De Gennes Theory Research Articles

Inhomogeneous stripe phase and tunable Majorana zero modes in Rashba s-wave superconducting systems in the presence of in-plane Zeeman field

Based on the extended Bogoliubov–de Gennes theory, the inhomogeneous stripe order and Majorana bound states are demonstrated in s-wave superconducting systems with Rashba spin–orbit interaction when the in-plane Zeeman field is applied. For a fully open square sample, topological phase transitions can be driven by the Zeeman field, and the stripe phase with spatially oscillating order parameter shows up across a critical field strength. The topological channels arise in this phase, which can be utilized to create and manipulate Majorana zero modes. Interestingly, (quasi-)one-dimensional channels with diminished pairing amplitude can be formed in narrow arms of a square loop, accompanied by the reconstruction of energy spectra of the condensate and the realization of robust Majorana zero-energy states at the ends of channels. The associated evolution of topological phases and the location of Majorana zero modes are highly sensitive to the field direction. Moreover, the effects of the rectangular aspect-ratio and the off-centered hole as well as the surface defects on Majorana end modes are explored in Rashba asymmetric loops. In comparison, the field-dependent evolution processes of low-energy levels behave more complicated because the emergence of confined topological channels can be effectively tuned by the length and width of loop arms as well as the size and position of an introduced small indentation at the outer edge. Rich patterns of Majorana corner-like states are generated for such asymmetric systems. Our theoretical predictions may shed new light on the tunability of Majorana zero modes and provide useful guidance for future experiments and applications.

Hidden symmetry of Bogoliubov de Gennes quasi-particle eigenstates and universal relations in flat band superconducting bipartite lattices

Unconventional flat band (FB) superconductivity, as observed in van der Waals heterostructures, could open promising avenues towards high-T_cc materials. In FBs, pairings and superfluid weight scale linearly with the interaction parameter, such an unusual behaviour justifies and encourages strategies to promote FB engineering. Bipartite lattices (BLs) which naturally host FBs could be particularly interesting candidates. Within Bogoliubov de Gennes theory and in the framework of the attractive Hubbard model in BLs, a hidden symmetry of the quasi-particle eigenstates is revealed. As a consequence, we demonstrate universal relations for the pairings and the superfluid weight that are independent of the characteristics of the hopping term. Remarkably, it is shown that these general properties are insensitive to disorder as long as the bipartite character is protected.

Theoretical models for the nematic twist-bend phase and the Landau de Gennes theory of Longa and Tomczyk

ABSTRACT We present a critical analysis of the existing phenomenological theories of the nematic twist-bend phase including the models proposed by Dozov and Selinger and the more general Landau de Gennes theory of L. Longa and W. Tomczyk. The role of the flexoelectric polarisation is considered in detail as well as the stabilisation of the nanoscale helical structure of the twist-bend phase. In particular we focus into the mechanism of the reduction of the bend elastic constant determined by negative flexoelectric corrections and discuss why the splay elastic constant is not reduced at all. A molecular theory of elasticity of bent-core nematics, which also enables one to describe the reduction of the bend elastic constant, is considered in detail as well. Finally we consider two different tensor orientational order parameters of the twist-bend phase which describe all kinds of uniaxial and local biaxial ordering of bent-core molecules in the twist-bend phase.

The radial hedgehog solution in the Landau–de Gennes theory: Effects of the bulk potentials

We study equilibrium configurations in spherical droplets of nematic liquid crystal with strong radial anchoring, within the Landau–de Gennes theory with a sixth-order bulk potential. The sixth-order potential predicts a bulk biaxial phase for sufficiently low temperatures, which the conventional fourth-order potential cannot predict. We prove the existence of a radial hedgehog solution, which is a uniaxial solution with a single isotropic point defect at the droplet centre, for all temperatures and droplet sizes, and prove that there is a unique radial hedgehog solution for moderately low temperatures, but not deep in the nematic phase. We numerically compute critical points of the Landau–de Gennes free energy with the sixth order bulk potential, with rotational and mirror symmetry, and find at least two competing stable critical points: the biaxial torus and split core solutions, which have biaxial regions around the centre, for low temperatures. The size of the biaxial regions increases with decreasing temperature. We also compare the properties of the radial hedgehog solution with the fourth-order and sixth-order potentials respectively, in terms of the Morse indices as a function of the temperature and droplet radius; the role of the radial hedgehog solution as a transition state in switching processes; and compare the bifurcation plots with temperature, with the fourth- and sixth-order potentials. Overall, the sixth-order potential has a stabilising effect on biaxial critical points and a de-stabilising effect on uniaxial critical points and we discover an altogether novel bulk biaxial critical point of the Landau–de Gennes energy with the sixth-order potential, for which the bulk biaxiality is driven by the sixth-order potential.

A novel strategy to predict the tensile strength of polymer/particle interphase based on De Gennes's self‐similar carpet theory

AbstractThe formation of polymer/particle interphase region besides affecting most of the mechanical characteristics of polymer nanocomposites also alters some of their physical properties. The physical/mechanical properties inside the interphase may change based on a specific pattern (e.g., linear, exponential, etc.). De Gennes's self‐similar carpet theory represents a unique definition for the concentration variation of the adsorbed polymer chains. Accordingly, this specific pattern was developed to predict the tensile strength of the polymer/particle interphase region based on the molecular characteristics of the matrix and polymer/particle compatibility. In addition, Pukanszky's model was developed to define the characteristics of the aggregates/agglomerates using the results of De Gennes's theory. Different high‐density polyethylene (HDPE) nanocomposite samples, containing compatibilized silica nanoparticles, were prepared and subjected to the tensile and field emission scanning electron microscopy (FE‐SEM) tests. Moreover, other data from the literature were used to further investigate the accuracy of the proposed model. It was revealed that considering the gyration radius of the polymer chains as the maximum interphase thickness and the scaling parameter as 0.49 provides accurate predictions. On the other hand, the obtained data regarding the average content and size of nanoparticle aggregates/agglomerates also were in good agreement with the actual results.

Pattern formation in Landau–de Gennes theory

We study the spherical droplet problem in 3D Landau–de Gennes theory with finite temperature. By rigorously constructing the biaxial–ring solutions and split–core–segment solutions, we theoretically confirm the numerical results of Gartland–Mkaddem in [15]. The structures of disclinations are also addressed.

Numerical simulation of the 2D lid-driven cavity flow of chiral liquid crystals

ABSTRACT In this study, the two-dimensional (2D), lid-driven cavity flow of chiral liquid crystals (LCs) was modelled using the Landau de-Gennes (LdG) theory. Parametric studies investigating the effect of Er (Ericksen number) and Θ (chiral strength) on the microstructure of chiral LCs were performed. In this study, we observed that an increase in Θ caused the chiral texture to have more striations and shorter pitch lengths as causes for an increased number of defects. When Θ was held constant, an increase in Er disrupted the chiral structure and even broke it at very high Er. Interestingly, a transition from low Er (10) to moderate Er (1,000) increased the number of defects; however, further increases in Er reduced the number of defects since much of the chiral structure was destroyed by the high viscous flow effects. In particular, even at very high Er, the chiral structure and defects were still present as a vortex was always present with lower velocities, where the viscous flow effect was smaller. We also found that a hexagonal structure with penta-hepta defects formed at high chiral strengths.

Boundary Effect and Critical Temperature of Two-Band Superconducting FeSe Films

Based on two-band Bogoliubov–de Gennes theory, we study the boundary effect of an interface between a two-gap superconductor FeSe and insulator (or vacuum). New boundary terms are introduced into two-band Ginzburg–Landau formalism, which modifies the boundary conditions for the corresponding order parameters of superconductor. The theory allows for a mean-field calculation of the critical temperature suppression with the decrease in FeSe film thickness. Our numerical results are in good agreement with the experimental data observed in this material.

Effects of asymmetric geometry and surface defects on vortical configurations in mesoscopic superconducting loops

In the framework of the microscopic Bogoliubov–de Gennes theory, we investigate the vortical patterns of dominant chiral p-wave orders in mesoscopic asymmetric superconducting loops. For the case of rectangular samples with a small ratio of length to height Ny/Nx, a coreless skyrmionic-like mode containing separated one-component vortices can be obtained. With further increasing Ny/Nx, such a spatial structure is broken due to the aspect-radio effect, and singular vortices with point-like cores tend to appear. In addition, when an indentation defect is introduced at the outer boundary of a mesoscopic square or rectangular loop, the order modulations are strongly affected by the locations of the surface defect, and several peculiar vortical configurations inside the sample can be presented.

Accurate modelling of the optics of high resolution liquid crystal devices: a reconfigurable liquid crystal grating

ABSTRACT An accurate method to model the optical behaviour of liquid crystal (LC) devices, particularly suited to devices where diffractive effects are present is described here. An accurate electromagnetic modelling programme that takes into account the full non-uniformity and anisotropy of the LC has been developed. This is combined with an existing in-house LC finite element modelling programme based on the Landau – De Gennes theory, that uses the order tensor representation of the LC orientation and allows an accurate descriptions of structures containing LC defects and small features. The electromagnetic model is based on the total field/scattered field (TF-SF) approach to electromagnetic scattering problems and is implemented using finite differences in the frequency domain (FDFD) in a form that can accommodate perfectly matched layers (PMLs) and periodic boundary conditions. The resultant matrix problem is solved efficiently using an especially adapted form of a sweeping preconditioner and the generalised minimum residual method (GMRes). This method has been implemented in 2D and is demonstrated here with the design and analysis of a reconfigurable blazed phase grating that utilises an LC defect to produce an abrupt fly-back, with the capability of short periods and high diffraction efficiency.

Stability of a split-core configuration induced by saddle-splay elasticity in a submicron nematic liquid crystal spherical droplet

ABSTRACT We use Landau–de Gennes theory to investigate the configurations of liquid crystals confined to a submicron spherical droplet subject to weak homeotropic anchoring. The results show that in a one-constant approximation, there are three equilibrium configurations: radial hedgehog, ring disclination and uniform, where the radial hedgehog configuration is a stable state. With increasing elastic anisotropy, which is essentially an increase in the splay elastic constant, the radial hedgehog configuration transforms into a split-core configuration, which is metastable. The split-core configuration becomes the lowest-energy state induced by saddle-splay elasticity within appropriate ranges of the radius and elastic anisotropy.

Hydrodynamics of fringing-field induced defects in nematic liquid crystals

Consumer demand for high resolution and high refresh-rate displays has naturally led to the fabrication of liquid crystal displays with ever smaller pixels. As a consequence, fringing fields between adjacent pixels grow in magnitude, leading to abrupt changes in orientation. Electric field strengths above some threshold can lead to order melting and, in turn, disclinations. This paper presents accurate modeling of disclinations induced by fringing fields due to interdigitated electrodes in a nematic liquid crystal calculated by means of the Landau–de Gennes theory. Disclination paths are determined while taking into account the flow of the liquid crystal. Making use of interdigitated electrodes, precise electrical control over the creation and positioning of defects is demonstrated for homeotropic, planar, hybrid, and in-plane surface alignments.

Endothelial glycocalyx detection and characterization by means of atomic force spectroscopy: Comparison of various data analysis approaches

In recent years, atomic force spectroscopy (AFS) has been used to detect and characterize the endothelial glycocalyx (eGlx) in in vitro and ex vivo experiments. Several analysis methods were proposed, which differ not only in the numerical implementations, but also in physical models of glycocalyx description. Therefore, it is difficult to directly relate the experiments performed by different groups. In this work, we compared different models used for quantitative analysis of atomic force spectroscopy datasets recorded for eGlx. To capture glycocalyx at various structural conditions, we used basic enzymatic protocols for glycocalyx removal and restoration in human aortal endothelial cells (HAEC). Nanoindentation experiments for this model system were performed for (i) untreated cells, (ii) for cells after heparinase incubation, which enzymatically removes glycocalyx, (iii) for cells with successive heparin treatment, which partially restores the glycocalyx layer. Analysis of nanoindentation data was performed using different models: (a) a single-layer contact mechanics, (b) a double-layer model contact mechanics, (c) a polymer “brush” two-layer model based on the Alexander – de Gennes theory and (d) a simple single-layer “mechanical spring” model. Although different physical parameters are evaluated in methods (a–d), we show that all approaches revealed similar qualitative changes of the glycocalyx layer, which reflected the processes of glycocalyx degradation and its partial restoration. This paper may facilitate a direct comparison of past and future glycocalyx oriented AFS experiments that are analysed with different approaches.

Tricritical Phenomena and Cascades of Temperature Phase Transitions in a Ferromagnetic Liquid Crystal Suspension

We consider temperature-driven phase transitions occurring in a liquid crystal suspension of ferromagnetic particles within the Landau–de Gennes theory. The temperature dependences of the order parameters in the uniaxial model with a vector order parameter for the magnetic subsystem are obtained. The dimensionless expression for the free energy density of the suspension has been used for the study of the phase behavior general regularities of the system. Phase state diagrams of the suspension and temperature dependences of the order parameters of the liquid crystal and the ensemble of magnetic particles for different values of the phenomenological expansion coefficients are constructed. It is shown that the considered model admits the existence of a cascade of temperature phase transitions: isotropic phase–superparamagnetic nematic phase–ferromagnetic nematic phase. We have shown that in the mesomorphic state of the liquid crystal, the spontaneous magnetization can appear in a continuous way or by a jump with decreasing temperature, which corresponds to the tricritical behavior. The values of temperature and expansion coefficients corresponding to the tricritical and triple points are numerically found.

Liquid crystal droplets under extreme confinement probed by a multiscale simulation approach

ABSTRACT In this work, we computationally investigate liquid crystal (LC) droplets in the size range 0.03–1 μm, confined within shells of combined anchoring conditions. Two different types of surface were defined to promote homeotropic and planar degenerate anchoring, respectively. We identified the LC behaviour within the nanoscale droplets using a bespoke multiscale simulation approach. To study 30 nm droplets, we used coarse grained simulations within the dissipative particle dynamics formalism; to study 0.1 μm and larger droplets, we used a finite element method based on the Landau–de Gennes theory. Good agreement between the two methods was observed in our prior analysis and was confirmed in the present work. We explicitly study droplets of size 0.1 and 1 μm by using continuum mechanics calculations. Our results for the largest droplet are consistent with those available in the literature, suggesting that the extension to smaller droplets presented here is realistic, and therefore can be helpful for innovations in which device intensification could be achieved using LC nanodroplets.

Effect of the Coupling Between Director and Mass Density Fluctuations on the Paranematic–Nematic Transition Temperature: A Mean-field Treatment

It has experimentally verified that the variation of the paranematic–nematic transition temperature follows the relation $$\Delta T_{PN}^{}=m E$$ for low electric fields. Values of the coefficient m are found to be positive and negative. Previous work, on the basis of a simple physical model supplemented by dimensional argument, attributed this behavior to the quenching of long-wavelength director fluctuations due to the field. In this paper, we show that the Landau–de Gennes theory predicts $$\Delta T_{PN}^{}\propto E^2$$ even when the quenching of director fluctuations is considered. Based on a recent generalization of the Landau–de Gennes theory that incorporates mass density fluctuations, we give a mean-field treatment of the coupling between director and mass density fluctuations and calculate the contribution of this mechanism to the Helmholtz free energy. Interestingly, the expression we obtain for $$\Delta T_{PN}^{}$$ is in good quantitative agreement with what is experimentally observed. In particular, it explains why m takes values of both signs.

Bogoliubov–de Gennes theory of the snake instability of gray solitons in higher dimensions

Gray solitons are a one-parameter family of solutions to the one-dimensional non-linear Schr\"odinger equation (NLSE) with positive cubic nonlinearity, as found in repulsively interacting dilute Bose-Einstein condensates or electromagnetic waves in the visible spectrum in waveguides described by Gross-Pitaevskii mean field theory. In two dimensions these solutions to the NLSE appear as a line or plane of depressed condensate density or light intensity, but numerical solutions show that this line is dynamically unstable to `snaking': the initially straight line of density or intensity minimum undulates with exponentially growing amplitude. To assist future studies of quantum mechanical instability beyond mean field theory, we here pursue an approximate analytical description of the snake instability within Bogoliubov-de Gennes perturbation theory. Within this linear approximation the two-dimensional result applies trivially to three dimensions as well, describing buckling modes of the low-density plane. We extend the analytical results of Kuznetsov and Turitsyn [Sov. Phys. JETP \textbf{67}, 1583 (1988)] to shorter wavelengths of the `snake' modulation and show to what extent the snake mode can be described accurately as a parametric instability, in which the position and grayness parameter of the initial soliton simply become dependent on the transverse dimension(s). We find that the parametric picture remains accurate up to second order in the snaking wave number, if the snaking soliton is also dressed by an outward-propagating sound wave, but that beyond second order in the snaking wave number the parametric description breaks down.

Dynamical Landau-de Gennes theory for electrically-responsive liquid crystal networks.

Liquid crystal networks combine the orientational order of liquid crystals with the elastic properties of polymer networks, leading to a vast application potential in the field of responsive coatings, e.g., for haptic feedback, self-cleaning surfaces, and static and dynamic pattern formation. Recent experimental work has further paved the way toward such applications by realizing the fast and reversible surface modulation of a liquid crystal network coating upon in-plane actuation with an AC electric field [Liu, Tito, and Broer, Nat. Commun. 8, 1526 (2017)10.1038/s41467-017-01448-w]. Here, we construct a Landau-type theory for electrically-responsive liquid crystal networks and perform molecular dynamics simulations to explain the findings of these experiments and inform on rational design strategies. Qualitatively, the theory agrees with our simulations and reproduces the salient experimental features. We also provide a set of testable predictions: the aspect ratio of the nematogens, their initial orientational order when cross-linked into the polymer network, and the cross-linking fraction of the network all increase the plasticization time required for the film to macroscopically deform. We demonstrate that the dynamic response to oscillating electric fields is characterized by two resonances, which can likewise be influenced by varying these parameters, providing an experimental handle to fine-tune device design.

Twist-Bend Nematic Phase from the Landau-de Gennes Perspective.

Generalized Landau–de Gennes theory is proposed that comprehensively explains currently available experimental data for the heliconical twist–bend nematic (NTB) phase observed in liquid crystalline systems of chemically achiral bent-core-like molecules. A bifurcation analysis gives insight into possible structures that the model can predict and guides in the numerical analysis of relative stability of the isotropic (I), uniaxial nematic (NU), and twist–bend nematic phases. An estimate of constitutive parameters of the model from temperature variation of the nematic order parameter and the Frank elastic constants in the nematic phase enables us to demonstrate quantitative agreement between the calculated and experimentally determined temperature dependence of the pitch and conical angle in NTB. Properties of order parameters also explain a puzzling lack of a half-pitch band in resonant soft X-ray scattering. Other key findings of the model are predictions of I–NTB and NU–NTB tricritical points and insight into biaxiality of NTB.

A Physics-Based TCAD Simulator for Superconducting Electronics Based on Josephson Junctions

Superconducting electronics are promising candidates for complementing semiconductor electronics. However, a comprehensive suite of superconducting design and simulation tools that is necessary to streamline and automate the design process is either nonexistent or very limited. Here, we describe Synopsys' advances in the technology computer-aided design (TCAD) front for physics-based simulation of superconducting electronics based on Josephson junctions (JJs). We employ Keldysh-Gor'kov-Nambu Green's function formalism within the context of the self-consistent mean-field theory of Bogoliubov and de-Gennes to simulate the equilibrium and transport properties of JJ-based devices in one dimension. Using these Green's functions, we calculate various electrical properties and effects, including order parameter and proximity effect, current-phase characteristics and current-density distribution for arbitrary temperatures, material properties and composition, and device dimensions. We further expand the TCAD simulator using the Floquet-Keldysh-Gor'kov-Nambu Green's function for calculating current-voltage characteristics of JJs in nonequilibrium and capture various quantum mechanical effects including multiple Andreev reflections (MAR) for constant applied voltage.