High-order electromechanical couplings in ferroelectrics THÈSE NO 6266 (2014) PRÉSENTÉE LE 15 JUILLET 2014 À LA FACULTÉ DES SCIENCES ET TECHNIQUES DE L'INGÉNIEUR LABORATOIRE DE CÉRAMIQUE PROGRAMME DOCTORAL EN SCIENCE ET GÉNIE DES MATÉRIAUX ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR Alexander KVASOV acceptée sur proposition du jury: Prof. H. Hofmann, président du jury Prof. A. Tagantsev, directeur de thèse Prof. J. Hlinka, rapporteur Prof. I. Kornev, rapporteur Prof. N. Marzari, rapporteur Suisse 2014 "Gravitation is not responsible for people falling in love." – Albert Einstein To Bárbara. Acknowledgements I would like to express my high gratitude to my thesis director Prof. Alexander Tagantsev who guided me throughout my thesis. Always available for discussion, always ready to help and explain, he taught me how to solve the problems and moreover how to solve difficult problems myself. His profound understanding and knowledge of almost all areas of physics in combination with his unique talent of explaining complex problems and ideas in a simple way makes him not only an excellent scientist, but also a very good supervisor. Without him this work would never have been accomplished. I would like to thank the members of my thesis jury: Prof. Heinrich Hofmann (EPFL), Prof. Nicola Marzari (EPFL), Prof. Igor Kornev (École Centrale Paris) and Prof. Jirka Hlinka (Institute of Physics, Academy of Sciences of the Czech Republic) I also would like to thank people from Ceramics Laboratory, especially Arnaud Crassous, Leo Mc Gilly, Maxim Gureev, Kaushik Vaideeswaran, Peter Yudin, many others, for interesting and useful discussions in the lab and even more interesting free time out of the lab. I want to acknowledge people from THEOS laboratory especially Dr. Giovanni Pizzi and Prof. Nicola Marzari for their valuable help with the first principals calculations. I am thankful to the secretary of the doctoral school Erika Menamkat and the secretary of LC Yuko Kagata who were always very friendly and ready to help. I am grateful to all my friends for their support in different aspects of my life during my PhD study. Finally, I would like to thank my family and, of course, Barbara, without whom my life would never be the same. Lausanne, 24 April 2014 A.K. v Abstract The thesis deals with electromechanical phenomena in ferroelectrics. High-order electrome- chanical couplings, in contrast to the ordinary low-order electrostrictive coupling (coupling which is linear in deformation and quadratic in polarization), were usually neglected in theo- retical consideration. This thesis will be focused on the investigation of the systems where these couplings are of importance, e.g. ferroelectric thin films, tunable Film Bulk Acoustic wave Resonators (FBARs), and problems involving the flexoelectric effect. This work was accomplished through a combination of two approaches: ab initio methods and phenomeno- logical Landau theory. We demonstrate that high-order electromechanical couplings are of importance for strained ferroelectric films. It is shown that theoretical methods can suffer from a substantial inaccu- racy unless these higher-order electromechanical interactions are taken into account. This statement is illustrated with an example of a temperature - misfit strain phase diagram of a BaTiO3 (BTO) thin film derived with high-order electromechanical interactions evaluated with the help of first principles calculations. Another domain where high-order electromechanical interactions are important is tunable FBARs, where the antiresonance frequency tuning is governed by the high-order electrome- chanical coefficient. The non-linear electrostrictive coefficient was for the first time calculated for BTO and SrTiO3 (STO) using ab initio methods. Then, taking into account the small dif- ference of obtained values for BTO and STO these results were linearly interpolated to the Bax Sr1 x TiO3 (BST) compositions. The obtained values are consistent with previously made ¡ order-of-magnitude estimates. Using parameters obtained with first principles calculations, we simulate the resonance behaviour of BST based tunable FBARs. Resulting antiresonance tuning is smaller than expected due to the compensation of two competing terms conditioned by linear and non-linear electrostrictions. Our calculations also confirm that, for tunable FBAR modeling, it is important to use a polarization-based Landau free energy expansion taking into account both non-linear electrostriction and background permittivity. It is demonstrated that for classical perovskites in the paraelectric phase, such as BTO, STO, and PbTiO3 (PTO), electrostrictive strain induced by an electric field may not obey traditionally considered "extension along the field and contraction perpendicular to it" behaviour if the ap- plied electric field is directed obliquely to the cubic crystallographic directions. A remarkable behaviour is predicted for bars of bulk BTO, STO, and PTO cut along the [111] crystallographic cubic direction. In this case, for an electric field parallel to the bar, some expansion along all its three dimensions is expected. The situation with the [110] cut is also unusual, the electrostric- vii Abstract tive effect from a field along the bar will result in a contraction in one transversal direction and an expansion in the other. In terms of the effective "polarization-strain" electrostrictive coefficients of the sample, this means a positive transversal electrostriction coefficient for some sample orientations. The related effects take place in differently oriented perovskite thin films. Here, we identify the manifestation of the angular dependance of electrostriction in the phase diagrams of thin films with the example of BST 30/70. We study the behaviour of the ferroelectric phase transition temperature of the film depending on its orientation. It was found that, for some film orientations, the phase transition temperature can be less than that in the bulk material. Ferroelectric properties of materials can be markedly changed through strain engineering, therefore the use of the thin films grown in different orientations gives more flexibility for engineering and provides a broader range of operating temperatures. The flexoelectric effect is a response of electric polarization to a mechanical strain gradient. It represents a higher-order effect with respect to piezoelectricity. Often this effect can be neglected, however, at the nanoscale, where large strain gradients are expected, the flexo- electric effect becomes appreciable. A significant number of recent publications devoted to flexoelectricity shows an increasing interest in the phenomenon. In this work, we demonstrate the possibility of the extraction of the bulk flexoelectric tensors from the phonon spectrum. For this, we exploit a model phonon spectrum of STO, which was calculated using first princi- ples methods. The dynamic flexocoupling coefficient, which was customarily not taken into account, but which gives a substantial contribution to the total flexocoupling coefficient, was for the first time obtained using ab initio calculations. Finally, antiphase domain boundaries (APB) in antiferroelectric PbZrO3 are studied using ab initio full relaxation calculations. Following transmission electron microscopy experiments, where the polarity in APB was originally found, our calculations confirm the presence of local polarity and, moreover, show the bistability of the APB structure. Taking into account the above results, one can conclude that the phenomenon observed at the APB is a signature of local ferroelectricity. Ferroelectric APBs in antiferroelectrics are of high interest since they can be potentially viewed as functional 1-10 nm wide elements, which carry the information. Key words: ferroelectrics, ferroelectricity, flexoelectricity, thin films, electrostriction, Landau theory, ab initio calculations viii Résumé Cette thèse traite des phénomènes électromécaniques dans les ferroélectriques. Les couplages électromécaniques d’ordre élevé, au contraire du couplage électrostrictif de bas ordre (cou- plage linéaire avec la déformation et quadratique avec la polarisation) sont généralement négligés dans les modèles théoriques. Cette thèse se concentre sur l’étude des systèmes dans lesquels ces couplages sont importants, tels que les films minces ferroélectriques et les résonateurs acoustiques ajustables, et les problèmes impliquant l’effet flexoélectrique. Ce travail a été réalisé en combinant deux approches : des calculs ab initio et la théorie phénoménologique de Landau. Nous avons démontré que les couplages électromécaniques d’ordre élevé sont importants pour les films ferroélectriques contraints, et que les modèles théoriques utilisés peuvent souffrir d’imprécisions substantielles à moins de les prendre en compte. Cette affirmation est illustrée par un diagramme de phases de la température en fonction de la contrainte épitaxiale de films minces de BaTiO3 (BTO). Celui-ci a été calculé à l’aide des couplages électromécaniques d’ordre élevé à partir de calculs des premiers principes. Un autre domaine dans lequel les interactions électromécaniques d’ordre élevé sont impor- tantes est les résonateurs acoustiques ajustables pour lesquels ils contrôlent la fréquence d’antirésonance. Le coefficient électrostrictif non-linéaire a été calculé pour la première fois pour BTO et SrTiO3 (STO) en utilisant des méthodes ab initio. En prenant en compte les faibles différences de valeurs pour BTO et STO, les résultats ont ensuite