論文 / 著書情報
Article / Book Information
題目(和文)

磁気的フラストレーションを有するクロムスピネル化合物およびパイ
ロクロアチタン酸化物におけるスピン−格子結合に関する熱力学的研
究Title(English)

Thermodynamic Studies of Spin-Lattice Effects in Magnetically
Frustrated Chromium Spinels and Titanium Pyrochlores

著者(和文)

気谷卓

Author(English)

Suguru Kitani

出典(和文)

学位:博士（理学）,学位授与機関:東京工業大学,
報告番号:甲第9733号,
授与年月日:2015年3月26日,
学位の種別:課程博士,
審査員:川路 均,大坂 武男,菅野

了次,松下

伸広,平山

Citation(English)

Degree:,Conferring organization: Tokyo Institute of Technology,
Report number:甲第9733号,
Conferred date:2015/3/26,
Degree Type:Course doctor,
Examiner:,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Tokyo Institute Research Repository)

雅章

Doctoral Thesis

Thermodynamic Studies of Spin-Lattice Effects
in Magnetically Frustrated Chromium Spinels
and Titanium Pyrochlores

Suguru Kitani
Department of Electronic Chemistry,
Interdisciplinary Graduate School of Science and Engineering,Tokyo Institute of Technology

2015
Supervisor Professor Hitoshi Kawaji

Contents
Chapter 1.

Introduction

1.1

Magnetic Frustration

1

1.2

Magnetic Systems with Pyrochlore lattice3

1.3

Spin-Lattice Coupling4

1.4

Thermodynamic relations

6

1.5

Outline of Present Study9

References

Chapter 2.
2.1

2.4

Experimental

Heat capacity measurement

20

2.1.1

Introduction

20

2.1.2

Thermal relaxation method

21

2.3.2.1 1 method

22

2.3.2.2 2 method

24

Quasi-adiabatic method

25

Thermal expansion measurement

27

2.2.1

Introduction

27

2.2.2

Capacitance method

29

2.1.3
2.2

18

Magnetic Measurement

33

References

38iChapter 3.

Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

3.1

Introduction

39

3.2

Experimental

40

3.3

Results and Discussion

41

3.3.1

Heat capacity

41

3.3.2

Thermal expansion

44

3.3.3

Magnetic properties

46

3.4

Conclusion

49

References

Chapter 4.

57

Spin-lattice coupling effect in strongly geometrically frustrated

spinel CdCr2O4
4.1

Introduction

59

4.2

Experimental

61

4.3

Results and Discussion

61

4.3.1

Magnetic Susceptibility

61

4.3.2

Heat capacity

62

4.3.3

Thermal expansion

62

4.3.4

Pressure dependence of the transition temperature

64

4.4

Conclusion

66

References

Chapter 5.
5.1

72

Lattice behavior in the conical spin state of CoCr2O4

Introduction

74
ii

5.2

Experimental

76

5.3

Results and Discussion

77

5.3.1

Magnetic Susceptibility

77

5.3.2

Heat capacity

78

5.3.3

Thermal expansion

79

5.3.4

Pressure dependence of the transition temperature

80

5.3.5

Magnetocaloric effect under pulsed magnetic field

83

5.4

Conclusion

85

References

94

Chapter 6.

Lack of spin-lattice coupling in spin liquid state of Tb2Ti2O7

6.1

Introduction

96

6.2

Experimental

97

6.3

Results and Discussion

98

6.3.1

Heat capacity

98

6.3.2

Thermal expansion

102

6.3.3

Discussion about low temperature anomaly at 2 K

104

6.4

Conclusion

106

References

Chapter 7.

112

Thermal expansion behavior in spin ice state of Dy2Ti2O7 andHo2Ti2O7
7.1

Introduction

115

7.2

Experimental

116
iii

7.3

7.4

Results and Discussion

116

7.3.1

Heat capacity

116

7.3.2

Thermal expansion

117

7.3.3

Magnetic contributions for Cp and 

117

7.3.4

Lattice behavior in spin ice state

118

Conclusion

122

References

128

Chapter 8.

Summary130

Appendix

List of published articles

136

Acknowledgments

137

iv

Chapter 1. Introduction

Chapter 1
Introduction

1.1 Magnetic Frustration
The term “frustration” generally refers to the state that it is impossible for all
requirements to satisfy each other. The frustration often appears in our lives, as you
would have experienced. Such frustrated state often induces unexpected situations torelieve a frustration. Similar states can, interestingly, be found in nature too. When amaterial includes a frustration, the ground state will not achieve a stable state. Instead,strong fluctuation emerges at low temperatures, and it is usually very sensitive for
perturbations. The strong fluctuation and perturbations are coupled with one another,which creates an exotic ground state. Thus, the frustration in the material science havebeen attracted much attention as an excellent playground in which to discover newstates and properties.
Magnetic materials are probably most typical and appropriate systems to study
frustration effects. The most accessible example of the magnetic frustration is theantiferromagnetic Ising model, where the spins can orient up or down. It is worthwhile
to start from considering the behavior of an unfrustrated case. Let the Ising spins with
only the nearest neighbor exchange interaction J1 arrange on the vertices of the squarelattice, as illustrated in Fig. 1.1(a). In this situation, all spins can antiparallelly alignwithout any problem. Then, consider the antiferromagnetic Ising spins on a triangularlattice as shown in Fig. 1.1(b). As opposed to the square lattice, spins cannot alignantiparallel to each other, resulting in the frustration. It is called “Geometrical
1Chapter 1. Introduction

frustration”. The earliest theoretical study on the geometrical frustration was performed
for two-dimensional triangular lattice by Wannier in 1950 [1]. and for pyrochlore lattice,
three-dimensional frustrated system, by Anderson in 1956 [2]. They suggested that thesespin systems show no phase transition down to the lowest temperature, and hence theground state is macroscopically degenerated. In other words, the ground state of
antiferromagnetic materials with such “frustratable” lattice is to be a spin-liquid state, inwhere spins strongly fluctuate without any transitions to long-range ordering.
And now, it is instructive to reconsider whether the square lattice model has apossible way to cause a magnetic frustration. Let the spins have an additional exchangeinteraction from next nearest neighbor (diagonal position) J2 besides nearest-neighbor J1,
which is so-called the J1-J2 model as illustrated in Fig. 1.2. The ground state of thismodel depends on the ratio J1/J2. For J1 > 2J2 > 0, neighboring spins are antiparallel,
resulting in ferromagnetic alignment of spins at the diagonal position. For 2J2 > J1 > 0,
spins are antiferromagnetically aligned with next nearest neighbors. These cases are
weakly frustrated. On the other hand, for J1=2J2, the two states mentioned above aredegenerated, leading to a strong frustration [3]. Such frustration from the competition
between the exchange interactions is called “bond frustration”.
The candidate materials for the spin-liquid state have been found in geometricallyfrustrated systems of two-dimensional triangular lattice [4], and kagome lattice [5], andthree-dimensional hyper-kagome lattice [6], and pyrochlore lattice [7]. Recent studies
have discovered geometrical frustration in cluster magnet [8], quasi-crystal [9], andorganic materials [10], which attract much attention due to their characteristic properties
that the conventional materials do not possess. In a system with the bond frustration, it
has also been shown an exotic ground state as in the geometrically frustrated systems.
2

Chapter 1. Introduction

Besides the J1-J2 model, the bond frustration has been known to arise in the Plaquettelattice, checkerboard lattice, and Shastry-Sutherland lattice, which are based on thesquare lattice. Further, it has been discovered that the bond frustration is appeared in the
spin-ladder system [11], diamond lattice [12], and honeycomb lattice [13], which
exhibit spin-liquid-like state.

1.2 Magnetic Systems with Pyrochlore lattice
The name “pyrochlore lattice” originates from a characteristic lattice in the mineral
pyrochlore, (Na,Ca)2Nb2O6(OH,F), the structure of which was ﬁrst reported by vonGaertner in 1930 [14]. Figure 1.3 shows a pyrochlore lattice, which forms acorner-sharing tetrahedral network. As in the triangular lattice, if the magnetic moments
are located on the vertices of each tetrahedron and exchange interaction between all
moments are equal, strong geometrical frustration will emerge. This geometry is oftheoretical and experimental interest.
In real compound, there are several crystal structures having a pyrochlore lattice.
The representative is pyrochlore compounds A2B2X7, where the A and B sites separately
form the pyrochlore lattice as presented in Fig. 1.4. The A-sites are eight coordinatedand located within distorted cubes, and the B-sites are six coordinated and are locatedwithin octahedron. The A and B ions are represented by metal cations and the X ismostly oxygen. It is known that the pyrochlore compound tolerates for the anion andcation vacancies. Hence, some defect pyrochlore can be present with the composition
A2B2X6X1-x (0<x<1) or AB2X6. The latter is recently referred as -pyrochlore, getting
attention due to the unconventional superconductivity induced by the rattling motion of
the A ion [15].
3Chapter 1. Introduction

Another typical compound which has the pyrochlore lattice is spinel AB2X4 shownin Fig. 1.4. The B-sites form the pyrochlore lattice, while the sub-lattice of the A-site is
the diamond lattice, as opposed to the pyrochlore compounds. The A-sites are four
coordinated and located within tetrahedron, and the B-sites are six coordinated and arelocated within octahedron. The A and B sites are occupied by metal cations and X isavailable to oxygen and chalcogen. Both pyrochlore and spinel belong to the cubic
space group Fd-3m.
A lot of studies have been discovered the effect of magnetic frustration in thesecompounds. As discussed in the previous section, the pyrochlore lattice can lead to thespin-liquid state. However, most of them undergo a magnetic phase transition with acomplex ordering or fall into a glassy state of spins. It is because the frustrated state ishighly unstable, and the macroscopic degeneracy can be easily lifted by a weakperturbation, such as further exchange interaction, dipole-dipole interaction, site dilution,
single-ion anisotropy, Dzyaloshinskii-Moriya interaction, quantum fluctuation, andspin-lattice coupling.

1.3 Spin-Lattice Coupling
In a pyrochlore antiferromagnet, the spin-lattice coupling often plays an importantrole for lifting the macroscopic degeneracy caused by the frustration. It means that thesystem is stabilized by distorting the lattice. However, there is, of course, a need to pay
the cost of the elastic energy to distort the lattice. The simplest situation is thebond-phonon model [16], where the bond length in the pyrochlore lattice can
independently expand or contract, and the energy gain through the spin-lattice couplingis proportional to the change of the bond length. From the elastic energy, this simple
4

Chapter 1. Introduction

model derives a biquadratic term in the spin Hamiltonian, which shows that the lattice
distortion induced by the spin-lattice coupling always minimize the energy of thesystem.
The effect of the spin-lattice coupling has been observed in real systems. The mostextensively studied materials are probably the chromium spinel oxides ACr2O4 (A=Zn,
Cd, and Hg). Here ZnCr2O4 is illustrated as a representative [17]. In ZnCr2O4, magnetic
Cr3+ ions form a pyrochlore lattice, and a strong antiferromagnetic interaction is present
between them, indicating the negative Curie-Weiss temperature of -390 K. Hence, thestrong frustration is induced. The paramagnetic phase remains well below theCurie-Weiss temperature. The system finally undergoes a magnetostructural transition
from a cubic paramagnetic phase to a tetragonal antiferromagnetic phase with acomplex spin order. As the cooperative phenomenon directly evokes the coupling
between spin and lattice degrees of freedom, the theoretical model including thespin-lattice coupling was phenomenologically in good agreement with several
experimental results related to the tetragonal distortion. It is noted that there is still notheory to simultaneously describe the observed lattice distortion and spin structure. To
shed light on the microscopic mechanism, the any additional interactions have been
considered to be required.
The idea of the spin-lattice coupling is further applied to the peculiar
magnetization behavior at high magnetic fields of the chromium oxides [18]. When an
external magnetic field is applied, they show a first-order transition following ahalf-magnetization plateau and lattice distortion. An interesting point is that despite thechromates have different ground states, their magnetic structures turn out to be the sameunder high magnetic fields, suggesting that a universal effect of the spin-lattice coupling5

Chapter 1. Introduction

is lying on the chromium spinels.
Besides the magnetic frustration, there are several routes to establish thespin-lattice coupling. One of recent interest is the multiferroic material, because thespin-lattice coupling deeply relates with the mechanism of the multiferroicity. When
spins form a spiral configuration, i.e., symmetry lowering of spin system, the latticesymmetry is reduced through the inverse effect of the Dzyaloshinskii-Moriya interaction,
resulting in an electric polarization [19]. In addition, recent theoretical development of
the spin-lattice coupling has explained anomalous spin-glass behaviors observed inpyrochlore magnets [20]. The spin-glass transition temperature generally depends on the
randomness. However, in the spinel oxide ZnCr2O4 and pyrochlore oxide Y2Mo2O7, theelement substation does not induce the change in the transition temperature, although
the randomness must be changed. For this puzzling behavior, the Heisenberg modelwith assuming the coupling between spin and local lattice distortion provides aquantitative explanation. As described in this section, the spin-lattice coupling is now
widely appeared in the material with novel physical properties.

1.4 Thermodynamic relations
In the study of the spin-lattice coupled system, heat capacity and thermal
expansion measurements can be of crucial importance for considering the spin behavior
in not only the macroscopic sense but also the microscopic sense. What makes it
possible is that microscopic properties of a solid are thermodynamically related to theheat capacity and thermal expansion. Heat capacity at constant volume denoted as CV isdefined as the quantity of the internal energy E required to increase the temperature T ofa unit quantity of a substance,
6

Chapter 1. Introduction

𝜕𝐸

𝐶𝑉 = (𝜕𝑇 ) .

(1.1)

𝑉

Here E is related to the partition function Z of system, given as𝜕 ln 𝑍𝐸 = 𝑘𝑇 2 ( 𝜕𝑇 ),

(1.2)

𝐸

𝑍 = ∑𝑖 exp (− 𝑘𝑇𝑖 ),(1.3)

where ∑𝑖 over all levels of the system, and Ei is an energy of the various possible levels
originated from such as lattice vibrations, electric conduction, magnetic moment, andlattice disorder. If the energies individually depend on the respective characteristicenergy scales, i.e., Ei = Elat + Eelec + ···, substituting Eq. (1.3) into Eq. (1.2) suggests thatthe internal energy is decomposed into several energy contributions, and hence the heat
capacity too. Therefore, the appropriate subtraction of a certain contribution in the heat
capacity from the measured heat capacity can provide the information of only theinterested system.
Volumetric thermal expansion coefficient  is defined as
1 𝜕𝑉

𝛽 = 𝑉 (𝜕𝑇 ) .

(1.4)

𝑃

The corresponding quantity for changes in one direction is called as linear thermal
expansion coefficient  written as
1 𝜕𝐿

𝛼 = 𝐿 (𝜕𝑇) ,

(1.5)

𝑃

where L is the length on a side. In the isotropic material, the relation  =3 holds.
The microscopic theory for the thermal expansion of solid was established by Mie
and Grüneisen [21], based on the consideration that the thermal expansion is induced by
a volume dependence of the anharmonicity of the interatomic potential. This idea led tothe particularly important finding that the ratio of the thermal expansion coefficient to7

Chapter 1. Introduction

heat capacity is effectively constant as a function of temperature, so called “Grüneisenparameter ”. It is widely known as the following expression:
𝛾=

𝛽𝐵𝑇 𝑉
𝐶𝑉

𝜕 ln 𝐸 ∗= − 𝜕 ln 𝑉 ,(1.6)

where BT is the isothermal compressibility, and E* is the relevant energy scale. The
original model in Grüneisen's work is based on Einstein model, where the crystal lattice
has only the same vibrational frequency v. In this case, the characteristic energy scale is
the Einstein temperature 𝜃𝐸 = ℎ𝑣/𝑘 , and hence 𝛾 = − 𝜕 ln 𝑣 ⁄𝜕 ln 𝑉 . Similar
discussion can be applied for any model with a single characteristic temperature, forexample, electronic, magnetic, and other non-vibrational contributions. Because thetotal thermal expansion is small in most solids, the ratio is expected to vary only slightly
with changing temperature. Thus, the whole behavior of the thermal expansion
coefficient will generally mimic that of the heat capacity, and typically ~1. One of thenotable exceptions is the case that the system exhibits negative thermal expansion.
Unlike the isothermal compressibility and heat capacity, the thermal expansion
coefficient can be either positive or negative. Equation (1.5) then results in the negativevalue for . Hence, an understanding of the origin of the negative Grüneisen parameter
will contribute to clarify the mechanism of negative thermal expansion. As in this way,assessing the  value often provides a useful insight on exotic properties.
Other important thermodynamic relations include information about the pressure
dependence of the phase transition, which allow a discussion about the phase stability.
Because of the difficulty in physical property measurements at low-temperature andhigh-pressure, the indirect way using the thermodynamic relation is convenient andrather accurate. The pressure effect for the first-order transition can be written as

8Chapter 1. Introduction

𝑑𝑇𝐶
𝑑𝑝

∆𝑉

= ∆𝑆 ,(1.8)

where TC is the transition temperature, and V and S denote the change in volume andentropy at the transition. This relationship is called the Clausius-Clapeyron relation. On
the other hand, the pressure effect for the second-order transition can be written as
𝑑𝑇𝐶
𝑑𝑝

∆𝛽

= 𝑇𝐶 𝑉𝑚 ∆𝐶 ,

(1.9)

𝑝

where Vm is the molar volume, and =3 and Cp are the jump in volume thermal
expansion coefficient and heat capacity at TC, respectively. Equation (1.9) is called the
Ehrenfest equation. These relations enable us to obtain the pressure dependence of thetransition temperature without applying pressure. Applying pressure is effectively
equivalent to isotropic volume change. Thus, if the system has a strong coupling
between lattice and other degrees of freedom, dTc/dp will be a large value.

1.5 Outline of this thesis
The driving force of this doctoral study is to understand the lattice behavior in
pyrochlore-based magnets with the strong spin-lattice coupling using heat capacity andthermal expansion measurements. Assuming the presence of the strong spin-latticecoupling, detailed study on the thermal expansion behavior should provide not only
insights into lattice properties but also deeper understanding of spin properties as seenin the previous section. Most of frustrated compounds are extensively investigatedthrough microscopic studies, e.g., x-ray and neutron scattering experiments. On theother hand, the lattice has been assumed to be a passive bystander. I will demonstratethat the investigated systems, chromium spinels ACr2X4 (A=Cd, Co, and X=O, S) andtitanium pyrochlores R2Ti2O7 (R=Tb, Dy, and Ho), possess the characteristic thermal
9Chapter 1. Introduction

expansion behavior originated definitely from their spin properties.
In chapter 2, the experimental background, basic knowledge and condition are
described. For heat capacity measurements, the thermal relaxation method has mainly
been employed, and the quasi-adiabatic method has been used for the low-temperature
heat capacity of the terbium titanates, Tb2Ti2O7. These two methods are suitable for themeasurement at low-temperature and on a few milligram sample. For thermal expansion
measurements, the capacitance method has been applied. Because of the surprisingly
high relative resolution (l/l~10-10), which is the highest among the dilatometers, tiny
lattice change can be detected. For magnetization measurements, a superconducting
quantum interference device (SQUID) magnetometer from Quantum Design Inc. wasused.
In chapter 3, spin-glass-like behavior in ferromagnetic spinel CdCr2S4 is presented.
This compound has been considered to be a typical three-dimensional Heisenberg
ferromagnet, and hence there is no frustration. However, recent studies show several
interesting behaviors below the ferromagnetic transition temperature, such as
multiferroicity, colossal magnetocapacitive effect, and negative thermal expansion.
These observations suggest the need to reinvestigate the basic properties of theferromagnetic phase. From the systematic thermodynamic investigation, the presence ofthe local lattice distortion will be indicated, and the relevant magnetic structure will be
discussed.
In chapter 4, the spin-lattice coupling effect in strongly geometrically frustrated
spinel CdCr2O4 is studied. This system is reminiscent of ZnCr2O4, as introduced in
section 1.4. Due to the strong frustration arising from the antiferromagnetic interaction
between Cr3+ spins, the system undergoes a magnetic transition with a structural
10Chapter 1. Introduction

transition well below a Curie-Weiss temperature. Interestingly, a spin-liquid-like statewith the low-energy excitations of antiferromagnetic hexagonal spin cluster wasobserved in the paramagnetic phase. This study will show the peculiar thermal
expansion behavior induced by the spin behavior. In addition, a different transition
behavior in CdCr2O4 and ZnCr2O4 will be considered through the pressure dependenceof the transition temperature.
In chapter 5, the lattice behavior of the conical spin state in spinel oxide CoCr2O4
is investigated. In an early study on the spinel magnet, Lyons et al. theoreticallysuggested that any magnetic ground state of CoCr2O4 becomes unstable due to thefrustration between Cr-Cr and Co-Cr exchange interactions. Indeed, a complex
sequence of magnetic transitions is observed. Of particular interest is deserved on aconical spin state, where the multiferroicity is emerged. The multiferroic mechanism has
been attributed to the inverse Dzyaloshinskii-Moriya interaction, where a spiralmagnetic structure induces the reduction of lattice symmetry, resulting in polarization.
Using thermodynamic relations, the stability of each phase, especially for the conicalspin state expected to have strong spin-lattice coupling, will be investigated
In chapter 6, the role of the lattice in spin-liquid material Tb2Ti2O7 is discussed.
The magnetic Tb3+ ion has an antiferromagnetic interaction with a Curie-Weisstemperature CW = -19 K. However, Tb2Ti2O7 does not show any conventional
long-range order down to 20 mK. For clues about the lack of the phase transition, thelow-lying first CEF excited state, which is separated by only 18 K from the ground state
doublet, is considered to play an important role. Additionally, recent studies impose thepresence of the spin-lattice effect, as indicated in several experiments. From the analysisof the heat capacity and thermal expansion coefficient, it will be shown that the thermal
11Chapter 1. Introduction

expansion behavior can be explained by the lattice vibration and CEF excitation, andhence the spin liquid ground state might be established with little relation with aspin-lattice coupling.
In chapter 7, the lattice behavior related to the spin-ice state in Dy2Ti2O7 andHo2Ti2O7 is studied. Due to the strong crystal-field anisotropy, the Dy3+ and Ho3+ ionsbehave as an Ising spin with the cubic <111> anisotropy. The effective interaction
resulted from the balance between the nearest-neighbor exchange interaction anddipolar-dipolar interaction between spins is weak ferromagnetic, indicating a small
positive Curie-Weiss temperature CW ~ 0.5 K and 1.9 K for Dy2Ti2O7 and Ho2Ti2O7,
respectively. This situation induces highly frustrated ground states with “two-in two-out”
spin configuration on each tetrahedron. Because the arrangement of spins is reminiscentof the proton positions in water ice, these compounds are called as spin ice. Fromthermal expansion measurements, an anomalous thermal expansion in Dy2Ti2O7 andHo2Ti2O7 will be shown, and its relationship with a spin ice state will be argued.
Finally, in chapter 8, summary and concluding remarks will be presented.

12Chapter 1. Introduction

Fig. 1.1. (a) Antiferromagnetic Ising spins on the square lattice. No frustration
occurs in this situation. (b) Antiferromagnetic Ising spins on the triangular lattice. Any
one spin cannot take a stable orientation, resulting in the geometrical frustration.

13

Chapter 1. Introduction

Fig. 1.2. J1-J2 model. (a) For J1 > 2J2 > 0. (b) For 2J2 > J1 > 0.

14

Chapter 1. Introduction

Fig. 1.3. Structure of pyrochlore lattice. The network of tetrahedral is formed.

15

Chapter 1. Introduction

Fig. 1.4. (a) The structure of pyrochlore, where the A-sites are eight coordinatedand located within distorted cubes, and the B-sites are six coordinated and are locatedwithin octahedron. (b), (c) The sub-lattice of the A-site and B-site, respectively. Bothsub-lattices form the pyrochlore lattice.

16

Chapter 1. Introduction

Fig. 1.5. (a) The structure of spinel, where the A-sites are four coordinated andlocated within tetrahedron, and the B-sites are six coordinated and are located withinoctahedron. (b) The sub-lattice of the A-site, forming the diamond lattice. (c) The
sub-lattice of the B-site, forming the pyrochlore lattice.

17

Chapter 1. Introduction

References[1] G. H. Wannier, Phys. Rev. 79, 357 (1950).
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[3] P. Chandra, and B. Doucot, Phys. Rev. B 38, 9335(R) (1988).
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Z. Tun, Phys. Rev. Lett. 82, 1012 (1999).
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3463 (1994).
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[15] S Yonezawa, Y Muraoka, Y Matsushita and Z Hiroi, J. Phys.: Cond. Mat. 16, L9(2004).
[16] Introduction to Frustrated Magnetism, edited by C. Lacroix, F. Mila, P. Mendels,
Springer, Berlin (2010).

18

Chapter 1. Introduction[17] S.-H. Lee, C. Broholm, T. H. Kim, W. Ratcliff, II, and S-W. Cheong, Phys. Rev.
Lett. 84, 3718 (2000).
[18] K. Penc, N. Shannon, H. Shiba, Phys. Rev. Lett. 93, 197203 (2004).
[19] I. A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434 (2006).
[20] H. Shinaoka, Y. Tomita, and Y. Motome, Phys. Rev. B 90, 165119 (2014).
[21] G. Mie, Ann. Phys. 11, 657 (1903); E. Grüneisen, Ann. Phys. 26, 393 (1908); E.
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19

Chapter 2. Experimental

Chapter 2
Experimental

2.1

Heat capacity measurement

2.1.1

Introduction

In recent material study, one becomes to investigate a complex system due to its
novel properties for fundamental study and also future application. As scientificdevelopments, numerous types of physical property measurement method are innovated
to investigate them. Especially, microscopic measurements, such as neutron, SR, and
NMR experiments, are considered to be the most powerful way. Nevertheless, heat
capacity has been placed on the most important physical property that should bemeasured before performing other experiments. It is because heat capacity
measurements are principally able to detect any internal energy change in theinvestigation system.
For heat capacity measurements, several methods have been developed. However,a suitable method is limited to some experimental conditions. In the study of novel
materials, one can hardly obtain the large size and large amount sample. The problem on
measuring a small sample is that the heat capacity of a measurement cell obscures thatof the sample, resulting in low accuracy and precision. In addition, such sample usuallyshows interesting properties at low temperature, which makes the situation more
complicated because the low-temperature measurements require a small experimental
cell due to the space limitation in a refrigerator. To overcome these constraints, thermal

20Chapter 2. Experimental

relaxation method is often used. In this study, heat capacity has measured mainly by therelaxation technique using a Quantum Design physical property measurement system(PPMS) for the temperature range from 2 to 300 K, and a homemade calorimeter with a3

He/4He dilution refrigerator (Kelvinox 300, Oxford Instruments plc.) for thetemperature range below 2 K. In addition, quasi-adiabatic method with a 3He/4He
dilution refrigerator

was also attempted for low-temperature heat capacity

measurements on the terbium titanate Tb2Ti2O7.

2.1.2

Thermal relaxation method

Thermal relaxation method was established by Bachmann in 1972 [1]. It allows us
to carry out heat capacity measurements on milligram order samples. Furthermore, themeasuring apparatus of thermal relaxation method can be so small that it will be
installed in most refrigerators with superconducting magnet. Despite these conveniences,
one should beware that the thermal relaxation method cannot measure the heat capacityof a first-order phase transition with latent heat and a non-equilibrium system due to its
measurement principle.
Here the measurement principle is introduced with using the schematic view of themeasurement apparatus. The apparatus consists of the platform with thermometer andheater, and thermal link connecting the platform and the thermal bath. The temperatureof the thermal bath is available for precise control. A measurement sample is attachedon the platform with Apiezon N grease (M&I Materials Ltd.), which exhibits no craze
down to very low temperatures. When the heater power applies to the platform and thenturns off, the applied heat will leak to the thermal bath through the thermal link. The
decaying of temperature, i.e., thermal relaxation, provides the heat capacity of the21Chapter 2. Experimental

sample through a thermal equation. There are two methods for the analysis of arelaxation curve: 1 method and 2 method. The former assume only the relaxation
process through the thermal link, which is able to analyze by a single exponentialfunction. The latter takes into account of an extra relaxation process, so called 2 effect,
which originates from the non-negligible heat capacity of components, the poor thermal
conductance between the sample and the platform, and so on. If there is no 2 effect,
both analysis leads to the same result. The followings provide the mathematical way to
calculate the heat capacity by each method.

2.1.2.1

1 method1 method is based on the following assumptions; i) The heat capacity of a thermal
link can be ignored compared to the heat capacity of a sample. ii) Thermal conductivity
and heat capacity keep a constant in a measurement. iii) The thermal conductivity
between a sample and platform is sufficiently larger than that of a thermal link. iv) The
heat leak from a sample is only a thermal link. A toy model of the measurement cell
including these assumptions is depicted in Fig. 2.1(a), where the sample with a heat
capacity C is attached onto the platform, connecting to the thermal bath with a fixed
temperature T0 by the thermal link with a weak thermal conductivity . Figure 2.2
shows the time dependence of a power P and a sample temperature T when a pulse
electrical power P is applied to the heater.
A part of a heat generated by P increases the internal energy of the system, andanother part flows into the thermal bath through the thermal link. The heat flow can be
written as

22

Chapter 2. Experimental

𝑇𝑑𝑇

𝑃 = ∫𝑇 𝜅(𝑇 ′ )𝑑𝑇′ + 𝐶 𝑑𝑡 ,(2.1.1)0

where T is the sample temperature and t is the elapsed time. When the power is
continuously applied, the system reaches to a steady state, resulting in zero of thesecond term in Eq. (2.1.1). The three assumptions that the sample temperature in thesteady state is T1, the temperature difference between T1 and T0 is negligible, and thetemperature dependence of the C and  undergoes very little change should be held.
Then, the integration of the first term in Eq. (2.1.1) can be replaced by a thermal
conductivity av. And the P is readily derived as
𝑃 = 𝜅av × ∆𝑇,(2.1.2)where ΔT denotes T1－T0.
As the next step, the heater power is turned off. This procedure brings about adecrease in the temperature from T1 to T0, which goes on as the red line in the Fig. 2.2.
The definite integration from t=0 to the arbitrary point in Eq. (2.1.1) with P=0 and T=T1
leads to𝑇(𝑡) − 𝑇0 = ∆𝑇exp(− 𝜅av 𝑡⁄𝐶 ).

(2.1.3)

An exponential decay is generally given as a form exp(-t/), where  is a relaxation time
constant. Hence, by defining𝜏 = 𝐶/𝜅av ,(2.1.4)Eq. (2.3) becomes𝑇(𝑡) − 𝑇0 = Δ𝑇exp(− 𝑡⁄𝜏).

(2.1.5)The slope of the single logarithm plot of Eq. (2.1.5) leads to a characteristic relaxation
time constant. Therefore, by determining the thermal conductivity av and the relaxation
time  from Eqs. (2.1.2) and (2.1.5), we can yield the heat capacity from Eq. (2.1.4).
This is 1 method. It is widely used for the case that the sample and platform have a23

Chapter 2. Experimental

good thermal conductivity and no heat gradient. The following section describes 2
method, which should be employed if a relaxation curve cannot be analyzed by a single
exponential function, that is, if there is 2 effect.

2 method

2.1.2.2

As mentioned above, 2 method analyzes a relaxation curve with taking into
account the second , which is explained by the model illustrated in Fig. 2.1(b). Thismethod allows the presence of a thermal link with a thermal conductivity a between asample with a heat capacity Ca and a platform with a heat capacity Cb. The timedependence of a sample temperature T will be similar with that of 1 method before
turning off the heater, as shown in Fig. 2.2. The relaxation process, however, issignificantly different from 1 method. The temperature decreases rapidly, followed bya slow decay, which is depicted as the blue dashed line in the Fig. 2.2. The thermal
equation for the sample is given as𝑑𝑇𝐶a 𝑑𝑡a = −𝜅a (𝑇a − 𝑇b ),(2.1.6)and for the platform as𝑑𝑇

𝐶b 𝑑𝑡b = −𝜅a (𝑇b − 𝑇a ) − 𝜅b (𝑇a − 𝑇0 ),(2.1.7)

where Ta is the temperature of the sample, Tb is the temperature of the platform, a is thethermal conductivity between the sample and platform, andb is the thermal
conductivity between the platform and thermal bath. Here the temperature dependenceof Ca, Cb, a, and b is regarded as vanishingly small. Using the following equations
𝜏a ≡ 𝐶a ⁄𝜅a , 𝜏b ≡ 𝐶b ⁄𝜅b , 𝜏 ≡ (𝐶a + 𝐶b )⁄𝜅b ,𝜏+𝜏a𝜏1 = (

2) (1 + √1 −4𝜏a 𝜏b(𝜏+𝜏a )2),

(2.1.8)(2.2.9)24

Chapter 2. Experimental

𝜏+𝜏

4𝜏 𝜏

𝜏2 = ( 2 b ) (1 + √1 − (𝜏+𝜏a b)2 ),(2.1.10)b

the Eqs. (2.1.6) and (2.1.7) is solved, and then the relaxation curve is obtained as
𝑃

𝜏−𝜏

𝑏

1

𝑡

𝜏 −𝜏

𝑡

1

1

2

𝑇𝑏 (𝑡) − 𝑇0 = (𝜅0 ) [(𝜏 −𝜏2 ) exp (− 𝜏 ) + (𝜏 1−𝜏 ) exp (− 𝜏 )].
2

2(2.1.11)

By replacing
𝑃

𝜏−𝜏2

𝜅𝑏

𝜏1 −𝜏2𝐴1 = ( 0 ) (

𝑃

𝜏 −𝜏

𝜅𝑏

𝜏1 −𝜏2), 𝐴2 = ( 0 ) ( 1), 𝐴1 + 𝐴2 = Δ𝑇 ,(2.1.12)

we obtain𝐴 𝜏 +𝐴 𝜏

𝐶a + 𝐶b = 𝜅b ( 1𝐴1+𝐴2 2 ).
1

(2.1.13)

2

In addition, Eqs. (2.1.9), (2.1.10), and (2.1.12) derives
𝐴1 + 𝐴2 = 𝑃0 ⁄𝜅b = Δ𝑇,(2.1.14)Substituting Eq. (2.1.14) into Eq. (2.1.13) will result in
𝐶a +𝐶b
𝜅b

𝐴 𝜏

𝐴 𝜏

𝐴 𝜏

1 1
= 𝐴 1+𝐴1 (1 + 𝐴 2+𝐴2 ) ⋍ ∆𝑇
.
1

2

1

(2.1.15)

2

If the platform heat capacity Cb has already known by the preliminarily measurementsof the heat capacity without mounting a sample, we can yield A1 and from thenon-linear fitting to the relaxation curve, leading to the sample heat capacity Ca.
Practically, 2 analysis is the most commonly used.

2.1.3

Quasi-adiabatic method

Quasi-adiabatic, also referred as semi-adiabatic or isoperibol, method is similar to
the adiabatic method [2], which is the most basic method for heat capacity
measurements. The definition of the heat capacity of∆𝑄

𝐶 = lim∆𝑇→0 ∆𝑇(2.1.16)means that under the adiabatic condition the heat capacity is obtained by measuring an
25

Chapter 2. Experimental

increase in the sample temperature T, caused by a supplied heat Q. Adiabatic method
is based on this principle, and it can provide the heat capacity with the highest accuracy.
On the other hand, quasi-adiabatic method allows the sample to exchange a heat withsurroundings, though a smaller heat exchange is absolutely favorable.
In order to consider a change of the sample temperature with applying a heat, it isbetter to start from introducing the schematic model presented in Fig. 2.1(b), which isthe same setup as used in the thermal relaxation method with 2 effect. A sample with aheat capacity Ca is attached onto the platform with a thermal conductivity a, and aplatform with a heat capacity Cb is connected to the thermal bath with a temperature T0
by the thermal link with a weak thermal conductivity b. In comparing with 2 method,
the difference is that the temperature of the thermal bath does not need to be fixed.
Hence one measurement cell can take a different method depending on the measurement
condition and purpose. The model assumes that the thermal conductivity between thesample and platform is much larger than that between the platform and thermal bath,a>>b. When a pulse of electrical power P is applied to the heater, the sample
temperature T will change as Fig. 2.3. Just after turning off the heater power, the sample
temperature can be considered to relax with a linear temperature dependence whena>>b. Thus, the extrapolation of the temperature before and after heating to the
midpoint of the heating period establishes T for the measurement. Substituting the Tand the applied power Q into Eq. (2.1.16), the heat capacity of the sample can beobtained. The great advantage of quasi-adiabatic method is shortening the measurement
time because it can perform heat capacity measurements with changing temperature. By
constructing the measurement cell with an appropriate condition and the measurement
system with FPGA hardware, the measurement time can be further shortened. These
26

Chapter 2. Experimental

attempts result in that a single point measurement only needs several hundred
milliseconds, which should be compared with that of relaxation method of tens ofseconds. The short measurement time enables the heat capacity measurement withsweeping temperature. Thus, heat capacity measurements for wide temperature range
can be accomplished by a significantly short time.

2.2

Thermal expansion measurement

2.2.1

Introduction

Thermal expansion measurement (dilatometry) is the measurement of the change inthe sample length with varying temperature. The close relationship between thermal
expansion and heat capacity suggests that the thermal expansion also relates to theinternal energy of a system. Thus, the knowledge of the thermal expansion is obviously
important for the study on solid state physics as well as that of heat capacity. It will
provide not only lattice property but also other information, such as electronic, magnetic,
and even nuclear properties. Furthermore, the combination of thermal expansion andheat capacity can characterize the phase transition through Clausius-Clapeyron relationor Ehrenfest equation, leading to fruitful insights on the many intriguing statesoriginated found in novel materials.
The measurement method of thermal expansion can be categorized into the absolutemeasurement that directly measure a sample length (or its correspondent), and relative
measurement that utilize a reference material. Typical examples for absolute expansion
measurementareopticalinterferometric27

method,x-ray

diffraction

method,Chapter 2. Experimental

telemicroscope method, and light scan method. Relative expansion measurement isrepresented by push-rod method, strain gauge method, thermo-mechanical analysis, andcapacitance method. Each dilatometry has its own benefits and drawbacks, and hence
one should take an appropriate way depending on the investigation sample andconditions. For the crystalline sample, x-ray diffraction method is the most common
way to obtain the thermal expansion. This method, however, has a difficulty ininvestigating the detailed temperature dependence of the thermal expansion due to theinability to measure with temperature scanning and its relative sensitivity of L/L~10-7.
The problem results in the insufficient information to a minute lattice change in anarrow temperature range, especially for the phase transition.
In contrast, capacitance method can offer very high relative sensitivity ofL/L~10-10, which is a few orders of magnitude better than the sensitivity of not only
x-ray method but of all other methods. Despite its outstanding performance, someexperimental difficulties had turned away the researchers. The earliest capacitancedilatometer with parallel plates, reported by White in 1961 [3], has the disadvantage on
the big size of the dilatometer, which was required to achieve a high resolution. Such
big size cell usually mismatch for the limited space in the most of cryostats with magnet
coil. In addition, thermal gradient in the cell also becomes a critical problem. A lot ofefforts have been devoted to solve these problems with improving its sensitivity.
However, there is another problem on sample handling. The parallel-type capacitancedilatometer needs a sample with strictly parallel surfaces.
The capacitance dilatometer used in this thesis research is designed by Rotter et al.
[4], which is based on the tilted plate principle. This system allows us to use nearly any
irregular shape of samples; only the base surface should be flat to give a stable sample
28

Chapter 2. Experimental

position. And of course, it has most advantages of the existing capacitive dilatometers,
avoiding their disadvantages.

2.2.2

Capacitance method

Capacitance method detects a change of a sample length as a change in the gap
between a pair of capacitor plates. For an ideal parallel-plate capacitor in vacuum,
capacitance C stored between capacitor plates depends on the gap d between and thearea A of the capacitor plates:
𝜀 𝐴

𝐶 = 0𝑑

(2.2.1)

where  is the permittivity. When a sample is placed between the capacitor plates,
varying temperature gives rise to a capacitance change due to thermal expansion. In
other words, the measurement of a capacitance change using capacitor plates with aknown area leads to the thermal expansion of a sample. This is the basic principle ofcapacitance dilatometer.
In the case of the tilted capacitance dilatometer, the situation turns to be a complex
problem. Before moving onto the derivation of the thermal expansion, it is helpful toconfirm the tilted plate construction made of silver. The schematic view of the crosssection of the measurement cell is presented in Fig. 2.4. The cell consists mainly of twoparts: the lower part, composed by the plate holder and the capacitor plate, which in thecenter has a depression to hold a sapphire slice used for the sample mounting space, andthe upper part, composed by the plate holder and the capacitor plate, which has a center
hole to house the sample. They are separated by two needle bearings. Both capacitor

29

Chapter 2. Experimental

plates as well as the sample place are strictly insulated from the holders. The detailed
drawing of the tilted plate capacitance dilatometer is found in Fig. 2 of Ref. [4].
The followings show how to obtain the thermal expansion coefficient from thecapacitance measurement [5]. For this tilted dilatometers, the gap d(T) is numericallycalculated by
2𝜀0𝐶(𝑇) = 𝑑(𝑇) [𝐴o (𝑇)1−√1−𝛾o2
𝛾o2− 𝐴i (𝑇)1−√1−𝛾i2
𝛾i2],(2.2.2)where 0 is the vacuum permittivity, C(T) is the capacitance, Ao and Ai are the area ofthe outer and inner capacitor plate, and  is the quantity determined by the angle of thetilted plates. o and i given as
𝑟

𝑘(𝑇)

𝑟

𝑘(𝑇)

𝛾o = 𝑏o [𝑑(𝑇) − 1], 𝛾i = 𝑏i [𝑑(𝑇) − 1],(2.2.3)where ro is the outer plate radius, ri is the inner plate radius, and b is the distance
between center of capacitor plate and pivot, which are the characteristic constants of theapparatus. k(T) is height of the cone bearing, which is defined as
∆𝑙

(𝑇) −
𝑘(𝑇) = 𝑘(𝑇0 ) + 2𝑑𝑆 [ Ag−Lit𝑙∆𝑙Sapphire−Lit
𝑙

(𝑇)],

(2.2.4)

where k(T0) is height of the cone bearing as T0=300 K determined from thecorresponding capacitance C0 keeping the capacitor plates parallel, dS is the thickness ofsapphire washers (0.8 mm), (lAg-Lit/l)(T) is the thermal expansion of Ag from literature,and (lSapphire-Lit/l)(T) is the thermal expansion of sapphire from literature. The
temperature dependence of the inner plate 𝐴𝑖 = 𝜋𝑟𝑖2 is given by
𝐴i (𝑇) = 𝐴i (𝑇0 ) [1 +∆𝑙Ag−Lit𝑙2(𝑇)] ,(2.2.5)and similarly that of the outer plate 𝐴𝑖 = 𝜋𝑟𝑖2 is given by𝐴o (𝑇) = 𝐴o (𝑇0 ) [1 +∆𝑙Ag−Lit𝑙2(𝑇)] ,(2.2.6)30

Chapter 2. Experimental

where Ai(T0) and Ao(T0) are the initial areas calculated from the apparatus constants.
Because d(T) is included in C(T) [Eq. (2.2.2)] and o and i [Eq. (2.2.3)], it requires a
numerical solution. In order to eliminate all influences of the expansion of silver,
Mehboob has modified the above equations by considering the schematic geometry of
the measurement cell [5]; the gap d(T) and the area of inner and outer capacitor platesAo(T) and Ai(T) are normalized as 𝑑̂ (𝑇), 𝐴̂𝑜 (𝑇), and 𝐴̂𝑖 (𝑇):𝑑̂(𝑇) ≡

𝑑(𝑇)(2.2.7a)∆𝑙Ag−Lit(𝑇)1+𝑙

implying 𝑑(𝑇) ≡ 𝑑̂ (𝑇) [1 +∆𝑙Ag−Lit𝑙(𝑇)].

𝐴̂o (𝑇) ≡𝐴o (𝑇)∆𝑙Ag−Lit(𝑇)
1+𝑙= 𝐴o (𝑇0 ) [1 +𝐴̂i (𝑇) ≡𝐴i (𝑇)∆𝑙Ag−Lit(𝑇)
1+𝑙= 𝐴i (𝑇0 ) [1 +∆𝑙Ag−Lit(𝑇)],(2.2.7b)(𝑇)].

(2.2.7b)𝑙∆𝑙Ag−Lit𝑙Using these new definitions, the thermal expansion the sample is obtained as∆𝑙Sample𝑙(𝑇) =∆𝑑̂𝑆 (𝑇)𝑙S[1 +∆𝑙Ag−lit𝑙(𝑇)] +∆𝑙Ag−Lit𝑙(𝑇),(2.2.8)where ∆𝑑̂𝑆 (𝑇) ≡ 𝑑̂(𝑇) − 𝑑̂ (𝑇0 ).
If a piece of silver is used as a sample, the thermal expansion obtained must agreewith literature data, that is,∆𝑙Sample𝑙(𝑇) =∆𝑙Ag−Lit𝑙(𝑇).

(2.2.9)

Accordingly, Eq. (2.2.13) get converted to∆𝑑̂𝑆 (𝑇) = 0,(2.2.10)indicating that the gap between the capacitor plates exhibits no temperature dependence.
However, in reality, Eq. (2.2.10) does not hold true. Due to mainly the imperfection andpurity of the silver, Ag sample and Ag capacitance plates do not expand or contract withsame ratio. Therefore, the resulting change of the gap is not zero. In order to31

Chapter 2. Experimental

compensate for these effects, the zero signal should be subtracted from the sample
signal through the calculation. Adding the correction term change Eq. (2.2.8) to∆𝑙Sample𝑙(𝑇) = [∆𝑑̂Sample
𝑙S

(𝑇) −

∆𝑑̂Ag−Sample𝑙S(𝑇)] [1 +∆𝑙Ag−lit𝑙(𝑇)] +∆𝑙Ag−Lit𝑙(𝑇),(2.2.11)

where ∆𝑑̂ = 𝑑̂(𝑇) − 𝑑̂(𝑇0 ) , ∆𝑑̂Sample (𝑇) is the measurement of the sample, and∆𝑑̂Ag−Sample (𝑇) is the measurement of an Ag for reference (calibration sample). The
pure value of the thermal expansion of the sample is eventually obtained as the left sideof Eq. (2.2.11).The thermal expansion coefficient of the sample can then be calculatedby differentiating ∆𝑙Sample⁄𝑙 (𝑇). In other words, the thermal expansivity is divided by
the difference between a certain measurement temperature and its next temperature.
In this study, the temperature control has been performed by PPMS as cryostat, thesample temperature has been measured by Cernox resistance thermometer(Model:CX-1050-SD-HT-1.4M, Lake Shore Cryotronics, Inc.) with the resistancebridge (Model 340, Lake Shore Inc.), and the capacitance has been measured by thecapacitancebridge(Model2500A 1kHz

Ultraprecision

Capacity

Bridge,Andeen-Hagerling, Inc.). The measurement is automatically controlled by Labview
program. In a lot of cases, the direct differentiating of the measured thermal expansivity
leads to highly scattered thermal expansion coefficient. Hence, the thermal expansivity
is smoothed by a Savitzky-Golay filter [6], and then differentiating. The results havebeen compared with those of point-by-point differentiation to confirm whether the sharp
features of the transitions were fully retained through the smoothing.

32

Chapter 2. Experimental

2.3

Magnetic Measurements
Magnetic measurements are explicitly important for the magnetic materials toinvestigate their macroscopic magnetic property. The analysis of the magnetic
susceptibility can provide the strength of the spin interaction and the effective momentof the component spins through the Curie-Weiss law. Furthermore, ac susceptibility
measurements, in which an ac magnetic field is applied to a sample, provides valuable
information about magnetization dynamics, which cannot be obtained in dc
measurements.
In this study, magnetic measurements have been carried out by a Quantum Design
magnetic property measurement system (MPMS-5S), using SQUID magnetometer.
SQUID are based on the principle of quantum interference in superconducting loops
containing Josephson junctions. Owing to the quantized state of the superconducting
loop and the non-linear behavior of the Josephson junction, this device can detect a very
low fields as low as 5×10−18 T. The measurement sample enclosed in a gelatin capsule
was fixed on the straw sample holder made of polypropylene, and installed into thesample chamber in both measurements of the temperature and magnetic fielddependence.

33

Chapter 2. Experimental

Fig. 2.1. Schematic model of thermal relaxation method with (a) 1 method and (b) 2
method. (b) is also used to introduce quasi-adiabatic method.

34

Chapter 2. Experimental

Fig. 2.2. Measurement procedure of the thermal relaxation method, showing the timedependence of a power P and a sample temperature T when a pulse electrical power P isapplied to the heater. If there is 2 effect, the thermal relaxation curve cannot be
described by a single exponential function.

35

Chapter 2. Experimental

Fig. 2.3. Measurement procedure of the quasi-adiabatic method, showing the timedependence of a power P and a sample temperature T when a pulse electrical power P isapplied to the heater. If there is 2 effect, the thermal relaxation curve cannot be
described by a single exponential function.

36

Chapter 2. Experimental

Fig. 2.4. Upper: Schematic model of capacitance dilatometer (a) Cone bearing (b)Capacitor plates (c) Silver spacer (d) Electrical shielding of sample space (e) Sample (f)Capacitor plate holder. Lower: Effect of the expansion of the sample, where the yellow
dashed line represents the original sample shape.

37

Chapter 2. Experimental

References[1] R. Bachmann, F. J. DiSalvo, Jr., T. H. Geballe, R. L. Greene, R. E. Howard, C. N.
King, H. C. Kirsch, K. N. Lee, R. E. Schwall, H. U. Thomas, and R. B. Zubeck,Rev. Sci. Instrum. 43, 205 (1972).
[2] G. K. White, Experimental Techniques in Low Temperature Physics, 3rd ed.
Clarendon, Oxford, 1979, Chap. VII.
[3] G. K. White, Cryogenics 1, 151 (1961).
[4] M. Rotter, H. Müller, E. Gratz, M. Doerr, and M. Loewenhaupt, Rev. Sci. Instrum.
69, 2742 (1998).
[5] N. Mehboob, Master Thesis, Vienna University of Technology, (2009).
[6] A. Savitzky and M. J. E. Golay, Anal. Chem. 36, 1627 (1964).

38

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

Chapter 3
Spin-glass-like behavior in
ferromagnetic phase of CdCr2S4

3.1

INTRODUCTION
In the sulfide spinel CdCr2S4, the 90o Cr3+-X2--Cr3+ ferromagnetic superexchangeinteraction becomes dominant, while the direct Cr-Cr exchange becomes considerably
weaker due to increasing the distance between the Cr3+ ions. The dominantferromagnetic superexchange leads to a ferromagnetic ground state (TC=84.5 K). Since
CdCr2S4 was found in 1960s, much attention has been attracted by the mutual coupling
between magnetic and electric properties. The most impressive findings are probablytherelaxorferroelectricity

above

TC,

and

multiferroicity

andcolossalmagnetocapacitance below TC [1, 2]. Although there has been a dispute whether themultiferroic behavior is intrinsic [3, 4], the possibility of the multiferroicity have been
supported by the findings of intrinsic local distortion (off-centering of the Cr ions) [5, 6,
7]. The local distortion leads to a non-centrosymmetric space group, and it provides the
explanation for the mechanism of the multiferroic behavior in CdCr2S4. That distortion
may be related to the observations of an unusual thermal expansion behavior [8] and abroadening of x-ray peaks [9]. Another interesting topic is an anomalous temperaturedependence of heat capacity below 20 K [8], which contradicts with results of powder
inelastic neutron scattering experiments showing a typical spectrum for conventional
Heisenberg ferromagnet [10]. This finding is probably related to sharp drop of the39

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

dielectric constant [1], and the emergence of magnetocrystalline anisotropy [6].
Those reports suggest the need to revisit the basic properties of the ferromagnetic
phase. In addition, the previous heat capacity measurements between 2 K and 300 K
found out two anomalous behaviors [11]. First is the peak of the magnetic heat capacity
around 10 K, at where there has been no report of an anomaly in other physicalproperties. Second is the significant shortfall of the estimated entropy for the
ferromagnetic phase transition. Hence, heat capacity, thermal expansion, and magnetic
measurements were carried out on the single crystal CdCr2S4. The heat capacity is able
to provide the information of the excitation spectrum and the entropy. Especially, theanalysis of low temperature heat capacity let us know what kind of excitations exist as
simple temperature dependences; e.g., T3 for phonons and T3/2 for ferromagnetic spinwaves. The coefficients of these terms provide the stiffness of each degree of freedom.
The thermal expansion is so sensitive for a structural change that some indication of thelocal lattice distortion would be captured. Several magnetization measurements havebeen carried out, though there is no detailed consideration of the magnetization behavior
under low magnetic field and the ac susceptibility measurements in ferromagnetic phase.
This study reports that those measurements show the sign of spin-glass-like behaviors in
CdCr2S4, which has been regarded as a conventional ferromagnet.

3.2

EXPERIMENTAL

Single crystals of CdCr2S4 used for the measurements were from the same batch as
in Ref. [8], which were grown by a chemical vapor transport technique with CrCl3 as
the transport agent. Heat capacity was measured by the relaxation technique with twocalorimeters for different temperature regions: from 0.10 to 2.5 K a homemade40Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

relaxation-type calorimeter with a 3He/4He dilution refrigerator [12], and from 2 to 300
K a Quantum Design PPMS. Thermal expansion measurements were carried out by thecapacitance dilatometer made from silver. a capacitance dilatometer constructed fromsilver. The measurements were performed on the same samples used in the heat capacity
measurements with the direction along the cubic <111> and the length of 1.13 mm. The
magnetization and ac susceptibility measurements were performed on collections of
single crystals over the temperature range 2 to 100 K with a Quantum Design MPMS.
The magnetization was measured on cooling and heating direction up to the magnetic
field of 5 T. The ac susceptibility was measured on heating direction in an ac field
Hac=3.8 Oe with the frequency range 1 to 997 Hz.

3.3

RESULTS AND DISCUSSION

3.3.1 Heat capacity
The heat capacity divided by temperature CpT-1 is shown in Fig. 4.1. The CpT-1 at 0
T showed a sharp peak at TC=84.5 K. In the magnetic field of 7 T, this peak was
broadened and shifted to around 110 K. This magnetic field response is general for a
simple Heisenberg ferromagnet. Each heat capacity curve overlapped only above 250 K,
which means that the spin fluctuation exists up to approximately 250 K. In order toextract the magnetic contribution to the heat capacity, the lattice contribution was
estimated by temperature scaling the heat capacity of the nonmagnetic referencecompound CdSc2S4 by a scaling factor of 1.02 to account for the difference in themolecular weight (solid line in Fig. 4.1). The validity of this estimation is supported
from that the estimated lattice heat capacity curve and the measured heat capacity

41

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

curves are almost in line above 250 K. By subtracting the lattice heat capacity from theheat capacity, the magnetic heat capacity Cm at 0 and 7 T was obtained as presented in
the inset of Fig. 4.1. The interesting thing is the existence of anomalous peaks at around10 and 35 K besides the peak of the ferromagnetic phase transition. However, the small
peak at 35 K did not show the temperature shift or intensity change with applying
magnetic field, which indicates that this peak would be resulted from an apparatus
problem. Actually, the subtraction of the heat capacity at 7 T from 0 T vanishes theanomaly of the Cm at zero field. Hence, this anomaly will be not touched more in thispaper. The peak at 10 K was suppressed by the magnetic field, and, surprisingly, still
remained under 7 T. It suggests the development of some kind of magnetic ordering
around 10 K. However, the spins must be completely ordered in this temperature region,
under the magnetic field of 7 T [1, 10]. Thus, the origin of this anomaly would beconsidered to be not only an unknown magnetic excitation but also the change of lattice
vibrations, suggesting the presence of the spin-lattice coupling. Integrating the CmT-1
gives the ferromagnetic transition entropy S=8.0 J K-1 mol-1 for either case, which is far
from the theoretical value S=2Rln4~23.1 J K-1 mol-1 for a spin system with S=3/2. The
anomalous excitation at low temperature gives only S=1.9 J K-1 mol-1. Accordingly, theshortfall of the transition entropy might also indicate the occurrence of anomalous
lattice vibrations over the wide temperature range.
In order to understand the low-temperature anomaly, the low-temperature heat
capacity was investigated. Figure 4.2 represents the low-temperature heat capacity Cp infull logarithmic scale. The low-temperature Cp under zero magnetic field was
proportional to T1.37, which essentially agrees with the previous report [13]. Although
the magnetization measurements and powder inelastic neutron scattering experiments
42

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

have been verified being a simple Heisenberg ferromagnet [1, 10], this temperaturedependence is obviously different from a conventional ferromagnet behavior Cp = T3/2+ T3, where the  and term corresponds to the contributions from the ferromagnetic
spin waves and the phonons. Based on this fact, it can be assumed that the existence ofan unknown excitation have a T-linear contribution for the Cp besides the ferromagnetic
spin wave and phonon excitations. The T-linear contribution in non-metallic systems is
usually observed in the glass system due to a continuous density of states. For the data
below 2 K, a least square fit with the equation, Cp = T3/2 + T3, was successfully
reproduced the measurement result with the coefficients =15.1 mJ K-2 mol-1, =36.7
mJ K-2.5 mol-1, =0.49 mJ K-4 mol-1, as shown in the inset of Fig. 2.
The value of the  corresponds to the Debye temperature D=303 K. On the other
hand, the analysis of the far-infrared spectroscopy measurements lead to the D=372 K
by using the data of paramagnetic phase [13]. The present result means the softening of
the lattice at low temperature indicating the existence of phonon softening as suggested
in Raman scattering experiments [5].
The value of the  is related to the spin-stiffness constant D through=0.113VmKB2.5/D1.5 [14], where Vm is the molar volume and kB is the Boltzmann
constant. Then, the present result leads to D=19.6 meV Å2. The D can also be evaluatedfrom the equation 𝐷 = (𝑆⁄3) ∑𝑛 𝑧𝑛 𝐽𝑛 𝑎𝑛 2 [15], where S is the quantum number, zn isthe number of the nth nearest neighbor atom, Jn is the exchange coupling constant
between the nth nearest neighbor pairs, and an is the nth nearest neighbor distance. The
exchange constants are available from inelastic neutron scattering experiments: the firstnearest neighbor J1/kB=13.25 K, the third nearest neighbor J3/kB=-0.915 K, and othercontributions are negligible [10]. The other terms, z1=6, a1=3.62 Å, z3=12, and a3=7.24
43

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

Å, are calculated from the crystal structure of the 𝐹𝑑3̅𝑚 space group with the lattice
constant 10.24 Å [16]. These values amount to D=20.1 meV Å2, which is in good
agreement with the values estimated from the heat capacity data. This consistency
convinces the presence of the ferromagnetic spin waves in the ground state, and hence
the existence of an unknown excitation in low-temperature region.
Under 7 T, the Cp below 2 K was well described by only the T3 term with =2.55 mJ
K-4 mol-1. The T3 dependence means the presence of only the phonon excitations due tothe suppression of the ferromagnetic spin wave excitations by the magnetic field. The
value of the coefficient  leads to the Debye temperature D=175 K, which issignificantly smaller than its zero field value. This result suggests that the lattice
vibrations are highly changed by the magnetic field. Although magnetic field isgenerally assumed to have little effect on lattice vibrations, the effect of magnetic fieldon the D has been observed in some compounds [17]. In CdCr2S4, magnetic fieldmight affect the lattice vibration due to strong spin-lattice coupling. Indeed, a large
magnetostriction has been found in the spin-lattice coupled system [18]. Consideringthat the change in magnetic field give rise to the huge change in an anomalous peakaround 10 K, large magnetostriction might be appeared at low temperature.

3.3.2 Thermal expansion
The thermal expansivity L/L0 of CdCr2S4 is shown in Fig. 3(a), which is in good
agreement with the report of the x-ray measurements [19], and essentially same with
Ref. [8]. With decreasing temperature from room temperature, the lattice contracts down
to near TC. However, from there negative thermal expansion appeared, and it kept down
to 2 K. The thermal expansion coefficient obtained from the L/L0 is depicted in Fig.
44

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

3(b). The  shows the large negative peak at the ferromagnetic transition, followed by
the gradual upturn, which tends to converge to zero at T = 0.
The pressure dependence of the ferromagnetic transition temperature dTC/dp was
estimated using the baselines shown in Fig. 4. For the second-order ferromagnetictransition, dTC/dp can be evaluated from the Ehrenfest relation, given as 𝑑𝑇C ⁄𝑑𝑝 =Δ𝛽𝑉𝑚 𝑇C ⁄∆𝐶𝑝 , where Vm = 8.1 × 10−5 m3 mol−1 is the molar volume [20], β = 3α isthe jump in volume thermal expansion coefﬁcient at TC, and Cp is the jump in Cp at TC.
The estimation in Fig. 4 leads to α = −2.2 × 10−6 K−1 and Cp = 5.9 J K−1 mol−1,
resulting in dTC/dp = −7.7 K/GPa, which reasonably agrees with dTC/dp = −5.8 K/GPaof obtained from the magnetic susceptibility measurements under applying pressure[19]. The negative sign of dTC/dp indicates that the system will be unstable with
contracting the lattice. The onset temperature of the negative thermal expansion is fairly
close to TC, which suggests that an unclear interaction competing to ferromagnetic
interaction is developed with forming the ferromagnetic ordering.
Negative thermal expansion generally appears to relieve a frustration in the system.
In particular, if the frustration stems from the magnetic property, negative thermal
expansion implies the presence of spin-lattice coupling. Among chromium chalcogenide
spinels, ZnCr2S4 and ZnCr2Se4 have exhibited negative thermal expansion [21]. Theyshow positive but rather small Curie-Weiss temperature than CdCr2S4, which indicatesthe presence of stronger antiferromagnetic interaction than in CdCr2S4. Hence, bothcompounds have the bond frustration between the superexchange interactions of nearest
neighbor ferromangetic Cr-S(Se)-Cr and next nearest neighbor antiferromagnetic
Cr-S(Se)-Zn-S(Se)-Cr, resulting in the antiferromagnetic state with a complex ordering.
This antiferromagnetic transition is accompanied by the structural transition, which has
45

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

been attributed to the strong spin-lattice coupling to relieve the frustration. On the other
hand, CdCr2Se4 and HgCr2Se4, which have a stronger ferromagnetic interaction thanCdCr2S4, do not exhibit any anomalous properties as noted above [22]. Thus, it isspeculated that the presence of the bond frustration in CdCr2S4 leads to negative thermal
expansion through the spin-lattice coupling. Indeed, recent far-infrared spectroscopy
indicate an evidence of the strong spin-lattice coupling [20]. The strong spin-latticecoupling may be originated from the local lattice distortion. For the nature of the local
lattice distortion, a second-order Jahn-Teller effect has been considered [5]. It should be
noted that the recent theoretical study on the pyrochlore-based systems reveals that the
spin-lattice coupling resulting from a local lattice coupling plays an important role forthe spin-glass behavior [23].

3.3.3 Magnetic properties
The magnetic field and temperature dependence of the magnetization are presented
in Fig. 3(a) and 3(b), respectively. The M-H curves exhibit a typical soft magnetic
behavior with the coercivity field less than 10 Oe in the whole temperature range in the
ferromagnetic phase. At 4.5 K, the magnetization almost saturated with a magnetic field
1 T to 5.8 B/f.u., which is consistent with the expected value 6 B/f.u.. The temperaturedependence of the magnetization M was obtained on cooling (FC) and heating (FCH)
procedure in a field of 30 Oe. Although the overall view of the M is quite similar to theprevious magnetization measurements [1], a temperature hysteresis was found as theinset of Fig. 4.5. With decreasing temperature, the M decreased through a broad
maximum around 40 K. In the normal ferromagnet, the M should increase withdecreasing temperature due to the suppression of thermal fluctuation. It was confirmed
46

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4that the decrease of the M and the hysteresis were gradually vanished with increasing
magnetic field, and could not be observed above 500 Oe. When the magnetic field wasincreased, the maximum magnetization of the FC curve shifted towards lower
temperature, whereas that of the FCH curve stayed around 20 K. These results suggest
the occurrence of an irreversible change of the magnetic state below 20 K.
In order to figure out this unusual magnetic property, ac susceptibility measurements
were performed with frequencies 1 to 997 Hz. The real (’) and imaginary (")components of ac susceptibility are shown in Figs. 4(a) and 4(b), respectively. (The "is standardized to zero by the value at 80 K.) The ’ sharply increased at TC, andgradually decreased below TC, which indicates very soft magnetically. The striking
feature of the ac susceptibilities are the frequency dispersion below 25 K. As theoccurrence of the frequency dispersion in the ’, the ’ decreased and the " increased
from near temperature. Similar behavior of the " was observed in the ac susceptibility
measurements with the frequency at 0.4 Hz for polycrystalline powders [24]. Becausethe " corresponds to the dissipative processes, the present results imply that an
irreversible movement of the spins occurs below 20 K.
The frequency dependence in the " appeared below 100 K might be resulted fromthe short-range magnetic clusters [7]. If the magnetic clusters remain down to the lowest
measurement temperature, some influence will be appeared in the frequency
dependence in the ’ and the increase of the " below 20 K. In addition, the frequency
dependence of the peaks was observed in the ", which follows an Arrhenius law withan activation energy Ea ~ 770 K. The simplest interpretation is that the activation energy
relates to the short range magnetic clusters. In that case, the clusters would consist of
about 60 spins. However, the nature of the peak shift is unclear so far.
47

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

The anomalous excitation was appeared in the magnetic heat capacity below about
20 K, which suggests a correlation to the unusual magnetic behaviors. In addition, either
the T-linear term in the low-temperature heat capacity and the frequency dependence inthe ac susceptibilities look like spin-glass-like behavior. Although the formation of
spin-glass state seems to be able to explain the current results, this hypothesis is deniedby the existence of the ferromagnetic spin wave excitations at low-temperature.
Considering that at low temperature the magnetization saturates relatively weakmagnetic field with showing very small magnetic hysteresis as seen in Fig. 3(a), thespin-glass-likebehavior

might

be

attributed

to

the

realizationofasuperparamagnetic-like state. Superparamagnetism is generally seen in nanoparticle
system, and known to show the frequency dependence in ac susceptibility. Similar
behavior has been observed in spinel oxides where the magnetic ions are placed on the
A-site and the B-site are replaced by the non-magnetic ion [25]. This
superparamagnetism stems from anisotropic magnetic interactions generated bynon-magnetic ion substitution. By analogy, it could be expected that the short-range
magnetic cluster in CdCr2S4 behaves as nanoparticles in ferromagnetic phase.
The superparamagnetic-like property in CdCr2S4 might be induced by a trigonal
distortion of the CrS6 octahedra below 20 K [6]. The trigonal distortion will orient themagnetic moment of a short-range magnetic cluster to a local easy axis. Because theferromagnetic interaction remains down to zero temperature, other clusters also tend toorient to same direction one after the other, and then a larger size of ferromagnetic spincluster could assemble with decreasing thermal fluctuations. This process would create
many cluster size depending on the condition of the local distortion around a cluster.
The strong anisotropy at lower temperature enhances the formation of larger
48

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

ferromagnetic spin cluster, which makes the spins less effective for magnetic field.
Therefore, with decreasing temperature, the magnetization decrease by trapping theferromagnetic spin cluster along several directions, and the existence of many cluster
size cause the frequency dependence behaviors in the ac susceptibility measurements.
The anomalous excitation observed in the zero-field low-temperature heat capacity
could also be explained as a continuous excitation of large size clusters of theferromagnetic spins.

3.4

CONCLUSION

The existence of anomalous excitation in CdCr2S4 at low temperature wasconfirmed by the magnetic heat capacity and the temperature dependence of the
low-temperature heat capacity. The low-temperature heat capacity can be well fitted by
the equation, Cp = T3/2 + T3, where an unknown excitation is included as a
T-linear contribution for the Cp besides the ferromagnetic spin waves and phonons.
Because even under high magnetic field the unexpected peak of the magnetic heat
capacity was observed at low temperature besides the ferromagnetic transition, theanisotropic distortion might be substantially unaffected by magnetic field and induce
low-energy lattice vibrations. The thermal expansion showed negative thermal
expansion behavior below 100 K, which indicates the presence of the magnetic
frustration. The magnetic susceptibilities exhibit the unusual behavior for aconventional ferromagnet corresponding to the anomalies in the heat capacity. The
magnetization decreased below 20 K and showed the hysteresis in the wholetemperature range. The frequency dependencies in the ac susceptibility suggest that two
relaxation processes exist in ferromagnetic phase of CdCr2S4. These results suggest that49

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

the short-range magnetic clusters gather and form larger spin clusters below 20 K. In
order to unveil the unusual properties in low-temperature region, the mesoscopic scalemeasurements, e.g., small-angle neutron and x-ray scattering experiments, would bedesired to be carried out. Furthermore, ac susceptibility measurements at lower
temperature range might also provide useful insights.

50

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

Fig. 4.1. Heat capacity divided by temperature CpT-1 of CdCr2S4 under 0 T (openblue circles) and 7 T (closed red triangles). The dashed black line shows the lattice heat
capacity calculated by scaling Cp of CdSc2S4. The inset displays the magnetic heat
capacity divided by temperature CmT-1 under 0 T (open blue circles) and 7 T (closed red
triangles) obtained by subtracting the lattice heat capacity from the Cp.

51

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

Fig. 4.2. Low-temperature heat capacity Cp of CdCr2S4 under 0 T (open blue
circles) and 7 T (closed red triangles) in full logarithmic scale. The black lines
correspond to 𝐶𝑝 ∝ 𝑇 1.37 for 0 T and 𝐶𝑝 ∝ 𝑇 3 for 7 T, respectively. The inset shows
the heat capacity divided by temperature CpT-1 below 2 K, where the red solid line
represents the best fit of the heat capacity with Cp = T3/2 + T3.

52

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4-

Fig. 4.3. (a) Thermal expansivity L/L0 and (b) Thermal expansion coefficient  of
CdCr2S4.

53

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

Fig. 4.4. (a) Heat capacity Cp (blue open circles) and thermal expansion coefficient of CdCr2S4 of the vicinity of the ferromagnetic transition temperature TC. The redlines are the baselines used to estimate the jumps in Cp and .

54

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

6-1

M (µB f.u. )

5
4

(a)

3

4K
12 K
25 K
40 K
55 K
80 K

2
1

0

0

1

2

3

4

5

H (T)

-1

M (emu mol )

400

300
410

200

(b)
100
80

FC
FCH

60

80

408
0

0

0

20

20

40

60

40

100

T (K)
Fig. 4.5. Magnetization of CdCr2S4 as a function of (a) magnetic field in the
ferromagnetic phase, and (b) temperature obtained on cooling (FC) and warming (FCW)
procedure in a field of 30 Oe. The inset in (b) shows the enlarged magnetization curve
below 80 K.

55

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4-1

' (emu mol )

19.2

19.0
(a)
1 Hz
10 Hz
100 Hz997 Hz

18.8

18.6

0

10

20

30

T (K)

(b)

-1

" (emu mol )

0.15

0.10

1 Hz
10 Hz
100 Hz997 Hz

0.05

0.00

0

20

40

60

80

T (K)

Fig. 4.6. The temperature dependence of (a) the real (’) and (b) imaginary (")components of ac susceptibility with frequencies 1 to 997 Hz. The imaginary part isstandardized to zero by the value at 80 K.
56Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4

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J. Hemberger, P. Lunkenheimer, R. Fichtl, H.-A. Krug von Nidda, V.Tsurkan, and A.
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J. Hemberger, P. Lunkenheimer, R. Fichtl, H.-A. Krug von Nidda, V.Tsurkan, and A.
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H. Göbel, J. Magn. Magn. Mater. 3, 143 (1976).

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K. Mizushima, Master thesis, Tokyo Institute of Technology, (2010).

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17

J. Lopez, O. F. de Lima, P. N. Lisboa-Filho and F. M. Araujo-Moreira, Phys. Rev. B
66, 214402 (2002).

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18

J. Hemberger, H.-A. Krug von Nidda, V. Tsurkan, and A. Loidl, Phys. Rev. Lett. 98,
147203 (2007).

19

V. C. Srivastavat, J. Appl, Phys. 40, 1017 (1969).

20

T. Rudolf, Ch. Kant, F. Mayr, J. Hemberger, V. Tsurkan, and A. Loidl, New J. Phys.
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21

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58

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

Chapter 4
Spin-lattice coupling effect in strongly
geometrically frustrated spinel CdCr2O4

4.1

INTRODUCTION
This chapter focus on the cubic spinel oxide CdCr2O4, in which the tetrahedral A

site is occupied by the non-magnetic ion Cd2+ (𝑒 4 𝑡26 ), and the octahedral B site by the3magnetic ion Cr3+ (𝑡2𝑔
𝑒𝑔0 ). The dominant exchange interaction is an antiferromagnetic

nearest neighbor interaction between Cr3+ ions due to the direct overlap of t2g orbitals.
Due to Cr3+ ions forming the pyrochlore network, CdCr2O4 becomes a strongly
frustrated Heisenberg antiferromagnet. Because of the strong frustration, theparamagnetic phase remains well below the Curie-Weiss temperature |ΘCW | = 70 K. In
addition, a spin-liquid like state with the low-energy excitation of an antiferromagnetic
hexagonal spin cluster has been observed in the paramagnetic phase, as shown in Fig.
4.1 [1, 2]. At 𝑇N = 7.8 K, CdCr2O4 undergoes a a magnetostructural transition from a
cubic paramagnetic phase to a tetragonal antiferromagnetic phase, which is elongated in
the c axis to relieve the frustration. Neutron scattering studies [1, 3] reveal that themagnetic structure of the antiferromagnetic state has an incommensurate wave vector
Q~(0,0.09,1). This long-range spiral ordering has been attributed to the weak
Dzyaloshinskii-Moriya interactions [4].
The magnetostructural transition indicates the presence of the strong spin-latticecoupling in this system. It is also suggested by far-infrared spectroscopy [5, 6, 7]

59

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

because a T1u phonon mode correlated with the modulation of the Cr-Cr distance showsa phonon softening below approximately 130 K. Furthermore, the phonon triplet splits
into singlet and doublet at TN, which indicates a separation of the vibrations along xy, yz,
and zx into two groups, corresponding to the tetragonal distortion [5]. For themechanism of the splitting, Tchernyshyov et al. proposed the model including thespin-lattice coupling [8]. This model first assumes Ising spins fixed along c axis. Due tothe geometrical frustration, spins will align parallelly or antiparallelly to its nearest
neighbors. Then, to reduce the magnetic energy between parallel spins, the elastic
energy to expand the distance between these spins arises. Thus, the nearest neighbor
antiferromagnetic interaction induces the magnetic anisotropy, which related to thephonon mode splitting at the magnetostructural transition. It is noteworthy that a similarmode splitting has been observed in ZnCr2O4, which has the same crystal structure andstronger antiferromagnetic interaction. However, the splitting occurs in a different
manner; the ratio of the intensity between singlet and doublet infrared phonon spectrum
is opposite [9], indicating that at TN the lattice elongates along the c-axis in CdCr2O4 [1],while contracts in ZnCr2O4 [10]. The nature of the different distortion has remained
unclear yet.
In the present study, the effect of the magnetic frustration on the thermal expansion
behavior has been investigated. Although several experiments suggest the presence ofthe spin-lattice coupling in CdCr2O4, the detailed thermal expansion data have not beenreported. The thermal measurements on CdCr2O4 will be especially valuable in figuring
out the lattice behavior in the paramagnetic phase, and how the lattice strains throughthe magnetostructural transition.

60

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

4.2

EXPERIMENTAL
Single crystals of CdCr2O4 were synthesized by a flux method using PbO as theflux [11]. To estimate the lattice heat capacity of CdCr2O4, single crystals of ZnGa2O4
were prepared by a flux method using PbO and B2O3 as the flux [12].
Magnetic susceptibility measurements were carried out using Quantum Design
MPMS. Heat capacity was measured by the relaxation technique using a Quantum
Design physical property measurement system (PPMS) for the temperature range from
2 to 300 K. Thermal expansion measurements were carried out with a capacitancedilatometer constructed from silver [13]. The specimens, which were the same samples
used for the heat capacity measurement were measured along the cubic <111> direction
with the length of 1.10 mm. Thermal expansivity data 𝛥𝐿/𝐿0 (𝛥𝐿 = 𝐿 − 𝐿0 ; L0 refers
to the value at 293 K) were taken on heating direction with a rate of 1 K/min between 2
and 300 K.

4.3

RESULTS AND DISCUSSION

4.3.1 Magnetic Susceptibility
Figure 4.2 shows the magnetic susceptibility  of CdCr2O4 under 1000 Oe, which isin good agreement with the previous reports [9]. The susceptibility curve exhibited abroad maximum around 10 K, followed by a sharp jump due to the magnetostructuraltransition at TN = 7.8 K, as shown in the inset of Fig. 4.2. The broad maximum can be
attributed to the strong spin fluctuation caused by the geometrical frustration. A linear
fit to the experimental data between 150 K and 300 K shown in Fig. 4.2 as the solid
curve, yields a Curie-Weiss temperature CW = −70 K and an effective magnetic
61

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

moment peff = 3.75 B/Cr. The latter value agrees with the theoretical spin-only value ofp = 3.87 B/Cr for Cr3+ resulting from a 3d3 state, which indicates that the spins behaveas the paramagnetic state above 150 K.

4.3.2 Heat capacity
The heat capacity Cp of CdCr2O4 is presented in Fig. 4.3, which shows a very sharp
peak due to the magnetostructural transition at TN = 7.8 K. The result agrees with theprevious experiments, except for a sharper peak [9]. The dashed line in Fig. 4.3 denotes
the estimated lattice heat capacity. The lattice heat capacity was determined from the Cpof ZnGa2O4 with its temperature axis scaled by 1.075 to overlap with the Cp of CdCr2O4
above 150 K. Figure 4.3 shows The calculated magnetic entropy of 24.0 J K-1 mol-1 is ingood agreement with the theoretical value 2Rln4 = 23.1 J K-1 mol-1. The large
contribution of the magnetic heat capacity above TN shows that spin fluctuations exist
up to approximately 150 K. Because the magnetostructural transition is first-order, therelaxation method could underestimate the latent heat at the transition; the validity ofthe present result was confirmed by comparing the transition entropy with the result of
the adiabatic method [14].

4.3.3 Thermal expansion
Fig. 4.4(a) and 4.4(b) present the thermal expansitvity Δ𝐿/𝐿0 and thermal
expansion coefficient 𝛼 of CdCr2O4, respectively. With decreasing temperature fromroom temperature, Δ𝐿/𝐿0 first decreases and then increase below 140 K, which results
in a broad minimum around 140 K. Such behavior, called negative thermal expansion

62

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4(NTE), is occasionally observed in frustrated materials [15, 16]. With further decreasing
temperature, positive thermal expansion returns at 45 K. At TN, there is a large jump inL/L0 and an extremely sharp peak in  showing a large lattice distortion.
The onset temperature of the NTE coincides with the onset of the magnetic
contribution in the Cp. Furthermore, the infrared spectra show softening of the phononmode corresponding to the modulation of the Cr-Cr distance below the same
temperature [5, 6, 7]. These behaviors have been attributed to the spin-lattice coupling[5], and this is also supported by ab initio calculation [17]. Thus, the NTE in CdCr2O4
should be related to the spin-lattice coupling. The mechanism of the NTE can beanalogized from the idea that Tchernyshyov et al.[8] have proposed for the tetragonal
distortion at TN. When an antiferromagnetic couple of spins are established in a single
tetrahedron, two other spins experiences the frustration. To reduce the frustration, thebond distance between the frustrated spins will be expanded. In this temperature range,
the spins thermally fluctuate to a large degree. Thus, a bond expansion is dynamicallyinduced in a tetrahedron rather than making a static structural distortion, and hence thevolume expansion is occurred.
However, positive thermal expansion returns at 45 K. This behavior is in contrast to
NTE observed in other spinel compounds with strong spin-lattice coupling that
continues down to the lowest temperature or at least down to the transition temperature[15, 16]. Moreover, this positive thermal expansion has large positive value of
𝛼 ~ 4.5 × 10−6 K −1 at 10 K, which is close to the value of 𝛼 ~ 5.0 × 10−6 K −1 atroom temperature. It indicates that the NTE may be correlated with a unique property in
CdCr2O4. For CdCr2O4, the formation of antiferromagnetic hexagonal Cr cluster has
been reported [1]. The formation of the hexagonal Cr clusters at low temperature
63

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

implies that the system will be stabilized by contracting the bond distance within theantiferromagnetic interaction to strengthen the hexagonal Cr cluster, which would
induce positive thermal expansion. Then, the results of the thermal expansion
measurements can be interpreted as the effect of the hexagonal Cr cluster overwhelming
the effect of the NTE below 45 K. This idea provides a key to clarifying the formation
mechanism of the hexagonal Cr clusters. Since the phonon mode related to the NTE
continues to soften below 45 K [5, 6, 7], other perturbations may significantly affect the
Cr ions in the hexagonal Cr cluster. Recently, Conlon and Chalker [18] have suggested
that further neighbor interactions beside an antiferromagnetic nearest neighborinteraction are important for the formation of spin cluster. However, because the NTE
expands the distance between the Cr ions, exchange interactions between the Cr ions
may be weaken with the decrease in temperature. It indicates that other perturbations(e.g., quantum fluctuations, Dzyaloshinskii-Moriya interaction, etc.) are also intimately
deeply related to the formation of the hexagonal Cr clusters. Thus, this result suggeststhat other perturbations should be taken into account for future theory of the formation
mechanism of the spin cluster.

4.3.4 Pressure dependence of the transition temperature
The magnetostructural transition at TN induces the large jump in Δ𝐿/𝐿0 , and an
extremely sharp peak appears in 𝛼. The pressure dependence of TN is calculated fromthe Clausius-Clapeyron relation. Taking the baselines as shown in the insets of Figs.
4(a) and 4(b) gives ∆𝑉 = 2.2 × 10−8 m3 mol−1 and Δ𝑆 = 3.2 J K −1 mol−1 , whichlead to 𝑑𝑇N ⁄𝑑𝑝 = 6.9 K⁄GPa . On the other hand, the result of the susceptibility
measurement under pressure shows 𝑑𝑇N ⁄𝑑𝑝~0.7 K⁄GPa, which is about a tenth of the64Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

present result [19].
The validity of the present result is confirmed from the following discussion. Δ𝑉
can be roughly estimated from the change of the lattice constant. In the cubic phase at
10 K > 𝑇N , the reported lattice constant is 8.58882 Å [1, 20]. In the tetragonal phase at
6 K < 𝑇N , no lattice constants have been reported, but Kim et al. [20] have reported thenearest neighbor Cr-Cr distances of 3.034 and 3.040 Å for the ab-plane and along thec-axis, respectively. Assuming that the fractional coordinates of atoms are scarcely
changed across the transition [21], ∆𝑉 is estimated to be ~3 × 10−8 m3 mol−1 , which
is close to the present measurements (∆𝑉 = 2.2 × 10−8 m3 mol−1 ). On the other hand,∆𝑆 cannot be 10 times larger than the estimated value because of the limitation of thetotal entropy change for the transition. These indicate that the discrepancy between thepresent result and that of Ref. [19] is intrinsic.
The dTN/dp of CdCr2O4 derived from thermodynamic relations corresponds to thevalue at zero magnetic field and ambient pressure, in contrast that the susceptibility
measurements in Ref. [19] was carried out under 1 T and at high pressures. Indeed, acomplex change of dTN/dp depending on the magnetic field and pressure is observed in
ZnCr2O4 shown in Fig. 4.5 [22], which has strong spin-lattice coupling and undergoes asimilar magnetostructural transition as in CdCr2O4. In particular, TN of ZnCr2O4 isstrongly affected by magnetic field at low pressures. The difference in TN between
ambient pressure and 0.35 GPa increases several times with increasing magnetic fieldfrom 0 to 1 T. Thus, a similar effect of the magnetic field on TN is expected in CdCr2O4
at low pressures.
However, the magnetic field effect shows the opposite tendency between CdCr2O4
and ZnCr2O4; with increasing the magnetic field, the dTN/dp of CdCr2O4 is reduced,
65

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

while the dTN/dp of ZnCr2O4 is increased. The difference may be related to the opposite
tetragonal distortion of CdCr2O4 with the elongated c-axis [1] compared to ZnCr2O4
with the contracted c-axis [10] at TN. Further research on the detailed effects of themagnetic field and pressure could help to clarify the reason of the opposite tetragonal
distortion.

4.4

CONCLUSION
The effect of the magnetic frustration on the thermal expansion behavior in

CdCr2O4 has been investigated by means of thermal expansion and heat capacity
measurements. Magnetic susceptibility and heat capacity measurements confirmed thatthe antiferromagnetic interaction is developed below 150 K. Below this temperature, theunusual thermal expansion behavior was revealed; with decreasing temperature from150 K, negative thermal expansion behavior was observed down to 45 K, and thenpositive thermal expansion returned without a phase transition. This behavior suggests
the competition between the bond expansion effect from the geometrical frustration andthe bond contraction effect from the hexagonal antiferromagnetic cluster. In addition,
this result provided an idea for that a certain perturbation except exchange interactions
may be needed to form antiferromagnetic hexagonal spin clusters. The pressure
dependence of the magnetostructural transition temperature indicates that the magnetic
field and pressure induce the opposite effect from ZnCr2O4. This difference might berelated to the opposite structural distortion at TN.

66

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

(a)

(b)

Fig. 4.1. Antiferromagnetic spin cluster formed in the spin-liquid state, taken from
Ref. [2].

67Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

Fig. 4.2. Temperature dependence of the bulk susceptibility . The red line shows
the result of the Curie-Weiss fit. The inset shows the in the vicinity of themagnetostructural transition TN.

68

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

Fig. 4.3. Heat capacity Cp of CdCr2O4.The dashed line shows the lattice heat
capacity calculated by scaling of the heat capacity of ZnGa2O4. The inset shows thevicinity of the magnetostructural transition at 𝑇N = 7.8 K. The line is used to obtain theexcess contribution.

69

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

0.0

10 L / L0

-0.1

3

-0.2
-0.3

-0.4

(a)

5

  



10



-0.5

(b)

100

0

4 5 6 7 8 9 10

0

-5
0

100

200

300

T (K)
Fig. 4.4. (a) The temperature dependence of the thermal expansivity 𝛥𝐿/𝐿0 of
CdCr2O4. Negative thermal expansion is seen in the temperature range from 140 K to 45
K. (b) The thermal expansion coefficient 𝛼 of CdCr2O4. The inset in the lower frameshows 𝛼 of the vicinity of the magnetostructural transition at 𝑇N = 7.8 K where theline is used to calculate the excess contribution.

70

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

Fig. 4.5. Magnetic field and pressure dependence of the magnetstructural transition
temperature TN of ZnCr2O4, from Ref. [22].

71Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

References1

J.-H. Chung, M. Matsuda, S.-H. Lee, K. Kakurai, H. Ueda, T. J. Sato, H. Takagi,
K.-P. Hong, and S. Park, Phys. Rev. Lett. 95, 247204 (2005).

2

S.-H. Lee, C. Broholm, W. Ratcliff, G. Gasparovic, Q. Huang, T. H. Kim, and S.-W.
Cheong, Nature (London) 418, 856 (2002).

3

A. B. Sushkov, O. Tchernyshyov, W. Ratcliff, II, S. W. Cheong, and H. D. Drew,
Phys. Rev. Lett. 94, 137202 (2005).

4

R. Valdés Aguilar, A. B. Sushkov, Y. J. Choi, S.-W. Cheong, and H. D. Drew, Phys.
Rev. B 77, 092412 (2008).

5

T. Rudolf, Ch. Kant, F. Mayr, J. Hemberger, V. Tsurkan, and A. Loidl, New J. Phys.
9, 76 (2007).

6

O. Tchernyshyov, R. Moessner, and S. L. Sondhi, et al., Phys. Rev. B 66, 064403(2002).

7

Ch. Kant, J. Deisenhofer, T. Rudolf, F. Mayr, F. Schrettle, A. Loidl, V. Gnezdilov, D.
Wulferding, P. Lemmens, and V. Tsurkan, Phys. Rev. B 80, 214417 (2009).

8

S.-H. Lee, C. Broholm, T. H. Kim, W. Ratcliff, II, and S-W. Cheong, Phys. Rev.
Lett. 84, 3718 (2000).

9M. Matsuda, M. Takeda, M. Nakamura, K. Kakurai, A. Oosawa, E. Lelièvre-Berna,
J.-H. Chung, H. Ueda, H. Takagi, S.-H. Lee, Phys. Rev. B 75, 104415 (2007).

10

G.-W. Chern, C. J. Fennie, and O. Tchernyshyov, Phys. Rev. B 74, 060405(R)(2006).

11

H. A. Dabkowska, J. Cryst. Grow. 54, 607 (1981).

12

Z. Yan, and H. Takei, J. Cryst. Grow. 171, 131 (1997).

13

M. Rotter, H. Müller, E. Gratz, M. Doerr, and M. Loewenhaupt, Rev. Sci. Instrum.
69, 2742 (1998).

14

K. Mizushima, Master Thesis, Tokyo Institute of Technology (2010).

15

M. Tachibana, N. Taira, and H. Kawaji, Solid Stat. Commun. 151, 1776 (2011).

16

J. Hemberger, H.-A. Krug von Nidda, V. Tsurkan, and A. Loidl, Phys. Rev. Lett. 98,
147203 (2007).

17

C. J. Fennie, and K. M. Rabe, Phys. Rev. Lett. 96, 205505 (2006).
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Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel CdCr2O4

18

P. H. Conlon, and J. T. Chalker, Phys. Rev. B 81, 224413 (2010).

19

H. Ueda, and Y. Ueda, Phys. Rev. B 77, 224411 (2008).

20

J.-H. Kim, M. Matsuda, H. Ueda, Y. Ueda, J.-H. Chung, S. Tsutsui, A. Q. R. Baron,and S.-H. Lee, J. Phys. Soc. Jpn. 80, 073603 (2011).

21

S.-H. Lee, H. Takagi, D. Louca, M. Matsuda, S. Ji, H. Ueda, Y. Ueda, T. Katsufuji,
J.-H. Chung, S. Park, S.-W. Cheong, and C. Broholm, J. Phys. Soc. Jpn. 79, 011004(2010).

22

Y. Jo, J.-G. Park, H. C. Kim, W. Ratcliff, II, and S-W. Cheong, Phys. Rev. B 72,
184421 (2005).

73

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

Chapter 5
Lattice behavior in the conicalspin state of CoCr2O4

5.1

INTRODUCTION
This chapter has treated the cubic spinel oxides CoCr2O4, in which the tetrahedral A

site is occupied by the magnetic ion Co2+ (𝑒 4 𝑡23 ) and the octahedral B site by the3magnetic ion Cr3+ (𝑡2𝑔
𝑒𝑔0 ). The dominant exchange interaction is an antiferromagnetic

nearest neighbor interaction between Cr3+ ions due to the direct overlap of t2g orbitals.
As opposed to the previous spinels, there are additional nearest neighbor interactions
between Co2+ ions, and between Co2+ and Cr3+ ions, which lead to a complex
magnetism in CoCr2O4. The ground-state spin structure of spinels was theoreticallyanalyzed by Lyons, et al. in 1962 using the Heisenberg model with the two exchangeinteractions JAB and JBB [1]. The model, called the “LKDM theory”, predicts that the
ground state depends on the parameter u, defined as 𝑢 = 4𝐽𝐵𝐵 𝑆𝐵 /3𝐽𝐴𝐵 𝑆𝐴 , where SA and
SB are the quantum number of the A ion and B ion, respectively. For u<8/9, the A-site
spins align parallel to each other, and they are antiparallel to the B-site spins, resulting
in the Néel state. For 8/9<u<1.3, the ferrimagnetic spiral long-range order arises. Foru<1.3, the magnetic structure will be locally unstable, and a short-range order will beestablished. The neutron scattering experiments have found u=2 [2], indicating that any
magnetic ground state of CoCr2O4 should be unstable.
Indeed, complex magnetic transitions are observed in CoCr2O4 [3]. First, a74

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

ferrimagnetic transition occurs at 𝑇C = 93 K. With further decreasing temperature, a
conical spin state with an incommensurate wave vector Q ~ (0.63, 0.63, 0) is formed at
𝑇S = 26 K, which is accompanied by spontaneous electric polarization. Finally, alock-in transition occurs at 𝑇L ~13 K with a slight change in Q. The magnetic structurein the ground state was determined to be the conical spin structure as shown in Fig. 5.1[3, 4]. The Co spins on the A-site have one magnetic sub-lattice, while the Cr ions on
the B-site make two magnetic sub-lattice. Each of them forms the conical spin structurewith same spiral wave vector Q and different cone angles.
For the conical spin state, some different behaviors have been reported [2, 4, 5, 6,
7]. The strong sample dependent could be attributed to the Cr-site mixing, which is
confirmed by Ref. [8], comparing the samples prepared by the chemical vapor method
and flux method. The sample grown by a chemical vapor method has no site disorderand exhibits the same behavior as Ref. [3]. Flux grown samples show the incorporationof the Co ion into the Cr site. The ferrimagnetic and conical spin transition temperaturesof the samples are decreased and the lock-in transition is faded away. These changes are
qualitatively same with the previous reports.
The current interest in CoCr2O4 has been focused on a “multiferroic” property
originated from the conical spin configuration [3, 7, 8]. The multiferroicicty originallyrefers to the state that possesses more than one ferroic properties, such as
ferromagnetism and ferroelectricity, though it includes the state with a magnetic
ordering and ferroelectricity. Why the multiferroicity is so attractive is that it can realize
the magnetoelectric effect, modulating the magnetization by an electric field and thepolarization by a magnetic field. The magnetoelectric effect was first predicted by Pierre
Curie in 19th century [9], and a century later, multiferroicity has been attracted renewed
75

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

attention due to the discovery of the practicable magnetoelectric effect in the frustrated
magnets TbMnO3 [10], TbMn2O5 [11], and Ni3V2O8 [12]. They are, however,antiferromagnet, resulting in no spontaneous magnetization. Then, CoCr2O4 has been
appeared as the first material with spontaneous magnetization and magnetically-induced
ferroelectricity. The magnetically-induced ferroelectricity below TS can be explained by
the spin-current model [13] or inverse Dzyaloshinskii-Moriya model [14], where aspiral spin configuration induces an electric polarization.
In the present study, the interplay between spin and lattice degrees of freedom on
the CoCr2O4 was investigated by the heat capacity and the thermal expansion
measurements. The mechanism of the multiferroicity is definitely related to both spin
and lattice degrees of freedom. Although CoCr2O4 have been focused on themultiferroic property, detailed thermal expansion data have not been reported. The
thermal expansion will provide the insights on the degree of distortion of the latticethrough the conical spin transition and lock-in transition. Furthermore, the combinationof the thermal expansion and heat capacity data can lead to the useful information in thephase stability.

5.2

EXPERIMENTAL

Single crystals of CoCr2O4 were grown by a chemical vapor transport techniquewith CrCl3 as the transport agent. Magnetic susceptibility measurements were carriedout by Quantum Design MPMS. Heat capacity was measured by the relaxation
technique using a Quantum Design physical property measurement system (PPMS) for
the temperature range from 2 to 300 K. Thermal expansion measurements were carriedout with a capacitance dilatometer constructed from silver [15]. The specimens, which76

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

were the same samples used for the heat capacity measurement were measured along
the cubic <111> direction with the length of 1.05 mm for CoCr2O4. Thermal expansivity
data 𝛥𝐿/𝐿0 (𝛥𝐿 = 𝐿 − 𝐿0 ; L0 refers to the value at 293 K) were taken on heating
direction with a rate of 1 K/min between 2 and 300 K. Magnetocaloric effect (MCE)
measurements were performed in pulsed high magnetic field with the 55-T magnet(duration: 36 ms) at the Institute for Solid State Physics, the University of Tokyo. To
measure accurately the sample temperature, which dynamically varies with changing
magnetic field, an Au-Ge thin film was directly deposited on the sample surface andused as the thermometer.

5.3

RESULTS AND DISCUSSION

5.3.1 Magnetic susceptibility
Figure 5.2(a) shows the magnetization below 120 K under 1000 Oe, which
exhibits the three distinct transition behavior being consistent with Ref. [3, 7]. The
strong upturn at TC=94 K arises from the ferrimagnetic transition. Two jumps in thelower temperature region are attributed to the formation of the conical spin ordering at
TS=26 K and the lock-in transition at TL=13 K. A linear fit to the experimental data
between 150 K and 300 K yields a Curie-Weiss temperature CW = −575 K, and an
effective magnetic moment peff = 7.43 B/f.u. The obtained peff is larger than thetheoretical spin-only value of ps = 6.71 B/f.u. Instead, the value is rather close to thevalue

expectedfortheunquenched

orbitalcontribution

from

Co2+

ion,

𝑝 = √4𝑆(𝑆 + 1) + 𝐿(𝐿 + 1) = 7.55𝜇𝐵 ⁄f . u. This is unexpected for the Co2+ ion in atetrahedral coordination, though the low-lying excited state (𝑒 3 𝑡24 ) could be the origin

77

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

for this difference [16]. The magnetic field dependence of the magnetization ispresented in Fig. 5.2(b), where the spontaneous magnetization MS = 0.07 B/f.u. can be
confirmed. The magnetization value at 0.5 T is about 0.1 B/f.u., which is consistent
with the report of the sample without the site-disorder showing M=0.09 B/f.u. at 0.5 T.
Thus, the sample used in this study has little Cr-site mixing, because the Co mixing
changes the effective moment due to the non-magnetic Co3+ ion at the octahedral site
6(𝑡2𝑔𝑒𝑔0 ).

5.3.2 Heat capacity
The heat capacity Cp of CoCr2O4 is presented in Fig. 5.3, which shows good
agreement with the results of previous studies [3, 7]. The ferrimagnetic transition at
TC=93 K is found as a -like peak. The most striking feature is a very sharp peak at
TS=26 K due to a transition to the conical spin state. The extremely sharp peak may
indicate the first-order nature of this transition. The lock-in transition at TL~13 K showsa small change in Cp, and there is a thermal hysteresis as seen in the inset of Fig. 5.3,
suggesting a first-order transition. It should be noted that there is a possibility of theunderestimation of the latent heat for a first-order transition by the relaxation method.
To exclude the possibility, heat capacity measurements with different thermal pulses
were performed. In each measurement, there is no anomaly in the relaxation curve. The
obtained heat capacity data shows little difference except for the peak temperature andheat capacity values just around the transition temperature at TS and TL. The transition
entropies estimated from these data is the same within 10%.
It is interesting that thermal hysteresis could not be observed for the conical spintransition at TS. In addition, the thermal expansion coefficient discussed later also
78

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

showed a second-order-like peak at TS, as the other experiments show asecond-order-like change at the conical spin transition [3, 6]. If the conical spintransition is actually a second-order transition, the critical behavior of the heat capacity
𝐴±

near the transition can be written by the commonly used function 𝐶 ± = ( 𝛼 ) |𝑡|−𝛼 +𝐵 + 𝐸𝑡, where 𝑡 = (𝑇 − 𝑇𝑆 )/𝑇𝑆 is the reduced temperature and the superscript + (−)
corresponds to t>0 (t<0). However, any fitting parameters could not describe the criticalbehavior using the same critical exponent  for t>0 and t<0, because of the sharp rise on
the high temperature side t>0. Without the consideration of the regular contribution
B+Et, the least-square fit leads to the critical exponents +=1.23 and −=0.57 as shownin Fig. 5.4. These critical exponents are extremely larger than the conventional
universality classes having  = −0.133 or −0.015 for standard XY or Heisenberg models[17]. It is known that the system having the chiral degrees of freedom shows relatively
large critical exponents  = 0.34 and 0.24 for the chiral XY and Heisenberg universality
classes [18], though these values are also much smaller than the present results. A
extremely large value was reported for La0.7Ca0.3MnO3 showing  = 0.93 [19], whichcharacterized by a strong charge-lattice-spin coupling. It may suggest the possibility thatin CoCr2O4 spin degrees of freedom couple to not only lattice but other degrees of
freedom.

5.3.3 Thermal expansion
Figure 5.5 shows the thermal expansion coefficient  of CoCr2O4, which is
obtained from the high-resolution linear thermal expansivity L/L0 data shown in theinset. At TC,  shows a -like peak as in Cp. The  shape of the peak is due to79

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

short-range fluctuations, which is typical for a second-order transition. With further
decrease in temperature, an upturn begins below 50 K, and large peaks are found at TS
and TL. In contrast to the results of Cp, the lock-in transition at TL exhibits a large peak
representing significant lattice distortion.
The upturn below 50 K is considered to be related to the development of spiralshort-range order observed in inelastic neutron scattering experiments [4]. The spiralshort-range order characterizes the conical spin structure below 26 K, which suggeststhat the growth of the short-range order should relate to the conical spin transition at 26
K. The spiral short-range order apparently grows below 50 K. Accordingly, some signof the conical spin transition could be appeared up to 50 K. Indeed, Lawes et al. [20]reported anomalous dielectric behavior around 50 K, which was attributed to thedevelopment of the spiral short-range order. That is because dielectric properties could
be affected by the spiral spin structure breaking spatial inversion symmetry. The spiralspin structure also induces with strong spin-lattice coupling through inverse
Dzyaloshinsky-Moriya interaction. Thus, it is natural to consider that the upturn of thethermal expansion coefficient below 50 K correlates to the development of the spiralshort-range order. This indicates that the effects of the conical spin transition are over awide temperature range. Furthermore, the shape of the transition peak seems to be asecond-order transition rather than a first-order transition.

5.3.4 Pressure dependence of transition temperatures
The pressure dependence of each transition temperature in CoCr2O4 was calculatedusing the baselines shown in Figs. 5.3 and 5.5. For the second-order ferrimagnetictransition at TC, the pressure dependence of the transition temperature dTC/dp can be
80Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

evaluated from the Ehrenfest relation 𝑑𝑇C ⁄𝑑𝑝 = Δ𝛽𝑉𝑚 𝑇C⁄∆𝐶𝑝 , where 𝑉𝑚 = 4.4 ×
10−5 m3 mol−1 is the molar volume [21], ∆𝛽 = 3∆𝛼 is the jump in volume thermal
expansion coefficient at TC, and Δ𝐶𝑝 is the jump in Cp at TC. The result𝑑𝑇C ⁄𝑑𝑝 = 2.6 K⁄GPa , estimated from Δ𝐶𝑝 = 10.5 J K −1 mol−1 and Δ𝛼 = 2.2 ×
10−6 K −1, is in good agreement with 2.5 K/GPa obtained from the magnetic induction
method measured between 0 and 1.15 GPa [22].
For the transition to the conical spin state at TS, dTS/dp can be calculated from theClausius-Clapeyron relation 𝑑𝑇S ⁄𝑑𝑝 = Δ𝑉⁄∆𝑆 with assuming the first-order transition,
where Δ𝑉 and Δ𝑆 denote the change in volume and entropy at the transition. The
integration of the anomalies ∆𝑉 = 𝑉𝑚 ∫ 3∆𝛼(𝑇)𝑑𝑇 and ∆𝑆 = ∫[∆𝐶𝑝 (𝑇)/𝑇]𝑑𝑇 leadsto∆𝑉 = 3.1 × 10−9 m3 mol−1

and

Δ𝑆 = 5.0 J K −1 mol−1 ,whichresultsin𝑑𝑇S ⁄𝑑𝑝 = 0.62 K⁄GPa. The baselines for the transition at TS are estimated by drawing
a line between 18 K to 40 K considered as clearly appearing the effect of the transition
on Cp and  data, which is a method generally used for estimating a baseline of afirst-order transition. Although the baselines for the transition at TS have a little arbitrary,
the validity can be supported from the following reasons. Below 40 K, the correlation
length of the short-range order apparently increases with decreasing error bar [4]. In
addition, the minimum value in Cp/T is approximately 40 K. On the other hand, thecorrelation length below the conical spin transition temperature 26 K increases withdecreasing temperature, and almost saturates at about 15 K. Moreover, the alpha data
between TL < T < TS show a minimum value at 18 K. In the case of the second-ordertransition, the dTS/dp can be evaluated from the Ehrenfest relation. Taking the samebaseline in the case of the first-order gives Δ𝐶𝑝 = 52.8 J K −1 mol−1 and Δ𝛼 = 6.31 ×
10−6 K −1 , which leads to 𝑑𝑇S ⁄𝑑𝑝 = 0.40 K⁄GPa . Both values are quantitatively
81

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

equivalent with each other obtained by assuming first-order and second-order.
For the first-order lock-in transition at TL, dTL/dp was calculated to be𝑑𝑇L ⁄𝑑𝑝 = 80 K⁄GPa

using the Clausius-Clapeyron relation with

∆𝑉 = 1.3 ×

10−9 m3 mol−1 and Δ𝑆 = 1.7 × 10−2 J K −1 mol−1. There is no published data for thepressure dependence of TS and TL.
The positive values for all of the pressure dependence of the transition temperaturein CoCr2O4 indicate that low-temperature phases prefer a smaller volume. The smallvalues of dTC/dp and dTS/dp indicate a weak influence of volume change to the phasestability. On the other hand, the large value of dTL/dp implies that the incommensurate(IC) phase between TL < T < TS is highly sensitive for pressure. Thus, a small difference
in atomic distances due to a slightly different sample quality may be the origin ofconfusion in reports such as various spin structures below TS [2, 4, 5, 6, 7] and thedifferent behavior of the multiferroic property at the lock-in transition [3, 7].
An IC phase is often found in systems with strong competition between exchangeinteractions as seen in CoCr2O4. An IC phase due to strong magnetic competition is also
observed in multiferroic compounds RMn2O5 (R = Ho, Dy, and Tb) [23] and MnWO4[24] , which have strong spin-lattice coupling. With decreasing temperature from the IC
phase, a lock-in transition is generally found. In RMn2O5 and MnWO4, the linear
expansion anomalies at the lock-in transition are highly anisotropic [23, 24]; the axis
with the change of a magnetic modulation shows a large expansion anomaly, and theother axis shows a relatively small expansion anomaly. These behaviors have been
attributed to the strong spin-lattice coupling. From this point of view, the thermal
expansion at the lock-in transition in CoCr2O4 should correspond to the magnetic
modulation characterized by a wave vector Q ~ (q, q, 0) [3], since the large value of82

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

dTL/dp indicates that the spin couples considerably to the lattice. The present result may
imply the possibility of the reduction of lattice symmetry from cubic at the lock-in
transition. Although CoCr2O4 is believed to remain cubic in its ground state [3, 6, 25], asmall signature of mode splittings in the infrared spectra has been reported [25].

5.3.5 Magnetocaloric effects under pulsed magnetic field
The representative MCEs for magnetic field along the [111] and [001] axis are
displayed in Fig. 5.6(a) and 5.6(b), respectively. Both cases showed roughly similar
response for the magnetic field, except for the low temperature and high magnetic fieldregion. The MCE curves at 30 K showed a reversible trace between field-increasing and-decreasing processes, indicating negligible heat exchange to the surroundings and fast
response in the thermometer, which supports the validity of the present results. Around
TS = 26 K, the present measurements could not find any sign of magnetic fielddependence of the conical spin transition up to 55 T, which means an extremely small
dTS/dH. It is interesting that the combination of two Clausius-Clapeyron relations withdTS/dp and dTS/dH leads to a large value of Δ𝑉⁄∆𝑀 , and hence a considerable
magnetostriction is expected at TS.
Further decreasing temperature, a small hysteresis was observed at high magnetic
fields, which could be attributed to the lock-in transition. This transition is of afirst-order, so the trace at zero field exhibited a small discrepancy between the initialand final state. The smallness suggests that a small latent heat is released at the lock-in
transition, which is consistent with the tiny peak in the zero field heat capacity shown in
the inset of Fig. 5.3. At lower temperature, another transition behavior besides thelock-in transition became to be observable at higher fields. The sample temperature was83Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

considerably increased from the conical spin state to the high field phase (HFP). Below
TL, the transition field gradually increased with decreasing temperature. Once entering
the HFP, the HFP continued metastable down to 10 T, and it did not to recover the
ground state at zero field due to the strong exotherm. The phase diagrams obtained for
H||[111] and H||[001] are shown in Figs. 5.7(a) and 5.7(b), respectively.
The large temperature increase at the HFP transition is related to the significant
release of entropy. Because the lattice entropy has almost vanished in this temperature
range, the released entropy should be of magnetic origin. In the heat capacity
measurements, the significant entropy change could be seen in the conical spintransition, which provides a simple idea that the conical spin configuration is broken at
the HFP transition. For the HFP, the results of the ultrasound and magnetization
measurements have been reported [26]. At the HFP transition, a step-like anomaly isobserved in the ultrasound experiments, while the magnetization did not exhibited any
anomaly, which suggests the structural phase transition. With the assumption of alowering of the lattice symmetry in the conical spin state due to the inverse DM
interaction, Ref. [26] speculated on the measurements results that while the longitudinalcomponent of the magnetization keeps a collinear long-range order, the transverse spiralcomponents are broken, leading to the disappearance of the inverse DM interaction andhence a structural distortion into cubic symmetry. Based on this speculation, it is likely
to agree with the present observation seen in Fig. 5.7 that the HFP transition
temperature is lower in applying field along the [001] axis than along the [111] axis,
though a field orientation dependence has not been reported in Ref. [26].
Then, the following question is why the diagram of the HFP has an inverted dome
shape. Generally, if a phase transition has a ferromagnetic character, the application of84

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

magnetic field leads to an increase in the transition temperature, and if a
antiferromagnetic character, a decrease in the transition temperature. Because the top of
the phase dome is located near 10 K, the dominant interaction changes fromferromagnetic in the IC state to antiferromagnetic in the C state. In the MCE curves, thehigher initial temperatures above 10 K resulted the higher temperature at 55 T, whichindicates that a ferromagnetic component drives the HFP transition. On the other hand,all curves converged to the same temperature of about 12 K for the initial temperature
below 10 K, i.e., finally the same magnetic state was established, which suggests that aferromagnetic component almost independent with respect to magnetic field and adeveloped antiferromagnetic component is destroyed with accompanying the
breakdown of the transverse spiral component. Thus, the crossover of the dominantinteraction might be occurred around 10 K. Its nature is considered to be the anisotropic
lattice distortion, as proposed in the section 5.3.4. This crossover would be related to thefinding of the magnetic irreversibility at low temperatures [27], and the short-range
magnetic ordering [7, 6].
It is finally noted that due to the strong exotherm at the HFP transition, it is hard to
study physical properties in the HFP below 12 K. Even if the sample is immersed inliquid helium, a strong rise in the sample temperature (over 10 K) was confirmed.

5.4

CONCLUSION

The effect of the interplay between spin and lattice degrees of freedom in CoCr2O4
has been studied by thermal expansion and heat capacity measurements. The largepressure dependence of the lock-in transition in CoCr2O4 indicates that the spins arestrongly coupled to the lattice in the incommensurate phase. Since the strong spin-lattice85

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

coupling can induce the anisotropic expansion along the magnetic modulation vector,
there may be lattice distortion below the lock-in transition. The magnetocaloric effectmeasurements in pulsed magnetic field up to 55 T reveals the field orientation
dependence for the high field phase, which shows that the lowest transition field is of H
= 30 T || [001] at 10 K. The obtained phase diagram suggests the crossover of thedominant magnetic interaction at low temperatures, which might be resulted from theanisotropic lattice distortion at the lock-in transition. In order to confirm the presence of
the anisotropic lattice distortion, thermal expansion measurements for the [110] and[001] axis should be carried out in future. Furthermore, polarization measurements inpulsed magnetic field will provide a valuable insight for the high field phase because
the spiral configuration might be broken and hence the polarization will be faded away.

86

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

Fig. 5.1. (a) Crystal structure of CoCr2O4, and the electronic configurations of the
A-site Co2+ ion and B-site Cr3+ ion. (b) The schematic view of the conical spin
configuration, where the spiral wave vector propagates along the [110] direction andthe polarization occurs along the [1̅10] axis. From Ref. [3, 4].

87

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

Fig. 5.2. (a) Temperature dependence and (b) field dependence of magnetization ofof CoCr2O4. The black solid line is a guide for the eye.

88Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

0.30

-2

-1

Cp/T (J K mol )-1-1

Cp (J K mol )

50

0.25
0.20
10 11 12 13 14 15
T (K)

0
0

50
T (K)

100

Fig. 5.3. Heat capacity Cp of CoCr2O4. The inset shows the heat capacity in thevicinity of transition temperature at the lock-in transition at 𝑇L = 13 K. The red lines
are used to obtain the jumps and the excess contributions in Cp.

89Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

Fig. 5.4. Critical behavior above and below TS as characterized by linear
fits of heat capacity data on a log-log plot.

90

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

3





  

10

10 L / L0

0

-1

0

100

200

300

T (K)

5

0
0

50
T (K)

100

Fig. 5.5. Thermal expansion coefficient 𝛼 of CoCr2O4 below 120 K. The lines are
the baselines used to obtain the jumps and/or the excess contributions in 𝛼. The inset
shows the linear thermal expansivity 𝛥𝐿/𝐿0 in the entire region.

91

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

Fig. 5.6. Magnetocaloric effects with the magnetic field along (a) the [111] axis and (b)the [001] axis, respectively. Here the representative curves were drawn.

92

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

Fig. 5.7. H-T phase diagram with the magnetic field along (a) the [111] axis and (b) the[001] axis. The open circles and closed circles the denote field-increasing and-decreasing processes.

93Chapter 5. Lattice behavior in the conical spin state of CoCr2O4

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Y. J. Choi, J. Okamoto, D. J. Huang, K. S. Chao, H. J. Lin, C. T. Chen, M. van
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I. Kim, Y. S. Oh, Y. Liu, S. H. Chun, J.-S. Lee, K.-T. Ko, J.-H. Park, J.-H. Chung,and K. H. Kim, Appl. Phys. Lett. 94, 042505 (2009).

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N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha, and S-W. Cheong, Nature(London) 429, 392 (2004).

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G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava, A. Aharony, O.
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H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).

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A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 (2002).

18

H. Kawamura, J. Phys.: Condens. Matter 10, 4707 (1998).

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J. A. Souza, Y.-K. Yu, J. J. Neumeier, H. Terashita, and R. F. Jardim, Phys. Rev. Lett.
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G. Lawes, B. Melot, K. Page, C. Ederer, M. A. Hayward, T. Proffen, and R.
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K. Ohgushi, Y. Okimoto, T. Ogasawara, S. Miyasaka, and Y. Tokura, J. Phys. Soc.
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C. R. dela Cruz, F. Yen, B. Lorenz, M. M. Gospodinov, C. W. Chu, W. Ratcliff, J. W.
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95

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

Chapter 6
Lack of the spin-lattice coupling in
spin liquid state of Tb2Ti2O7

6.1

INTRODUCTION
Tb2Ti2O7 has a strong Ising magnetic anisotropy along the local <111> axes. Tb3+

ions has an antiferromagnetic interaction with a Curie-Weiss temperature CW = -19 K[1, 2, 3], which should induce a transition into a long-range ordered state at TN ~ 1 K [4].
However, Tb2Ti2O7 does not show any conventional long-range order down to 20 mK [2,3]. In addition, the existence of competing reports gives rise to the confusion in thesystematic comprehension for the spin liquid state [5]. To elucidate the nature of theground state, several theoretical models have been proposed [6, 7, 8, 9, 10], though notheory has succeeded in simultaneously accounting for the results of neutron scatteringexperiments and other macroscopic magnetic properties over a decade. Recently,
neutron scattering experiments for single crystalline sample found out the diffuse
magnetic scattering at (1/2, 1/2, 1/2) and the pinch-point-like scattering, suggesting therealization of a quantum spin ice state [11, 12, 13].
Through the investigation of Tb2Ti2O7, it has been considered that the spin-latticecoupling plays a crucial role as manifested by an anomalous temperature dependence ofYoung’s modulus [14, 15], elastic constants [16], giant magnetostriction [17, 18], andanisotropic structural Bragg-peak broadening [19]. Strong spin-lattice coupling in thefrustrated material can lead to an anomalous thermal expansion and finally result in a96Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

phase transition with a structural distortion as reported in the chromium spinel oxides[20, 21]. The spin-lattice effects observed in Tb2Ti2O7 have been attributed to aprecursor of a cooperative Jahn-Teller transition as firstly suggested by Mamsurova [14].
Such transition and even a static distortion have not been observed [22, 23], though
recent x-ray scattering measurements on a single crystalline sample reported dynamic
distortion, resulting in negative thermal expansion below 20 K [19].
In this study, the low-temperature lattice behavior of the pyrochlore titanate,
Tb2Ti2O7, has been investigated by heat capacity and thermal expansion measurements.
The initial motivation is to obtain detailed thermal expansion data of Tb2Ti2O7 at low
temperature using a capacitance dilatometer, because a recent x-ray study on apolycrystalline sample showed only positive thermal expansion [24], which is strikingly
conflict with the previous x-ray result on a single crystalline sample [19]. Our results
show that Tb2Ti2O7 has a normal thermal expansion as interpreted by the lattice
vibrations and the crystal field contributions.

6.2

EXPERIMENTAL
Single crystals of Tb2Ti2O7, which were from the same batch as in Ref. [25], weregrown by a flux method with PbO, PbO2, PbF2, and MoO3 as the flux. Single crystal of
non-magnetic Lu2Ti2O7 was also prepared by the same method. The sample of Lu2Ti2O7
is used to estimate the lattice contribution without magnetic interaction to the heat
capacity and thermal expansion coefficient. The obtained single crystals weretranslucent brown with octahedral shapes, and their crystal structures were confirmed bypowder x-ray diffraction within the space group Fd-3m.
Heat capacity was measured by the relaxation technique with a Quantum Design
97Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

PPMS down to 2 K. In order to investigate the heat capacity of Tb2Ti2O7 below 2 K, theheat capacity measurements by the quasi-adiabatic technique with a home-madecalorimeter in a 3He/4He dilution refrigerator were carried out. Thermal expansion
measurements were carried out with a high-resolution capacitance dilatometer made ofsilver [20, 26]. The measurements were performed on the cubic <111> axis with thelength of 3.03 mm.

6.3

RESULTS AND DISCUSSION

6.3.1 Heat capacity [27, 28, 29, 30, 31, 32, 33, 34, 35, 36] [37]Figure 6.1 shows the heat capacity Cp of Tb2Ti2O7. The two shoulders around 8 K
and 2 K in the Cp have been observed in the previous heat capacity measurements,
though the shapes of the anomalies are considerably different from report to report[27-37]. Most resemble data to our results can be seen in Ref. [30, 32]. The criticaldifference between the reports is the presence or absence of a sharp peak around 0.5 K.
Taniguchi et al. [35] figured out that the strong sample dependence is accounted for the
non-stoichiometry x of the sample Tb2+xTi2−xOy. The value of x changes the ground statefrom a spin liquid state (x<0.05) to an ordered state (x>0.05). Although it is generally
difficult to control the non-stoichiometry of single crystal, Helean et al. [38] reported
that the flux-grown sample has Tb-poor composition (x<0). The negative value of x
suggests that the flux grown sample belongs to the spin liquid regime. A pyrochlore
sample grown with Pb-flux may have a small impurity of lead [38], which mightmodulate the low-temperature magnetism. Nevertheless, our sample would belong to aspin liquid compound, because the Cp did not show any indication for the phase98

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

transition down to 0.7 K [29, 35, 37].
The crystal-electric-field (CEF) in Tb2Ti2O7 affects the low-temperature Cp.
Leaving aside the intriguing possibility of the splitting of the ground state doublet, one
can accept the presence of the low-lying excited CEF doublet at around 18 K [1, 3, 28,39]. When the ground doublet and excited doublet states is separated by an energy ,
the thermal population of the excited state leads to a contribution to the molar heat
capacity, so called Schottky heat capacity with the form
∆ 2

𝐶Sch (𝑇) = 𝑘𝐵 𝑁𝐴 (𝑇)𝑒 −∆⁄𝑇(1+𝑒 −∆⁄𝑇 )

2

,

(1)

where kB is the Boltzmann constant and NA is the Avogadro constant. Because theequation resulted in a poor fit for the experimental data, we considered that the CEF
splitting energy  has a distribution due to the inhomogeneity. This assumption might
be reasonable for the strong sample dependence of the Cp of Tb2Ti2O7. Applying a
Gaussian distribution to  derives a possibility distribution of :
1𝑃(∆′ ) = √2𝜋𝜎2 exp (−(∆′ −∆0 )2
2𝜎2

),

(2)

where 0 is the median of the CEF splitting energy, and  is its variance. The CEF
contribution to the heat capacity is then given by
𝐶CEF (𝑇) = ∫ 𝑃(∆′)𝐶Sch (𝑇, ∆′)𝑑∆′.

(3)

The experimental data were fitted by numerical integration of this equation and thelattice contribution estimated from scaling of the Cp of Lu2Ti2O7. The curve with theparameters 0 = 18 and = 7.6 fits reasonably well above 7 K, as shown in Fig. 6.1. The
slight deviation at higher temperature is due to the higher CEF level at around 120 K.
Further insight on the 2 K anomaly is provided from the magnetic field dependenceof the Cp below 8 K, as presented in Fig. 6.2. It should be noted that the values of heat99

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

capacity of Tb2Ti2O7 obtained by the two calorimeters reasonably agree in thetemperature region from 1.8 K to 2.5 K, but their agreements got worse with increasing
temperature, and the deviation of their value reaches to ~10% above 4 K. The reason of
the deviation has not been figured out at present. Hence, the absolute values of the heat
capacity and entropy will not be discussed.
Applying magnetic fields along the <111> direction have induced noticeable effects
on the shoulder-like peak around 2 K. The peak was enhanced by magnetic field up to 1
T with a little suppression in the Cp below 1 K. With further increasing field, theshoulder shape gradually collapsed and changed the shape into a broad hump. At 17 T,
the hump had faded away and a sharp rise appeared below 1 K, which is attributed to anuclear Schottky anomaly, resulting from the splitting of the energy levels of the 159Tb
nuclear spin (I = 3/2) due to the hyperfine field. As seen in Fig. 6.2, the Cp under highmagnetic fields still had much higher values than the lattice heat capacity, whichindicates that Tb3+ spins has not aligned all along the field direction.
The magnetic field along the [111] axis (threefold axis) breaks down the magnetic
structure of the cubic Fd-3m symmetry and leads to the new magnetic structure of therhombohedral R-3m [40]. The symmetry conversion results in a splitting of one Tb
position in Fd-3m into two crystallographically nonequivalent 3b and 9e Wyckoff
positions in R-3m, which gives rise to symmetry constrains for the Tb magnetic
moments. As a results, the magnetic structure under magnetic field along the [111] axis
is to be the so-called “3-in, 1-out / 1-in 3-out” structure; in a single tetrahedron, one
moment (3b position) points along the field direction [111], and the other three moments(9e position) point toward the center of the tetrahedron or the reverse depending on the
direction of the moment on the 3b position. The three moments are hard to rotate their100

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

orientations against the field direction due to the symmetry constrains. Neutron
diffraction experiments demonstrate that these moments still deviate from the field
direction by about 40o even at 12 T [40]. Thus, the field dependence of the Cp stems
from this spin behavior, and the deviation is arguably not to be resolved yet at 17 T. The
temperature dependence of the Cp under 17 T was estimated by subtracting the nuclear
Schottky contribution, obtained from fitting the upturn in the Cp below 1 K by theequation AT -2, which resulted in Cp ~ T 1.75 below 4 K. At low temperature, the strong
magnetostriction has been observed [17, 18]. However, the lattice entropy would not beremained at low temperature if a considerable lattice distortion is induced by themagnetic field. Thus, the anomalous temperature dependence might be the characteristicfeature of the spin excitation.
Assuming that the Cp linearly decreases down to lowest temperature from the lastmeasurement point, the entropy under various magnetic fields was calculated as shownin Fig. 6.3, where the lattice and nuclear contributions would be partially included. The
entropy under 1 T was higher than that of zero field at 8 K. Even if the Cp under 1 T will
be decreases with the T 3 manner below 0.7 K, this difference could not be overcome.
The contribution of the 2 K anomaly does not persist above 8 K, as seen in the later.
Thus, applying magnetic field of 1 T (~1.5 K) is not able to gain additional entropy
from above 8 K. The low-lying CEF state at 18 K may be affected by the magnetic field.
However, the Cp of the 2 K shoulder had already increased at 0.5 T, while the Cp at 8 K
was not changed. Therefore, the low-lying CEF contribution can also be excluded. The
3b site moment saturates (~6 B) above 1 T, which indicates the unique configuration of
the 3-in, 1-out / 1-in 3-out state [40]. The presence of the saturated 3b site moment
should lead the entropy loss. If the Tb3+ ions under magnetic field still have Ising101

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

character, the entropy loss is to be 0.5Rln2. Thus, the increase in the Cp with applying
magnetic field indicates that the ground state of Tb2Ti2O7 has higher entropy than the
0.5Rln2.
In the pyrochlore analogue of spin ice Dy2Ti2O7, applying magnetic field along the[111] axis induces the kagome ice state, where the “2-in, 1-out / 1-in 2-out” spin
structure emerges in the kagome plane piled in the [111] direction [41]. Further applying
field leads to the 3-in, 1-out / 1-in 3-out state. Due to the formation of such spin state,
the residual entropy in zero field, 1/2Rln(3/2), decreases by 60% in the 2-in, 1-out / 1-in
2-out state, and by 75% in the 3-in, 1-out / 1-in 3-out state. Note that no residual
entropy is theoretically expected for the latter state because the spin configuration in thetetrahedron is of the unique 1-in, 3-out state. However, as shown in the above, theentropy of Tb2Ti2O7 was nearly constant up to 1 T field that is expected to induce the3-in, 1-out / 1-in 3-out state. Furthermore, the magnetization plateau, which isconsidered to be the evidence for the quantum kagome ice state, has not been observed
below 1 T. Accordingly, the spins continuously modify the dynamical correlation withincreasing magnetic field, and the moments at the 9e site might be correlated as strong
as in the zero field state even at 1 T.

6.3.2 Thermal expansion
Figure 6.4 provides the thermal expansion coefficient  of Tb2Ti2O7, where only
positive thermal expansion was observed down to 2 K. Previous x-ray experiments havebeen provided conflict reports for the low-temperature thermal expansion that a single
crystalline sample exhibited negative thermal expansion below 20 K [19], while apowder sample showed only normal positive thermal expansion down to 4 K [24]. Our
102

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

results with the single crystalline sample well agree with the powder x-ray results. Notethat the Cp of the polycrystalline sample used in Ref. [24] showed a clear peak at 0.5 K,
suggesting that the ground state differs from our sample.
Unlike the Cp, the  shows only one shoulder around 7 K, which would be
attributed to the effect of the low-lying CEF doublet at around 18 K. Then, we estimatedthe CEF contribution to the thermal expansion coefficient CEF. According to theMaxwell’s relations, we can obtain the relationship between heat capacity and thermal
expansion as
𝛼(𝑇)
𝐶(𝑇)

1 𝜕 ln 𝐸

= 3𝑉

𝑚

𝜕𝑝

,

(4)

where Vm denotes the molar volume, and E is the energy for a relevant system. Thus, the'CEF can be expressed as
1 𝜕 ln ∆𝛼CEF (𝑇) = 3𝑉𝑚𝜕𝑝

𝐶CEF (𝑇).

(5)

When Eq. (5) was used to fit the experimental data, we took the green line in Fig. 6.1 as
CCEF(T), and ∂ ln ∆ /𝜕𝑝 was used as a fitting parameter. The lattice contribution was
estimated from scaling of the  of Lu2Ti2O7. The fit with ∂ ln ∆ /𝜕𝑝=0.031/GPa
adequately reproduces the  for the whole temperature range below 20 K. Therefore, it
indicates that the origin of the Cp anomaly below 7 K has little influence on the thermal
expansion.
The combination of ∂ ln ∆ /𝜕𝑝 and bulk modulus 𝐵 = −𝑉 ∂𝑝/𝜕𝑉 derives thevolume dependence of the CEF splitting energy 𝛾CEF = − ∂ ln ∆ /𝜕 ln 𝑉 , called
Grüneisen parameter. Taking the bulk modulus of 46 GPa at 4.2 K [16], our result leads
to CEF ~1.4. The CEF can also be roughly estimated from the change in  with a lattice
constant change. From Tb2Ti2O7 to Tb2Sn2O7, the lattice constant a increases about103

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

2.7% [3, 42], resulting in the decrease of  about 9.2%,[IM07] i.e., ∆∝ ~𝑎−3.6 .
Because V = a3, we have 𝛾CEF = −(1/3)(∂ ln ∆ /𝜕 ln 𝑎) = 1.2, being consistent with
our estimation. The magnitude of the evaluated CEF is comparable to CEF ~ 5/3, whichstems from the point charge model for the cubic CEF field [43]. Similar discussion canbe applied for the lattice contribution to the Cp and . For them, Eq. (5) can be rewrittenas
1 𝜕 ln 𝛩D

𝛼lat (𝑇) = 3𝑉𝑚𝜕𝑝𝐶lat (𝑇),(6)

whereD is the Debye temperature. Using the black lines in Figs. 6.1 and 6.4 as the Clatand lat, we obtain ∂ ln 𝛩D /𝜕𝑝=0.030/GPa, which results in lat ~ 1.4. The lattice
Grüneisen parameter is typically of the order of unity for many solids. Thus, theestimated CEF and lat are both moderate, indicating that a considerable coupling
between CEF and lattice as suggested by the previous measurements seems to be absent
in the thermal expansion.

6.3.3 Discussion about low temperature anomaly at 2 K
The correspondence of the CEF contribution in the Cp and  allows us to yield theclear component of the low-temperature Cp anomaly below 7 K, which has been debated[28, 32, 44]. The subtraction of the CEF and lattice contribution from the Cp gives theexcess heat capacity Cex shown in Fig. 6.5 (Blue circles). It is worth noting thatmultilevel Schottky models could not adequately describe the Cex. Below 1 K, the Cex is
proportional to T. The T-linear dependence of the low-temperature heat capacity has
been observed in some spin liquid materials [45]. Assuming that the Cex linearlydecreases down to lowest temperature, we evaluated the excess entropy Sex by104

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

integrating Cex/T, which leads to the total excess entropy Sex~7.5 J/Kmol. Although the
splitting of the ground state doublet into two singlets gives rise to the entropy of 2Rln(2),the Sex is about 65% of 2Rln(2). Additionally, further decrease in the Sex is expectedfrom the presence of the nuclear contribution, as seen in Fig. 6.2. The shortfall of theentropy is still larger compared with the spin-ice entropy S=2R[ln2-1/2Rln(3/2)]=8.16
J/Kmol. Similar shortfall of the entropy can be observed in Pr2Sn2O7, a candidate forquantum spin ice materials [46]. Naively, the Sex implies the quantum spin state assuggested in several theories for the ground state of Tb2Ti2O7, or at least reflects astrong dynamic nature of the ground state.
The thermodynamic equations lead to further interesting insight on the relevantenergy scale of the low temperature anomaly below 7 K. It is obvious that equation (4)results in ∂ ln 𝐸 /𝜕𝑝 → 0 with 𝑇 → 0, and hence ∂ ln 𝐸 /𝜕𝑉 → 0 with 𝑇 → 0, becausethe vanishes at low temperature and the Cp remains a significant value. These
convergences indicate that the low-temperature Cp anomaly is independent from apressure and volume. Therefore, the ground state without a long range ordering might
be established with little relation to the spin-lattice coupling. We speculate that somesamples possibly show a different thermal expansion from our results and Ref. [24]. It is
known that the Grüneisen parameter diverges at a quantum critical point, where thecharacteristic energy scale vanishes [47]. This divergence corresponds to the divergenceof the thermal expansion coefficient with decreasing temperature. Hence, a sample
located near the quantum critical point between a spin liquid state and an ordered state
will exhibit an anomalous thermal expansion. We expect that thermal expansion
measurements on well-characterized samples leads to more systematic comprehension
on the spin-lattice coupling, and even the enigmatic ground state in Tb2Ti2O7.
105

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

6.4

CONCLUSION

The present chapter has focused on the lattice behavior in the spin liquid state of the
pyrochlore titanate Tb2Ti2O7. Spin liquid compound, Tb2Ti2O7, showed theshoulder-like anomaly around 8 K in both heat capacity and thermal expansion
coefficient, and the anomalous contribution below 7 K could be observed only in theheat capacity. The 8 K anomaly in both heat capacity and thermal expansion coefficient
can be accounted for the Shottky effect, which is resulted from the presence of the
low-lying first excited CEF state at 18 K for which a Gaussian distribution due to theinhomogeneity was applied. The Grüneisen parameters of the lattice and CEF were thenormal value. In addition, the characteristic energy scale of the heat capacity anomaly
below 7 K is independent from a pressure and volume. These results indicate theabsence of a spin-lattice effect, and the spin-lattice coupling is not need to establish a
non-ordered ground state. The combination of our heat capacity and thermal expansion
data provided the pure component for the heat capacity anomaly below 7 K. The excess
heat capacity led to the excess entropy Sex~7.5 J/Kmol, which is rather small comparedto the expected entropy for the ground CEF doublet splitting S=2Rln(2)=11.5 J/Kmol.
The large shortfall in entropy indicates the strong fluctuation in the ground state, andpossibly relates with a realization of a quantum spin ice state.

106Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

Fig. 6.1. Heat capacity Cp of Tb2Ti2O7. The CEF contribution estimated from =18
K with a Gaussian distribution (GD) is indicated by the green line, while that without a
GD by the purple dashed line. The lattice contribution estimated by scaling the Cp of
non-magnetic Lu2Ti2O7 is shown as the black line. Above 10 K, the total heat capacity
obtained by the summation of the CEF and lattice contribution is in good agreementwith the experimental data, though the lower temperature data considerably depart fromthe estimated heat capacity.
107Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

Fig. 6.2. Magnetic field dependence of the heat capacity Cp of Tb2Ti2O7. The redline indicates the summation of the lattice contribution being the same with that in Fig.
6.1 and the nuclear contribution, which was estimated by fitting the Cp at 17 T using theequation AT-2 with A = 0.38 J K mol-1.

108Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

Fig. 6.3. Entropy S under various magnetic fields of Tb2Ti2O7, which were obtainedby integrating the Cp/T without any subtraction from Cp.

109

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

Fig. 6.4. Thermal expansion coefficient  of Tb2Ti2O7. The CEF contribution
indicated by the green line is obtained from scaling the CEF heat capacity (see text),
while the lattice contribution was estimated from the  of Lu2Ti2O7 as in the samemanner in the Cp. The total thermal expansion coefficient fairly describes theexperimental data, which is totally different from the situation in the heat capacity.

110

4

10

2

5

0

0
0

2

4

6

8

Sex (J K-1 mol-1)

CexT-1 (J K-2 mol-1)

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

10

T (K)
Fig. 6.5. Excess heat capacity divided by temperature CexT-1 (blue open circles) andexcess entropy Sex (red closed circles) of Tb2Ti2O7. The Cex showed T-linear temperaturedependence below 1 K. This curve cannot be reproduced by a multilevel Schottky
model. The Sex reached to ~7.5 J K-1 mol-1, which is just 2/3 of Rln(2), expected for the
ground doublet splitting.

111

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7

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114

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

Chapter 7
Thermal expansion behavior in spin ice
state of Dy2Ti2O7 and Ho2Ti2O7

7.1

INTRODUCTION
In Dy2Ti2O7 and Ho2Ti2O7, the strong crystal-field anisotropy makes the rare-earth

ion an Ising spin. Because the local Ising spins align with the cubic <111> direction, thespins on the pyrochlore lattice can only point in or out of each tetrahedron. The effective
interaction resulted from the balance between the nearest-neighbor exchange interaction
and dipole-dipole interaction is weak ferromagnetic, indicating a small positive
Curie-Weiss temperature CW ~ 0.5 K and 1.9 K for Dy2Ti2O7 and Ho2Ti2O7,
respectively [1]. In the case of antiferromagnetic Ising spins on a pyrochlore lattice, theground state will form the “all-in all-out” structure. In contrast, ferromagnetic ones
induce highly frustrated ground states with “two-in two-out” spin configuration on eachtetrahedron. Because the situation reminds us of the proton positions in water ice, as canbe seen in Fig. 7.1, the spin state in Dy2Ti2O7 and Ho2Ti2O7 are called as “spin ice” state.
In addition, both compounds show a zero-point entropy 1/2Rln(3/2) [1, 2, 3, 4], which isconsistent with Pauling’s entropy of water ice.
In this study, the thermal expansion behavior in the spin ice state of pyrochlore
titanates, Dy2Ti2O7 and Ho2Ti2O, investigated by heat capacity and thermal expansion
measurements. Less attention has been paid on the lattice behavior of Dy2Ti2O7 andHo2Ti2O7. It is speculated that the lattice will show an anomalous behavior in response115

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

to the spin ice state. Dy2Ti2O7 and Ho2Ti2O7 exhibit an anomalous thermal expansion
corresponding to the formation of the spin ice state.

7.1

EXPERIMENTAL
Single crystals of Dy2Ti2O7 and Ho2Ti2O7, which were from the same batch as inRef. [5], were grown by a flux method with PbO, PbO2, PbF2, and MoO3 as the flux.
Single crystal of non-magnetic Lu2Ti2O7 was also prepared by the same method. The
sample of Lu2Ti2O7 is used to estimate the lattice contribution without magnetic
interaction to the heat capacity and thermal expansion coefficient. The obtained single
crystals were translucent brown with octahedral shapes, and their crystal structures wereconfirmed by powder x-ray diffraction within the space group Fd-3m.
Heat capacity was measured by the relaxation technique with a Quantum Design
PPMS down to 2 K. Thermal expansion measurements were carried out with ahigh-resolution capacitance dilatometer made of silver [6, 7]. The measurements were
performed on the cubic <111> axis with the length of 1.71 mm, and 1.73 mm for
Dy2Ti2O7, and Ho2Ti2O7, respectively.

7.3

RESULTS AND DISCUSSION

7.3.1 Heat capacity
The Cp of Dy2Ti2O7 and Ho2Ti2O7 is shown in Fig. 7.2, which are in good
agreement with the previous studies [1, 4]. Ho ion has a large nuclear moment (I=7/2),
and hence a nuclear hyperfine effect becomes considerable below 10 K. Employing thesame manner in Ref. [1], the nuclear Schottky contribution from a splitting of the eight116Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

nuclear levels with a level spacing of 0.3 K was estimated. The measured Cp of
Ho2Ti2O7 and estimated nuclear Schottky heat capacity Cnuc of Ho are shown in theinset of Fig. 7.2. The heat capacity subtracted the Cnuc from the Cp of Ho2Ti2O7 isplotted in the main part of Fig. 7.2, and it exhibits the similar behavior with Dy2Ti2O7.

7.3.2 Thermal expansion
Figure 7.3 presents the  of Dy2Ti2O7 and Ho2Ti2O7. With decreasing temperature,both of the  decreases without any anomaly down to 10 K. It is pointed out that ourdilatometer with the high relative resolution l/l~10-10 could not find any evidence of a
local distortion around 80 K, which is considered to give rise to the anomaly in thetemperature dependencies of the elastic constants [8], permittivities [9], and Raman
spectra [10]. The anomalous behavior of the  was found for both compounds below 10
K. Interestingly, they exhibited the opposite thermal expansion; the  of Dy2Ti2O7 is
negative, while that of Ho2Ti2O7 is positive.

7.3.3 Magnetic contributions for Cp and 
In order to extract the magnetic contribution from the Cp, the normal latticecontribution was estimated by the temperature scaling of the Cp of non-magnetic
reference compound Lu2Ti2O7 (black solid and red dashed line in Fig. 7.2 for Dy2Ti2O7
and Ho2Ti2O7, respectively). The subtraction of the lattice heat capacity gives themagnetic heat capacity Cm shown in Fig. 7.4(a). In the same manner, the magnetic
thermal expansion coefficient m was estimated from the subtraction of the latticecontribution obtained from scaling the  of Lu2Ti2O7, which is presented in Fig. 7.4(b).

117

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

Both compounds showed similar behavior except the opposite sign in the m. The broad
upturn of the Cm is resulted from the formation of the spin ice state, which has been
evidenced from that the magnetic entropy obtained by integrating Cm/T shows a missing
entropy close to Pauling’s prediction for water ice [4]. On the other hand, the mappeared at a lower temperature than the Cm. Near the onset temperature of the m, themagnetic correlation length develops [11], and the spin-relaxation time slows down, i.e.,
spin freezing [12, 13]. Thus, the m is probably induced by the spin ice correlation.
Similar temperature dependence can be seen in the magnetic thermal conductivity [14].
Although the anomalous thermal conductivity was attributed to the dynamics ofmagnetic monopole, our results suggest the need to account a contribution from thelattice dynamics originated from the magnetic excitation. It should be noted that thetime scale of our thermal expansion measurements (heating rate: 0.3 K/min) is arguably
enough to capture this slow dynamics (at 2 K,  ~ 10-3 s for Dy2Ti2O7 [12], and 10-8 s
for Ho2Ti2O7 [13], respectively).

7.3.4 Lattice behavior in spin ice state
In Dy2Ti2O7 and Ho2Ti2O7, the low-temperature magnetic properties stem from thenearest neighbor exchange interaction Jnn and dipole-dipole interaction Dnn. The dipolar
spin ice model proposed by den Hertog and Gingras [15] indicates that the magnetic
state depends on the ratio of the nearest-neighbor exchange and dipole-dipole
interactions, Jnn/Dnn, where the spin-ice phase appears for Jnn/Dnn > -0.91, while theantiferromagnetic phase with all-in/all-out spin configuration emerges for Jnn/Dnn <-0.91. Based on this theory, the heat capacity peak height and temperature at which it
occurs provide Jnn/Dnn as -0.53 for Dy2Ti2O7 [15], and Ho2Ti2O7 [1], respectively.
118

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

Hence, the phase stability of spin-ice state in Dy2Ti2O7 is lower than in Ho2Ti2O7. Thisdifference in the strength of the exchange interactions is considered to be a reason forthe different thermal expansion. As opposed to Tb2Ti2O7, the crystal field effect isnegligible at low temperature because of the large separation between the ground andlowest excited crystal field state with the energy separation of about 380 K for Dy2Ti2O7
and ~230 K for Ho2Ti2O7 [16].
Further insight can be obtained from the study of chemical pressure effects on themagnetic properties through the substitution of the B-site Ti4+ ion with Ge4+ ion, as
shown in Fig. 7.5 [17]. Dy2Ge2O7 and Ho2Ge2O7 with pyrochlore structure also show
the spin-ice state at low temperature. Due to the smaller ionic radius of Ge4+ ion than
Ti4+ ion, both Dy2Ge2O7 and Ho2Ge2O7 have the contracted lattice constant comparedwith Dy2Ti2O7 and Ho2Ti2O7. The dipole-dipole interaction in the pyrochlore lattice isinversely proportional to the cube of the distance between the nearest neighbor magnetic
ions, while the exchange interaction is generally more sensitive to the distance change.
Indeed, Dy2Ge2O7 and Ho2Ge2O7 exhibit the larger negative Jnn/Dnn than Dy2Ti2O7 andHo2Ti2O7, meaning that the ratio Jnn/Dnn approaches to the phase boundary between spin
ice and antiferromagnetic phases. It suggests that the lattice contraction with decreasing
temperature destabilizes the spin ice state. Thus, negative thermal expansion may beable to stabilize a spin ice state by weakening the exchange interaction.
The pressure dependence of the magnetic energy scale of the spin ice state∂ ln 𝐸𝑚 /𝜕𝑝 can also lead to useful information for the phase stability, which isevaluated from the following equation:𝛼(𝑇)1 𝜕 ln 𝐸m

= 3𝑉
𝐶(𝑇)

𝑚

𝜕𝑝

,

where Vm denotes the molar volume, Em is the energy for a relevant system, and the Cm
119

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

and m were used for (T) and C(T), respectively. Here Em is considered to correspondto Jnn/Dnn. With decreasing temperature, the value ∂ ln 𝐸m /𝜕𝑝 for Dy2Ti2O7 andHo2Ti2O7 gradually increases with the opposite sign as shown in Fig. 7.4(c). At 2 K,
their values reach to -0.034/GPa and 0.009/GPa for Dy2Ti2O7 and Ho2Ti2O7,
respectively. The negative value of ∂ ln 𝐸m /𝜕𝑝 means that Jnn/Dnn will shift tonegative side by applying pressure. This effect is consistent with the change in Jnn/Dnnfrom Dy2Ti2O7 to Dy2Ge2O7, which is equivalent to applying a pressure to Dy2Ti2O7.
However, Ho2Ti2O7 shows a small but positive value of ∂ ln 𝐸m /𝜕𝑝, as opposed to thedecrease in Jnn/Dnn from Ho2Ti2O7 to Ho2Ge2O7. The absolute value of ∂ ln 𝐸m /𝜕𝑝 for
Ho2Ti2O7 is rather small than that for Dy2Ti2O7. Because the peak temperature of theheat capacity due to the formation of the spin ice state is rather close to 2 K in Ho2Ti2O7
than in Dy2Ti2O7, Ho2Ti2O7 is expected to possess stronger coupling between spin andlattice degrees of freedom at 2 K, resulting in higher ∂ ln 𝐸m /𝜕𝑝. Hence, there might
be some kind of contribution for the thermal expansion coefficient.
Then, two possibilities could be proposed for the absence of negative thermal
expansion in Ho2Ti2O7. The first hypothesis is that the nuclear hyperfine effect obscuresa negative thermal expansion. A nuclear contribution to the thermal expansion has beenreported for some rear-earth compounds [18, 19, 20]. Hirschkoff and Wolcott [21]suggested that the nuclear thermal expansion can be emerged at low temperature as in
the heat capacity. They estimated the nuclear contribution for the thermal expansion
coefficient of holmium metal, nuc ~ 0.6 × 10-6 K-1 at 2 K. The nuclear hyperfine
coupling constant of holmium metal a’~0.3 K is nearly equal to with that of Ho2Ti2O7,
and hence the  of Ho2Ti2O7 would include the nuc with a similar magnitude toholmium metal. If negative contribution to the thermal expansion is induced by the spin120

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

ice state in Ho2Ti2O7 and its magnitude is comparable with that of Dy2Ti2O7, the nuc
can exceed the negative thermal expansion, providing the positive thermal expansion.
Another possibility is that the lattice contraction has an opposite effect on theinternal energy of each compound. The chemical substitution study found out the
non-linear effect of the chemical pressure on Jnn/Dnn, as shown in Fig. 7.5 [17]. In theDy systems, the substitution of the Ti4+ ions with the Ge4+ ions increases by 82% and
15% in Jnn and Dnn, respectively. Meanwhile, in the Ho systems, the same substitution
increases by 55% and 15% in Jnn and Dnn, respectively. This result suggests that thevariation of Jnn with the volume contraction is grater in Dy2Ti2O7 than in Ho2Ti2O7. In
addition, the Jnn of Dy2Ti2O7 is more than double for that of Ho2Ti2O7. Thus, at low
temperature Dy2Ti2O7 is rather unstable than Ho2Ti2O7. On the other hand, the system
has to pay the elastic energy to distort the lattice. This cost is expected to be nearly
identical in Dy2Ti2O7 and Ho2Ti2O7 because the difference in the type of the rare-earth
ion leads to subtle influence on the lattice energy. For these reasons, the opposite
thermal expansion might be resulted from the balance between the energy gain by
weakening the exchange interaction and the elastic energy cost to contract the lattice.
Hence, Dy2Ti2O7 lower the internal energy by expanding the lattice to avoid
strengthening Jnn, while Ho2Ti2O7 by contracting the lattice due to the insensitivity of
Jnn. This scenario would be verified by performing thermal expansion measurements on
B-site substituted systems. Further investigations on this property in spin ice material
need to reveal its mechanism. The emergence of negative thermal expansion isreminiscent of the thermal expansion of water just before freezing. Although thedifferent mechanism induces negative thermal expansion, the comparison of twomaterials might provide an unexpected aspect of “ice state” in spin ice material.
121

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

7.4

CONCLUSION
This chapter studied the lattice behavior in the spin ice compounds, Dy2Ti2O7 andHo2Ti2O7. Both compounds exhibited the anomalous behavior in both the heat capacity
and thermal expansion, which are associated with a spin ice state. While the magnetic
heat capacity was observed below 15 K, the magnetic thermal expansion coefficientswere appeared below 5 K, indicating the correlation between the lattice degrees of
freedom and the development of static spin ice state. The discussion of the stability ofspin ice system suggests that negative thermal expansion could be common behavior inspin ice state. The positive thermal expansion at low temperature in Ho2Ti2O7 may be
due to a large nuclear hyperfine effect of the Ho ion, or the difference of the requiredenergy to expand the lattice between Ho2Ti2O7 and Dy2Ti2O7.

122Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

Fig. 7.1. The ordered configuration of (a) water ice and (b) spin ice. From [12].

123Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

Fig. 7.2. Heat capacity of Dy2Ti2O7 (Blue circles) and Ho2Ti2O7 (Green triangles). The
Cp of Ho2Ti2O7 has been subtracted the nuclear hyperfine contribution from theexperimental data. The solid black line and dashed red line show the lattice heat
capacity estimated by scaling the Cp of Lu2Ti2O7. The inset displays the measured heat
capacity data (Green triangles) and the nuclear Schottky heat capacity, assuming a
splitting of the eight nuclear levels with a level spacing of 0.3 K.

124

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

Fig. 7.3. Thermal expansion coefficient  of Dy2Ti2O7 and Ho2Ti2O7. The solid blackline and dashed red line show the lattice heat capacity estimated by scaling the  ofLu2Ti2O7.

125

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

Fig. 7.4. (a) Magnetic heat capacity Cm. (b) Thermal expansion coefficient m. They
are obtained from the subtraction of the lattice contribution from the experimental data.
(c) The pressure dependence of the magnetic energy scale of the spin ice state∂ ln 𝐸𝑚 /𝜕𝑝 evaluated from Eq. (4) using the Cm and m.
126

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

Table 7.1. Lattice parameters and selected magnetic parameters for all six
pyrochlore spin ices. From [17].

Fig. 7.5. Dependencies of the speciﬁc heat peak position Tpeak/Dnn on Jnn/Dnn ratio.
The symbols are experimental results, and the solid lines are the theoretical calculations
from the dipolar spin ice model. From [15, 17].

127

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7

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S. Rosenkranz, A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan, and B. S.
Shastry, J. Appl. Phys. 87, 5914 (2000).

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H. D. Zhou, J. G. Cheng, A. M. Hallas, C. R. Wiebe, G. Li, L. Balicas, J. S. Zhou, J.
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R. Gonano and E. D. Adams, Bull. Am. Phys, Soc. 13, 667 (1968)
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129Chapter 8. Summary and Conclusion

Chapter 8
Summary and Conclusion

The concept of “frustration” has been attracted much attention as the playgroundfor novel physical properties. In most cases, the resulting macroscopic degeneracy will
be lifted by coupling to other degrees of freedom, leaving peculiar behaviors. In the caseof magnetically frustrated materials, as discussed in Chapter 1, the spin degrees of
freedom often couple to the lattice degrees of freedom, i.e., spin-lattice coupling. In this
doctoral thesis, the spin-lattice effects in systems with the pyrochlore lattice have been
investigated by thermodynamic measurements, as introduced in Chapter 2. Their
exciting properties have been discussed from the thermodynamic point of view.

Chapter 3. Spin-glass-like behavior in ferromagnetic phase of CdCr2S4
Spin-glass-like behaviors in the ferromagnetic phase of CdCr2S4 have investigated.
Although the term “ferromagnetic” seems to be out of place in the context of frustration,
CdCr2S4 has exhibited anomalous behaviors, which is not expected for a typical
Heisenberg ferromagnet. The anomalous behaviors in the low-temperature heat capacity
suggest that a glassy state grows below 20 K, coexisting with the ferromagnetic state.
Thermal expansion measurements have shown negative thermal expansion below 100 K,
which indicates the presence of the magnetic frustration resulted from the local lattice
distortion. Both the dc and ac susceptibility measurements have exhibited the glass-likebehavior. Assuming the presence of a local lattice distortion, arising from a second
Jahn-Teller effect, these results have thus been concluded as follows; First, the130Chapter 8. Summary and Conclusion

short-range magnetic clusters is formed due to the local anisotropic interaction, and it
performs as a nanoparticle. With decreasing temperature, some clusters will be trapped
into a metastable state. The trapped clusters work as nucleus for nucleation. Finally theclusters will gather and form larger spin clusters below 20 K. The spin configuration ofthis anomalous ferromagnetic state should be clarified in future. As can be expectedfrom the present results, the search for the low-temperature spin structure will be thecentral issue for the future works, which might be achieved by the apparatus that can
detect mesoscopic scale of about hundreds of nanometer, such as small-angle neutron
scattering.

Chapter 4. Spin-lattice coupling effect in strongly geometrically frustrated spinel
CdCr2O4
The spin-lattice coupling in strongly geometrically frustrated spinel CdCr2O4 hasstudied. Strong frustration due to Cr3+ ions forming the pyrochlore network is known tobe relieved by distorting the lattice, and hence the spin-lattice coupling should play animportant role in CdCr2O4. The magnetic susceptibility and heat capacity evidenced thatthe magnetic interaction is effective below 150 K. The thermal expansion behavior was
impressive; it exhibited negative thermal expansion below 150 K, and returned topositive thermal expansion below 45 K without a phase transition. This peculiar
behavior was attributed to the competition of the effect between the magnetic frustrationand the antiferromagnetic hexagonal spin cluster in the paramagnetic regime. This idea
further provided the important clue for the formation of that spin cluster. The heat
capacity and thermal expansion data derived the pressure dependence of the131Chapter 8. Summary and Conclusion

magnetostructural transition temperature. The obtained value contradicted to theprevious report, though the difference must not be an experimental error. The contracted
results might be resulted from the presence or absence of magnetic field. The
comparison of the family compound ZnCr2O4, which shows a helpful response topressure and magnetic field, was instructive. Future works need to deal with the detailed
pressure and magnetic field dependence of the magnetostructural transition temperature,
which might give an important insight on the different structural distortion of CdCr2O4
and ZnCr2O4 at the transition. Thermal expansion measurements on ZnCr2O4 and also
HgCr2O4, which have a weaker antiferromagnetic interaction than CdCr2O4, are also
considered to be important. Because they have the different strength of not only the
frustration but also the spin-lattice coupling, thermal expansion behavior will be muchdifferent from CdCr2O4.

Chapter 5. Lattice behavior in the conical spin state of CoCr2O4
This chapter has focused on the lattice behavior in the conical spin state of thespinel CoCr2O4. Due to the magnetic Co2+ ions in the A-site, this compound shows
highly complex magnetic orderings. Although CoCr2O4 has been known to show thestrong sample dependent due to the site mixing, the present sample was very purebecause each ferrimagnetic, conical spin, lock-in transitions were clearly observed, andthe heat capacity at the conical spin transition was significantly sharp. At the conicalspin transition, the thermal expansion coefficient was broadened above 20 K higher than
the transition temperature, indicating the growth of the short-range ordering. Theinteresting behavior of the thermal expansion has been observed at the lock-in transition,
where the heat capacity peak was very small, while the thermal expansion coefficient132Chapter 8. Summary and Conclusion

showed the large peak, as opposed to the conical spin transition. The pressure
dependence of the transition temperatures calculated from these results led to theintriguing insights. Especially, the considerable pressure dependence of the lock-in
transition temperature proposed the possibility of the symmetry reduction from the
cubic at the lock-in transition. From the magnetocaloric effect measurements under thepulse magnetic field, the precise H-T phase diagram was presented for the <111> and<001> axis. There was a first-order phase transition with huge latent heat at high field,
which transition temperature apparently depends on the field direction. As for future
works, the crystal orientation dependence of the thermal expansion should be performed
to certify the anisotropic lattice distortion at the lock-in transition. To know what
happened at the high field phase transition in more detail, it is interested to conduct
other experiments under pulsed magnetic field. Heat capacity measurement inevitably
has to be carried out to get the information on the entropy of the high field phase. Other
promising measurements are the capacitive and thermal expansion measurement; theformer will show a sudden drop of the capacitance at the phase transition due to thecollapse of the conical spin state, and the latter will confirm the lattice reconstruction to
the cubic phase due to lacking the inverse DM interaction.

Chapter 6. Lack of the spin-lattice coupling in spin liquid state of Tb2Ti2O7
The effect of the spin-lattice coupling in the spin-liquid compound Tb2Ti2O7 has
been studied. It has been a long-standing problem in Tb2Ti2O7 why it shows the
spin-liquid behavior, and what stabilize the spin-liquid state? The spin-lattice couplinghas been considered to be one of the key factors for those issues, because anomalous
temperature dependences have been found in several lattice properties. The present
133Chapter 8. Summary and Conclusion

study has performed heat capacity and thermal expansion measurements. The heat
capacity showed two shoulders at around 8 K and 2 K. With applying magnetic field,
the 2 K anomaly was enhanced up to 1 T without large suppression in the heat capacity
below 2 K. Although at 1 T, the 3-in, 1-out / 1-in, 3-out spin structure is formed, themagnetic entropy did not show a large change expected for the unique spin state. Thus,
the puzzling correlation providing the 2 K anomaly in the ground state at zero fieldmight be rearranged by magnetic field without losing the entropy. Thermal expansion
coefficient exhibited only one shoulder-like anomaly around 8 K, as opposed to the heat
capacity. This behavior was well reproduced by the lattice contribution and the Schottky
anomaly, resulted from the presence of low-lying ﬁrst excited CEF doublet with theseparated energy of 18 K. No anomalous behavior below 7 K indicates the absence of aspin-lattice effect, and the spin-lattice coupling is not need to establish a non-orderedground state. The combination of our heat capacity and thermal expansion data provided
the pure component for the heat capacity anomaly below 7 K. The excess heat capacity
led to the excess entropy Sex~7.5 J/Kmol, which is rather small compared to the
expected entropy for the ground CEF doublet splitting S=2Rln(2)=11.5 J/Kmol. The
large shortfall in entropy indicates the strong fluctuation in the ground state, andpossibly relates with a realization of a quantum spin ice state. Future works for the spin
liquid state in Tb2Ti2O7 need to explain the present results. For the confirmation of thelack of negative thermal expansion, the crystal orientation dependence of the thermal
expansion should be confirmed.

Chapter 7. Thermal expansion behavior in spin ice state of Dy2Ti2O7 and Ho2Ti2O7
This chapter dealt with the lattice behavior of the spin ice compound Dy2Ti2O7 and134

Chapter 8. Summary and Conclusion

Ho2Ti2O7. They are known to show the “two-in two-out” spin structure, which isreminiscent of the proton positions in water ice. Dy2Ti2O7 and Ho2Ti2O7 exhibited theanomalous behavior in both the heat capacity and thermal expansion, which areassociated with a spin ice state. The interesting findings were negative thermal
expansion below 5 K in Dy2Ti2O7. The discussion of the stability of spin ice systemsuggests that negative thermal expansion could be common behavior in spin ice state.
However, Ho2Ti2O7 exhibited the positive thermal expansion below 5 K, which may be
due to a large nuclear hyperfine effect of the Ho ion, or the difference of the requiredenergy to expand the lattice between Dy2Ti2O7 and Ho2Ti2O7. To determine which ideas
make sense, thermal expansion measurements on the other spin ice materials will need
to be performed. In addition, it will be needed to make clear whether the nuclear
thermal expansion due to 165Ho nuclear spin is appeared or not.

As described above, the present thesis has studied on the frustrated spinel andpyrochlore systems with the spin-lattice coupling. It has demonstrated that the
spin-lattice coupled systems exhibit the characteristic lattice behaviors corresponding totheir spin behaviors. Furthermore, the combination of the thermodynamic studies
provided valuable insights on the spin behavior and phase stability as well as the latticeproperty. I hope that the present results and suggestions become the cornerstone fordeveloping a comprehensive mechanism of the novel properties in the materials studied
in this thesis.

135Appendix
List of published articles
1.

Suguru Kitani, Makoto Tachibana, Naoya Taira, Hitoshi Kawaji, “Thermal study ofthe interplay between spin and lattice in CoCr2O4 and CdCr2O4”, Phys. Rev. B 87,
064402-1 – 064402-6 (2013).
(Including in chapter 4 and 5)

2.

Suguru Kitani, Makoto Tachibana, Hitoshi Kawaji, “Spin-glass-like behavior in
ferromagnetic phase of CdCr2S4”, Solid State Communications 179, 16-19 (2014).
(Including in chapter 3)

3.

Suguru Kitani, Makoto Tachibana, Hitoshi Kawaji, “Low-temperature lattice propertiesof frustrated pyrochlore magnets R2Ti2O7 (R=Tb, Dy, and Ho)”, to be submitted to Phys. Rev. B.

(Including in chapter 6 and 7)136

Acknowledgements
This work had accomplished in Kawaji Laboratory, Materials and Structures
Laboratory, Tokyo Institute of Technology, Japan, from the April 2010 to the December
2014. First of all, I especially thank Professor Hitoshi Kawaji for invaluable guidanceand discussions during the course of this work. I wish to thank the late Professor Tooru
Atake for many encouragements. I also want to thank the members of Kawaji
Laboratory for their help and discussions, and I am indebted to my senior colleagues,
Kyohei Mizushima and Naoya Taira, in this regard.
I have a special thanks to Dr. Makoto Tachibana for fruitful discussions, thedilatometer and the samples through all of this work. I would also like to thankResearch Associate Yoshimitsu Kohama for useful advices on the low-temperature
measurements and the pulsed magnetic field measurements in the research of CoCr2O4(Chapter 5).
I also thank Professor Takeo Ohsaka, Ryoji Kanno, Nobuhiro Matsushita, andMasaaki Hirayama for their review of this thesis.
Finally, I would like to give my greatest appreciation to my family and relative.
Without their understanding and support, I could never have completed this work.

December, 2014
Suguru Kitani

137