Thesis
Reference
Local probe studies of the role of surface adsorbates in polarization switching, screening and domain wall functionalities in ferroelectric
thin films
GAPONENKO, Iaroslav
Abstract
La présente thèse se consacre à l'étude par microscopie à force atomique des propriétés fonctionnelles des domaines et parois de domaines ferroélectriques et leur interaction avec les adsorbats de surface. En particulier, des couches minces de titano-zirconate de plomb sont utilisées comme système modèle pour l'étude de la conduction des parois de domaine.
Ces dernières, révélées conductrices dans les échantillons vierges lors d'études précédentes, montrent un comportement bien plus complexe gouverné par la présence ou l'absence des adsorbats de surface. Les études présentées ici donnent non seulement un aperçu fondamental de la physique des domaines et parois de domaines, mais ont également donné lieu à des développements instrumentaux majeurs tels qu'un contrôleur d'humidité à bas niveau de bruit et un algorithme de correction de distorsions d'images basé sur la vision par ordinateur. Ces développements ont permis des études poussées du rôle spécifique de l'eau sur les propriétés des domaines ferroélectriques.
GAPONENKO, Iaroslav. Local probe studies of the role of surface adsorbates in
polarization switching, screening and domain wall functionalities in ferroelectric thin films . Thèse de doctorat : Univ. Genève, 2017, no. Sc. 5097
DOI : 10.13097/archive-ouverte/unige:112566 URN : urn:nbn:ch:unige-1125668
Available at:
http://archive-ouverte.unige.ch/unige:112566
Disclaimer: layout of this document may differ from the published version.
de la Matière Quantique
Local probe studies of the role of surface adsorbates in polarization
switching, screening and domain wall functionalities in ferroelectric
thin films
THÈSE
présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de docteur ès Sciences, mention Physique
par
Iaroslav Gaponenko
Novosibirsk (USSR)de
Thèse n◦ 5097
GENÈVE
and left raw nerve endings to touch the world outside.
Of parallel and irreconcilable lives,
this mix of happiness, pain, love, despair, exhilaration and terror.
Feeling intensely, unbelievably alive,
and at the same time barely able to face the effort of breathing, and putting one foot in front of the other.
I can talk of where it leads,
or at least one kind of ending - since all of us in the end can only talk with full insight about out own experience.
-A friend
Introduction et motivation
Jusqu’à présent, la miniaturisation a guidé les progrès de l’électronique moderne. Le nombre de transistors n’a cessé d’augmenter dans les circuits intégrés en doublant d’année en année, comme décrit par la loi de Moore.
Cependant, cette quête du spectaculairement petit arrive peu à peu à sa fin et bientôt fera face au mur insurmontable des lois de la physique, la production des composants à base de silicium commençant à se heurter à des limitations technologiques. C’est ainsi que la recherche s’est graduellement focalisée sur des matériaux dits « intelligents », capables de more-than-Moore, qui ne seraient pas que de simples transistors améliorés, mais qui incorporeraient de nouvelles fonctionnalités à même le composant. Parmi ces matériaux, les oxydes complexes ont un potentiel non négligeable et leurs fonctionnalités – intrinsèques ou d’interface – ont fait l’objet ces dernières années d’études approfondies dans la quête de nouveaux paradigmes de l’électronique. [1–4]
Les oxydes complexes comprennent divers matériaux avec pour seul dénomi- nateur commun la présence d’oxygène. Parmi ceux-ci se trouvent des métaux, diélectriques, isolants topologiques ou supraconducteurs. Leurs propriétés sont toutes aussi diverses et peuvent inclure la ferroélectricité, les transi- tions métal-isolant, les couplages optoélectroniques ou magnétoélectriques, la commutativité résistive, la magnétorésistance, ainsi que d’autres propriétés topologiques non-triviales. En se basant sur ces fonctionnalités, de nouvelles applications comme des convertisseurs énergétiques, des nano-composites ou des mémoires ont été développés utilisant ces matériaux et leurs interfaces.
En particulier, les matériaux ferroélectriques peuvent présenter des régions de polarisation uniformément orientée, appelés « domaines », qui sont sé-
absentes dans les domaines ferroélectriques environnants grâce à la présence de symétries ou corrélations fortes au sein même ou à proximité de la paroi.
En se basant sur ces propriétés fortement localisées, un nouveau paradigme a été proposé: domain wall nanoelectronics / nanoélectronique des parois de domaines [5, 6] – ou l’utilisation de composants à base de parois de domaine à l’échelle nanométrique comme éléments pour l’électronique du futur. Parmi les fonctionnalités découvertes aux parois de domaine dans les ferroélectriques, la conduction électrique est d’un intérêt technologique particulier de par sa nature reconfigurable et intrinsèquement nanométrique.
Cependant, bien que son existence ait été démontrée comme un phénomène général dans les oxydes ferroélectriques, l’origine microscopique de cette conduction peut s’expliquer par différents mécanismes qui dépendent du matériau – avec une contribution non-négligeable des conditions environ- nementales. Une meilleure compréhension des propriétés fonctionnelles de ces parois de domaine – en particulier leur relation avec les effets internes (structure, stœchiométrie, topologie) et externes (adsorbats, conditions de bord électrostatiques et de contrainte) – est un des défis majeurs dans ce champ de recherche en pleine effervescence.
Résumé et structure du manuscrit
La présente thèse se consacre à l’étude par microscopie à force atomique des propriétés fonctionnelles des domaines et parois de domaines ferroélectriques et leur interaction avec les adsorbats de surface. En particulier, des couches minces de titano-zirconate de plomb (PbxZr1−xTiO3 ou PZT) – matériau ferroélectrique le plus commun dans les applications industrielles – sont utilisées comme système modèle pour l’étude de la conduction des parois de domaine. Ces dernières, bien que révélées conductrices dans les échantillons vierges lors d’études précédentes [7], montrent un comportement bien plus complexe gouverné par la présence ou l’absence des adsorbats de surface [8]. Les études présentées dans ce manuscrit donnent non seulement un aperçu fondamental de la physique des domaines et parois de domaines, mais l’étude sur les adsorbats de surface par microscopie à force atomique a également donné lieu à des développements instrumentaux majeurs sous la forme d’un contrôleur d’humidité à bas niveau de bruit [9] et d’un algorithme de correction de distorsions d’images basé sur la vision par ordinateur [10].
Ces développements ont permis des études poussées du rôle spécifique de l’eau sur les propriétés des domaines ferroélectriques – que ces derniers soient intrinsèques ou générés artificiellement.
Ce manuscrit adopte la structure suivante. Après une brève introduction
macroscopique, en focalisant sur la dynamique de l’inversion de la polarisa- tion et sur la physique des domaines et parois de domaine. Les propriétés et fonctionnalités de ces dernières sont brièvement discutées en mettant l’accent sur la conduction électrique. Une présentation des propriétés de l’eau et de son interaction avec les surfaces vient ensuite, avec en parti- culier le rôle des surfaces polaires sur la structure à longue échelle des molécules d’eau adsorbées. Ensuite, le Chapitre3 présente les deux matéri- aux utilisés lors des études présentées dans les chapitres suivants : le PZT, ferroélectrique universellement utilisé dans des applications industrielles, et le graphène, matériau semi-conducteur quasi-bidimensionnel formé d’une couche monoatomique d’atomes de carbone. La deuxième partie du Chapitre 3se concentre sur la microscopie à force atomique (AFM) utilisée dans ce travail afin de caractériser localement les propriétés des matériaux étudiés, et plus particulièrement la microscopie à force piézoélectrique, une tech- nique AFM en mode contact permettant de déterminer la polarisation des matériaux ferroélectriques à l’échelle nanométrique. Dans le Chapitre4, un algorithme de correction de distorsions de séries temporelles d’images de microscopie à sonde locale est présenté avec deux exemples d’application : la quantification de la relaxation des plis sur du graphène mesuré sous con- trainte mécanique, et l’évolution temporelle de l’inversion de polarisation de domaines ferroélectriques sous champ électrique à partir d’une série d’images.
Cet algorithme nous a permis de suivre la nucléation de ces domaines sur des sites aléatoirement distribués, leur croissance radiale, puis leur fusion lors d’évènements de type avalanche. Afin de faciliter l’étude de l’interaction des ferroélectriques avec l’eau sur leur surface, un contrôleur d’humidité innovant a été développé. Décrit dans le Chapitre5, ce dernier est basé sur un flux de gaz continu en utilisant un atomiseur ultrasonique afin de générer de l’air humide, donnant lieu à des niveaux de bruit et une performance sans précè- dent lors de mesures in-operando. Dans le Chapitre6, un contrôle réversible de la conduction électrique aux parois de domaine dans des couches minces de PZT est démontré. Lorsque les couches minces sont soumises à des cycles de recuit sous ultra-haut vide et d’exposition à l’atmosphère ambiante, la conductivité peut être « allumée » ou « éteinte » à cause de la redistribution des lacunes d’oxygène ainsi que de la dynamique des adsorbats de surface.
Cette interaction des domaines ferroélectriques avec les adsorbats, et en particulier avec l’eau de surface, est explorée plus en détail dans le Chapitre 7, lors d’études en fonction de l’humidité ambiante. Une accélération de la dynamique des charges de surface est constatée lorsque le contenu d’eau dans l’air et sur la surface est augmenté – confirmant le rôle essentiel de l’humidité lors de mesures AFM sur les matériaux ferroélectriques. Finale- ment, le Chapitre8 présente les divers projets collaboratifs auxquels j’ai
la validation des propriétés des matériaux. Une conclusion générale couronne le manuscrit, proposant de nouveaux travaux scientifiques se basant sur les résultats des recherches menées lors de cette thèse.
Résumé iii
Introduction et motivation . . . iii
Résumé et structure du manuscrit . . . iv
1 General Introduction 1 1.1 Introduction and motivation . . . 1
1.2 Synopsis . . . 2
2 Background 5 2.1 Ferroelectrics . . . 5
2.1.1 Definition of Ferroelectricity . . . 6
2.1.2 Phenomenological description . . . 7
2.1.3 Microscopic considerations . . . 11
2.1.4 Emergence of domains . . . 12
2.1.5 Domain walls . . . 17
2.2 Adsorbates . . . 21
2.2.1 Water . . . 21
2.2.2 Screening and charge dynamics . . . 25
2.2.3 Interaction of adsorbates with ferroelectrics . . . 26
3 Materials and Methods 27 3.1 Lead zirconate titanate . . . 27
3.1.1 Materials growth . . . 28
3.1.2 Characterization . . . 30
3.2 Graphene . . . 31
3.2.1 Structure and properties . . . 32
3.2.2 Fabrication . . . 32
3.3 Atomic Force Microscopy . . . 33
3.3.1 Operating principle . . . 34
3.3.2 Operation modes . . . 36
3.3.3 Techniques for functional measurements . . . 40
3.3.4 Instrumentation . . . 47
4 Distortion correction of AFM time series 51 4.1 Need for correction . . . 52
4.2 Distortion correction algorithm . . . 53
4.2.1 Correction validity . . . 53
4.2.2 Implementation . . . 55
4.3 Results and discussion . . . 56
4.3.1 Nanoscale polarization reversal tracking . . . 56
4.3.2 Wrinkle relaxation in strained graphene . . . 59
4.4 Conclusion . . . 61
5 Humidity control 63 5.1 Introduction . . . 63
5.2 Humidity controller . . . 65
5.2.1 General design . . . 65
5.2.2 Generating humidity . . . 67
5.2.3 Integration and Characterization . . . 71
5.2.4 Control and measurement . . . 73
5.3 Performance characterization . . . 75
5.3.1 Bubbler design . . . 75
5.3.2 Atomiser design . . . 77
5.4 Conclusion . . . 78
5.4.1 Next generation . . . 78
6 Reversible control of PZT domain wall conduction 81 6.1 Domain wall conduction in ferroelectrics . . . 81
6.2 Effect of substrate on domain wall conduction . . . 83
6.3 Measuring domain wall conduction . . . 86
6.3.1 Thickness dependence . . . 88
6.4 Effects of vacuum annealing . . . 90
6.5 Post-annealing conductance . . . 92
6.5.1 Conduction vs displacement . . . 92
6.5.2 Effect of thickness on conduction . . . 92
6.5.3 Currents at a-domains . . . 96
6.6 Towards a qualitative understanding . . . 96
6.7 Reversible control of domain wall conduction . . . 98
6.7.1 Effect of annealing on sample stoichiometry . . . 100
6.8 Conclusion . . . 102
7 Effect of surface water and ferroelectric polarization on sur- face charge dissipation 105 7.1 Introduction . . . 105
7.2 AFM-based measurements . . . 108
7.2.1 Surface charge dissipation . . . 108
7.2.2 Topographic signature . . . 111
7.2.3 Polarization dependent KPFM . . . 113
7.3 Insights . . . 122
7.4 The path forward . . . 123
8 Lab on a tip 125 8.1 Investigating graphene . . . 125
8.1.1 Chemical species adsorption . . . 126
8.1.2 Terahertz magnetoplasmons . . . 127
8.1.3 Strain properties . . . 127
8.2 Probing biomaterials . . . 129
8.2.1 Reptile epithelial microstructures and surface adhesion 129 8.2.2 Snake scale reconstruction . . . 131
8.3 Overview . . . 132
9 Conclusion 133 9.1 Overall conclusion . . . 133
9.2 Perspectives . . . 135
A List of publications 137
B Distortion correction codes 139
Acknowledgements 157
Bibliography 159
General Introduction
1.1 Introduction and motivation
Progress in electronics over the last decades has been governed by Moore’s Law - with the aerial density of transistors in integrated circuits approxi- mately doubling every year. However, as smaller and smaller components have been developed, further miniaturization of silicon-based devices has been hitting ever-increasing technological roadblocks - and will ultimately be challenged by fundamental limitations imposed by the laws of physics.
Thus, the focus has been gradually shifting towards smarter materials - capable ofmore-than-Moore- incorporating novel functionalities directly at the device level, for which complex oxides hold significant promise. Indeed, recent research in this direction has ushered in a new era of functionalities, both intrinsic and interfacial, giving these novel materials a place in the spotlight on the stage of future generations of electronics. [1–4]
This new field encompasses multiple classes of materials - with the pres- ence of oxygen in the crystal lattice as the common denominator. Amongst these can be found metals, dielectrics, topological insulators or supercon- ductors. Their properties, equally diverse, can include phenomena such as ferroelectricity, metal-to-insulator transitions, optoelectronic or magnetoelec- tric couplings, resistive switching, magnetoresistance and other nontrivial topological properties. Building upon these, novel applications such as energy converters, nanocomposites, and memories have been developed, exploiting the properties of these materials or their interfaces.
In particular, domains in ferroelectric materials - regions naturally gen- erated to counteract depolarizing fields, or artificially written by voltage application - are bounded by one such type of complex interfaces, their domain walls. These have only recently been shown to host a plethora of functionalities absent from the parent ferroelectric phase, but allowed at the domain walls through the presence of different symmetries or strong correlations. Based on these highly localised functional properties, a new paradigm was recently proposed: domain wall nanoelectronics [5, 6] - the idea that devices composed of nanoscale domain wall elements would be the foundation of a new electronics. Amongst all the functionalities discovered at ferroelectric domain walls, electrical conduction is of particular interest due to its intrinsically nanoscale and reconfigurable nature. However, although shown to be a general phenomenon in oxide ferroelectrics, its microscopic origins can be traced to quite different mechanisms in different materials, with also a significant role of environmental conditions.
A better understanding of the functional properties of domain walls, their microscopic relation to internal (structure, stoichiometry, topology) and external (adsorbates, electrostatic and strain boundary conditions) effects is therefore one of the key challenges in this exponentially growing field.
1.2 Synopsis
In this work, we investigate the functional properties of ferroelectric domains and domain walls by means of atomic force microscopy, as well as their interplay with surface adsorbates. In particular, lead zirconate titanate - the most commonly used piezoelectric in industrial applications - is used as a model system to study the conductivity of domain walls. The latter, although found to be conductive in as-grown samples in previous studies [7], demonstrate a much more complicated behavior mediated by the presence or absence of surface adsorbates [8]. As well as providing fundamental insights, our atomic force microscopy investigation of these adsorbates has also led to significant instrumental developments in the form of a low-noise humidity controller [9] and a computer-vision image distortion correction algorithm [10]. These developments enable in-depth studies of the specific role of water in the surface properties of ferroelectric domains, both as grown and artificially switched.
This manuscript is structured as follows. After the present brief introduc- tion and motivation, Chapter2sets out the scientific context in which our work is situated. Fundamentals of ferroelectric materials are discussed from the macroscopic point of view, with a focus on the physics and dynamics of domains and domain walls. The properties of such nanoscale domain
walls are briefly highlighted, focusing in particular on electrical conduction.
A discussion on the properties of water and its interactions with surfaces, specifically the role of polar surfaces on the long range structuring of over- lying water molecules, follows. Then, the two materials discussed in the remainder of the manuscript are described in the first part of Chapter3:
the ubiquitous ferroelectric lead zirconate titanate (PZT) and graphene - a two-dimensional semiconducting material consisting of a single sheet of carbon atoms. Following this presentation, the second part of Chapter 3 focuses on atomic force microscopy, the technique used to locally character- ize the properties of materials. Special attention is given to piezoresponse force microscopy, a contact mode AFM-based technique which allows us to characterize ferroelectric materials at the nanoscale. A computer vision based algorithm for the distortion correction of scanning probe microscopy image time series is presented in Chapter4. It is shown to not only quantify the relaxation of wrinkles in graphene strained in-situ, but is also used to track the evolution of ferroelectric domain reversal under applied voltage during a continuously acquired series of images - from nucleation at randomly distributed sites, through conventional radial domain growth, towards finally a large scale merging of the growing domains in avalanche-like events. As part of our study of the interaction of surface water with ferroelectrics, a novel humidity controller was designed and constructed in-house, enabling the precise control of the sample environment. Described in Chapter5, it uses a continuous flow design with an ultrasonic atomiser based humidity generation mechanism, providing unprecedented performance forin-operandohumidity control. In Chapter6, a reversible control of domain wall conduction in PZT thin films is demonstrated. As the films are cycled between ultrahigh vacuum annealing and ambient exposure, the observed domain wall conduction is switchedon andoff, respectively, as a result of the redistribution of oxygen vacancies and surface adsorbate dynamics. The interaction of ferroelectric domains with adsorbates, and in particular surface water, is further dis- cussed in Chapter 7. Humidity-dependent studies are presented, showing the accelerated surface charge dynamics with increasing water content in the air and on the surface, highlighting the key importance of humidity during AFM experiments on ferroelectrics. Finally, in Chapter8, some of the diverse collaborative projects in which I participated during my thesis are highlighted - demonstrating the multidisciplinary role and increasing importance of scanning probe microscopy as a technique for characterization and validation of materials properties .
Background
2.1 Ferroelectrics
The discovery of ferroelectricity by Joseph Valasek in 1921 in potassium sodium tartrate (KNaC4H4O6·4H2O) - also known as Rochelle salt - cast the light on a new class of compounds whose understanding brought about significant advances in the physics of materials, stimulating innovation both in fundamental research and applications. In the decades following Valasek’s breakthrough, multiple ferroelectric materials have been discovered and integrated in military, industrial, and medical applications thanks to their diverse properties - which has molded them into the backbone of modern technology. Their high dielectric constants make them indispensible in multi- layer capacitors or as high-k gate dielectrics in transistor applications. Their high piezoelectric coefficients have enabled a new class of transducers or actuators both for civilian and military applications. Their pyroelectric prop- erties allow them to be integrated as infrared detectors in a new generation of night vision cameras. Their optical properties, being of a highly nonlinear nature, serve as the basis of current electro-optical modulators or frequency doublers. Finally, their reversible remanent polarization - property defining ferroelectrics - has been put to use in non-volatile random access memories which have been built into products as popular as the Sony Playstation 2 or the PASMO card used for contactless payment in the Tokyo metropolitan area.
Figure 2.1:Illustration of the crystallographic constraints governing the existence of ferroelectricity. Amongst the 32 crystal point groups, themselves containing 230 space groups of possible crystal symmetries, 20 are piezoelectric. Out of these, 10 are pyroelectric - and only some of these can exhibit ferroelectricity, restricting heavily the number of possible ferroelectric materials.
2.1.1 Definition of Ferroelectricity
As defined by the IEEE Standards,[a] ferroelectric phase is one in which the spontaneous electric polarization can be reoriented between possible equi- librium directions (determined by the crystallography of the system) by a realizable, appropriately oriented electric field [11]. This definition contains several important aspects, the combination of which gives rise to a ferroelec- tric material, and which will be discussed below. First, the crystallographic symmetries of the system give constraints as to which materials can become ferroelectric. Secondly, the polarization, which can be reoriented by the application of an electric field, gives ferroelectric materials a rich functional behavior. Lastly, the required existence of two or more equilibrium states of electric polarization can imply their coexistence in the same materials in the form of domains.
From the pure crystallographic point of view, materials that can be ferroelectric are heavily restricted. In the realm of crystals, there exist 230 crystallographic space groups that are categorized by their symmetries into 32 crystalline point groups. Amongst these 32 groups, there are 21 that do not exhibit centrosymmetry. Amongst these, 20 can be piezoelectric, generating electrical charge in response to applied stress. Amongst these, even fewer - just 10 point groups - can acquire a temperature-dependent spontaneous polarization - pyroelectricity. Finally, only some of these pyroelectrics can be ferroelectric. This crystallographic relationship amongst the point groups is illustrated in Fig.2.1.
The above constraints imposed on the existence of a ferroelectric material are a double-edged sword. They limit the number of possible materials due to the presence - or absence - of symmetries, yet each ferroelectric can (in theory) inherit the properties of the more general point groups. Thus, it will not only posess the unique property of ferroelectricity, but should also be pyroelectric and piezoelectric; usually more strongly so than most simple pyroelectrics or piezoelectrics. This explains the wide use of ferroelectrics in today’s electronics for properties other than the reversible spontaneous polarization.
Most ferroelectrics do not posess their eponymous quality at all temper- atures, but undergo a temperature-dependent phase transition between a high-temperature, high-symmetry paraelectric phase - usually cubic - and a low-temperature low-symmetry phase - for instance tetragonal or rhombohe- dral. This critical threshold is called the Curie temperatureTc, by similarity with magnetic materials that undergo a transition between the paramagnetic and ferro-/antiferro-/ferri-magnetic states, and in fact even the "ferro-" in ferroelectrics has been chosen due to their similarities with ferromagnetics.
Under the application of an electric field of the appropriate polarity and magnitude, the polarization of a ferroelectric can be switched reversibly.
This is illustrated in Fig.2.2(a), where an experimentally acquired hysteresis loop for a Pb(Zr0.2Ti0.8)O3 thin film is shown 1. As the field - or in this case the voltage - is swept from zero towards positive values, the remanent polarization will switch from itsPr−state to thePr+state at a specific value of voltage/field, called the coercive voltageVc+ (or coercive field Ec+). If the voltage is increased further, then an increase in polarization is observed up to a certain saturation polarizationPs+ - with magnitude depending on the dielectric response of the material to the applied field. As the field is swept back down to zero, the polarization decreases to its final remanent value Pr+. The same occurs for negative values of the voltage/field, and the polarization-voltage (or polarization-field) curve thus forms a closed hysteresis loop. As the polarization is switched, a flow of charge can be observed as a current from one side of the crystal to the other due to the dipole reversal, as shown in Fig.2.2(b).
2.1.2 Phenomenological description
The ferroelectric phase transition is not only temperature-dependent but is also accompanied by a symmetry lowering with the emergence of two or more equienergetic states between which the order parameter - in this case the polarization - can be switched. This type of physics is well described by the Landau-Ginzburg-Devonshire phenomenological phase transition theory
1The author would like to acknowledge Dr. Stefano Gariglio, DQMP, University of Geneva for this measurement.
Figure 2.2: (a) Experimentally acquired ferroelectric hysteresis loop from a Pb(Zr0.2Ti0.8)O3thin film, demonstrating switching between the two possible polar- ization states.Pr+/−is the remanent polarization,Ps+/−is the saturation polariza- tion, andEc+/−is the coercive voltage. (b) Current flowing observed between the two sides of the material during the switching measurement due to the reversal of polarization. Measurement performed by Dr. Stefano Gariglio, DQMP, University of Geneva.
2. The starting point for this macroscopic theory is that the stable state of the ferroelectric material around the ferroelectric transition is the one that minimizes the Gibbs free energy:
G=U−T S−ησ−ED=F−ησ−ED (2.1) withU the internal energy,T the temperature,S the entropy,η the strain, σ the external stress, E the external electric field and D =0E+P the dielectric displacement taking into account both the spontaneous polarization and the dielectric behavior. In the case of a ferroelectric phase transition in absence of an electric field and with no external strain applied (E= 0 and σ= 0), this expression reduces to the Helmholz free energy:
F=U−T S (2.2)
In the simple uniaxial ferroelectric case, the free energy can be expanded in terms of an order parameter, in this case the polarizationP, using the classic Landau-Ginzburg approach:
F =F0+α 2P2+β
4P4+γ
6P6 (2.3)
withF0the free energy of the high-symmetry paraelectric phase, andα,β,γ temperature and pressure-dependent materials coefficients. The choice of even powers in the expression above stems from the fact that any expansion has to remain invariant under the symmetry operations both above and belowTc 2The following discussion is inspired by the development inPrinciples and Applications of Ferroelectric and Related MaterialsbyM.E. Lines and A.M. Glass[12]
in the high-symmetry paraelectric and lower-symmetry ferroelectric phases, respectively. By using the Landau-Ginzburg expansion, the properties of the ferroelectric material can thus be calculated by minimizing the expression with respect to the polarization:
∂F
∂P = 0, ∂F
∂P2 >0 (2.4)
In practice, two types of ferroelectric phase transition have been observed:
first-order for materials such as BaTiO3, (Ba,Sr)TiO3, PbTiO3 and KNbO3
with a discontinuous change of polarization, and second-order for materials such as triglycine sulphate, Rochelle salt and dihydrogen phosphate with a continuous change of polarization across the transition temperatureTc. Second-order phase transition
In the second-order phase transition, the expansion coefficients have the following form:
α= 1
0C(T−T0), β <0, γ= 0 (2.5) withC the material-dependent Curie-Weiss constant,T0 the Curie temper- ature, and0 the permittivity of free space. By taking equation 2.3 and minimizing it with the conditions in equation2.4, the transition temperature Tcand the saturation polarizationPsin the ferroelectric state can be trivially calculated to be:
Tc=T0
Ps= 0 forT > Tc
Ps=± r 1
β0C(T0−T)1/2 forT < Tc
(2.6)
As can be seen from equation 2.6, there is no discontinuity in the polar- ization - as this is a second-order phase transition - and the polarization magnitude depends only on the difference of temperature from the phase transition temperature Tc =T0. There are two possible (and equivalent) polarization orientations, denoted by the±sign, as we are in the case of a uniaxial ferroelectric. The free energy, polarization, specific heat, and inverse permittivity are plotted in Fig.2.3.
First-order phase transition
In most ferroelectric materials, the phase transition is of first order. Such a transition is characterized by a discontinuity in the first derivative of the thermodynamic potential with respect to the main order parameter, the
Figure 2.3:Illustration of the properties of a ferroelectric with a second order phase transition. The free energy (a) exhibits one single state forα >0 corresponsing to the paraelectric state, and two possible states forα <0 corresponding to the ferroelectric state, withα∝T−Tc. The saturation polarizationPs in (b) is non-zero belowTc
and the inverse permittivityκX,T (c) has a drop atTc. From [12].
polarizationP. The equation2.3is still valid, with a nonzero constant for the sixth-order term. The expansion coefficients have the following form:
α= 1
0C(T −T0), β <0, γ >0 (2.7) The transition temperatureTc is now different fromT0, as it acquires a shift:
Tc=T0+3β20C
16γ (2.8)
Moreover, the spontaneous polarization jumps discontinuously to a non-zero value at the transition temperature:
Ps= 0 forT > Tc
Ps=± s
−β 2α
1 +
r
1−4αγ β2
forT < Tc (2.9) The discontinuity in polarization is shown in Fig. 2.4 alongside the free energy, specific heat, and inverse permittivity.
Field and strain dependence
The Landau-Ginzburg-Devonshire formalism can also describe the coupling of the ferroelectric polarization with strain as well as with external electric field or stress through the additional terms in the Gibbs free energy:
G=F−ησ−ED (2.10)
This is particularly important when the material undergoes a structural transition - for example cubic-to-tetragonal - which generates the relaxation
Figure 2.4:Illustration of the properties of a ferroelectric with a first order phase transition. The Free energy (a) shows two possible states forT < Tc, and a single energetically favorable state forT > Tc. The saturation polarizationPsin (b) exhibits a discontinuity atTcalongside the inverse permittivityκX,T in (c). From [12].
of the unit cell; or when it is subjected to electric fields - which may gen- erate a polarization reversal. The added terms allow for instance for the field-dependent ferroelectric hysteresis to be modeled.
Although not developed further, it is noteworthy to specify that a full phenomenological description of a ferroelectric within the Landau-Ginzburg- Devonshire framework will give an accurate description of the variation of the elastic and dielectric properties [13], given the appropriate experiment- deduced parameters. Moreover, it can even be extended towards the de- scription of time-dependent processes such as domain kinetics within the framework of the time-dependent Ginzburg-Landau model [14–16].
2.1.3 Microscopic considerations
A nonzero polarization of materials implies the existence of either permanent or induced electric dipole moments. Although treated abstractly in the phenomenological description above, they can in fact be attributed to a microscopic origin, related to the atomic and electronic displacements.
In ferroelectrics, the polarization can appear through two separate mech- anisms which are determined by the primary order parameter that brings about ferroelectricity. Proper ferroelectrics are distinguished by the ferroelec- tric distortion, which results in a cationic movement - effectively generating a permanent dipole through the motion of ions in opposite directions (in reality also strongly influenced by the changes in electronic density of the partially covalent bonds in the unit cell). Improper ferroelectrics, however, acquire their polarization as a secondary effect through coupling with an- other primary order parameter - such as lone pair hybridization (BiFeO3), magnetic ordering or structural changes (hexagonal rare-earth manganates).
A cartoon depicting a unit cell of PbTiO3 illustrates the onset of proper
Figure 2.5:Cartoon representing the ferroelectric transition in PbTiO3. The cubic high temperature state in (a) undergoes a phase transition to a ferroelectric low- temperature tetragonal state (b) with two equivalent polarization states denoted by PupandPdown. Image adapted from Prof. Matt Dawber.
ferroelectricity through cationic displacement with respect to the oxygen octahedra as the material undergoes a cubic-to-tetragonal transition below Tc in Fig.2.5. The polarization is oriented along a single axis and can thus acquire one of two possible values, represented in Fig. 2.5(b) asPup and Pdown.
2.1.4 Emergence of domains
As real ferroelectric crystals are finite in size, a discontinuity arises at the interface between the material and its surrounding medium - another solid, liquid, gas or vacuum. This, alongside with the presence of the polarization- bound charges in the ferroelectric, will give rise to a depolarizing field.
This is due to the fact that the polarization of the material not only comes from the field-dependent dielectric response 0E but also from the presence of a spontaneous polarizationP:
D=0E+P (2.11)
In order to satisfy the Poisson equation, any spatial variation ofD will give rise to a free charge density,
∇ ·D=ρ (2.12)
And therefore,
∇ ·E= 1
0(ρ−∇ ·Ps) (2.13)
Inside the ferroelectric, the polarization is supposed to be uniform, and therefore∇ ·E= ρ0 like in a classical dielectric. However, at the surfaces and other polarization discontinuities (such as defects or domain walls),∇·P will give rise to a depolarizing field, which will act against the polarization.
Multiple processes may occur in order to compensate this depolarizing field and are illustrated in Fig.2.6[17]. These include:
• Conduction that may occur inside the material, effectively compensat- ing the field by the flow of chargeσ, asρ=Rt
0σ·Edt.
• Charge may accumulate on the surfaces - coming either from surface adsorbates, atmospheric in origin at ambient conditions, or from the finite conductivity of the material.
• The polarization may be suppressed or rotate in a more favorable direction.
• In the case of ferroelectric/dielectric superlattices, a polarization may be induced in the dielectric layers to ensure continuity of the displace- ment field.
• Domains with different polarization orientations will form, with shapes and sizes depending on the geometry and material properties, as well as the electrostatic and strain boundary conditions.
The latter being of prime importance to the work described in the following chapters, domains and domain walls will be discussed in some detail below.
Kittel law
When the ferroelectric breaks down into domains of different polarization orientation in presence of a depolarizing field, the Gibbs free energy can be rewritten as an integral of the contributions of the displacement fieldD over the volumeV of the ferroelectric crystal - with the additional contributions of the depolarizing energyWE and domain wall energyWw:
G=G0+Z α 2D2+β
4D2+γ 6D6
dV +Ww+WE (2.14) As the physical properties of the material will depend on the various deriva- tives of the Gibbs free energy, it is safe to assume that the presence of domains and of the depolarizing field will have an effect on properties such as the specific heat and susceptibility. In the case of a uniaxial ferroelectric with the thickness much lower than the lateral dimensions, a periodic domain geometry is expected. In that case, the two additional energy contributions take the following form [18, 19]:
WE= ∗dP02V t Wd= σ
dV
(2.15)
Figure 2.6:In a finite freestanding ferroelectric, a uniform polarization is destabilized by the depolarizing field. Multiple processes can act in order to compensate this field, from the suppression of the polarization to a finite conductivity of free charge carriers for subsurface screening. The most common mechanisms will however be the screening by the presence of surface adsorbates, as well as the breaking of the material into domains. Image adapted from [17].
Figure 2.7:Kittel law scaling in multiple ferroic materials. The width of the domains, squared, as a function of film thickness (a) obeys a linear behavior as predicted by the Kittel law. If the periodicity of the width of the domains is renormalized by the width of the domain walls themselves, all curves fall on the same universal scaling law in (b). Image adapted from [20].
withdthe width of the periodic domains,∗a material constant that depends on the dielectric constant,P0 the remanent polarization within the domain, σthe domain wall energy per unit area, as well asV andt, the volume and thickness of the crystal, respectively. By minimizing the energy contribution to the Gibbs free energy ofWE+Wdwith respect to the domain widthd, it follows trivially that the equilibrium width scales as the square root of the crystal thickness:
d= σt ∗P02
1/2
(2.16) Domains larger than the equilibrium width will not be stable due to the contribution of the depolarizing energy, and conversely, smaller domains will not be allowed due to the high cost of the domain wall energy. This scaling of domain periodicity with thickness, called the Kittel law, has been found to be a good approximation for experiemental measurements of periodic domain width in multiple real ferroelectrics, as demonstrated in Fig.2.7(a), where both diverse ferroelectrics and a ferromagnetic material are shown to follow this general scaling. Moreover, if the domain size is renormalized by the effective domain wall thickness, all curves collapse onto a universal scaling law, shown in Fig.2.7(b) [20].
Domain reversal dynamics
If an appropriately oriented electric field of a sufficient magnitude is applied to a ferroelectric crystal, polarization reversal occurs. Following the mean field description of ferroelectrics we have discussed above, during this process an initially oriented crystal will simply reverse its polarization instantaneously towards the new field-dependent orientation. In practice, however, such
Figure 2.8:Domain reversal dynamics in ferroelectrics. A single domain unpoled crystal (a) will reverse polarization if a sufficient electric field is applied. The reversal will start with the nucleation (b) of small nuclei in the vicinity of defects. If above a critical size, the nuclei will be stable - enveloped by thin domain walls on their sides, and thick charged domain walls on their forward end to screen the polarization discontinuity. If the field is further applied, the growth of the domains will proceed further through forward growth (c) until a complete needle is formed throughout the crystal thickness. Sideways growth (d) will then occur, until the full polarization reversal (e) is complete. Image adapted from [25].
a homogeneous reversal seldom happens, as other polarization reversal mechanisms take over at lower field intensities due to the presence of defects or other inhomogeneities [21–26]. Thus, polarization reversal in real materials goes through the following stages, from the uniformly polarized state in Fig.
2.8(a):
1. Nucleation of small domains will happen inhomogeneously, at the positions of defects or thermodynamic fluctuations, as shown in Fig.
2.8(b).
2. Above a certain critical size, such a nuclei will be stable and exist in the surrounding material with thin domain walls on its sides and thicker charged domain walls at its forward end to suppress the polarization discontinuity.
3. Under sufficient field, these nuclei will then extend, as shown in Fig.
2.8(c), like needles in a process calledforward growth throughout the whole thickness of the crystal
4. Lateral growth will then occur, as illustrated in Fig. 2.8(d), with the domain walls travelling outwards from the initial nucleation site across the potential energy landscape of the crystal.
Finally, if the electric field is applied during a sufficient amount of time, complete polarization reversal will occur, as shown in Fig.2.8(e). Once the field is removed, the new configuration will be stable.
The kinetics of the above domain growth process have been observed by molecular dynamics [27,28] and described numerically with two distinct models. TheKolmogorov-Avrami-Ishibashi model (KAI) [29] describes the nucleation and growth of ferroelectric domains in a domain-wall-motion- limited switching process, with a constant domain wall velocity. The polar-
ization change as a function of time is described as:
∆P(t) = 2PS
1−exp−(t/t0)n
(2.17) withPS the remanent polarization magnitude,t0the characteristic switching time, andnthe geometric dimensionality of domain growth. This model has been successfully applied to bulk and thin film single crystals, but has failed to produce good results in switching studies of polycrystalline ferroelectrics, which have much slower dynamics. This incompatibility is explained by the more significant presence of defects such as grain boundaries, which in turn changes the behavior from a motion-limited to a nucleation-limited switching.
Thus, a new model was proposed byTagantsev et al, called theNucleation- limited-switchingmodel (NLS) [30]. In the latter, a distributionF(log(t0)) of local switching times models the inhomogeneous nucleation rate due to the presence of defects:
∆P(t) = 2PS
Z ∞
−∞
1−exp−(t/t0)n
F(log(t0))d(log(t0)) (2.18) The NLS model has been successfully applied to polycrystalline ceramics where polarization behavior is governed by nucleation.
An example of experimental validation of the above models is shown in Fig.
2.9, where a direct imaging of the polarization reversal of a ferroelectric capacitor by means of piezoresponse force microscopy shows a KAI behavior at short switching times and an NLS-like behavior at longer times due to the higher spread of switching times induced by the presence of defects [31].
2.1.5 Domain walls
When multiple domains coexist in a ferroelectric material, they are sepa- rated by domain walls. In fact, the latter - delimiting different polarization orientations - are not simple mathematical constructs, but have a physically meaningful nonzero width. In ferroelectric materials, the typical domain wall, spanning over a few unit cells, and thus of nanometer-scale width, is viewed to be Ising-like, with the polarization magnitude decreasing and changing sign across the boundary.3 This contrasts with ferromagnets, where a free rotation of the magnetic dipoles allows for smoother Néel or Bloch type transitions over broader length scales - with domain wall thicknesses in the range of 10-100 nanometers.
3In fact, while domain walls in ferroelectrics are of Ising type, they have both been predicted and shown experimentally to also present in some cases a smaller but measurable mixed Néel and/or Bloch character. [32–35]
Figure 2.9:Polarization reversal in ferroelectric capacitors imaged by piezoresponse force microscopy, with the switching dynamics fitted to the KAI/NLS models through the switching area as a function of time. The KAI model was found to be a good fit for short switching times, with an NLS-like behavior at longer times due to the higher spread of switching times induced by the presence of defects. Image adapted from [31].
Figure 2.10:Density functional theory calculations for a domain wall in the pro- totypical ferroelectric PbTiO3. Two unit cells with opposite polarization are shown in (a). The atomic displacements are indicated by green arrows. The calculation in (b) shows that the polarization changes orientation over several unit cells, giving an
effective domain wall width of a 1-2 nanometers. Image adapted from [32].
Observation of domain walls
Ferroelectric domain walls have been observed both by modeling and experi- mentally, confirming their spatial dimension. From density functional theory (DFT) calculations performed on a 180◦ domain wall in the prototypical ferroelectric PbTiO3, the width of the domain wall was found to be of the order of a few nanometers, as shown in Fig. 2.10 [32]. Moreover, at the position of the domain wall, the polarization locally drops to zero, implying the existence of an effective atomic scale paraelectric region which should have different properties.
These observations have been confirmed experimentally by means of
Figure 2.11: High resolution transmission electron micrograph acquired in the vicinity of a 180◦domain wall in a Pb(Zr0.2Ti0.8)O3 thin film. The atomic positions in (a) allow the polarization orientation to be determined, as shown in the inset cartoons. The domain wall is shown as a dashed line. A map of the relative cationic displacements in (b) makes it possible to extract the effective polarization orientation and magnitude, showing that the polarization decreases and locally vanishes at and around the location of the domain wall. Image adapted from [36].
high-resolution transmission electron microscopy, which allows the atomic positions in a lamella of material to be probed with subatomic precision. The micrograph shown in Fig.2.11has been acquired in the vicinity of a 180◦ domain wall in a Pb(Zr0.2Ti0.8)O3thin film [36]. The polarization orientation shown in Fig.2.11(a) is deduced from the cationic structure (indicated by the inset cartoon). A more precise analysis of the displacement of the atoms reveals the magnitude of the polarization, shown by yellow arrows in Fig.
2.11(b). As predicted by DFT, the domain wall spans over a few unit cells only, with the polarization dropping to zero at the location of the domain wall. Finally, a non-straight behavior of the domain wall can be observed along the thickness of the film - indicating the possible presence of charged head-to-head or tail-to-tail portions - that,while intuitively not stable, imply the presence of structural or electronic defects energetically favoring such a configuration, as domain wall energy is optimised in the presence of defects [37–40]. Defects in the ferroelectric can thus act as pinning sites for domain walls, and when rendered mobile by thermal activation or electric fields, they can also segregate at the domain walls [41–43].
Novel functionalities at domain walls
Their unique nature as an intrinsic interface within the parent ferroelectric phase can also confer on domain walls some unusual physical properties - resulting for example from the highly localised symmetry changes/breaking, off-stoichiometry, competing order parameters, and strain gradients - that are not present in the surrounding domains. These can give rise to functional responses which can be of interest not only for fundamental research but also potentially for novel devices and applications in the next generation electronics - in particular as a result of the intrinsically nanoscale width of the domain walls. The observed properties of ferroelectric domain walls can
be classified into four broad categories: structural, magnetic, optical and electrical. 4
As mentioned above, the domain wall and its immediate surroundings differ from the bulk of the neighboring domains. This manifests itself as a structural distortion in which the domain wall tends towards the paraelectric high symmetry phase, with locally a zero polarization state at its center (for a purely Ising type domain wall), or alternatively a rotation of the polarization component either in or orthogonal to the plane of the domain wall (in Néel and Bloch type domain walls, respectively). Such structural distortion was confirmed by high-resolution TEM studies [36,44–47]. In thin films with a high density of domain walls this can have significant effects on the overall properties of the material, for example decreasing the overall tetragonality in PbTiO3thin films [48]. In materials with further competing or coexisting order parameters, these locally different structural properties can moreover allow magnetism or optical couplings to emerge at the domain walls [49].
Magnetism has been postulated to exist from symmetry consideration at non-ferroelastic domain walls [50], a view that was backed by Landau theory calculations for the multiferroic BiFeO3[51]. An experimental confirmation of the possibility of existence of magnetism at ferroelectric domain walls has been shown in TbMnO3, where thin films of the material were grown under strain on SrTiO3 substrates. Thanks to the strain at the location of ferroelectric domain walls, a two-dimensional ferromagnetic phase was generated [52]. Moreover, domain walls have been shown to exhibit a strong exchange bias coupling with magnetic materials, as evidenced by experiments of their effects in BiFeO3 ferroelectric and Co0.9Fe0.1ferromagnet epitaxial heterostructure. [53]
The optical properties of domain walls have been known for decades and have been applied to industrially available devices. These properties have been traditionally attributed to the symmetry breaking at and in the vicinity of the domain wall due to the structural considerations already discussed above. For example, non-trivial second harmonic generation signals have been shown to arise at their locations, and were even linked to a more complex structure within the domain wall [33–35]. Moreover, recent studies have also demonstrated an anomalously high photovoltaic effect due to the presence of domain walls [54].
Finally, and of most relevance to this work, the domain walls can present novel and unusual electronic properties. As nanoscale objects, conductive domain walls carry the promise of reconfigurable electronic channels for device applications [6,55]. First observed by Seidel et al, domain walls in
4In most cases, as the ferroelectric thin films get thinner, more domain walls are present, as predicted from the Kittel law, and confirmed by experimental observations.
Thus, domain wall properties can become important and even determine the overall material properties [20].
BiFeO3 thin films have been found to exhibit a much higher conductivity than the surrounding bulk domain regions [56]. Several mechanisms have been proposed to explain this conduction - which since its original discovery has been observed in a large set of perovskite oxide ferroelectrics - a subject further developed in Chapter6.
2.2 Adsorbates
While the presence of polarization, complex dynamics of switching, and the functional properties of domain walls discussed in the previous sections are in a large part related to the fundamental physics of ferroelectric materials, the impact/contribution of the sample environment cannot be neglected. Of particular interest to this work are surface adsorbates - which have been found to play a key role in the screening of ferroelectric polarization, thus influencing switching dynamics, domain size and shape, and, in some cases, the functional response of the domain walls [57–59].
Although many chemical species can adsorb5 on surfaces, in ambient conditions surface water is regarded as the most prominent and omnipresent adsorbate - forming layers on all exposed materials.
These layers of water influence the properties of the surface and its interactions, particularly important for scanning probe microscopy studies.
Below, we therefore briefly discuss the adsorption of water and related processes in the context of such studies.
2.2.1 Water
Water is one of the simplest chemical substances surrounding us and yet has one of the most complex behaviors. It is essential to life and covers approximately 71% of the Earth’s surface.6 It is also the most prominent of atmospheric adsorbates, coating all surfaces exposed to ambient conditions with a molecular/nanoscopic layer.
Water is composed of one oxygen and two hydrogen atoms, and has the
5Adsorption of chemical species onto a surface has to be separated into two distinct processes: physisorption and chemisorption. In the case of physisorption, the interaction between the adsorbed chemical species and the surface depends on van der Waals interac- tions, driven by the permanent (in case of polar molecules such as water) or induced dipoles in the adsorbate and on the surface. For chemisorption, changes in the electronic structure of the surface and adsorbate are playing a role - leading to the formation of stronger chemical bonds. Due to the local nature of this interaction, chemisorbed adsorbates can only exist in the first molecular layer above the surface, and are usually involved in more complex chemical processes such as intercalation or dissociation.
6While accurate as of 2016, this number is likely to increase thanks to the accession to coal power and greenhouse-gas-friendly policies of the 45th president of the United States of America.
Figure 2.12:(a) Schematic illustration of water in a non-solid configuration. The H2Omolecules are randomly oriented, and decomposed charged species,OH−and H+ are present - with a pH of 7, or 10−7mol/l in pure water at 25◦C. (b) The effective charge of each species is illustrated, showing thatH2Ois a neutral dipole, H+a simple charge (proton), andOH−both dipolar and charged.
chemical formula H2O. Under normal conditions, a very small fraction of these molecules - whether in air, on surfaces or in liquid form - can be found decomposed intoH+ hydrogen cations andOH− hydroxyl anionic groups.
The coexistence of these species is shown in Fig.2.12(a). TheH2O,OH− andH+ molecules and radicals exhibit an electric dipole for the two former and a net charge for the two latter, as illustrated in Fig.2.12(b). Thus, they can be attracted onto polar surface as well as act as a screening mechanism for uncompensated surface charges.
Water in air
In order to understand its interaction with surfaces, a measurement of water content in the air is essential. The physical parameter used to quantify the amount of water in the surrounding air is given by the temperature- dependentrelative humidity(RH). It is expressed as a percentage defined as follows:
RH= Pw Pws
·100% (2.19)
withPw the water vapor pressure andPws the saturation water pressure, which represents a dynamic steady state between evaporation and conden- sation. When the relative humidity reaches 100%, condensation therefore occurs by means of the formation of water droplets. The above formula is valid for all temperatures, with physical restrictions as to the maximum
values of relative humidity above 100◦C(condensation cannot occur in an unpressurised vessel) and below 0◦C (where ice formation occurs). Thus, condensation will occur when the water vapor pressure in a gas will reach the saturation water pressurePws- a defining temperature-dependent property of water that for calculations can be approximated by a functional valid from 0◦C up to 373◦C:
lnPws
Pc
=Tc
T C1θ+C2θ1.5+C3θ3+C4θ3.5+C5θ4+C6θ6 (2.20) withT the temperature in K,θ= 1−TTc the reduced temperature,Pc = 220640 hP athe critical pressure,Tc= 647.026K the critical temperature, andC1toC6the empirical coefficients7fitted to measured properties of water [60]. Although this gives the saturation pressure above which condensation will occur, the exact quantity of water in a certain mass of gas is dependent on both the gas and the pressure, and is given by the mixing ratioX, which is the ratio of the mass ofH2O per unit mass of dry gas:
X =B Pw
Ptot−Pw (2.21)
with Ptot the total pressure andB a gas-dependent constant. For air,B= 621.9907g/kg, but is in general calculated by the molecular weight ratio of H2O to gas molecules. This definition of water content is not practical and more usually the absolute humidityAis preferred, as it represents the mass of water in a given volume of gas:
A=CPw
T (2.22)
with C= 2.16679gKJ andT the temperature inK. With this definition, it becomes possible to approximate the effective humidity in the environment of an object which is maintained at a different temperature than the surrounding volume.
Thanks to the above equations, it is possible to relate the relative humidity measured by a sensor to the exact water content in the volume of air above the investigated surface - paving the way towards a quantitative understanding of the effect of water on the properties of materials.
Water on surfaces
In presence of ambient water, all surfaces will adsorb some water molecules.
Although depending on multiple processes, to a large extent this behavior
7For the sake of completeness, the numerical values of the coefficients are:C1 =
−7.85951783,C2= 1.84408259,C3=−11.7866497,C4= 22.6807411,C5=−15.9618719, C6= 1.80122502
