The properties and growth of multiferroic BiMnO3 Model formation & evaluation and growth studies

Master thesis, 01-04-2003 – 18-08-2004

G.W.J. Hassink

TN s9802835 Master student of Applied Physics

University of Twente

The figure on the front page shows the orbital ordering in BiMnO3. The solid lines denote the dz² orbi-tals of the manganese ions. The manganese ions are shown in orange, the bismuth ions are not shown [Moreira dos Santos, 2002-2].

The properties and growth of multiferroic BiMnO3 Model formation & evaluation and growth studies

Master thesis: 01-04-2003 – 18-08-2004 Master student: G.W.J. Hassink TN s9802835 Master student of Applied Physics Master committee: Prof.dr.ing. D.H.A. Blank Dr.ing. A.J.H.M. Rijnders Dr.ir. J.W.M. Hilgenkamp Ir. A.E. Molag Institution: Inorganic Materials Science Faculty of Science and Technology MESA+ Institute of Nanotechnology University of Twente The Netherlands

Abstract Based on the individual mechanisms for ferroelectricity and ferromagnetism in BiMnO3 a model for its multiferroic behaviour is proposed. The orbital ordering is found to be important for establishing these mechanisms individually and for coupling them together resulting in multiferroic behaviour. The model is evaluated by both mean-field approximation theory and Monte-Carlo simulations. The results of these evaluations agree with the experimental data from literature. Thin films of BiMnO3 have been grown on SrTiO3, SrRuO3 buffered SrTiO3 and KTaO3 substrates. These thin films are not single-phase, although only a single crystallographic BiMnO3 phase is de-tected. The optimal substrate temperature range was found to be narrow, due to the interplay between crystal formation and bismuth evaporation. The initial growth mechanism seems to be dominated by the energy of the arriving plasma species which determines whether the thin film grows aligned with the substrate lattice or more randomly oriented. The best thin films of BiMnO3 are grown on SrRuO3 buffered SrTiO3 at low fluency. These thin films are insulating and have a Curie temperature of 93 K, close to the bulk value of 105 K.

Index

Abbreviations & symbols ................................................................................................................ 7 Introduction ..................................................................................................................................... 8

1 Bismuth manganite......................................................................................................................... 10 1.1 Structure................................................................................................................................. 10 1.2 Properties ............................................................................................................................... 11

1.2.1 Ferroelectricity................................................................................................................. 11 1.2.2 Ferromagnetism ............................................................................................................... 12 1.2.3 Multiferroism................................................................................................................... 13 1.2.4 Electronic properties ........................................................................................................ 14

1.3 Fabrication ............................................................................................................................. 15 1.3.1 Bulk ................................................................................................................................. 15 1.3.2 Thin film .......................................................................................................................... 15

1.4 Conclusions............................................................................................................................ 16 2 Theoretical investigation................................................................................................................ 18

2.1 Modified Ising-model ............................................................................................................ 18 2.1.1 Original Ising-model........................................................................................................ 18 2.1.2 Extended Ising-model ...................................................................................................... 19 2.1.3 Special aspects of the extended Ising-model ................................................................... 20

2.2 Mean-field approximation calculations ................................................................................. 21 2.2.1 Mean-field approximation theory .................................................................................... 21 2.2.2 Application of MFA theory to the extended Ising-model................................................ 23 2.2.3 Fitting of model parameters............................................................................................. 24 2.2.4 Results ............................................................................................................................. 25

2.3 Monte-Carlo simulations ....................................................................................................... 28 2.3.1 Metropolis Monte-Carlo algorithm.................................................................................. 28 2.3.2 Application of the Monte-Carlo Metropolis algorithm to the extended Ising-model ...... 29 2.3.3 Fitting of simulation parameters ...................................................................................... 30 2.3.4 Results ............................................................................................................................. 33

2.4 Conclusions............................................................................................................................ 37 3 Experimental investigation............................................................................................................. 38

3.1 Thin film growth .................................................................................................................... 38 3.2 Pulsed laser deposition........................................................................................................... 38

3.2.1 PLD process..................................................................................................................... 38 3.2.2 Advantages of PLD.......................................................................................................... 39

3.3 Growth experiments............................................................................................................... 40 3.3.1 Initial choice of parameters.............................................................................................. 40 3.3.2 Crystallographic structure................................................................................................ 41 3.3.3 Chemical composition ..................................................................................................... 45 3.3.4 Manganese ion valency.................................................................................................... 45 3.3.5 Surface morphology......................................................................................................... 46

3.4 BiMnO3 thin films.................................................................................................................. 50 3.4.1 Best settings ..................................................................................................................... 50 3.4.2 Structural properties......................................................................................................... 50 3.4.3 Electric and magnetic properties...................................................................................... 52

3.5 Conclusions............................................................................................................................ 52 Conclusions ................................................................................................................................... 54 Recommendations ......................................................................................................................... 56 Literature ....................................................................................................................................... 58 Dankwoord .................................................................................................................................... 60

A Appendix: derivations.................................................................................................................... 62 B Appendix: calculations................................................................................................................... 64 C Appendix: simulations ................................................................................................................... 72 D Appendix: experiments.................................................................................................................. 92

Abbreviations & symbols Abbreviations AFM atomic force microscopy EDX energy dispersive X-ray MFA mean-field approximation PLD pulsed laser deposition RMS root-mean-square SEM scanning electron microscope XRD X-ray diffraction Symbols B magnetic induction [T] χ magnetic susceptibility [-] Cme magneto-electric coupling constant [J] D electric displacement [C/m2] ∆K relative change in the dielectric constant [-] E electric field [V/m] ε dielectric permittivity [F/m] Hm/e Hamiltonian for either the magnetic or dielectric state [J] H magnetic field [A/m] Jij coupling constant in the general spin-Hamiltonian [J] Jm/e coupling constant in the Ising-Hamiltonian for respectively the magnetic or dielectric state [J] kb Boltzmann constant [1.381·10-23 J/K] K dielectric constant [-] m relative magnetization [-] µ magnetic permeability [H/m] P(a) probability of the state a [-] Pi dipole moment at location o [-] p relative polarization [-] Si magnetic moment at location i [-] T absolute temperature [K] Tc,e ferromagnetic critical temperature [K] Tc,m ferroelectric critical temperature [K] θc Curie temperature [K]

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Introduction ‘If we knew what we were doing, it wouldn’t be called research.’ The above quote by Albert Einstein has always struck me as funny. And appropriate. In the later years of my study, where I started to do my own research, I saw it very often at and in my work. Sometimes you just try and try, without some real logic behind it, and then see what you get. But with those re-sults you take a little step forward and you learn a bit more; both about your subject and about how to do research. In fact, I learned that this is how you do research. After the first three years at the University of Twente I started my specialization within the Low Tem-perature group of Professor Rogalla. As I got to know the group better, one contact, that with then Dr.Ing. Dave Blank, became increasingly important. Through him I managed to arrange my external training and his enthusiasm for his field of research was and is one of the reasons I choose this spe-cialization. The other reason is that material science fascinates me in its own right. Through the research at the MESA+ institute that Dave Blank coordinated I slowly drifted away from the superconductors that are the main research topic of the Low Temperature group to dielectric mate-rials. These materials were the subject of both a short project here at university and my external train-ing abroad. When I was looking for a Master thesis assignment again Dave Blank, a full Professor of Inorganic Materials Science by then, had a very interesting proposal. In relation with a just-awarded VICI grant he was starting research into artificial materials that combine materials with different properties to create new functionalities. An example would be the combination of ferroelectricity and ferromagnet-ism, often called multiferroism. An example of the use of multiferroic materials will be in multiple-state memory elements, where in-formation is stored in both the magnetization and the polarization. If the coupling between the ferro-electricity and ferromagnetism is strong, such materials can be used to create devices where the elec-tric behaviour can be tuned by applying a magnetic field or vice-versa. In general it will offer more degrees of freedom when designing devices. Before making any artificial materials, the study of materials where this combination occurs naturally helps in understanding the behaviour of or even the design of the artificial materials. My Master as-signment thus became to study the properties of BiMnO3, a material that exhibits both ferroelectricity and ferromagnetism, and to try and make thin layers of BiMnO3. Its structure can be thought of as a distorted perovskite and this relative simple structure makes identifying the underlying mechanisms easier. Therefore BiMnO3 can be used of as a model system for the research into multiferroism. In Chapter 1 of this report an overview of the properties of BiMnO3 will be given. This provides the basis for the further chapters in which these properties are researched and discussed more extensively. Chapter 2 describes the theoretical investigation; the formulation of a quantitative model describing the behaviour of BiMnO3 and the subsequent calculations and simulations. Chapter 3 then contains the experimental investigation; the production and analysis of thin films of BiMnO3. Finally in the conclu-sions and recommendations the main results and suggested further research are presented. Gerwin Hassink 05-08-2004

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1 Bismuth manganite The study of bismuth manganite, BiMnO3, could yield useful information for the study and creation of artificial multiferroic materials. This chapter presents the known literature and theory about BiMnO3. First the crystallographic structure is discussed. Then the properties of BiMnO3 are presented in the second and main part of this chapter. Finally an overview of the production of BiMnO3 both in bulk form and as thin films is given and the chapter closes with some conclusions. 1.1 Structure At room temperature BiMnO3 has a monoclinic crystal structure, space group C2. The unit cell is shown in Figure 1.1(a). Note that this is the crystallographic unit cell, containing two formula units.

(a) (b)

Bi3+

Mn3+

O2-

Bi3+

Mn3+

O2-

Figure 1.1: Unit cell of BiMnO3, (a) monoclinic [Atou, 1999] and (b) pseudo-perovskite. Its cell parameters are a = 9.532 Å, b = 5.606 Å, c = 9.854 Å and β = 110.7º [Atou, 1999]. To have more insight into the physics of this material, it is often helpful to view it as having a perovskite struc-ture. Indeed, the triclinic distortion from the ideal perovskite structure is rather small. The cell parame-ters of this pseudo-perovskite phase are a = 3.950 Å, b = 3.995 Å, c = 3.919 Å, α = 90.7º, β = 90.9º and γ = 91.0º [Faqir, 1999]. The pseudo-perovskite unit cell, containing one formula unit, is shown in Figure 1.1(b). The coinciding crystal directions between the two descriptions are given in Table 1.1.

Triclinic description (100) (010) (001)

Monoclinic description (121) (101) (111) Table 1.1: Coinciding directions between triclinic and monoclinic descriptions of BiMnO3.

Upon cooling from high temperatures BiMnO3 undergoes several phase transitions. The different phases and transition temperatures are shown in Table 1.2.

Temperature Phase information > 600 ºC In air BiMnO3 decomposes into Bi2O3 and Bi2O3·Mn2O3+δ. In other atmospheres

this could be different. 600 ºC BiMnO3 has an orthorhombic Pbnm structure above 500 ºC. 500 ºC Around 500 ºC there is a phase transition from the high-temperature orthorhom-

bic Pbnm phase to the low-temperature monoclinic C2 phase. The C2 phase allows ferroelectricity to occur (it is non-centrosymmetric), but it is unsure whether the ferroelectric ordering also occurs at this temperature.

180 ºC There is another phase transition at 180 ºC and there are indications that this (second-order) transition is the paraelectric-to-ferroelectric transition. At this transition the crystallographic phase does not change, but the lattice parameters change abruptly, possibly indicating ordering of the bismuth ions.

-168 ºC (105 K) Finally, at the ferromagnetic critical temperature of -168 ºC (105 K) magnetic ordering of the magnetic moments of the manganese ions occurs [Faqir, 1999; Kimura, 2003; Moreira dos Santos, 2002-1].

Table 1.2: BiMnO3 phases and transition temperatures.

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Mn3+ is a Jahn-Teller-active ion, so in pure BiMnO3 all the oxygen octahedra around the manganese ions are distorted. This results in a lowering of the energy of the dz² orbital relative to that of the dx²-y² orbital. Orbital ordering of these Jahn-Teller distorted oxygen octahedra has also been observed in BiMnO3 [Moreira dos Santos, 2002-2]. Since the same crystal structure is observed above and below 180 ºC, this orbital ordering persists at least up to 500 ºC. As discussed in the next paragraph, the or-bital ordering is of great influence on both the ferroelectric and ferromagnetic behaviour. 1.2 Properties To understand the multiferroic behaviour of BiMnO3 we have to understand the mechanisms behind the individual ferroelectric and ferromagnetic properties. In this paragraph these properties and their interaction are discussed. 1.2.1 Ferroelectricity A ferroelectric is a material that can exhibit a spontaneous polarization. A stereotypical example is BaTiO3 where the off-center titanium ion causes the formation of an electric dipole and a net overall polarization. This off-center position is energetically stabilized by hybridisation of the empty 3d-levels of titanium with the occupied 2p-levels of the surrounding oxygen ions. One electron from the 2p-levels is shared with titanium. The Ti-O bond can then be interpreted similarly to the H2 molecule; two charged cores with one electron each that are near each other. Just as in H2 there is a lowering of en-ergy when the two ions move closer to each other. In the case of BiMnO3, the mechanism is a bit different and more similar to the lead-containing Pb(Zr,Ti)O3. While zirconium and titanium also contribute to the ferroelectric effect in this material, the main component is due to lead. Just as with titanium the lead ion moves off-center due to interac-tions with the surrounding oxygen ions that lower the energy. In addition, lead has a ‘lone pair’ of electrons in its 6s level. This lone pair of electrons is repulsed by the bonding pairs in the lead-oxygen bonds and will thus be oriented away from them, similar to the way the 2s lone pair of nitrogen points away from the bonds with hydrogen in ammonia (see Figure 1.2) [O’Leary, 2000]. But oriented like that the electron pair will also push against the bonding pairs of other ions, effectively pushing the lead ion even more off-center.

Figure 1.2: Lone pair 2s level in ammonia as an example for lone pair formation in BiMnO3.

Since lead and bismuth ions, Pb2+ and Bi3+, have the same electron configuration the same mechanism for stabilization of the off-center position can occur. In BiMnO3 bismuth is solely responsible for the ferroelectric behaviour. The off-center displacement of the bismuth ions is due to their bond with their surrounding oxygen ions. If the latter are not ordered, the former most likely will not be ordered either. So for ferroelectric ordering to occur, the oxygen ions have to be ordered as well. In BiMnO3 there is a mechanism that orders the oxygen ions specifically, namely the orbital ordering of the Jahn-Teller distorted oxygen octahedra. Above 500 ºC the crystal structure does not allow dipoles and thus there is no ferroelectric-ity. Between 500 and 180 ºC the crystal structure does allow dipoles, but they are not ordered (com-

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pare with a paramagnetic state). Below 180 ºC the dipoles are ordered and ferroelectricity occurs. In Figure 1.3 some P-E hysteresis loops for BiMnO3 are shown, both above and below the ferromagnetic critical temperature.

Figure 1.3: P-E hysteresis loops for BiMnO3 [Moreira dos Santos, 2002-1].

1.2.2 Ferromagnetism In BiMnO3 the magnetism is due to the unpaired spins of the manganese ions; it is an example of a material with localized spins. According to Hund’s rules all the spins of the Mn3+ ion are expected to point in the same direction. The expected magnetic moment for three up-spins is 3µB, which is also found in experiments [Faqir, 1999]. The most interesting aspect of the ferromagnetism in BiMnO3 is the mechanism for the ordering of the magnetic moments. Here we can draw on work done on other manganites, for example (La,Sr)MnO3. A very important contribution was made by Goodenough. He discussed the possible orbital orienta-tions within a manganite and the corresponding coupling between the magnetic moments of the man-ganese ions [Goodenough, 1955]. Focussing on BiMnO3 there are three possible orbital orientations and magnetic interactions for the Mn3+-O-Mn3+ complexes [Goodenough, 1961]. These are given in Table 1.3.

Orbital orientation of Mn3+-O-Mn3+ Magnetic interaction [Goodenough, 1961]

empty dx²-y² to empty dx²-y² weak antiferromagnetic half-filled dz² to empty dx²-y² ferromagnetic half-filled dz² to half-filled dz² strong antiferromagnetic

Table 1.3: Orbital orientations and magnetic interactions for Mn3+-O-Mn3+. From neutron-diffraction experiments it has been found that in BiMnO3 only dz² - dx²-y² and dx²-y² - dx²-y² orientations occur. Thus the magnetic interactions in BiMnO3 are only ferromagnetic or weakly anti-ferromagnetic, giving an overall ferromagnetic behaviour. As an example the magnetic hysteresis loops below the magnetic critical temperature for BiMnO3 are shown in Figure 1.4 on the next page.

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Figure 1.4: M-H hysteresis loops for BiMnO3 [Moreira dos Santos, 2002-1].

The fact that only these two orbital orientations occur in BiMnO3 is due to orbital ordering. For exam-ple, in LaMnO3 the orbitals order differently, yielding ferromagnetic planes that are antiferromagneti-cally coupled to each other, so that overall LaMnO3 is antiferromagnetic [Moreira dos Santos, 2002-2]. Thus the orbital ordering of BiMnO3 is essential for both the ferroelectric behaviour (ordering of the bismuth ions) and for the ferromagnetic behaviour (ordering of the manganese magnetic moments). 1.2.3 Multiferroism Of great interest is the mechanism for the interaction between ferroelectricity and ferromagnetism in BiMnO3. A key element in the mechanisms driving both properties is the oxygen ion. It cooperates in both the Bi-O bond and the Mn3+-O-Mn3+ interaction. In the case of ferromagnetic interaction the orbital orientation is half-filled dz² pointing to empty dx²-y². Here the oxygen ion is more or less covalently bonded to the manganese ion with the dx²-y² orbital. Be-low the magnetic critical temperature the bonding electron with its spin parallel to the magnetic mo-ment of the manganese ion is slightly pulled away from the oxygen ion due to Hund’s rules [Goode-nough, 1955]. Oxygen is strongly electronegative, so it will try to compensate for this ‘loss’. The manganese ion on the other side of the oxygen ion is bonded ionically and cannot give up its electrons. But the bismuth ion is bonded covalently to the oxygen. The oxygen ion can pull these electrons closer to itself. The bismuth ion moves closer to the oxygen ion accordingly, enhancing the ferroelectric or-dering. This enhancement has been seen in literature. See Figure 1.5 for a dielectric constant versus tempera-ture profile. Below the magnetic critical temperature the dielectric constant decreases faster. Since the dielectric constant is related to the disorder of the dipoles, below the Tc,m the ferroelectric order should increase.

Figure 1.5: Dielectric constant versus temperature for BiMnO3 [Kimura, 2003].

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The reverse effect, influencing the magnetic properties by applying an electric field, has not been ob-served, but could easily fit within this model. With the applied electric field the bismuth ions are either pushed closer to or pulled away from their bonded oxygen ions. This will influence the effective elec-tronegativity of the oxygen ion. In turn this will influence the electrons the oxygen ion shares with the manganese ions, increasing respectively decreasing the strength of the magnetic coupling. Another observed effect is the linear relation between the change in the dielectric constant and the square of the magnetization (see Figure 1.6) [Kimura, 2003]. The model so far is only qualitative and cannot explain the form of this relation. In the next chapter the model will be developed further into a quantitative theory. There this linear relationship will be used for fitting the model to the experimental data.

Figure 1.6: Relationship between the change of the dielectric constant and magnetization squared [Kimura, 2003].

1.2.4 Electronic properties BiMnO3 is a dielectric material and an insulator. The resistivity versus temperature plot in Figure 1.7 the typical increase in resistivity as the temperature decreases. Also, doing a full cycle of both cooling and heating hysteresis at 180 ºC and 500 ºC can be observed. These anomalies coincide with the prob-able paraelectric-to-ferroelectric transition and the Pbnm-to-C2 transition.

Figure 1.7: Resistivity versus temperature for BiMnO3 The optical properties of BiMnO3 have been invehave been observed, which can also be influenceproperties could be affected by external magnetic

[Kimura, 2003].

stigated [Sharan, 2003]. Large optical nonlinearities d by external electric fields. Possibly these optical fields as well.

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1.3 Fabrication The properties described above make BiMnO3 an interesting candidate for experiments and applica-tions. However BiMnO3 is a difficult material to fabricate, both in bulk form and as thin film, limiting the possibilities. 1.3.1 Bulk The bulk preparation of BiMnO3 most often follows the standard solid-state reaction process. A mix-ture of individual oxide powders, Bi2O3, Mn2O3 and sometimes MnO2, are mixed, calcined and sin-tered. An overview of the parameters used during the sintering is given in Table 1.4.

Temperature (ºC) Pressure (GPa) Time (hr) Reference 900 5 Troyanchuk, 1996 600 4 1 Chiba, 1997 600-610 6 3 Faqir, 1999 700 6 3 Atou, 1999 800 2.5 Woo, 2001 750 6 3 Moreira dos Santos, 2002-1 700 6 Moreira dos Santos, 2002-2 700 3 0.5 Kimura, 2003

Table 1.4: Sintering parameters for bulk BiMnO3. These process parameters, especially the high pressure, are not suitable for either large volume production or device fabrication. 1.3.2 Thin film Thin film deposition processes are another way to fabricate BiMnO3. The working pressures are often much lower and additionally the fabrication as a thin film simplifies device fabrication. However, there has not been much work done on thin films of BiMnO3. An overview of what has been done, all with pulsed laser deposition (PLD), is given in Table 1.5 on the next page.

Reference Ohshima, 2000

Te Riele, 2003

Moreira dos Santos, 2003

Son, 2004

Laser ArF (193 nm) KrF (248 nm) KrF (248 nm) Nd:YAG (355 nm) Bi:Mn target 1:1 1.2:1 1.2:1 Fluency (J/cm2) 3 1.5-2 2 Frequency (Hz) 1 5 4 Gas composition O2 O2 or Ar/O2 O2 + 10% O3 O2 Pressure (mbar) 5.0·10-4 1.0·10-2 2.7·10-2 2.7·10-1 Target-substrate (mm) 65 Substrate SrTiO3 SrTiO3 SrTiO3 LaAlO3 Pt/TiO2/SiO2/SiMismatch (%) -0.8 -0.8 -0.8 -4.3 -0.9 Temperature (ºC) 450 450-500 695 800 800 Crystallinity good none good good good Growth rate (Å/pulse) 0.5 0.1

Table 1.5: PLD parameters for BiMnO3 thin film growth. As can be seen from literature often a target with a surplus of bismuth is chosen to compensate for the fast evaporation of bismuth at the high temperatures during growth. Similar problems occur with lead in for example Pb(Zr,Ti)O3.

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SrTiO3 has a good lattice match with BiMnO3, about -0.8%, so it is often chosen as the substrate. Also high-quality SrTiO3 substrates are readily available and a simple preparation results in well-defined surfaces. However, in all these cases the surface morphology of the BiMnO3 thin film was not good enough for device fabrication. 1.4 Conclusions BiMnO3, considered a model system for the research of multiferroism, already shows a lot of complex behaviour. Its ferroelectricity is due to the displacement of the bismuth ions, while its ferromagnetism is due to ordering of the magnetic moments of the manganese ions. The orbital ordering is essential in establishing these behaviours individually and, in the proposed model, it is responsible for tying them together in the magnetoelectric behaviour of BiMnO3. This makes the study of orbitals essential for understanding the behaviour of BiMnO3, as it forms the link between several separate mechanisms coming together to form this unique material. Detailed study is complicated by the difficulty of fabricating high-quality (single-phase) samples of BiMnO3. The bulk preparation method produces only small amounts of material and is incompatible with device fabrication. BiMnO3 has been fabricated as thin films using PLD, but there is little work done on this material. The model developed for the multiferroic behaviour of BiMnO3 is probably only applicable to a small class of materials. An A-cation with a ns lone pair of electrons is needed, as is a Jahn-Teller distorted magnetic B-cation. Only few other materials with these requirements are known. For some of these materials, such as SeCuO3 and TeCuO3, the linear relation between the change in the dielectric con-stant and magnetization squared has been observed [Lawes, 2003]. Other materials like PbCrO3 show magnetic ordering and strong indications for ferroelectricty, but the magnetoelectric effect has not been investigated [Jaya, 1992].

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2 Theoretical investigation In the previous chapter some qualitative models for the ferroelectric, ferromagnetic and multiferroic behaviour of BiMnO3 were presented. It would be of interest to study these models and to see whether they indeed reproduce the actual measurements. To do this, the qualitative models have to be imple-mented in a more quantitative (if still phenomenological) model of which we can calculate or simulate the behaviour. In this chapter such a more quantitative model will be developed and evaluated. The model itself will be constructed in the first paragraph. The second paragraph will discuss the calculations that have been done with this model and their results. Another approach of the same model by simulation is the sub-ject of the third paragraph. Finally some conclusions about the theoretical work conclude this chapter. 2.1 Modified Ising-model The Ising-model is a simple model used to describe the magnetic behaviour of isolated magnetic mo-ments. It uses one of the fundamental theorems of physics, energy minimalization, to describe (anti)ferromagnetism as a result from the interaction between large numbers of individual magnetic moments. 2.1.1 Original Ising-model The start of the Ising-model is the spin-Hamiltonian. It describes the energy of a system of magnetic moments in a magnetic field. Derivation of this equation is given in Appendix A.

,ij i j i

i j i

J S S B S= − ⋅ − ⋅∑ ∑mH (Equation 2.1)

Here Hm is the Hamiltonian, Jij is the coupling strength between the magnetic moments Si and Sj. B is the magnetic induction. The summation is over all magnetic moments. If the coupling constant Jij is positive, the energy is lower if two magnetic spins are aligned parallel. If Jij is negative, the energy is lower if they are aligned anti-parallel. So Jij greater than zero gives ferromagnetic behaviour and Jij less than zero gives antiferromagnetic behaviour. The Ising-model simplifies this equation in three ways:

1) The magnetic moments Si and Sj only point in two directions. Localized magnetic moments are often created by electron spins which do only have two orientations, called ‘up’ and ‘down’;

2) The coupling constant Jij is constant for all magnetic moments Si and Sj. In a homogenous ma-terial this, by symmetry, must be the case;

3) Only the nearest-neighbour interaction is taken into account. In manganites the direct interac-tion seems to be nearest-neighbour in nature.

The spin-Hamiltonian above is then simplified into the Ising-Hamiltonian: (Equation 2.2)

,m i j

i j i

J S S B< >

= − −∑ ∑Hm iS

Where Jm is the constant coupling constant, Si and Sj are the spins either +1 (‘up’) or -1 (‘down’). The summation of the spins is over nearest-neighbours only, while the summation over the field is still over all spins.

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With the energy of the magnetic system now given by Equation 2.2 and the Boltzmann probability function (Equation 2.3) in theory the most probable configuration of magnetic spins can be calculated.

( )

( )

b

b

a k T

s k T

s

ee

−

−=∑

H

HPm

m(a) (Equation 2.3)

Here P(a) is the probability of state a, kb is the Boltzmann constant, T is the absolute temperature and the summation over s is over all possible states. It is this last summation that makes Equation 2.3 so difficult to evaluate since the total number of states is enormously large. The one-dimensional case of the Ising-model, where all spins are along an infinite line, has been solved analytically in 1924 by Ising himself. However, the 1D case does not show any ordering of spins and thus there is no phase transition from a disordered to ordered state. The two-dimensional case where the spins reside on an infinite square lattice and there is no external magnetic induction has been solved in 1944 by Onsager (see comment in Appendix A). This case does show a phase transition from a disordered to an ordered state. The Onsager solution gives the magneti-zation as a function of temperature (Equation 2.4) and a relation between the coupling constant Jm and the ferromagnetic critical temperature Tc,m (Equation 2.5) [Gonsalves, 2002; Gottschalk, 1998; Tuck-ermann, 1999].

1/84

,

,

21 sinh

0

( )m

c mb

c m

J T Tk T

T T

m T

− − < = >

(Equation 2.4)

( )

, 2 2.269..ln 1 2

b c m

m

k TJ

= ≈+

(Equation 2.5)

2.1.2 Extended Ising-model Using the Ising-model and the qualitative models for the ferromagnetic, ferroelectric and multiferroic properties of BiMnO3 an extended Ising-model is constructed. Since there are actual measurements of the influence of the magnetic field on the electric properties [Kimura, 2003] we will focus on that be-haviour. To describe the magnetic behaviour of BiMnO3 the Ising-model can be used as is. The manganese ions form localized magnetic moments with spin ‘up’ or ‘down’ and they interact only directly with their nearest-neighbours. In fact, each manganese ion has four ferromagnetic couplings with its neighbours, the last two being weakly anti-ferromagnetic or even paramagnetic. Thus the choice of a square lattice of spins with four nearest neighbours is especially appropriate. (Equation 2.6)

,m i j

i j i

J S S Hµ< >

= − −∑ ∑Hm iS

The theory to describe the ferroelectric behaviour has not been discussed. However, by analogy again the Ising-model can be used to describe this behaviour as well. A dipole-Hamiltonian similar to Equa-tion 2.1 can be written to describe a system of interacting electric dipoles in an electric field (see Ap-pendix A for more info). From there the same simplifications as for the magnetic Ising-model can be made to arrive at a Hamiltonian similar to Equation 2.2 describing the ferroelectric behaviour. Since

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the ferroelectric ordering is due to the orbital ordering, which is independent of the interaction be-tween actual dipoles itself, a square lattice is appropriate enough here as well.

,

e i ji j i

DJ PPε< >

= − −∑ ∑He iP (Equation 2.7)

As described in Paragraph 1.2.3 there is stronger ferroelectric ordering when the spins in BiMnO3 be-come aligned. The coupling constant is thus dependent on the spin orientation of the neighbouring spins. The easiest way to put that into a formula is to state that there is an additional constant coupling factor when the spins on two neighbouring sites are parallel and there is no additional coupling when they are anti-parallel. Equation 2.7 can then be extended to describe multiferroic behaviour as well.

, ,

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i je i j me i j

i j i j i

S SJ PP C PP E

< > < >iP

+= − − −∑ ∑He ∑ (Equation 2.8)

A simplification can be made by replacing the spin values with their average values over the entire system. This makes it possible to use mean-field approximations and it simplifies simulations. How-ever, this simplification is only possible for ferromagnetic materials. In that case the magnetization is a good parameter for the overall order of spins. In the antiferromagnetic case this is not so; the magneti-zation is zero for the random lattice and for the ordered lattice. So Equation 2.9, the simplified version of Equation 2.8, is only applicable to ferromagnetic systems.

2

, ,

2

,

12

12

e i j me i ji j i j i

e me i j ii j i

mJ PP C PP E

mJ C PP E P

< > < >

< >

+= − − −

+= − + −

∑ ∑

∑ ∑

He iP∑ (Equation 2.9)

In the extended model the ferromagnetic behaviour of BiMnO3 is modelled by a square lattice of spins ‘up’ or ‘down’ whose energy is described by Equation 2.2. Similarly the ferroelectric behaviour is de-scribed by a square lattice of dipoles either ‘up’ or ‘down’ with an energy given by Equation 2.8 or 2.9 depending on the simplification used. Since the coupling in both Equation 2.8 and 2.9 is dependent on the magnetic state of the system, multiferroic behaviour is also described. 2.1.3 Special aspects of the extended Ising-model The extended Ising-model as formulated above has another interesting feature. In theory it is possible for the electric coupling constant to change from ferroelectric (positive value) to antiferroelectric (negative value) or vice-versa depending on the magnetization. Such a transition requires that either the material is ferroelectric but the additional magnetization-induced ordering is antiferroelectric or the reverse, the material is antiferroelectric and the additional magnetization-induced ordering is ferroelec-tric. More precisely, for a switch of the overall electric coupling constant it is necessary that:

20

mee m

e me

C J C

J C

< <

⋅ <

e (Equation 2.10)

The shape of the dielectric constant versus temperature profile greatly depends on the sign of the mag-netoelectric coupling constant Cme. Using the Onsager solution and the values Tc,m = 100 K and Tc,e = 200 K this dependence is shown in Figure 2.1 on the next page.

20

70 80 90 100 110 120 1301

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2detail dielectric constant for different CME values

temperature (K)

diel

ectri

c co

nsta

nt (-

)CME < 0CME = 0CME > 0

Figure 2.1: Cme sign dependence of dielectric constant temperature profile.

As the extended Ising-model is a further development of the physical model from Paragraph 1.2.3 it should be equally applicable to the sister-compounds of BiMnO3: SeCuO3, TeCuO3 and PbCrO3. Note however that both TeCuO3 and PbCrO3 are antiferromagnetic [Lawes, 2003; Jaya, 1992], so Equation 2.9 is not valid for these materials. Equation 2.8 should be applicable to all these materials. The model of two coupled Ising-models describing the magnetic and electric properties of materials is more generally applicable. In all cases the form of the coupling has to come from a physical model. For example, in YMnO3 the physical model is that the magnetic transition causes magnetostriction, which changes the ferroelectric coupling constant [Huang, 1997]. The coupled Ising-model can then be evaluated to check whether the chosen physical model can represent the observed data. Again anti-ferromagnetic systems (such as YMnO3) require great care, as the magnetization cannot be used as a parameter for the magnetic order within the system. 2.2 Mean-field approximation calculations The extended Ising-model created in the previous paragraph can be evaluated either by calculations or by simulations. In this paragraph the calculations using mean-field approximation theory will be dis-cussed. 2.2.1 Mean-field approximation theory The basis of mean-field approximation (MFA) theory is the assumption that the influence on spin i of all other spins can be approximated by a mean field. That means that besides the spin at location i no other spins need to be evaluated. This is a severe approximation but it has two big advantages:

1) It works for non-zero magnetic induction. This as opposed to the Onsager solution which only works for zero magnetic induction;

2) Its results are relatively simple to work with and the main constants of the Ising-model are easily recognized.

The derivation of MFA theory has been discussed extensively in literature and text books. For an out-line specifically applied to the Ising-model see Tuckerman [1999]. For a more general equation appli-cable to systems with a magnetic moment other than one-half see Kittel [1986] or Núñez-Reguiro [1996]:

21

3( , )

1JJ m bm B T B

J t+ = +

(Equation 2.11)

Where BJ is the Brillouin function, J for this formula only is the total magnetic moment, m is the rela-tive magnetization M/MS where MS is the saturation magnetization, b is the relative magnetic induc-tion B/λMS where λ is the MFA exchange coefficient that indicates the strength of the coupling be-tween the magnetization and the spins and t the relative temperature T/Tc,m. For the Ising-model to total magnetic moment is J = ½. Substituting this into Equation 2.11 and simplifying yields (see Ap-pendix B):

( , ) tanh m

b

J m Hm H Tk T

µ +=

(Equation 2.12)

In addition there is a relation between the magnetic coupling constant Jm and the magnetic critical temperature Tc,m given by:

, 1b c m

m

k TJ

= (Equation 2.13)

Equations 2.12 and 2.13 describe the magnetization. An equation for the magnetic susceptibility can be derived from Equation 2.12 (see Appendix B).

( )

2

2

11 m

mT m

χ −=

− − J (Equation 2.14)

In Figure 2.2 the Onsager solution and the MFA theory are compared. Note that the Onsager solution is only derived for zero magnetic induction.

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

mag

netiz

atio

n (-)

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

temperature (K)

norm

aliz

ed s

usce

ptib

ility

(-)

OnsagerMFA - H=0MFA - H=5

Figure 2.2: Comparison of Onsager solution and MFA theory.

Both the Onsager solution and the MFA theory predict the onset of magnetic ordering, as evidenced by the non-zero magnetization below the critical temperature of 100 K. Applying a magnetic induction induces ordering above the critical temperature. The applied magnetic induction also broadens the transition, which can be seen by the broadening of the magnetic susceptibility.

22

2.2.2 Application of MFA theory to the extended Ising-model Equations 2.12 to 2.14 are readily applicable to the magnetic part of the extended Ising-model of Para-graph 2.1.2. Luckily, those Equations are equally applicable to the ferroelectric Ising-model if you use Equation 2.9. Equation 2.8 is not amendable to MFA theory because it depends on the exact orienta-tion of the neighbouring spins. However, in Equation 2.9 the influence of the spins is felt through the magnetization only, which can be obtained separately by first solving the ferromagnetic Ising-model. So for a given temperature and magnetic field the electric coupling constant Je is constant and the ferroelectric equivalents of Equations 2.12 to 2.14, Equations 2.15 to 2.17, can be solved (see Appen-dix B for comment on the term K-1 in Equation 2.17).

( )21

2( , ) tanhm D

e me

b

J C pp D T

k Tε

+ + + =

(Equation 2.15)

( )2

,

12

1b c e

me me

k TJ C +

=+

(Equation 2.16)

( )( )2

2

2 12

111r m

e me

pKT p J C

ε+

−− = =

− − + (Equation 2.17)

The procedure for calculating the response of the extended Ising-model would be (for more info on solving Equations 2.12 and 2.15 see Appendix B):

calculate magnetic response (Equation 2.12 & 2.14)

calculate electric response (Equation 2.15 & 2.17)

calculate electric model parameters(Equation 2.16)

end

calculate magnetic model parameters(Equation 2.13)

start

Figure 2.3: Flowchart MFA calculation of the extended Ising-model.

This procedure has been implemented in a series of Matlab 6.5 M-files, the results of which will be discussed below. The source code of these files is given in Appendix B. A short discussion of the error estimation is also given in appendix B.

23

2.2.3 Fitting of model parameters Looking at the extended Ising-model (Equations 2.12 to 2.17) there are five fitting parameters. These parameters describe the behaviour of BiMnO3 and should be derived from its properties, such as the measured critical temperatures. Table 2.1 gives an overview of these parameters.

Fitting parameter Description Jm coupling constant for the magnetic state [J] µ magnetic permeability [H/m] Je coupling constant for the dielectric state [J] Cme magneto-electric coupling constant [J] ε dielectric permittivity [F/m]

Table 2.1: Overview of fitting parameters for the extended Ising-model. A numerical constant in all the equations is the term 1/kb. Since it is present as a factor with all the fitting parameters and because of its size it is easier to incorporate it into the numerical value of the fitting parameter. So we will talk about the numerical value of, for example, Je/kb instead of Je. Magnetic coupling constant The magnetic coupling constant Jm can easily be derived from Equation 2.13.

,m

c mb

J Tk

= (Equation 2.18)

The magnetic critical temperature Tc,m of BiMnO3 is 105 K, so the value of the magnetic coupling con-stant for the MFA theory would be 105 K as well. Magnetic permeability & dielectric permittivity For both the magnetic permeability µ and the dielectric permittivity ε it was not possible to derive nu-merical values. No suitable experiments on BiMnO3 were done to do so. Their role in the model is to normalize the strength of respectively the magnetic field and the electric displacement. So they can be chosen to be equal to one without destroying the model. However, the absolute strength of the mag-netic field and the electric displacement will not correspond to those used in actual measurements. Electric and magneto-electric coupling constant The electric coupling constant Je can be derive from Equation 2.16.

2

,1

2e me

c eb b

J C mTk k

+= − (Equation 2.19)

For BiMnO3 the electrical critical temperature Tc,e is 450 K, which is much higher than the Tc,m of 105 K. A good approximation is to consider the magnetization at Tc,e to be zero. Je then becomes:

,12

e mc e

b b

J CTk k

= − e (Equation 2.20)

Je is thus a function of magneto-electric coupling constant Cme. This last parameter is the true indicator for multiferroic behaviour. To fit it, data on this multiferroic behaviour is needed. Kimura [2003] shows that the relation between the relative change in the dielectric constant and the square of the magnetization is a straight line (see Figure 1.6). By calculating, plotting and comparing the slope of

24

this linear relation the value of Cme can be determined. This yields Cme = 170 K, which in turn gives Jc,e = 365 K. Table 2.2 shows the numerical values of the fitting parameters as discussed above. The determination of the Cme will be discussed in detail in the next paragraph.

Fitting parameter Value Description Jm/kb 105 coupling constant for the magnetic state [K] µ/kb 1 magnetic permeability [Km/A] Je/kb 365 coupling constant for the dielectric state [K] Cme/kb 170 magneto-electric coupling constant [K] kbε 1 dielectric permittivity [C/Km2]

Table 2.2: Overview of fitting parameters and their values for the extended Ising-model. 2.2.4 Results There are two important graphs that show the behaviour of the model: the response of the model ver-sus temperature and the response versus applied magnetic field. Versus temperature Figure 2.4 shows the magnetic response of the extended Ising-model versus temperature for different applied magnetic fields. The source code of the M-file for this calculation is given in Appendix B.

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1MFA temperature profile - m & chi vs. T, variable H

mag

netiz

atio

n (-)

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

temperature (K)

susc

eptib

ility

(-)

H=~0H= 2H= 4

Figure 2.4: Magnetic response versus temperature of the extended Ising-model, calculations.

It shows most of the aspects of an experimentally observed magnetic transition. A sharp transition in the magnetization at low fields and a more gradual transition at higher fields. A coincident broadening and lowering of the magnetic susceptibility with higher fields is also seen. The field at zero was cho-sen not to be exactly zero but slightly positive. This simulates the presence of the earth magnetic field or a remnant magnetization induced during the fabrication process. Figure 2.5 on the next page shows the electric response of the extended Ising-model versus tempera-ture for different applied magnetic fields.

25

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1MFA temperature profile - p & K-1 vs. T, D=~0, variable H

pola

rizat

ion

(-)

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

temperature (K)

norm

aliz

ed d

iele

ctric

con

stan

t (-)

H=~0H= 2H= 4

Figure 2.5: Electric response versus temperature of the extended Ising-model, calculations.

The effect of the different applied magnetic fields cannot be seen in this plot. This effect is most prominent around the magnetic critical temperature where the values of the polarization and the dielectric constant are very close to one respectively zero relative on the scale of Figure 2.5. To compare the extended Ising-model with actual measurements a close-up of the dielectric constant around the magnetic critical temperature is shown in Figure 2.6 along with experimental data. (a)

Figure 2.6: Comparison of the dielectric constant profile from (a) calculations and (b) literature [Kimura, 2003]. Both the calculation and the measurement show the bend in the dielectric constant at zero applied magnetic field. This bend disappears for higher applied magnetic fields because the transition in the magnetization becomes more gradual. It is the sharp transition from a paramagnetic to a ferromagnetic state at zero field that gives rise to this discontinuity in the slope. Versus applied magnetic field From the response versus applied magnetic field the value of the magneto-electric coupling constant Cme can be determined. Figure 2.7 on the next page shows the magnetic response versus magnetic field.

26

-4 -3 -2 -1 0 1 2 3 4-1

-0.5

0

0.5

1MFA magnetic field behaviour - m & chi vs. H, variable T

mag

netiz

atio

n (-)

T=100T=110T=130

-4 -3 -2 -1 0 1 2 3 40

0.05

0.1

0.15

0.2

applied magnetic field (A/m)

susc

eptib

ility

(-)

(a)

Figure 2.7: Magnetic response versus magnetic field of the extended Ising-model, (a) calculations and (b) literature [Kimura, 2003].

The difference in the magnetization versus magnetic field graphs for the calculation and experiment is due to the fact that the calculation considers one single magnetic and crystallographic domain, while the experiment measures a piece of material with all sizes, types and orientations of domains. For ex-ample, a single magnetic domain will switch immediately from one orientation (+1) to the other (-1) at a certain critical point. However, the spread of these critical parameters in a bulk sample causes the magnetization, which is an average over all domains, to go through zero.

-4 -3 -2 -1 0 1 2 3 40.998

0.9985

0.999

0.9995

1

1.0005MFA magnetic field behaviour - p & K-1 vs. H, D=0.1, variable T

pola

rizat

ion

(-)

T=100T=110T=130

-4 -3 -2 -1 0 1 2 3 40

1

2

3

4x 10-5

applied magnetic field (A/m)

diel

ectri

c co

nsta

nt (-

)

Figure 2.8: Electric response versus field of the extended Ising-model, calculations.

Figure 2.8 shows the electric response versus applied magnetic field. Both the polarization and the dielectric constant are dependent on the magnetic field through the coupling with the magnetization. However, there is no similar data from literature so no fitting of Cme can be done. A linear relation was observed between the relative change in the dielectric constant and the square of the magnetization.

2( ) ( 0)( )( 0)

K H K HK H mK H− =

−∆ = − ∝=

(Equation 2.21)

27

Kimura [2003] explained this from Ginzburg-Landau theory. As shown in Figure 2.9 the extended Ising-model also reproduces this behaviour.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.05

0.1

0.15

0.2

0.25

MFA magnetic field behaviour - -dK vs. m2, D=0.1, variable T

magnetization squared (-)

rela

tive

chan

ge in

die

lect

ric c

onst

ant (

-)

T=100T=110T=130

(a) (b)

Figure 2.9: Relative change in dielectric constant versus magnetization squared, (a) calculations and (b) literature [Kimura, 2003]. From the literature data we can determine the average slope of the -∆K versus m2 lines. After correct-ing for the fact that the calculation uses the relative magnetization and the literature gives the absolute magnetization we find that the slope is about 1.47 at 100 K. Iteration of the calculation then yields the value of Cme = 170 K as given in Table 2.2. All calculations are carried out using this value. 2.3 Monte-Carlo simulations Besides calculations using MFA theory simulations with Monte-Carlo techniques are a tool commonly used to study systems as the Ising-model. This paragraph describes these simulations applied to the extended Ising-model of Paragraph 2.1.2. 2.3.1 Metropolis Monte-Carlo algorithm The basis of any Monte-Carlo simulation is a probability function. In the case of the Metropolis algo-rithm this distribution is simply the Boltzmann function.

( )

( )

b

b

a k T

s k T

s

ee

−

−=∑

H

HPm

m(a) (Equation 2.3)

Together with the spin- and dipole-Hamiltonian (Equations 2.6 and 2.8 or 2.9) this would in principle describe the complete system. However, as mentioned before, the calculation would be impossible due to the enormous number of different states to be summed in the denominator of the function. From Equation 2.3 we can easily get a relation for the relative probability of one state over another.

( )( ) ( ) ba b ke− −= H HPP

m m(a)(b)

T (Equation 2.22)

Now consider a square lattice of vectors, either pointing up or down. For each vector we can calculate the energy associated with the current orientation (considering only nearest neighbours) and with its orientation reversed. From that we can deduce relative probability of the two states using the above equation.

28

Comparing a random number to this probability we decide whether or not to flip this particular vector because the other state is more probable. In this manner we consider all the vectors on the lattice and we start again with the first one. Repeating this procedure over and over again equilibrium is reached. Using averaging techniques we can calculate the average sum over and the correlation between several samplings of the lattice. These averages and correlations correspond to actual physical quantities. For the magnetic Ising-model these values are given by Equations 2.23 and 2.24 [Gonsalves, 2002; Gottschalk, 1998; Tuck-erman, 1999]. im S∑= (Equation 2.23)

( )22i iS S

Tχ −∑ ∑m

1= (Equation 2.24)

These equations are readily applicable to the electric part of the extended Ising-model as well. In that case they read: ip P∑= (Equation 2.25)

( )221 i iK PT

ε− = −∑ ∑r

1= P (Equation 2.26)

2.3.2 Application of the Monte-Carlo Metropolis algorithm to the extended Ising-model The structure of the simulation programs is given in the flowchart in Appendix C. The core of the pro-gram is the ‘flipping’ of the lattices. During one ‘flip’ all vectors on the spin and dipole latticed are scanned and possibly flipped, using the procedure described in the paragraph above (see Appendix C for a discussion of the importance of the random number generator). In considering Equation 2.22 we use the averaged-magnetization Hamiltonian from Equation 2.9. First we ‘flip’ the spin lattice. Then we calculate the average magnetization and use that value for all dipoles when ‘flipping’ the dipole lattice. For a given set of parameters, both internal (Jm, Je, etc.) and external (T, H), the simulation consists of three main parts:

1) The equilibrium phase; 2) The sampling phase; 3) The calculation phase.

Since the simulation is based on the averaging over a series of samples, we have to be sure that start-ing conditions, such as the initial configuration of the lattice, no longer influence the sampling of the simulated system. During the equilibrium phase the program just goes through the lattice flipping pro-cedures to make sure equilibrium is reached before the actual sampling takes place. Then during the sampling phase a number of samplings are taken, each separated from the other by an interval of lat-tice flippings. Finally in the calculation phase these samplings are taken and the values of m, χ , p and K-1 are calculated. The rest of the program is used to vary the temperature and/or magnetic field to obtain a profile of the magnetization m, susceptibility χ, polarization p and dielectric constant K-1 versus either temperature T or magnetic field H. This profile is then, together with all the parameters used, written to a file.

29

2.3.3 Fitting of simulation parameters The program accepts a number of input parameters and outputs a number of variables, detailed below in Table 2.3. These input parameters can be divided into three groups. First the calculation parameters that determine the number of operations, the size of the lattices and such. Second the internal parame-ters. These are the actual model fitting parameters, such as Jm, µ and Je. They describe the system just as they do for the MFA calculations. Thirdly there are the external parameters that describe the envi-ronment. This includes the temperature and applied fields.

Calculation parameters Description n_overall number of profiles over which is averaged [-] n_equilibrium number of flips during equilibrium phase [-] n_measurement number of samplings taken [-] n_interval number of flips between samplings [-] size size of both the spin and dipole lattice [-] hotstart initial lattice is either random or ordered [-] Internal parameters strength absolute size of the spin/dipole [-] jm coupling constant for the magnetic state (Jm/kb) [K]mu magnetic permeability (µ/kb) [Km/A] je coupling constant for the dielectric state (Je/kb) [K] epsilon dielectric permittivity (εkb) [C/Km2] cme magneto-electric coupling constant (Cme/kb) [K] External parameters (temperature profile) t_start start temperature [K] t_end end temperature [K] t_step temperature step [K] h applied magnetic field [A/m] d applied electric displacement [C/m2] External parameters (magnetic field profile) t temperature [K] h_max maximum applied magnetic field [A/m] h_step magnetic field step [A/m] d applied electric displacement [C/m2]

Table 2.3: Input parameters for the extended Ising-model simulation, both temperature and magnetic field profile. Each of these parameters will be discussed shortly and a value will be assigned to it. Just as for the MFA calculations the factor 1/kb will be incorporated into the numerical value of the internal parame-ters. Calculation parameters The overall repetition of the measurement is not too useful since the measurements are already aver-aged for each temperature. Therefore this parameter is set to 1. The number of flips during the equilibrium phase is mainly important for the results at low tempera-ture (see Figure C.1 in Appendix C). At low temperature the probability to flip is very low, so equilib-rium will be reached very slowly, necessitating a long equilibrium phase. Switching to a ‘cold start’, meaning the initial lattices are ordered instead of randomly oriented, removes this need. Comparing ‘hot started’ to ‘cold started’ simulations shows that for higher temperatures (above 80 K) the two simulations coincide, while below 80 K the ‘cold started’ simulation gives better results; the magneti-zation approaches one respectively the susceptibility approaches zero as required from theory (Figure C.2). So for the temperature profile a ‘cold start’ with a moderate number of equilibrium steps (around

30

20.000) are good choices. For the magnetic field profile on the other hand a ‘hot start’ is needed since the applied magnetic field makes sure that ordering is induced. The lattice size has a great influence on the simulation (see Figure C.3). The larger the lattices, the sharper the transition in both the magnetization and the susceptibility. In general the larger the lattices, the better the simulation simulates the real problem. For example, as the lattices becomes larger, the transition starts to resemble the Onsager-solution for zero field. Therefore we choose the lattices to be as large as possible, which in this implementation is 100 by 100. The number of samplings greatly influences the smoothness of the profile (see Figure C.4). This num-ber should be at least 1000, but often a larger number is chosen for more samplings and thus better averaging. The size of the interval between the samplings does not seem to influence the simulation very much. Basically these two parameters are determined by the allotted time for a simulation (once the minimum of 1000 samplings is accounted for). With the other calculation parameters fixed be-cause of other considerations, these two can be made as large as allowable by time considerations. It turned out that the values of 6000 samplings with a 10-flip interval were convenient. Internal parameters An important factor for the determination of the internal parameters is the factor kbTc/J. For the On-sager-solution this factor is equal to 2.269.., but in MFA theory it is equal to 1. So the question is what value it would be in the simulation and on what parameters it depends. It turns out that this factor only depends on the size of the lattices. By determining the critical tempera-ture for different settings of je/jm at different lattice sizes we can determine the values of this factor for that lattice size. Figure 2.10 shows these values versus lattice size.

0 20 40 60 80 100 1202.26

2.28

2.3

2.32

2.34

2.36

2.38

2.4

2.42

2.44kbTc/J relation

lattice size (-)

kbTc

/J ra

tio (-

)

dataexp. fitOnsager

Figure 2.10: kbTc/J factor versus lattice size.

Figure 2.10 also shows an excellent exponential fit to these data points. For an infinite lattice this fit yields the same value for kbTc/J as the Onsager-solution. Which it should do, since the Onsager-solution is an exact solution of the Ising-model for an infinite lattice. From this fit we calculate that the kbTc/J factor for a 100x100 lattice is 2.270... The magnetic coupling constant jm can be derived from an equation similar to Equation 2.13, but the kbTc/J factor is now 2.270.. instead of 1.

31

, 2.270..b c m

m

k TJ

= (Equation 2.27)

For Tc,m = 105 K this gives jm = Jm/kb = 46 K. Just as for MFA theory the values of the magnetic permeability mu and dielectric permittivity epsilon cannot be determined from literature. Thus both are set equal to one. For the electric coupling constant je and magneto-electric coupling constant cme we would have liked to go through the same procedure as for MFA theory; that is to obtain the value for cme by comparing the relative change in dielectric constant versus magnetization squared for both the simulation and lit-erature and then iterate of the simulation procedure. However, as we shall see in the next paragraph, it was not possible to do simulations with both the correct ferroelectric critical temperature of 450 K and a noticeable multiferroic effect. It turned out it was only possible to get a proof of principle from the simulation. It shows that the ex-tended Ising-model does show multiferroic behaviour when evaluated using a simulation, but it was not possible to exactly fit the parameters to the BiMnO3 system. With Tc,e = 200 K it was possible to observed the multiferroic effect in the simulation, while the mag-netization at that temperature was close enough at zero to neglect it when calculating Je/kb from Equa-tion 2.28.

( )2

,

12

2.270..b c e

me me

k TJ C +

=+

(Equation 2.28)

Comparing the temperature profiles from simulation to the figure in literature shows that cme = 50 K gives good comparable results. This then yields je = Je/kb = 63 K. External parameters The external parameters vary very much, depending on the kind of profile (temperature or magnetic field) you simulate, on the precision of the profile and on the time span for the simulation. Probably the only constant was an electric displacement of zero. We are only interested in the behaviour versus magnetic field. The external parameter settings used for these simulations are shown in Table 2.4.

External parameter Temperature profile Detailed temperature profile Magnetic field profile t_start [K] 5 70 t_end [K] 300 130 t_step [K] 5 1 h [A/m] 0-2 0-2 t [K] 100-130 h_max [A/m] 3 h_step [A/m] 0.1 d [C/m2] 0 0 0

Table 2.4: External parameter settings.

32

An overview of all the simulation parameters is given in Table 2.5.

Calculation parameters Value Description n_overall 1 number of profiles over which is averaged [-] n_equilibrium 20000 number of flips during equilibrium phase [-] n_measurement 6000 number of samplings taken [-] n_interval 10 number of flips between samplings [-] size 100 size of both the spin and dipole lattice [-] hotstart 0 / 1 initial lattice is either random or ordered [-] Internal parameters strength 1 absolute size of the spin/dipole [-] jm 46 coupling constant for the magnetic state (Jm/kb) [K]mu 1 magnetic permeability (µ/kb) [Km/A] je 63 coupling constant for the dielectric state (Je/kb) [K] epsilon 1 dielectric permittivity (εkb) [C/Km2] cme 50 magneto-electric coupling constant (Cme/kb) [K] External parameters (temperature profile) t_start 5 / 70 start temperature [K] t_end 300 / 130 end temperature [K] t_step 5 / 1 temperature step [K] h 0-2 applied magnetic field [A/m] d 0 applied electric displacement [C/m2] External parameters (magnetic field profile) t 100-130 temperature [K] h_max 3 maximum applied magnetic field [A/m] h_step 0.1 magnetic field step [A/m] d 0 applied electric displacement [C/m2]

Table 2.5: Input parameters and their values for the extended Ising-model simulation, both temperature and magnetic field profile. A short discussion of the error estimation for the Monte-Carlo simulations is given at the end of Ap-pendix C. 2.3.4 Results Figure 2.11 shows the results from simulations for values of jm, je and cme, which are equivalent to those values used in the MFA theory calculations.

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1simulation temperature profile, settings comparable to MFA calculations

mag

netiz

atio

n /

susc

eptib

ility

(-)

mχ

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

temperature (K)

pola

rizat

ion

/di

elec

tric

cons

tant

(-)

pK-1

70 80 90 100 110 120 130-6

-4

-2

0

2

4

6x 10-6 detail of simulation dielectric constant temperature profile

temperature (K)

diel

ectri

c co

nsta

nt (-

)

(a) (b)

Figure 2.11: Simulations for settings equivalent to MFA theory parameters (a) overview and (b) detail.

33

As can be seen from Figure 2.11(b) for these settings the simulation output of the dielectric constant around Tc,m is so close to zero that no features can be distinguished. This is due to the fact that the simulation only uses a finite set of dipoles. For temperatures far below the Tc,e there is so little varia-tion in the dipole orientation that the correlation between samplings, which describes the dielectric constant, is essentially zero when averaged over a finite lattice. The MFA theory is derived for an infi-nite lattice, so what little variation is still present can be calculated. Where MFA theory is capable of exactly calculating the value of the dielectric constant, the simulation does not have enough samplings for the averaging to give a single, well-defined trend. Hence, the noise at these low temperature. The necessary increase in size of the simulation (either in lattice size or number of samplings) would in-crease the time each simulation requires too much to make their use practical within this research. The focus of the simulations became whether the simulation yielded results that were similar in behav-iour to both experiments and the MFA theory. To do this, new values for the je and cme parameters were needed. The value for jm was kept fixed (corresponding to Tc,m = 105 K) to have a starting point. First je was set just above jm and then increased (with cme being zero), while the simulation was still showing a clear trend around 105 K. This way Tc,e of 200 K was found to both fulfil the need for suffi-cient ‘signal’ at 105 K and for the need to have Tc,e as far from Tc,m as possible so the magnetization can be neglected when calculating je for non-zero cme. Once Tc,e was fixed, cme was varied to obtain a bend in the temperature profile at zero field similar to that from Figure 1.5. With these values, as seen in Table 2.5, the rest of the simulations were carried out. Versus temperature Figure 2.12 shows the magnetic response of the extended Ising-model versus temperature for different applied magnetic fields, as simulated using the Monte-Carlo Metropolis algorithm.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1simulation temperature profile - m & chi vs. T, variable H

mag

netiz

atio

n (-)

H=0H=1H=2

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

temperature (K)

susc

eptib

ility

(-)

Figure 2.12: Magnetic response versus temperature of the extended Ising-model, simulations.

This figure is very comparable to Figure 2.4 where the same quantities are plotted. The discussion of the features seen in Figure 2.4 applies here as well. One of the differences is that the transition in the magnetization is much sharper as for the calculations; the change occurs in a much smaller tempera-ture range. This is due to the fact that the simulations resemble the Onsager-solution much more then the MFA theory. No additional simplifications are used so the simulations resemble the actual Ising-model (and its analytical solution) much better than the MFA theory. The electric response versus temperature, as shown in Figure 2.13 on the next page, is also very com-parable to the output of the MFA theory calculations as seen in Figure 2.5. The main difference is the scale of the x-axis, showing that the electric critical temperature of the simulation is 200 K instead of

34

450 K. Another difference is that for the calculations the profiles for different applied magnetic field exactly overlap, while for the simulations this is not the case. This is due to reliance of the Monte-Carlo algorithm on random numbers. No run is ever identical to another, so there will not be a perfect overlap.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1simulation temperature profile - p & K-1 vs. T, D=0, variable H

pola

rizat

ion

(-)

H=0H=1H=2

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

temperature (K)

norm

aliz

ed d

iele

ctric

con

stan

t (-)

Figure 2.13: Electric response versus temperature of the extended Ising-model, simulations.

Finally we compare the dielectric constant around the magnetic critical temperature of the simulations with that of the calculations and that from literature. In Figure 2.14 these three profiles are shown next to one another.

70 80 90 100 110 120 130-0.5

0

0.5

1

1.5

2

2.5

x 10-4 simulation temperature profile detail - K-1 vs. T, D=0, variable H

temperature (K)

diel

ectri

c co

nsta

nt (-

)

H=0H=1H=2

70 80 90 100 110 120 1300

0.5

1

1.5

2

2.5

3

3.5x 10-5 MFA temperature profile - detail K-1 vs. T, D=~0, variable H

temperature (K)

diel

ectri

c co

nsta

nt (-

)

H=~0H= 2H= 4

3(a) (b) (c)

Figure 2.14: Comparison of the dielectric constant profile from (a) simulation, (b) calculation an All three sub-figures show the bend at the magnetic critical temperature, althodegree. For the simulations the bend is much sharper than for the calculations.the fact that the simulations have a sharper transition in the magnetization thanthe change induced by the magnetoelectric coupling on going through Tc,m is min the simulations, resulting in a sharper bend. Versus applied magnetic field Since it was not possible to fit the internal parameters to the properties of BiMested in finding the correct value of cme. Instead we use the value that results temperature profile. Figure 2.15 on the next page shows the magnetic response of the extended Isinfrom the Monte-Carlo simulations.

35

d (c) literature [Kimura, 2003].

ugh not all to the same This is probably due to the calculations. Thus uch more pronounced

nO3 we are not inter-in a clear bend in the

g-model as obtained

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1simulation magnetic field behaviour - m & chi vs. H, variable T

mag

netiz

atio

n (-)

-3 -2 -1 0 1 2 30

0.5

1

1.5

applied magnetic field (A/m)

susc

eptib

ility

(-)

T=100T=110T=130

(a)

Figure 2.15: Magnetic response versus field of the extended Ising-model, (a) simulation and (b) literature [Kimura, 2003]. One difference with both experiment and calculations is the fact that the magnetization does not switch direction when the magnetic field is applied in the negative direction. This is because the simu-lation program only returns the absolute value of the magnetization. Taking that into consideration, the output of the simulations resembles the actual measurements from literature a bit better than the calculations do. The simulations show a continuous transition from one orientation to the other instead of the discontinuous transition seen in MFA theory. The fact that the magnetization does not reach zero for zero field is not due to the fact that we consider one single mag-netic domain, as it was in the case of the MFA calculations, but because with a finite lattice it is prac-tically impossible to get an average value of zero.

-3 -2 -1 0 1 2 30.985

0.99

0.995

1

1.005simulation magnetic field behaviour - p & K-1 vs. H, D=0, variable T

pola

rizat

ion

(-)

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3x 10-4

applied magnetic field (A/m)

diel

ectri

c co

nsta

nt (-

)

T=100T=110T=130

Figure 2.16: Electric response versus field of the extended Ising-model, simulations.

The electric response obtained from the simulations, shown in Figure 2.16, looks even more similar to that obtained from the calculations. However, the best way to compare the simulated, calculated and experimental data is to look at the linear relation between the relative change in the dielectric constant and magnetization squared.

36

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-2.5

-2

-1.5

-1

-0.5

0

0.5

1simulation magnetic field behaviour - -dK vs. m2, D=0, variable T

magnetization squared (-)

rela

tive

chan

ge in

die

lect

ric c

onst

ant (

-)T=100T=110T=130

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0

0.05

0.1

0.15

0.2

0.25

MFA magnetic field behaviour - -dK vs. m2, D=0.1, variable T

magnetization squared (-)

rela

tive

chan

ge in

die

lect

ric c

onst

ant (

-)

T=100T=110T=130

(a) (c) (b)

Figure 2.17: Relative change in dielectric constant versus magnetization squared, (a) simulations, (b) calculations and (c) litera-ture [Kimura, 2003].

Just as the calculations, the simulations show the linear relation between the relative change in dielec-tric constant versus the magnetization squared as observed in experiment (Figure 2.17). For tempera-tures below the Tc,m the dielectric constant contains a lot of noise. This is probably due to the fact that for those temperatures the output from the simulation is at the lowest possible end of the numerical range it can evaluate. See for example Figure 2.14(a) where for temperatures below 100 K the dielec-tric constant is only fluctuating around zero. Lowering je, and thus Tc,e, in hope of increasing the size of the dielectric constant just below Tc,m does not help. Below Tc,m the induced magnetoelectric cou-pling is so strong that je increases so much that the dielectric constant is forced to zero. 2.4 Conclusions Extending the original Ising-model to multiferroic behaviour is readily done. Equations 2.6 and 2.8 or 2.9 translate the qualitative physics from Paragraph 1.2.3 into a quantitative phenomenological model. This extended Ising-model is only applicable to BiMnO3 and its sister-compounds. However, the model can easily be adapted to other materials such as YMnO3. Using both mean-field approximation theory calculations and Monte-Carlo Metropolis simulations the extended Ising-model constructed in Paragraph 2.1.2 was evaluated. Key features of the multiferroic behaviour of BiMnO3, such as the bend in the dielectric constant versus temperature profile and the linear relation between the relative change in the dielectric constant and the magnetization squared were reproduced by both methods. The MFA theory calculations allow exact fitting of the parameters to the properties of BiMnO3. The Monte-Carlo simulations cannot be fit exactly because of the limit of a finite number of dipoles in the simulation. However, the Monte-Carlo simulations reproduce the sharp transition in both the magneti-zation and the dielectric constant at Tc,m better than the MFA theory calculations do.

37

3 Experimental investigation Physical models such as those constructed in the previous two chapters have to agree with actual ex-periments. It is experimental verification of theories that is one of the fundamental basics of physics. However, for experiments one needs samples made of BiMnO3 and as seen in Paragraph 1.3 it is a dif-ficult material to fabricate. This chapter focuses on the growth of thin films of BiMnO3. The first paragraph discusses why we want to grow thin films of BiMnO3 instead of fabricating bulk samples for the research. The second paragraph focuses on the choice for pulsed laser deposition (PLD) as the thin film growth technique. In the next paragraph the initial process parameters are chosen and the influence of varying those pa-rameters on the properties of the thin films of BiMnO3 are studied. The fourth paragraph then deals with the properties of the best obtained thin films and compares them with results from literature. The chapter closes with discussion of these results and some conclusions. 3.1 Thin film growth The fabrication of bulk samples of BiMnO3 is a difficult and high-pressure process, which makes it unsuitable for device fabrication. Growing thin films of BiMnO3 has two main advantages over bulk fabrication. Epitaxial stabilization Growing thin films of BiMnO3 on lattice-matched substrates can ease the fabrication of the desired crystal structure by lowering the energy of that structure relative to the individual compounds. In the case of BiMnO3 both Bi2O3 and Mn2O3 individually are a more stable phase than the mixed BiMnO3 phase, as evidenced by the extreme fabrication parameters needed to produce BiMnO3 as bulk material. However, the lattice mismatch energy of Bi2O3 and Mn2O3 with a substrate that is lat-tice-matched to BiMnO3 is much larger than this phase separation energy. Thus, on a substrate lattice-matched to BiMnO3 the BiMnO3 phase is stabilized over the Bi2O3 and Mn2O3 phases [Flynn, 1986]. Device fabrication Thus growing BiMnO3 as a thin film can change the fabrication parameters, bringing them into a range that allows the fabrication of devices. In fact a lot of devices nowadays are made from stacks of thin films. Because of this, growing BiMnO3 as thin films would greatly facilitate its application. 3.2 Pulsed laser deposition There are several techniques available for the fabrication of thin films. Of these different techniques pulsed laser deposition (PLD) was the technique of choice for the fabrication of thin films of BiMnO3. 3.2.1 PLD process The concept behind PLD is rather simple. An intense laser pulse is used to evaporate material from a bulk piece, called the target. The ablated material forms a high-pressure plasma near the target surface that expands in the near-vacuum between the target and the substrate. On the - relatively - cold sub-strate this plasma ‘condenses’ and forms a thin film. Figure 3.1 on the next page shows a schematic of this process.

38

Figure 3.1: Schematic of the PLD process.

There are a lot of parameters that influence the growth of the thin film. Table 3.1 gives an overview of some of the most important ones and the chief influence these parameters have.

Parameter Influence Fluency Droplet formation, kinetic energy of plasma Pulse frequency Growth kinetics Target composition Chemical composition of thin film Gas composition & pressure Chemical composition of thin film, kinetic energy of plasma Target-to-substrate distance Kinetic energy of plasma Substrate type & temperature Growth kinetics Annealing procedure Precipitate formation, oxygen content

Table 3.1: Overview of important PLD parameters and their influence. 3.2.2 Advantages of PLD The three main advantages of PLD are considered to be:

1) the stoichiometric transfer; 2) the high supersaturation during the deposition pulse; 3) the tuneable kinetics of the deposited species.

We shall see why these properties of the PLD system make it especially advantageous for use with BiMnO3. When growing thin films of BiMnO3 there are four properties that are of great importance:

1) the crystallographic structure; 2) the chemical composition; 3) the valency of the manganese ions; 4) the surface morphology.

As mentioned above epitaxial stabilization provides a means to grow the correct crystallographic phase of BiMnO3 without the extreme conditions required for bulk preparation. The supersaturation during the deposition pulse enhances the formation of the mixed BiMnO3 phase over the individual bismuth and manganese oxide phases, as the arriving species do not have the time to ‘unmix’ and clus-ter by species. During growth they are then surrounded by an environment of random other species facilitating the formation of the mixed BiMnO3. In growth processes without such supersaturation it is possible for the arriving species to unmix and form the individual compounds.

39

Bismuth is an element that is easily evaporated. So much so that to grow BiMnO3 with the desired chemical composition often targets with extra bismuth have to be used. This ensures that there will be enough bismuth available at the substrate for film growth, while excess bismuth will evaporate easily. This self-limiting behaviour is observed in experiments and often used to obtain films with the correct chemical composition [for example Migita, 1999]. It is then very useful that the transfer from target to substrate occurs stoichiometric, so the actual deposition process itself does not complicate the control of the bismuth content of the thin film. The oxygen content of the thin film can also be controlled easily by adjusting the (partial) oxygen pressure during deposition and annealing. This is especially important as the manganese ions easily reduce further from the desired Mn3+ to the unwanted Mn4+ ionization. Any Mn4+ content would make the thin film conducting and non-ferromagnetic. By controlling the oxygen content of BiMnO3 we can control the ionization of the manganese and prevent the occurrence of Mn4+. Finally the control of the growth kinetics should allow us to force the correct phase to grow by induc-ing epitaxial stabilization. Ideally control of the growth kinetics will also allow us to obtain smooth and single-phase thin films. 3.3 Growth experiments As described in the previous paragraph PLD is a useful tool for the growth of complex oxides such as BiMnO3. Finding the optimum parameters for the growth of high-quality (meaning single-phase and smooth) thin films of BiMnO3 still requires a lot of experiments. A short list of the equipment used in these experiments is given in Appendix D. 3.3.1 Initial choice of parameters Table 1.5 gave an overview of the PLD parameters used for the growth of thin films of BiMnO3 as found in literature. From these settings and the requirements formulated in the previous paragraph an initial choice of parameters for the PLD experiments was made. Chemical composition Scanning electron microscopy (SEM) examination of the target surface after ablation at different flu-encies shows that for fluencies above 2 J/cm2 the pillar formation is greatly reduced [Te Riele, 2003]. This means that all material is ablated, leading to a plasma with the same chemical composition as the target. In literature a fluency close to 2 J/cm2 is used with good results. The chosen fluency of 2.2 J/cm2 is above the ablation threshold, but still close to the values from literature. A set of four BiMnOx targets was provided by Prof. Schlom from Penn State University, Pennsyl-vania. All four targets contain an excess amount of bismuth: Bi1.2MnOx, Bi1.8MnOx, Bi2.4MnOx and Bi3MnOx. All targets were prepared using a solid-state reaction, as described in the article by Moreira dos Santos [2003]. Of these four targets the 1.8-target with the Bi:Mn ratio of 1.8:1 was chosen for the initial experiments. This was done because the lack of ozone in the PLD set-up, which would other-wise react with bismuth and prevent it from evaporating from the thin film. To compensate for the lack of this binding a target richer in bismuth was chosen as compared to the deposition parameters of Moreira dos Santos [2003]. The 1.8-target was chosen because the experiments by Te Riele [2003] showed that for lower temperatures the stoichiometric transfer of material was almost perfect, so no large increase in bismuth content was needed. To grow BiMnO3 there has to be enough oxygen available. Since the PLD set-up does not have an ozone supply, pure oxygen was used as the gas during deposition. After the deposition the thin film has to cool down as quickly as possible to prevent bismuth from evaporating from the thin film. For that we need a high anneal pressure. But we also want to prevent

40

the formation of Mn4+ ions because of too much or too little oxygen. The PLD set-up is not very pre-cise in controlling the anneal pressure, so more than a rough setting of 100 mbar is not possible. From the temperature-oxygen pressure phase diagram for manganese oxides (courtesy of Prof. Schlom from Penn State University, Pennsylvania, Figure D.2) a pressure of 100 mbar would put the anneal proce-dure in the regime for the formation of Mn3+ down to about 420 ºC, which hopefully is enough to maintain the Mn3+ ionization. Crystallographic structure For the reflective high-energy electron diffraction system work properly, the target-to-substrate dis-tance was fixed at 45 mm. A pressure of 0.05 mbar was chosen. This is higher than the value used by Moreira dos Santos [2003], intended to thermalize the plasma species over the shorter target-to-substrate distance. As a substrate SrTiO3 was chosen. These substrates are readily available, easy to clean and prepare [Koster, 1998] and have a lattice mismatch with BiMnO3 of only -0.8 %. This close lattice match is required for epitaxial stabilization. The substrate temperature greatly influences the crystallographic structure. A substrate temperature of 700 ºC was chosen, reminiscent to Moreira dos Santos [2003] who was able to grow good films at that temperature. The same pulse frequency of 4 Hz from Moreira dos Santos [2003] was adopted, again because of the quality of the films obtained. Manganese ion valency & surface morphology Neither the manganese ion valency nor the surface morphology were taken into account for the initial parameters. Both were thought to be controllable by, for example, (partial) oxygen pressure and pulse frequency after the crystallographic structure and chemical composition conditions were satisfied. Table 3.2 gives an overview of the initial PLD parameters.

Parameter Setting Fluency 2.2 J/cm2 Pulse frequency 4 Hz Target composition Bi1.8MnOx Gas composition O2 Gas pressure 0.05 mbar Target-to-substrate distance 45 mm Substrate type SrTiO3 Substrate temperature 700 ºC Annealing procedure ‘quench’ at 100 mbar

Table 3.2: Overview of the initial PLD parameters. 3.3.2 Crystallographic structure Given the influence of the substrate temperature on the crystallographic phase it is the first parameter that was investigated. Figure 3.2 on the next page shows the X-ray diffraction (XRD) patterns of thin films grown at respectively 600, 650 and 700 ºC. The scans are all along the (001) axis of SrTiO3. The marks indicate literature values of identified crystallographic phases. Moreira dos Santos [2003] ob-served the epitaxial growth of BiMnO3 on SrTiO3 with twinned orientations. These orientations were used to identify the BiMnO3 peaks in the XRD patterns.

41

0 20 40 60 80 100 120100

102

104

106temperature dependence of crystallographic phase

(a)SrTiO3Bi2O3Bi12MnO20

0 20 40 60 80 100 120100

102

104

106

inte

nsity

(a.u

.) (b)SrTiO3BiMnO3

0 20 40 60 80 100 120100

102

104

106

2θ (º)

(c)SrTiO3

Figure 3.2: XRD patterns for BiMnO3 thin films grown with the 1.8-target at (a) 600, (b) 650 and (c) 700 ºC.

Only the film at 650 ºC shows the formation of crystalline BiMnO3. Further experimentation showed that for a substrate temperature of 679 ºC no crystalline BiMnO3 was present, while for a substrate temperature of 625 ºC both BiMnO3 and the Bi2O3, Bi12MnO20 and other phases were present. This seems to indicate that there is a very narrow temperature window for the growth of only crystalline BiMnO3. For further experimentation a substrate temperature of 650 ºC was used. The chemical composition of the target also influences the crystallographic phase. If there is too little bismuth available the growth of BiMnO3 will not be optimal. This process is complicated by the fact that bismuth easily evaporates, making control of the bismuth content of the thin film a difficult mat-ter. Figure 3.3 shows the XRD patterns for the thin films grown at 600, 650 and 700 ºC, using the 1.2-target instead of the 1.8-target (as used for the rest of the experiments).

0 20 40 60 80 100 120100

102

104

106temperature dependence of crystallographic phase, 1.2-target

(a)SrTiO3BiMnO3Bi2O3Bi12MnO20

0 20 40 60 80 100 120100

102

104

106

inte

nsity

(a.u

.) (b)SrTiO3

0 20 40 60 80 100 120100

102

104

106

2θ (º)

(c)SrTiO3

Figure 3.3: XRD patterns for BiMnO3 thin films grown with the 1.2-target at (a) 600, (b) 650 and (c) 700 ºC.

42

The crystalline phase of BiMnO3 was not observed at 650 ºC. In fact, for the 1.2-target the XRD pat-terns at 650 and 700 ºC are comparable to the XRD pattern for the 1.8-target at 700 ºC. In the XRD pattern at 600 ºC crystalline BiMnO3 is observed, but the signal is lower and all the other phases (Bi2O3, Bi12MnO20 and others) are already present. The chemical composition obtained from energy dispersive X-ray measurements (EDX) seem to indi-cate that the overall Bi:Mn ratios of the films grown with the 1.8- and the 1.2-target are comparable. So even with the 1.2-target there is still an excess of bismuth at the substrate, after which the self-limiting behaviour of bismuth ensures a constant Bi:Mn ratio. Still, the smaller amount of excess bis-muth in the 1.2-target shifts the optimal substrate temperature to lower values compared to the 1.8-target. Because less bismuth arrives at the substrate, the equilibrium between arriving and evaporating bismuth is reached at a lower substrate temperature. Thus the optimal growth conditions are shifted to lower temperatures. The optimal substrate temperature for growth with the 1.2-target lies probably between 600 and 650 ºC. From the XRD patterns we can also determine the cell parameter of the thin film of BiMnO3 perpen-dicular to the substrate. Also, since we use (001)-oriented SrTiO3 substrates we can determine the c-axis cell parameter of SrTiO3 and use that to estimate the error in the calculation of the cell parameter. See Appendix D for an overview of the cell parameter calculation. Figure 3.4 shows the observed cell parameter for the BiMnO3-containing thin films. The different pseudo-perovskite cell parameters of BiMnO3 (see Paragraph 1.1) are indicated by the red dotted lines.

0 2 4 6 8 10 12 14 16 18 20 223.9

3.92

3.94

3.96

3.98

4

4.02b-axis of BiMnO3 from XRD patterns

sample (-)

b-ax

is (Å

)

Figure 3.4: b-axis cell parameter for several BiMnO3 thin films as calculate From considering the lattice mismatch we expect the pseudo-pethe substrate as it is the most mismatched cell parameter. As cabe the case as the observed cell parameters lie closest to the expcompressive in-plane stress (lattice mismatch -0.8 %) a slightlypected. The lower value observed in Figure 3.4 and Table 3.3 iis not only stretched, but also buckled. Unfortunately no in-planbecause of the overlap between BiMnO3 and SrTiO3 peaks in thtion of the BiMnO3 films could not be determined. The average value of the observed c-axis cell parameter for SrTBiMnO3 are compared to their bulk value in Table 3.3 on the nis in good correspondence to the bulk value, indicating that thethe cell parameters of the thin films.

43

b-axis

s

s

c-axi

a-axi

d from the XRD patterns.

rovskite b-axis to be perpendicular to n be seen from Figure 3.4 this seems to ected value of 3.995 Å. Because of the larger b-axis cell parameter is ex-

ndicates that the pseudo-perovskite cell e XRD measurements were possible e XRD patterns, so the exact orienta-

iO3 and the b-axis cell parameter for ext page. The cell parameter of SrTiO3 XRD analysis can be used to determine

Cell parameter Observed (Å) Bulk value (Å) c-axis SrTiO3 3.903 ± 0.004 3.905 b-axis BiMnO3 3.98 ± 0.01 3.995

Table 3.3: Cell parameter comparison for SrTiO3 and BiMnO3. Although the lattice mismatch between SrTiO3 and BiMnO3 is small, there are other materials that have an even smaller mismatch, most importantly SrRuO3 and KTaO3. Their crystal type, cell parame-ters and lattice mismatch with BiMnO3 are given in Table 3.4.

Material Crystal type Cell parameter (Å) Lattice mismatch (%) SrTiO3 cubic 3.905 -0.8 SrRuO3 pseudo-cubic 3.93 -0.1 KTaO3 cubic 3.988 0.4

Table 3.4: Comparison of several possible substrate materials. However, these materials are less readily available as SrTiO3 and their preparation is more difficult and yields less ideal surfaces. Still it is interesting to investigate the influence of different substrate types on the crystallographic phase of BiMnO3 thin films. KTaO3 substrates were courtesy of Dr. Leca, former Ph.D. student at the University of Twente, cur-rently at the University of Tübingen. For the experiments with SrRuO3 a buffer layer of SrRuO3 was deposited on SrTiO3 substrates to obtain a relaxed and smooth surface [Rijnders, 2001]. Figure 3.5 below shows the XRD patterns for thin films of BiMnO3 grown with the 1.8-target on re-spectively SrTiO3, SrRuO3 buffered SrTiO3 and KTaO3.

0 20 40 60 80 100 120100

102

104

106influence of substrate type on crystallographic phase

(a) SrTiO3SrTiO3BiMnO3

0 20 40 60 80 100 120100

102

104

106

inte

nsity

(a.u

.)

(b) SrRuO3SrTiO3BiMnO3SrRuO3

0 20 40 60 80 100 120100

102

104

106

2θ (º)

(c) KTaO3KTaO3BiMnO3

Figure 3.5: XRD patterns for BiMnO3 thin films grown on (a) SrTiO3, (b) SrRuO3 buffered SrTiO3 and (c) KTaO3.

These XRD patterns show that the PLD parameter settings easily carry over to other substrate types. In all three cases single-phase crystalline BiMnO3 was grown. From the lattice mismatch it was expected that on both SrTiO3 and on SrRuO3 buffered SrTiO3 the b-axis of BiMnO3 would be perpendicular to the surface, while on KTaO3 the c-axis would be perpendicular to the surface. This was indeed ob-served by calculation of the cell parameters of the thin films. These values are shown in Table 3.5 on the next page.

44

Substrate Cell parameter Observed (Å) Bulk value (Å) SrTiO3 SrTiO3 c-axis 3.904 ± 0.003 3.905 BiMnO3 b-axis 3.978 ± 0.007 3.995 SrRuO3//SrTiO3 SrTiO3 c-axis 3.902 ± 0.006 3.905 SrRuO3 c-axis 3.94 ± 0.01 3.93 BiMnO3 b-axis 3.985 ± 0.007 3.995 KTaO3 KTaO3 c-axis 3.98 ± 0.01 3.988 BiMnO3 c-axis 3.87 ± 0.01 3.919

Table 3.5: Cell parameter comparison for thin films grown on SrTiO3, SrRuO3 buffered SrTiO3 and KTaO3. Again the cell parameters of the substrates are in excellent agreement with the bulk value for each ma-terial. The SrRuO3 buffer layer has a slightly larger c-axis cell parameter, which is expected since the lattice mismatch between the SrRuO3 film and the SrTiO3 substrate is compressive. Remarkable is that in all three cases the BiMnO3 cell parameter is smaller than the bulk value. It seems that on all cubic substrates the BiMnO3 pseudo-perovskite unit cell undergoes buckling in addition to stretching. 3.3.3 Chemical composition The chemical composition of several thin films was studied using EDX. The most important observa-tion was that the Bi:Mn ratios of films grown with the 1.2- and 1.8-target and respectively 600 and 650 ºC were nearly identical. Table 3.6 shows the results for these two films. The film grown with the 1.2-target has a thickness of about one-third of the film grown with the 1.8-target.

Grown with the 1.2-target at 600 ºC Grown with the 1.8-target at 650 ºC Element Percentage (%) Element Percentage (%) Sr 48 Sr 45 Ti 50 Ti 45 Bi 0.8 Bi 4 Mn 1.2 Mn 6

Table 3.6: EDX results for films grown with the 1.2- and 1.8-target. For both films the Bi:Mn ratio is about 2:3. Unfortunately the EDX measurements are not accurate enough to state whether this ratio is the actual ratio in the films. It can be stated however that the ac-tual Bi:Mn ratio in both films is comparable. This seems to indicate that the self-limiting behaviour of bismuth is at work during the growth with both the 1.2- and 1.8-target. The EDX measurements and the complementary SEM images indicate that there are a lot of precipi-tates and off-stoichiometric areas present in the films grown with respectively the 1.2- and 1.8-target. So while the XRD pattern for the film grown with the 1.8-target (Figure 3.2(b)) does not show the presence of other crystallographic phases, there are other chemical phases present; either in very small amounts or in an amorphous form. The XRD pattern for the film grown with the 1.2-target (Figure 3.3(a)) does show other crystallographic phases, so the presence of other chemical phases is not unex-pected. 3.3.4 Manganese ion valency The main focus during this Master research was on the fabrication of crystalline films of BiMnO3 and the manganese ion valency was not investigated. The valency is probably controllable by adjusting the (partial) oxygen pressure and the anneal conditions. For this the temperature-oxygen pressure phase diagram for manganese oxides shown in Figure D.2 could be used to determine the optimal anneal procedure.

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3.3.5 Surface morphology The investigation of the crystallographic structure and the chemical composition has assured us that we actually grow BiMnO3. The study of the surface morphology of the thin films can then help to de-termine the growth process of BiMnO3. This in turn can lead to the fabrication of smooth layers of BiMnO3 for device applications. Figure 3.6 shows the surface of BiMnO3 thin films grown at different substrate temperatures as ob-tained with an atomic force microscope (AFM). In all cases the area displayed is 5x5 µm. The target used was the 1.8-target. The height scale for Figure (a) is 20 nm, for Figures (b) and (c) it is 50 nm.

(a) 600 ºC, 1.8-target

(b) 650 ºC, 1.8-target

(c) 700 ºC, 1.8-target

Figure 3.6: AFM surface images for BiMnO3 thin films grown at different substrate temperatures. Figure 3.6 shows three different morphologies corresponding to the different crystallographic struc-tures from Figure 3.2. At 600 ºC a mixture of island and spiral growth is observed, not unlike that ob-served in YBa2Cu3O7-x [Dam, 1998]. However, from the XRD pattern the crystallographic phase re-sponsible for this growth could not be identified. Equally curious is the morphology at 700 ºC. Large triangular shapes, intermixed with ribbon-like shapes were observed. Again, from the XRD pattern the corresponding crystallographic phase could not be identified. The XRD patterns indicated that for both 625 and 650 ºC BiMnO3 is formed, although a lot of other phases are also present at 625 ºC. From these AFM images it looks like BiMnO3 grows in small is-lands. This morphology was also observed in thin films of BiMnO3 grown on LaAlO3 [Son, 2004] and is possible due to the two in-plane orientations on SrTiO3 that were observed by Moreira dos Santos [2003]. At the first stage of the growth a cluster on the substrate can chose its in-plane orientation ran-domly as no other clusters are nearby to give a preferred orientation. As the clusters continue to grow, they will encounter other clusters with possibly the other in-plane orientation. Between these clusters a grain boundary will form, while on top of the clusters the growth will continue with the same in-plane orientation as the cluster below it, thus forming islands of BiMnO3 separated from each other by grain boundaries. From the XRD patterns of thin films grown with different targets, the optimal substrate temperature shifted to lower temperatures for the target with less excess bismuth. Figure 3.7 on the next page shows that the occurrence of the different surface morphologies also shift to lower temperatures. Again the area is 5x5 µm. In Figure (a) the height scale is 20 nm. In the other sub-figures it is 50 nm.

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(a) 625 ºC, 1.8-target

(c) 600 ºC, 1.2-target

(b) 700 ºC, 1.8-target

(d) 650 ºC, 1.2-target

Figure 3.7: AFM surface images for BiMnO3 thin films grown with different targets at different substrate temperatures. As can be seen from Figure 3.7(c) and d the films grown with the 1.2-target show similar morpholo-gies as the films grown with the 1.8-target, but at lower temperatures. Also, in the thin film at 600 ºC for the 1.2-target there seems to be a lot more material in-between the BiMnO3 islands and it is more comparable to the thin film grown at 625 ºC for the 1.8-target (shown in Figure 3.7(a)) than to the thin film grown at 650 ºC (shown in Figure 3.6(b)). This other material is probably made up of the other phases observed in the XRD pattern for the film grown at 600 ºC with the 1.2-target. As mentioned in Paragraph 3.3.2 both SrRuO3 and KTaO3 have a smaller lattice mismatch with BiMnO3 than SrTiO3. The initial stress in the thin film is often of great influence on the surface mor-phology. To study the effect of the lattice mismatch we compare the surface of a thin film of BiMnO3 grown on SrRuO3 buffered SrTiO3, which has the smallest lattice mismatch with BiMnO3, and a thin film grown directly on SrTiO3. This comparison is shown in Figure 3.8. The displayed area is 5x5 µm with a height scale of 50 nm.

(a) SrRuO3//SrTiO3

(b) SrTiO3

Figure 3.8: Comparison between growth (a) on SrRuO3 buffered SrTiO3 and (b) directly on SrTiO3.

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From Figure 3.8 we can see that the film on SrRuO3 buffered SrTiO3 is more smooth than the film grown directly on SrTiO3. This is confirmed by the roughness calculation done on the areas in-between the precipitates. For the buffered film the RMS roughness value is 24 Å, while for the film directly on SrTiO3 it is 33 Å. Since the deposition conditions beside the substrate type were identical, the thickness is thought to be identical. Comparing the two images in Figure 3.8 it looks as if on the buffered film the islands grow together; they coalesce. This can be explained by the small lattice mismatch of SrRuO3 with BiMnO3. Because of this the BiMnO3 film is less stressed and the island can grow larger. Also, when two islands meet, there is less stress at the interface between the islands, reducing the number of grain boundaries. To further investigate the effect of different substrates, thick films (about three times as thick as the films shown in Figure 3.8) were grown on SrTiO3, on SrRuO3 buffered SrTiO3 and on KTaO3. AFM images of these films are shown in Figure 3.9. The area displayed is 10x10 µm. For both the film grown directly on SrTiO3 (a) and the film on KTaO3 (c) the height scale is 150 nm. For the thin film on SrRuO3 buffered SrTiO3 (b) it is 100 nm.

(a) SrTiO3 (b) SrRuO3//SrTiO3 (c) KTaO3

Figure 3.9: Comparison between thick films on different substrates. Again the film on the buffered substrate is the smoothest. Measurement of the roughness in-between the precipitates shows that the surface is almost as smooth as for the thin film shown in Figure 3.8(a): the RMS roughness value is 25 Å. The other two films are so rough that measuring the roughness in-between the precipitates is not possible. Another way to influence the surface roughness is to adjust the fluency on the target. Often an increase in the fluency is beneficial for the smoothness because of the prevention of droplets. However, for BiMnO3 decreasing the fluency seems to be beneficial as well. Figure 3.10 shows the AFM images for a fluency of 1.8, 2.2 and 2.5 J/cm2. The displayed area is 10x10 µm. The height scale is 100 nm.

(a) 1.8 J/cm2 (a) 2.2 J/cm2 (c) 2.5 J/cm2

Figure 3.10: Influence of the fluency on the surface morphology.

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The density of precipitates seems to be constant. It is unknown whether the precipitates are droplets or excess bismuth, so identifying the cause of this is difficult. In between the precipitates the roughness seems to increase with the fluency. For low fluency the overlap between the different grains is easy to see, while for increasing fluency the grains seem to become more distinct, showing less coalescence. In all three cases the XRD patterns only show the presence of SrTiO3 and BiMnO3, indicating that the individual grains are BiMnO3. It is most probable that the lower energy of the plasma species arriving at the substrate causes the grains to grow more smoothly. It is well-known that the best film growth occurs when the arriving plasma species are thermalized [Strikovski, 1998]. Thus a lower fluency would promote the growth of smooth films. Comparing thin and thick films of BiMnO3 grown at different fluencies can help to determine the ini-tial growth of BiMnO3. Figure 3.11 shows films grown at 1.8 and 2.2 J/cm2. The thick films are three times as thick as the thin films. The displayed area is 5x5 µm. The height scale is respectively 20, 100, 60 and 100 nm for Figures (a), (b), (c) and (d).

(a) 1.8 J/cm2, thin film

(c) 2.2 J/cm2, thin film

(b) 1.8 J/cm2, thick film

(d) 2.2 J/cm2, thick film

Figure 3.11: Thin and thick films for different fluencies, 1.8-target. The density of precipitates is about equal for the thin films and for the thick films. The difference lies in-between the precipitates. For the thin film grown at 1.8 J/cm2 the islands are large, while for a flu-ency of 2.2 J/cm2 the islands are small. Because of this there are more grain boundaries in the film grown at 2.2 J/cm2, which increases the overall roughness of the film. For the low fluency the plasma species arriving at the substrate have little energy to move around. Thus they do not have the freedom to find their optimum position in the BiMnO3 crystal lattice. In-stead they will adopt the lattice of the SrTiO3. This causes the formation of square and in-plane aligned BiMnO3 islands. This alignment makes it easy for different islands to grow together, thus forming large islands.

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For high fluency the plasma species are free to move about and more randomly oriented BiMnO3 is-lands are grown. These different orientations make it difficult for the islands to grow together, result-ing in many small islands. For the thin film at low fluency the cell parameter is b = 3.998 Å, close to the expected bulk value of 3.995 Å. This is due to the alignment between the substrate and the thin film, which causes the b-axis to point perpendicular to the substrate. For the thin film at high fluency the cell parameter is b = 3.974 Å, lower than the expected bulk value. The grains are more randomly oriented, so the axes are not ori-ented perpendicular to the substrate. Since the a- and c-axis cell parameters of BiMnO3 are smaller than the b-axis a lower measured cell parameter is expected. For thicker films the crystallographic structure of the films has the chance to relax, so the in-plane alignment is lost. For both high and low fluency the plasma species are free to find their optimum crystallographic positions. The cell parameters calculated from the XRD patterns are smaller than the expected bulk value, indicating that the grains are more randomly oriented for both the low and high fluency. The smoothness of the initial growth still influences the roughness of the thick films, but the difference becomes smaller. 3.4 BiMnO3 thin films In the previous paragraph the growth conditions for BiMnO3 thin films were studied. Now that single-phase BiMnO3 thin films can be grown, their properties can be studied. 3.4.1 Best settings The best settings for the growth of thin films of BiMnO3, as obtained in the previous paragraph, are shown in Table 3.7. Note that these settings are not the optimal settings, as the films can still be greatly improved, for example by increasing the fluency and pressure to - hopefully - reduce the num-ber of precipitates while still having a thermalized plasma.

Parameter Setting Fluency 1.8 J/cm2 Pulse frequency 4 Hz Target composition Bi1.8MnOx Gas composition O2 Gas pressure 0.05 mbar Target-to-substrate distance 45 mm Substrate type SrRuO3 buffered SrTiO3 Substrate temperature 650 ºC Annealing procedure ‘quench’ at 100 mbar

Table 3.7: Overview of the best obtained PLD parameters. 3.4.2 Structural properties Figure 3.12 on the next page shows the XRD pattern of the thin film of BiMnO3 grown with the best settings described above. The pattern of the actual multilayered film is in blue, green shows the pattern for BiMnO3 directly on SrTiO3 and the red pattern is SrRuO3 directly on SrTiO3. Because the diffrac-tion peaks lie close together, there is a lot of overlap between the different peaks and the determination of the crystallographic structure is complex. However, the peaks in front of the third and fourth SrTiO3 peaks can clearly be identified as BiMnO3 respectively SrRuO3.

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

1 0 ^4

1 0 ^2

1 0 ^0

1 0 ^2

1 0 ^4

1 0 ^6XR D p a t t e rn o f B iM n O 3 o n S rR u O 3 b u ffe re d S rT iO 3 , b e s t s e t t in g s

2 θ ( º )

inte

nsity

(-)

m u l t i l a y e rB iM n O 3S rR u O 3

Figure 3.12: XRD pattern of BiMnO3 thin film on SrRuO3 buffered SrTiO3, best settings.

The calculated cell parameters for the different materials perpendicular to the substrate are given in Table 3.8.

Cell parameter Observed (Å) Bulk value (Å) SrTiO3 c-axis 3.902 ± 0.003 3.905 SrRuO3 c-axis 3.950 ± 0.005 3.924 BiMnO3 b-axis 3.97 ± 0.02 3.995

Table 3.8: Calculated cell parameters for the thin film, best settings. The cell parameter for SrTiO3 is very close to the bulk value. As observed before the cell parameter for SrRuO3 is larger than in bulk, while the cell parameter for BiMnO3 is smaller. For the (60-6) peak of BiMnO3 (the peak just in front of the third SrTiO3 peak at 2θ = 71.12º) a rock-ing curve was measured to study the epitaxy of the BiMnO3 thin film. The full width at half maximum of the rocking curve was 0.25º. As a rule of thumb this indicates that the BiMnO3 film grows epitaxi-ally on the SrRuO3 buffer layer. Figure 3.13 shows several AFM scans of this film. In all pictures the height scale is 50 nm.

(a) 5x5 µm (b) 10x10 µm (c) 16.5x16.5 µm

Figure 3.13: AFM scan of BiMnO3 film for different scan areas, best settings. The overall roughness is in the same range as for the other films. This is probably due to the many precipitates. In-between these precipitates this film is the smoothest of all films.

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3.4.3 Electric and magnetic properties Besides the structure of the thin films of BiMnO3 the electric and magnetic properties are also of inter-est, as these are the active properties in most thin-film devices. Some measurements to determine these properties have been carried out on the best thin films of BiMnO3. Electrical properties We expect the BiMnO3 thin film to have a high resistivity, mainly because there is no conduction mechanism, such as hopping electrons, in pure BiMnO3. It is important to check this condition because a low resistivity can indicate the presence of a significant amount of Mn4+ ions. The Mn3+-O-Mn4+ complex allows electron-hopping conduction by the double exchange mechanism. The presence of Mn4+ ions would destroy the orbital ordering necessary for all the ferroelectric, ferromagnetic and multiferroic behaviour. A 4-probe resistance measurement carried out from room temperature down to 5 K showed no notice-able leakage current whatsoever, indicating that the BiMnO3 thin films are non-conducting in-plane. A film without a SrRuO3 buffer layer was used since SrRuO3 is a conducting oxide. Magnetic properties Magnetic measurements of the first and second derivative of the magnetic susceptibility were carried out on a thin film of BiMnO3. From these temperature profiles a magnetic critical temperature Tc,m of 93 K was found. The measurements above Tc,m were also fit to the Curie-Weiss law (see Figure D.4). From this a Curie temperature θc of 93 K was determined, confirming the Tc,m. These values are lower than the bulk value of 105 K, probably due to the stress in the unit cell of the thin film which distorts the interaction between the manganese magnetic moments. 3.5 Conclusions The use of epitaxial stabilization to grow thin films of BiMnO3 on lattice-matched substrates was suc-cessful. The substrate temperature is limited by the evaporation of bismuth, allowing only a narrow temperature window for growing crystalline films. From the XRD patterns no other crystallographic phases were discovered. However, judging from the AFM, EDX and accompanying SEM measure-ments other phases are still present in the film. As these extra phases do not show up in the XRD pat-tern they are either amorphous or only present in very small amounts. From the EDX measurements it can be concluded that the excess bismuth evaporates from the thin film. As there are other phases present besides the desired BiMnO3 phase the chemical composition of the thin films is not exactly known. When more optimal settings for the growth of BiMnO3 are found, this behaviour of bismuth could insure the correct chemical composition. The surface morphology seems to indicate that BiMnO3 exhibits island growth. It is proposed that for low fluency the thin films adopt the lattice orientation of the substrate. For higher fluency the plasma species have enough energy to find their optimum positions in the BiMnO3 crystal structure and more randomly oriented islands are grown. With increasing thickness this relaxation also occurs so the ini-tial orientation at low fluency is lost. A better lattice match with the substrate reduces the surface roughness by reducing the in-plane stress in the thin film. The large density of precipitates, due to both excess bismuth and other phases, still makes the surface very rough. The best PLD parameters from this research were used to fabricate a number of thin films of which the electrical and magnetic properties were measured. These films were isolating and have a Curie tem-perature of 93 K, which agrees with the properties from literature for bulk samples.

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53

Conclusions Understanding BiMnO3 Based on the individual mechanisms for the ferromagnetism and ferroelectricity in BiMnO3 a model for the multiferroic behaviour is proposed. In this model the interaction between the magnetic mo-ments of the individual manganese ions cause an enhancement of the electronegativity of the interme-diate oxygen ion. This in turn strengthens the bismuth-oxygen bond, leading to a stronger ferroelectric coupling. Using the two-dimensional Ising-model on a square lattice and introducing it to the ferroelectric case makes it possible to transform the qualitative model into a quantitative if semi-phenomenological model. This extended Ising-model has been evaluated using both mean-field approximation theory and Monte-Carlo Metropolis simulations. Both the MFA theory calculations and the Monte-Carlo simulations, after fitting of the model parame-ters, reproduce experimental data such as the distinct shape of the dielectric constant versus tempera-ture profile and the linear dependence between the change in the dielectric constant and the magnetiza-tion squared. The extended Ising-model has in this research been implemented for a ferromagnetic and ferroelectric material such as BiMnO3 or SeCuO3. Adaptation of the model to other types of materials, such as PbCrO3 or YMnO3 which are antiferromagnetic and ferroelectric, is certainly possible but a new im-plementation has to be developed. Growing thin films of BiMnO3 Using epitaxial stabilization thin films of BiMnO3 have been grown on different substrates. The inter-play between growing the correct crystallographic structure and the evaporation of bismuth results in a narrow temperature range for the optimal growth of BiMnO3. Here the self-limiting behaviour of bis-muth plays a role as well. Although at optimum temperature no other crystallographic phases are de-tected, chemical and structural analysis shows that the films are not purely BiMnO3 but also contain other compounds. Growth seems to occur in islands that for low fluencies tend to align with the substrate lattice but for higher fluencies form individually oriented grains. The fluency seems to determine whether the arriv-ing plasma species have the energy to move around and find their relaxed positions or whether they stick where they land and conform to the substrate lattice. The best thin films of BiMnO3 are grown on SrRuO3 buffered SrTiO3 substrates at low fluency. These films are isolating and have a Curie temperature of 93 K, which is close to the properties found in lit-erature for bulk samples. Application to other (artificial) materials The physical model developed for the multiferroic behaviour of BiMnO3 is probably only applicable to a small class of materials. It is based on the Mn3+-O-Mn3+ interaction. If you want to construct a multiferroic material by stacking layers of a magnetic perovskite with a ferroelectric non-manganese perovskite the model predicts, at most, some interaction at the interface. The physical model is perhaps more suited for the design of a bulk material with inherent multiferroism. The general concept of two coupled Ising-models describing the ferromagnetic and ferroelectric prop-erties of a material can be applied more generally to other materials. A different physical model for the

54

magnetoelectric coupling should then provide the form of the coupling, as it is in this research from considering the orbital ordering in BiMnO3. Of course BiMnO3 itself could also be applied in artificial materials. A possibly interesting combina-tion would be with a (non-manganese) magnetoresistive material. Here the interaction between mag-netic and electric properties would be an interesting subject. Another possibility would be the combi-nation with a purely magnetic material for the creation of tunable magnetic wave-guides. In these cases the quantitative extended Ising-model could be used to predict the behaviour of such an artificial material.

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Recommendations Extended Ising-model

- The Monte-Carlo simulations have been carried out using the simplified Equation 2.9. It should be possible to rewrite the simulation program to use the spin-specific Equation 2.8. First, this would allow us to check whether the simplification is justified by comparing the new results with the results obtained in this research. Second, the simulation could then also be used to simulate antiferromagnetic multiferroic materials such as YMnO3.

- It would be interesting to apply the extended Ising-model to other materials, for example the sister-compounds of BiMnO3 or the previous mentioned YMnO3, and to study the magneto-electric coupling in more detail.

- The extended Ising-model is at most a semi-phenomenological model; the model describes ionic interaction with model parameters that are derived from experimental results. It should be investigated whether it is possible to calculate these parameters ab-initio from atomic prop-erties.

- The simulations could be expanded to allow the coupling constants to change over the simula-tion lattice. That way an interface or even a multilayer of different materials could be simu-lated. This could be especially useful for the design of artificial multiferroic materials.

BiMnO3 thin films

- The exact orientation of the BiMnO3 unit cell on the substrate could not be determined. The low calculated b-axis cell parameter indicates that the unit cell is deformed more strongly by the lattice mismatch than for example SrRuO3. This deformation should be investigated as it can influence the orbital ordering and consequently all magnetoelectric properties of BiMnO3.

- The chemical composition of the thin films should also be further investigated. It is very likely that films grown with the 1.2-target can also have only BiMnO3 in its XRD pattern, if the cor-rect substrate temperature can be found. Then the target composition could be used to reduce the precipitate density on the films.

- Related to the chemical composition would be the study of the manganese ion valency in the films. X-ray photo-spectroscopy should be able to determine this. Since the manganese valency is closely related to the oxygen content of the thin film, it could be controlled by ad-justing the (partial) oxygen pressure during deposition or by changing the anneal procedure. The temperature-oxygen pressure phase diagram in Appendix D could offer a starting point for this investigation

- The fluency used to produce the best films is below the threshold for clean ablation from the target. This could be the source of some of the precipitates found on the thin films. By adjust-ing the overall deposition pressure, but keeping the partial oxygen pressure constant, the flu-ency can probably be increased while still having a thermalized plasma arriving at the sub-strate.

- The isolating nature of the best thin films show that it is possible to measure the dielectric constant of the thin film in a capacitance-measurement set-up. These films also exhibit a mag-netic phase transition at 93 K. It would be interesting to try and measure the dielectric constant versus temperature profile to see whether these thin films are multiferroic and show coupling between the magnetic and ferroelectric orders.

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Woo, H. et al. (2001). Correlations between the magnetic and structural properties of Ca-doped BiMnO3. Physical Review B 63. art.no. 134412.

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Dankwoord* The research presented in this report would not have been possible without the help and participation of a lot of people. I would especially like to thank Professor Dave Blank for this assignment and the discussions about the research (and for all the other things he has done for me during my study at the University of Twente). Guus Rijnders as my supervisor was often indispensable for discussion and showing me how research is really done. Arjen Molag was the person who taught me how to use all the machinery in the laboratory. Hans Hilgenkamp rounded out the Master committee and was a great help earlier on in my study. Related to the VICI project I would like to thank Arjen Janssens and Thang Duc Pham, whom it was a great pleasure to work with. I must not forget Frank Roesthuis for his many hours of work to keep the PLD system up and running, even with all the punishment we as users put it through. Gerrit van Hummel, Sybolt Harkema, Louise Vrielink, Gerard Willering and Mark Smithers were very helpful when it came to the analysis of my thin films. Professor Darrell Schlom from Penn State University, Pennsylvania, provided the BiMnO3 targets used in the experimental work. Victor Leca from the University of Tübingen, Germany, pro-vided the KTaO3 substrates. Of course I have to thank all the students who worked in ‘Ut Escherhofje’: there were many during my long stay there. The entire group of Inorganic Material Science must not be forgotten either. I have seen much of the work done there and learned a lot. And thanks for all the tea and cake we have shared in the last sixteen-and-a-half months. I would also like to thank my family and friends, here in Enschede, elsewhere in the Netherlands and abroad, for all the support, friendship and fun that I had during the six years of my study. I thank God for this beautiful world He created, which we all live and work in (in that order!). Gerwin Hassink

* I realize that the title of this page is the only Dutch word in the entire report, but I couldn’t find a decent English translation, so I stuck with it.

60

61

A Appendix: derivations Spin-Hamiltonian In principle we want to describe the energy of a magnetic moment in an magnetic field. From electro-magnetism there is a relation for that energy: B S= − ⋅Hm (Equation A.1) Now this magnetic induction can be due to two factors: the external applied magnetic induction and the magnetic induction induced by the other magnetic moments in the material. ( )other moments externalB B S= − + ⋅Hm (Equation A.2)

The magnetic induction due to the other magnetic moments could be described easily by a summation over all these moments, with an appropriate weighting factor:

(Equation A.3) j j externalj

J S B S

= − + ⋅ ∑Hm

Equation A.3 is for a single magnetic moment. If we want to know the energy for the entire system, we have to sum Hm over all moments.

(Equation A.4) j j external ii j

J S B S

= − + ⋅

∑ ∑Hm

Note that the weighting factor J can depend on both location i and location j of the system. Together with some simplification this yields the spin-Hamiltonian:

,ij i j i

i j i

J S S B S= − ⋅ − ⋅∑ ∑Hm (Equation 2.1)

From quantum mechanics a similar term for the spin-spin energy eigenvalues can be determined. See for example Ashcroft & Mermin, pp. 679-681 [Ashcroft, 1976]. Add the term to describe the interac-tion with the external magnetic induction and the result is again Equation 2.1. Onsager solution The derivation of the Onsager solution of the 2D Ising-model is highly complex. An outline is given by Tuckerman [Tuckermann, 1999]. The relation between the coupling constant Jm and the ferromag-netic critical temperature Tc,m follows from calculating the temperature for which the magnetization goes to zero. Analogy of Ising-model for ferromagnetism and ferroelectricity. Just as for a magnetic spin the energy of an electric dipole in an electric field can be given by an equa-tion of a form similar to that of Equation A.1. (Equation A.5) E P= − ⋅He

62

Following the same derivation as above for the spin-Hamiltonian we can write down a dipole-Hamiltonian with a form identical to Equation 2.1. By then applying the same simplifications as for the magnetic Ising-model we end up with an Ising-model describing the ferroelectric behaviour.

63

B Appendix: calculations Simplification of the general MFA relation Equation 2.11 states the general relation of the MFA theory. This can be simplified for the Ising-model with J = ½.

( , ) tanh m bm H Tt+ =

(Equation B.1)

Expanding b and t and simplifying yields:

( , ) tanhc

S

Tc MT m B

m H TTλ +

=

(Equation B.2)

For the Ising-model we can also easily calculate the Curie-constant C of the Curie-Weiss relation.

c

CT

χθ

=−

(Equation B.3)

This constant is given by:

2 2 ( 1

3B

b

Ng J JCk

µ +=

) (Equation B.4)

For an Ising-model we have a density of magnetic moments N of 1 because the distance between lat-tice points is chosen as the unit of length. The electronic g-factor g is equal to 2. The Bohr magneton µB, the ‘unit’ of the magnetic moment, can be chosen to be 1. The total magnetic moment J for the Ising-model is ½. Together this gives:

2 2 1 1

2 21 2 1 ( 1) 13 b b

Ck k

⋅ ⋅ ⋅ ⋅ += = (Equation B.5)

The MFA exchange coefficient is then given by:

cb c

T k TC

λ = = (Equation B.6)

With this Equation B.2 can be simplified to:

1

( , ) tanh b Sc k MT m Bm H T

T+

=

(Equation B.7)

Choosing MS to be 1 and with the relation between J and Tc in MFA theory we arrive at Equation 2.12.

( , ) tanh m

b

J m Hm H Tk T

µ +=

(Equation 2.12)

64

Derivation of the equation for the susceptibility The susceptibility is defined as:

1Hm

mH

χ∂∂

∂= =∂

(Equation B.8)

Solving Equation 2.12 for H and substitution into Equation B.8 yields Equation 2.14:

( )

2

2

11 m

mT m

χ −=

− − J

p

(Equation 2.14)

Origin of K-1 In Equation 2.17 we see that the dielectric constant K is related to the relative dielectric permittivity εr; the equivalent of the magnetic susceptibility. This relation can be easily seen from: 0D Eε= + (Equation B.9) The polarization is often a function of the electric field: (Equation B.10) ( )0 0 1r rD E Eε ε ε ε= + = + E The dielectric constant K is given by the term in between the brackets, while the right side of Equation 2.17 gives εr. Solving Equation 2.12 The easiest way of solving Equation 2.12 is probably by finding the zero points of

tanh 0m

b

J m H mk T

µ +− =

(Equation B.11)

Equation B.11 has either one or three solutions. See Figure B.1 for an example with J/kb = 105 K, µ/kb = 1 Km/A, H = 1 A/m en T = 80 K.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15solving Equation 2.12

magnetization (-)

tanh

((J*m

+ µ*H

)/T)-m

(-)

Figure B.1: Solving Equation 2.12.

65

It should be noted that for H=0 solving Equation B.11 yields identical results for J or –J. This indicates that MFA theory cannot calculate the antiferromagnetic case of the Ising-model. Which is expected, since for antiferromagnetism the magnetization is not a parameter for the magnetic order. Thus de ba-sic assumption of MFA theory, the replacement of the influence of all other spins by the average mag-netization, is not applicable to antiferromagnetic materials. Matlab M-file source code Matlab M-file implementation of the Onsager solution. % Onsager solution to 2D Ising model % [m,chi] = onsager(J,T) function [m,chi] = onsager(J,T) f=2/log(1+sqrt(2)); Tc=f*J; m=(1-sinh(2*J./T).^(-4)).^(1/8); for i=1:length(T) if T(i)>Tc m(i)=0; end; end; chi=abs(1-sinh(2*J./T).^(-4)).^(-7/4); if chi(1)==Inf chi(1)=2*chi(2); end for i=2:length(chi)-1 if chi(i)==Inf chi(i)=chi(i-1)+chi(i+1); end end if chi(length(chi))==Inf chi(length(chi))=2*chi(length(chi)-1); end Matlab M-file implementation of the mean-field approximation theory. % mean-field approximation to 2D Ising model using fzero % [m,chi] = mfa(J,H,T) function [m,chi]=mfa(J,H,T) % spin value S=1/2; % temperature (K) below zero excluded if ~(T(1)>0) disp('Temperature must not be below or at zero') return end warning off MATLAB:fzero:UndeterminedSyntax % calculation vs. temperature if length(T)>1 c=1; J=J.*ones(size(T)); H=abs(H).*ones(size(T)); for i=1:length(T)

66

if f(1e-6,S,J(i),H(i),T(i))>0 m(i)=fzero(@f,+1,[],S,J(i),H(i),T(i)); else m(i)=0; end end % calculation vs. field elseif length(H)>1 c=2; J=J.*ones(size(H)); T=T.*ones(size(H)); for i=1:length(H) zp=fzero(@f,0,[],S,J(i),H(i),T(i)); if zp*H(i)>0 m(1,i)=zp; m(2,i)=zp; else m(1,i)=fzero(@f,+1,[],S,J(i),H(i),T(i)); m(2,i)=fzero(@f,-1,[],S,J(i),H(i),T(i)); end end % error handling else disp('J, H & T are not compatible.') return end warning on MATLAB:fzero:UndeterminedSyntax % calculation of correlation for r=1:c chi(r,:)=(1-m(r,:).^2)./(T-(1-m(r,:).^2).*J); if chi(r,1)==Inf chi(r,1)=2*chi(r,2); end for i=2:length(chi(r,:))-1 if chi(r,i)==Inf chi(r,i)=chi(r,i-1)+chi(r,i+1); end end if chi(r,length(chi(r,:)))==Inf chi(r,length(chi(r,:)))=2*chi(r,length(chi(r,:))-1); end end % MFA function for zero finding function d=f(n,S,J,H,T) d=brillouin(S,3*S/(S+1)*(J.*n+H)/T)-n; % Brillouin function for MFA function y=brillouin(s,x) warning off MATLAB:divideByZero y=(2*s+1)/(2*s)*coth((2*s+1)/(2*s).*x)-1/(2*s)*coth(1/(2*s).*x); %Ashcroft & Mermin warning on MATLAB:divideByZero for i=1:length(x) if x(i)==0 y(i)=0; end end

67

Figure 2.1: Comparison of Onsager solution and MFA theory. % compare Onsager solution and MFA theory % Figure 2.1 % external parameters Tc=100.5; T=1:200; H0=0; H5=5; % model ratio kb*Tc/J fOns=2/log(1+sqrt(2)); fMFA=1; % Onsager solution JOns=Tc/fOns; [mOns,chiOns]=onsager(JOns,T); % MFA theory JMFA=Tc/fMFA; [m0,chi0]=mfa(JMFA,H0,T); [m5,chi5]=mfa(JMFA,H5,T); % plot figure(1); subplot(2,1,1); plot(T,mOns,T,m0,T,m5); axis([0 200 0 1]); legend('Onsager','MFA - H=0','MFA - H=5'); ylabel('magnetization (-)'); subplot(2,1,2); plot(T,chiOns/max(chiOns),T,chi0/max(chi0),T,chi5/max(chi5)); xlabel('T (K)'); ylabel('normalized susceptibility (-)'); Figure 2.2: Magnetic response of the extended Ising-model, calculations. Figure 2.3: Electric response of the extended Ising-model, calculation. Figure 2.4: Comparison of dielectric constant from calculation (a) and literature (b) [Kimura, 2003]. % mean field approximation calculation for BiMnO3 % temperature profile % Figure 2.2, 2.3 & 2.4 function mfa_temp % material parameters Tcm=105; Tce=450; Cme=170; % external parameters H0=0.1; H1=2; H2=4; T=1:600; % model ratio kb*Tc/J f=1;

68

% calculation Jm Jm=Tcm/f; % calculation magnetization [m0,chi0]=mfa(Jm,H0,T); [m1,chi1]=mfa(Jm,H1,T); [m2,chi2]=mfa(Jm,H2,T); % display magnetization figure(1) subplot(2,1,1) plot(T,m0,T,m1,T,m2) title('MFA temperature profile - m & chi vs. T, variable H') legend('H=~0','H= 2','H= 4') ylabel('magnetization (-)') subplot(2,1,2) plot(T,chi0,T,chi1,T,chi2) xlabel('temperature (K)') ylabel('susceptibility (-)') % calculation Je(m) Je=Tce/f-Cme/2; Je0=Je+Cme*(1+m0.^2)/2; Je1=Je+Cme*(1+m1.^2)/2; Je2=Je+Cme*(1+m2.^2)/2; % calculation polarization [p0,K0]=mfa(Je0,0.1,T); [p1,K1]=mfa(Je1,0.1,T); [p2,K2]=mfa(Je2,0.1,T); % display polarization figure(2) subplot(2,1,1) plot(T,p0,T,p1,T,p2) title('MFA temperature profile - p & K-1 vs. T, D=~0, variable H') legend('H=~0','H= 2','H= 4') ylabel('polarization (-)') subplot(2,1,2) plot(T,K0/max(K0),T,K1/max(K1),T,K2/max(K2)) xlabel('temperature (K)') ylabel('normalized dielectric constant (-)') % display detail dielectric constant t=T(70:130); k0=K0(70:130)/max(K0); k1=K1(70:130)/max(K1); k2=K2(70:130)/max(K2); figure(3) plot(t,k0,t,k1,t,k2) title('MFA temperature profile - detail K-1 vs. T, D=~0, variable H') legend('H=~0','H= 2','H= 4',2) xlabel('temperature (K)') ylabel('normalized dielectric constant (-)')

69

Figure 2.5: Magnetic response versus field of the extended Ising-model, calculation (a) and literature (b) [Kimura, 2003]. Figure 2.6: Electric response versus field of the extended Ising-model. Figure 2.7: Relative change in dielectric constant versus magnetization squared, calculation (a) and literature (b) [Kimura, 2003]. % mean field approximation calculation for BiMnO3 % magnetic field behaviour % Figure 2.5, 2.6 & 2.7 function mfa_field % material parameters Tcm=105; Tce=450; Cme=170; % external parameters T0=60; T1=100; T2=110; H=-4:0.05:4; % model ratio kb*Tc/J f=1; % calculation Jm Jm=Tcm/f; % calculation magnetization [m0,chi0]=smfa(Jm,H,T0); [m1,chi1]=smfa(Jm,H,T1); [m2,chi2]=smfa(Jm,H,T2); % display magnetization figure(1) subplot(2,1,1) plot(H,m0,H,m1,H,m2) title('MFA magnetic field behaviour - m & chi vs. H, variable T') ylabel('magnetization (-)') legend(' 60','100','110',2) subplot(2,1,2) plot(H,chi0,H,chi1,H,chi2) xlabel('applied magnetic field (A/m)') ylabel('susceptibility (-)') % calculation Je(m) Je=Tce/f-Cme/2; Je0=Je+Cme*(1+m0.^2)/2; Je1=Je+Cme*(1+m1.^2)/2; Je2=Je+Cme*(1+m2.^2)/2; % calculation polarization [p0,K0]=smfa(Je0,0.1*ones(size(H)),T0); [p1,K1]=smfa(Je1,0.1*ones(size(H)),T1); [p2,K2]=smfa(Je2,0.1*ones(size(H)),T2); % display polarization figure(2) subplot(2,1,1) plot(H,p0,H,p1,H,p2)

70

title('MFA magnetic field behaviour - p & K-1 vs. H, D=0.1, variable T') ylabel('polarization (-)') legend(' 60','100','110') subplot(2,1,2) plot(H,K0,H,K1,H,K2) xlabel('applied magnetic field (A/m)') ylabel('dielectric constant (-)') % calculate relative difference in dielectric constant dK0=(K0-K0(find(H==0)))./K0(find(H==0)); dK1=(K1-K1(find(H==0)))./K1(find(H==0)); dK2=(K2-K2(find(H==0)))./K2(find(H==0)); % display Je(m) and relative difference in dielectric constant figure(3) subplot(2,1,1) plot(H,Je0,H,Je1,H,Je2) title('MFA magnetic field behaviour - Je & dK vs. H, D=0.1, variable T') legend(' 60','100','110') ylabel('Je (a.u.)') subplot(2,1,2) plot(H,dK0,H,dK1,H,dK2) xlabel('H (a.u.)') ylabel('dK (-)') % fit -dK vs. m2 c1=polyfit(m1.^2,-dK1,1); c2=polyfit(m2.^2,-dK2,1); % display -dK vs. m2 and fits figure(4) plot(m0.^2-min(m0.^2),-dK0,m1.^2-min(m1.^2),-dK1,m2.^2,-dK2,m1.^2-min(m1.^2),c1(1).*m1.^2+c1(2),m2.^2,c2(1).*m2.^2+c2(2)) title('MFA #4 - -dK vs. m^2, D=0.1, variable T') xlabel('magnetization squared (-)') ylabel('relative change in dielectric constant (-)') legend(' 60','100','110') axis([0 0.18 0 0.3]) % give output of slopes and standard deviation c=c1(1) stdev=std(dK1+c1(1).*m1.^2+c1(2)) c=c2(1) stdev=std(dK2+c2(1).*m2.^2+c2(2)) % additional simplification of magnetic-dependent MFA output function [m,chi]=smfa(J,H,T) [m,chi]=mfa(J,H,T); m=[-sign(H(1))*m(2,1:(length(H)-1)/2) m(1,(length(H)-1)/2+1:length(H))]; chi=(1-m.^2)./(T-(1-m.^2).*J); Error estimation Matlab does all its computing with double-precision floating-point values of 64 bits. The relative accu-racy of these variables is of the order of 2-52 or about 2.22·10-16.

71

C Appendix: simulations Flowchart simulation program structure

no

no

no

yes

yes

yes

end

store sampling

sampling averaging vs. n

increase n

increase T

sampling averaging vs. T

increase i

data sampling

interval flipping

i<isample?

equilibrium flipping

initialize variables

T<Tmax?

n<noverall?

store parameters

input parameters

start

72

Random considerations The simulation requires a very large number of random numbers to be generated. We can calculate how many quite easily.

( ) 21 2 2 2end startoverall equilibrium measurement interval

step

t tn hotstart n n n sizet

−⋅ + ⋅ ⋅ + ⋅ + ⋅ ⋅ ⋅

(Equation C.1)

Using Equation C.1 and the parameter values from Table 2.5 we need at least 1011 random numbers. The problem with some programmed random number generators is that they are only pseudo-random; they are periodic (with a very long period, for sure). We want to be sure that the random numbers used in the simulation are non-repetitive and uncorrelated. So, to be sure that the random numbers are truly random, an attempt was made to determine a) whether the random number generator was periodic and b) if so, with which period. It is sufficient if the period is larger than the required number of random numbers. Testing showed that if the random number generator has a period, it is at least larger than 37·1011, which is sufficient for these simula-tions. Hot versus cold start The lattices can be initialised in two different ways. First, with a ‘hot’ start, the orientations of the vec-tors on the lattice can be chosen randomly. This situation would correspond to a high temperature where the spins and dipoles are randomly distributed. Second, with a ‘cold’ start, the orientations are determined by the implied ordering, whether antiparallel or parallel. In the former case the vectors are oriented antiparallel to their nearest neighbours, in the latter case they are oriented parallel to their nearest neighbours. This would correspond to a low temperature where the spins and dipoles are or-dered. Calculation parameters

0 100 200 300 400 500 6000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1m - equilibrium steps (hot start)

temperature (K)

mag

netiz

atio

n (-)

01e42e43e44e45e4

0 100 200 300 400 500 600

0

0.01

0.02

0.03

0.04

0.05

0.06chi - equilibrium steps (hot start)

temperature (K)

susc

eptib

ility

(-)

01e42e43e44e45e4

Figure C.1: The effect of the number of equilibrium steps, hot start.

0 100 200 300 400 500 6000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1m - equilibrium steps (hot & cold start)

temperature (K)

mag

netiz

atio

n (-)

hot startcold start

0 100 200 300 400 500 600

0

0.01

0.02

0.03

0.04

0.05

0.06chi - equilibrium steps (hot & cold start)

temperature (K)

susc

eptib

ility

(-)

hot startcold start

Figure C.2: Comparison between hot and cold started simulations.

73

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1m - lattice size

temperature (K)

mag

netiz

atio

n (-)

3 5101520

0 100 200 300 400 500 600

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09chi - lattice size

temperature (K)

susc

eptib

ility

(-)

3 5101520

Figure C.3: The influence of the lattice size on the simulation.

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1m - measurement & interval steps, i = 10

temperature (K)

mag

netiz

atio

n (-)

1 10 1001000

0 100 200 300 400 500 600

0

0.005

0.01

0.015

0.02

0.025

0.03chi - measurement & interval steps, i = 10

temperature (K)

susc

eptib

ility

(-)

1 10 1001000

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1m - measurement & interval steps, i = 100

temperature (K)

mag

netiz

atio

n (-)

1 10 1001000

0 100 200 300 400 500 600

0

0.005

0.01

0.015

0.02

0.025

0.03chi - measurement & interval steps, i = 100

temperature (K)

susc

eptib

ility

(-)

1 10 1001000

Figure C.4: The effect of different length equilibrium phase for interval 10 and 100.

C source code The extended Ising-model has been implemented in C using the Monte-Carlo Metropolis algorithm. The source code was compiled using the GCC 3.3.1 compiler running under Cygwin 1.5.7 and Win-dows 2000 Professional. No specific compiler options were given. The simulations were run on an AMD Athlon 1.4 GHz processor with 256 Mb internal memory. Temperature profile Metropolis Monte-Carlo simulation program. /* program name: tbmo.c author: Gerwin Hassink (g.w.j.hassink@student.utwente.nl) [01-09-1998 – 18-08-2004] Master student at Inorganic Materials Science group, faculty of Science and Technology, University of Twente, the Netherlands description: Ising model of magnetism and ferroelectricity in multiferroic BiMnO3 (temperature profile) history: [2004-02-20] first conception of the program; program outline & first implementation.

74

[2004-02-22] adjusted flip function after separate testing. flipping probability calculations checked out. [2004-02-23] applied prev/next speedup to flipping probability calculation. added hot or cold start parameter. pretty much wrote out the rest of the program. changed the calculation of the correlation function (giving chi and epsilon). [2004-02-24] changed default numbers as to get a simulation that finishes in one day. added overall loop indication during execution (y=2/(1+x)). changed definition of epsilon to be consistent (eps = 1+chi_e). [2004-02-26] changed equilibrium steps to 40000. removed srand's except first one. changed definition of epsilon back to old (without +1). [2004-03-08] new set of default parameters. changed chi/eps output to exponential format. [2004-03-11] changed sum parameters to double due to strange behaviour at lattice size 60. rearranged internal parameters into simulation function. [2004-03-18] adapted the program so now the size/strength of the spin/dipole can be varied. [2004-03-31] changed the electric flipping probability calculation from external field-like magnetization-dependent term to magnetization-dependent J. added checking of temperature step (> 0). [2004-04-01] changed equilibrium step to 20000 to speed up simulation. [2004-04-26] changed default parameters to more general settings. added output of t-loop step. [2004-04-29] changed effective dipole coupling from Je-cme*m to Je+cme*m*m. [2004-05-14] checked effective dipole coupling at Je+cme*(1+m*m)/2. */ //includes #include <stdio.h> #include <stdlib.h> #include <math.h> #include <time.h> //global calculation parameters int n_overall = 1; //number of overall repeats (> 0, <= 100) [-] int n_equilibrium = 20000; //number of flippings for thermal equilibrium [-] int n_measurement = 6000; //number of measurements for averaging (> 0) [-] int n_interval = 10; //number of lattice flippings between measurements (> 0) [-] int size = 100; //dipole/spin lattice size (> 2, <= 100) [-] int hotstart = 0; //hot or cold start of simulation (1 -> hot, 0 -> cold) [-] //global internal parameters int strength = 1; //dipole/spin strength (> 0) [-] float jm = 46; //spin-spin coupling constant (Tc,m = 105 K) [K] float mu = 1; //spin-external magnetic field coupling constant (roughly magnetic permeability) [Jm/A] float je = 63; //dipole-dipole coupling constant (Tc,e = 450 K) [K] float epsilon = 1; //dipole-external electric displacement coupling constant (roughly dielectric constant) [C/Jm2] float cme = 50; //dipole-magnetisation coupling constant [K] //global external parameters float t_start = 5; //start temperature (> 0) [K]

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float t_end = 300; //end temperature [K] float t_step = 5; //temperature step (> 0, maximum 120 steps) [K] float h = 0; //external magnetic field [A/m] float d = 0; //external electric displacement [C/m2] //Ising simulation void simulation(void) { //other parameters time_t clock; //time variable FILE *file; //file handle //lattices int s_lattice[100][100]; //magnetic spin lattice [-] int p_lattice[100][100]; //electric dipole lattice [-] //model parameters float m = 0; //magnetization [-] float chi = 0; //susceptibility [-] float p = 0; //polarization [-] float eps = 0; //dielectric constant [-] float t = 10; //current temperature [K] float fpm[9][3]; //magnetic flipping probabilities [-] float fpe[9][3]; //electric flipping probabilities [-] double sum = 0; //general sum parameter [-] double sum_s = 0; //sum of spins [-] double sum_s2 = 0; //sum of spins squared [-] double sum_p = 0; //sum of dipoles [-] double sum_p2 = 0; //sum of dipoles squared [-] double smap[2][120][101]; //magnetic properties storage array [-] double pmap[2][120][101]; //electric properties storage array [-] //loop parameters int i_overall = 0; //overall loop counter int i_measurement = 0; //measurement loop counter int i = 0; //general loop counter int prev[100]; //'previous-index' array int next[100]; //'next-index' array //function declarations //magnetic flipping probability float hm(int si, int ssum) { return exp(2*(-jm*ssum-mu*h)*si/t); } //electric flipping probability float he(int pi, int psum) { return exp(2*(-(je+cme*(1+m*m)/2)*psum-d/epsilon)*pi/t); } //initialize flipping probablity matrix void calculatefp(float (*fp)[9][3], float (*ham)(int x, int y)) { //loop parameter int i = 0; //calculate flipping probabilities for (i=-4; i<=4; i+=2) { (*fp)[i+4][0] = (*ham)(-1, i); (*fp)[i+4][2] = (*ham)(+1, i); } } //initialize lattice

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void newlattice(int (*lattice)[100][100], float J) { //loop parameters int x = 0; //lattice site coordinates int y = 0; float rnd = 0; //random value //coordinates loop start if (hotstart) { for (y=0; y<size; y++) { for (x=0; x<size; x++) { //random choice of initial spin (high temperature start) rnd = rand()/((double)RAND_MAX); if (rnd<0.5) (*lattice)[x][y] = -strength; else (*lattice)[x][y] = +strength; } } } else { for (y=0; y<size; y++) { for (x=0; x<size; x++) { //either F or AF ordering (zero temperature start) if ((J<0) && ((x+y)%2)) (*lattice)[x][y] = -strength; else (*lattice)[x][y] = +strength; } } } } //lattice flipping void flip(float (*fp)[9][3], int (*lattice)[100][100]) { //loop parameters int x = 0; //lattice site coordinates int y = 0; float rnd = 0; //random value int sum = 0; //spin/dipole sum //coordinates loop start for (y=0; y<size; y++) { for (x=0; x<size; x++) { //random flip sum = (*lattice)[next[x]][y]+(*lattice)[prev[x]][y]+(*lattice)[x][next[y]]+(*lattice)[x][prev[y]]; rnd = rand()/((double)RAND_MAX); if (rnd<(*fp)[sum/strength+4][(*lattice)[x][y]/strength+1]) (*lattice)[x][y] = -(*lattice)[x][y]; } } } //calculate the sum of spins/dipoles of a lattice int sumlattice(int (*lattice)[100][100]) { //loop parameters

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int x = 0; //lattice site coordinates int y = 0; int sum = 0; //spin/dipole sum //coordinates loop start for (y=0; y<size; y++) { for (x=0; x<size; x++) { sum += (*lattice)[x][y]; } } return abs(sum); } //simulation code //initialization time(&clock); srand(clock); //write parameters to file file = fopen("bmo.csv", "a"); fprintf(file, "BiMnO3 simulation program,,,\n"); fprintf(file, "\n"); fprintf(file, "time stamp,,,%s", ctime(&clock)); fprintf(file, "\n"); fprintf(file, "overall repeat,,,%d\n", n_overall); fprintf(file, "thermal equilibrium step,,,%d\n", n_equilibrium); fprintf(file, "number of measurements,,,%d\n", n_measurement); fprintf(file, "measurement step,,,%d\n", n_interval); fprintf(file, "lattice size,,,%d\n", size); if (hotstart) fprintf(file, "hot start,,,TRUE\n"); else fprintf(file, "hot start,,,FALSE\n"); fprintf(file, "\n"); fprintf(file, "strength,,,%d\n", strength); fprintf(file, "J_M,,,%f\n", jm); fprintf(file, "mu,,,%f\n", mu); fprintf(file, "J_E,,,%f\n", je); fprintf(file, "epsilon,,,%f\n", epsilon); fprintf(file, "C_ME,,,%f\n", cme); fprintf(file, "\n"); fprintf(file, "start temperature,,,%f\n", t_start); fprintf(file, "end temperature,,,%f\n", t_end); fprintf(file, "temperature step,,,%f\n", t_step); fprintf(file, "external magnetic field,,,%f\n", h); fprintf(file, "external electric displacement,,,%f\n", d); fprintf(file, "\n"); fprintf(file, "T,m,chi,p,K-1\n"); fclose(file); //define previous- and next-index arrays for (i=0; i<size; i++) { prev[i] = i-1; next[i] = i+1; } prev[0] = size-1; next[size-1] = 0; //overall loop start for (i_overall=0; i_overall<n_overall; i_overall++) { printf(" loop number %d\n", i_overall+1); //temperature loop start

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for (t=t_start; t<=t_end; t+=t_step) { printf("temperature step %6.1f\n", t); //initialize parameters sum = 0; sum_s = 0; sum_s2 = 0; sum_p = 0; sum_p2 = 0; //initialize per temperature step calculatefp(&fpm, hm); calculatefp(&fpe, he); //initialize lattices newlattice(&s_lattice, jm); newlattice(&p_lattice, je); //flip lattices to reach thermal equilibrium for (i=0; i<n_equilibrium; i++) { flip(&fpm, &s_lattice); m = sumlattice(&s_lattice)/((float)(size*size*strength)); calculatefp(&fpe, he); flip(&fpe, &p_lattice); } //measurement loop start for (i_measurement=0; i_measurement<n_measurement; i_measurement++) { //flip lattices between measurements for (i=0; i<n_interval; i++) { flip(&fpm, &s_lattice); m = sumlattice(&s_lattice)/((float)(size*size*strength)); calculatefp(&fpe, he); flip(&fpe, &p_lattice); } //take measurement sum = sumlattice(&s_lattice); sum_s += sum; sum_s2 += sum*sum; sum = sumlattice(&p_lattice); sum_p += sum; sum_p2 += sum*sum; //measurement loop end } //average measurement vs. measurements m = sum_s/((float)(size*size*n_measurement*strength)); chi = (sum_s2/((float)(size*size*n_measurement))-(size*size*strength*strength)*m*m)/t; p = sum_p/((float)(size*size*n_measurement*strength)); eps = (sum_p2/((float)(size*size*n_measurement))-(size*size*strength*strength)*p*p)/t; //'remember' measurements i = (int)((t-t_start)/t_step); smap[0][i][i_overall+1] = m; smap[1][i][i_overall+1] = chi; pmap[0][i][i_overall+1] = p; pmap[1][i][i_overall+1] = eps; //temperature loop end }

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//overall loop end } printf("\n"); //average measurements vs. T for (t=t_start; t<=t_end; t+=t_step) { //clear summation variables sum_s = 0; sum_s2 = 0; sum_p = 0; sum_p2 = 0; //sum over all temperature runs i = (int)((t-t_start)/t_step); for (i_overall=0; i_overall<n_overall; i_overall++) { sum_s += smap[0][i][i_overall+1]; sum_s2 += smap[1][i][i_overall+1]; sum_p += pmap[0][i][i_overall+1]; sum_p2 += pmap[1][i][i_overall+1]; } //calculate average values smap[0][i][0] = sum_s/((float)n_overall); smap[1][i][0] = sum_s2/((float)n_overall); pmap[0][i][0] = sum_p/((float)n_overall); pmap[1][i][0] = sum_p2/((float)n_overall); } //write measurements to file file = fopen("bmo.csv", "a"); for (t=t_start; t<=t_end; t+=t_step) { i = (int)((t-t_start)/t_step); fprintf(file, "%f,%f,%e,%f,%e\n", t, smap[0][i][0], smap[1][i][0], pmap[0][i][0], pmap[1][i][0]); } fprintf(file, "\n"); fclose(file); } int main(void) { //switch parameter char a = ' '; char b = ' '; char c = ' '; //main menu do { printf("---------- menu ----------\n"); printf(" calculation [c]\n"); printf(" internal [i]\n"); printf(" external [e]\n"); printf("--------------------------\n"); printf(" start [s]\n"); printf(" exit [x]\n"); printf("--------------------------\n"); printf("? > "); scanf("%c", &a); if (a!='x') printf("\n"); scanf("%c", &c);

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switch(a) { //input calculation parameters case 'c': do { printf("overall repeat? [%4d] > ", n_overall); scanf("%d", &n_overall); } while ((n_overall<1) || (n_overall>100)); printf("thermal equilibrium step? [%7d] > ", n_equilibrium); scanf("%d", &n_equilibrium); do { printf("number of measurements? [%5d] > ", n_measurement); scanf("%d", &n_measurement); } while (n_measurement<1); printf("interval between measurements? [%4d] > ", n_interval); scanf("%d", &n_interval); do { printf("lattice size? [%3d] > ", size); scanf("%d", &size); } while ((size<3) || (size>100)); do { printf("hot start? [%1d] > ", hotstart); scanf("%d", &hotstart); } while ((hotstart<0) || (hotstart>1)); printf("\n"); scanf("%c", &c); break; //input internal parameters case 'i': do { printf("strength? [%2d] > ", strength); scanf("%d", &strength); } while (strength<0); printf("J_M? [%7.2f] > ", jm); scanf("%f", &jm); printf("mu? [%7.2f] > ", mu); scanf("%f", &mu); printf("J_E? [%7.2f] > ", je); scanf("%f", &je); printf("epsilon? [%7.2f] > ", epsilon); scanf("%f", &epsilon); printf("C_ME? [%7.2f] > ", cme); scanf("%f", &cme); printf("\n"); scanf("%c", &c); break; //input external parameters case 'e': do { do { printf("start temperature? [%7.2f] > ", t_start); scanf("%f", &t_start); } while (t_start<=0);

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printf("end temperature? [%7.2f] > ", t_end); scanf("%f", &t_end); do { printf("temperature step? [%7.2f] > ", t_step); scanf("%f", &t_step); } while (t_step<=0); } while (((int)((t_end-t_start)/t_step)>=120) || ((int)((t_end-t_start)/t_step)<=0)); printf("external magnetic field? [%7.2f] > ", h); scanf("%f", &h); printf("external electric displacement? [%7.2f] > ", d); scanf("%f", &d); printf("\n"); scanf("%c", &c); break; //parameter overview and simulation case 's': printf("---------------------------------------------\n"); printf(" overall equi. measure intv. size hot\n"); printf("calculation: %4d %7d %5d %4d %3d %1d\n", n_overall, n_equilibrium, n_measurement, n_interval, size, hotstart); printf("\n"); printf(" strength J_M mu J_E epsilon C_ME\n"); printf("internal: %2d %7.2f %7.2f %7.2f %7.2f %7.2f\n", strength, jm, mu, je, epsilon, cme); printf("\n"); printf(" T_start T_end T_step H D\n"); printf("external: %7.2f %7.2f %7.2f %7.2f %7.2f\n", t_start, t_end, t_step, h, d); printf("\n"); printf("ok[o] or cancel[c]? > "); scanf("%c", &b); if (b=='o') { printf("\n"); simulation(); } else printf("\n"); scanf("%c", &c); break; //program end case 'x': break; //'error' handling default: break; } } while(a != 'x'); }

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Magnetic field profile Metropolis Monte-Carlo simulation program. /* program name: hbmo.c author: Gerwin Hassink (g.w.j.hassink@student.utwente.nl) [01-09-1998 – 18-08-2004] Master student at Inorganic Materials Science group, faculty of Science and Technology, University of Twente, the Netherlands description: Ising model of magnetism and ferroelectricity in multiferroic BiMnO3 (hysteres loop simulation) history: [2004-02-20] first conception of the program; program outline & first implementation. [2004-02-22] adjusted flip function after separate testing. flipping probability calculations checked out. [2004-02-23] applied prev/next speedup to flipping probability calculation. added hot or cold start parameter. pretty much wrote out the rest of the program. changed the calculation of the correlation function (giving chi and epsilon). [2004-02-24] changed default numbers as to get a simulation that finishes in one day. added overall loop indication during execution (y=2/(1+x)). changed definition of epsilon to be consistent (eps = 1+chi_e). [2004-02-26] changed equilibrium steps to 40000. removed srand's except first one. changed definition of epsilon back to old (without +1). [2004-03-08] new set of default parameters. changed chi/eps output to exponential format. [2004-03-11] changed sum parameters to double due to strange behaviour at lattice size 60. rearranged internal parameters into simulation function. [2004-03-18] adapted the program so now the size/strength of the spin/dipole can be varied. [2004-03-31] changed the electric flipping probability calculation from external field-like magnetization-dependent term to magnetization-dependent J. added checking of temperature step (> 0). [2004-04-01] changed equilibrium step to 20000 to speed up simulation. [2004-04-08] changed simulation from K vs. T to M vs. H. [2004-04-22] because of for-loop using double counter the stop criterium had to be changed from <=h_max to <h_max+h_step/2. [2004-04-26] put newlattices inside h-loop. changed default parameters to more general settings. added output of h-loop step. [2004-04-29] changed effective dipole coupling from Je-cme*m to Je+cme*m*m. [2004-05-14] checked effective dipole coupling at Je+cme*(1+m*m)/2. */ //includes #include <stdio.h> #include <stdlib.h> #include <math.h> #include <time.h> //global calculation parameters

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int n_overall = 1; //number of overall repeats (> 0, <= 100) [-] int n_equilibrium = 20000; //number of flippings for thermal equilibrium [-] int n_measurement = 6000; //number of measurements for averaging (> 0) [-] int n_interval = 10; //number of lattice flippings between measurements (> 0) [-] int size = 100; //dipole/spin lattice size (> 2, <= 100) [-] int hotstart = 1; //hot or cold start of simulation (1 -> hot, 0 -> cold) [-] //global internal parameters int strength = 1; //dipole/spin strength (> 0) [-] float jm = 46; //spin-spin coupling constant (Tc,m = 105 K) [K] float mu = 1; //spin-external magnetic field coupling constant (roughly magnetic permeability) [Jm/A] float je = 63; //dipole-dipole coupling constant (Tc,e = 200 K) [K] float epsilon = 1; //dipole-external electric displacement coupling constant (roughly dielectric constant) [C/Jm2] float cme = 50; //dipole-magnetisation coupling constant [K] //global external parameters float t = 100; //simulation temperature (> 0) [K] float h_max = 5; //maximum magnetic field strength (> 0) [A/m] float h_step = 0.2; //magnetic field step (> 0, maximum 120 steps) [A/m] float d = 0; //external electric displacement [C/m2] //Ising simulation void simulation(void) { //other parameters time_t clock; //time variable FILE *file; //file handle //lattices int s_lattice[100][100]; //magnetic spin lattice [-] int p_lattice[100][100]; //electric dipole lattice [-] //model parameters float m = 0; //magnetization [-] float chi = 0; //susceptibility [-] float p = 0; //polarization [-] float eps = 0; //dielectric constant [-] float h = 0; //current magnetic field strength [A/m] float fpm[9][3]; //magnetic flipping probabilities [-] float fpe[9][3]; //electric flipping probabilities [-] double sum = 0; //general sum parameter [-] double sum_s = 0; //sum of spins [-] double sum_s2 = 0; //sum of spins squared [-] double sum_p = 0; //sum of dipoles [-] double sum_p2 = 0; //sum of dipoles squared [-] double smap[2][120][101]; //magnetic properties storage array [-] double pmap[2][120][101]; //electric properties storage array [-] //loop parameters int i_overall = 0; //overall loop counter int i_measurement = 0; //measurement loop counter int i = 0; //general loop counter int prev[100]; //'previous-index' array int next[100]; //'next-index' array //function declarations //magnetic flipping probability float hm(int si, int ssum) { return exp(2*(-jm*ssum-mu*h)*si/t); } //electric flipping probability float he(int pi, int psum)

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{ return exp(2*(-(je+cme*(1+m*m)/2)*psum-d/epsilon)*pi/t); } //initialize flipping probablity matrix void calculatefp(float (*fp)[9][3], float (*ham)(int x, int y)) { //loop parameter int i = 0; //calculate flipping probabilities for (i=-4; i<=4; i+=2) { (*fp)[i+4][0] = (*ham)(-1, i); (*fp)[i+4][2] = (*ham)(+1, i); } } //initialize lattice void newlattice(int (*lattice)[100][100], float J) { //loop parameters int x = 0; //lattice site coordinates int y = 0; float rnd = 0; //random value //coordinates loop start if (hotstart) { for (y=0; y<size; y++) { for (x=0; x<size; x++) { //random choice of initial spin (high temperature start) rnd = rand()/((double)RAND_MAX); if (rnd<0.5) (*lattice)[x][y] = -strength; else (*lattice)[x][y] = +strength; } } } else { for (y=0; y<size; y++) { for (x=0; x<size; x++) { //either F or AF ordering (zero temperature start) if ((J<0) && ((x+y)%2)) (*lattice)[x][y] = -strength; else (*lattice)[x][y] = +strength; } } } } //lattice flipping void flip(float (*fp)[9][3], int (*lattice)[100][100]) { //loop parameters int x = 0; //lattice site coordinates int y = 0; float rnd = 0; //random value int sum = 0; //spin/dipole sum //coordinates loop start for (y=0; y<size; y++)

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{ for (x=0; x<size; x++) { //random flip sum = (*lattice)[next[x]][y]+(*lattice)[prev[x]][y]+(*lattice)[x][next[y]]+(*lattice)[x][prev[y]]; rnd = rand()/((double)RAND_MAX); if (rnd<(*fp)[sum/strength+4][(*lattice)[x][y]/strength+1]) (*lattice)[x][y] = -(*lattice)[x][y]; } } } //calculate the sum of spins/dipoles of a lattice int sumlattice(int (*lattice)[100][100]) { //loop parameters int x = 0; //lattice site coordinates int y = 0; int sum = 0; //spin/dipole sum //coordinates loop start for (y=0; y<size; y++) { for (x=0; x<size; x++) { sum += (*lattice)[x][y]; } } return abs(sum); } //simulation code //initialization time(&clock); srand(clock); //write parameters to file file = fopen("bmo.csv", "a"); fprintf(file, "BiMnO3 simulation program,,,\n"); fprintf(file, "\n"); fprintf(file, "time stamp,,,%s", ctime(&clock)); fprintf(file, "\n"); fprintf(file, "overall repeat,,,%d\n", n_overall); fprintf(file, "thermal equilibrium step,,,%d\n", n_equilibrium); fprintf(file, "number of measurements,,,%d\n", n_measurement); fprintf(file, "measurement step,,,%d\n", n_interval); fprintf(file, "lattice size,,,%d\n", size); if (hotstart) fprintf(file, "hot start,,,TRUE\n"); else fprintf(file, "hot start,,,FALSE\n"); fprintf(file, "\n"); fprintf(file, "strength,,,%d\n", strength); fprintf(file, "J_M,,,%f\n", jm); fprintf(file, "mu,,,%f\n", mu); fprintf(file, "J_E,,,%f\n", je); fprintf(file, "epsilon,,,%f\n", epsilon); fprintf(file, "C_ME,,,%f\n", cme); fprintf(file, "\n"); fprintf(file, "simulation temperature,,,%f\n", t); fprintf(file, "maximum magnetic field,,,%f\n", h_max); fprintf(file, "magnetic field step,,,%f\n", h_step); fprintf(file, "external electric displacement,,,%f\n", d); fprintf(file, "\n"); fprintf(file, "\n"); fprintf(file, "H,m,chi,p,K-1\n");

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fclose(file); //define previous- and next-index arrays for (i=0; i<size; i++) { prev[i] = i-1; next[i] = i+1; } prev[0] = size-1; next[size-1] = 0; //overall loop start for (i_overall=0; i_overall<n_overall; i_overall++) { printf(" loop number %d\n", i_overall+1); //temperature loop start for (h=-h_max; h<h_max+h_step/2; h+=h_step) { printf("magnetic field step %6.2f\n", h); //initialize parameters sum = 0; sum_s = 0; sum_s2 = 0; sum_p = 0; sum_p2 = 0; //initialize lattices newlattice(&s_lattice, jm); newlattice(&p_lattice, je); //initialize per temperature step calculatefp(&fpm, hm); calculatefp(&fpe, he); //flip lattices to reach thermal equilibrium for (i=0; i<n_equilibrium; i++) { flip(&fpm, &s_lattice); m = sumlattice(&s_lattice)/((float)(size*size*strength)); calculatefp(&fpe, he); flip(&fpe, &p_lattice); } //measurement loop start for (i_measurement=0; i_measurement<n_measurement; i_measurement++) { //flip lattices between measurements for (i=0; i<n_interval; i++) { flip(&fpm, &s_lattice); m = sumlattice(&s_lattice)/((float)(size*size*strength)); calculatefp(&fpe, he); flip(&fpe, &p_lattice); } //take measurement sum = sumlattice(&s_lattice); sum_s += sum; sum_s2 += sum*sum; sum = sumlattice(&p_lattice); sum_p += sum; sum_p2 += sum*sum; //measurement loop end } //average measurement vs. measurements

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m = sum_s/((float)(size*size*n_measurement*strength)); chi = (sum_s2/((float)(size*size*n_measurement))-(size*size*strength*strength)*m*m)/t; p = sum_p/((float)(size*size*n_measurement*strength)); eps = (sum_p2/((float)(size*size*n_measurement))-(size*size*strength*strength)*p*p)/t; //'remember' measurements i = (int)((h+h_max)/h_step); smap[0][i][i_overall+1] = m; smap[1][i][i_overall+1] = chi; pmap[0][i][i_overall+1] = p; pmap[1][i][i_overall+1] = eps; //temperature loop end } //overall loop end } printf("\n"); //average measurements vs. T for (h=-h_max; h<h_max+h_step/2; h+=h_step) { //clear summation variables sum_s = 0; sum_s2 = 0; sum_p = 0; sum_p2 = 0; //sum over all temperature runs i = (int)((h+h_max)/h_step); for (i_overall=0; i_overall<n_overall; i_overall++) { sum_s += smap[0][i][i_overall+1]; sum_s2 += smap[1][i][i_overall+1]; sum_p += pmap[0][i][i_overall+1]; sum_p2 += pmap[1][i][i_overall+1]; } //calculate average values smap[0][i][0] = sum_s/((float)n_overall); smap[1][i][0] = sum_s2/((float)n_overall); pmap[0][i][0] = sum_p/((float)n_overall); pmap[1][i][0] = sum_p2/((float)n_overall); } //write measurements to file file = fopen("bmo.csv", "a"); for (h=-h_max; h<h_max+h_step/2; h+=h_step) { i = (int)((h+h_max)/h_step); fprintf(file, "%f,%f,%e,%f,%e\n", h, smap[0][i][0], smap[1][i][0], pmap[0][i][0], pmap[1][i][0]); } fprintf(file, "\n"); fclose(file); } int main(void) { //switch parameter char a = ' '; char b = ' '; char c = ' '; //main menu do

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{ printf("---------- menu ----------\n"); printf(" calculation [c]\n"); printf(" internal [i]\n"); printf(" external [e]\n"); printf("--------------------------\n"); printf(" start [s]\n"); printf(" exit [x]\n"); printf("--------------------------\n"); printf("? > "); scanf("%c", &a); if (a!='x') printf("\n"); scanf("%c", &c); switch(a) { //input calculation parameters case 'c': do { printf("overall repeat? [%4d] > ", n_overall); scanf("%d", &n_overall); } while ((n_overall<1) || (n_overall>100)); printf("thermal equilibrium step? [%7d] > ", n_equilibrium); scanf("%d", &n_equilibrium); do { printf("number of measurements? [%5d] > ", n_measurement); scanf("%d", &n_measurement); } while (n_measurement<1); printf("interval between measurements? [%4d] > ", n_interval); scanf("%d", &n_interval); do { printf("lattice size? [%3d] > ", size); scanf("%d", &size); } while ((size<3) || (size>100)); do { printf("hot start? [%1d] > ", hotstart); scanf("%d", &hotstart); } while ((hotstart<0) || (hotstart>1)); printf("\n"); scanf("%c", &c); break; //input internal parameters case 'i': do { printf("strength? [%2d] > ", strength); scanf("%d", &strength); } while (strength<0); printf("J_M? [%7.2f] > ", jm); scanf("%f", &jm); printf("mu? [%7.2f] > ", mu); scanf("%f", &mu); printf("J_E? [%7.2f] > ", je); scanf("%f", &je); printf("epsilon? [%7.2f] > ", epsilon); scanf("%f", &epsilon); printf("C_ME? [%7.2f] > ", cme);

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scanf("%f", &cme); printf("\n"); scanf("%c", &c); break; //input external parameters case 'e': do { printf("simulation temperature? [%7.2f] > ", t); scanf("%f", &t); } while (t<=0); do { do { printf("maximum magnetic field? [%7.2f] > ", h_max); scanf("%f", &h_max); } while (h_max<=0); do { printf("magnetic field step? [%7.2f] > ", h_step); scanf("%f", &h_step); } while (h_step<=0); } while ((int)(2*h_max/h_step+1)>120); printf("external electric displacement? [%7.2f] > ", d); scanf("%f", &d); printf("\n"); scanf("%c", &c); break; //parameter overview and simulation case 's': printf("---------------------------------------------\n"); printf(" overall equi. measure intv. size hot\n"); printf("calculation: %4d %7d %5d %4d %3d %1d\n", n_overall, n_equilibrium, n_measurement, n_interval, size, hotstart); printf("\n"); printf(" strength J_M mu J_E epsilon C_ME\n"); printf("internal: %2d %7.2f %7.2f %7.2f %7.2f %7.2f\n", strength, jm, mu, je, epsilon, cme); printf("\n"); printf(" T_sim H_max H_step D\n"); printf("external: %7.2f %7.2f %7.2f %7.2f\n", t, h_max, h_step, d); printf("\n"); printf("ok[o] or cancel[c]? > "); scanf("%c", &b); if (b=='o') { printf("\n"); simulation(); } else printf("\n"); scanf("%c", &c); break; //program end case 'x': break; //'error' handling default:

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break; } } while(a != 'x'); } Error estimation Instead of only calculating the desired quantities at the end of the sampling phase, it is also possible to divide the total sampling phase into smaller blocks and calculate the desired quantities for each of these blocks. The advantage of this approach is that besides the desired output, you can now also cal-culate the standard deviation of those values. This gives you an estimate for the error in your simula-tion [Gonsalves, 2002]. Figure C.5 below shows examples of these error estimations for the temperature profile with the input parameters as given in Table 2.4. The block size is 100 samplings, with overall 60 blocks for the cal-culation of the standard deviation.

0 50 100 150 200 250 300-0.2

0

0.2

0.4

0.6

0.8

1

simulation temperature profile with error bars

mag

netiz

atio

n (-)

0 50 100 150 200 250 300

0

0.5

1

1.5

2

temperature (K)

susc

eptib

ility

(-)

H=0H=2

0 50 100 150 200 250 300-0.2

0

0.2

0.4

0.6

0.8

1

simulation temperature profile with error bars

pola

rizat

ion

(-)

0 50 100 150 200 250 300-0.2

0

0.2

0.4

0.6

0.8

temperature (K)

diel

ectri

c co

nsta

nt (-

)

H=0H=2

(a) (b)

Figure C.5: Temperature profile with error estimates, (a) magnetic response and (b) electric response. As can be seen from Figure C.5 the error in the simulation is relatively small except at the critical temperature. At high temperatures the orientation of the vectors is always random, averaging to zero for each sampling. At low temperatures the orientation is fixed by the interaction, averaging to about one. But at the critical temperature you can have either situation, ordered or random, and thus the spread of calculated values is larger than for higher or lower temperatures. Figure C.6 shows the error in the dielectric constant around the magnetic critical temperature. As can be seen from this figure the error is small enough to clearly show the bend in the zero-field graph.

70 80 90 100 110 120 130-0.5

0

0.5

1

1.5

2

2.5

3x 10

-4 simulation temperature profile with error bars

temperature (K)

diel

ectri

c co

nsta

nt (-

)

H=0, detailH=0, overallH=2, detailH=2, overall

Figure C.6: Detail of dielectric constant temperature profile around magnetic critical temperature with error indication.

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D Appendix: experiments Equipment Laser: Lambda Physik LPX200 (KrF) PLD setup: custom build, Twente Solid State Technology RHEED: Oxford Applied Research EGC04 (power supply) STAIB Instrumente EK300R1 (electron gun) k-Space Associates kSA400 4.11 (capture software) DataTranslation DT3155 (frame grabber) Atomic force microscope: Digital Instruments NanoScope IV X-ray diffractometer: Enraf-Nonius Diffractis 586 Scanning electron microscope: Jeol JSM5600 Optical path & fluency calculation

⊗laser

window target

455 (measured value)

1279±10

150 1355

lens rail A

long rail B

short rail C

+

1950

0

289±

1

470 730

Figure D.1: Optical path of PLD setup.

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Constants Value Error offset A 455 mm offset B 1994 10 mm offset C 4233 11 mm zero A 470 mm zero B 0 mm max C 1355 mm loss at lens 0,09 - loss at window 0,15 0,05 - focal length 447 3 mm Input energy at mask 16,0 0,1 mJ mask area 14,85 0,08 mm2 lens position on A 544 0,5 mm mask position on B 1758 0,5 mm mask position on C mm Output fluency 2,2 0,8 J/cm2 =(1-loss at window)*energy at lens/spotsize Temporary image length 529 1 mm =lens position on A-zero A+offset A object length 3223 11 mm =mask position on B-zero B-image length+offset B

OR =mask position on C-max C+image length-offset C

magnification 0,164 0,001 - =image length/object length energy at lens 14,6 0,1 mJ =(1-loss at lens)*energy at mask spot size 0,57 0,01 mm2 =mask area*(magnification)^2*square root of 2

Table D.1: Fluency calculation sheet for PLD setup. The above table is a representation of the Excel spreadsheet that was used to calculate the fluency and related parameters. Some lens and mask positions that give a focussed laser spot on the target are given in Table D.2.

Lens 544 mm (lens rail A) 542 mm (lens rail A) Mask 1758 mm (long rail B) 1880 mm (long rail B)

Table D.2: Focussed lens and mask positions. The loss at the lens represents all losses between the mask and the window, which is mainly the loss when the laser light passes through the lens. This value was copied from other users of the PLD set-up. The loss at the window represents the loss when the laser light passes through the window of the vac-uum chamber. Because the vacuum side of the window is subject to the deposition process as well, during use it is slowly covered by a thin layer of material which (sometimes greatly) reduces its transmittance. The value for this loss was guesstimated on basis of several measurements of the trans-mittance after normal use of the system. As the transmittance was often found to be (much) lower than 90 %, the value of 85 % transmittance was chosen. The factor of square root of two in the calculation of the spot size is due to the fact that the target sur-face is at an angle of 45º to the incident laser beam. The image of the mask is then projected onto a rotated surface, resulting in a larger area being illuminated.

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Temperature-pressure phase diagram for manganese oxides

Figure D.2: Temperature-pressure phase diagram for manganese oxides (courtesy of Prof. Schlom, Penn State University, Penn-sylvania).

The light red arrow indicates the annealing process used by Moreira dos Santos [2003]. During the annealing they cross over from the Mn3+ into the Mn4+ state at a temperature of about 490 ºC. The gray arrow represents the annealing as described in chapter 3. The Mn3+/Mn4+ border is crossed at about 420 ºC. Cell parameter calculations The lattice spacing can easily be determined from the XRD pattern using Bragg’s law: ( )2 sind nθ λ= (Equation D.1) For SrTiO3 and BiMnO3 the reflections chosen to determine the cell parameters were respectively the (001) and (010) reflections. Here the lattice spacing corresponds directly to the c- and b-axis cell pa-rameter. Several orders of these reflections were used to calculate the cell parameters. These values were then averaged to give a single cell parameter per sample. Error estimates were calculated by de-termining the standard deviation. The main source for errors was the uncertainty in the positions of the BiMnO3 peaks. Especially the low-order peaks are often not much more than shoulders on the (strong) SrTiO3 peaks and an error in determining their position is not unlikely.

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Table D.3 gives an overview of the different thin films and their cell parameters. Only thin films that had crystalline BiMnO3 are taken into account here. Correcting the b-axis cell parameter of BiMnO3 using the measured and bulk c-axis cell parameter for SrTiO3 yielded the same average cell parameter, indicating again the quality of the XRD measurement.

Sample BiMnO3 crystal quality c-axis SrTiO3 (Å) b-axis BiMnO3 (Å) 0915-BMO1.8 o 3.912 3.977 0924-BMO1.8 ++ 3.900 3.993 1112-BMO1.2 -- 3.900 3.984 0122-BMO1.8 + 3.903 3.978 0305-BMO1.8 - 3.900 4.005 0319-BMO1.8a -- 3.911 4.000 0319-BMO1.8b + 3.898 3.984 0507-BMO1.8a +++ 3.903 3.974 0513-BMO1.2a - 3.901 3.985 0601-BMO1.8a ++++ 3.909 3.998 0601-BMO1.8b +++ 3.901 3.980 0608-BMO1.8a + 3.901 3.962 0608-BMO1.8b +++ 3.904 3.978 0621-SRO_BMO1.8 ++++ 3.902 3.985 0628-BMO1.8 +++ 3.904 3.973 0629-BMO1.8a +++ 3.902 3.982 0629-BMO1.8b +++ 3.901 3.979 0713-BMO1.8 ++++ 3.913 3.977 0719-SRO_BMO1.8 ? 3.902 3.968 0719-BMO1.2 +++ 3.898 3.973 0721-BMO1.8 +++ 3.906 3.976 average cell parameter 3.903 3.98 standard deviation 0.004 0.01

Table D.3: Cell parameters of SrTiO3 and BiMnO3. The BiMnO3 crystal quality was assigned based on the number of peaks and shoulders on substrate peaks in the XRD pattern.

Crystal quality Number of shoulders/peals -- 2 shoulders - 1 shoulder, 1 peak 0 2 peaks + 2 shoulders, 1 peak ++ 1 shoulder, 2 peaks +++ 1 shoulder, 3 peaks ++++ 4 peaks

Table D.4: Crystal quality assignment. The dependence of the cell parameter of BiMnO3 on the PLD parameters was investigated, but no clear dependencies were found. Plots of the cell parameter versus several PLD parameters are given in Figure D.3 on the next page. In every plot the top data corresponds to the b-axis cell parameter of BiMnO3 and the bottom data to the c-axis cell parameter of SrTiO3. The red lines represent the respec-tive bulk values.

95

1.5 2 2.5 33.85

3.9

3.95

4

4.05

fluency (J/cm2)

axis

(Å)

1 1.2 1.4 1.6 1.8 23.85

3.9

3.95

4

4.05

target composition Bi:Mn (-)

550 600 650 700 7503.85

3.9

3.95

4

4.05

substrate temperature (ºC)

axis

(Å)

0.05 0.1 0.15 0.23.85

3.9

3.95

4

4.05

substrate miscut (º)

0 1000 2000 30003.85

3.9

3.95

4

4.05

deposition time (s)

axis

(Å)

Figure D.3: Cell parameters as a function of PLD parameters.

Magnetic measurements Figure D.4(a) shows the measured magnetic susceptibility data for the best BiMnO3 thin films. Note that actually the first and second derivative versus temperature are measured. The green line indicates the Curie temperature of 93 K. A change in curvature can clearly be observed. Figure D.4(b) shows fitting of the susceptibility data just above the Curie temperature. From these fits the Curie tempera-tures of 92 K respectively 95 K for the first and second derivative, averaged to 93 K as well.

80 90 100 110 120 130 140-8

-6

-4

-2

0

x 10-4 χ' and χ'' versus temperature

χ'

80 90 100 110 120 130 140-3

-2

-1

0

1

2x 10

-5

temperature (K)

χ''

92 94 96 98 100 102 104 106 108 110 112-10000

-8000

-6000

-4000

-2000

0

1/χ'

1/χ' and 1/χ'' versus temperature

92 94 96 98 100 102 104 106 108 110 1120

0.5

1

1.5

2

2.5x 105

temperature (K)

1/χ'

'

2(a) (b)

Figure D.4: Magnetic susceptibility measurements for BiMnO3 thin films, (a) temperature profiles and (b) fitted curves.

96