On the non-analytic behaviour of thermodynamic potentials at first order phase transitions

This work is devoted to the study of the analyticity properties of thermodynamic potentials (free energy, pressure) for classical lattice systems at low temperature. The central topic of our analysis, in this framework, is to show rigorously the absence of analytic continuation at points of first order phase transition. Our first result applies to the general class of two phase models considered in the Theory of Pirogov-Sinai. The analysis reveals that the Peirls condition, which is the basic hypothesis of the theory, suffices to show the absence of analytic continuation of the pressure at the transition point. In a second part, we study a particular two body potential, of the form γdJ(γx), where γ > 0 is a small parameter and J a function with bounded support (in the limit γ —> 0, this potential gives a rigorous justification of the "equal area rule" of the van der Waals-Maxwell Theory). For all small strictly positive values of the parameter γ, we show that the free energy has no analytic continuation at the transition points. These results confirm early conjectures stating that the finiteness of the range of interaction is responsible for the presence of singularities in the thermodynamic potentials.

Pulsed laser deposition of (SrBa)Nb2O6 thin films and their properties

Anna Infortuna

Strontium barium niobate (SrxBa1-xNb2O6, shortly SBN) is a solid solution system with tetragonal tungsten bronze crystal structure. It exhibits a ferroelectric phase with only one polar axis and a transition temperature depending on the Sr/Ba ratio. This thesis studies the growth of SBN thin films, in order to obtain a ferroelectric thin film model system for the study of 180° domain switching close to the ferroelectric phase transition. SBN has been chosen not only because uniaxial, but because it is possible to tune the temperature of phase transition varying the Sr/Ba ratio. It is thus possible to study electric properties of the phase transition close to room temperature, where conduction phenomena related to defects are normally lower. Thin films of SBN were grown by Pulsed Laser Deposition in a vacuum system constructed during this thesis work. For the formulation of a model it is desirable to have a system as close as possible to a single crystal. Therefore deposition parameters have been optimized to grow SBN thin films with the polar axis oriented perpendicular to the film plane. The films are integrated in parallel plate capacitor structures, in order to measure dielectric properties. The stability of the ferroelectric phase is found to be sensitive to impurities; contaminants with concentration below 1% induce a lowering of the transition temperature of about 100°C. Strain as well has influence on the phase transition; in the case of substrate with thermal expansion coefficients different from the SBN ones, the SBN film is under stress. A 0.6% tensile strain induces a lowering of the transition temperature of about 100°C, this is the case of silicon single crystal substrate. Grown on strontium titanate single crystals (STO), whose thermo-mechanical properties are close to the one of SBN, the films exhibit a phase transition temperature compatible with the one measured on single crystals. The obtained films suffer from leakage that is reduced by thermal treatment in oxygen rich atmosphere. The oxygen vacancies, created during the deposition process, are considered to be responsible of leakage. Vacancies recovery upon annealing does not eliminate the problem, and measurement of polarization at room temperature is difficult. However, piezoelectric measurements performed on individual grains, with atomic force microscope, prove that these have ferroelectric properties; by these experiment has been proved that it is possible to switch the polarization. Films with a disordered structure of grains are much less leaky than the ones with columnar grains spanning the whole cross section. These observations lead to the conclusion that conduction goes trough the grain boundaries. On STO single crystals, SBN grows epitaxial in Volmer-Weber mode. With a suitable preparation of the substrate surface, it is possible to induce the growth of films with different orientations of the polar axis: in the film plane, perpendicular to it or inclined of 45°. A model is proposed, explaining the epitaxial relation with the substrate on the basis of the perovskite kernel contained in the unit cell of the tetragonal tungsten bronze structure. The film grows from the nucleation of the perovskite structure that rules the orientation of the film; the high temperature at which the substrate is kept during the deposition, assure the necessary mobility for the arriving atoms to organize in the SBN structure around these nuclei. The study of literature data allows to state that such a model is valid not only for substrates with perovskite structure, but can be extended to substrates with a cubic crystal structure, like magnesium oxide.

EPFL2005

Etude par microscopie électronique à transmission à haute résolution de l'élargissement thermique des parois de domaines ferroélectriques dans un système de premier ordre

The macroscopic properties of ferroelectric materials depend directly on the domain configuration and the structure of the domain boundaries. For this reason, their study is of great scientific and technological interest. Several models have been developed to describe the properties of domain walls (structure, thickness, stress at the interface, mobility, …). However, few quantitative experimental observations of ferroelectric domain walls at an atomic scale have been reported, and even less results exist at higher temperatures. The ferroelectric domain-wall thickness is an important parameter whose behavior as a function of temperature is directly related to the order of the phase transition. In a second-order system, the temperature dependence of the domain-wall thickness follows a power law L ~ (Tc - T)-ω where ω is the critical exponent characterizing the divergence of L near the critical temperature Tc The predictions concerning the numerical value of are different depending on the theoretical model considered. Nevertheless, the majority of ferroelectric materials are known to undergo a first-order transition. In these systems the domain-wall thickness increases when the transition temperature is approached, but without diverging as for a second-order phase transition. The major objective of this work was to measure this broadening predicted by the theory for the first time at the approach of the phase transition temperature. We have shown that high resolution transmission electron microscopy is powerful technique for the study of ferroelectric domain walls not only at room temperature but also at higher temperatures. This method allows a direct and local observation of interfaces at an atomic scale which is a considerable advantage over diffraction techniques (X-ray, neutron, electron) and most of the other electron microscopy methods. However, the requirements of this method concerning the performance of the optical system of the microscope, the quality of the specimen and its stability in the column are very high, and today high resolution microscopy at high temperature remains a major challenge. The domain-wall thickness is obtained from the measurement of the crystal lattice distortion near the interface using two different numerical techniques of image analysis. The first method is based on the accurate determination of the center of bright peaks which appear on high resolution images for a judicious choice of the experimental parameters (delocalization and crystal thickness) and which represent the crystal lattice nodes. The second technique consists in selecting a circular region of the power spectrum around a reflection peak, centering the Fourier space on this reflection and performing the inverse Fourier transform. The information about the local displacement of atomic planes corresponding to the selected reflection is extracted form the phase component of the obtained complex image. Both techniques have been successfully applied to the measurement of the thickness of the 90° ferroelectric domain walls in PbTiO3 single crystals. This perovskite presents a first order phase transition at Tc = 492.2°C. which corresponds to the transformation of the crystal from the high-temperature paraelectric cubic phase to the low-temperature ferroelectric tetragonal phase. For the first time, the thermal broadening of domain walls has been measured quantitatively using HRTEM. The results are compared with the predictions made by phenomenological theories. We find that the domain wall thickness increases continuously from 0.7 nm at room temperature up to 5.5 nm near the transition temperature. Our results are consistent with those obtained by other authors at room temperature and those of a very recent study performed at higher temperatures using weak-beam microscopy. The measured domain-wall broadening is greater than the one predicted by a three-dimensional Ginzburg-Landau model where the polarization which is the primary order parameter is coupled to the elastic strain which plays the role of secondary order parameter. Our results are better described by a one-dimensional Ginzburg-Landau model without secondary order parameter. In this case the gradient coupling coefficient κ of the Ginzburg-Landau free energy is found to be 1.3 · 10-10 m3F-1. The methods developed for the measurement of the domain-wall thickness can by applied not only to other first-order ferroelectric crystals but also to ferroelectric systems presenting a second-order phase transition thus allowing the determination of the critical exponent ω. Characteristics only present in 2-D systems could also be detected by studying very thin specimens. Moreover, the same techniques can be adapted to the study of domain walls in purely ferroelastic materials.

EPFL2000

Anomalous solitons

Matthias Schmid

Solitons are stable, non-singular solutions to the classical equations of motion of non-linear field theory. Their energy is localized and finite and their shape remains unaltered during propagation. For this reason they represent particle-like states in field theory. Their mass and their size can be very large compared to those of the elementary particles in the theory. Therefore, a soliton can be viewed as a single particle-like object containing a large number of individual particles. The chiral Abelian Higgs model contains an interesting class of non-topological solitons, that carry a non-zero fermion number NF or Chern-Simons number NCS, which is the same because of the chiral anomaly. They consist of a bosonic configuration of gauge and Higgs fields characterized by NCS and are stable for sufficiently large NCS. In the first part of this thesis we study the properties of these anomalous solitons. We find that their energy-versus-fermion-number ratio is given by E ∼ NCS3/4 or E ∼ NCS2/3 depending on the structure of the scalar potential. For the former case we prove, using some inequalities from functional analysis, that there is a lower bound on the soliton energy, which reads E ≥ c NCS3/4 , where c is some parameter expressed through the masses and coupling constants of the theory. We construct the anomalous solitons numerically for two different choices for the potential accounting both for Higgs and gauge dynamics. Solutions are obtained as a function of NCS and the Higgs mass mH and we find that they are not spherically symmetric. In addition, we outline a relation between the structure of anomalous Abelian solitons and the intermediate state observed in type-I superconductors in external magnetic fields. In the limit of large NCS anomalous solitons can be described in the thin wall approximation, which allows us to remove the Higgs field from consideration. For absolute stability of anomalous solitons, it is essential that the gauge group is Abelian. If the gauge group is non-Abelian, fermions can always be converted to a gauge vacuum configuration with an arbitrary integer NCS. Therefore, if anomalous non-Abelian solitons exist, they could only be metastable. Interestingly, anomalous solitons can potentially exist in the electroweak theory, because this theory contains all necessary ingredients, namely chiral fermions and an Abelian gauge symmetry. In the second part of this thesis we investigate this possibility. Using the numerical solutions for anomalous Abelian solitons as a starting point, we construct the corresponding numerical solutions in electroweak theory. These solutions have a similar structure as the Abelian solitons with the Abelian gauge field replaced by the Z boson field. The charged boson fields W± vanish identically. However, for weak mixing angle θω > 0, the solutions have an associated magnetic field as well, that can be characterized by a magnetic dipole moment mem. Furthermore, the shape of the solutions and the structure of the gauge fields depend on θω. In the last part of this work we analyze the classical stability of the numerical solutions in the electroweak case. It is clear that the solutions are stable in the semilocal limit sin θω → 1, where the Abelian case is reproduced exactly. For arbitrary θω, we consider perturbations in the Higgs field and in the gauge fields Z and A and show that the solutions are stable with respect to these perturbations. For small θω however, the solutions are unstable with respect to the formation of a condensate of charged boson fields W± in the centre of the solution. This W-condensation instability is essentially the same, which also destabilizes the Z-string solution of electroweak theory.

EPFL2008