
out any upper bound combining high-frequency phonons, strong electron-phonon coupling
and high density of states [27]. Such conditions are satisfied for compounds dominated
by hydrogen [28, 29] since it has such characteristics, predicting compounds with Tc in
the range of 50−235 K for numerous hydrides [30], since the hydrogen mass is low and it
would increase the angular frequency w and thus the critical temperature Tc. Yet, the
hydrogen sulfide compound H2S was discovered to be the Tc record history (203 K) under
high-pressure [31].

Another fascinating discovery was that a double-stranded DNA [32] can superconduct.
Regarding the electrical transport properties of DNA, it was predicted that it would be
conducting [33,34] and insulating [35,36].

Several thousands superconductors are known in the literature and, as new systems
emerge, new theories are being developed and not always are in accordance with the ex-
istent ones. Every single new theory that explains one system or a gama of systems is
one step closer to the unification of them all into a single theory that could definitively
explain superconductivity. Besides all discoveries regarding superconductivity, there are
very much to understand how this quantum effect works and how it can be fully un-
derstood. The search for a room-temperature superconductor happens every day in all
over the globe due to its magnificent application in technology that will result in a huge
advance in science.

3.2 Materials of interest

Regarding electron systems, with the discovery of superconductivity in the last decades
in heavy fermions systems [37], copper oxides (cuprates) [38] and organic conductors


CHAPTER 3. SUPERCONDUCTIVITY 31

[26], it became evident that spectacular forms of collective behavior emerge in strongly
correlated electronic systems.

For example, when the Coulomb energy interaction is comparable or higher than the
kinetic energy of a study system, a Mott insulator state can emerge. In such systems,
the interplay between the charge degrees of freedom (spin), lattice and orbitals originates
very rich phase diagrams. Stands out the emergence of a superconducting phase from
a Mott insulator state [39], applying external pressure or doping, as observed in organic
conductors [40] and cuprates [41], respectively.

There is a very interesting class of materials discovered by the scientific community
in the last years: iron-based superconductors [42]. Even more interesting is the fact that
since the discovery of superconducting cuprates, this class of materials constitutes the first
generation of superconductors with critical temperatures above 50 K [43]. In particular,
the binary alloy FeSe1−x offers unique opportunities to study superconductivity in this
class of materials [44] and is the alloy focused in this Master Thesis.

3.3 The binary alloy FeSe1−x

The observation of superconductivity in alloys containing the element iron (Fe) is
counterintuitive since the description of the pairing mechanism based on the BCS [21]
theory refers the formation of triplets (originated from the 3d electrons moment of the
Fe atoms) would annihilate the Cooper pairs. In this regard, FeSe1−x represents an alloy
of great interest, since its Tc can be raised by external applied hydrostatic pressure from
8.5 K to 36.7 K [45]. Also, thin films of FeSe1−x deposited in SrTiO3 substrates showed
superconductivity with Tc ∼ 100 K [46]. In this context, there are two phases of interest
for the FeSe1−x alloy: the tetragonal and hexagonal crystalline structures.

Figure 3.2: Phase diagram temperature versus atomic concentration for the binary alloy FeSe1−x. It is
possible to observe the various phases of FeSe including the ones of interest: δ’ (green - tetragonal unit
cell) and δ (blue - hexagonal unit cell) [47].

As the temperature is reduced, the electrical resistivity of the tetragonal phase sample
goes to zero near Tc, i.e., it superconducts below Tc, as can be seen in Figure 3.4.

However, for the hexagonal phase, a semiconducting behavior is predicted (dR/dT <
0), as can be seen in Figure 3.5.

Also, the crystallography planes of the tetragonal phase of FeSe1−x are separated by
the so-called van der Waals gap (Figure 3.3), which allows the material to be extremely


32 3.3. THE BINARY ALLOY FESE1−X

Figure 3.3: Unit cells of the tetragonal (up) and hexagonal (lower) phases of FeSe1−x. Note that in the
tetragonal unit cell there is empty space between the atoms of Se (purple) denominated as van der Waal’s
gap.

Figure 3.4: Main panel: electric resistance measurement as a function of temperature for the δ’-FeSe0,96
with Tc = 8.5 K. Inset: electric resistance measurements at low temperatures as a function of external
hydrostatic pressure applied, note that there is a significantly increase of Tc with application of pressure
[48].

Figure 3.5: Electric resistivity as a function o temperature for the δ phase (continuous line) and for the
δ’ one (dotted line) of the FeSe1−x as theoretically predicted in Ref. [49].

compressible. Such property is strongly associated with the increase of Tc from 8.5 K to
36.7 K under hydrostatic pressure [45].


CHAPTER 3. SUPERCONDUCTIVITY 33

Figure 3.6: Without applied pressure, the binary alloy FeSe1−x is structurally distorted from orthogonal
to orthorhombic at 90 K. The maximum value of Tc is observed at 36.7 K under 8.9 GPa applied pressure.
At high pressures, the alloy is uniquely at the hexagonal phase and presents a semiconducting behavior [45].

3.3.1 The synthesis of FeSe1−x

The main goal in this Master Thesis was to synthesize single-crystals of δ-FeSe1−x in
order to perform systematic transport investigations in this phase comparing with the
δ’-FeSe1−x, which is already intensively investigated in literature, aiming to understand
the fundamental concepts that annihilates superconductivity in the δ-FeSe1−x.

a) b)

Fe + Se

Figure 3.7: a) After the pump and purge with nitrogen, the ampoule was sealed employing a torch.
Special thanks to Geraldo Aparecido de Lima Sobrinho for the technical support. b) After the sealing the
ampoule is ready to be inserted into the furnace for the proper thermal treatment.

In order to synthesize single crystals of δ-FeSe1−x, the following stoichiometry was
employed regarding the phase diagram in Figure 3.2:

0.52Fe + 0.48Se→ 1.00δ-FeSe1−x.

From the molar mass of both Fe (55.84 g/mol) and Se (78.96 g/mol), it was possible
to calculate the mass of the compound to be reacted. Such appropriate mixture of Fe and


34 3.3. THE BINARY ALLOY FESE1−X

Se was inserted into a glass ampoule since it can withstand temperatures until ∼ 400oC
without any significant changes in their mechanical resistance. A pump and purge with
nitrogen was performed in order to maintain an inert atmosphere inside the ampoule.
Then, the ampoule was carefully sealed employing a torch (Figure 3.7a)).

a) b)

single 

crystals

Figure 3.8: a) After the ampoule was cooled to room-temperature, it was possible to observe the
formation of single crystals in the solidified mixture of Fe and Se. b) The single crystals were collected
and stored properly.

The glass ampoule containing the mixture was then inserted into a horizontal furnace
and the temperature is varied from room-temperature to ∼ 400oC, which remains at such
temperature for a couple of hours. After this process, the ampoule was cooled to room-
temperature, carefully broken and the single crystals were collected, as can be seen in
Figure 3.8.

Thereon, a couple of samples were sent to x-ray measurements to determine whether
or not a pure phase of δ-FeSe1−x was achieved. The x-ray measurements results are
discussed in the next section.

3.3.2 Experimental Results

a) b)

Figure 3.9: a) Room-temperature intensity versus 2θ for one sample synthesized, aiming to achieve an
homogeneous hexagonal phase of the hexagonal phase of FeSe1−x. One of the Miller indices (011) is shown
in order to compare with X-ray results from the tetragonal structure. The X-ray experiment was carried
out by M. Sc. Paulo Eduardo Menegasso Filho at the State University of Campinas; b) Room-temperature
intensity versus 2θ for the tetragonal phase of FeSe0.95 [48] with their respective Miller indices. The base
lines are shown for comparison between the X-ray results and the calculated ones. Inset: cut of the
tetragonal FeSe1−x crystal structures.

After employing the solid-state physics reaction in order to obtain single crystals of the
hexagonal phase of FeSe1−x, the ampoules were opened and the single crystals collected.


CHAPTER 3. SUPERCONDUCTIVITY 35

In order to confirm the homogeneity of the hexagonal structure, x-ray experiments were
carried out in such samples.

Comparing the X-ray results from the synthesized samples with the tetragonal one
already reported in literature [48], it is possible to observe the Miller indices (011) in the
synthesized sample X-ray, indicating that a homogeneous phase of the hexagonal structure
was not achieved since it indicates traces of the tetragonal structure into the sample.

Although a homogeneous phase of the hexagonal binary alloy FeSe1−x was not achieved,
the understanding of both solid-state synthesis and the thermodynamics therein con-
tributed expressively for my scientific career. Such knowledge can be extended to projects
to be performed in a near future.


Chapter 4

Correlation phenomena in the
(TMTTF)2X (X = PF6-H12, PF6-D12)
Fabre-salts

In this section are presented the theoretical aspects of the exotic properties of the
(TMTTF)2X salts, which are crucial to understanding both theoretical and experimental
studies. The goal is to understand the theory, namely the extended Hubbard model,
which enables the understanding of the ferroelectric Mott-Hubbard phase, which is the
main topic of the experimental results in this thesis. The physical properties of the
TMTTF-based molecular conductors are discussed.

4.1 Relevant fundamental literature and theoretical aspects

Nowadays, correlation phenomena are one of the most investigated topics in the sci-
entific community, since several exotic phenomena emerge due to the interaction between
electrons. Some examples relies on the Mott-insulator phase [12, 50], superconductiv-
ity [26, 51, 52] and charge-ordering [53]. In this regard, molecular systems are very crys-
talline materials, which enables a proper investigation of electron-electron interactions in a
clean environment. In particular, the molecular conductors (TMTTF)2X, where TMTTF
is the tetramethyltetrathiafulvalene molecule and X a monovalent counter-anion since it
represents one of the most appropriate playgrounds to explore correlation effects in low
dimensions. Also, the high-purity and tunability of such systems represent a rich play-
ground for the exploration of fundamental Physics.

In such system, there is one electron per two molecules of TMTTF and they are
organized in chains in the a-axis (Figure 4.3), which are dimerised since the counter-anions
X (e.g.X = PF6, AsF6, SbF6) have a particular way to order, occasioning inequivalent
charge distribution in the system.

4.2 The Extended Hubbard model

A proper physical model employed in strong correlation systems with many particles
is the extended Hubbard model [56]. The hamiltonian of the physical system is expressed
by the form:

36


CHAPTER 4. CORRELATION PHENOMENA IN THE (TMTTF)2X (X = PF6-H12, PF6-D12) FABRE-SALTS 37

530 Naturwissenschaften (2007) 94:527–541

S S

S S

C C

CCH3

CH3

CH3

CH3

C

C

(a)

(b)

(c)

C

Fig. 2 a TMTTF molecule. b View along the stacks of TMTTF
(a-direction) and c perpendicular to them (b -direction). Along
the c-direction the stacks of the organic molecules are separated
by the AsF−

6 anions, for instance. In the case of the TMTSF salts,
S is replaced by Se

the itinerant charges of a one-dimensional metal are
not homogeneously distributed. However, the trans-
lational symmetry can be broken if electron–phonon
interaction and electron–electron interaction become
strong enough; later we will also consider spin–phonon
coupling. Energy considerations then cause a charge
redistribution in one or the other way, leading to CDWs
or CO. Indeed, these ordering phenomena affect most
thermodynamic, transport, and elastic properties of the

crystal, and in some cases also its structure; in this re-
view, we want to focus on the electrodynamic response,
i.e., optical properties in a broad sense.

Any sort of charge disproportionation implies a
partial localization of the electrons. The density of
states at the Fermi level is reduced, which has se-
vere consequences for the metallic state. In certain
cases the material can even become totally insulat-
ing with a complete gap open. First of all, there will
be single-particle electron-hole excitations, which re-
quire an energy of typically an electron volt, like in
a band insulator. But, in addition, collective modes
are expected. There is a rather general argument by
Goldstone (1961) that whenever a continuous sym-
metry is broken, long-wavelength modulations in
the symmetry direction should occur at low fre-
quencies. The fact that the lowest energy state
has a broken symmetry means that the system is
stiff: Modulating the order parameter (in ampli-
tude or phase) will cost energy. In crystals, the
broken translational order introduces a rigidity to
shear deformations, and low-frequency phonons. These
collective excitations are expected well below a
millielectron volts.

10

100

1

(TMTSF) PF

(TMTTF) Br(TMTTF) AsF (TMTSF) ClO

Pressure

SP

loc

CO

SC

AFMAFM SDW

metal

~5 kbar

T
e
m

p
e
ra

tu
re

 (
K

)

(TM)2X
1D

2D

3D

(TMTTF) PF(TMTTF) SbF

Fig. 3 The phase diagram of the quasi-one-dimensional TMTTF
and TMTSF salts, first suggested by Jérome (1991) and fur-
ther developed over the years. For the different compounds
the ambient-pressure position in the phase diagram is indicated.
Going from the left to the right, the materials get less one-
dimensional due to the increasing interaction in the second
and third direction. loc Localization, CO charge ordering, SP
spin–Peierls, AFM antiferromagnet, SDW spin-density-wave, SC
superconductor. The description of the metallic state changes
from a one-dimensional Luttinger liquid to a two- and three-
dimensional Fermi liquid. While some of the boundaries are clear
phase transitions, the ones indicated by dashed lines are better
characterized as a crossover. The position in the phase diagram
can be tuned by external or chemical pressure

Figure 4.1: Schematic general P − T phase diagram for the Fabre-Bechgaard salts. The position of the
different compounds under ambient pressure are indicated above with the black arrows, such difference
is directly connected with the counter-anion chemical pressure over the molecule. The phases shown are
localized electrons (loc), charge-ordering (CO), antiferromagnetic (AFM), spin-Peierls (SP), spin-density
wave (SP) and superconductivity (SC) [54,55].

H =
∑
i,j,σ

ti,jc
†
i,σcj,σ + U

∑
i

ni,↑ni,↓ + V
∑
i,j

ninj , (4.1)

where ti,j are the transfer integrals between the crystal lattice sites’ (also known as the

hopping term) and are associated with the kinetic energy of the electrons, c†i,σ is the
creation operator, which “creates” one electron with spin σ in the i site, while the cj,σ
“destroys” one electron with spin σ from the j site (such operators simulate an electron
“hopping” from one site j to another i and that is why the ti,j is called the hopping term),
U is the Coulomb electronic repulsion of two electrons occupying the same site, V is the
Coulomb repulsion between two electrons at neighbor sites and ni,↑ and ni,↓ represents
the number operator of electrons with spin up and down, respectively.

The correlation phenomena between electrons (namely the inter-site U term) cause a
splitting between the energy bands, characterizing the system as a semiconductor [54].
Such phenomenon is the so-called Mott-insulator phase or Mott metal-insulator phase
transition [50].

4.3 The charge-ordered phase

Whenever the TMTTF-based molecular systems are at low temperatures (between
50−150 K) a charge-ordered phase can be achieved coexisting with a ferroelectric one [57].
Such phenomena can also be explained through the Hubbard model. In this case, when
the hopping term ti,j ≈ U the minimal energy configuration of the system is encountered
by ordering the charges. However, as the V term increases, a critical value of it can be
found where the minimal energy configuration is not a homogeneous charge distribution,
but a charge disproportion one.

Above the charge-ordering phase temperature (Tco) electrons are localized without the
formation of electrical dipoles, while below Tco there is a finite charge disproportion [58]
between the molecules (above the Vc), creating electric dipoles. From such electrical


38 4.3. THE CHARGE-ORDERED PHASE

Figure 4.2: Dependence of the charge transfer σ as a function of V /t2. After a critical inter-site value
(Vc) is achieved, an inhomogeneous charge distribution takes place [56].

configuration, the ferroelectricity phase emerges, concomitant with the charge-ordering
one. It is worth mentioning that a metal to Mott-insulating phase transition can be
associated directly with the charge-ordering transition and can be experimentally observed
in electrical resistivity measurements for several systems [53,54,59,60].

H
C S

a

o

P
F

c c*
b

Figure 4.3: Molecular stacks structure of the (TMTTF)2X, where the anion X (X = PF6, AsF6 or
SbF6) is represented in the octahedral structure. The triclinic unit cell is represented by the blue lines.
The a, b, c and c∗-axis (dotted lines) are represented. Adapted with permission from [61].

The charge-ordering phase plays an important role in the comprehension of some
exotic phenomena [62–64] and recently was recognized as well in the Physics of some
molecular conductors [65,66], such as the previously-mentioned (TMTTF)2X salts.

Another strong evidence of ferroelectricity is an anomaly (maximum) in the dimension-
less dielectric constant ε′ measurements for such systems, reported in the literature [57]
and reproduced in this Master Thesis. Since the dielectric constant is directly associ-
ated with the polarization through the Clausius-Mossoti relation, such anomaly reveals a
change in the polarization of the system, directly associated with the ferroelectric phase.

All the previously mentioned phases in this Chapter (namely, charge-ordering, and
ferroelectricity) occurs simultaneously at Tco, in the so-called Mott-Hubbard ferroelectric
phase. The exploration of this exact phase was the motivation for all the experiments
carried out in this Master Thesis.


CHAPTER 4. CORRELATION PHENOMENA IN THE (TMTTF)2X (X = PF6-H12, PF6-D12) FABRE-SALTS 39

Figure 4.4: Resistivity versus temperature for some of the Fabre-Bechgaard salts. The change in the
dρ/dT for curves 1 (AsF6 − 102 K), 2 (SbF6 − 157 K) and 3 (PF6 − 67 K) is directly associated with a
Mott metal-insulator transition [54].

4.4 The dielectric constant and the Clausius-Mossoti rela-
tion

The so-called dielectric constant ε relatively to vacuum is described as a function of
the electrical field E by the expression [8]:

ε =
ε0E + P

ε0E
= 1 + χ, (4.2)

where ε0 is the vacuum permittivity, P the polarization and χ the magnetic susceptibility.
The polarizability α of a single atom is directly proportional to the local dielectric

field Elocal by the form:

p = α · Elocal, (4.3)

where p is the electric dipole moment (p = q · l).
The polarizability is defined in terms of a single atom and the dielectric constant

defined in how such atoms are assembled to form a crystal. The polarization P of a
crystal is the sum of all the atomic polarizabilities α, given by the form:

P =
∑
j

Njpj =
∑
j

NjαjElocal(j), (4.4)

where Nj the number of j-site atoms, αj the polarizability of j-site atoms and Elocal(j)
is the local electrical field at atoms j sites.

To relate the dielectric constant to the polarizability it is needed to relate the macro-
scopic electric field to the local one, by the form:

P =

∑
j

Njαj

 (
E +

P

3ε0

)
︸ ︷︷ ︸
Lorentz relation

. (4.5)


40 4.5. THE SOLID STATE PHYSICS LABORATORY IN RIO CLARO, SP − BRAZIL

Solving Equation 4.5 for P :

P =

∑
j

Njαj

 · (3ε0E + P

3ε0

)
,

3ε0P − P

∑
j

Njαj

 = 3ε0E

∑
j

Njαj

 ,

P

3ε0 −

∑
j

Njαj

 = 3ε0E

∑
j

Njαj

 ,

χ =
P

ε0E
=

3
∑

j Njαj

3ε0 − (
∑

j Njαj)
=

�3
∑

j Njαj

�3[ε0 − 1/3(
∑

j Njαj)]
.

So,

χ =

∑
j Njαj

[ε0 − 1/3(
∑

j Njαj)]
. (4.6)

Combining Equations 4.6 and 4.2:

ε− 1 =

∑
j Njαj

[ε0 − 1/3(
∑

j Njαj)]
=

3
∑

j Njαj

[3ε0 − 1/3(
∑

j Njαj)]
,

(ε− 1)

3ε0 −
∑
j

Njαj

 = 3
∑
j

Njαj ,

3ε0ε− ε
∑
j

Njαj − 3ε0 +
∑
j

Njαj = 3
∑
j

Njαj ,

3ε0(ε− 1)− ε
∑
j

Njαj = 2
∑
j

Njαj ,

3ε0(ε− 1) = (ε+ 2)
∑
j

Njαj ,

Thus:

ε− 1

ε+ 2
=

∑
j Njαj

3ε0
. (4.7)

Equation 4.7 is the so-called Clausius-Mossoti relation, connecting a macroscopic phys-
ical quantity (dielectric constant ε) with a microscopical one (polarizability α). Thus, the
dielectric constant is directly proportional to the polarizability/polarization of the system.
With such motivation, dielectric constant measurements were carried out, in molecular
systems of the TMTTF family in order to explore the ferroelectric behavior below the
charge-ordering transition temperature.


CHAPTER 4. CORRELATION PHENOMENA IN THE (TMTTF)2X (X = PF6-H12, PF6-D12) FABRE-SALTS 41

Figure 4.5: Panoramic view of the Solid State Physics laboratory showing the Teslatron PT cryostat
and all the equipments employed to perform the electrical measurements.

Figure 4.6: All the measuring equipments of the Solid State Physics Laboratory: a) Keithley 2182A
nanovoltimeter; b) Keithley 617 programable electrometer (voltage, electrical resistance, charge and cur-
rent); c) LakeShore 350 temperature controller; d) Keithley 2182A nanovoltmeter; e) Keithley 6220 pre-
cision current source (µA currents); f) Oxford MercuryiTC temperature controller; g) Andeen-Hagerling
ultra-precision fixed 1 kHz capacitance bridge with resolution of 10−6 pF; h) Computer interfaced with all
equipments for the data acquisition and i) MercuryiPS electrical current source for the superconducting
magnet (up to 110 A).

4.5 The Solid State Physics laboratory in Rio Claro, SP −
Brazil

The cryostat employed for the dimensionless dielectric constant and resistivity mea-
surements was a Teslatron-PT (1.4 K < T < 300 K) supplied by Oxford Instruments with


42 4.6. EXPERIMENTS PERFORMED IN THE FRAME OF THIS MASTER THESIS

4He closed cycle, i.e., it operates with a fixed amount of Helium gas without the need to
refill it for several periods of time.

There is a NiTi (Niobium Titanate) superconducting magnet with Tc ≈ 4.6 K that
can apply magnetic fields up to 12 T, remaining in the so-called permanent mode [67].
Also, there is an outer vacuum chamber (OVC) of the cryostat that can reach pressures
of 10−7 mbar, which is an excellent thermal isolation.

The 4He is circulated through the system cooling the sample space through the ex-
change gas (also 4He). Then, a compressor attached to the cryostat through ≈ 12 m of
cryogenic lines (in order to maximize the efficiency of the thermal machine) cools back the
helium (employing the expansion and compression method) and dissipates such potency
(≈ 12 kVA) in another external water-based cooling system (Chiller). Such cycle keeps
happening until the system is powered off. Yet, all the electrical signal of the laboratory
is reconstructed by two no-breaks, guaranteeing that the oscillations of the power supply
do not affect in any way the electrical measurements performed.

4.6 Experiments performed in the frame of this Master
Thesis

A preliminary systematic investigation on the ε′(T,B), R(T,B) and P (T,B) was
performed in our research group in the frame of the Master Thesis of Paulo Menegasso [68].

All the experiments carried out in this Master Thesis in the molecular salts were
performed in the c∗-axis (see Figure 4.3). Since the polarization at Tco occurs in the a-
axis (electronic polarization), by measuring the c∗-axis the finite polarization observed
indirectly by the dimensionless dielectric constant as a function of temperature was the
ionic contribution for the total polarization.

It is worth mentioning that all experiments shown in this Master Thesis were per-
formed in the Solid State Physics Laboratory in Rio Claro, SP - Brazil.

The samples were prepared using the 2-points method to accomplish all the bulk
measurements described below. It starts in the choosing of the sample, aiming the ones
that present the most parallel planes in the b∗-axis. Such planes are painted with a
conductive paint (carbon paste in our case) to achieve an electrical configuration as close
to a parallel plates capacitor. Later on, two gold wires of about 20µm were attached
in both opposite planes with carbon paste as well and then, the samples were fixed in
the so-called mask (semiconducting to isolate the contact) to be attached in our system,
allowing the electrical contact between them.

It is worth mention that for the ε′(T , B) it is crucially relevant that the ratio between
the sample thickness and its areas on the b∗-axis should be appropriate, otherwise the
experimental results of ε′ would not be possible to achieve. Also, this impacts directly in
the capacitance of the prepared sample, since the capacitance depends on the area of the
sample and the distance between the plates (thickness of the sample in this case), so it
was able to be measured in the capacitance bridge of the laboratory. In the cases that the
capacitance of the sample was too low or too noisy, a commercial capacitor was employed
in series or in parallel (depending on the case) with the sample to reduce the so-called
loss of the bridge and to achieve a clean measure for C x T , for example.

The parallel surfaces of the c*-axis of the sample were revested with a thin layer
of carbon paste in order to achieve a configuration similar to a parallel plates capacitor.
Then, the capacitance C of the sample could be measured as a function of temperature T ,
aiming to obtain the dimensionless dielectric constant ε′. Such capacitance measurements
were performed by employing an Andeen-Hagerling 2550A capacitance bridge with a
resolution of 10−6 pF.

The dielectric constant ε could be determined by employing the well-known parallel
plates capacitor mathematical expression, given by:

C =
ε ·A
d
−→ ε =

C · d
A

, (4.8)


CHAPTER 4. CORRELATION PHENOMENA IN THE (TMTTF)2X (X = PF6-H12, PF6-D12) FABRE-SALTS 43

A = 1 x 2 mm²
d = 0.2 mm

Sample #1 (TMTTF)2PF6 (H12)

A = 1  x 2mm²
d = 0.6 mm

Sample #2 (TMTTF)2PF6 (H12)

A = 1 x 2 mm²
d = 0.6 mm

Sample #3 (TMTTF)2PF6 (D12)

gold wire (Ø = 20µm)

carbon paste
sample

gold wire

Figure 4.7: Electrical contacts employed in the c∗-axis with carbon paste (conductive) and golden
wires attached in each parallel surface of the c∗ axis of the samples, in order to achieve a nearly parallel
capacitor plates configuration. The samples were chosen carefully in order to have the most parallels and
homogeneous surfaces possible.

where C is the capacitance of the sample, A is the capacitor’s plates area and d is the
distance between them.

Thus, the dimentionless dielectric constant was determined by employing:

ε′ =
ε

ε0
, (4.9)

where ε0 is the electrostatic permittivity of the vacuum (ε0 = 8.854187817×10−12 F/m)
[69].

The Table 4.1 shows the samples in which experiments of resistivity and dielectric
constant were carried out in this dissertation thesis.

System Sample Dimensions (height x length x width) Batch

(TMTTF)2PF6 (H12) #1 1 × 2 × 0.2 mm3 EII 93 2

(TMTTF)2PF6 (H12) #2 1 × 2 × 0.6 mm3 EII 93 2

(TMTTF)2PF6 (D12) #3 1 × 2 × 0.6 mm3 IIE 107

Table 4.1: Samples employed in the experiments of this thesis with their proper dimen-
sions. The batches were provided by the French collaborators Dr. Jean-Paul Pouget and
Dr. Pascale Foury-Leylekian (Laboratoire de Physique des Solides, Université Paris Sud,
CNRS UMR 8502, Orsay, France).

It is worth mentioning that the dielectric constant measurements in molecular salts
already reported in literature [57] were not properly reproduced with such resolution
as showed in this Master Thesis. Also, all the experiments carried out were performed
employing a fixed frequency of 1 kHz. In what follows, electric resistivity R(T ) and


44 4.6. EXPERIMENTS PERFORMED IN THE FRAME OF THIS MASTER THESIS

dielectric constant ε′(T,B) were performed for both the fully hydrogenated and the 97.5%
deuterated variant of the system (TMTTF)2PF6. The results are discussed below.


CHAPTER 4. CORRELATION PHENOMENA IN THE (TMTTF)2X (X = PF6-H12, PF6-D12) FABRE-SALTS 45

4.6.1 (TMTTF)2PF6 (H12)

The dielectric constant was measured employing the method described in the last
section. The voltage applied by the capacitance bridge, in order to measure the capaci-
tance of the sample, was obtained to achieve an electrical field (E = U/d) of 50 mV/cm.
However, specifically with Sample #1, the dielectric constant was measured employing
an electrical field of 150 mV/cm.

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0
1 0

2 0

3 0

4 0

5 0

 
ε’

T  ( K )

( T M T T F ) 2 P F 6  ( H 1 2 )
S a m p l e  # 1

Figure 4.8: Dielectric constant versus temperature for the fully hydrogenated (TMTTF)2PF6 showing
a broad maximum behavior at Tco ∼ 57 K, indicating a finite polarization of the molecular salt below such
temperature. A hysteretic behavior in the dielectric constant is observed upon heating and cooling. The
data were measured employing a rate of ±6 K/h and applied electric field of 150 mV/cm. The red and
blue arrows indicate that the dielectric constant was measured upon heating and cooling, respectively.

1 5 3 0 4 5 6 0 7 5 9 0
1 0

2 0

3 0

4 0

5 0  B  =  0 T
 B  =  1 0 T
 B  =  0 T

 
ε’

T ( K )

( T M T T F ) 2 P F 6  ( H 1 2 )
S a m p l e  # 1  

Figure 4.9: Dielectric constant versus temperature for the fully hydrogenated (TMTTF)2PF6. After
10 T of applied magnetic field, the dielectric constant was attenuated. However, upon removing the
magnetic field the dielectric constant did not return to its original configuration. Such effect is associated
with permanent disorder inserted by the magnetic field in the methyl groups. The data were measured
employing a rate of +6 K/h and applied electric field of 150 mV/cm.

Such anomaly is associated with a finite polarization of the system below Tco, where


46 4.6. EXPERIMENTS PERFORMED IN THE FRAME OF THIS MASTER THESIS

a charge disproportion occurs and a ferroelectric phase takes place. The maximum in the
dielectric constant is associated with a finite polarization of the system at Tco and also
to the maximum variation of the free-energy. A multiferroic behavior was not observed
in this system. Instead, permanent disorder inflicted in the methyl groups due to the
magnetic field permanently attenuated the dielectric constant.

The same investigation was carried out in another sample for the fully hydrogenated
system of (TMTTF)2PF6. Resistivity measurements as a function of temperature were
carried out employing the 2-points method since dielectric constant measurements were
planned as well.

0 4 0 8 0 1 2 0 1 6 0 2 0 0 2 4 01 0 2

1 0 3

1 0 4

1 0 5

1 0 6

1 0 7

1 0 8

1 0 9

 
ρ (
Ω

.cm
)

T ( K )

( T M T T F ) 2 P F 6  ( H 1 2 )

Figure 4.10: Electrical resistivity versus temperature for the fully hydrogenated (TMTTF)2PF6 showing
and insulating behavior (dρ/dT < 0) both upon heating and cooling with an employed rate of ±10 K/h.
A hysteretic behavior is observed upon cooling and heating. The resistivity measurements were carried
out employing the two-points method.

Since the system is very metallic, there is an expressive difficulty in measuring the
dielectric constant. In order to be able to measure the sample’s capacitance, an auxiliary
capacitor of ∼ 200 pF was employed in parallel with the sample.

In order to investigate a possible multiferroic [70] character of the system, an external
magnetic field of 10 T was applied aiming to investigate the effect of the magnetic field in
the charge-ordered phase.

In this system, discontinuities in the dielectric constant were associated with ferro-
electric clusters being thermally activated in different critical temperatures. Such clusters
provide a broader dielectric response for the fully-hydrogenated salt. Also, the permanent
disorder was inserted into the system by applying external magnetic field and a successive
attenuation of the dielectric constant was observed, characterizing the fully hydrogenated
(TMTTF)2PF6 as a relaxor ferroelectric, i.e., a ferroelectric system with disorder.


CHAPTER 4. CORRELATION PHENOMENA IN THE (TMTTF)2X (X = PF6-H12, PF6-D12) FABRE-SALTS 47

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 00 . 0

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5

ε’ (
x10

3 )

T ( K )

( T M T T F ) 2 P F 6  ( H 1 2 )
S a m p l e  # 2

Figure 4.11: Dielectric constant versus temperature for the fully hydrogenated (TMTTF)2PF6 showing
a broad anomaly at Tco = ≈ 57 K. The difference between the heating and cooling is due to an hysteretic
behavior in the dielectric constant. The discontinuities observed below Tco are due to thermally activated
ferroelectric clusters. The noise above 65 K is due to the relatively low resistivity of the system. The data
were measured employing a temperature rate of ±7 K/h and an electric field of 50 mV/cm.

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0
0 . 0
0 . 4
0 . 8
1 . 2
1 . 6
2 . 0
2 . 4  B  =  0 T

 B  =  1 0 T
 B  =  0 T
 B  =  0 T
 B  =  0 T

 
ε’ (
x10

3 )

T ( K )

( T M T T F ) 2 P F 6  ( H 1 2 )
S a m p l e  # 2  

Figure 4.12: Dielectric constant versus temperature for the fully hydrogenated (TMTTF)2PF6 showing
a broad maximum anomaly at Tco ∼ 57 K. The red data set was measured employing a rate of −7 K/h,
while it was employed a rate of −10 K/h for the others datasets and an electric field of 50 mV/cm was
applied. With an applied magnetic field of 10 T the dielectric constant was attenuated. By removing
the magnetic field, the dielectric constant did no return to its original configuration and subsequent
attenuations were observed in the successive thermal cycles. Such effect is due to permanent disorder
inserted in the methyl groups by applying the magnetic field.


48 4.6. EXPERIMENTS PERFORMED IN THE FRAME OF THIS MASTER THESIS

4.6.2 (TMTTF)2PF6 (D12)

In order to perform a systematic investigation, a 97.5 % deuterated sample was mea-
sured in order to compare with the fully hydrogenated one. This is the very first-time
dielectric constant measurements for the 97.5 % deuterated variant of (TMTTF)2PF6

were carried out.

2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 01 0 3

1 0 4

1 0 5

1 0 6

1 0 7

1 0 8

 
ρ (

Ω
.cm

)

T  ( K )

( T M T T F ) 2 P F 6  ( D 1 2 )

Figure 4.13: Electric resistivity versus temperature for the 97.5 % deuterated variant of (TMTTF)2PF6,
showing an insulating behavior. The data were measured employing a rate of +9 K/h.

6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 1 0 5

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0
 

ε’

T ( K )

( T M T T F ) 2 P F 6  ( D 1 2 )

Figure 4.14: Dielectric constant versus temperature for the 97.5 % deuterated variant of (TMTTF)2PF6

showing a peak-like anomaly at Tco ∼ 87 K. The anomaly differs from the fully hydrogenated salt (bump-
like). The data were measured employing a rate of −12 K/h with 50 mV/cm of applied electric field.

A hysteretic behavior was also observed upon heating and cooling. In order to also
explore the possible multiferroic character of this system, a magnetic field of 4 T was
applied. This is also due to the inserted disorder in the methyl groups by the magnetic


CHAPTER 4. CORRELATION PHENOMENA IN THE (TMTTF)2X (X = PF6-H12, PF6-D12) FABRE-SALTS 49

field. A capacitor of ∼ 120 pF in parallel with the sample was employed in order to
measure the capacitance of this sample due to its relatively low electrical resistance.

6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 1 0 50

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0
 B  =  0  T
 B  =  4  T
 B  =  0  T

 
ε’

T ( K )

( T M T T F ) 2 P F 6  ( D 1 2 )

Figure 4.15: Dielectric constant versus temperature for the 97.5 % deuterated variant of (TMTTF)2PF6

showing a peak-like anomaly at Tco ∼ 87 K. A hysteretic behavior was observed upon heating (pink
data set) and cooling (orange). The dielectric constant measured upon heating (blue) was permanently
attenuated by the magnetic field, due to inserted disorder in the methyl groups. The data were measured
with a rate of ±12 K/h and 50 mV/cm of applied electric field.

A permanent attenuation of the dielectric constant was also observed due to the
applied magnetic field. The 97.5 % deuterated variant (TMTTF)2PF6 show a peak-like
anomaly at Tco instead of a broad-like one for the fully hydrogenated one.

4.6.3 The mean-field theory

Regarding the permanently inserted defects in the methyl groups, an analysis in the
so-called mean-field theory can be performed. In this regard, it is possible to plot the
dielectric constant in the so-called Curie-Weiss behavior. The Curie-Weiss behavior is
given by the expression:

1

ε′
= A± · |T − Tco|, (4.10)

where A± are the expected slopes for such behavior and the relation between them is
given by:

A−
A+

= 2. (4.11)

From Equation 4.11 it is clear a factor 2 between the slopes of the solid lines (θ =
2θ). In this context, the Curie-Weiss behavior was analyzed for the fully hydrogenated
and the 97.5% deuterated (TMTTF)2PF6, in order to compare the measured dielectric
constants in the scope of the mean-field theory. The solid lines in Figure 4.16 represent
the expected behavior of the data accordingly to the mean-field theory. The more the
data diverges from the mean-field behavior (θ 6= 2θ), the greater the number of intrinsic
defects in the system. In this context, the fully hydrogenated salt diverges expressively
from the mean-field behavior indicating the presence of intrinsic defects in the system. As
shown in Figure 4.12, the dielectric constant was permanently attenuated due to external
magnetic field and a successive attenuation of it was observed. Such results indicate


50 4.6. EXPERIMENTS PERFORMED IN THE FRAME OF THIS MASTER THESIS

that not only the fully hydrogenated salt present intrinsic defects in its structure (relaxor
ferroelectric [71]), but such defects can be inserted by the application of external magnetic
field and/or thermal cycling.

Following discussion with Prof. Roberto E. Lagos Monaco during the qualify exam,
it is worth mentioning that the mean-field approach does not work for low-dimension
systems. Hence, the deviation from the mean-field theory observed in the present in-
vestigations for the TMTTF salts (Figure 4.16) could not only be associated with de-
fects/disorder, but also with the low-dimension character of such systems.

0 10 20 30 40 50 60 70 80 90 100
0.0

0.2

0.4

0.6

0.8

1.0

1
0

2
/ε

’

 
T(K)

(TMTTF)
2
PF

6 D
12

H
12  

Figure 4.16: Curie-Weiss plots for the fully hydrogenated and 97.5% deuterated (TMTTF)2PF6. The
solid lines represent the expected behavior for the data by the mean-field theory and the dotted line
represents the smoothed data for both salts to show the factor two between the left and right slopes.
For an aesthetic question, the dashed line was suppressed for the 97.5% deuterated salt data. It is clear
that the fully hydrogenated salt diverges expressively from the theory, indicating the presence of intrinsic
defects in this system, being considered as a relaxor ferroelectric, i.e. a ferroelectric with disorder [71].

Recently, it was shown that there is a predominance of valence electrons almost ex-
clusively in the methyl groups of the TMTTF molecule [72], as shown in Figure 4.17.

Figure 4.17: Valence electrons distribution in the TMTTF molecule, the red regions denote more valence
electrons. Note there is a significant higher distribution of the valence electrons in the methyl groups [72].


CHAPTER 4. CORRELATION PHENOMENA IN THE (TMTTF)2X (X = PF6-H12, PF6-D12) FABRE-SALTS 51

Since the counter-anions are located in the cavity formed by the methyl groups and
there is a significant density of valence electrons in it, by applying external magnetic field
it is possible to change the methyl’s groups degrees of freedom and thus the position
of the counter-anion. Therefore, the result shown in Figure 4.17 are totally in line with
the experimental results shown in Figure 4.12, since disorder is inserted by application of
external magnetic field in the methyl groups and it can be observed by the fact that the
dielectric constant was attenuated by the magnetic field and subsequent thermal cycling.

It is worth mentioning that Prof. Dr. Ricardo Paupitz (UNESP, Rio Claro, SP −
Brazil) also performed DFT calculations in the hydrogenated variant of (TMTTF)2PF6

and obtained a similar result from Figure 4.17 by observing a more expressive concentra-
tion of magnetic moments in the methyl groups of the TMTTF molecule.

During the final period of my master degree, I had the opportunity to perform ab initio
DFT calculations, with support from Prof. Dr. Ricardo Paupitz and my advisor Prof.
Dr. Mariano de Souza, in the hydrogenated variant by optimizing both the molecular
structure of the system and the unit cell, in order to find the configuration of minimal
energy at T = 0 K. Some perspectives lies on performing systematic DFT calculations
in order to investigate the electronic density in the TMTTF molecule and the effects of
rotating the methyl groups with the total energy of the system, in order to corroborate
the experimental results shown in this Master Thesis, since magnetic field can change the
degrees of freedom of such groups and insert disorder into the system.


Chapter 5

Summary and Conclusions

In order to make a more clear and comprehensive summary and conclusions, the three
main chapters of this Master Thesis were divided to provide a better understanding of
the fundamental aspects here discussed. Thus, it follows:

The magnetocaloric effect (the magnetic Grüneisen parameter): the classical magnetic
model Brillouin and the 1D−Ising were solved in details following discussions presented
in classical textbooks. Next, a discussion regarding the Grüneisen parameter was per-
formed to demonstrate its importance as being the only mathematical tool to determine
experimentally a quantum critical point (divergence of the Grüneisen parameter indicates
a QPT). In this regard, the Grüneisen parameter was calculated for the Brillouin model
considering both J = 1/2 and arbitrary J . The Grüneisen parameter for such model
diverges at T = 0 K and B = 0 T. This indicates that even that the Brillouin model
is purely classic (it does not incorporate quantum mechanics) it does show a quantum
critical behavior intrinsically at T = 0 K. Such results are part of a published manuscript
in the journal Physical Review B (see file in the end of this Master Thesis)

The binary alloy FeSe1−x: a detailed discussion of the evolution of superconductivity
over the years was performed followed by the characterization of the physical properties
of the binary alloy FeSe1−x. The synthesis method was described and several samples
were obtained (aiming the hexagonal phase). X-ray measurements were performed in the
synthesized samples and concluded that we did not have a homogeneous phase of the
hexagonal one. Further systematic investigations will be carried out to synthesize such
phase and completely characterize it (as well as the tetragonal one already is) in order to
connect such properties with the theoretical model of the 3d orbitals of Fe, proposed by
L. Craco and Leoni.

Correlation phenomena in the (TMTTF)2X (X = PF6-H12, PF6-D12) Fabre-salts: a
proper description of all the relevant fundamental literature was performed aiming to
focus on the Mott-insulator, charge-ordering and ferroelectric phases, in order to connect
directly such theoretical aspects with the experiments carried out in this Master Thesis.
The systems investigated were the hydrogenated (TMTTF)2PF6 and the 97.5% deuter-
ated variant of (TMTTF)2PF6. Regarding the hydrogenated variant, measurements of
electrical resistivity and dielectric constant were performed. The dielectric behavior was
found to have a maximum at Tco indicating a finite polarization of the system at this tem-
perature. Several “jumps” were observed in the dielectric constant of one sample of this
system. Such behavior is directly connected with the presence of ferroelectric clusters

52


CHAPTER 5. SUMMARY AND CONCLUSIONS 53

being thermally activated upon varying the temperature, explaining the broad dielec-
tric response in this system. Regarding the 97.5% deuterated variant, it was observed
a peak-like anomaly in the dielectric constant measurements and such results are not
yet published in literature. Both systems were measured under external magnetic field
and an attenuation of the dielectric constant was observed, such result is connected with
both magnetoelectricity and insertion of defects into the systems due to thermal cycling
or magnetic field. Yet, the deviation between the Curie-Weiss plots and the mean-field
theory provided crucial physical information regarding intrinsic defects present in the hy-
drogenated system. Part of the experimental results in this Master Thesis contributed to
a manuscript published in Physical Review B (see file in the end of this Master Thesis).


Chapter 6

Perspectives and Outlook

Further systematic investigations regarding this Master Thesis includes new synthesis
methods (as well as a more systematic investigation of the electrochemical growth method)
aiming not only to learn how the methods work, but to synthesize samples of systems
of interest in order to continue such studies, specially regarding the FeSe1−x hexagonal
phase and the theoretical proposal to understand the difference between the two phases
of interest.

Also, our group can perform thermal expansion measurements via dilatometric cell
with ultra-high resolution (∆l ∼ 0.05−0.1 Å), which I got acquainted during my Master’s
degree. Thermal expansion measurements of the investigated molecular conductors in this
Master Thesis, as a function of temperature and magnetic field (magnetostriction as well),
are to be performed in a near future aiming to connect such results with the dielectric ones
reported here. Such investigation will provide crucial information regarding the influence
of the magnetic field on the ferroelectric Mott-Hubbard phase and will contribute to its
fundamental understanding.

54


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