Investigation Into the Magnetoelectric Effect and Magnetic Properties of Iron-Doped Cobalt Molybdate

Investigation into the Magnetoelectric Effect and Magnetic Properties of Iron-doped Cobalt Molybdate

By: Mathew C. Pula, B.Sc.

Physics (Condensed Matter & Materials Science)

Submitted in partial fulﬁllment of the requirements for the degree of: Master of Science

Faculty of Mathematics and Science Brock University St. Catherines, Ontario

c 2019 Dedication

To my parents.

Thank you for giving me my life, your unconditional love, and limitless support over the years, ﬁnancially and otherwise. You taught me how to laugh when I might otherwise cry.

To my siblings.

Your resilience, drive, and creativity never cease to amaze me. You are a source of constant inspiration. Thank you for providing me with wisdom on questions for which I have no answers and teaching me the importance of family.

To my friend.

You taught me the most important lesson of my life and I will forever be in your debt. I will never forget you.

i Abstract

The plausibility of revealing linear magnetoelectric coupling is investigated in the cobalt molybdate (Co2Mo3O8). Recently, Wang et al.[Scientific Reports. 2015;Vol. 5:Article 12268] showed that iron molybdate(Fe2Mo3O8) can be induced into a ferrimagnetic state from a nominal antiferromagnetic state via application of a magnetic field. As such, it may be possible that cobalt molybdate exhibits a similar effect intrinsically or with addition of iron dopant. Single crystals of the hexagonal molybdate (Co1−xFex)2Mo3O8 (x=0, 0.25, 0.5, 0.75, 1) were synthesized via chemical vapour transport. The magnetic properties were investigated along the polar axis and in the basal plane. Despite doping with iron, no metamagnetic phase transition was present in (Co1−xFex)2Mo3O8 (x=0.25, 0.5, 0.75). Low field measurements of the susceptibility reveal the presence of an anisotropic ferromagnetic-like moment, which is suppressed at moderate fields. This is believed to be a product of an exchange-bias-like phenomena, which is not fully understood. The magnetocapacitance was measured along the c-axis for x=(0.25, 0.5). Co1.5Fe0.5Mo3O8 exhibits the conventional magnetodielectric effect, with a proportionality constant of (5.1±0.3) ×10−14 Oe−2 at 40K, while the capacitance of

−9 −1 Co1Fe1Mo3O8 shows linear dependency on H, with slope (-6.99 ±0.07)×10 Oe at 49K.

Keywords: magnetoelectric, magnetocapacitance, Co2Mo3O8, cobalt molybdate, multiferroic

ii Contents

Abstract ii

List of Tables v

List of Figures vi

Acknowledgements ix

1 INTRODUCTION 1

1.1 Properties of Magnetism and Magnetic Orders ...... 2 1.1.1 The Exchange Interaction ...... 3 1.1.2 The Zeeman Interaction and Magnetostatic Energy ...... 5 1.1.3 Demagnetizing Field ...... 6 1.1.4 The Magnetic Anisotropy Energy ...... 9 1.1.5 Domains and Bloch Walls ...... 10 1.1.6 Coercivity and Hysteresis ...... 12 1.1.7 Magnetic Ordering- Landau Theory ...... 16 1.1.8 Curie-Weiss Paramagnetism, Magnetic Susceptibility, and the Molecular Field ...... 17 1.2 Antiferromagnets and Ferrimagnets ...... 19 1.2.1 Spin-ﬂop and Linear Rotation ...... 21 1.3 Ferroelectricity ...... 25 1.4 The Magnetoelectric Effect ...... 26 1.5 The Dzyaloshinskii-Moriya Interaction ...... 29 1.6 Kamiokite and The Symmetric Exchange Striction ...... 35

2 METHODS 40

2.1 Chemical Vapour Transport ...... 40 2.2 X-ray Diffraction ...... 43 2.3 The Magnetization and The Magnetic Susceptibility ...... 45

iii 2.3.1 The SQUID Magnetometer ...... 46 2.4 The Speciﬁc Heat ...... 49 2.5 The Dielectric Function and The Magnetocapacitance ...... 51 2.6 Magnetoelectric Coupling ...... 54

3 RESULTS AND DISCUSSION 57

3.1 Sample Synthesis and Characterization ...... 57 3.2 Magnetic Properties ...... 68 3.2.1 Cobalt, Iron, and the Cobalt-Iron Molybdates- Part I ...... 68 3.2.2 Part II- Bifurcation ...... 80 3.2.3 Part III- Cobalt-Zinc Molybdate and Cobalt-Manganese Molybdate . . 90 3.3 The Speciﬁc Heat ...... 91 3.4 The Dielectric Function, the Magnetoelectric Capacitance, and the Magnetoelectric Coupling ...... 96 3.4.1 The Dielectric Function ...... 96 3.4.2 The Magnetocapacitance and The Magnetoelectric Coupling ...... 102

4 CONCLUSION 107

REFERENCES 108 List of Tables

Table 1 Chemical vapour transport synthesis details and results...... 60 Table 2 Rietveld reﬁnement factors for cobalt and the cobalt-iron molybdates. . . . 62 Table 3 Summary of the transition temperatures for the speciﬁc heat capacity

(denoted Cp) and the magnetic susceptibility (χ) of Co2−2xFe2xMo3O8...... 66 Table 4 Calculation of the Curie constant (C), Weiss constant (θ), and effective

moment (µeff )...... 78 Table 5 Comparison of the qualities deduced in table 4...... 78

v List of Figures

Fig. 1 A simple 2-D model depicting the domain-wall shifting in an initially unmagnetized (A) material...... 11 Fig. 2 Ferromagnetic hysteresis loops of idealized and real magnets...... 15 Fig. 3 The possible magnetic structures in the hexagonal molybdates...... 20 Fig. 4 Linear rotation of an antiferromagnetic spin-pair under a perpendicular applied field and spin-flop of an antiferromagnetic spin-pair under a parallel applied field ...... 22 Fig. 5 The DMI interaction in a 1-D chain of antiferromagnetic spins...... 30 Fig. 6 The DMI interaction in a 1-D chain of ferromagnetic spins...... 31 Fig. 7 A spin-cycloid induced by the DMI...... 32

Fig. 8 Polarization induced by the spin-cycloid described by Sn = S(sin(kx)ex −

cos(kx)ey)...... 34 Fig. 9 The crystal structure of cobalt molybdate along the c-axis. The crystal symmetry is retained for any of the following 3-d transition metals: Fe, Zn, Mn, Ni. As such, it is to be understood that the cobalt atoms represent M2+ sites. For instance, replacing the cobalt atoms with iron atoms results in the crystal

structure of Fe2Mo3O8...... 38 Fig. 10 The crystal structure of cobalt molybdate along the a-axis...... 39 Fig. 11 The chemical vapour transport method for growing single crystals...... 41 Fig. 12 Diagram depicting Bragg’s law...... 44 Fig. 13 The second-order gradiometer utilized in Quantum Design’s MPMS. . . . . 48 Fig. 14 The speciﬁc heat capacity sample holder ...... 50 Fig. 15 Anomalous behaviour due to the presence of apiezon in the speciﬁc heat capacity of cobalt molybdate...... 51 Fig. 16 Error in the heat capacity measurement for the cobalt, iron and cobalt-iron molybdates...... 52 Fig. 17 The probe used to measure the dielectric function...... 53 Fig. 18 The background magnetic moment of the constructed probes...... 55

vi Fig. 19 The ﬁrst iteration of the constructed magnetoelectric probes...... 56 Fig. 20 The second and ﬁnal iterations of the constructed magnetoelectric probes. . . 56 Fig. 21 Pieces of the quartz ampoule post-synthesis...... 58 Fig. 22 The cobalt, iron and cobalt-iron molybdate single crystals...... 59 Fig. 23 collage of auxiliary crystals and byproducts ...... 59 Fig. 24 The lattice parameters of the cobalt and cobalt-iron molybdates...... 62

Fig. 25 X-Ray diffraction patterns of Co2Mo3O8 and Co1Zn1Mo3O8...... 63

Fig. 26 The x-ray diffraction patterns for the Co2−2xFe2xMo3O8 series, shifted along the y-axis for clarity...... 64 Fig. 27 Approximate composition of cobalt-iron molybdates based on the speciﬁc heat. 66 Fig. 28 Approximate composition of cobalt-iron molybdates based on magnetic ordering temperature from the magnetic susceptibility...... 67

Fig. 29 The magnetic susceptibility of Co2Mo3O8...... 69 Fig. 30 The enhancement of the perpendicular susceptibility at the Neel´ temperature in cobalt molybdate...... 70

Fig. 31 The susceptibility of Co2Mo3O8 under a 350 Oe parallel field...... 70 Fig. 32 M-H curves under a parallel field for cobalt, iron, and the cobalt molybdates. 71 Fig. 33 M-H curves under a perpendicular field for cobalt molybdate and the cobalt- iron molybdates, showing the temperature independence of the susceptibility. . . 72 Fig. 34 Parasitic ferromagnetism in iron molybdate and the cobalt/ cobalt-iron

molybdates synthesized utilizing PtCl2 as a transport agent...... 74 Fig. 35 The parallel susceptibility of the cobalt, iron, and cobalt-iron molybdates in the vicinity of the Neel´ temperature...... 76 Fig. 36 The derivative of the zero-ﬁeld cooled magnetic susceptibility in the vicinity of the Neel´ temperature...... 76

Fig. 37 Magnetic susceptibility of Co1.5Fe0.5Mo3O8...... 77

Fig. 38 Magnetic susceptibility of Co1Fe1Mo3O8 at 3KOe...... 77

Fig. 39 The magnetic susceptibility of Co0.5Fe1.5Mo3O8...... 78 Fig. 40 The magnetic susceptibility of iron molybdate...... 79

vii Fig. 41 Ferromagnetic-like enhancement in the perpendicular ﬁeld-cooled susceptibility at the Neel´ temperature for single crystals of the cobalt-iron molybdates...... 81

Fig. 42 The behaviour seen in the M-H curves of Co1Fe1Mo3O8 in the low-H-ﬁeld range of |H| < 1KOe...... 83

Fig. 43 M-H curves of Co1.5Fe0.5Mo3O8 and Co1Fe1Mo3O8 at 5K, poled under a negative ﬁeld...... 84 Fig. 44 The parallel M-H behaviour at low ﬁelds and temperature...... 85

Fig. 45 The magnetic susceptibility of Co1.5Fe0.5Mo3O8 under a 10KOe magnetic ﬁeld...... 86 Fig. 46 The ferromagnetic moment in the low-ﬁeld parallel susceptibility of

Co1.5Fe0.5Mo3O8...... 87 Fig. 47 The magnetic susceptibility of cobalt-zinc molybdate...... 90 Fig. 48 The specific heat capacity of cobalt molybdate...... 91 Fig. 49 The specific heat capacity of cobalt, iron, zinc-iron, and the cobalt-iron molybdates...... 92 Fig. 50 The specific heat as a function of T3...... 93 Fig. 51 The magnetic contribution to the specific heat and the entropy associated with the magnetic contribution...... 94 Fig. 52 The dielectric function and loss in cobalt and iron molybdate...... 97

Fig. 53 Co1Fe1Mo3O8 showing the temperature lag of the sample compared to the system...... 98 Fig. 54 The overarching exponential decay of the capacitance with time...... 99 Fig. 55 The dielectric function of the four measured samples...... 101

Fig. 56 The magnetocapacitance of Co2Mo3O8 (5K), Co1.5Fe0.5Mo3O8 (40K), and

Co1Fe1Mo3O8(49K)...... 103

Fig. 57 The dielectric loss in Co2Mo3O8, Co1.5Fe0.5Mo3O8, and Co1Fe1Mo3O8. . . . 106

viii Acknowledgements

First, I must thank my parents for their countless sacriﬁces. I would also thank Dr. F. Razavi for supervising and mentoring me through both my Bachelor’s degree and Master’s degree. I would also thank all of the other faculty members of the physics department, for their dedication to providing excellent education.

Thank you to all the staff members of Brock University for keeping everything running. Particular gratitude is expressed to the electronic and machine shop, without which much of this work would not be possible.

Thank you to each and every student I had the pleasure of teaching throughout this endeavor. Your love of learning reinvigorated me whenever I lost my own in frustration.

Lastly, thank you to all the friends I made throughout my time at Brock. You offer a form of distraction that has no equal. Especial thanks to Sara Hateﬁ Monfared, with whom I shared many enlightening conversations.

ix 1 INTRODUCTION

There exists materials- known as multiferroics- which exhibit multiple simultaneous orders. These materials may order in four ways: magnetically, electrically, toroidally, and elastically. Magnetic ordering manifests itself as a spontaneous magnetic moment, electric as a spontaneous polarization, and elastic as a spontaneous strain. Toroidal order refers to the ordering of magnetic vortices[1]. The latter two orders- elastic and toroidal- are mentioned here for completeness sake only; they will not be discussed in detail.

Multiferroics are of interest due to the interaction of their multiple orderings. For instance, piezoelectricity results from the coupling between elastic and electric orderings. This phenomena allows energy to be transformed from one physical form to another, i.e., a piezoelectric material transforms mechanical energy into electrical energy. This property makes these materials viable transducers, e.g., the piezoelectric effect is used to measure static pressure, acceleration, weight, etc;[2] however, not all multiferroics, e.g., BiMnO3 [3], exhibit coupling between their orders. Multiferroics which order magnetically and electrically may exhibit a coupling between these orders. This phenomenon is known as the magnetoelectric effect and is the focus of this study. Materials that exhibit this effect are of particular relevance to the ﬁeld of spintronics, especially for computer memory. For instance, the work of M. Gajek et al. shows promise for the development of four-state logic[4]. The mechanics of this phenomena will be investigated in due time, but ﬁrst, some fundamentals need to be discussed. We will begin by discussing general properties and mechanisms of magnetism, including a brief discussion on ferroelectricity, and then discuss the candidate sources of the magnetoelectric effect in the hexagonal molybdates.

Note that, throughout this work, several models will be invoked. These models are not meant to be rigorous; rather, they are constructed for illustrative purposes and to give the reader a rough idea of the mechanisms at play. Some of these models (namely, the exchange interaction in a two-electron system [section 1.1.1] and the Dyzaloshinski-Moriya interaction in a 1-D spin-chain [section 1.5]) are inspired by literature, others were constructed by the author.

1 1.1 Properties of Magnetism and Magnetic Orders

Magnetic ordering is a phenomenon in which the magnetic moments associated with atoms align in a predictable, i.e., periodic, manner. Magnetic order is long-range and, much like a crystal lattice, is describable via space groups. Three main types of magnetic ordering exist: ferromagnetism, ferrimagnetism, and antiferromagnetism. For simplicity, we will examine magnetic phenomena under the guise of ferromagnetism, though many of the properties ferromagnets exhibit exist in the other types.

Ferromagnetism describes a state in which adjacent magnetic moments spontaneously align to be parallel (the source of this behaviour is described in section 1.1.1). Due to perturbations of spin orientation via thermal energy (in other words, competition between magnetic and thermal energy), this state occurs only below a material dependent temperature: the Curie temperature. Above this temperature, paramagnetism, i.e., disorder, is observed. Strictly speaking, Curie paramagnetism will be observed for ferromagnetic materials. Curie paramagnetism is investigated in section 1.1.8. We will begin by ﬁrst investigating the various internal energies that exist in a ferromagnet.

2 1.1.1 The Exchange Interaction

The ordering of spins is largely driven by the exchange interaction, a phenomena described by the Heisenberg exchange Hamiltonian[5]:

X Hexch = − Jij(Si · Sj) (1) ij

where Si and Sj are spin operators and J is known as the exchange integral (the meaning of this is discussed below).

This effect is typically strong in magnitude and quantum mechanical in nature. It originates from symmetry considerations of the wavefunction when two spins are exchanged[6]. It can be

seen that the energy of two interacting spins, Si and Sj is minimized when the two spins are either parallel (for J > 0) or antiparallel (J < 0). As such, the exchange interaction prefers to align all spins in a single direction. It should be noted that this phenomenon is independent of crystal structure[6]. This is an important property when one considers the formation of domains.

A simple example of this can be found in J. M. D. Coey’s Magnetism and Magnetic

Materials[5], which examines a molecule of H2. For this two-electron system, the total spin,

1 S, equals S = S1 + S2, where S1=S2= 2 . There exists two values of S: S = 1 and S = 0. These correspond to the spin-triplet and spin-singlet states, respectfully. The spin-triplet

occurs when S1 and S2 are parallel and its wave function is symmetric under exchange of the

two spins, i.e., ψ(S1,S2) = ψ(S2,S1). The spin-singlet occurs when S1 and S2 are antiparallel, and has an antisymmetric wave function under the same exchange:

ψ(S1,S2) = −ψ(S2,S1). By the Pauli exclusion principle, the total wave function, which is

the product of the spatially dependent wave function, φ(r1, r2), and the spin wave function,

ψ(S1,S2), must be antisymmetric under exchange of the two electrons:

Ψ(1, 2) = φ(r1, r2)ψ(S1,S2) = −φ(r2, r1)ψ(S2,S1) = −Ψ(2, 1) (2)

Therefore, for the spin-triplet state, φ(r1, r2) must be antisymmetric to satisfy the Pauli

3 exclusion principle. Likewise, φ(r1, r2) for the singlet state must be symmetric. The triplet state is said to be antibonding, since the Pauli exclusion principle keeps the identical electrons from overlapping orbitals (the antisymmetric φ(r1, r2) necessarily has a node mid-way between the hydrogen atoms). The singlet state is said to be bonding, since it allows the two orbitals to overlap.

The energy difference between the spin-triplet and singlet is proportional to J[5]:

1 J ∝ (E − E ) (3) 2 s t

where Et is the energy of the triplet state and Es is the energy of the singlet-state. If J < 0, the energy of the singlet will be less than that of the triplet, and the interaction between the spins will be antiferromagnetic (the singlet state is favourable). If J > 0, the energy of the triplet will be less than that of the singlet, and the interaction is ferromagnetic (the triplet state is favourable).

There is another type of exchange known as the superexchange interaction. This exchange can occur between two cations separated by a non-magnetic anion, typically oxygen. Superexchange is expressed with the same mathematical form as the exchange interaction (sometimes called the direct exchange to distinguish it from the superexchange) but conventionally lacks a negative sign[7]. As such, antiferromagnetic superexchange corresponds to J>0 and ferromagnetic to J<0.

4 1.1.2 The Zeeman Interaction and Magnetostatic Energy

The Zeeman interaction describes the interaction between spins-in general, any magnetic dipole moment- and external magnetic ﬁelds. The Zeeman energy for a single dipole moment, m, is[5]:

EZ = −m · B (4)

This energy tends to align m parallel to B, since EZ is minimal when m · B is maximal. Let us examine a collection of dipoles and deﬁne the magnetization as:

dm M = (5) dV

Here, we take M to be a macroscopic ﬁeld, such that it does not vary in space. Equation (4) can be expressed in its differential form as:

1 dE = − M · BdV (6) Z 2

It should be noted that this is usually expressed in terms of the H-ﬁeld. Substituting

B = µ0(H + M) into equation (6) and integrating both sides, we arrive at:

Z Z 1 1 EZ = − 2 µ0 H · MdV − 2 µ0 M · MdV (7)

1 The factor of 2 is introduced in equation (6) to account for the self-counting[5, 9], i.e., since each small increment dm contributes to both M and B. The second integral behaves like a macroscopic exchange interaction, i.e., it has the form M 2 sin θ; however, the magnitude is found to be negligible compared to its microscopic counterpart[5]. It will thus be omitted. Finally, we arrive at: Z 1 EZ = − 2 µ0 H · MdV (8)

Equation (8) has the same form as the magnetostatic energy (if we imagine the the B-ﬁeld as being produced by the material, this would be the magnetostatic energy). Here, it is to be

5 understood that H contains many terms, and it is often useful to break it down into its constituents. One can, for instance, explain the formation of domains by examining the magnetic energy of the demagnetizing ﬁeld.

1.1.3 Demagnetizing Field

The Demagnetizing field, Hd, is an internal magnetic field that arises when a material is magnetized. Consider Maxwell’s equations for magnetostatics[10] and the magnetic field inside a magnetized material[10]: ∇ × H = 0 (9)

∇ · B = 0 (10)

B = µ0(H + M) (11)

Here, it is to be understood that B and H refer to the ﬁelds generated by the sample, rather than an external source. Substituting equation (11) into equation (10), we have:

∇ · H = −∇ · M (12)

It can be seen from equation (12) that a uniform M makes no volume contribution to H. However, the divergence of M at the surface, where M is discontinuous, is non-zero. As such, there is an H-ﬁeld generated by the surfaces. By equation (9), H can be written as the gradient of some scalar. Substituting for H in equation (12) for this arbitrary scalar-gradient results in Poisson’s equation[10]. This relation is often described in an analogy to polarization, where the ﬁeld is generated by surface and bulk charges densities. Utilizing this idea, one can show[5]:

Z 0 0 Z 0 1 (∇ · M)(r − r ) 3 0 (M · n)(r − r ) 2 0 H(r) = − 0 3 d r + 0 3 d r (13) 4π V |r − r | S |r − r |

Here, n is the surface normal. As described above, the volume term is zero for any uniform M and H is generated purely by the surfaces. Note that, inside a uniformly magnetized material,

6 the H-ﬁeld opposes the direction of the magnetization, hence the ﬁrst term of equation (13) is zero for a uniform M and the second term is negative because |r0| > |r| . For this reason, we call

the H-field the demagnetizing field, Hd, inside the magnet. Outside the magnet, as previously mentioned, M = 0 and B and H are proportional to each other. This external field is known as

the stray ﬁeld. In general, Hd is difﬁcult to compute. However, for an ellipsoid[5, 10]:

Hd = −NM (14)

Where N is a shape dependent constant and is known as the demagnetization factor. In practice, one typically approximates a given shape as an ellipsoid and uses equation (14) to calculate Hd.

The magnetic energy, EM , contained within the demagnetization H ﬁeld is[9]:

Z 1 2 3 EM = 2 µ0 Hd d r (15) all space

or equivalently[5]: Z 1 3 EM = − 2 µ0 Hd · Md r (16) magnet vol.

One can arrive at equation (16) by substituting H for Hd in equation (8), or by substituting B Hd for − M. Using equation (10), B may be written as the curl of some vector potential, A, µ0 resulting in the integral:

Z Z 1 3 1 3 EM = 2 µ0 Hd · (∇ × A)d r − 2 µ0 Hd · Md r (17) all space all space

Using the vector identity[5, 10] H · (∇ × A) = ∇ · (A × H) + A · (∇ × H), equation (17) can be written as:

Z Z 1 3 1 3 EM = 2 µ0 (∇ · (A × Hd) + A · (∇ × Hd))d r − 2 µ0 Hd · Md r (18) all space all space

7 The second term in the first integrand is zero by equation (9). The first term can be evaluated via Gauss’ theorem[5]. This term tends to zero for an infinite surface, as A varies as

−2 −3 2 r and Hd varies as r , while the area of the surface of integration varies as r only[5]. The second integral is reduced to the integration over the material’s volume, since M=0 outside the material.

EM is minimized when M·n is zero (when M is parallel to the surface). This leads to some competition between the volume contribution and the surface contribution, hence the volume contribution prefers M to be uniform, but the surface prefers M ⊥ n. This competition leads to the formation of domains, and is known as the pole-avoidance principle[9].

It is also worth noting that the pole-avoidance principle leads to shape anisotropy. Imagine a single domain particle which is uniaxial. The magnetization will prefer to point in the direction that minimizes the the surface charge, which is along the long-axis of the particle[5].

8 1.1.4 The Magnetic Anisotropy Energy

The magnetic anisotropy (here, I speciﬁcally refer to magnetocrystalline anisotropy; other forms exist, but they will not be considered in this work) energy arises from crystallographic symmetry considerations. For uniaxial crystals (crystals which have two equivalent directions [denoted a and b] and one unique direction [denoted c]), the magnetic anisotropy energy is approximately given by[5, 6, 8]:

2 4 Uani = K1 sin θ + K2 sin θ (19)

Where K1 and K2 are known as the anisotropy constants and θ is the angle between the magnetization vector, M (deﬁned by equation (5)), and the z-axis (the c-axis). In general, equation (19) contains higher order terms and trigonometric relations involving φ, the angle between M and the x-axis (the a-axis), but these are not necessary for illustrative purposes.

Let us assume K1 K2, such that equation (19) becomes:

2 Uani ≈ K1 sin θ (20)

It is easily seen that, if K1 > 0, then Uani is minimized when θ = 0, i.e., when M is

parallel to the z-axis (the c-axis). This axis is then known as the easy-axis. If K1 < 0, Uani is minimized when M is constrained to the xy-plane (the ab-plane). In this case, the direction of the easy-axis is determined by higher-order terms.

It should be noted that there can exist multiple easy-axes. For instance, in the case of a cubic ferromagnet, all three axes (x,y,z) can be easy-axes [6].

9 1.1.5 Domains and Bloch Walls

Naively, one might conclude that a ferromagnetic material would exhibit spontaneous magnetization, hence, by deﬁnition, the spins align constructively. However, were a ferromagnetic material to spontaneously magnetize, it would need to contend with the energy stored in the corresponding magnetic ﬁeld (see section 1.1.3 for a discussion of the

R 2 pole-avoidance principle). The energy stored in a magnetic field is Em ∝ allspace B dV [10]. As a result, the energy is minimized if the magnetic field is localized to a small region of space. One way to achieve this, specifically when the magnet is anisotropic, is via the formation of magnetic domains: pockets of aligned spins which have a net moment but form in a pseudo-random direction.

The formation of domains is mediated by the competition of the various internal energies that exist within a magnet. The exchange interaction prefers to keep all the spins aligned in a single direction (see section 1.1.1). The anisotropy energy is minimized when the spins align along the easy-axes (see section 1.1.4). The demagnetizing field favours aligning spins perpendicular to the magnet’s surface (see section 1.1.3). The resulting domain structure is one which find the minimum of all these energies. However, in general, domains will form in pockets of uniform magnetization and the distribution of their direction will ensure that no stray fields exist. In other words, that the net magnetization is zero (this is, of course, in the absence of an external magnetic field).

Domains are separated by what are known as domain walls. These are regions between domains in which the spins gradually rotate from one domain to the next. A common type of domain wall is the 180◦ Bloch wall, in which domains, which have antiparallel neighbours, are separated by a region of spins rotated in the plane of the wall, i.e., transversely. Bloch walls have the property of ensuring that[5] ∇ · M = 0. The thickness of these walls is determined by the anisotropy energy and the exchange interaction[5, 8]. The former prefers the wall to be thin, as to minimize the number of spins deviating from the easy-axes. The latter has a proclivity towards thick walls, since this allows for smaller adjacent spin angles. The typical thickness of

10 these walls is approximately 50nm[8].

Additionally, there is a type of domain wall known as the Neel´ wall, in which the rotations of spins occurs normal to the plane of the wall, i.e., longitudinally. Neel´ walls have a non-zero divergence of M, making them energetically unfavourable in bulk materials. Neel´ walls are only observed in thin-ﬁlms, when the domain wall thickness is greater than the thickness of the ﬁlm[5].

When a magnetic field is applied to a ferromagnet, there is an associated change to the energy, due to interaction of each spin with the external field. This interaction, the Zeeman interaction, generates a torque on each spin and tries to align it parallel to the external field, thereby increasing the magnetization. This occurs through two mechanism[5]: domain wall motion and domain rotation. The first entails the movement of domain walls, such that the size of the domains shift: domains aligned with the field become energetically favourable, increasing their size, at the expense of less favourable domains, thus leading to a net magnetization. The second type only occurs in magnets with magnetic anisotropy. This type occurs when an applied magnetic field is off-axis w.r.t. the easy-axis and has sufficient strength to overcome the anisotropic field[5]. In a process known as coherent rotation, the moments of whole domains rotate towards the applied field in order to minimize the magnetostatic energy.

Fig. 1: A simple 2-D model depicting the domain-wall shifting in an initially unmagnetized (A) material. The application of a magnetic ﬁeld in the positive x-direction alters the size of the domains, increasing the area of domains with moments parallel to the x-axis and decreasing those with moments antiparallel, leading to a net magnetization in the positive x-direction (B).

11 Evidently, the magnetic moment will only increase so long as there remains unaligned domains; the maximum value obtainable is known as the saturation magnetization. It should be noted that: often, ferromagnetic compounds contain non-magnetic ions, which will contribute to the magnetization through Curie paramagnetism (see section 1.1.8). The magnetization of such compounds will follow a paramagnetic behaviour above the saturation ﬁeld.

1.1.6 Coercivity and Hysteresis

Hysteresis describes the ability of ferromagnets to retain their magnetization after exposure to a magnetic field. The magnetization a ferromagnet retains after exposed to a saturating field is known as the remanent magnetization. If a field of sufficient magnitude is applied antiparallel to the moment of a magnetized ferromagnet, the material will become demagnetized; in the case that the ferromagnet was saturated, this field is known as the coercivity or the coercive field.

The origin of coervicity and hysteresis lies in the restrictions on the formation and propagation of reversal domains. Imagine a idealized uniaxial magnet is placed inside an initial magnetic field of sufficient magnitude to saturate the magnet and with direction parallel to the easy-axis. For simplicity, we will take the direction of this applied field to be fixed, and vary the magnitude. We will take the initial magnitude to be positive. If the external field is subsequently reduced past a critical magnitude, known as the nucleation field, Hn, reversal domains- domains which have magnetization parallel to the change in the external field- are able to nucleate. Since the energy required to create a domain is equal only to the energy required to create the domain wall[8], the reversal domains can easily propagate throughout the magnet, but may become pinned due to the crystal defects, i.e., the domain walls are unable to propagate through the strong fields created by the defects.

Typically, grain boundaries (planar defects) offer the largest energy barrier to domain wall propagation[8], but clusters of line or point defects can also act as energy barriers[5]. These energy barriers are known as pinning centres. Pinned domains can still increase in size by

12 bowing around the pinning centre, but the volume they can occupy is limited. Whether or not

a magnet experiences pinning depends on the relative magnitudes of the nucleation field, Hn, and the depinning field, Hdp. If |Hn| < |Hdp|, the reversal domains are easily trapped by the pinning centres. The reversal domains will remain pinned until the applied field satisfies the

condition H ≤ Hdp. At this point, the reversal domains are able to propagate through the magnet and reverse its magnetization[8]. This is known as depinning. A magnet that displays

this behaviour is referred to as a pinning-type magnet. If Hn > Hdp, nucleation will occur freely and the magnet will reverse its magnetization at the nucleation field. This magnet is known as the nucleation-type. In these idealized magnets, the saturation field, the coercive field, and the nucleation or depinning field (depending on the magnet’s type) are identical, and the magnetization forms a square Hysteresis loop (see figure 2). Of course, in reality, magnets display a combination of both nucleation and pinning, as well as other effects, like coherent rotations of the domains due to the anisotropic energy. Thermal energy must also be accounted for. As a result, the propagation of reversal domains is not nearly as instantaneous as described above.

The dependence of the magnetization, as a function of applied magnetic ﬁeld, is, in practice, nearly indistinguishable for nucleation and pinning-type magnets. The exception lies in the virgin curve, which has a large susceptibility in nucleation-type magnets, since there is almost no pinning, and a small susceptibility in the pinning-type magnets, hence the reversal domains become pinned and can not expand readily. Here, it should be noted that the magnet type refers to the dominant mechanism, not the exclusive mechanism as described above.

Consider now a realistic uniaxial magnet that has been saturated by an external magnetic field, which points in an arbitrary direction. Again, let this applied field subsequently decrease in magnitude. Reversal domains are able to nucleate in regions with strong internal fields. These strong internal fields are due to surface roughness, grain boundaries (planar defects), line defects, and point defects[5, 8] and lower the energy required to form reversal domains locally. Again, these reversal domains can become pinned by very same defects that help form them. However, not all defects are created equally, and some domains will depin before others.

13 The randomness of thermal energy also contributes to the depinning and formation of reversal domains. Additionally, the magnet will experience a reversal of any coherent rotation, once the magnetostatic energy falls below the anisotropic energy, if the applied ﬁeld is off-axis from the easy-axis. The result is a gradual decay of the magnetization above the coercive ﬁeld, in contrast to the sharp reversal of the magnetization seen in the idealized cases.

Some discontinuity remains, however. As the force created on a reversal domain by a given pinning centre is suddenly alleviated upon depinning, the reversal domain propagate suddenly in discontinuous jumps. This leads to a phenomena known as the Barkhausen effect, where the magnetization of the magnet changes in discrete amounts, known as Barkhausen jumps[5]. Barkhausen jumps are associated with cascade movement of domains, where the sudden change in the local energy proﬁle, due to the depinning of domains and coherent rotation of entire domains, allows for clusters of domains to propagate and rotate. Further decreases to the ﬁeld result in complete propagation of the reversal domains and coherent rotation of off- axis domains becomes the dominant source of the magnetic susceptibility (see section 1.1.8 for a brief discussion of magnetic susceptibility)[5].

14 M Idealized

Ms Realistic A C MR B -Hc

Hc H

Fig. 2: Ferromagnetic hysteresis loops of idealized and real magnets. For the idealized case, the remanent magnetization (the magnetization when the field is reduced to zero after saturation), MR, is equal to the saturation magnetization, Ms. The coercive field, Hc, is equal to the saturation field, Hs (not depicted here), and either the nucleation field (A), Hn, or the depinning field (B), Hdp. Label A shows the susceptibility of the virgin curve for a nucleation-type magnet, while label B shows that of a pinning-type magnet. For a real magnet, the remanent magnetization is less than the saturation magnetization due to coherent rotation and nucleation of reversal domains. The virgin curve’s, labeled as C, susceptibility has features of both nucleation and pinning. This is the depiction of a real pinning-type magnet. The susceptibility (the slope) is initially small as the reversal domains become pinned. Once H > Hdp, the susceptibility increases as the reversal domain propagate through the magnet. In a real nucleation-type magnet, the susceptibility would initially resemble that seen for the idealized nucleation-type magnet, but eventually decrease once the domains propagate completely and coherent rotation becomes dominant (this also occurs in pinning-type magnets as is roughly at the point of inflection of the virgin curve labeled C).

15 1.1.7 Magnetic Ordering- Landau Theory

Thermodynamically, the phase transition from a disordered to ordered state can be modeled phenomenologically-for second-order phase transitions- via Landau theory. The free energy is expressible as a power series of the order parameter; for magnetic ordering, the order parameter is the magnetization, M. The free energy is then:

1 1 1 F (M,T ) = F + α(T )M + β(T )M 2 + γ(T )M 3 + δ(T )M 4 + ··· 0 2 3 4

Above the critical temperature Tc, where the phase transition occurs, the system is

disordered (say, paramagnetic); thus M = 0. Below Tc, M will take ﬁnite values. Since M breaks time-inversion symmetry, and F must be invariant under space-and-time-inversion (F has units of joules, which is even in both space and time), F can not contain odd powers of M (hence F would invert sign under time-inversion, which is forbidden). Thus:

1 1 F (M,T ) = F β(T )M 2 + δ(T )M 4 + ··· 0 2 4

If β and δ are positive, the minimum of the free energy occurs at M = 0. If β is negative,

1 the minimum of the free energy occurs when |β(T )/δ(T )| 2 = |M| and thus, M is spontaneously non-zero. Therefore, the phase transition is concomitant with the change in sign of β. Typically, one takes δ to be constant and positive, and β to equal:

β(T ) = (T − Tc)β (21)

where Tc is the curie temperature for a ferromagnet. β is thus positive for T > Tc and negative

for T < Tc. Macroscopically, this magnetization averages to zero, due to the formation of domains (see section 1.1.5), and is only revealable with the application of a magnetic ﬁeld.

Above Tc, i.e., when β is positive, paramagnetism- disorder- is observed. The Curie-Weiss law describes this disorder in ferromagnetic materials.

16 1.1.8 Curie-Weiss Paramagnetism, Magnetic Susceptibility, and the Molecular Field

The Curie-Weiss law has the form[12]:

C χ = (22) (T − Tc) where χ is the magnetic susceptibility and C is a material dependent constant, known as the Curie constant.

The Curie-Weiss law originates when one considers a material to generate an internal field when magnetized. This field, Hm, known as the molecular field, or Weiss field, is an example of a mean-field: all the internal interactions of a magnet are averaged into one a single quantity, the Weiss field. The Weiss field is expressed as[8]:

Hm = λM

Where λ is a material dependent constant and M is the magnetization. The total H-ﬁeld is then:

Htot = Hext + HM

CH Assuming M has the form[8] M = tot , where C is, again, the Curie constant: T

C(H + λM) M = ext T

C M = H T − Cλ ext

The Magnetic susceptibility is deﬁned as[6]:

dM χ = (23) dH

17 dM If the material is linear, such that is constant for constant temperature: dH

C C χ = = (24) T − Cλ T − θ

For ferromagnetic materials, λ > 0, hence θ > 0. Moreover, the curie temperature, Tc, is equal to θ. Often, one is interested in χ−1, since one can determine the Curie temperature

−1 T −θ when χ = 0, hence 0 = C ⇒ T = θ. Additionally, from C, the effective moment µeff is calculable[12]:

1 1 2 2 2 µeff = g[J(J + 1)] µB = (3kBC/nNAµB) (25)

where g is the Lande´ factor, J is the total angular momentum quantum number, kB is

Boltzmann’s constant, nNA is the number of magnetic ions per mole, and µB is the Bohr magneton. Thus:

µeff 1 = (3kB/nNA)C 2 (26) µB

This relation may be used to calculate the effective moment from the magnetic susceptibility, which provides information on the total angular momentum, such as orbital angular momentum quenching. Angular momentum quenching describes a discrepancy in the effective moment: the effective moment follows the spin contribution rather than the total angular momentum. In other words, the orbital angular momentum, despite being non-zero, does not contribute to the effective moment[5].

18 1.2 Antiferromagnets and Ferrimagnets

In antiferromagnetics and ferrimagnetics, the magnetic structure may be viewed as two ferromagnetic-like sub-lattices with opposing magnetization (see ﬁgure 3). In antiferromagnets, the magnitude of the magnetization of each sub-lattice is equal. In ferrimagnets, the magnitude

is not necessarily equal. In the antiferromagnetic case, there are two Weiss fields[5]: λ1 and λ2, where λ1 denotes the inter-sublattice Weiss field, and λ2 denotes the intra-sublattice Weiss field. It should be noted that there exists four λ parameters but, in the case of an ideal antiferromagnet, only two are unique (hence each sublattice will have equal but opposite Weiss fields). Here,

λ1 < 0 and λ2 > 0.

The Curie-Weiss law of an antiferromagnet and a ferrimagnet has the same form as a ferromagnet, but has an important distinction: because there exists two molecular ﬁelds, θ has the form[5]:

1 θ = 2 C(λ1 + λ2) (27)

Often, in the antiferromagnetic case, θ is found to be negative, indicating that the magnitude of the inter-sublattice ﬁeld is larger than the intra-sublattice ﬁeld. The transition temperature, known as the Neel´ temperature, is equal to:

1 TN = 2 C(λ2 − λ1) (28)

The susceptibility above TN is equivalent to the ferromagnetic case, seen in equation (24). λ1 and λ2 are therefore deduced by measuring the susceptibility and calculating θ. TN is observable in the susceptibility by inspection.

Antiferromagnets differ from ferromagnets and ferrimagnets in that they generate no demagnetizing ﬁeld, hence the two sub-lattices cancel exactly[5]. This suggests that antiferromagnets should not form domains, since there is no need to minimize the stray ﬁeld when it does not exist. However, domains do form, but their formation is complex and will not be discussed here, other than to say entropy is the typical culprit[5].

19 Fig. 3: The possible magnetic structures in the hexagonal molybdates. The arrangement (A) depicts the structure when the intralayer interaction is antiferromagnetic and interlayer interaction is ferromagnetic. (B) shows antiferromagnetic intralayer interaction and antiferromagnetic interlayer interaction. (C) has ferromagnetic intralayer and antiferromagnetic interlayer. (D) illustrates ferromagnetic interaction in both intra-and-interlayers. Flipping every other layer in (B) results in the structure presented in (A).

20 1.2.1 Spin-ﬂop and Linear Rotation

When two antiferromagnetic coupled spins are exposed to an external magnetic field, the spins will develop a new energy minimum that depends on the magnitude and direction of the external field and is prescribed by the Zeeman interaction (see section 1.1.2). We can construct a simple model to explore this behaviour. Suppose two spins are aligned antiparallel- due to the exchange interaction- and are exposed to an external B-field which has an arbitrary direction but is constrained to the plane formed by the two spins, i.e., the model is 2-D. The Zeeman interaction for the two spins is:

ˆ ˆ EZ = −µB(S1 + S2) · B (29)

ˆ ˆ Where S1 and S2 are unit vectors in the direction of the spins. B can be broken into two terms: B⊥ and Bk. Here, it is to be understood that perpendicular refers to the direction ˆ ˆ perpendicular to S1 and S2. Consequently, equation (29) can be expanded to:

⊥ ˆ ˆ EZ = −µB(S1 + S2) · B⊥ (30)

k ˆ ˆ EZ = −µB(S1 + S2) · Bk (31)

Let us now examine these terms in isolation and include the exchange interaction (we use 1 ˆ Si = 2 Si, ~ = 1 for simplicity):

J U ⊥ = − µ (Sˆ + Sˆ ) · B + (Sˆ · Sˆ ) (32) B 1 2 ⊥ 4 1 2

J U k = − µ (Sˆ + Sˆ ) · B + (Sˆ · Sˆ ) (33) B 1 2 k 4 1 2

Here, the anisotropy energy that would occur in a real crystal is absent. In truth, the

21 B∥

θ1 θ1

B α ⊥ β

θ2

Fig. 4: [Left]α: Linear rotation of an antiferromagnetic spin-pair under a perpendicular applied field (blue arrow). The black arrows are the original spin directions in absence of field. The grey arrows are the direction after rotation towards B⊥. [Right] β: Spin-flop of an antiferromagnetic spin-pair under a parallel applied field (blue arrow). Again, black arrows indicate initial spin directions. Gray arrows indicate the spin directions after the spin-flop transition.θ1 and θ2 are the angles between the initial spin direction and the rotated/flopped direction in both cases. The spin-flopped case, β, mimics the linear rotation case, α, after the transition.

anisotropy is necessary for equation (33); however, we can constrain our model for equation (32) to simulate this, as we will see shortly. Equation (32) may be written in terms of the trigonometric functions:

J U ⊥ = − µ (cos( π − θ ) + cos ( π − θ ))B + cos(π − θ − θ ) (34) B 2 1 2 2 ⊥ 4 1 2

î ˆ î Where θ1 is the angle between the initial S1 direction and S1 and θ2 is the angle between S2 ˆ and S2 (see figure 4 for a diagram of the model). Using the constraint that θ1 = θ2 (this prevents the entire system from rotating in space, much like the anisotropy would) and J = −|J|, equation (34) becomes:

22 |J| U ⊥ = − 2µ sin θ B + cos(2θ ) (35) B 1 ⊥ 4 1

⊥ Taking the derivative of U w.r.t θ1 and setting it equal to zero, we ﬁnd that the minimum of U ⊥ occurs when:

2µ B sin θ = B ⊥ (36) 1 |J|

π Constraining θ1 to the range of 2 ≥ θ1 ≥ 0, it can be seen from equation (36) that the π spin-pair will align antiparallel (θ1 = 0) when B⊥ = 0 and parallel (θ1 = 2 ) to B⊥ when |J| |J| B⊥ = . For > B⊥ > 0, the spin-pair will rotate towards B⊥ by an amount determined 4µB 2µB by equation (36). This model can be extended to a large number of spins by utilizing mean field theory. This has the additional benefit of including the anisotropy. It can be shown [8] that the susceptibility of an uniaxial antiferromagnet in a perpendicular field is temperature independent, and its magnitude is equal to the negative reciprocal of the inter-sublattice Weiss constant. This can be seen from the above model by realizing the components of the spins

perpendicular to B⊥ are antiparallel and therefore cancel. The magnetic moment of the system is thus:

M⊥ = 2µB sin θ1 (37)

Plugging in equation (37) into equation (36), we arrive at:

4µ2 B M = B ⊥ (38) ⊥ |J|

Since M⊥ is linear in B⊥, the susceptibility is constant. I will refer to this phenomena as linear rotation, to distinguish it from spin-canting and spin-ﬂopping. We can also examine the

k effects of a parallel ﬁeld. U has the same form as U⊥:

23 J U k = − µ (cos θ + cos(π − θ ))B + cos (π − θ − θ ) (39) B 1 2 k 4 1 2

However, the lack of anisotropy is no longer sufﬁcient to extract any useful information. Therefore, we will consider the competition between the Zeeman interaction energy and the anisotropy energy, assuming that the spins exist in a uniaxial crystal and the easy axis is along

Bk. Initially, the Zeeman interaction energy for the spin-pair is held to zero by the exchange

interaction and the anisotropy energy. However, with application of a sufﬁciently strong Bk ﬁeld, the Zeeman interaction will overcome the anisotropy energy and the spin-pair will rotate

perpendicular[13] to Bk, without drastically changing their relative angle, to minimize both the Zeeman interaction and the exchange interaction. This behaviour is known as a spin-ﬂop transition and occurs when[5] U⊥ = Uk. For an uniaxial crystal with a single easy-axis, this p occurs when [13] H⊥ = 2HExchangeHAnisotropy. After this rotation, the magnetic moment of the system increases linearly until saturation [5], as expected from our results for linear rotation, described by equation (37).

24 1.3 Ferroelectricity

Ferroelectricity describes a state in which electric dipoles spontaneously order. Ferroelectricity- being the ordering of analogous dipoles- shares many of the properties of ferromagnetism, such as: the existence of hysteresis, saturation polarization, remanent polarization, coercivity, and a phase transition from a paraelectric state to a ferroelectric state at the Curie temperature; the formation of domains and associated domain walls; etc. The origin of these properties can be explained using similar methodology to ferromagnetism. One notable exception, with regards to domains, is: ferroelectric domains are dominated by the anisotropy energy, since there is no analogous exchange interaction type energy[6].

The phase transition from paraelectricity to ferroelectricity can result-other sources exist, such as charge ordering[14]- from a structural phase transition, i.e, a non-polar lattice transitions into a polar lattice (polar point groups are a subset of non-centrosymmetric point groups). Consider the dipole moment,p, for an array of n point charges[10]:

n X p = qiri (40) i=1 where q is the charge of each point and r is the location of each point w.r.t. the origin. It is clear that spontaneous polarization can arise from a structural phase transition when one

◦ considers lead titanate(PbTiO3). PbTiO3 undergoes a structural phase transition at 490 C , tetragonally distorting the perovskite-like lattice. With respect to the oxygen atoms, Pb atoms shift by 0.47 Aand˚ Ti atoms shift by 0.30 A;˚ the shifts occur in the same direction[27]. Evidently, spontaneous polarization occurs because cation and anions develop non-equivalent centres of charge, i.e., over a unit-cell, equation (40) does not vanish.

Because ferroelectric materials form spontaneous polarization, they are necessarily electrical insulators. The dielectric function, in absence of DC ﬁelds, is describable above the

Curie temperature, Tc, by[15]: C r = (41) T − Tc

25 , where C is the Curie-Weiss Constant. Below Tc[15]:

C r = (42) 2(Tc − T )

1.4 The Magnetoelectric Effect

We have now covered some basics about magnetism and the properties of magnetically ordered systems. We will now examine the magnetoelectric effect and its sources in intrinsic (intrinsic meaning both orders pertain to a single compound) multiferroics.

The magnetoelectric effect is the coupling between magnetic(electric) fields and electric polarization(magnetization). Materials may-it is necessary but not a sufficient condition- exhibit this effect if the magnetic point group breaks both spatial and time-inversion symmetry. Magnetic ordering guarantees temporal symmetry is broken, hence magnetic moment is a pseudovector, i.e, L(−t) = r × v(−t) = −L(t). Spatial symmetry may be broken by complex magnetic behaviour, such as magnetic spiral structure[3]. In polar crystallographic phases, spatial inversion is broken by definition. Coupling can occur in a linear and non-linear fashion.

One can express linear coupling-in fact, higher order coupling is also described by Landau theory, but we will focus on linear- phenomenologically via Landau theory[16]. Take the total electric ﬁeld to equal:

T a s Ei = Ei + Ei (43) where the subscript i denotes the ith direction, i.e., x,y, and z, and the superscripts a and s denote the applied field and the spontaneous field, i.e., the internal field due to ordering, respectfully. The superscript i denotes the induced field, as we will see later. Similarly, let the total magnetic field equal:

T a s Hi = Hi + Hi (44)

Which follows the same convention described above. The free energy density (units of J/m3) of

26 an electric and magnetic sample, in the presence of external Ea and Ha ﬁelds, may be written as (this follows the Einstein summation convention):

1 1 G(Ea,ET ,Ha,HT ,T ) = G − EaET − µ HT Ha − α ET HT − · · · (45) o 2 ij i j 2 ij i j ij i j

The polarization (magnetization) is equal to the negative derivative of G(E,H,T ) w.r.t E (H), at a constant temperature[6]. Therefore, in zero applied electric (magnetic) ﬁeld:

dG T T − s = Pi = αijHj + ··· (46) dEi

dG T T − s = µ0Mj = αijEi + ··· (47) dHj

And, in zero applied magnetic (electric) ﬁeld:

s s Pi = αijHj (48)

s s µ0Mj = αijEi (49)

Thus, we can conclude that, in equations (46) and (47):

i a Pi = αijHj (50)

s a µ0Mj = αijEi (51)

Here, P s, P i, M s, and M i are the spontaneous polarization, the induced polarization, the spontaneous magnetization, and the induced magnetization, respectfully. It can be seen that, if

s s the term αijEi Hj in the free energy is non-zero, coupling to applied ﬁelds is guaranteed by virtue of non-zero αij[16],i.e., equations (50) and (51) are guaranteed non-zero if equations (46) and (47) are non-zero. Thus, linear coupling is guaranteed in materials which have non-

s s zero αij, M , and P .

Non-zero α is determined by the symmetry conditions of the magnetic point group, hence

27 α is localized at the magnetic ion sites[16]. In order for magnetoelectric coupling to exist, by Neumann’s principle[17], the magnetic space group must break spatial and time-inversion symmetry. That is to say, since α is localized to the magnetic ion sites, the local symmetry condition of the magnetic ion sites must allow for the simultaneous existence of electric and magnetic dipole moments, which requires that spatial and inversion symmetry is broken[16].

One expects to observe a change in P in a crystal which is ordered electrically and undergoes a subsequent magnetic ordering, provided α 6= 0. Such is the case for Co2Mo3O8

(see section 3.4.1) and Fe2Mo3O8[18], which orders ferroelectrically above 300K and magnetically at approximately 40K and 60K, respectfully. This is also the source of the strong magnetoelectric coupling in the vicinity of the Neel´ temperature and critical ﬁeld in

Fe2Mo3O8: As iron molybdate undergoes a metamagnetic transition from antiferromagnetism to ferrimagnetism, the sharp changes to M leads to large changes in P.

Magnetoelectric materials which exhibit tangible coupling are rare. Ferroelectric materials tend to be transition metal oxides and favour empty d-orbitals[3, 19]. This is referred to as “d0-ness.” Contrary to this, magnetic ordering in 3d transition metals favours half-ﬁlled orbitals(d5-ness). Also, simultaneous, strong, spontaneous polarization and magnetization does not guarantee strong magnetoelectric coupling. BiMnO3 shows simultaneous magnetization and polarization, yet the magnetoelectric coupling is exceedingly weak because the polarization and magnetization are associated with different ionic sites[3] (see above).

The magnitude of the magnetoelectric response, i.e., the magnitude of the coupling constant α, is limited by the magnetic and electric phases. W. F. Brown, Jr. et al.[20, 21] showed that the upper limit of the coupling constant is given by:

1 α ≤ (µ) 2 (52)

This relation ensures that antiferromagnetic materials will limit the coupling constant, hence they have low magnetic susceptibilities. Ideally, one strives to construct a material that

28 orders ferroelectrically and ferromagnetically. The mutual exclusivity of d0-ness and d5-ness prevents this. Additionally, the units of alpha are clearly seconds per metre, since the speed of

−0.5 light is equal to c = (µ00) and the relative permittivities are unitless.

1.5 The Dzyaloshinskii-Moriya Interaction

A source of magnetoelectric coupling in intrinsic multiferroics is the so-called inverse Dzyaloshinskii-Moriya interaction (IDMI). Theoretically, the Dzyaloshinskii-Moriya (DMI) arises from the superexchange phenomena[22] when one considers spin-orbit coupling. It has the form[22, 23]:

HDM = Dn,n+1 · [Sn × Sn+1] (53)

Here, n corresponds to individual spins. The vector D, the Dzyaloshinskii vector, is proportional to the spin-orbit coupling constant λ[3] and is a geometric quantity[22], i.e., it depends on the local symmetry conditions (see below). This interaction favours spin-canting of otherwise collinear-spin structures[3], e.g., it is the source of weak ferromagnetism in antiferromagnetic

α-Fe2O3[23].

The DMI is a necessary component in the formation of some spin-cycloid structures[24], such as in BiFeO3[3, 25, 26]. However, this requires spatial inversion symmetry breaking throughout the crystal[24]; the DMI between two spins requires only local symmetry breaking. It was shown by I. Sosnowska and A.K. Zvezdin that the formation of these spin-cycloids is characterized via the Lifshitz invariant[3, 24, 26]:

P ∝ ((M · ∇)M − (∇ · M)M) (54)

To examine how the DMI induces ferromagnetism or ferroelectricity, one can modify the nearest-neighbour exchange Hamiltonian to include the DMI:

H = Jn,n+1(Sn · Sn+1) + Dn,n+1 · [Sn × Sn+1] (55)

29 and apply it to a 1-D chain of spins. Take Jn,n+1 to be positive and constant- so that the superexchange interaction results in antiferromagnetism- and set the easy-axis along the

x-axis. Dn,n+1 is proportional to[3] dn,n+1 × rn,n+1, where dn,n+1 is the displacement of the exchange path ion, e.g., oxygen, perpendicular to the chain and rn,n+1 is the displacement between successive magnetic ions (see ﬁgure 5). This ensures that, for two ions coupled via superexchange, D will be perpendicular to the plane formed by the magnetic ions and their connecting ligand. Note: this relation only determines the sense of Dn,n+, not the orientation,

i.e., Dn,n+ can be parallel or antiparallel to dn,n+1 ×rn,n+1. For the proceeding work, excluding

ﬁgure 8, the orientation will follow parallel to dn,n+1 × rn,n+1.

If one takes the axis of the chain- rn,n+1 is parallel to this axis, by deﬁnition- to be along

the x-axis and dn,n+1 to be along the y-axis, then Dn,n+1 is constrained to the z-axis. The

sign of Dn,n+1 depends on dn,n+1; in other words, to the position of the oxygen atoms. If

dn,n+1 = −dn+1,n+2 (we will take d1,2 to lie along the positive y-axis), then Dn,n+1 alternates

direction between z and -z and Dn,n+1 · [Sn × Sn+1] is minimized when Sn × Sn+1 alternates

oppositely, e.g., if D12 k z, S1 × S2 k −z, and the angle between Sn and Sn+1 is π/2. This

y Oxygen Ion Magnetic Ion Spin

-z x M

d θ θ1 23 4 1 2 3 4 d12 d34

Fig. 5: The DMI interaction in a 1-D chain of antiferromagnetic spins. The interaction induces a weak ferromagnetic moment, in the positive y-direction, (indicated here as M) in the otherwise antiferromagnetic chain. Black arrows indicate the initial directions of the spins, grey arrows indicate the canted spins due to the DMI. θ shows the change in the angle between the canted and initial spins. θ is exaggerated for clarity. Here, Dn,n+1 alternates direction; for spin pair [1,2], it is into the page and thus [S1 × S2] must point out of the page. For spin pair [2,3], D2,3 is out of the page and [S1 × S2] is into the page. The DMI is thus minimized when the angle π between any two spins in , which results in canting of the spins towards one another. 2

30 results in a canting of the spins: the minimizations of Sn ·Sn+1 and D12 ·Sn ×Sn+1 are mutually exclusive, i.e., Sn · Sn+1 is minimized when Sn is antiparallel to Sn+1, but D12 · Sn × Sn+1 is minimized when Sn is perpendicular to Sn+1. The alternation of the sign and the proclivity of the DMI towards perpendicular spins causes all spins to cant towards the positive y-axis, thus inducing a ferromagnetic moment in the direction of the positive y-axis (see ﬁgure 5).

Consider now if Jn,n+1 is negative and constant, so that the superexchange interaction is ferromagnetic. Let the 1-D chain be aligned along the x-axis and let the ferromagnetic moment be parallel to the positive x-axis. Again, let dn,n+1- and thus Dn,n+1- alternate in sign for each successive n, i.e., dn,n+1 = −dn+1,n+2, and let dn,n+1 be along the positive y-axis for odd n. As before, Dn,n+1 will alternate direction between −z and z. Then, to minimum HDM , Sn × Sn+1 must alternate between z and −z(θn alternates sign). Since the exchange interaction aligns the spins ferromagnetically, the spins will cant to increase the angle between each spin pair. This induces an antiferromagnetic moment(see ﬁgure 6).

The previous two cases are examples of systems in which spatial inversion symmetry is broken only locally, i.e., between the superexchange paths. With respect to the magnetic ions, inversion symmetry holds. Take an arbitrary nth magnetic ion, which is surrounded by two

1 1 oxygen atoms, located at roxy1 = − 2 rn−1,n + dn−1,n and roxy2 = 2 rn,n+1 + dn,n+1, with respect

y Oxygen Magnetic Ion Spin

-z x M

d 23 θ4 1 2 3 4 θ1 d12 d34

r12

Fig. 6: The DMI interaction in a 1-D chain of ferromagnetic spins. This ﬁgure follows the same convention as outlined in ﬁgure 5. The induced moment is referred to as antiferromagnetic because it tends to align spins antiparallel, which decreases the ferromagnetic moment.

31 d

Fig. 7: A spin-cycloid induced by the DMI. This ﬁgure follows the same convention establish in ﬁgure 5. Here, Dn,n+1 is out of the page. The DMI is thus minimized when [Sn × Sn+1] is into the page, causing a clockwise rotation of each spin.

to the magnetic ion. By deﬁnition, −dn−1,n = dn,n+1 and rn−1,n = rn,n+1. Thus inversion

symmetry is not broken, hence roxy1 = −roxy2 . This type of interaction is sometimes called the microscopic DMI.

Instead of a dn,n+1 vector which alternates sign, one can construct dn,n+1 to be constant. Again, we will examine the 1-D spin-chain. Let the direction of the chain be the positive x- axis and allow the easy-axis to be along the y-direction. Let dn,n+1 be parallel to the positive y-axis; Dn,n+1 is then in the −z direction. In order for HDM to be negative-and thus producing a new minimum energy of the system- [Sn × Sn+1] must always be in the z direction, implying that θn is always positive. If Jn,n+1 is negative (ferromagnetism), this results in a spin-cycloid state, hence the spins will cant away from their neighbours and always in the same angular direction(see ﬁgure 7). The direction of the cycloid depends on the sign of dn,n+1. However, if

Jn,n+1 is positive (antiferromagnetism), the minimum of HDM = 0, since the condition that θn is positive for all pairs is not satisﬁable (each subsequent spin begins out of phase by π).

Spin canting, resulting from frustration, can form spin-spiral states[3]. Consider the exchange interaction Hamiltonian for a 1-D system, including next-nearest-neighbour

32 coupling:

0 H = Jn,n+1Sn · Sn+1 + Jn,n+2Sn · Sn+2 (56)

0 = −|Jn,n+1| cos θn + Jn,n+2 cos(θn + θn+1) (57)

0 where Jn,n+1 < 0 and Jn,n+2 > 0. If we ignore the next-nearest-neighbour interaction, the

0 spins align collinearly (speciﬁcally, ferromagnetically). However, when we consider Jn,n+2, the additional coupling will induce a canting of the spins from their collinear state, hence:

dH(θ) = 0 = |J | sin θ − 2J 0 sin(2θ) (58) dθ n,n+1 n,n+2

Here, we apply the condition that θn = θn+1 = θ, which would describe a spin cycloid. Using the small angle approximation:

|Jn,n+1| 4 = 0 (59) Jn,n+2

This gives one some idea of how the spiral can form. In reality, the condition to stabilize a J0 spin cycloid state is that[3] the magnitude of n,n+2 > 1 . |Jn,n+1| 4

Ferroelectricity can result from the spin-spiral states mentioned above due to the inverse Dzyaloshinskii-Moriya interaction. Consider the previous 1-D spin-chain where a spin-cycloid has been stabilized via frustration. Take the spin-cycloid to propagate along the x-axis and to be describable as:

Sn = S(sin(kxn)ex − cos(kxn)ey) (60)

where xn is the coordinate of each spin and k is the magnitude of the wave vector. The vector

product Sn × Sn+1 is non-zero and is parallel to the positive z-axis for π > k(xn+1 − xn) > 0. J0 This constraint on k is guaranteed by the condition n,n+2 > 1 , hence equation (58) becomes |Jn,n+1| 4

2 sin 2θ > sin θ, which is satisﬁed for 0 < θ < 1.318. Thus, HDM is minimized when Dn,n+1 is

parallel to the negative z-axis. Take dn,n+1 to have an initial magnitude of zero. By deﬁnition,

33 y Oxygen Ion Shifted Oxygen Ion Magnetic Ion

-z x

Δd P

Fig. 8: Polarization induced by the spin-cycloid described by Sn = S(sin(kx)ex − cos(kx)ey). th x0 corresponds to the central magnetic ion (the 4 magnetic ion from the left or right). Here, Dn,n+1 is out of the page. The presence of non-zero Sn × Sn+1 leads to a shift in the position of each oxygen ion in order to minimize HDM . Since every oxygen ion shifts in the same direction, a net displacement between the magnetic ion and oxygen ion sublattices develops, leading to a net polarization.

rn,n+1 is parallel to x; thus, shifting dn,n+1 along the y direction minimizes HDM . Thus, all, e.g., oxygen ions, shift in the same direction, generating a polarization along the -y-axis(see ﬁgure 8). The magnitude of the shift is limited by competition with the crystal lattice. In other words, the lattice relaxes in order to minimize the magnetic energy. This phenomena is known as the antisymmetric exchange striction: antisymmetric because the effect is due to the antisymmetric portion, i.e., the vector product, of the superexchange phenomena. The Lifshitz invariant for

2 2 2 2 this spin cycloid predicts P ∝ S [sin (kx) + cos (kx)]ey = S key. Note: the difference in sign is due to the chosen direction of D.

In the case It is worth noting that the DMI acts as a source for a change in the magnetic state of as system (non-zero D induces non-zero Sn × Sn+1), while the IDMI is a response to a change in the magnetic state of a system (non-zero Sn × Sn+1 induces non-zero D).

While the antisymmetric exchange striction mechanism is invokable in systems with complex magnetic behaviour, it is also possible for magnetoelectric coupling to arise in system with collinear, or near collinear, spin states via symmetric exchange striction[29, 30]. Fundamentally, the mechanisms of the symmetric exchange are similar to the antisymmetric exchange, i.e., lattice relaxation in response to an increase in magnetic energy; however, as the name implies, symmetric exchange is associated with the conventional superexchange:

34 H = Jn,n+1(Sn · Sn+1). Recently, Y. Wang et al.[18] observed this phenomena in iron molybdate (Fe2Mo3O8), known colloquially as kamiokite.

1.6 Kamiokite and The Symmetric Exchange Striction

Iron molybdate is a polar magnet,i.e., it exhibits magnetic ordering in a crystallographic polar phase, with space group P63mc (see ﬁgures 9 and 10 for the crystal structure). It belongs to a family of transition metal hexagonal molybdates, with the general formula M2Mo3O8, where M is a transition metal in the 2+ oxidation state. This material forms in layers of alternating iron and molybdenum. The molybdenum atoms occur in a 4+ oxidation state and do not contribute to the magnetization: they occupy the singlet state[18]. The layers of iron atoms have a honeycomb-like structure. The iron atoms form alternating octahedral and tetrahedral arrangements with the surrounding oxygen ligands[34]. The octahedral and tetrahedral sublattices are displaced by 0.614 A˚ along the c-axis[18, 34]. Below 60K, iron molybdate orders antiferromagnetically. With application of a moderate magnetic ﬁeld along the c-axis, i.e, perpendicular to the hexagonal layers, a metamagnetic transition, into a ferrimagnetic state, can be induced. In this state, Y. Wang et al. report linear, giant

dM magnetoelectric coupling (µ0 dE = −5700ps/m) in the vicinity of the Neel´ temperature.

The magnetoelectric coupling in kamiokite is attributed to- though this may be disputed[35]- symmetric exchange striction[18]. As the magnetic structure magnetically orders into an antiferromagnetic state, the lattice undergoes a distortion- from the paramagnetic state- due to exchange striction; characterized mostly via oxygen atom displacement[18]. During the metamagnetic transition, induced via applied ﬁeld in the antiferromagnetic state, into the ferrimagnetic state, the lattice partially relaxes the aforementioned distortion; again, this is due to exchange striction. In the vicinity of this transition, giant, nearly linear magnetoelectric coupling is observable.

This metamagnetic transition is associated with the ﬂipping of the moment of every other Fe layer[18]. Each Fe layer has an intraplanar ferrimagnetic structure and each layer is aligned

35 along the c-axis- the easy-axis- antiparallel to its neighbouring planes, resulting in a net antiferromagnetic moment. This implies a ferromagnetic interplanar superexchange coupling[18]. Application of a moderately large ﬁeld (≈3.5T) along the c-axis realigns the antiparallel planes, i.e., along the negative c-axis, resulting in a ferrimagnetic state. Wang et al. found that the this is concomitant with an increase in the Fe-O-Fe angle (from 109◦ to 111◦) involved in intraplanar superexchange. They associate this with more favourable orbital overlapping. This can be understood by examining Anderson-Goodenough-Kanamori rules[48, 49], which states: superexchange is antiferromagnetic (J>0) for collinear (θ=180◦) ion-cation-ion bonds and ferromagnetic (J<0) for perpendicular(θ=90◦) bonds. M. A. Subramanian et al.[48] discovered that J varies continuously with the bonding angle, θ. Therefore, an increase to θ leads to an increase in the intraplanar J and a decrease in the magnetic energy.

The resulting magnetoelectric coupling is evidently due to the change in spontaneous magnetization,i.e., the Hs term discussed in the Landau free energy (see section 1.1.7), induced via the metamagnetic transition. Similarly, it is expected that the dielectric function of kamiokite shows an anomaly under a transition into the antiferromagnetic state from the paramagnetic state, hence Hs is altered under the magnetic ordering. Indeed, this was observed in kamiokite, both in this work and by Y. Wang et al[18].

The ferrimagnetic state may also be stabilized via zinc doping[18, 36, 47]. Zn2Mo3O8 exists in an equivalent crystal symmetry as Fe2Mo3O8 and, hence, zinc doping of Fe2Mo3O8 will not alter the crystal symmetry[36]. Fe2+ ions prefer to occupy the octahedral sites, with a ratio of 9:1 (octahedral to tetrahedral) in FeZnMo3O8. FeZnMo3O8 has a Curie temperature of 20K.

Mn2Mo3O8 is another member of the hexagonal molybdates. Mn2Mo3O8 is a polar ferrimagnet, making it a candidate for the magnetoelectric effect. In fact, T. Kurumaji et al. determined that Mn2Mo3O8 does exhibit the magnetoelectric effect, and the magnitude of the response may be increased by a factor of 4 via iron doping[37].

36 Cobalt molybdate (Co2Mo3O8) is another potential material of this type. Experiments in X-ray and neutron diffraction[32] suggest that cobalt molybdate has similar magnetic and crystal structures as iron molybdate, making it a promising candidate for the multiferroic magnetoelectric effect. However, the effect, in this material, has not yet been investigated.

Similar to iron molybdate, cobalt molybdate has an antiferromagnetic ground-state[31, 32] which occurs below 40.8K[31]. It may be possible that cobalt molybdate would exhibit a similar metamagnetic transition, under application of an applied ﬁeld, to that observed in iron molybdate. However, Mossbauer¨ spectroscopy has not been performed on cobalt molybdate, so one can not be sure if the intraplanar moment is ferrimagnetic or antiferromagnetic, i.e., if the octohedral and tetrahedral sites have distinct magnetic moments. Regardless, this state may also be achieved via doping if certain elements, with magnetic moments distinct from Co2+, show preference towards certain sites, as Zn does in iron molybdate.

With this idea at the zenith, various samples of cobalt molybdate, with and without dopants, were synthesized and tested. Dopants were chosen from the family of 3-d transition metal hexagonal molybdates, which includes Mn, Fe, Ni, and Zn. Zinc, in the 2+ oxidation state, has a full valence shell and should not be magnetic, while nickel molybdate does not magnetically order[31] at all temperatures. Between these two, Zn was chosen as the non-magnetic dopant. Of the remaining two elements (Fe and Mn), iron was chosen as the focus.

37 Fig. 9: The crystal structure of cobalt molybdate along the c-axis. The crystal symmetry is retained for any of the following 3-d transition metals: Fe, Zn, Mn, Ni. As such, it is to be understood that the cobalt atoms represent M2+ sites. For instance, replacing the cobalt atoms with iron atoms results in the crystal structure of Fe2Mo3O8.

38 Fig. 10: The crystal structure of cobalt molybdate along the a-axis. Of note is the displacement between adjacent cobalt atoms, which is indicative of the polar nature of the crystal structure. The crystal symmetry is retained for any of the following 3-d transition metals: Fe, Zn, Mn, Ni. As such, it is to be understood that the cobalt atoms represent M2+ sites. For instance, replacing the cobalt atoms with iron atoms results in the crystal structure of Fe2Mo3O8.

39 2 METHODS

2.1 Chemical Vapour Transport

(Co(1−x)Mx)2Mo3O8 (M=Fe, Zn, Mn) samples were synthesized via chemical vapour transport. Chemical vapour transport is a technique used to grow crystalline materials, namely single crystals. Non-volatile precursor compounds, contained within a sealed ampoule, are subjected to a temperature gradient in the presence of a transport agent, e.g., gaseous chlorine. The purpose of the transport agent is to volatilize, via reaction (the precursor ’dissolves’ into the gaseous transport agent), the precursors. The volatilized precursors migrate across the temperature gradient via diffusion and convection. Once across the gradient, the precursor-agent association decomposes into its original constituents, and the precursor is deposited on the surface of the ampoule. The transport agent then migrates back across the gradient and the process repeats (see ﬁgure 11). The precursor-agent reaction is favourable in the source zone and unfavourable in the sink (growth) zone, the locations, with respect to the temperature gradient, of which depends on the energy proﬁle. A precursor-agent reaction which is endothermic is favourable in the hot-zone (Tsnk) and unfavourable in the cold-zone

(Tsrc)[33]. Thus, for an endothermic reaction, the source-zone is located in the hot-zone and the growth-zone is located in the cold-zone.

Several considerations are required to ensure successful crystal growth. The free enthalpy of the precursor-agent reaction can not be too large in magnitude, or transport can not occur. This is because the reaction will tend to be too one sided at temperatures that are conventionally available (one has a range of approximately 103K). If the dissolution of the precursor is highly exergonic[33](∆G/-100Kj/mol), the precursor will easily dissolve, but it will not deposit: the equilibrium point is too in favour of the forward reaction. Conversely, highly endergonic reactions[33] (∆G'100Kj/mol) will easily deposit, but transport will not occur because the material will not dissolve. If the transport rate is too high, the crystals will be prone to defects[33]: an atom may be placed in a location that is metastable and be covered by another layer of atoms before it can be dissolved back into the transport agent, thus

Fig. 11: The chemical vapour transport method for growing single crystals. The diagram depicts a sealed quartz ampoule which is exposed to a temperature gradient (provided by an external source, e.g., a multi-zone furnace). The precursor-transport agent reaction is endothermic in this example. The process is as follows: the precursor and transport agent are placed in the source zone of a quartz ampoule, which is subsequently sealed. The transport agent can take the form of a solid that decomposes before Tsnk and Tsrc are reached, releasing the agent into a gaseous phase; or as a gaseous charge. The former was chosen for this work. At a sufficiently high temperature, the transport agent begins to volatilize the precursor. The transport agent, along with the now volatile precursor, diffuse across the ampoule, into the sink zone. Due to the change in thermal energy, the precursor deposits onto the surface of the ampoule. The now free transport agent diffuses back across the ampoule, and the cycle repeats. creating a defect in the crystal structure. Furthermore, high transport rates can lead to crystal agglomeration-clustering of small crystals. If the number of nucleation sites is large, the material will deposit diversely and individual crystal growth will be minimal. The number of nucleation sites can be reduced by performing a back-transfer: temporarily reversing the orientation of the temperature gradient prior to the intended transport. For a given growth temperature, many crystal structures may be stable and crystals may form in multiple phases. In high temperature systems, undesirable reaction (referred to as ‘attack’) may occur between the precursor and the ampoule, as seen with FeO in the presence of chlorine gas[28]. Recrystallization of quartz, a process known as devitrification, also poses a challenge at high ( 1000◦C) temperatures (see figure 21). The dominate mention of transport -diffusion or convection- is determined by pressure, convection favouring relatively high(>3bar) pressures[33]. These complications are, in general, dependent on similar quantities (temperature, pressure, etc) and, as such, optimization of these systems is inherently difficult.

This is especially true of the transport rate, as it is dependent on eight (Tsrc,Tsnk, P, Psrc,Psnk,

41 cross-sectional area, transport length, stoichiometric coefﬁcients) variables[33].

Due to the above mentioned complication of ampoule-precursor reaction, cobalt monoxide stoichiometry was achieved by using a one-to-one molar ratio of Co2O3(99.9985%) and Co(99.998%). Similarly, iron monoxide stoichiometry was achieved by using a one-to-one ratio of Fe3O4(99.998%) and Fe(99.999%). A 1:1 combination of MnO2(99.999%) and elemental Mn(99.95%) was used for manganese monoxide stoichiometry. ZnO(99.99%) was used directly. PtCl2(99.9%) and TeCl4(99.9%) were used as transport agents.

MoO2(99.98%) was used as a molybdenum source. The ratios of MO:MoO2

(M=Co,Fe,Zn,Mn) and pressure of Cl2 are tabulated in table 1 (section 3.1) and largely followed the suggestion of P. Strobel and Y. Le Page[28].

Quartz ampoules were used to create a closed system. The ampoules (14mm outer diameter, 12mm inner diameter) were cleaned in a 1:3 HNO3:HCl aqua regia solution for two day (exchanging the solution once, after twenty-four hours) prior to evacuation of the ampoules via a turbo-molecular pump. The ampoules were held at approximately 10−6 torr for 24h. The ampoules were then brieﬂy heated (using a standard blow torch) to evaporate any water, with an especial focus on the source and sink zones. The ampoules were then, again, held at 10−6 torr for 24h. Carbon coating occurred at this stage for trials which included it. The interior of the ampoules were coated in acetone. The ampoule was then heated until it glowed a dull red (about 500 ◦C), leaving behind the carbon coating. This was repeated until the ampoule became opaque (thereby ensuring that no quartz was exposed). The transport agent and precursor (the precursor was mixed in a mortar and pestle for approximately 5 minutes prior to installation) were inserted into the source zone of the ampoule by means of a thin glass funnel. The process described above was, again, prescribed. However, the source zone was not heated to prevent the transport agent from decomposing. Following this evacuation process, the ampoules were sealed and placed into a three-zone furnace. The details of the ﬁnal temperatures, back transfers, and ramping time can be found in table 1 (section 3.1).

The grown crystals were separated from the spent ampoules mechanically. Separated

42 crystals were rinsed with 95% ethanol and, in the event that tellurium coated the crystals, aqua regia was employed to remove it. A simple bar magnet was utilized to separate the spinel phase crystals from the desired hexagonal molybdates, hence the spinel phases are ferrimagnetic at room temperature. The grown crystals took three main macroscopic shapes: acicular (elongated along the c-axis), plate-like (elongated ab-plane), and equidimensional. Of the three, acicular was the most common.

2.2 X-ray Diffraction

X-Ray diffraction is describable via Bragg’s law, which states[55]:

2d sin θ = nλ (61)

Where d is the distance between reflection planes (which are lattice planes in a crystal), λ is the wavelength of the incident photons, n is an integer, and theta is the angle between the incident photons and the lattice plane. Bragg’s law may easily be derived by examining the condition of constructive interference for elastically scattered photons off a crystal lattice (see figure 12). If the photons elastically scatter off a surface, the angle of reflection must be equal to the angle of incidence. Letting the distance between the planes be d, the path difference is therefore equal to 2d sin θ (again, see figure 12). Bragg’s law results when one considers the condition for constructive interference: constructive interference of two waves occurs when the path difference is an integer multiple of the wavelength. Thus, 2d sin θ = nλ.

The goal of an x-ray diffraction experiment is to obtain a ’ﬁngerprint’ of a given material and cross-reference this to known diffraction patterns. This is made possible by the fact that only a ﬁnite number of crystal space groups exist, 230 to be exact[17].

X-ray diffraction was performed on a Rigaku SmartLab X-ray diffractometer, which utilizes a copper source. Single crystals of the materials were ground into a ﬁne powder for use in powder diffraction. The Bragg-Brentano geometry was employed for this purpose.

43 θ θ

θ θ

Fig. 12: Diagram depicting Bragg’s law. The red arrows denote the path of the incident and elastically scattered photons. θ is the angle between the incident(reﬂected) photons and the lattice planes (which are separated by a distance d). The path difference between the two rays is highlighted in blue and equal to 2d sin θ.

Monocrystalline silicon was used as a zero-background sample holder. The materials were cross-referenced against the Inorganic Crystal Structure Database (ISCD) and Rietveld reﬁnements were performed on the X-Ray diffraction analysis software PDXL2.

Crystallographic information ﬁles (CIF’s) of Co2Mo3O8 were used in the reﬁnement for

Co2Mo3O8, Co1.5Fe0.5Mo3O8, and Co1Fe1Mo3O8, while Fe2Mo33O8 and Co0.5Fe1.5Mo3O8

used a CIF of Fe2Mo33O8. CIF’s were obtained via the Materials Project[57, 58].

44 2.3 The Magnetization and The Magnetic Susceptibility

The magnetization as a function of temperature, which is proportional to the magnetic susceptibility (see section 1.1.8), was measured in zero-field cooled (ZFC) and field-cooled (FC) format- that is, cooling the sample in the absence and presence, respectfully, of a magnetic field. The susceptibility was measured both parallel and perpendicular (within the ab-plane) to the c-axis. The random error in the susceptibility is about one in one thousand (This is true for all measurements-M(H),M(T ),M(E)- performed using the SQUID).

The magnetization as a function of magnetic ﬁeld was measured using the format of either a full hysteresis loop (Happ = 0Oe→ Happ = Hmax → Happ = −Hmax → Happ = Hmax) or a quarter hysteresis loop (Happ = 0Oe→ Happ = Hmax → Happ = 0Oe). M(H) measurements were performed along both the c-axis and within the ab-plane. Parallel measurements refer to measurements performed along the c-axis of the crystal; perpendicular naturally refers to the ab-plane. This convention is used for all measurements performed.

The demagnetization factor was not considered for any of the magnetization measurements. However, acicular crystals were used to minimize the demagnetizing ﬁeld along the c-axis (see section 1.1.3), which is the axis of most interest to this research.

45 2.3.1 The SQUID Magnetometer

Magnetic susceptibility and M(E) measurements were performed using Quantum Design’s Magnetic Property Measurement System (MPMS- model XL). This magnetometer is capable of generating DC magnetic ﬁelds up to 5.5T, reaching temperatures as low as 1.7K, and has a detection resolution approaching 10−15T[38]. The large operation range and high resolution is made possible by the application of superconductivity in the three key components: the superconducting magnet, the superconducting detection coil and the Superconducting Quantum Interference Device (SQUID).

The superconducting magnet is capable of producing a stable, homogeneous magnetic ﬁeld. The superconducting magnet is created by wrapping superconducting wire into a solenoid[38]. The solenoid is a closed circuit, any current present in the solenoid will persist until a section of the solenoid, known as a persistent-current switch, is heated past the critical temperature. This electronically opens a portion of the solenoid circuit, allowing the current in the solenoid to be increased or decreased[38]. This exact process can be found in Quantum Design’s Fundamentals of Magnetism and Magnetic Measurements Featuring Quantum

Design’s Magnetic Property Measurement System, written by M. McElfresh of Purdue University.

The superconducting detection coil consists of three pick-up coils, which are located inside the superconducting magnet. Two of these coils are wound e.g., counter-clockwise(CCW), and have a single turn. The third coil consists of two turns in the opposite direction of the first two and is placed, equidistant, between them (see figure 13). For the purposes of simplicity, the first two coils will be referred to as the CCW coil and the last coil will be the clockwise (CW) coil. This arrangement forms a second-order gradiometer[38]. When a sample is measured, the sample is placed inside the centre of the gradiometer. This inductively couples the gradiometer to the sample: changes to the magnetic flux seen at the coils due to the sample changing, e.g., position, induces a current in the gradiometer. The gradiometer exists in a closed, superconducting circuit. As such, any current induced in the

46 gradiometer is persistent and the motion of the sample need not be continuous, i.e., only the displacement is important. This allows one to take multiple readings at a single position.

Since the gradiometer consists of two coils wound clockwise and two wound counter-clockwise, current will only be induced in the gradiometer, assuming each winding’s enclosed area is equal, if there is a relative change in the ﬂux through the CCW coil and the CW coil. consequently, the gradiometer will not detect objects that are long enough, and uniformly magnetized, to pass completely through the gradiometer. The DC magnetic ﬁeld will also not be detected, i.e., because it is homogeneous.

The detection coil is connected to the SQUID input coil. As such, changes in the current in the detection coil induce a voltage response in the SQUID. The magnetic moment of the sample is determined by moving it through the gradiometer and ﬁtting the corresponding voltage output to that of an ideal dipole. To reduce systematic error, due to geometric consideration, the system must be calibrated with a standard which is close in shape and size to the samples being measured.

The SQUID employed by the MPMS is an rf SQUID[38], which operates on a single Josephson junction. A detailed account on the mechanics of the SQUID’s operation can be found in The SQUID Handbook[39]. The SQUID is magnetically shielded from the superconducting magnet and, as such, requires an additional input coil to detect the current in the detection coil. The input coil and the SQUID are inductively coupled. As such, the SQUID produces a voltage that is proportional to the current in the detection coil.

47 Fig. 13: Left: (A) the second-order gradiometer utilized in Quantum Design’s MPMS. Right: the current induced in the gradiometer as a paramagnetic sample passes through it. The dotted axis corresponds 1:1 to the position of the sample w.r.t. the gradiometer. The amplitude is the current produced by the gradiometer. (B) The sample holder. (C) The sample. The typical scan path of the sample is shown by the dotted line. (D) The amplitude of the current when the sample is at the location of the top/bottom coil. (E) is the current when the sample is in the middle of the gradiometer. (F) The summation of the amplitude of the current at each coil centre, showing that each loop contributes equally. This is a necessary condition for the gradiometer to function, for it eliminates contributions from homogeneous ﬁelds.

48 2.4 The Speciﬁc Heat

The speciﬁc heat was measured via Quantum Design’s Physical Property Measurement System (PPMS- model 6000). The PPMS uses the standard relaxation method[44] to determine the heat capacity, which is similar to deducing capacitance by measuring the time-dependence of the voltage of an RC circuit. The relaxation method is detailed by A. N. Medina et al. but I will describe it here for convenience. The sample rests on a substrate that is attached to a thermal bath (a copper puck) by thin conductive wires (see ﬁgure 14). The entire system is

allowed to equilibrate to a temperature T0, which is the bath temperature. A constant power is applied to substrate, of which a portion is transferred to the heat bath via the conductive wires. The power obeys the following relation[45]:

d∆T P = C + K ∆T (62) dt eff

Where C is the systems (the substrate and the sample) total heat capacity, ∆T is the temperature

difference between the system and the thermal bath and Keff is the thermal conductance of the wires.

Equation (62) is a ﬁrst-order differential equation. The solution is:

K P −t eff ∆T = [1 − e C ] (63) C

P Keff Where C is the steady-state temperature and the ratio C acts as a time constant. After a sufﬁcient amount of time has passed (the maximum is roughly 10 time constants [44]) for the system to reach the steady-state, the power input is removed, and the solution to equation (62) becomes:

K P −t eff ∆T = [e C ] (64) C

49 D

C F

A B E

Fig. 14: The specific heat capacity sample holder and thermal bath. Left side: Aerial view of the apparatus. A: The sample, which is affixed to the substrate via apiezon grease. B: The substrate. Apiezon is applied to the surface of the substrate prior to the measurement of the addenda. C The conductive wires that thermally and structurally connect the substrate to the copper puck. D The copper puck, which acts as a thermal bath. Right side: Canted view of the cross section, showing the flow of energy through the system. E The power input into the system (blue arrow), which is applied to the substrate-and therefore to the sample as well. F: Power output into the thermal bath (red arrows), which travels through the conductive wires.

The systems heat capacity can therefore be determined by measuring the time dependence of ∆T . However, this contains both the sample and the substrate; thus, two measurements are required to deduce the heat capacity of the sample. One, known as the addenda, is performed in absence of a sample (but includes everything else: the substrate and the adhesive used to attach the sample, typically apiezon), the second includes the sample. The heat capacity of the sample is therefore the difference between the pair.

The measurements of the specific heat were performed using a 2% temperature raise and 5 repetitions per measurement temperature. Measurements were performed below 200K, to avoid the complications (see figure 15) associated with low temperature apiezon[46]. Addenda measurements were performed using a logarithmic scale, with 60 points collected between 200K and 5K. 5K was chosen as a lower limit hence the random error in the specific heat rapidly increased below approximately 5K. 5K has an error of approximately 1.5% to 4.2%, while measurements performed at 2.7K had up to 8% error (see figure 16).

50 A 4 0 0 ) K 3 0 0 e l o m / J (

t a

e 2 0 0 H

c i f i c e p

S 1 0 0

0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 T e m p e r a t u r e , K

Fig. 15: Anomalous behaviour due to the presence of apiezon in the specific heat capacity of cobalt molybdate. The anomaly marked ’A’ occurs due to insufficient sampling of the addenda- the background. In the event the reader suspects that this anomaly may be a single bad data point, it should be noted: each recorded point is plotted individually, i.e., the point labelled ’A’ consists of five separate data points.

2.5 The Dielectric Function and The Magnetocapacitance

The dielectric function of cobalt molybdate and the magnetocapacitance of cobalt

molybdate, Co1.5Fe0.5Mo3O8, and Co1Fe1Mo3O8 were measured. The measurements were performed using an Andeen-Hagerling Electrometer- model 2700A. Quantum Design’s PPMS- model 6000 was used to apply fields and for temperature control. This was done by coating the single crystals in conductive epoxy and treating the system as a parallel plate capacitor. As such, plate-like crystals were used for this purpose. The epoxy was applied to the faces of the crystal such that the applied electric field would be parallel to the c-axis. If necessary, the samples were first polished flat. The surface area and thickness, for use in

A calculating the dielectric function(C = d ), were measured using a micrometer. The surface area was calculated by taking the average of the square of each side length of the hexagonal crystals, i.e., each side length was measured and the average of the side length squared was

π 2 used to calculate Ahex = 3 sin( 3 )a , where Ahex is the area of an ideal hexagon and a is the average side length. The random error of the dielectric measurements was about one part in one hundred thousand to one part in one million.

51 C o M o O 1 4 2 3 8 C o F e M o O 1 . 5 0 . 5 3 8 C o F e M o O 1 2 1 1 3 8 C o F e M o O 0 . 5 1 . 5 3 8 F e M o O 1 0 2 3 8

) 8 % (

C / c 6 σ

0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 T e m p e r a t u r e , K

Fig. 16: Error in the heat capacity measurement for the cobalt, iron and cobalt-iron molybdates. The exponential increase at low temperature is a feature of the apparatus, i.e., it is due to the method of cooling the system, rather than the sample.

52 D

B A C F

Fig. 17: The probe used to measure the dielectric function. A: The inner stainless steel rod. B: The outer copper sleeve, providing thermal uniformity. Attached to this is the outer stainless steel sleeve. C: The copper ﬁtting. D: The coaxial cable. Thick lines represent two parallel cables, as shown in the inset. E: The sample, which is bridged to the coaxial lines via gold wire. F: The insulating ceramic rod.

Gold wire was used to bridge the sample to the probe. To minimize stray fields generated by currents sent to the sample, the probe utilized coaxial cable. The coaxial lines were paired, i.e., the input ran parallel to the output, to minimize any fields generated by the input signal. The probe consisted of a stainless steel rod and an aluminum oxide ceramic rod, which were bridged by a copper fitting. The sample was fixed to the insulating ceramic rod and acted as a capacitor. This probe was inserted into a hollow copper and stainless steel sleeve. The sample was completely surrounded by the copper sleeve. This provided a region of high thermal conductivity, ensuring thermal uniformity. The sleeve allowed the sample space to be evacuated and filled with an inert -helium was used- gas, if desired. This was done to increase the thermal coupling of the sample to the sample probe. For a diagram of the probe, see figure 17.

53 2.6 Magnetoelectric Coupling

The magnetoelectric coupling may be determined by measuring the magnetization as a function of applied electric ﬁeld, or the polarization as a function of applied magnetic ﬁeld. The former was chosen in this work.

To measure the magnetoelectric coupling, the sample was treated, like in the dielectric function, as a parallel plate capacitor. A DC voltage was applied to the sample, utilizing a Hewlett-Packard power supply- model E3631A, such that the resulting electric ﬁeld would be along the c-axis. The magnetization was then measured using the SQUID magnetometer, discussed in section 2.3.1. Again, the random error in these measurements was about one part in one thousand.

Since the electric ﬁeld would be E = V/d, where d is the sample thickness, one can easily dM determine the magnetoelectric coupling constant α, hence α = . dE

The Magnetoelectric coupling proved difﬁcult to measure. Several iterations of probes were constructed in an effort to reduce the background magnetization. Initially, the probe used in the capacitance measurement was used. However, this proved insufﬁcient, as the background magnetization hid that of the sample’s.

In an effort to reduce the background, a new probe was constructed (see figure 19). The focus of the design was to remove as many discontinuities as possible, as these generate anisotropic magnetic fields. It consisted of two concentric, hollow polypropylene plastic straws and coaxial cable. The input and output coaxial lines were twisted together and fed through the inner straw. Two slits were made in the inner straw, on either side of the sample. The input and output lines were fed through these slits. Silver filled epoxy was used to connect gold wires to the (001) faces of the sample. The gold wires were soldered to the coaxial leads. Kapton was used to secure the leads of the coaxial cable to the inner straw. This served two purposes: to prevent the leads from vibrating and to shield the plastic beneath the leads from exposure to high temperatures while soldering the gold wires to the leads. The plastic straws

54 C o a x i a l P r o b e ( F i g . 1 9 ) T w i s t e d C o p p e r P r o b e - 2 1 0 W i t h S a m p l e ( F i g . 2 0 ) T w i s t e d C o p p e r P r o b e ( F i g . 2 0 ) 2 . 5 X 1 0 - 3

1 0 - 3 - 4

) 7 . 5 X 1 0 u m e (

t n e m

o - 4

M 1 0

3 . 5 X 1 0 - 5

1 0 - 5 0 1 2 3 4 5 6 7 8 V o l t a g e , V

Fig. 18: The background magnetic moment of the constructed probes. The y-axis is logarithmically scaled. The measurements were conducted under a 3000 Oe field at 40K. The magnitude of the magnetic moment of the first constructed probe, utilizing coaxial cable, was sufficient to completely mask the sample. This is because the magnetic moment of individual crystals is fairly small, hence the mass of individual crystals is on the scale of milligrams and they order antiferromagnetically. The second probe’s magnetic moment was significantly smaller (approximately 3.5 × 10−5emu) than the first ( 2.5 × 10−3emu) and did not mask the sample (which has a moment of 7.5 × 10−4emu).

55 C

A B E

Fig. 19: The first iteration of the constructed magnetoelectric probes, referred to as the coaxial probe. A: The stainless steel rod. B: The Teflon fitting, which has two diameters. C: The outer and inner polypropylene straws, which fit over the Teflon fitting. They are held in place by friction. D The coaxial lines. E: The sample, which is connected to the coaxial lines via gold wire. It is located between the inner and outer straw.

A B E

Fig. 20: The second and ﬁnal iterations of the constructed magnetoelectric probes, referred to as the twisted copper probe. A,B,C: The same as ﬁgure 19. D The copper wire. The wire is twisted upon itself and splits around the sample. The wire is held in place by Kapton tape. E: The sample, which is connected to the copper via gold wires and colloidal silver. It is located between the inner and outer straws.

were fixed to a stainless steel rod by means of a Teflon fitting.

This probe ultimately failed. It was determined that the relatively large magnetization (2.5 × 10−3emu at 40K, see ﬁgure 18 for a comparison of the probes from ﬁgures 19 and 20) of the probes was due to either the solder used to join the sample to the coaxial lines or the coaxial lines themselves.

A second probe was constructed utilizing insulated copper wire in lieu of coaxial cable. The wire was constructed using a single copper wire, folded over and twisted together. The wire was separated only at the very bottom of the probe, well outside the detection coil. The wire was split around the sample and the insulation removed from this section, allowing the sample to be connected without creating a discontinuity. Colloidal silver was used to attach the gold wires to the copper wire. This probe was successful in reducing the background moment sufﬁciently to detect the sample (the moment of the probe at 40K was approximately 3.5 × 10−5emu, while the largest sample’s moment was about 7.5 × 10−4emu).

56 3 RESULTS AND DISCUSSION

The results section will be broken down by purpose and measurement type, and is presented in order of relevance to the topic of the magnetoelectric effect. The purpose of each section is to justify the sections which follow. As such, the ﬁrst section will focus on the sample synthesis and characterization and the last will discuss the magnetoelectric coupling-or lack thereof- itself. Intermediate sections will present evidence on whether or not the effect should exist.

3.1 Sample Synthesis and Characterization

Initial trials were performed using PtCl2 as a chlorine source. However, the amount of material transferred in trials containing iron was negligible (trials 5, 18, 19, and 20 in table 1).

Even the spinel phase failed to grow, which TeCl4 commonly grew to dimensions exceeding 2mmx2mmx2mm (see ﬁgure 23). A large- the long axis measures almost a full centimetre (see ﬁgure 23)- crystal of CoO was produced in trial 3, though this is undesired. Additionally, M-

H curves measured for Co2Mo3O8 and Co1.5Fe0.5Mo3O8 revealed ferromagnetic behaviour in the otherwise antiferromagnetic regime (see ﬁgure 34). Trials performed utilizing TeCl4 as a transport agent experienced no such ferromagnetic response. It is concluded that use of PtCl2 lead to impurities in the resulting crystals. Furthermore, P. Strobel and Y. Le Page[28] found that neither Te nor Cl were incorporated into any of the hexagonal molybdates synthesized via

TeCl4. As a result, TeCl4 was chosen as the preferred transport agent. It should be noted that, despite the use of TeCl4, the sample of iron molybdate produced using TeCl4 was determined to contain ferromagnetic impurities (see ﬁgure 34 and section 3.2).

Of the cobalt-iron molybdates, the samples with a high concentration of iron proved most difﬁcult to synthesis. Speciﬁcally, in the samples containing a concentration of iron equal to or exceeding 75%, silica attack was so severe, the ampoules were often breached sometime throughout the growth period, exposing any grown crystals to an abundance of oxygen (see

57 Fig. 21: Pieces of the quartz ampoule post-synthesis. Left: Quartz impregnated with iron. In this trial, the quartz ampoule was breached; however, it is not clear if the impregnation occurred before or after failure. Note: The iron oxide seen on the edge of the quartz was not present upon initial examination. Right: Devitriﬁcation of the quartz ampoule from trail 29. The affected surface was in contact with refractory alumina.

ﬁgure 21). The attack was so severe that even double tubing was insecure. Reducing the allowed growth period was necessary to combat this complication, for carbon coating proved equally as ineffective as the double tubing. However, the size of crystals grown under a constrained growth period are naturally reduced in size. Many of the grown crystals were unsuitable for measurement due to their size (see ﬁgure 22). This is especially true of iron

−4 molybdate and Co0.5Fe1.5Mo3O8, for which only minuscule- the typical mass was 10 g- acicular and equidimensional crystals were ever obtained. These samples were unsuitable for dielectric and M(E) measurements.

Trails containing a composition of iron equal to or less than 50% experienced significantly less attack, though, in the case of trial 9 and 10, it was significant enough to destroy the ampoule. However, this complication was overshadowed by the proclivity of the system to grow spinel crystals (M3O4) of a combination of Fe and Co (see figure 23). These spinel crystals consumed much of the available precursor and placed an additional constraint on the size of the desired crystals. Spinel crystals were found to form in all trails, excluding pure cobalt molybdate. However, despite this complication, fairly large, plate-like crystals of

58 Fig. 22: The cobalt, iron and cobalt-iron Fig. 23: In clockwise order from the molybdates. In clockwise order from the top left (trial number in brackets): top left (trial numbers are in brackets): Underside of image seen to the right(29); Co2Mo3O8(3,4), Co1.5Fe0.5Mo3O8(8), Co1Fe1Zn3O8(30); crust showing large Co1Fe1Mo3O8(12), Large CoO Crystal(3), crystallites(8); crust showing the Fe2Mo3O8(25), Co0.5Fe1.5Mo3O8(15). agglomeration of minute crystals(12); The bottom scale is metric, with major spinel-type crystals, determined to ticks of 5mm, intermediate ticks of 1mm be magnetite, showing pitting due to and minor ticks of 0.5mm. The scale on oxidation(16); Unknown rhombohedral the top is in inches. and octahedral shaped crystals(29). The bottom scale is metric, with major ticks of 5mm, intermediate ticks of 1mm and minor ticks of 0.5mm. The scale on the top is in inches.

Co1.5Fe0.5Mo3O8 and Co1Fe1Mo3O8 were obtained (see figure 22). This was only possible after observing that the precursor tended to form a plug in the ampoule if allowed to span the entire cross-section of the growth zone. This was due to the formation of microscopic crystals in the growth zone which formed a crust over the remaining precursor. Naturally, this prevented any further material from diffusing into the growth zone once established. This effect could not be completely eliminated, but it could be reduced by spreading the precursor over a larger surface area (see figure 23 for an image of the crust). This also allowed the growth duration to be significantly reduced (from a typical duration of 30 days to as low as 8).

59 Back- Ramp Pt Pressure T (K) T (K) Duration Double- Carbon- Ampoule Compound Trial N /N src snk transfer Duration or Phases MO MoO2 (105 Pa) σ = ±1 ±1 (days) tubed? coated? Breached? (hours) (hours) Te?

Co2Mo3O8 1 4.825 ±0.004 1.54 ±0.04 968 817 30 24 24 - - Pt Co2Mo3O8* - Co2Mo3O8 2 1.9989 ±0.0003 1.30 ±0.03 968 817 30 24 24 - - Pt Co2Mo3O8* - Co2Mo3O8 3 1.9979 ±0.0003 1.28 ±0.03 968 817 30 24 24 - - Pt Co2Mo3O8 + CoO - Co2Mo3O8 4 2.0214 ±0.0003 1.04 ±0.03 968 817 10 - 14 - Y Pt Co2Mo3O8 -

Co1.5Fe0.5Mo3O8 5 2.0020 ±0.0008 1.07 ±0.03 968 837 30 24 24 - - Pt Co1.5Fe0.5Mo3O8 - Co1.5Fe0.5Mo3O8 6 2.0019 ±0.0008 1.04 ±0.03 943 811 30 24 24 - - Te Spinel - Co1.5Fe0.5Mo3O8 7 2.0041 ±0.0008 1.08 ±0.03 958 853 30 24 24 - Y Te Co1.5Fe0.5Mo3O8* - Co1.5Fe0.5Mo3O8 8 2.001 ±0.002 1.17 ±0.04 958 833 14 - 14 - - Te Spinel + Co1.5Fe0.5Mo3O8 -

Co1Fe1Mo3O8 9 2.0014 ±0.0006 1.03 ±0.04 963 847 30 24 24 - - Te - Y Co1Fe1Mo3O8 10 2.0025 ±0.0006 1.05 ±0.04 958 853 30 24 24 - Y Te - Y Co1Fe1Mo3O8 11 2.003 ±0.001 1.13 ±0.04 958 853 30 - 14 - - Te Co1Fe1Mo3O8* - Co1Fe1Mo3O8 12 2.000 ±0.001 1.12 ±0.04 958 833 14 - 14 - - Te Spinel+Co1Fe1Mo3O8 -

Co0.5Fe1.5Mo3O8 13 1.994 ±0.001 1.04 ±0.04 950 812 30 24 24 - - Te - Y Co0.5Fe1.5Mo3O8 14 1.994 ±0.001 1.04 ±0.04 958 853 12 - 14 - Y Te Spinel - Co0.5Fe1.5Mo3O8 15 1.993 ±0.002 1.07 ±0.04 958 855 8 - 14 - - Te Spinel + Co0.5Fe1.5Mo3O8* - Co0.5Fe1.5Mo3O8 16 1.991 ±0.002 1.06 ±0.04 956 837 8 - 14 - - Te N/A Y**

Fe2Mo3O8 18 0.9827 ±0.0001 1.98 ±0.04 957 867 11 24 24 - - Pt - Y Fe2Mo3O8 19 1.0020 ±0.0001 1.96 ±0.04 957 869 30 24 24 - Y Pt - Y Fe2Mo3O8 20 0.9985 ±0.0001 2.04 ±0.04 957 869 16 24 24 - Y Pt - Y Fe2Mo3O8 21 1.9079 ±0.0004 1.01 ±0.04 955 848 30 24 24 Y Y Te N/A Y Fe2Mo3O8 22 1.9100 ±0.0002 1.10 ±0.04 955 848 14 - 14 - Y Te N/A Y Fe2Mo3O8 23 1.9056 ±0.0004 1.09 ±0.04 960 855 11 - 14 Y Y Te N/A Y Fe2Mo3O8 24 1.9083 ±0.0004 1.10 ±0.04 958 853 10 - 14 - - Te N/A Y Fe2Mo3O8 25 1.9150 ±0.0004 1.12 ±0.04 958 853 7 - 14 - - Te Spinel + Fe2Mo3O8 - Fe2Mo3O8 26 1.9110 ±0.0004 1.00 ±0.04 958 853 7 - 14 - - Te Spinel + Fe2Mo3O8* - Fe2Mo3O8 27 1.9116 ±0.0004 1.04 ±0.04 956 837 8 - 14 - - Te Spinel + Fe2Mo3O8* - Fe2Mo3O8 28 1.9109 ±0.0004 1.10 ±0.04 956 855 7 - 14 - - Te Spinel + Fe2Mo3O8* -

CoMnMo3O8 29 1.992 ±0.002 1.28 ±0.03 930 830 14 24 14 - - Te CoMnMo3O8 + unknown -

CoZnMo3O8 30 1.999 ±0.001 1.06 ±0.03 930 830 14 24 14 - - Te CoZnMo3O8 - Table 1: Chemical vapour transport synthesis details and results. The ’trials’ column is not in chronological order. Pressure was calculated assuming ideal gas law, that chlorine gas took the form Cl2, and that the average temperature was the average between the source and sink zones. Partial pressures of Te and Pt were not considered. The double asterisk marks the breached ampoule in trial 16. This is noted because this ampoule had a defectively thin wall thickness. The breach was likely due to the defect rather than any new phenomena not present in trial 14 and 15. Under Phases: ’N/A’ indicates that material was transported but oxidized due to ampoule failure (breaching). ’-’ indicates that no material was transported. Asterisks mark samples that were too small to perform dielectric and M(E) measurements on. Samples highlighted in red are concluded to contain magnetic impurities. Samples highlighted in blue were used for measurements.

60 Samples of Co1Zn1Mo3O8 were readily synthesized into relatively large

-2mmx2mmx2mm- hexagonal crystals (see figure 23). Samples of Co1Mn1Mo3O8 had the additional complication of forming large- the octahedral crystals were as large as 3mmx3mmx3mm- rhombohedral and octahedral crystals in the source zone. These, like the spinel crystals, consumed much of the available precursor and limited the size of the cobalt-manganese crystals (see figure 21). The composition of these crystals could not be identified, as they were not present in the ICSD.

Chemical vapour transport produces a finite number of products and the macroscopic shape of a single crystal reflects its microscopic crystal symmetry. Since all the 3d transition metals used as dopants exist as a molybdate with the same crystal symmetry as cobalt molybdate, and their transport occurs at similar temperatures, one can use unconventional methods to determine their composition. The phase may be determined definitely via X-ray diffraction.

To determine if the correct phase was obtained, X-ray diffraction was performed on iron, cobalt, the cobalt-iron, and cobalt-zinc molybdates. The samples of cobalt molybdate show no impurity phases, suggesting that the synthesis method is appropriate for the creation of pure crystals (see figure 25). The sample of cobalt-zinc molybdate likewise exhibits no evidence of an impurity phase. As shown later (see section 3.2.3), Co1Zn1Mo3O8 lacks a magnetic ordering temperature, and is therefore suitable for use in determining the magnetic contribution to the specific heat. Iron molybdate, however, does show evidence of an impurity phase, though the candidate phase could not be determined (see figure 26).

No impurity phases were detected in intermediate- Co1.5Fe0.5Mo3O8, Co1Fe1Mo3O8, and

Co0.5Fe1.5Mo3O8- samples. The values of d, i.e., the d associated with each peak, of the intermediate samples varies positively with composition between the patterns of cobalt molybdate and iron molybdate (see inset of ﬁgure 26). This is expected for substitution-type doping, hence the lattice distorts in all directions due to the change in average ionic radius. Were intercalation to occur, one might expect to see distortion primarily along the c-axis, hence that is the axis along which the layers stack. Conversely, when examining the lattice

61 5 . 7 8 5 a b - P l a n e 1 0 . 0 6 c - A x i s 1 0 . 0 4

) 5 . 7 8 0 1 0 . 0 2 ) m m 0 0 1 1 - - 0 0 1 1 0 . 0 0 1 ( (

, 5 . 7 7 5 , c a

r r 9 . 9 8 e e t t e e m m a a 9 . 9 6 r r 5 . 7 7 0 a a P P

e e 9 . 9 4 c c i i t t t t a a 5 . 7 6 5 L L 9 . 9 2

9 . 9 0 5 . 7 6 0 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 I r o n C o n c e n t r a t i o n , [ N / ( N + N ) ] F e C o F e

Fig. 24: The lattice parameters of the cobalt and cobalt-iron molybdates. The lattice parameters reported by Bertrand and Kerner-Czeskleba[32] for cobalt molybdate and Kanazawa and Sasaki[59] for iron molybdate are shown as open circles. parameters, one find that the c-axis continually increases with doping concentration, while the ab-plane initially increases but subsequently remains approximately constant. This would imply intercalation is possible; however, it is difficult to draw any conclusions from the data, due to the presence of exceptionally large error for the case of Co0.5Fe1.5Mo3O8 and the relatively small data set. The parameters of the Rietveld refinements for cobalt and the cobalt-iron molybdates are tabulated in table 2, iron molybdate is absent since an acceptable refinement could not be found.

2 Compound Rwp Re Chi Co2Mo3O8 0.77 0.54 3.6775 Co1.5Fe0.5Mo3O8 1.27 0.82 4.136 Co1Fe1Mo3O8 1.09 0.78 3.3037 Co0.5Fe1.5Mo3O8 0.92 0.66 4.2149

Table 2: Rietveld reﬁnement factors for cobalt and the cobalt-iron molybdates. The reﬁnements were performed from a set of ranges of 17◦ < 2θ < 35◦ and 38◦< 2θ <55◦. This range was chosen to avoid clustered peaks, whose accuracy was affected by sampling error.

62 C o M o O 2 3 8 ( 0 0 2 ) C o Z n M o O ( 1 0 2 ) 1 1 3 8 1 . 2

( 2 0 1 ) 1 . 0 ( 0 0 4 )

y 0 . 8 t i

s ( 2 0 0 ) ( 2 0 3 ) n e

t ( 1 0 3 ) n

I 0 . 6

e ( 2 0 2 ) ( 2 0 4 ) v i t ( 1 0 4 ) ( 2 1 1 ) a ( 1 0 1 ) l

e 0 . 4 ( 2 1 0 ) R ( 1 1 4 )

0 . 2

0 . 0 2 0 3 0 4 0 5 0 6 0 7 0 2 θ, d e g r e e s

Fig. 25: X-Ray diffraction patterns of Co2Mo3O8 and Co1Zn1Mo3O8. The blue lines correspond to the ISCD entry for cobalt molybdate, from reference [32].

63 C o M o O 2 3 8 C o F e M o O 1 . 5 0 . 5 3 8 C o F e M o O 3 1 . 0 1 . 0 3 8 C o F e M o O 0 . 5 1 . 5 3 8 F e M o O 2 3 8

∆d ( p m ) , d n - d n x C o x

a 0 1 2 3 4 5 6 m I /

) 2 n i ( 2 1 1 ) m I

- ( 1 1 4 ) I (

, ( 2 1 0 ) y t i ( 2 0 3 ) s

n ( 2 0 2 ) e t n

i ( 2 0 1 )

e 1 ( 0 0 4 ) v i t ( 1 1 2 ) a l

e ( 1 0 3 ) R ( 1 0 2 )

( 1 0 1 )

( 0 0 2 ) 0 2 0 3 0 4 0 5 0 6 0 7 0 2 θ, d e g r e e s

Fig. 26: The x-ray diffraction patterns for Co2−2xFe2xMo3O8 series, shifted along the y-axis for clarity. The arrows above the pattern of iron molybdate denote the peaks belonging to the impurity phase. Dotted lines represent where the data was interpolated due to absent peaks. Inset: The variation of the lattice spacing, dhkl, from the parent compound Co2Mo3O8. The y-axis is unitless and represents the peak number n. n varies from 0 to 12 (this corresponds to a 2θ range of approximately 17-50 degrees), with n=0 occurring at the bottom of the y-axis. The diffraction patterns of cobalt and iron molybdate were taken from the ISCD under references [32] and [59], respectfully.

The reader may wish to proceed to section 3.2 and 3.3 before reading the remainder of this section, as it utilizes results which are presented in the aforementioned sections. Sample compositions can be approximated by examining the transition point in the magnetic and thermal properties, using the assumption- no new type of interaction arises, so the internal interactions average out- that the magnetic phase transition will vary linearly with iron concentration. That is to say, one can interpolate between the Neel´ temperatures of cobalt and iron molybdate to predict the Neel´ temperature of a mixture of the two. To do this, one can

64 construct a straight line using:

Co F e Co T = TN + (TN − TN )x (65)

Co Where T is the predicted transition temperature, TN is the Neel´ temperature of cobalt

F e molybdate, TN is the Neel´ temperature of iron molybdate, and x is the ratio of Fe atoms to M atoms- M being the sum of Co and Fe and therefore x takes values between 0 and 1.

From the magnetic susceptibility (see section 3.2), it can be concluded that the sample composition follows the initial ratios of Fe:M used in the sample synthesis, to within experimental error, (see ﬁgure 28) with the exception of Co1.5Fe0.5Mo3O8, which has a transition temperature corresponding to x = 0.29 ± 0.015 (calculated using equation (65). The sample composition is better predicted by ﬁtting to our results for the transition temperature of cobalt and iron molybdate than to P. Strobel and Y. Le Page’s results[28]. However, our results

Co F e Co (TN = (40.5 ± 0.3)K, TN = (60.3 ± 0.3) and Strobel and Le Page’s (TN = (40.8 ± 0.3)K,

F e TN = (59.5 ± 0.5)K) agree to within experimental error. The location of the Neel´ temperature gives some insight as to the magnitude of J, the exchange constant (see section 1.2, equation 28). Higher Neel´ temperatures imply larger J’s, hence the Neel´ temperature is the point in which the thermal energy is approximately equal to the magnetic energy. For instance, this can explain why frustrated magnetic systems tend to have low transition temperatures, since the competing interactions reduces the magnetic energy. It is therefore concluded that the exchange constant increases with iron concentration.

Interestingly, the speciﬁc heat capacity shows another result: every sample but

Co1Fe1Mo3O8 agrees with the interpolated composition to well within experimental error (see ﬁgure 27). Based on interpolation between cobalt and iron molybdate transition temperatures,

Co1Fe1Mo3O8 has a predicted Fe:M ratio of x = 0.33 ± 0.13, signiﬁcantly lower than the anticipated x = 0.5. A summary of the observed transition temperatures from the magnetic susceptibility and the speciﬁc heat capacity can be found in table 3.

65 x Cp TN χ TN 0 41.8 ±0.6 40.5 ±0.3 0.25 47 ±3 46.3 ±0.3 0.5 49 ±3 50.6 ±0.5 0.75 58 ±3 55.3 ±0.3 1 64 ±3 60.3 ±0.3

Table 3: Summary of the transition temperatures for the speciﬁc heat capacity (denoted Cp) and the magnetic susceptibility (χ) of Co2−xFe2xMo3O8. Cells highlighted in red indicate that the two transition temperatures do not agree within the experimental error, blue indicates that they do.

7 0

6 5

6 0 K

, e

r 5 5 u t a r e p 5 0 m e T

4 5

4 0 0 . 3 3

0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 I r o n c o n c e n t r a t i o n , x [ N / ( N + N ) ] F e F e C o

Fig. 27: Approximate composition of cobalt-iron molybdates based on the speciﬁc heat. The ordering temperature was taken to be the maximum of the peak in the heat capacity. The error is half the difference between the two adjacent points, i.e., the resolution. The black line represents the expected transition temperatures of each composition based on the transition F e Co temperature of the cobalt and iron molybdates, i.e., the slope is TN − TN . The red line is the same but from the magnetic susceptibility results (Also shown in ﬁgure 28).

66 6 0

5 5 K

, e r u

t 5 0 a r e p m e

T 4 5

4 0 0 . 2 9

0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 I r o n c o n c e n t r a t i o n , x [ N / ( N + N ) ] F e C o F e

Fig. 28: Approximate composition of cobalt-iron molybdates based on magnetic ordering temperature from the magnetic susceptibility. The ordering temperature was taken to be the point at which the magnetic susceptibility sharply increases in the ﬁeld-cooled perpendicular susceptibility. The error is taken to be half the difference between the adjacent points, i.e., the resolution. The black line shows the expected transition temperature for each composition based on the Neel´ temperatures of the produced compounds of iron and cobalt molybdate (see equation (65). The red line shows the same but using the results of P. Strobel and Y. Le Page.

67 3.2 Magnetic Properties

In the following discussion, there are two properties (parasitic ferromagnetism and the low-field magnetic behaviour) with very similar behaviours. To avoid confusion, this section will be broken into multiple sections. The first will focus on the general magnetic behaviour and the presence of parasitic ferromagnetism in the Co2−2xFe2xMo3O8 series. The second will focus on a peculiar behaviour seen at low fields. The last section will comment on the magnetic properties of cobalt-zinc molybdate.

3.2.1 Cobalt, Iron, and the Cobalt-Iron Molybdates- Part I

The magnetic susceptibility of cobalt molybdate exhibits typical antiferromagnetic behaviour. No variations between ZFC and FC were observed in a parallel (along the c-axis) 10000 Oe field and only slight variation (a possible source of this is discussed in part II) in the susceptibility was observed, in the low temperature region of <25K, in a parallel 3000 Oe field (see figures 29 and 33). The magnitude of the parallel susceptibility greatly exceeds that of the perpendicular (in the ab-plane) susceptibility, even above the Neel´ temperature. This reflects the strong uniaxial anisotropy present in this family of materials [31]. The absence of a sharp discontinuity in the susceptibility with the onset of magnetic ordering makes it difficult to determine the exact Neel´ temperature. However, this is not a problem for the perpendicular susceptibility.

Under a perpendicular field of 3000 Oe, pronounced enhancement of the susceptibility reveals a Neel´ temperature of TN ≈ 41K (see figure 29 and 30). Interestingly, S.P. McAlister and P. Strobel report a similar enhancement, for both ZFC and FC measurements, in the presence of a perpendicular and parallel field of 42 Oe. A small step at TN is also reported for a parallel field of 1000 Oe. They attribute these enhancements to parasitic ferromagnetism, due to iron impurities[31]; however, this seems unlikely to explain the behaviour observed in cobalt molybdate, hence the susceptibility under parallel fields shows no ferromagnetic behaviour nor enhancement at the Neel´ temperature, down to 350 Oe (see figure 31). Indeed, it

68 Z F C P a r . 3 K O e F C P a r . 3 K O e A Z F C P a r . 1 0 K O e F C P a r . 1 0 K O e 0 . 0 5 Z F C P e r p . 3 K O e F C P e r p . 3 K O e

| | 8 4 ) 3

0 . 0 4 m

c 5 6 / e l o

m 2 8

) B ( m e

l 0 . 0 3 χ / o

⊥ 1

m 0 /

3 - 1 0 0 0 1 0 0 2 0 0 3 0 0 m

( 0 . 0 2

, m χ

0 . 0 1

0 . 0 0 - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 T e m p e r a t u r e , K

Fig. 29: The magnetic susceptibility of Co2Mo3O8. A: A small anomaly is detected in field- cooled susceptibility under application of a 10KOe field, parallel to the c-axis. This is attributed to a gas leak (it is the contribution of oxygen). Note that the parallel susceptibility, under a 3KOe field, shows minor bifurcation at low temperature, while almost no bifurcation is present under a 10KOe field. B: The ferromagnetic-like enhancement at the Neel´ temperature. Inset: The parallel and perpendicular inverse-susceptibility under a 3KOe field. The x-axis is in Kelvin. is concluded that the enhancement in the perpendicular susceptibility has a different origin, which will be discussed in Part II.

The magnetization, below the ordering temperature, as a function of applied field (referred to as ’M-H’) is linear for both parallel and perpendicular fields (see figures 32 and

33) and no metamagnetic transition was present, up to 5.5T, as was observed in Fe2Mo3O8

[18] and Mn2Mo3O8[37]. This is true despite iron doping (again, see ﬁgure 32). However, in preliminary synthesis trials utilizing PtCl2 as a transport agent, single crystals of Co2Mo3Mo8 and Co1.5Fe0.5Mo3Mo8 (these were the only two sample successfully synthesized using PtCl2, see table 1) displayed a ferromagnetic moment in the M-H curves of approximately equal magnitude along both the c-axis and the basal plane. This moment is present even above the Neel´ temperature (see ﬁgure 34). This implies that this moment is independent of the ordering of the overall magnetic structure, since, otherwise, one would expect the moment to follow the anisotropy of the crystal and only appear below the Neel´ temperature. Furthermore, further synthesis trials yielded crystals which did not exhibit this effect, suggesting it is extrinsic in

69 0 . 0 3 5 ) e l o m / 3 0 . 0 3 0 m c (

, m χ

0 . 0 2 5 0 2 0 4 0 6 0 T e m p e r a t u r e , K

Fig. 30: The enhancement of the perpendicular susceptibility at the Neel´ temperature in cobalt molybdate. Measured under a 3000 Oe H-field. The blue data points correspond to the ZFC measurement, while red belong to the FC measurement. The bifurcation is indicative of a metastable state in the ZFC measurement, where the spins are not able to align towards the applied field until there is sufficient thermal energy.

Z F C P a r . 3 5 0 O e F C P a r . 3 5 0 O e 0 . 0 6 0

0 . 0 5 5

0 . 0 5 0 ) e l o 0 . 0 4 5 m / 3 m c (

m 0 . 0 4 0 χ

0 . 0 3 5

0 . 0 3 0 0 2 5 5 0 7 5 1 0 0 T e m p e r a t u r e , K

Fig. 31: The susceptibility of Co2Mo3O8 under a 350 Oe parallel field. Contrary to the results McAlister and Strobel, no enhancement at the Neel´ temperature is observed at low field. This is an indication that the enhancement observed under a perpendicular field is not due to the same parasitic ferromagnetic observed in some of the aforementioned author’s cobalt molybdate samples.

70 0 . 5 0 . 6 C o M o O a t 4 1 K C o F e M o O 2 3 8 1 . 5 0 . 5 3 8 4 6 K P a r . 0 . 4 P e r p . | | 0 . 5

) P a r . ) 4 5 K P a r . . .

u u 0 . 4 4 4 K P a r . . . f f / 0 . 3 / 4 3 K P a r . B B µ µ

( ( 0 . 3 4 2 K P a r .

, , t 0 . 2 t n n 0 . 2 e ⊥ e m m

o 0 . 1 o 0 . 1 M M 0 . 0 0 . 0 0 1 0 2 0 3 0 4 0 5 0 6 0 0 1 0 2 0 3 0 4 0 5 0 6 0 0 . 5 H , K O e 1 . 0 H , K O e C o F e M o O C o F e M o O 1 1 3 8 0 . 5 1 . 5 3 8 5 0 K P a r . 5 5 K P a r . 0 . 4 0 . 8 5 4 K P a r .

) 4 9 K P a r . ) . .

u 4 8 K P a r . u 5 3 K P a r . . . f f / 0 . 3 4 7 K P a r . / 0 . 6 5 2 K P a r . B B

µ µ 5 1 K P a r .

( 4 6 K P a r . (

, , 5 0 K P a r . t 0 . 2 t 0 . 4 n n 4 9 K P a r . e e m m

o 0 . 1 o 0 . 2 M M 0 . 0 0 . 0 0 1 0 2 0 3 0 4 0 5 0 6 0 0 1 0 2 0 3 0 4 0 5 0 6 0 H , K O e H , K O e

F e M o O 0 . 8 2 3 8 6 0 K P a r .

) 5 9 K P a r . .

u 0 . 6 5 8 k P a r . . f

/ 5 7 K P a r . B

µ 5 6 K P a r . (

, 0 . 4

t 5 5 K P a r . n e

m 0 . 2 o M 0 . 0 0 1 0 2 0 3 0 4 0 5 0 6 0 H , K O e

Fig. 32: M-H curves under a parallel ﬁeld for cobalt, iron, and the cobalt molybdates. All measurements performed from 0→+ve→0. Top Left: M-H curves for single crystals of Co2Mo3O8 under perpendicular and parallel ﬁelds at 41K. The anisotropy of the system is evident in the difference of slopes. No hysteresis was observed in either orientations. No evidence of a ferromagnetic moment or metamagnetic transitions, as was observed in iron molybdate and manganese molybdate, is present. This does not change with the introduction of iron. Bottom: The metamagnetic transition from an antiferromagnetic state to a ferrimagnetic state in iron molybdate.

71 C o M o O 0 . 5 C o F e M o O 2 3 8 1 . 5 0 . 5 3 8 0 . 4

0 . 4

0 . 2 ) ) u u

. . 0 . 3 f f / / B B µ µ

( 0 . 0 (

t t n n

e 4 1 K e 0 . 2

m 4 0 K m o - 0 . 2 3 9 K o M M 3 8 K 4 6 K 2 5 K 0 . 1 4 5 K 5 K 4 4 K - 0 . 4 1 . 7 K F C 4 3 K ( 6 0 0 O e ) 4 2 K 0 . 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 0 1 0 2 0 3 0 4 0 5 0 6 0 H , K O e H , K O e 0 . 5 C o F e M o O 0 . 5 C o F e M o O 1 1 3 8 0 . 5 1 . 5 3 8

0 . 4 0 . 4 ) ) u u

. 0 . 3 . 0 . 3 f f / / B B µ µ ( (

t t n n

e 0 . 2 e 0 . 2 m m o o M M 0 . 1 5 0 K 0 . 1 4 9 K 4 8 K 5 4 K 4 7 K 5 3 K 4 6 K 5 2 K 0 . 0 0 . 0 0 1 0 2 0 3 0 4 0 5 0 6 0 0 1 0 2 0 3 0 4 0 5 0 6 0 H , K O e H , K O e Fig. 33: M-H curves under a perpendicular field for cobalt molybdate and the cobalt-iron molybdates, showing the temperature independence of the susceptibility. This is particularly true for pure cobalt molybdate (Top left), which shows very little variation over a large temperature range, even after poling- that is, cooling through the Neel´ temperature in the presence of a magnetic field, also referred to as field-cooled- in a 600 Oe H-field. nature. Therefore, it is believed to be due to parasitic ferromagnetic of the same type reported by McAlister and Strobel.

Additionally, we know that the perpendicular susceptibility of a uniaxial antiferromagnet is temperature independent from mean-field theory. It is therefore expected that the M-H curves under a perpendicular field would be largely temperature invariant. This is indeed observed in cobalt molybdate and the cobalt-iron molybdates (see figure 33). However, iron molybdate did not exhibit a near-constant perpendicular susceptibility. Upon inspection of the magnetization under a parallel field, it is noted that the weak ferromagnetic moment is also present in this sample (see figure 34). It is thus concluded that the iron molybdate sample synthesized via

TeCl4 exhibited parasitic ferromagnetism, probably of the stoichiometric nature- that is to say,

72 it has an excess of iron[43]. Furthermore, impurity peaks were detected in this sample’s XRD. However, the associated compound could not be identiﬁed within the ISCD.

The values of the Curie constant, the Weiss constant, the effective moment are tabulated in table 4. All inverse susceptibilities were fit to a straight line (y = mx + b) from 125K to 275K (see figures 29-40). This was done to avoid the non-linearity of the inverse-susceptibility near the Neel´ temperature and in the high-T region. As a note: the presence of this high-T anisotropy will affect the quantitative accuracy of the Curie-Weiss law, as it is expected to vanish in the very high-T limit (as the anisotropy energy becomes negligible compared to the thermal energy). In other words, the parallel susceptibility should converge towards the perpendicular susceptibility at some high-T and then run the curve equivalent of collinear. It is therefore expected that 1/χ should exhibit non-linear behaviour near both the Neel´ temperature and at high-T. This is observed in e.g., cobalt molybdate (see figure 29).

As there was no observed difference between the FC and ZFC susceptibilities above the Neel´ temperature, either may be ﬁtted to the Curie-Weiss law. Comparisons of the parallel and perpendicular quantities are shown in table 5, along with the expected moment for the total

J S angular moment (µeff ) and the spin contribution only (µeff ) and the effective moment expected

2 ⊥ 1 k for a polycrystalline sample, which is calculated using[31] µeff = 3 µeff + 3 µeff .

McAlister and Strobel found that iron molybdate has a parallel, perpendicular and

polycrystalline effective moment of (4.4±0.2)µB, (5.8±0.3)µB, and (4.9 ±0.2)µB [31]. The relatively large difference in the directional effective moment is indicative of anisotropy, even in the paramagnetic regime. Additionally, they conclude that iron molybdate experienced complete orbital quenching, hence the polycrystalline effective moment is exactly the spin-only effective moment. We found that iron molybdate has an effective moment of (6.23

±0.09)µB and exhibits little difference between the directional effective moments, with a

∆µeff of 0.16; almost a full order of magnitude smaller than what McAlister and Strobel report.

Of cobalt molybdate, they report that it has a perpendicular and parallel effective moments

73 0 . 3 0 F e M o O 2 3 8 ) )

e 0 . 0 8 l e l o

o 0 . 2 4 m m / /

u 0 . 0 6 u

0 . 1 8 m m e e ( (

6 0 K t t 0 . 0 4 n

n 0 . 1 2 5 9 K e e 5 8 K m m 5 7 K o

o 0 . 0 2 0 . 0 6 5 6 K M M 5 5 K H , K O e 5 4 K H , O e 0 . 0 0 0 . 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0

1 . 0 0 . 2 0 )

C o F e M o O ) e 1 . 5 0 . 5 3 8 l e l o

0 . 8 o 0 . 1 6 m

/ m 0 . 3 6 / u u

m 0 . 6 0 . 1 2 m e e (

( 0 . 1 8 t t

n 4 7 K 0 . 4 n 0 . 0 8 e

4 6 K e m

4 5 K m 0 . 0 0 o 0 . 2 4 4 K o 0 . 0 4 0 1 0 2 0 3 0 M

4 3 K M H , K O e 4 2 K H , K O e 0 . 0 0 . 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 0 1 0 2 0 3 0 4 0 5 0 6 0

C o M o O 0 . 0 1 5 2 3 8

) 0 . 0 4 e l ) 0 . 0 1 0 o e l m o / 0 . 0 2 0 . 0 0 5 u m / m u e m

( 0 . 0 0 0 . 0 0 0

e t ( n

e - 0 . 0 0 5 - 0 . 0 2 n m e

o 6 0 K m - 0 . 0 1 0 M - 0 . 0 4 4 0 K o

H , O e 1 . 7 K M H , O e - 0 . 0 1 5 - 6 0 0 0 - 4 0 0 0 - 2 0 0 0 0 2 0 0 0 4 0 0 0 6 0 0 0 - 1 5 0 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0 1 5 0 0

Fig. 34: Parasitic ferromagnetism in iron molybdate and the cobalt/ cobalt-iron molybdates synthesized utilizing PtCl2 as a transport agent. Top left: the perpendicular magnetization of iron molybdate showing a weak, hysteresis-free ferromagnetic moment at low fields. Top right: the parallel magnetization of iron molybdate displaying the same weak ferromagnetic moment at low fields. Middle left: parallel magnetization of Co1.5Fe0.5Mo3O8 synthesized using PtCl2, showing the weak ferromagnetic moment, without hysteresis. Middle right: saturation magnetization of the Co1.5Fe0.5Mo3O8 sample shown on the left at 47K. Inset: perpendicular magnetization of the same sample measured at 55K, above the Neel´ temperature. Again, this displays the weak ferromagnetic moment and also shows that the weak ferromagnetic moment occurs independent of the antiferromagnetic order. This is consistent with the observed isotropic behaviour. Bottom left: magnetization of powdered cobalt molybdate, synthesized using PtCl2, displaying the same parasitic ferromagnetism as seen in the previous two samples. Again, the weak ferromagnetic moment is present, even at 60K, far above the Neel´ temperature. Bottom right: magnetization of the same cobalt molybdate sample at 1.7K, showing a slight hysteresis at low fields. The measurement was performed from 0 →+ve[not shown]→- ve→+ve. This hysteresis exists up to 60K (implying that this moment is independent of the overall order in the system), but is most marked at 1.7K.

74 of (5.9±0.2)µB and (5.8± 0.2)µB, respectfully; however, we found that the directional effective

moments are (4.14 ± 0.02)µB and (4.49 ± 0.02)µB. This shows a paramagnetic anisotropy and orbital angular momentum quenching, similar to the behaviour seen in iron molybdate by

⊥ k McAlister and Strobel. Furthermore, we ﬁnd that µeff > µeff , which further agrees with the results of McAlister and Strobel regarding iron molybdate.

In the intermediate materials, no discernible pattern is present in the effective moment.

µk − µ⊥ is initially small and positive (Co1.5Fe0.5Mo3O8), then moderate and negative

(Co1Fe1Mo3O8), and ﬁnally large and positive (Co0.5Fe1.5Mo3O8).

The magnetization of the cobalt-iron molybdates (Co1.5Fe0.5Mo3O8, Co1Fe1Mo3O8,

Co0.5Fe1.5Mo3O8,) is largely unchanged in comparison to cobalt molybdate. The Neel´ temperature increased with increasing iron concentration (see figure 41). This may be used to determine the approximate composition of the material (see section 3.1). The magnitude of the parallel magnetic susceptibility at the Neel´ temperature, i.e., the maximum in the susceptibility, also increased with increasing iron concentration (see figure 35). This is expected, since iron molybdate has a maximum susceptibility 3 times that of cobalt molybdate and one expects the maximum of the susceptibility to reflect this with iron doping. However, the exact relationship between this maximum and the sample composition is difficult to determine, since the Curie-Weiss law is known to break-down near the ordering temperature. This is the justification for choosing the Neel´ temperature to deduce the sample composition rather than the susceptibility (one might, for instance, deduce the sample composition by predicting the effective moment, which would result in vastly different compositions).

The broadness in the susceptibility in the vicinity of the Neel´ temperature seen in cobalt molybdate (see ﬁgure 31) under a parallel ﬁeld is replaced by a sharp discontinuity with iron

doping. This is evident in the derivative of χm, whose magnitude at the Neel´ temperature increases with iron concentration. However, the enhancement in the perpendicular ﬁeld still clearly marks the onset of magnetic ordering in all samples and was used to determine the Neel´ temperature (see ﬁgure 41) over the parallel susceptibility.

75 0 . 1 5 C o M o O 2 3 8 C o F e M o O 1 . 5 0 . 5 3 8 C o F e M o O 0 . 1 2 1 1 3 8 C o F e M o O 0 . 5 1 . 5 3 8 F e M o O 2 3 8 0 . 0 9 ) e l o m / 3 m

c 0 . 0 6 ( m χ

0 . 0 3

0 . 0 0 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 T e m p e r a t u r e , K

Fig. 35: The parallel susceptibility of the cobalt, iron, and cobalt-iron molybdates in the vicinity of the Neel´ temperature.

0 . 0 5 C o M o O 2 3 8 C o F e M o O 1 . 5 0 . 5 3 8 0 . 0 4 C o F e M o O 1 1 3 8 C o F e M o O 0 . 5 1 . 5 3 8

) 0 . 0 3 F e M o O 2 3 8 K

e l o m

/ 0 . 0 2 3 m c ( T d

/ 0 . 0 1 m χ d

0 . 0 0

- 0 . 0 1 3 5 4 0 4 5 5 0 5 5 6 0 6 5 T e m p e r a t u r e , K

Fig. 36: The derivative of the zero-ﬁeld cooled magnetic susceptibility in the vicinity of the Neel´ temperature. Note the secondary feature in the behaviour of Co1.5Fe0.5Mo3O8 (the red curve), which is not present in any other the other cobalt-iron molybdates.

76 Z F C P a r . 1 0 K O e F C P a r . 1 0 K O e Z F C P e r p . 1 K O e 0 . 0 6 F C P e r p . 1 K O e 2 0 0 ) 3 1 6 0 m c

0 . 0 5 /

e 1 2 0 l o

m 8 0 ( m ) χ /

e 0 . 0 4

l 4 0 1 o

m 0 /

3 - 1 0 0 0 1 0 0 2 0 0 3 0 0

m 0 . 0 3 c (

, m χ 0 . 0 2

0 . 0 1

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 T e m p e r a t u r e , K

Fig. 37: Magnetic susceptibility of Co1.5Fe0.5Mo3O8. Note the pronounced bifurcation in the perpendicular susceptibility as the ferromagnetic-like enhancement at the Neel´ temperature, spanning almost 0.025cm3/mole, and the smaller bifurcation of the parallel susceptibility at low T (this is suppressed due to the high ﬁeld, see graphs 37 - 39 for a better example). Inset: Inverse susceptibility of the FC perpendicular and FC and ZFC parallel susceptibilities. The x-axis is in Kelvin. The ﬁtted lines represent the Curie-Weiss law extrapolation.

Z F C P a r . 3 K O e F C P a r . 3 K O e Z F C P e r p . 3 K O e 0 . 0 8 F C P e r p . 3 K O e 6 0

0 . 0 7 ) 3 5 0 m c

/ 4 0 e 0 . 0 6 l o 3 0 m (

m 2 0 χ 0 . 0 5 / 1 1 0 ) l

o 0

m - 1 0 0 - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0

/ 0 . 0 4 3 m c ( 0 . 0 3 m χ

0 . 0 2

0 . 0 1

0 . 0 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 T e m p e r a t u r e , K

Fig. 38: magnetic susceptibility of Co1Fe1Mo3O8 at 3KOe. Inset: Inverse susceptibility of the FC parallel and perpendicular susceptibilities. The x-axis is temperature, in Kelvin.

77 3 3 3 Compound m (mole/cm K) σm b (mole/cm ) σb µeff (µB) σµ C (Kcm /mole) σC θ (K) σθ Co2Mo3O8 k 0.19690 0.000865 4.20504 0.16497 4.49 0.02 5.08 0.02 -21.36 0.8 Co2Mo3O8 ⊥ 0.23213 0.000915 17.95620 0.17449 4.14 0.02 4.31 0.02 -77.35 0.8

Co1.5Fe0.5Mo3O8 k 0.14794 0.000275 7.55157 0.05661 5.18 0.01 6.75 0.01 -51.0 0.4 Co1.5Fe0.5Mo3O8 ⊥ 0.14952 0.001460 20.36792 0.29708 5.16 0.05 6.69 0.07 -136.22 2.4

Co1Fe1Mo3O8 k 0.19010 0.000423 2.78416 0.08813 4.57 0.01 5.26 0.01 -14.65 0.5 Co1Fe1Mo3O8 ⊥ 0.15431 0.000490 17.29589 0.10309 5.08 0.02 6.48 0.02 -112.09 0.8

Co0.5Fe1.5Mo3O8 k 0.08055 0.000169 6.89221 0.03454 7.03 0.01 12.41 0.03 -85.56 0.5 Co0.5Fe1.5Mo3O8 ⊥ 0.12901 0.000516 12.61176 0.10567 5.55 0.02 7.75 0.03 -97.76 0.9

Fe2Mo3O8 k 0.10258 0.001440 6.97417 0.29425 6.23 0.09 9.7 0.1 -67.99 3.0 Fe2Mo3O8 ⊥ 0.10794 0.000485 17.06778 0.09917 6.07 0.03 9.26 0.04 -158.12 1.2

Table 4: Calculation of the Curie constant (C), Weiss constant (θ), and effective moment (µeff ). The red division marks the start of the tabulation of the ﬁtting parameters taken from ﬁgures 29 and 37 - 40. The black division indicates the beginning of the deduced quantities mentioned above. σi indicates the error in the quantity ’i’ and has the same units as ’i’. The effective moment was calculated from equation (26) with n = 2.

3 J s Compound ∆C (Kcm /mole) ∆µ (µB) µeff (µB) σµ Ion µeff (µB) µeff (µB) 2+ Co2Mo3O8 0.77 0.36 4.26 0.02 Co 6.63 3.87 2+ Co1.5Fe0.5Mo3O8 0.07 0.03 5.17 0.04 Fe 6.71 4.90 Co1Fe1Mo3O8 -1.22 -0.50 4.91 0.01 Co0.5Fe1.5Mo3O8 4.66 1.47 6.04 0.02 Fe2Mo3O8 0.48 0.16 6.12 0.05 Table 5: Comparison of the qualities deduced in table 4. ∆ indicates the difference was taken in the proceeding quantity between the parallel and perpendicular values.

Z F C P a r . 3 K O e F C P a r . 3 K O e 0 . 1 2 Z F C P e r p . 3 K O e F C P e r p . 3 K O e 6 0 ) 0 . 1 0 3 5 0 m c / 4 0 e l o 3 0 m ( 0 . 0 8 m 2 0 χ / 1 ) 1 0 e l

o 0 0 . 0 6 - 1 0 0 - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 m / 3 m c ( m

χ 0 . 0 4

0 . 0 2

0 . 0 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 T e m p e r a t u r e , K

Fig. 39: The magnetic susceptibility of Co0.5Fe1.5Mo3O8. Inset: Inverse χ of Co0.5Fe1.5Mo3O8. The horizontal-axis is in Kelvin.

78 Z F C P a r . 4 K O e F C P e r p . 4 K O e 0 . 1 5 Z F C P a r . 4 K O e F C P e r p . 4 K O e

1 4 0 0 . 1 2 1 2 0 )

3 1 0 0 m c / 8 0 e l o ) 0 . 0 9 m 6 0 e (

l , m o

χ 4 0 / 1 m /

3 2 0

m 0 c

( 0 . 0 6 - 1 5 0 - 1 0 0 - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0

, m χ

0 . 0 3

0 . 0 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 T e m p e r a t u r e , K

Fig. 40: The magnetic susceptibility of iron molybdate. Inset: Inverse χ of iron molybdate. The horizontal axis is in Kelvin.

79 3.2.2 Part II- Bifurcation

There is a second ferromagnetic-like behaviour seen in all these compounds: bifurcation of the zero-field-cooled and field-cooled measurements in the perpendicular susceptibility at the Neel´ temperature. Furthermore, there is a sharp increase in the susceptibility (which will be referred to as a ”ferromagnetic-like enhancement”) at the Neel´ temperature, which is typically indicative of a ferromagnetic moment. Furthermore, the gradual convergence, as T increases, of the zero-field-cooled susceptibility to that of the field-cooled implies there exists some metastable state in which the magnetic energy suspends the system until the thermal energy reaches a sufficient magnitude (see figure 30).

These behaviour are unexpected, since the acquired behaviour of the perpendicular susceptibility measurement at low temperatures, i.e., when the susceptibility is constant, is concomitant with the magnetic ordering observed in Fe2Mo3O8, where the intraplanar coupling is ferrimagnetic but the interplanar coupling is antiferromagnetic; hence a near-constant susceptibility implies that every-other-layer has an equal but opposite moment, which rotates an equal but opposite amount towards the applied ﬁeld. Molecular ﬁeld theory predicts the perpendicular susceptibility to be constant in this scenario[8] (see section 1.2.1).

This is expected, hence D. Bertrand and H. Kerner-Czeskleba report Fe2Mo3O8 and

Co2Mo3O8 share the same magnetic structure[32] and Mossbauer¨ spectroscopy reveals that iron molybdate has intraplanar ferrimagnetic moment. However, even iron molybdate exhibits a sudden jump in the perpendicular susceptibility at the Neel´ temperature (see ﬁgure 41).

This behaviour can be explained, in part, by observing the low-field magnetic properties. When measured at low field values (B < 0.1T ), an interesting property presented itself in the M-H curves (see figure 42). Normally, the value of the magnetization of an antiferromagnet at zero applied magnetic field is history independent and exclusively zero in magnitude. However, the field at which the magnetization is zero is not only finite, it is temperature dependent and its sign is determined by an external poling field (in this case, poling refers to cooling the material though the Neel´ temperature in the presence of an applied magnetic field).

80 C o M o O 2 3 8 C o F e M o O 1 . 4 0 1 . 5 0 . 5 3 8 C o F e M o O 1 1 3 8 C o F e M o O 1 . 3 5 0 . 5 1 . 5 3 8 F e M o O 2 3 8 1 . 3 0

1 . 2 5 ) K 5 6

= 1 . 2 0 T ( m χ / 1 . 1 5 m χ

1 . 1 0

1 . 0 5

1 . 0 0 2 0 3 0 4 0 5 0 6 0 T e m p e r a t u r e , K

Fig. 41: Ferromagnetic-like enhancement in the perpendicular ﬁeld-cooled susceptibility at the Neel´ temperature for single crystals of the cobalt-iron molybdates. For clarity, the susceptibility is taken as a ratio of the measured susceptibility and the susceptibility at 65K for each data set. The region of constant susceptibility is a feature of mean-ﬁeld theory in a uniaxial antiferromagnet(see section 1.2.1).

81 A kink in the magnetization is seen at the point in which M = 0 (see figure 42). This behaviour is seen under both parallel and perpendicular fields. The parallel required field is smaller in magnitude than the perpendicular, but the associated kink, especially at low temperatures, is much larger in both magnitude and range. The kink under a perpendicular field is almost discontinuous, regardless of temperature. Conversely, the kink under a parallel field spans a range of about 50 Oe at 40K, and increases that range to roughly 700 Oe at 5K The required field is temperature dependent, as is most evident under a perpendicular field, where the magnitude of the required field changes by a factor of 1.5 (approximately 500 Oe to 750 Oe) between the temperatures of 40K to 5K. No significant hysteresis is observed in the three samples that were explored, namely Co2Mo3O8, Co1.5Fe0.5Mo3O8, and Co1Fe1Mo3O8, except in Co2Mo3O8 (measurement performed on an early sample and the hysteresis was determined-via XRD [not shown], to likely be contribution from cobalt oxide impurities)

Co1Fe1Mo3O8 when exposed to relatively high fields (2T at 5K). This apparent hysteresis is likely due to a decrease in the required field. Co1.5Fe0.5Mo3O8 showed no evidence of hysteresis, even when exposed to fields up to 7T at 5K (see figure 43).

The magnitude of the required field decreases with iron concentration (see figure 44). This is not unexpected, since the bifurcation in iron molybdate is minimal relative to the other samples (see figure 40 for the susceptibility of iron molybdate, see below for an argument of the connection between the bifurcation and the required field).

As mentioned previously, the sign of the required field can be inverted by inverting the poling field. When the poling field is negative (-10 Oe was used), the required field is positive, and vise versa. This is true in all recorded cases excluding the case of a parallel field at 40K. In this scenario, the required field was simply reduced in magnitude (see the inset of figure 42).

It should be noted that the required field was positive in absence of any poling. This is due to an effect known as flux-pinning, which occurs in the superconducting magnet used to produce homogeneous magnetic fields. Flux-pinning results in magnet remanence [60]; for practical purposes, this leads to small negative fields when a positive field is first created and subsequently removed, i.e., if a positive field of 10 KOe is applied and then removed, so that

82 4 0 1 0 4 0 K P e r p . 4 0 K i n 1 0 O e , P e r p . 2 5 K P e r p . 3 0 4 0 K i n - 1 0 O e , P e r p . ) e l 0 5 K P e r p . o 2 0 m

/ - 1 0 u 1 0 m e

( - 2 0 3

0 0 1 *

- 3 0 n

o - 1 0 i t

a - 4 0 z i

t - 2 0 e

n - 5 0 g - 3 0 a

M - 6 0 - 4 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0 - 5 0 0 0 5 0 0 H , O e 4 0 K i n 1 0 O e , P a r a . 5 K i n 1 0 O e , P a r a . 4 0 4 0 K i n - 1 0 O e , P a r a . 2 0 5 K i n - 1 0 O e , P a r a .

2 0 1 0 4 0 0 2

- 2 0 0 - 1 0

- 2 - 4 0 - 2 0 - 4 2 0 0

- 6 0 C 1 - 3 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0

Fig. 42: The behaviour seen in the M-H curves of Co1Fe1Mo3O8 in the low-H-field range of |H| < 1KOe. Note that, while the overall shape of the M-H curves is that of an antiferromagnet, the magnetic field at which the magnetic moment equals zero is non-zero, and depends on the external poling field. The point in which M = 0 is accompanied by a small kink. Top Left: M-H curves at several temperatures, under a perpendicular field. The field required to reach zero magnetization- This will be referred to as the ”required field”- increases as temperature decreases, to a point. At 25K, the required field is approximately 750 Oe, while at 40K, the requirement is approximately 500 Oe. Note that this data was not collected after poling the sample, unlike the others shown along it. However, due to the behaviour exhibited, it is concluded that a small, negative field must have existed, the origin of which is discussed in the accompanying section. Top Right: Reversal of the required field after poling. Bottom Left: M-H curve under a parallel field at 40K. Note that the magnitude of the required field is significantly less than under a perpendicular field. Poling in a positive field only reduced the required field, rather than inverting its sign. Inset: A magnified view of the kink at M=0. Bottom Right: M-H curves under a parallel field at 5K. Note that the kink present at previous temperatures becomes significant in magnitude, compared to the overall magnetization (which, being antiferromagnetic, decreases with temperature), and in range, occurring over almost a 700 Oe interval.

83 3 6 C o F e M o O 1 . 5 0 . 5 3 8 C o F e M o O 1 1 3 8

1 8 ) . u . f / B µ 3 -

0 0 8 1 (

, t

n 4 e m

o 0

M - 1 8 - 4

- 8 - 4 - 2 0 2 4 6 - 3 6 - 2 0 - 1 0 0 1 0 2 0 H , K O e

Fig. 43: M-H curves of Co1.5Fe0.5Mo3O8 and Co1Fe1Mo3O8 at 5K, poled under a negative field (Co1.5Fe0.5Mo3O8 measured clockwise [-ve → +ve → -ve], Co1Fe1Mo3O8 counter-clockwise). Interestingly, the slope and vertical shift of both samples are nearly identical. Inset: magnified image of the low-field dependence. Note that Co1.5Fe0.5Mo3O8 shows no change to the required field, despite exposure to a 7T field (not shown). Conversely, the required field of Co1Fe1Mo3O8 decreases from about 1000 Oe to 0 Oe upon exposure to a 2T field. This generates an apparent hysteresis.

84 2 . 0 C o M o O 1 . 7 K 2 3 8 C o F e M o O 5 K 1 . 5 1 . 5 0 . 5 3 8 C o F e M o O 5 K 1 1 3 8 1 . 0

0 . 5

0 . 0

- 0 . 5

- 1 . 0

- 1 . 5

- 2 . 0 0 2 0 0 4 0 0 H , O e

Fig. 44: The parallel M-H behaviour at low ﬁelds and temperature. The y-axis is in arbitrary units.

the intended field is zero, the actual field will be -18 Oe [60]. As the magnet was charged with a positive field when calibrating the position of each sample, a small, negative, remnant field would exist within the magnet, which would pole the material.

This is believed to be the source of the bifurcation seen in the perpendicular susceptibility and, to a lesser degree, in the parallel susceptibility at low temperatures. For the same reason discussed above, the measurements of the zero-field-cooled susceptibility were actually field- cooled in a small negative field. This resulted in the magnetization being irregularly small, or right shifted. The field-cooled measurements were irregularly large, or left shifted. This resulted in a displacement between the two. This displacement is proportional to the required field, as it indicates the magnitude of the offset, which initially decreases as temperature decreases and then remains constant. This is consistent with the observed convergence of the zero-field-cooled and field-cooled susceptibilities as the temperature approaches the Neel´ temperature. It also explains the bifurcation of the parallel susceptibility at low temperature, hence the inversion of the poling field at 40K affects the required field insignificantly, while at 5K, the difference is quite marked.

This behaviour is reminiscent of exchange-bias. Exchange-bias arises from the

85 1 0 K O e , Z F C 0 . 0 5 0 1 0 K O e , F C 3 K O e , Z F C 3 K O e , F C 0 . 0 4 5

0 . 0 4 0 )

e 0 . 0 3 5 l o m / 3 0 . 0 3 0 m c ( m χ 0 . 0 2 5

0 . 0 2 0

0 . 0 1 5 1 0 2 0 3 0 4 0 5 0 T e m p e r a t u r e , K

Fig. 45: The magnetic susceptibility of Co1.5Fe0.5Mo3O8 under a 10KOe magnetic field. Note that the behaviour of the susceptibility under a 10KOe field is reminiscent of the ZFC behaviour under a 3KOe field. interaction of ferromagnetic and antiferromagnetic phases, and results in a horizontal shift in the coercivity of said ferromagnetic[63]. However, no evidence is present to support a two-phase system. Due to the magnetic structure, even uncompensated spins at the surface would average to zero. Even if clusters of purely iron atoms could form throughout the material, iron molybdate isn’t ferrimagnetic until exposed to an external field of approximately 3.5T [18]. As such, it is unlikely exchange-bias is the source of the observed offset.

Interesting, measurements on the perpendicular susceptibility at 10 KOe in

Co1.5Fe0.5Mo3O8 show that the field-cooled enhancement following the Neel´ temperature is suppressed and follows the zero-field-cooled behaviour (see figure 45). This is unexpected, since, intuitively, the field-cooled behaviour should be favoured at the higher magnetic energies associated with a larger magnetic field. Furthermore, the resulting behaviour recorded at 3 KOe suggest the opposite would occur- the ZFC should converge to the FC, hence this is what occurs near the Neel´ temperature.

Upon further investigation, a similar ferromagnetic-like behaviour in the magnetic susceptibility occurs under a parallel ﬁeld (see ﬁgure 46). This moment is most present at very

86 ) - 1

e 1 0 l o m / 3 m c (

, m χ 1 0 - 2

0 . 6 2 0 O e Z F C 2 0 O e F C 0 . 5 1 0 0 O e 2 5 0 O e ) 5 0 0 O e 2 0 0 0 O e u .

f 1 0 0 0 0 O e / 0 . 4 B µ

2 −

0 0 . 3 1 (

, t 0 . 2 n e

m 0 . 1 o M 0 . 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 T e m p e r a t u r e , K

Fig. 46: The ferromagnetic moment in the low-field parallel susceptibility of Co1.5Fe0.5Mo3O8. These measurements were performed by poling the sample in a very small positive field (as opposed to a positive and negative field). Top: The magnetic susceptibility at different fields (see legend, solid markers correspond to ZFC measurements, Hollow to FC) Bottom: Suppression of the ferromagnetic-like moment as field increases from 20Oe to 500 Oe.

87 low ﬁelds (20 Oe) and is suppressed as the ﬁeld is increased, being almost completely suppressed at 500 Oe.

The change in the magnetic moment at 20 Oe is approximately one third of the increase

−3 in the moment seen in the perpendicular susceptibility (approximately 2 × 10 µB/f.u versus

−4 6 × 10 µB/f.u). This is consistent with the idea that the bifurcation is due to the magnitude of the required ﬁeld. With an assumption, we can perform a back of the envelope calculation

to show this. The perpendicular ferromagnetic-like enhancement in Co1.5Fe0.5Mo3O8 is about three times larger than the parallel, while the perpendicular required ﬁeld is about 5 times the

parallel required ﬁeld at 40K in Co1Fe1Mo3O8. Assuming Co1.5Fe0.5Mo3O8 has a similar ratio, the parallel enhancement will be about three times smaller than the perpendicular, hence the parallel magnetic susceptibility is roughly 1.5 times smaller than the perpendicular.

The observed behaviour is difﬁcult to explain. Below will be a discussion on some ﬂawed but potential candidates.

A possible explanation to the ferromagnetic-like enhancement was discussed by A.N. Bogdanov et al. [41] in their paper on noncentrosymmetric uniaxial antiferromagnets. Bogdanov found that, under application of a parallel field, noncentrosymmetric uniaxial antiferromagnets will either undergo a spin-flop transition, as was observed in Mn2Mo3O8 [37], or remain antiferromagnetic. Application of an oblique field results in spin-canting and, if the magnitude of the perpendicular component is sufficient, weak ferromagnetism[41]. This spin-canting origin of weak ferromagnetic moment could explain the ferromagnetic-like enhancement that is observed in cobalt/cobalt-iron molybdate. While one might expect to see

a ferromagnetic moment in the perpendicular magnetization, S. Lee et al. conclude that CoF3 exhibits spin-canting, yet it lacks any apparent ferromagnetic moment [42]. However, this does not account for the parallel direction, only the perpendicular. This could only be plausible if the crystals deviated from the measurement axis. Since the perpendicular required ﬁeld is roughly 5 times the parallel ﬁeld, this would require a deviation of 10 degrees. This would be perfectly reasonable for a ferromagnet. However, the crystallographic and shape anisotropy would prevent this, and the magnetic moment is inherently low. Furthermore, each sample was

88 inspected both before and after the measurements, and no deviation was ever detected.

One can’t help but wonder if the behaviour is due to the parasitic ferromagnetism due to impurities previously observed. However, it is noted that McAlister and Strobel observe this behaviour is ﬁelds as high as 1000 Oe in cobalt molybdate, while it was found in this work that 350 Oe was sufﬁcient to completely suppress the enhancement. Furthermore, one would expect that non-linear behaviour would be detected in the M-H curves, and that this behaviour would reverse at H=0. This was not observed.

The last explanation is phenomenological in nature. The observed behaviour is similar to that of iron molybdate. However, it is in reverse. While iron molybdate experiences a metamagnetic transition from an antiferromagnetic state to a ferrimagnetic state, it is possible that cobalt molybdate and the cobalt-iron molybdates begin in the ferromagnetic state and transition into an antiferromagnetic state. This is observed in figure 46. This would explain the enhancement in the perpendicular susceptibility, as the system would still be ferrimagnetic under a perpendicular field. It also explains why the perpendicular FC susceptibility decays to the ZFC, hence at small deviations from the measurement axis are inevitable. At 10 KOe, only about 3 degrees of deviation are required to generate a 500 Oe field along the C-axis, which is enough to suppress the ferromagnetic enhancement. However, this explanation can not account for the low-field M-H behaviour (the offset from M=0 at H=0), nor the presence of the perpendicular enhancement at the Neel´ temperature in iron molybdate-it was observed to favour the ferrimagnetic state at high fields. Lastly, it lack physical basis, since transforming from a ferromagnetic state to an antiferromagnetic state has an unfavourable Zeeman energy and thus is not expected to occur after application of a magnetic field (unlike the antiferromagnetic to ferrimagnetic transition).

None of the proposed solutions are without ﬂaws and, as such, more investigation is needed to describe the phenomena at play.

89 3.2.3 Part III- Cobalt-Zinc Molybdate and Cobalt-Manganese Molybdate

This section will brieﬂy present the parallel susceptibility of cobalt-zinc molybdate, as

it is used in section 3.3, as well as comment on cobalt-manganese molybdate. Co1Zn1Mo3O8 was chosen as a potential non-magnetic material to be used to determine the electronic and phononic contributions to the speciﬁc heat. Indeed, it was found that cobalt-zinc molybdate does not magnetically order (see ﬁgure 47).

0 . 4 5

0 . 4 0

0 . 3 5

0 . 3 0

) 0 . 2 5 e l o m

/ 0 . 2 0 3 m c ( 0 . 1 5 m χ

0 . 1 0

0 . 0 5

0 . 0 0

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 T e m p e r a t u r e , K

Fig. 47: The magnetic susceptibility of cobalt-zinc molybdate. No ordering temperature is detected, down to 1.7K. As such, this sample is suitable for use in determining the magnetic contribution to the speciﬁc heat in the Co2−2xFe2xMo3O8 series.

Cobalt-manganese was found to be uninteresting to the goals of this project. Manganese doping showed no sign of changing the magnetic order (not shown) and had the additional complication of an equivalent Neel´ temperature as cobalt molybdate [31], making it difﬁcult to determine the chemical composition. As such, it was abandoned and is only mentioned here for completeness sake.

90 3.3 The Speciﬁc Heat

The specific heat of cobalt molybdate shows an anomaly in the temperature range of 43K to 41K (see figure 48). This is concomitant with the magnetic phase transition observed in the magnetic susceptibility. The introduction of iron into the system pushes this anomaly to higher temperatures, a behaviour which is also seen in the magnetic susceptibility (see figure 49 and 35).

A 4 0 0 ) K 3 0 0 e l o m /

J 2 4 (

t a

e 2 0 0 2 2 H

i 2 0 f i c

e 1 8 p

S 1 0 0 1 6

1 4 3 8 4 0 4 2 4 4 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 T e m p e r a t u r e , K

Fig. 48: The speciﬁc heat capacity of cobalt molybdate. The point labeled ’A’ is due to sampling error. Inset: a close-up of the anomaly. The axes have the same scale as the host plot. The maximum of the peak occurs at (41.8±0.6)K.

To find the magnetic contribution to the heat capacity, two methods may be utilized. The first, and preferable, is to find a material that has an identical crystal structure but does not magnetically order. Using this method, one should be able to approximate the lattice contribution and any electronic contribution to the specific heat. Luckily, it was determined

that Co1Zn1Mo3O8 lacks an ordering temperature (see section 3.2.3) and may be used for this purpose. The second method is to utilize the Debye model, which predicts that the speciﬁc heat should vary as T3. This method is not practical, since magnons in 3-D antiferromagnetic systems also vary as T3 [56], making it difﬁcult to differentiate the lattice contribution from the magnetic contribution. This is further complicated by the induced polarization seen at the

91 C o M o O 1 2 0 2 3 8 C o F e M o O 1 . 5 0 . 5 3 8 C o F e M o O 1 1 3 8 1 0 0 C o F e M o O 0 . 5 1 . 5 3 8 F e M o O 2 3 8 C o Z n M o O 8 0 1 1 3 8 ) K

e l o 6 0 m / J (

C 4 0

2 0

0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 T e m p e r a t u r e , K

Fig. 49: The speciﬁc heat capacity of cobalt, iron, zinc-iron, and the cobalt-iron molybdates. Of note are the anomalous behaviour of Co1Zn1Mo3O8 at approximately 20K and the convergence of Co1.5Fe0.5Mo3O8, Co1Fe1Mo3O8, and Fe2Mo3O8 to Co1Zn1Mo3O8 at temperatures above their respective transition temperatures. All that may be said of the anomaly seen in Co1Zn1Mo3O8 is it is not magnetic in origin, as the magnetic susceptibility shows no indication of an accompanying anomaly.

Neel´ temperature, which implies a corresponding distortion of the crystal lattice. This affects not only the lattice contributions, but also the electronic contributions.

Noting that these materials are ferroelectric, the electronic contribution to the specific heat- which normally has a linear dependence in T- should be negligible. However, since the ferroelectricity is a product of the crystal structure, this is only necessary along the polar axis- the c-axis. This was first alluded to by McAlister and Strobel [31], who noted that the resistance of M2Mo3O8 (M=Co, Fe, Mn, Ni) is that of a high-resistivity semiconductor, with much larger resistance along the polar axis. The specific heat of Co1Zn1Mo33O8 and

Co2Mo3O8- the electronic contribution in the remaining 4 materials is discussed below- reveal that the electronic contribution need not be negligible in this family of materials, despite their ferroelectric nature (see figure 50). As a result, Co1Zn1Mo33O8 was fit to a polynomial of order three to properly interpolate its specific heat data (see figure 50).

Above T≈125K, the behaviour of Co1.5Fe0.5Mo3O8, Co1Fe1Mo3O8, and Fe2Mo3O8

92 C o M o O 2 3 8 C o Z n M o O A b o v e T C o F e M o O 1 1 3 8 N 1 . 5 0 . 5 3 8 F i t B e l o w T N 1 0 0

8 0

) K

6 0 e l o

m 4 0 / J (

C 2 0

C o F e M o O C o F e M o O F e M o O 1 1 3 8 0 . 5 1 . 5 3 8 2 3 8

0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 T 3 / 1 0 0 0 ( K 3 )

Fig. 50: The specific heat as a function of T3. If the Debye model holds, the dependence of the specific heat should be linear in T3. The scale of each axis is identical to that shown on the bottom-left plot of Co1Fe1Mo3O8. The temperature range displayed is from 0 to about 90K. The specific heat of Co1Zn1Mo3O8 was fit to a polynomial of order three and used as a baseline for the lattice and electronic contributions in all materials except Co2Mo3O8. This fit is displayed alongside each data sets. Each data set was fit to the Debye model above (grey line) and below (red line) their respective transition temperatures. Note that the specific heat of Co1Zn1Mo3O8 exceeds that of Co0.5Fe1.5Mo3O8.

93 ) 7 0 A K

l 6 0 o

m 5 0 / J (

4 0 t a

e 3 0

i 2 0 f i c 1 0 e p

S 0 1 0 5 B 4

1 0 ) K

3 e 1 0 l o

C o M o O 2 m 2 3 8 /

1 0 J C o F e M o O 1 . 5 0 . 5 3 8 m 1 ( 1 0 C o F e M o O y 1 1 3 8 p

0 o C o F e M o O r 0 . 5 1 . 5 3 8 1 0 t n

F e M o O - 1 E 2 3 8 1 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 T e m p e r a t u r e , K

Fig. 51: A:The magnetic contribution to the specific heat. The x-axis scale is equivalent to that of B. The range selected for each sample was taken to be the point of minimal positive distance between the data in question and the function- either a polynomial of order three, fit to the specific heat of Co1Zn1Mo3O8, or the Debye model- used as the lattice contribution. B: The entropy associated with the magnetic contribution, in logarithmic scale. Note that: the magnetic contribution, and therefore the entropy, of Co0.5Fe1.5Mo3O8 is underestimated.

follow that of Co1Zn1Mo3O8. Since it was determined that Co1Zn1Mo3O8 does not magnetically order, one may use its speciﬁc heat to ﬁnd the magnetic contribution in the aforementioned materials. However, it is worth noting that Co1Zn1Mo3O8 displays anomalous behaviour below approximately 20K, and this method is therefore valid only above this

temperature. This method also can not be used for Co2Mo3O8, since it does not converge with

Co1Zn1Mo3O8. As such, the Debye model was used in this case. Co0.5Fe1.5Mo3O8 also does not converge, but at temperatures around its transition temperature, this method is still preferable- for reasons explained below- over the Debye model.

In the case of Co0.5Fe1.5Mo3O8, application of the Debye model resulted in two unique slopes above and below the anomalies (see figure 50). In fact, this property was exhibited in all of the iron doped materials, and in iron molybdate. As mentioned previously, magnons in 3-D antiferromagnetic systems also contribute to the specific heat as T3. This complicates matters, especially when one considers the magnetic contribution of iron molybdate (see figure 51), which is non-zero well above its Neel´ temperature. This makes it unclear whether

94 the heat capacity of Co0.5Fe1.5Mo3O8 contains magnetic contribution above its Neel´

temperature. Due to this, Co1Zn1Mo3O8 was again used to approximate the lattice and electronic contributions, with the understanding that this would underestimate the magnetic contribution: the speciﬁc heat of Co1Zn1Mo3O8 exceeds that of Co0.5Fe1.5Mo3O8 above the

Neel´ temperature of Co0.5Fe1.5Mo3O8.

In the case of cobalt molybdate, the slopes of the lines ﬁt to T 3 dependence above and below the Neel´ temperature of cobalt molybdate were similar enough that the Debye model could be used to estimate the lattice contribution. The region above the Neel´ temperature of cobalt molybdate was used for this purpose.

As noted previously, the electronic contribution is non-negligible in these materials. This

3 is evident in Co1Zn1Mo3O8 and Co2Mo3O8 as a root-cubed dependence on T (in other word, linear in T) of the specific heat. However, it is also apparent in the remaining compounds as a finite y-intercept above the Neel´ temperature, which should otherwise be zero in the Debye model. This offset is eliminated below the Neel´ temperature (see figure 50),i.e., the y-intercept is zero. Since both the magnon and phonon contributions should produce a straight line in T3 with a zero y-intercept, this is an indication that electronic contributions may exist above the Neel´ temperature but not below it, hence the non-zero y- intercept above the Neel´ temperature can not be attributed to magnon nor phonons. Wang et al. report that the polarization of iron molybdate increases through the Neel´ temperature[18]. This would increase the materials resistivity and therefore suppress the electronic contribution.

The entropy was calculated as:

Z dQ Z C S = = m dT (66) m T T

The trapezoidal rule was utilized to calculate equation (66) analytically. The entropy of the transition and the limits of the integration can be found in ﬁgure 51. No surprising features are present in the resulting entropy curves, except that the entropy of Co1.5Fe0.5Mo3O8 exceeds that of Co1Fe1Mo3O8 at all temperatures. This is unexpected, since the underestimation of

95 Co0.5Fe1.5Mo3O8 likely puts its true value somewhere between Co1Fe1Mo3O8 and Fe2Mo3O8.

3.4 The Dielectric Function, the Magnetoelectric Capacitance, and the

Magnetoelectric Coupling

In this section, the dielectric functions, magnetoelectric capacitance and the magnetoelectric coupling will be discussed. All three quantities are intimately related in both their physical qualities and in their frustrating persistency as sources of experimental difﬁculty- this is especially true for the direct measurements of the magnetoelectric coupling. As such, it seems appropriate to discuss them together. This section will be broken into three parts, one for each topic.

3.4.1 The Dielectric Function

The dielectric function of cobalt molybdate (see figure 52) was investigated to determine if an anomaly was present at the Neel´ temperature. This would indicate that magnetoelectric coupling is possible, as at least one of the coupling coefficients must be non-zero (see section 1.4). Indeed, a small anomaly was present around the Neel´ temperature, in absence of applied magnetic field. This was expected, as iron molybdate exhibited the same phenomena[18] (see figure 55). In fact, the anomaly was present in every measured sample (one sample was excluded, for reasons explained below).

However, the performed measurements suffered from a fatal issue, which jeopardizes the quantitative potential of the data: it was determined that temperature of the sample lagged behind that of the system. This was initially believed to be possible hysteresic behaviour, but variable field measurements revealed an overarching inverse exponential behaviour, which was time dependent (see figure 54). This indicates there is a temperature gradient between the system and the sample, since we know from equation (64) that the temperature difference between two bodies decays exponentially. This was confirmed by measurements of the

96 C o M o O ε 8 0 0 2 3 8 r 8 x 1 0 5 C o M o O L o s s 2 3 8 6 0 0 6 x 1 0 5 ) m / S µ

5 ( r 4 0 0 ε 4 x 1 0 s s o 2 0 0 2 x 1 0 5 L

0 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 1 0 0 T e m p e r a t u r e , K 2 . 5 x 1 0 5 F e M o O ε 2 3 8 r F e M o O L o s s 5 8 0 2 3 8 2 . 0 x 1 0

6 0 1 . 5 x 1 0 5

4 0 1 . 0 x 1 0 5

2 0 5 . 0 x 1 0 4

0 0 . 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0

Fig. 52: The dielectric function and loss in cobalt and iron molybdate. The units of each axis are equivalent for both plots. The region where both the dielectric function and loss are constant indicate when the thermal energy is insufﬁcient to free dipoles. These measurements were performed with a 5V amplitude and 20Khz frequency. dielectric function as a function of temperature (see ﬁgure 53), which showed that there was a systematic error in temperature as large as 30K between measuring on cooling (in this case, 300K to 5K) and on warming (5K to 300K).

Several attempts at remedying this issue were made, including ﬁlling the probe with helium gas, in an attempt to increase the thermal coupling between the sample and the probe. This was somewhat successful: in vacuum, when measured on cooling from 70K to 5K and then on warming from 5K to 70K immediately afterward, the transition temperature of

Co1Fe1Mo3O8 varied by 15K between the two measurements. This was performed at a rate of 0.25K/minute- a rate which is demanded by only the highest resolution measurements. This was reduced to 5K by charging the system with 2 torr of helium at room temperature, again at the same rate. This however is of questionable utility when the temperature difference between the Neel´ temperatures of pure cobalt molybdate and iron molybdate is only 20K. Furthermore, it is far from practical to attempt to calculate what the systematic error would be at a given temperature, as, in these kinds of measurements, the temperature of the thermal bath is always in ﬂux (in equation (64), ∆T is time dependent). As such, the data from the dielectric function

97 1 . 0

0 . 8

0 . 6 e c n a t i c a p

a 0 . 4

C ~ 3 0 K

0 . 2

0 . 0 2 0 0 2 5 0 3 0 0 T e m p e r a t u r e , K

Fig. 53: Co1Fe1Mo3O8 showing the temperature lag of the sample compared to the system. The y-axis here is in arbitrary units of capacitance. When measuring from 5K to 300K at 0.25K/min, the sample reached only 270K when the system reaches 300K, resulting in a lag of 30K. This presents a problem when measuring the magnetocapacitance and the magnetization as a function of electric ﬁeld [M(E)], as these quantities vary minimally over the ranges available. Variation in temperature due to an unequilibrated system can potentially mask the desired effects. As such, when measuring these quantities, the sample was allowed to equilibrate with the system for at least twenty-four hours before collecting data.

98 y 0 = 0 . 3 4 1 7 3 , x 0 = 6 9 2 5 . 3 3 6 2 4 A 1 = 2 . 1 9 4 3 9 E - 4 , t 1 = 1 9 2 1 4 . 5 5 2 3 8 e c n a t i c a p a e C

c n a t i

c 4 0 0 0 0 6 0 0 0 0 8 0 0 0 0 1 0 0 0 0 0 a p

a T i m e , s C

0 2 0 0 0 0 4 0 0 0 0 6 0 0 0 0 8 0 0 0 0 1 0 0 0 0 0 T i m e , s

Fig. 54: The overarching exponential decay of the capacitance with time. The Y-axis has arbitrary units in both the host plot and the inset. The ’waves’ in the capacitance correspond to changes in the magnetic ﬁeld, from -9T to 9T. The minimum of each ’wave’ occurs when there is zero magnetic ﬁeld. Inset The capacitance after removal of the exponential decay. This measurement was performed after cooling the system from 300K to 40K and allowing the system to equilibrate for 10 hours and had a duration of roughly 30 hours. should be examined for its qualitative properties only. Luckily, for measurements performed at a constant temperature, e.g., the magnetoelectric capacitance, this is not a problem, as one can simply wait for the entire system to equilibrate at the desired temperature.

As mentioned in section (1.3), the dielectric function of ferroelectrics shows a counter-intuitive property: the magnitude of r increases with increasing temperature. This is peculiar, as one expects the ferroelectric to become more conductive with increasing temperature, hence thermal energy would disorder the system and decrease the polarization (here, and in the preceding discussion, I am referring to polarization as the polarization of e.g., a single domain, or as the average magnitude of the dipole moment for a single dipole; the net polarization is, of course, zero). This is reflected in the dielectric loss, which increases with increasing temperature (see figure 52). These conflicting results can be explained by realizing that r increases with temperature because the strong internal electric fields prevents the electric dipoles from rotating towards an external electric field.

99 The dielectric function is typically measured with AC fields of relatively high frequency (20Khz was used in these experiments). In order to see a response, the electric dipoles need to be able to follow the external field. This is not possible in ferroelectric at low temperatures, hence the system will be highly ordered in the absence of thermal energy. However, as the thermal energy increases, some of the dipoles are freed and able to rotate with the external field. These freed dipoles then can act like a conventional insulator and contribute to the dielectric function. However, the polarization decreases, since the freed dipoles have a time-average dipole moment of 0 (they are randomly oriented). As such, the conductivity increases, as seen in figure 52.

The measurements of the dielectric function are presented in ﬁgure 55. The expected anomaly is present in all of the measured samples, indicating that magnetoelectric coupling occurs in these samples. The samples of Co1.5Fe.5Mo3O8 proved too small in cross-section to measure reliably, and thus it has been excluded from this section. Interestingly, below the anomaly- which occurs at the Neel´ temperature, the slope of the dielectric function increases for all the measured samples, excluding Co2Mo3O8, for which the slope remains almost unchanged.

This can be explained by examining equation (42). When T is small, i.e., TTC , the binomial expansion may be used to acquire the relation:

C CT r ' + 2 (67) 2TC 2TC

C As such, one expects linear behaviour at low temperature with slope m = 2 and 2TC C intercept b = . If the slope decreases below the Neel´ temperature, this implies that TC must 2TC increase, hence the intercept below the Neel´ temperature both increases or decreases and the slope is equal to m = b . This implies that the polarization is increased by the magnetic TC ordering, since the value of TC is roughly when the thermal energy is equal to that of the electrical energy. This is a reﬂection of the fact that the onset of magnetic ordering further polarizes the material and therefore further orders it. It is thus expected to require more

100 Fig. 55: The dielectric function of the four measured samples. The units on each axis are equivalent to that of the lower left plot of Co1.5Fe0.5Mo3O8. The grey lines correspond to a linear ﬁt above the Neel´ temperature, while the red lines are ﬁt below it. r was calculated assuming the samples acted as perfect parallel plate capacitors. Note that the slope below TN decreases for all the sample excluding Co2Mo3O8.

101 thermal energy to disorder it. One may also picture this via molecular ﬁeld theory as an

increase the the molecular ﬁeld, hence TC is proportional to λ. This is consistent with the result obtained for iron molybdate by[18] Wang et al., who showed that the polarization increases as iron molybdate is cooled through the Neel´ temperature.

Since an anomaly was detected in the dielectric function at the Neel´ temperature, it is expected that the dielectric function should also exhibit some ﬁeld dependence. This is referred to as magnetocapacitance, and is the topic of the proceeding section.

3.4.2 The Magnetocapacitance and The Magnetoelectric Coupling

Since none of the doped samples exhibited the desired metamagnetic transition into a ferrimagnetic state, linear coupling should not occur. The magnitude of any magnetoelectric coupling will be limited by the antiferromagnetic order. Nevertheless, non-linear coupling can occur and it may be valuable to measure this phenomena.

Direct measurements of the magnetoelectric coupling could not be performed in this work. Despite reducing the background magnetic moment to at least 1/10 of the typical antiferromagnetic sample’s magnet moment, in measurements of the magnetization versus applied electric field, the variation of the magnetization was on the order of the noise in the system (one part in one thousand). This is a technological limitation that has only a monetary solution, in the form of either a larger applied electric field (The maximum applied field is limited to just 250 V/cm), so that the variation between points is larger, or a SQUID with a larger signal-to-noise ratio.

Furthermore, only three ﬁfths of the samples were even suitable for either the magnetocapacity or the direct magnetoelectric coupling. Most single crystals were simply too

small. This is particularly true for Fe2Mo3O8 and Co0.5Fe1.5Mo3O8, for which no suitable

samples were ever obtained. The magnetocapacitance was measured for Co2Mo3O8,

Co1.5Fe0.5Mo3O8, and Co1Fe1Mo3O8.

102 0 . 0 8 0 . 0 6 0 . 0 4 0 . 0 2 0 . 0 0 - 0 . 0 2 C o M o O - 0 . 0 4 2 3 8 0 . 0 3

0 . 0 2

0 . 0 1

0 . 0 0

- 0 . 0 1 C o F e M o O 1 . 5 0 . 5 3 8 - 0 . 0 2 0 . 0 0 C o F e M o O 0 1 1 3 8 =

H - 0 . 0 1 ε /

) - 0 . 0 2 0 =

Η - 0 . 0 3 ε - Η

r - 0 . 0 4 ε ( - 0 . 0 5

- 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 H , K O e

Fig. 56: The magnetocapacitance of Co2Mo3O8 (5K), Co1.5Fe0.5Mo3O8 (40K), and Co1Fe1Mo3O8(49K).

The measurements were performed at 20KHz frequency and 5V amplitude. The results of the three measured samples are displayed in figure 56. Due to the inability to induce a metamagnetic transition in these materials, the magnitude of the magnetoelectric coupling is limited by the antiferromagnetic order. This is reflected in the magnetocapacitance. This caused some difficulties when measuring cobalt molybdate, whose surface area was about 1/5 of that of the Co1Fe1Mo3O8. In fact, the level of noise was to such a magnitude that the exact dependence of the capacitance on the external magnetic field could not be determined. Nevertheless, fitting the data to a linear dependence reveals there is a net increase in the capacitance with the magnitude of H. This indicates that the system does exhibit magnetoelectric coupling.

The coupling is more apparent in the remaining two samples. Co0.5Fe1.5Mo3O8 exhibits a positive quadratic dependence, while Co1Fe1Mo3O8 shows linear dependence with a negative

103 slope. The behaviour shown may be explained phenomenologically via Landau theory. The free energy of the system up to terms with quadratic dependence in M and P is:

2 2 2 2 2 2 ]E = E0 + aP + bM + αMP + βP M + γP M + δP M + ... (68)

The inverse of the electrical permittivity is equal to the second partial differential of E in equation 68) w.r.t. P [61]. Thus:

1 = 2a + 2γM + 2δM 2 (69) χE and 1 H = 1 + (70) r 2a + 2γM + 2δM 2

If γ is negligible and δM 2 2a, equation 70 can be binomially expanded to[61]:

1 δM 2 H = 1 + (1 − ) (71) r 2a a

If δ is negligible and γM 2a, equation 70 becomes:

1 γM H = 1 + (1 − ) (72) r 2a a

Equation (71) is the conventional magnetodielectric effect [62]. The deﬁnition of the H − H=0 magnetocapacitance is MC = [62], which, using equation (71), simpliﬁes to r H=0 −γM 2 MC = . Since Co Fe Mo O shows a linear dependence of M w.r.t H, is r 2a2 + a 1.5 0.5 3 8 r therefore proportional to H2.

−δM If instead equation (72) is used, one arrives at MC = . It should be noted that, r 2a2 + a similar to α, this term can only occur in the free energy if δ breaks time inversion symmetry.

MC This results in linear dependence of r w.r.t. H.

104 MC The linear regime of r begins above approximately |H| = 10KOe in Co1Fe1Mo3O8 and magnitude of this linear magnetocapacitance is surprising (NiCr2O4 exhibits a similar magnitude despite ordering ferrimagnetically). Conventionally, M for an antiferromagnet exactly cancels over a magnetic unit cell. However, this is only true at zero kelvin, as evident by the temperature dependence of the magnetic susceptibility. As such, it is optimal to measure antiferromagnets close to the the Neel´ temperature, where their susceptibility is the largest. It is thus expected that the magnetocapacitance would decrease with decreasing temperature. This can only be commented on anecdotally, hence the data sets at lower temperatures were excessively noisy (not shown).

−γ(χT =40K )2 −δχT =49K The coefﬁcients m and m are equal to (5.1 ±0.3)×10−14 Oe−2 and 2a2 + a 2a2 + a (-6.99 ±0.07) ×10−9 Oe−1.

The behaviour exhibited by Co1Fe1Mo3O8 may be of interest to application hence it exhibits linear response over a large range of ﬁelds. However, the caveat to this is the response is expected to be highly temperature dependent, since the magnetic susceptibility rapidly decreases below the Neel´ temperature. Unfortunately, this could not be conﬁrmed rigourously, as repeated measurements at lower temperatures proved too erratic to extract any useful information (not shown).

While conducting these types of experiments, one must always be cautious of extrinsic results. In this case, G. Catalan has shown that magnetocapacitance can result from a combination of magnetoresistivity and the interface between two materials with unique dielectric functions[64]. This effect is characterized by a dielectric loss (loss here refers to the conductivity π/2 radians out of phase) which varies with the opposite sign as the capacitance.

In this case, the dielectric either shows no dependence on ﬁeld (Co2Mo3O8 and

Co1.5Fe0.5Mo3O8) or follows the dependence of the capacitance (Co1Fe1Mo3O8), which rules out this effect (see ﬁgure 57).

105 C o M o O 2 3 8 0 . 4 C o F e M o O 1 . 5 0 . 5 3 8 C o F e M o O X 1 0 1 1 3 8

0 . 2 ) 0 = H L / ] 0 =

H 0 . 0 L - H L [ (

s s o

L - 0 . 2

- 0 . 4 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 H , K O e

Fig. 57: The dielectric loss in Co2Mo3O8, Co1.5Fe0.5Mo3O8, and Co1Fe1Mo3O8.

106 4 CONCLUSION

Single crystals of the hexagonal molybdates were synthesized via chemical vapour transport. These included: Co2Mo3O8, Co1.5Fe0.5Mo3O8, Co1Fe1Mo3O8, Co0.5Fe1.5Mo3O8,

Fe2Mo3O8, Co1Zn1Mo3O8, and Co1Mn1Mo3O8. All synthesis were successful and the compositions were conﬁrmed via X-ray diffraction. However, the yield of some samples- namely Co0.5Fe1.5Mo3O8, Fe2Mo3O8- and the properties of others- Co1Mn1Mo3O8- ensured that these crystals were of no use in this project.

The desired metamagnetic transition in cobalt molybdate, which was the basis of this project, could not be achieved via iron doping. Single crystals of Co1.5Fe0.5Mo3O8,

Co1Fe1Mo3O8, and Co0.5Fe1.5Mo3O8 fail to show evidence of an antiferromagnetic to ferrimagnetic transition, as was observed in Fe2Mo3O8 as a metamagnetic transition [Wang et al. Scientiﬁc Reports. 2015;Vol. 5:Article 12268] induced by applied magnetic ﬁeld and as a change in the ground-state via Zn doping [Kurumaji et al. Physical Review X. 2015;Vol.5:Article 031034]. This eliminate the possibility for tangible linear magnetoeletric coupling, due to the persistence of the antiferromagnetic state.

However, the M-H behaviour in low magnetic ﬁelds showed an interesting exchange-bias-like behaviour, which resulted in a horizontal- or equivalently vertically, since the magnetization is linear in H, shift to the magnetization. The origin of this phenomena is not fully understood. The effect occurs along both the polar c-axis and the basal plane, but is anisotropic, with larger shifts occurring in the basal plane. This behaviour was measured only in Co2Mo3O8, Co1.5Fe0.5Mo3O8, and especially in Co1Fe1Mo3O8. However, a signature bifurcation in the magnetic susceptibility is attributed to this effect and present in every iron containing compound. It is therefore likely to exist in Co0.5Fe1.5Mo3O8 and Fe2Mo3O8.

Measurements of the dielectric function showed an anomaly at the Neel´ temperature in

Co2Mo3O8, Co1.5Fe0.5Mo3O8, Co1Fe1Mo3O8, and Fe2Mo3O8. This conﬁrmed the presence of magnetoeletric coupling. This coupling was shown to have a quadratic dependence in H in

−14 −2 Co1.5Fe0.5Mo3O8 with a proportionality constant of (5.1 ±0.3)×10 Oe at 40K, and a

107 −9 −1 linear dependence in H in Co1Fe1Mo3O8 of (-6.99 ±0.07) ×10 Oe at 49K. The magnetocapacitance of Co0.5Fe1.5Mo3O8 and Fe2Mo3O8 could not be successfully measured, due to their small crystallite size.

More work is required to successfully synthesis crystals of Co0.5Fe1.5Mo3O8 and

Fe2Mo3O8 of a usable dimension for dielectric measurements, as well as to categorize the exchange-bias-like phenomena seen in the magnetic properties. A Mossbauer¨ study would be of great utility for this purpose.

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