Domain imaging of multiferroic K3Nb3B2O12 by transmission electron microscopy

(1)

NTNU Norwegian University of Science and Technology Faculty of Natural Sciences Department of Physics

Oskar Ryggetangen

Domain imaging of multiferroic

K ³ Nb ³ B ² O

¹²

by transmission electron microscopy

Master’s thesis in Applied Physics and Mathematics Supervisor: Antonius T. J. van Helvoort

Co-supervisor: Dennis Gerhard Meier June 2021

Master ’s thesis

(2)

(3)

Oskar Ryggetangen

Domain imaging of multiferroic

K ³ Nb ³ B ² O

¹²

by transmission electron microscopy

Master’s thesis in Applied Physics and Mathematics Supervisor: Antonius T. J. van Helvoort

Co-supervisor: Dennis Gerhard Meier June 2021

Norwegian University of Science and Technology Faculty of Natural Sciences

Department of Physics

(4)

(5)

Abstract

The imaging and characterization of domain structures in ferroic compounds is a pre- requisite for their applications in functional nanoscale devices. This thesis presents the results of the first transmission electron microscopy (TEM) study of ferroelectric–

ferroelastic K3Nb3B2O12(KNBO). The aim of of the work was the examination of its domain structure in detail, beyond what has been achieved by other methods so far.

Employing a wide array of TEM techniques to conduct high spatial resolution studies of the domain structure of KNBO requires preparation of pristine TEM specimens, and a study of their basic crystal structure. Finally, correlated microscopy studies must be carried out to identify and image domain walls by TEM.

Mechanical tripod polishing was found to be an excellent routine for the preparation of high-quality KNBO TEM specimens, and an optimal polishing scheme was established. Superb control over specimen orientation and site-selectivity was achieved when tripod polishing was accompanied with domain imaging by polarized light microscopy (PLM), ensuring the presence of interesting regions in the finished specimens.

Wedge-shaped KNBO specimens with out-of-planec-axes were successfully prepared.

The wedge specimens were of high quality, with electron transparent regions of sizes

∼1000 µm×10 µm which were stable under an electron beam. Thus, the viability of KNBO for TEM investigations was confirmed.

The single orthorhombic crystal phase of KNBO exhibited in the [001] projection a striking hexagonal symmetry, with nearly identical spacings d₄₀₀ and d₂₃₀. Further- more, a clear 3msymmetry along zone axis [001] was found through convergent-beam electron diffraction (CBED) and scanning precession electron diffraction experiments.

Off-zone CBED experiments depicted a symmetry in seeming agreement with the proposed orthorhombic structure of KNBO, where mirror planes along directions [100] and [010] were observed. Furthermore, pronounced cleavings along [100], [120] and [120]

were found, and a number of interesting contrast phenomena were observed along the thin edges of the KNBO specimens.

PLM imaging produced clearly discernable domain contrast in KNBO specimens of thicknesses down to 40 µm. In addition to being invaluable during specimen preparation, it was found that identifying regions of interest through PLM images prior to TEM inspection is a viable method for locating domain walls in the TEM. Domain contrast in the thinnest regions of finished wedge samples was obtained with piezore- sponse force microscopy (PFM) imaging, and this technique proved to be a promising method for further correlated studies of KNBO.

Although no conclusive evidence for planar defects or domain walls were found through the present TEM studies, a viable method for identifying regions of interest in the wedge samples was established. As such, a proof-of-concept of correlated PLM–PFM–

TEM studies is presented. Lastly, subtleties in the imaging of ferroelectric and, espe- cially, ferroelastic domain walls in KNBO were addressed, and a method for charac- terizing domain walls through acquisition of on-zone and off-zone CBED patterns is proposed.

(6)

Sammendrag

En forutsetning for anvendelser av ferroiske forbindelser i funksjonelle apparater på nanoskala er avbildning og karakterisering av deres domenestrukturer. Denne avhand- lingen presenterer resultatene av den første studien av ferroelektrisk–ferroelastisk K3Nb3B2O12 (KNBO), som er et mindre forstått multiferroisk system. Målet ved arbeidet var å undersøke materialets domenestruktur i større detalj enn hva som tid- ligere har blitt gjort. For å anvende et vidt spekter av TEM-teknikker for å granske KNBOs domenestruktur kreves det at TEM-prøver av høy kvalitet prepareres, og at disse prøvenes krystallstruktur blir studert rigorøst. Først når dette har blitt gjen- nomført kan korrelerte mikroskopistudier utføres for gjenkjenning ov avbildning av domenevegger ved TEM.

Mekanisk tripodpolering ble fastslått å være en fremragende metode for preparer- ing av KNBO-prøver til bruk i TEM, og en optimal poleringsprosedyre ble bestemt.

Utmerket kontroll over prøveorientering ble oppnådd der tripodpolering ble supplert med domeneavbildning gjennom polarisert lys-mikroskopi (PLM), hvilket gjorde det mulig å sikre interessante områder i ferdigpolerte prøver. Kileformede KNBO-prøver med c-akser som plannormaler og kanter normalt på observerte domenevegger ble la- get. Kileprøvene var av eksemplarisk kvalitet med elektrontransparente områder på størrelser av orden ∼ 1000 µm, stabile under elektronstråler. Altså ble KNBOs an- vendbarhet under TEM-eksperimenter bekreftet.

KNBOs ortorombiske enkrystallfase lot til å ha en slående heksagonal symmetri langs [001]-projeksjonen, med nær identiske planavstanderd₄₀₀ogd₂₃₀. Videre ble en tydelig 3m-symmetri langs soneakse [001] funnet gjennom konvergent-stråle elektrondiffraks- jon (CBED) og skannende presesjonselektrondiffraksjon. Av-sone CBED-eksperimenter viste symmetrier i overensstemmelse med den påståtte ortorombiske strukturen til KNBO, da antydninger til speilplan langs retninger [100] og [010] ble funnet. Tydelige kløyveplan ble også funnet, disse parallelt med retninger [100], [120] og [120].

PLM produserte tydelig domenekontrast i KNBO-prøver med tykkelser ned mot 40 µm.

I tillegg til å være en uvurderlig komplementær teknikk under prøvepreparering, gav forsøk på identifisering av domenevegger ved hjelp av PLM i forkant av TEM-studier lovende resultater. Domenekontrast i selv de tynneste områdene av ferdigpolerte prøver ble oppnådd gjennom piezoresponskraftmikroskopi (PFM), og PFM ble dermed fast- slått å være en lovende kandidat for videre korrelerte studier av KNBO.

Selv om ingen utvetydige tegn på plandefekter eller domenevegger ble funnet i dette arbeidet, ble en levedyktig metode for å lokalisere interessante områder i kileprøvene fastslått. Altså vil et “proof-of-concept” for korrelerte PLM–PFM–TEM-studier bli presentert. I tillegg ble utfordringer knyttet til avbildning av ferroelektriske og især ferroelastiske domenevegger påpekt, og en mulig prosedyre for å karakterisere domenevegger gjennom innhenting av av- og på-sone CBED-mønstre blir foreslått.

(7)

Preface

With this thesis, I conclude my M.Sc. degree in Applied Physics from the Department of Physics at the Norwegian University of Science and Technology (NTNU). The work has been conducted at the TEM Gemini Centre at NTNU, under the supervision of Prof. Antonius T.J. van Helvoort at the Department of Physics and co-supervision of Prof. Dennis Meier at the Department of Materials Science and Engineering. A small portion of the work was also carried out at NTNU NanoLab. The thesis is a con- tinuation of a specialization project which I concluded in the Fall of 2020. As such, a portion of Chapter 2 consists of reprints from earlier writings of mine. Acquisition and processing of the data presented in this work was conduced by me, unless otherwise specified.

I would like to extend my sincere thanks to Ton for his invaluable support and guid- ance. Thank you for the many insightful and enlightening discussions, your inspiring presence at the TEM lab, and for always being able to take time out of your schedule to offer your help. Bjørn Gunnar Soleim and Emil Christiansen also deserve a big thanks for the immaculate TEM-training i recieved, and for always showing up on short notice whenever help was needed. Thanky you also to Ivan, who acquired PFM data for me – the results were immensely useful. Lastly, thank you to all the people of the TEM group for welcoming me into your inclusive and inspiring research environment.

Oskar Ryggetangen Trondheim

June 25, 2021

(8)

Abbreviations

KNBO = K₃Nb₃B₂O₁₂

TEM = Transmission electron microscopy/Transmission electron microscope

HOLZ = Higher-order Laue zone

ZOLZ = Zeroth-order Laue zone

FEG = Field-emission gun

CCD = Charge-coupled device

SAED = Selected-area electron diffraction CBED = Convergent beam electron diffraction

SA = Selected-area

BF = Bright-field

OA = Objective aperture

DF = Dark-field

HRTEM = High-resolution transmission electron microscopy

HR = High-resolution

SED = Scanning electron diffraction

HAADF-STEM = High-angle annular dark-field scanning electron transmission microscopy S(P)ED = Scanning (precession) electron diffraction

DLF = Diamond lapping film

VDF = Virtual dark-field

VA = Virtual aperture

WKc = Weickenmeier-Kohl (phonon and core absorption)

PRDW = Peng-Ren-Dudarev-Whelan

(9)

1 Introduction

Following the rise in development of complex technical devices, the demand for novel functional materials has been steadily increasing [1, 2]. Today, innovation beyond Si based devices seems necessary to accommodate the growing need for efficient comput- ing and dense integrated circuits, and exotic two-dimensional systems have garnered much attention in this regard [3]. In particular, recent studies of the planar boundaries separating domains in ferroic materials have paved the way for a conceptually new field of inquiry; domain wall engineering [4–6].

Ferroic is a generic term used to describe compounds in which an order parameter may exist in more than one orientation, switchable by some external field (e.g. electric, magnetic). States of differently oriented order parameters are energetically degener- ate, and the compound may therefore divide into distinct homogeneous regions – or domains – separated by domain walls [4, 7]. Particularly important in the following thesis are ferroelectric and ferroelastic materials. Ferroelectric compounds exhibit a non-zero switchable polarization in the absence of an external field. The special cases where the polarization is perpendicular to the ferroelectric domain walls are of special interest. In these head-to-head or tail-to-tail domains, the domain walls display electrical conductance even in non-conducting compounds [8]. Ferroelastic materials, which constitute the largest species of ferroic compounds, may accommodate a nonzero strain tensor in the absence of any external stress, and have applications ranging from piezoelectric devices to mechanical switches [9]. Interestingly, some compounds exhibit both ferroelectric and ferroelastic ordering, the coupling of which leads to many interesting properties. These multiferroic materials, intensely studied over the last 15 years, present great opportunities for technological applications [10, 11].

Applications of novel multiferroic materials are predicated on a detailed understanding of their domain structure. Studying new candidate systems for application in domain wall engineering using atomic resolution imaging techniques is immensely advantage- ous, as this grants the possibility of studying interactions between domain walls and the lattice structure of the material. For the conventional ferroelectrics, like BaTiO₃ and PbTiO₃, transmission electron microscopy (TEM) has been crucial in determining their domain structure [12]. In this work, the first TEM study of the less studied, but potentially fundamentally interesting ferroelectric–ferroelastic K3Nb3B2O12 (KNBO) has been conducted. After its discovery in 1977 [13], KNBO has been studied to a very limited extent, using only relatively low-resolution imaging techniques, such as visible light microscopy, X-ray diffrcation and atomic force microscopy [14–21]. As such, many open questions pertaining to its structure and ferroic properties exists, and no clear consensus has been reached regarding the nature of this compound. Specifically, dis- agreements concerning the crystal structure and ferroic ordering of KNBO are present in the literature [14, 18, 19].

As TEM offers a superb spatial resolution and the opportunity of extracting informa- tion about lattice structure and crystallographic directions, this technique holds great promise in resolving some of the apparent conflicts pertaining to KNBO. As a stepping stone towards an exhaustive understanding of the properties of KNBO, the aim of the

(12)

following study has been to conduct a TEM based domain imaging of KNBO. As no such studies have been published in the past, this aim may be divided into three main goals. First, a suitable method for preparing high quality, electron transparent KNBO specimens must be established. The specimens need to be prepared in a manner which allows for domain imaging, and they should be well behaved in the TEM. Second, a structural characterization must be carried out on the prepared specimens. Here, it is necessary to map out the structure of the KNBO specimens with respect to crystallographic orientations and symmetries, as this is crucial for the eventual characterization of domain walls. Lastly, the ferroelectric or ferroelastic domains may be imaged in the TEM, correlated with data obtained from complementary lower-resolution imaging techniques to gain a richer picture of the domain structures in the prepared KNBO samples.

(13)

2 Theoretical background

The following chapter contains the theoretical background necessary for the reading of Chapters 4 and 5. Large portions of this chapter are re-prints from earlier work on KNBO [22], but some important additions are included. The chapter starts with a detailed description of three-dimensional crystal structures, introducing the many important aspects of crystallography. Following this, the theoretical background of electron diffraction is presented in Section 2.2, outlining both kinematical and dynamical diffraction theory. Section 2.3 will take the reader through the basic principles of TEM operation, and a selected number of operational modes will be presented. Fer- roelectricity and ferroelasticity will be described in Section 2.4, before a survey of the current literature on KNBO is given in Section 2.5.

2.1 Crystallography

2.1.1 A description of three-dimensional crystal structures

In this section, a description of the ideal crystal will be laid out. The equilibrium state of a solid compound made up of a large number of identical units tends to be a periodic arrangement of said units [23]. These states are referred to as crystalline, and in the ideal case, they are made up of an infinite number of units placed on a grid that is periodic in all directions. Formally, one may describe an ideal crystal in terms of a crystal latticeand abasis. These terms will be defined in the following section, starting with the crystal lattice.

A crystal lattice is a specific set of points, defined in such a way that any lattice point may be reached from any other through a lattice translation T, given by

T =ua₁+va₂+wa₃. (2.1)

Here, (u, v, w) are integers anda₁,a₂anda₃ are three non-parallel vectors that do not lie in the same plane. If an initial lattice point is placed at the origin, then any other point reached through T for some combination of integers (u, v, w) is also a lattice point. As such, Equation (2.1) generates a set of equivalent points invariant under any translation of the type T, referred to as a Bravais lattice. The Bravais lattice may be described in terms of its primitive axes a_i, (i = 1,2,3): Given a specific lattice, its primitive axes may be identified as three vectors that may be combined to form a translation T under which the lattice is invariant. The choice of primitive axes is not unique; the single constraint being that a linear combination of the axes must yield a lattice translation. This is illustrated in two dimensions in Figure 2.1, where two attempts are made at choosing primitive axes.

By varying the lengths of and angles between the primitive axes a_i, different types of Bravais lattices are generated. The Bravais lattices are best described in terms of theirunit cells, a unit cell being a smaller part of a lattice having the same symmetry as the entire lattice. Under repeated translations through a lattice vector T, the unit cells must be such that they fill all space without overlap. In a given Bravais lattice, several choices of unit cells are possible, and the cells may contain one or more lattice

(14)

(a) (b)

Figure 2.1: Illustration of primitive and non-primitive vectors. (a) All lattice points can be reached through a linear combination of a₁ and a₂, and these may therefore be chosen as primitive axes. (b) All lattice points cannot be reached through a linear combination of a⁰₁ and a⁰₂. Thus, they arenot primitive lattice vectors.

points. Unit cells containing only one lattice points are referred to as primitive unit cells, while those containing more than one lattice point are non-primitive. A simple choice of unit cell is the volume defined by the primitive axes ai [23, 24].

In three dimensions, there exist 14 Bravais lattices belonging to one of seven types of cells – referred to as crystal systems. The conventional choices of unit cells are not necessarily primitive, as will become apparent. In classifying the different crystal systems, the relevant parameters are the lengths of the crystal axes ai and the angles between them, as illustrated in Figure 2.2. The crystal systems and the associated parameters are listed in Table 2.1.

Table 2.1: The seven crystal systems and respective values for the unit cell parameters [24].

System Cell axes Cell angles

Triclinic a₁ 6=a₂, a₁ 6=a₃, a₂6=a₃ α6=β, α6=γ, β 6=γ Monoclinic a₁ 6=a₂, a₁ 6=a₃, a₂6=a₃ α=γ = ^π₂ 6=β Orthorhombic a₁ 6=a₂, a₁ 6=a₃, a₂6=a₃ α=β =γ = ^π₂ Tetragonal a₁ =a₂6=a₃ α=β =γ = ^π₂ Cubic a₁ =a₂=a₃ α=β =γ = ^π₂

Trigonal a₁ =a₂=a₃ α=β =γ (6= ^π₂

> π Hexagonal a₁ =a₂6=a₃ α=β= ^π₂, γ=π

(15)

Figure 2.2: Specification of the unit cell parameters necessary for describing a Bravais lattice.

In the orthorhombic system, there are four Bravais lattices. These are the primit- ive, base-centered, body-centered and the face-centered lattices, often denoted by P, C, I and F, respectively. Only one of these are primitive, as the C, I and F cells all contain more than one lattice point. The four orthorhombic unit cells are illustrated in Figure 2.3.

(a) (b) (c) (d)

Figure 2.3: The four standard unit cells in the orthorhombic crystal system. (a) Or- thorhombic primitive cell. (b) Orthorhombic base-centered cell. (c) Orthorhombic face- centered cell. (d) Orthorhombic body-centered cell.

All lattice points are occupied by identical units. The units may consist of one or more atoms, ions or molecules, and one such unit is referred to as the crystal basis. As such, a crystal is constructed by “placing” a basis unit at every lattice point. In mathematical terms, a crystal is obtained by convolving the basis unit with the Bravais lattice, as illustrated in Figure 2.4.

The positions of the various basis constituents may be described by the cell axes a_i. Relative to some lattice point, the position of basis constituent j is described by

r_j =x_ja₁+y_ja₂+z_ja₃, (2.2) where 0 ≥ x_j, y_j, z_j ≤ 1. Thus, through Equations (2.1) and (2.2), every basis constituent on any lattice site may be reached from any other.

Miller indices for crystal planes and directions

In the description of physical properties of crystalline structures, establishing an appropriate coordinate system is necessary. One way of obtaining this is by “fixing” to

(16)

Figure 2.4: Two-dimensional illustration of the construction of a crystal through a convolution between a basis and a Bravais lattice.

the crystal a coordinate system defined by its three crystal axes ai. This provides a reference frame in which crystal directions may be unambiguously defined. Crystal directions are described in terms of their Miller indicesh,kandl, and are denoted by [hkl]. The Miller indices of a crystal direction are obtained by finding a set of integers that have the same ratios as the components along the crystal axes of a vector in the same direction. The axes a₁,a₂ and a₃ then correspond to the directions [100], [010]

and [001], respectively. Some crystallographic directions are equivalent by symmetry, and a set of equivalent directions is denoted byhhkli.

Crystallographic planes that intercept certain sets of lattice points are also relevant for the treatment of crystal structures. These planes are also defined in relation to the crystal axes, and are denoted by (hkl). The indices are calculated by taking the reciprocals of the plane’s intercepts – in terms of a₁, a₂ and a₃.h,k and l are then three integers with the same ratios as these reciprocals. If a plane intercepts the axesa₁,a₂ and a₃ of a given lattice at 2a₁, 1a₂ and 4a₃, respectively, the reciprocals are ¹₂,1,¹₄. Thus, the plane is described by the indices (214). Negative intercepts are denoted by an overline over the corresponding index, for instance (hkl). Similarly to the crystal directions, some planes are symmetrically equivalent, and a family of equivalent planes is denoted by{hkl}. For illustration, Figure 2.5 shows a cubic unit cell where the (111) plane and the [111] direction are indicated.

Figure 2.5: A (111)-plane in a cubic system, intercepting the crystal axes ata₁=a₂ = a₃ = 1. The [111]-direction is indicated by a vector normal to the plane.

(17)

2.1.2 Point groups and space groups in crystallography In the following, crystal axes (a₁,a₂,a₃) will be denoted by (a,b,c).

The different crystal systems specified in Table 2.1 are best viewed as a set ofsymmetry systems. Starting with a completely general polygon, the crystal systems are obtained by imposing different constraints on the parameters (a, b, c) and (α, β, γ). This is analogous to starting with a system of no specific symmetry, and imposing different spatial transformations under which the system must remain invariant. Such transformations, which transform a specific system into itself, are referred to as symmetry operations. An elegant classification of crystallographic structures may be formulated in terms of these operations. As a preface to this, a language of finite symmetry groups must be developed.

A set G of elements gi, for which a rule of composition ? is defined, is said to be a group if all of the following criteria are fulfilled [25]:

1. If g₁ ∈ G and g₂ ∈ G, then g₁ ? g₂ = g₃ ∈ G. This is the property of closure, stating that the combination of any two elements inGmust give another element, also in G.

2. If g₁, g₂, g₃ ∈G, then (g₁? g₂)? g₃ =g₁?(g₂? g₃). This is called the associative property.

3. InG, there must exist an elementethat satisfiese ? g=g ? e=g for anyg∈G.

eis then referred to as the identity element.

4. For any element g∈G, there must exist an element ˜g, also inG, which satisfies g ? ˜g = ˜g ? g = e. In other words, every element in G must have an inverse element, which is also in G. The combination of an element with its inverse yields the identity element.

Sets of symmetry operations may satisfy these group axioms, and thus constitute groups – the rule of composition being the consecutive application of two operations.

Systems that are invariant under operations corresponding to elements of a symmetry group are said to belong to ortransform as that particular group.

A first step towards a group-theoretical classification of crystallographic structures is the introduction of crystallographic point groups. Consider an object which trans- forms as some group G. If the symmetry operations corresponding to the elements of G are such that they leave at least one point on the object fixed, then G is said to be a point group. Symmetries corresponding to elements of point groups are one of two types of operations: proper or improper rotations about specific axes [26]. A proper rotation is simply the rotation about an axis of some angle 2π/n. If an object is invariant under such a rotation, the axis of the rotation is called an n-fold axis.

The axis and the symmetry element corresponding to the rotation are denoted by n. Improper rotations – or rotoinversions – are the combination of a proper rotation and an inversion through a point. If an object is invariant under an n-fold rotoinversion, the rotation axis and the corresponding symmetry element are denoted by ¯n. Because the unit cells of crystal structures must fill all of space through regular repetition, the possible point symmetries under which they can transform are limited to the rotations 1, 2, 3, 4 and 6, and the rotoinversions ¯1, ¯2, ¯3, ¯4 and ¯6 [26].

(18)

The point symmetries of crystal structures may be combined in various ways to form 32 groups consistent with the group axioms. Thus, any crystal structure belongs to one of these crystallographic point groups – or crystal classes. As an example, a crystal belonging to the orthorhombic crystal system will belong to one of the following point groups: 222, mm2 or mmm. The notation here needs some deciphering: For the orthorhombic system, the first, second and third entry in the point group symbol correspond to symmetries of the axes a,b and c, respectively [26]. So, an orthorhombic crystal belonging to point group 222 will have three two-fold rotation axes, one along each of the crystal axes. Generally, m is used instead of ¯2 to denote a two-fold rotoinversion axis, as this is equivalent to the operation of mirroring through a plane perpendicular to that axis. Crystals belonging to point group mm2 will then have two mirror planes perpendicular to a and band one two-fold rotation axis alongc. The concept of crystallographic point groups may be expanded to reflect the transla- tional invariance of crystal lattices. Symmetry elements corresponding to screw-axes and glide planesare composed through the combination of translations with rotations and rotoinversions, respectively. A screw-axis symmetry operation contains a rotation through an angle 2π/n together with a translation along the axis of the rotation. The translation is such that after n rotations, each lattice point is brought to the equivalent point in the neighboring unit cell. Screw axes are denoted by n_m, where n is the

“rotational order” of the axis, and m is an integer, 1 ≤ m < n. A glide plane is the combination of a mirroring through a plane and a subsequent translation parallel to the mirror plane, usually by half the lattice parameter in the relevant crystal direction.

Glide planes are denoted bya,borc, and the translation occurs along the corresponding crystal axis.

These operations may be applied to infinite crystal lattices, and sets of elements corresponding to the transformations form space groups, of which there exist 230 [26].

Any crystal structure may be classified according to one of 32 crystal classes, and one of the 230 space groups. Table 2.2 shows the “group hierarchy” for the tetragonal and orthorhombic crystal systems.

Table 2.2: Example of the classification of crystal structures according to crystal systems, crystal classes and space groups. Within every crystal system, there are several crystal classes (point groups), each of which contains a number of space groups.

Crystal system Crystal class Space group Tetragonal 4/m

42m

P41 I41

P42c P42₁m

Orthorhombic 222 mm2

P212121 F222 P ca2₁ P ba2

(19)

2.1.3 The reciprocal lattice

The theoretical construction of thereciprocal lattice will be described in the following.

The starting point of the construction of a reciprocal lattice is the Fourier expansion of a crystal’s electron density, which follow the same periodicity as the crystal lattice itself [24]. Every real-space lattice has associated with it a reciprocal lattice, and the reciprocal lattices are completely analogous to their real-space counterparts: They consist of an infinite array of equivalent points which may all be reached through linear combinations of the primitive reciprocal lattice vectors a^∗_i. The reciprocal primitive vectors need to satisfy the relation

a^∗_i ·a_j = 2πδ_ij,

where a_j is a real-space primitive vector and δ_ij is the Kronecker-delta. In terms of the primitive vectorsa,band c, the corresponding reciprocal primitive vectors a^∗,b^∗ and c^∗ are defined as follows:











a^∗ = 2π_a·b×c^b×c b^∗ = 2π_a·b×c^c×a c^∗ = 2π_a·b×c^a×b

(2.3)

[24]. Linear combinations of the vectors defined in Equation (2.3) form the reciprocal lattice translations

g_hkl =ha^∗+kb^∗+lc^∗,

under which the reciprocal lattice is invariant. Distances in reciprocal space have units inverse length, and reciprocal lattice vectors ghkl are orthogonal to real-space crystal planes with the same indices. The length ofg_h⁰_k⁰_l⁰ corresponds to the spatial frequency of the planes (h⁰k⁰l⁰) [24]. As an example, the reciprocal vectorsg₁₀₀ and g₂₀₀ correspond to planes in the families{100}and{200}, respectively. Because{100}and{200} are families of parallel planes, g₁₀₀ k g₂₀₀. However, |g₂₀₀| = 2|g₁₀₀| due to the fact that the spatial frequency of the {200}-planes are twice that of the{100}-planes.

2.1.4 Grain boundaries

Conformity to the perfect periodicities described in Section 2.1.1 is not a feature of the real world. Real crystals, however carefully grown, will necessarily deviate from this absolute order to some extent. Such deviations from the ideal crystal lattice, crystal defects, affect the physical properties of compounds to various degrees – depending on the type of defect, its severity and the property in question. Crystal planes in the vi- cinity of a defect will be distorted, so there is generally some strain associated with any crystal defect. Defects may be classified with respect to the way in which they extend through the material, and typically fall into one of three categories:point defects,line defects orplanar defects [23, 27]. In the following, a description of the specific case of planar defects called grain boundaires will be presented.

A planar defect is a disruption of the long-range order of a single crystal which is confined to a nearly two-dimensional plane. A grain boundary is a specific kind of such a defect that separates two adjacent regions of a crystalline solid in which the periodicity and symmetry of the crystal structure is preserved. These regions of local homogeneous orientation are referred to asgrains. A crystal may consist of several such grains having slightly different orientations (but identical composition), in which case

(20)

its long-range order is interrupted by the surfaces that separate adjacent grains [27]. A grain boundary may be characterized by its orientation and the relationship between the grains on either side of it through the parameters n,R(r) and ϑ. Here, n is the plane normal of the grain boundary, and R(r) and ϑ denotes the relative translation and rotation between adjacent grains, respectively. As such, two adjacent grains may be transformed into one another through a given translationR and a rotationϑ [28].

There are several special cases of the grain boundary, all leading to a symmetry break- ing in the crystalline structure. Anti-phase boundaries, for instance, are grain boundaries withϑ= 0 and a translationRsuch that two adjacent grains are “out of phase”

in the sense that they are oppositely ordered when viewed in the projection parallel to the boundary [28]. Another special case of a grain boundary, which is illustrated in Figure 2.6, is the twin boundary. Twins are grains that may be brought into one another through either a rotation (rotation twins) about a twin axis or a reflection (re- flection twin) through the twin boundary [29]. Furthermore, twin boundaries may be classified according to their origin: Growth twinsarise in the growth process of a crystal,deformation twinsarise as a result of externally applied stress and transformation twins appear when a compound undergoes a structural phase transformation lowering its symmetry [27, 29]. Figure 2.6 shows a rotation twin boundary with ϑ= 120°.

Figure 2.6: Illustration of a rotation twin boundary where adjacent twins are rotated by 120° with respect to each other.

(21)

2.2 Electron diffraction

Electron diffraction is the central phenomenon on which the operation of a TEM is predicated. Concepts ranging from diffraction patterns to contrast mechanisms will be explained by the theory laid out in this section, where the theory of both kinematical and dynamical electron diffraction will be developed. The following discussion is, unless explicitly stated otherwise, based on the bookTransmission Electron Microscopy and Diffractometry of Materials by B. Fultz and J. M. Howe [30].

2.2.1 Kinematical theory of electron diffraction Electron scattering on a single atom

The discussion on electron diffraction begins with considering the scattering of an electron on a single atom. The incident electron may be regarded as a plane wave if its source is far away from the scattering object:

Ψ_i=eⁱ⁽^k⁰^·r⁰^−ωt⁾. (2.4) Here,r⁰ is the coordinate of the center of the scattering object,k₀ is the incident wave vector and ω is the angular frequency of the wave. The outgoing scattered wave may be described as a spherical, anisotropic wave with wave vector k. Omitting the time dependence e^−iωt, the scattered wave may be written on the form

ψ_s =f(k₀,k)e^ik|r−r⁰^|

|r−r⁰|, (2.5)

where the wave is observed at r, or a distance |r−r⁰| from the center of the atom.

f(k₀,k) is a scattering factor that depends on the directions of the incident and scattered wave, and may be obtained by solving the Schrödinger equation for an electron in a potential V(r⁰) originating from the atom on which it scatters:

∇²+2mE

¯ h²

₂!

ψ(r⁰) = 2mV(r⁰)

¯

h² ψ(r⁰), (2.6)

or

∇²+k₀²ψ(r⁰) = ˜V(r⁰)ψ(r⁰), (2.7) where k²₀ ≡ 2mE/¯h² and ˜V ≡ 2mV /¯h², and m and E are the mass and energy of the incident electron, respectively. Equation (2.7) is solved by introducing a Green’s function

G(r,r⁰) =− e^ik|r−r⁰^|

4π|r−r⁰| (2.8)

and integrating over the coordinates of the atom:

ψ(r) =e^ik⁰^·r+^Z V˜(r⁰)ψG(r,r⁰)dr⁰. (2.9) The total wave ψ has contributions from both the incident and scattered wave: The first term describes the incident wave, while the second term describes the scattered wave ψs. In order to solve the integral, the first Born approximation is invoked. By this approximation,ψis replaced by the incident plane wavee^ik⁰^·r⁰ within the integral.

The first Born approximation relies on the assumption that the incident wave is not diminished and that it is scattered only once. This assumption lies at the heart of

(22)

kinematical diffraction, and its validity rests on whether the scattering is appreciably weak. As will become apparent, the conditions for kinematical diffraction are generally not met in TEM diffraction experiments. The results of the following discussion will, however, be of great use in interpreting results from certain diffraction experiments.

The integral in Equation (2.9) may, in the limiting case where |r| |r⁰|, be written on the form

−

Z V˜(r⁰)e^ik⁰^·r⁰e^ik·⁽^r−r⁰⁾

4π|r| dr⁰, (2.10)

which, after some rearrangement, gives

− m 2π¯h²

1

|r|e^ik·r Z

V(r⁰)eⁱ⁽^k⁰^−k⁾^·r⁰dr⁰. (2.11) This is the scattered wave ψ_s on the same from as in Equation (2.5), where the scattering factorf(k₀,k) has been identified as

f(∆k) =− m 2π¯h²

Z

V(r⁰)e^−i∆k·r⁰dr⁰. (2.12) The vector ∆k is defined as the difference between the wave vectors of the incident and scattered wave, ∆k≡k−k₀. Thus, the scattered wave may be written as

ψ_s= e^ik·r

|r| f(∆k). (2.13)

The factor f(∆k) – referred to as the atomic form factor – determines the manner in which the amplitude of the scattered wave field decreases along different directions after being scattered. In the first Born approximation, f is recognized as the Fourier transform of the scattering potential V. The intensity of the scattered wave may be obtained by taking the absolute square of ψ_s:

I_s=ψ_s^∗ψ_s=|f(∆k)|² 1

r². (2.14)

Kinematical diffraction from monoatomic crystals

Diffraction as a phenomenon arises due to interference between waves scattered on multiple scattering objects. Suppose that a plane electron wave is incident on an array of scattering objects whose centers lie at coordinates R_j. In the same manner as for the single-atom case, the scattered wave may be written as

ψ_s(∆k,r) = e^ik·r

|r|

−m 2π¯h²

Z

V(r⁰)e^−i∆k·r⁰dr⁰, (2.15) by the first Born approximation. An appropriate choice for the potential V will be a sum of atomic scattering potentials, one for each atom, with origins at the atomic positions R_j. Using this, Equation (2.15) may be written as

ψ_s(∆k,r) = e^ik·r

|r|

−m 2π¯h²

Z X

j

Vj(r⁰−Rj)e^−i∆k·r⁰dr⁰, (2.16) where the sum is taken over every atom and a potential Vm is assigned to the mth atom. Now, defining the vector r ≡ r⁰−R_j will be convenient. This allows for the rewriting of Equation (2.16) into the following form:

Ψ_s= e^ik·r

|r|

X

j

−m 2π¯h²

Z

V_j(r)e^−i∆k·rdr

e^−i∆k·R^j. (2.17)

(23)

The scattered wave may now be expressed as follows:

ψ_s = e^ik·r

|r|



 X

j

fj(∆k)e^−i∆k·R^j



, (2.18)

where the atomic form factor f_j is defined as f_j(∆k) = −m

2π¯h² Z

V_j(r)e^−i∆k·rdr

for each atom j. As with the single-atom case, the form factor fj describes the loss in amplitude of the scattered wave at different angles to the axis of incidence. The factor e^−i∆k·rdescribes the relative phase of the wave function scattered from a single atom. The relative phase differences between such wavelets determine their interference and thus the resulting observable intensity distribution, given by Equation (2.14).

Consider now a monoatomic crystal with crystal axes a, b and c. As stated in Sec- tion 2.1.1, the coordinates for atomic positions Rj now take on the special form of R_uvw=ua+vb+wc. The phase factors in Equation (2.18) then become

X

j

e^−i∆k·R^j = ^X

u,v,w

e^−i∆k·⁽^ua⁺^vb⁺^wc⁾. (2.19)

The exponential, and thus the amplitude of the scattered wave, will be maximized when

∆k·(ua+vb+wc) = 2πn, (2.20) where nis integer. Vectors ∆k for which Equation (2.20) is satisfied are on the form

∆k=ha^∗+kb^∗+lc^∗, where

a^∗= 2π b×c a·b×c, b^∗= 2π c×a

a·b×c, c^∗= 2π a×b

a·b×c,

which are precisely the primitive vectors of the reciprocal lattice. Thus, by imposing an invariance under translations of the type R_uvw on the system, the Laue condition for diffraction may be formulated: The condition states that diffraction from a periodic material will occur if ∆kis a reciprocal lattice vector,

∆k=g_hkl. (2.21)

Structure and shape factors

As stated in Section 2.1.1, a crystal structure may be described as the convolution between a lattice and some basis. The coordinates of the atoms may be “split” into two parts: one coordinate rl for the appropriate lattice site and a coordinate rb describing the position of the basis constituent on a specific site. Using this, the scattered wave in Equation (2.18) may be written as follows:

ψ_s=^X

rl

X

rb

fb(rb)e⁻ⁱ²^π∆k·⁽^r^l⁺^r^b⁾, (2.22)

(24)

where the factor 2π has been factored out of ∆k for convenience and the prefactor e^ik·r/|r| has been omitted for brevity. Equation (2.22) may be rearranged into two factors; one dependent onrland the other dependent onrb. In doing this, thestructure factor F and theshape factor S are defined:

ψ_s=^X

rl

e^{−i2π∆k·r}^l^X

rb

f_b(r_b)e^{−i2π∆k·r}^b ≡S(∆k)F(∆k), (2.23) where the sum in F is taken over all basis atoms and the sum in S is taken over the whole lattice. The structure factor F pertains to interference between wavelets scattered from the different basis atoms. Under certain conditions, these interferences may lead to theextinctionsof reflections otherwise allowed by the Laue condition. The shape factorSdetermines the geometry of the diffracted intensities. In the limiting case of an infinitely large crystal, the shape factor intensities S^∗S become delta functions centered in k-space at ∆k = g_hkl. For crystals of finite size, the intensity peaks are modulated according to the geometry of the material. These modulated intensity peaks are referred to asrelrods.

The Ewald sphere, diffraction patterns and Laue zones

The Laue condition may be visualized in an elegant manner using the Ewald sphere construction. The construction goes as follows: The incident wave vectork₀is “placed”

such that it terminates at a reciprocal lattice point. A spherical shell of radius k₀ is then constructed such that its center coincides with the origin of k₀. In the case of elastic scattering, the scattered wave vectorkwill have the same magnitude ask₀ and will, as such, also terminate at the spherical surface if placed tail-to-tail withk₀. The Laue condition is now equivalent to the statement that all points ghkl intercepted by the Ewald sphere correspond to ∆k’s for which diffraction may occur.

Because the reciprocal lattice points are modulated by the shape factor, the Ewald sphere only needs to intercept the relrods in order for intensity to occur. This is ac- counted for by introducing a deviation vector s. ∆k may then be expressed as the deviation sfrom a reciprocal lattice vectorg:

∆k=g−s. (2.24)

An illustration of the Ewald sphere is provided in Figure 2.7.

The observable intensity distribution resulting from the scattered wave ψ_s will be proportional to the absolute squares of the shape and structure factor,

I_s=ψ^∗_sψ_s ∝ |S|²|F|².

Barring shape modulations and possible extinct reflections, the resulting diffraction pattern is a map of the sample’s reciprocal lattice projected along the direction of the wave vector k₀. Its peaks – or diffractionspots– are labelled by the indices (hkl). The spots represent wave vectors, and a reflection (hkl) may be interpreted as the spatial frequency of corresponding crystal planes. In a TEM, electron waves are incident on the specimen with wave vectors k₀ along the optical axis. As high-symmetry crystal directions often are of special interest, the specimen may be tilted such that the optical axis coincides with low-index crystal axes. The crystal axis along which the electron waves are incident are referred to as zone axes and are denoted by the appropriate indices [uvw] for crystallographic directions. An indexed diffraction pattern from the

(25)

Figure 2.7: A two-dimensional illustration of the geometrical construction of the Ewald sphere. Here, its surface intersects only two relrods in addition to the central point. In a TEM, the curvature of the surface is far smaller, and several reciprocal lattice points will be intersected.

Figure 2.8: An indexed diffraction pattern from the [110] zone axis of KNBO. Only two diffraction peaks are indexed, and the indices of the remaining peaks are determined by linear combinations of (001) and (440).

[110] zone axis of KNBO is provided as an example in Figure 2.8.

Another effect arising from the curvature of the Ewald sphere is the occurrence of higher-order Laue zones (HOLZ). The zeroth-order Laue zone (ZOLZ) contributions to the diffracted wave come from the plane normal to k₀, on which the origin of the reciprocal lattice – the transmitted beam – lies. Under certain conditions, the inter- ception of the Ewald sphere with reciprocal lattice points “above” the reciprocal origin allows for the contribution from points outside this normal plane to the diffracted intensity. The resulting diffraction pattern will consist of “rings” of diffraction intensity peaks around the ZOLZ pattern. These rings will contain diffraction spots corresponding to crystal planes not orthogonal to the zone axis. An illustration of this effect is provided in Figure 2.9.

2.2.2 Dynamical theory of electron diffraction

The first Born approximation, and thus kinematical diffraction, is generally not valid for electron diffraction in TEM. The strong electron–matter interaction leads to strong diffraction, and the assumption that an electron wave scatters only once in the sample does not hold. Although kinematical theory is sufficient for interpreting diffraction spot patterns, dynamical effects must be taken into account when discussing contrast

(26)

Figure 2.9: Diffraction pattern obtained from a sample of KNBO. FOLZ diffraction peaks are clearly visible along a circle around the ZOLZ. Weaker intensity peaks may also be seen outside the FOLZ, belonging to the second order Laue zone (SOLZ).

mechanisms and other types of diffraction experiments. In the following section, an outline of the dynamical theory of electron diffraction will be presented.

Mathematical outline of dynamical theory

This section provides some background for the discussion of dynamical theory. This is relevant for the present study, as it turned out that dynamical diffraction experiments and simulations were necessary for describing the crystal structure of KNBO in the [001] projection. By considering the Fourier expansion of the crystal potential, it will be shown that it leads to an amplitude transfer between diffracted beams. By the end of this outline, the Howie-Whelan-Darwin equations will be derived, and the extinction distance ξg−g⁰ and excitation error sg defined.

The description of dynamical electron diffraction starts by considering an electron entering a crystalline solid from vacuum. Before taking into account the periodic crystal potential, the effects of an average potential U₀₀ on the electron wave vector is considered. This average potential is attractive, and serves to increase the kinetic energy of the electron as it enters the solid. If χ and kdenotes the electron wave vector in vacuum and inside the solid, respectively, then the change in kinetic energy arising from the average potential U₀₀ may be written as

¯ h²

2m(χ²−k²) =U₀₀, (2.25)

where U₀₀ <0. Moving on from this, the periodicity of the crystal potential must be included. In anticipation of the periodicity of the potentialV(r) sharing the periodicity of the crystal lattice, it is written as a Fourier series in reciprocal lattice vectors g.

The same is done for the electron wave function Φ(r):

V(r) =^X

g6=0

Uge^ig·r+U₀₀, (2.26)

(27)

Φ(r) =^X

g

φg(z)eⁱ⁽^k⁺^g⁾^·r. (2.27)

As before, the Schrödinger equation

− ¯h²

2m∇²Φ(r) +V(r)Φ(r) =EΦ(r) (2.28) is to be solved. By substituting in the Fourier expansions for V and Φ, using the expression for U₀₀ from Equation (2.25) and multiplying by 2m/¯h², it may be shown that the Schrödinger equation can be rewritten as

X

g

((k_x+g_x)²+ (k_y+g_y)²+k²_z−χ²+2m

¯

h² U₀₀)φ_g(z)eⁱ⁽^k⁺^g⁾^·r

−^X

g

(2ik_z∂φ_g

∂z +∂²φ_g

∂z² )eⁱ⁽^k⁺^g⁾^·r +2m

¯ h²

X

g⁰

X

g⁰⁰6=0

φ_g⁰(z)eⁱ⁽^g⁺^g⁰⁾^·rU_g⁰⁰e^ig⁰⁰^·r = 0.

(2.29)

By multiplying Equation (2.29) with eⁱ⁽^k⁺^g⁾^·r and integrating over all of space, the orthogonality relation

Z l/2

−l/2eⁱ²^π⁽ⁿ⁻ⁿ⁰⁾^k/ldk=

(lifn⁰ =n

0 otherwise (2.30)

is imposed. The only terms in the sums that survive are those whose exponentials have the same argument k +g. That is, the condition for a nonzero term will be k +g = k+g⁰ +g⁰⁰, or g⁰⁰ = g −g⁰. Using this, together with Equation (2.25), Equation (2.29) may be written as

(2k_xg_x+g_x²+ 2k_yg_y+g_y²)φ_g

−2ik_z∂φ_g

∂z −∂²φ_g

∂z² + 2m

¯ h² (^X

g⁰6=g

φ_g⁰(z)Ug−g⁰) = 0.

(2.31)

Lastly, by assuming that the Fourier coefficients of the beam Φ vary slowly with pen- etration depth z, the term _∂z^∂²2φg can be disregarded, allowing for the rewriting of Equation (2.31) as

∂φg

∂z =i k_x²−(k_x+g_x)²+k²_y−(k_y+g_y)² 2k_z

! φg

− 2im 2kz¯h²



 X

g⁰6=g

φ_g⁰Ug−g⁰



.

(2.32)

Equation 2.32 represents a set of coupled differential equations for every diffracted beam, and this coupling between diffracted beams g and g⁰ lies at the heart of dynamical theory. From the second term in Equation (2.32), it is clear that the coupling

Domain imaging of multiferroic K3Nb3B2O12 by transmission electron microscopy

Oskar Ryggetangen

Domain imaging of multiferroic

K 3 Nb 3 B 2 O

by transmission electron microscopy

Master ’s thesis

Oskar Ryggetangen

Domain imaging of multiferroic

K 3 Nb 3 B 2 O

by transmission electron microscopy

Abstract

Sammendrag

Preface

Abbreviations

Contents

1 Introduction

2 Theoretical background

2.1 Crystallography

2.2 Electron diffraction

K ³ Nb ³ B ² O

K ³ Nb ³ B ² O