Ginzburg–Landau theory

Summary

In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. One GL-type superconductor is the famous YBCO, and generally all Cuprates. Later, a version of Ginzburg–Landau theory was derived from the Bardeen–Cooper–Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given. This general setting then extends to quantum field theory and string theory, again owing to its solvability, and its close relation to other, similar systems.

Official source

https://en.wikipedia.org/wiki/Ginzburg%E2%80%93Landau_theory

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High Tc superconductivity in cuprates is still an unresolved topic, and one particular challenge is why in-plane compressive strain enhances Tc in ultra thin LSCO-214 films (< 40nm), whilst tensile strain diminishes it. Especially intriguing is whether altering this strain at a given doping has an effect on T* (pseudogap) or Tc (condensation/phase coherence). As the ratio between T* and Tc is so high, we systematically grew a series of BSCCO-2201 films by in house pulsed laser deposition (PLD), and then thoroughly analysed these films with X-Rays, four point resistivity measurements and XPS. We succeeded to induce strain into the films; however, the strain was lost in the annealing process necessary for superconductivity. Subsequently, we focused on LSCO-214 films, and grew a series of strained ultra thin films of this compound. While the films were superconducting, none of our samples showed enhanced superconduc- tivity, probably due to growth disorder.Therefore, we eventually studied high quality films grown by MBE from BNL. We used the low energy muon beamline at PSI to perform μSR studies on two different com- pressive strain states. The successful measurements of the penetration depth and Ginzburg- Landau parameter κ on ultra thin films of this type using low energy muons showed the benefits of this technique. As the 2 films were of different thicknesses, further studies need to be done to disentangle which observed effects are due to strain and/or variation in c-axis, and which are due solely to thickness. Our accumulated results and those of other groups, (notably Naito et. al. at NTT) on strain in LSCO-214 films, and the ensemble of data, including our study, indicate that most likely it is a phase coherence enhancement due to the in-plane compressive strain that raises the Tc. In summary, in addition to the succesful implementation of PLD heteroepitaxy, XRD analysis and ρ(T) studies of LSCO-214 films we were able to establish that: 1. Phase coherence is improved by in plane compressive strain, hence Tc enhancement 2. Tc also seems to scale with apical oxygen, as previously reported by H. Keller et. al., and local distortions around the copper-oxygen tetrahedra 3. Last but not least, we were even able to induce strain into BSCCO-2201 films, yet at present the full electronic properties of this system are neither measured nor under- stood. Finally, further systematic studies on μSR are proposed, and are indeed underway at PSI, to further elucidate the microscopic role of strain in the mechanism of high Tc superconductivity.

EPFL2013

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

This thesis consists of a theoretical analysis of charged domain walls in ferroelectrics based on Landau theory and the theory of semiconductors. First, the internal structure of a 180-degree charged domain wall is considered. It is shown that different regimes of screening and different dependences for width of charged domain walls on the temperature and parameters of the system are possible, depending on spontaneous polarization and concentration of carriers in the material. In typical perovskites the screening is provided by a degenerate electron gas in the nonlinear screening regime. The corresponding width of charged domain walls is about one order of magnitude larger than the width of neutral domain walls. The formation energies of "head-to-head" walls in an isolated sample under different regimes of screening are derived, neglecting the poling ability of the surface. In the nonlinear regimes of screening, this energy is equal to the energy necessary for the creation of electron-hole pairs in a sufficient amount to screen the spontaneous polarization, which is proportional to the band gap of the ferroelectric. It is shown that if the poling ability of the surface is large enough, either "head-to-head" or "tail-to-tail" configurations can be energetically favorable in comparison with the monodomain state of the ferroelectric. For the case of a surface with weak poling ability the existence of charged domain walls in bulk ferroelectrics is merely a result of domain-growth kinetics. It is shown that, at large enough sample thicknesses, a charged domain wall can be energetically favorable in comparison to other possible states: a multidomain state with antiparallel domains separated by neutral walls and a state with zero polarization. This size effect provides a possible explanation of the reason for experimental observation of 180-degree charged domain walls in bulk lead titanate but not in barium titanate. The results obtained for the case of an isolated ferroelectric sample are compared with the results for a sample with electrodes. It is shown that a charged domain wall in an electroded sample can either be metastable or stable, depending on the work function difference between the electrodes and the ferroelectric and the poling ability of the surface. The interaction of an electric field with charged domain walls in ferroelectrics is also addressed. A general expression for the force acting per unit area of the a charged domain wall carrying free charge is derived. It is shown that in proper ferroelectrics, the free charge carried by the wall dependeds on the size of the adjacent domains. As a result, it is found that the mobility of such a domain wall (with respect to the applied field) is sensitive to the parameters of the domain pattern containing this wall. The problem of the force acting on a planar charged 180° domain wall normal to the polarization direction in a periodic domain pattern in a proper ferroelectric is solved analytically. It is shown that, in the linear regime under small applied field, the forces acting on walls in such a pattern increase with decreasing wall spacing. The direction of the forces coincides with those for the case of the corresponding neutral walls. At the same time, for large enough wall spacings and large enough fields, these forces can be of the opposite sign. It is shown that the domain pattern under consideration is unstable in a defect-free ferroelectric. The poling of a crystal containing such pattern, stabilized by the pinning pressure, is also considered. It is shown that, except for a special situation, the presence of charge domain walls can make poling more difficult. It is demonstrated that the results obtained are also applicable to "zig-zag" walls under the condition that the "zig-zag" amplitude is much smaller than the domain wall spacing. A mechanism that leads to a strong enhancement of the dielectric and piezoelectric properties in ferroelectrics with increasing density of charged domain walls is proposed. It is demonstrated that an incomplete compensation of bound polarization charge at these walls creates a stable built-in depolarizing field across each domain leading to increased electromechanical response. This model clarifies a long-standing unexplained effect of domain wall density on macroscopic properties of domain engineered ferroelectrics. It is demonstrated that lead-free ferroelectrics like barium titanate with dense patterns of charged domain-walls are expected to have strongly enhanced piezoelectric properties, thus suggesting a new route to high-performance, lead-free ferroelectrics. The problem of the screening of ferroelectric bound charge on a ferroelectric-semiconductor interface is considered. It is shown that the screening ability of the interface can be drastically improved due to the presence of a built-in potential in the semiconductor, which depends on the electron affinities and surface state density and can be controlled by a judicious choice of materials.

EPFL2012

EPFL2013

EPFL2000