Landau Energy Research Articles

A Similar Change in the Kinetic-Energy Components of Longitudinal Continuous Motion of Magnetized Electrons within the Landau Levels and Determination of the Characteristics of a Degenerate System

Account is taken of a similar change in the kinetic-energy components of a longitudinal electron motion along the magnetic field within each Landau energy level. The maximum electron energy is found as well as the kinetic energy of a transverse electron motion in the magnetic field. The electron concentrations are determined depending on the magnetic-field strength, whose intrinsic magnetic moments are directed along and opposite the field. The magnetic-field strengths for which a certain number of Landau quantum levels is realized in an electron degenerate system of a specified concentration are also found.

Ginzburg-Landau equations and boundary conditions for superconductors in static magnetic fields

We derive the Ginzburg–Landau equations for superconductors in static magnetic fields. Instead of the square of the gauge-invariant gradient of the order-parameter wave function, we consider the quantum-mechanical expression for the kinetic energy in the Ginzburg–Landau energy functional. We introduce a new surface term in the free energy functional which results in the de Gennes interface conditions. The phenomenological Ginzburg–Landau theory thus contains three length scales which must be determined from microscopic theory: the Ginzburg–Landau coherence length, the London penetration depth, and the de Gennes length.

Analysis of Solutions to the Lawrence--Doniach System for Layered Superconductors

We consider a coupled Ginzburg--Landau system, the so-called Lawrence--Doniach system, which models layered superconductors as a stack of nonlinearly coupled, parallel two-dimensional superconducting layers, separated by an insulating material or vacuum, in an applied magnetic field. We prove that weak solutions (e.g., energy minimizers) in an appropriate divergence-free gauge are uniformly bounded and continuous and satisfy a priori estimates based on elliptic theory and single layer potentials. Moreover, we show the existence of an upper critical field ${\bar{h}}$ such that when the modulus of a constant applied magnetic field $\vec{H}=h\vec{v}$ in a direction $\vec{v}$ nontangential to the layers (where $|\vec{v}|=1$) is greater than ${\bar{h}}$, the normal (nonsuperconducting) state is the only solution to the Lawrence--Doniach system. It follows from these results and methods developed earlier by Chapman, Du, and Gunzburger [SIAM J. Appl. Math., 55 (1995), pp. 156--174] that under certain assumptions on the relative values of parameters in the model, minimizers of the Lawrence--Doniach energy converge, as the interlayer spacing tends to zero, to minimizers of an appropriate anisotropic Ginzburg--Landau energy in three dimensions. Finally, we derive that ${\bar{h}} \leq C\kappa/\mu$ for all $\kappa$ sufficiently large and all unit vectors $\vec{v}$ satisfying $\vec{v}\cdot\vec{e_3} \geq \mu > 0$ for the Lawrence--Doniach system, where $\kappa$ is the Ginzburg--Landau constant for the superconducting material and C is independent of $\kappa$ and $\mu$.

APPROXIMATIONS WITH VORTICITY BOUNDS FOR THE GINZBURG–LANDAU FUNCTIONAL

We propose an approximation scheme for complex-valued functions defined on a smooth domain Ω: the approximating functions have a Ginzburg–Landau energy of the same magnitude as the initial function, but they possess moreover improved bounds on vorticity. As an application, we obtain a variant of a Jacobian estimate first established by Jerrard and Soner. This variant was conjectured by Bourgain, Brezis and Mironescu.

New estimates for the Laplacian, the div–curl, and related Hodge systems

We establish new estimates for the Laplacian, the div–curl system, and more general Hodge systems in arbitrary dimension, with an application to minimizers of the Ginzburg–Landau energy. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Addition to the Solution of the Problem on the Motion of a Free Nonrelativistic Electron in a Magnetic Field

In addition to the solution of the quantum mechanical problem on the motion of a free electron in a magnetic field, given by L. D. Landau in his pioneering work, this quantum mechanical problem is solved in view of the fact that the sum of the components of the free kinetic energy of an electron along two axes is a periodic quantity varying identically within the boundaries of every Landau energy level. This periodicity is a consequence of the quantized motion of an electron in a plane normal to the direction of the external magnetic field.

Magneto-volume coupling constant in itinerant-electron metamagnets

A pressure P dependence of the critical field B C of the metamagnetic transition from the paramagnetic state to ferromagnetic one in itinerant-electron metamagnets is discussed by the Landau energy expanded up to the sixth power of the magnetization, including the effect of the volume magnetostriction. It is shown that the magneto-volume coupling constant is given by the ratio between d B C/d P and the induced magnetic moment M ind by the magnetic field. From the observed values of these quantities, the magneto-volume coupling constant is estimated for Co(S,Se) 2, Y(Co,Al) 2, Lu(Co,M) 2 (M=Al, Ga), and U(Co,Fe)Al.

New oscillation in terahertz magneto-optical effect of single-walled carbon nanotubes film

The magneto-optical Kerr effect of single-walled carbon nanotubes (SWNTs) film in the Terahertz region is theoretically studied by means of the Drude model. The calculation shows a new oscillation occurring near the plasma frequency as the origin of the magneto-optical Kerr effect for the single-walled nanotubes film. We propose that the amplitude of this near plasma frequency oscillation is directly proportional to the external magnetic field and its period increases with increasing cyclotron frequency. We consider these features of this oscillation of SWNTs are related to the Landau energy of SWNTs. The reflectivity and figure of merit (FOM) spectrum also reveal that this oscillation is the dominant mechanism for the magneto-optical Kerr effect of SWNTs film. Meanwhile, the shift and large enhancement of Kerr rotation under different external magnetic fields is explained.

Itinerant-electron metamagnetism and giant magnetocaloric effect

An isothermal magnetic entropy change in itinerant-electron systems is discussed on the theory of itinerant-electron metamagnetism. The effect of spin fluctuations is taken into account based on the phenomenological Ginzburg-Landau theory. By the Clausius-Clapeyron relation, the magnetic entropy change depends not only on the magnetization jump at the Curie temperature ${T}_{\mathrm{C}},$ but also on the temperature dependence of the critical field of the metamagnetic transition. It is shown that the magnetic entropy change at low fields becomes maximum when ${T}_{\mathrm{C}}$ is about half of the temperature ${T}_{\mathrm{max}}$ where the susceptibility reaches a maximum. The isothermal magnetic entropy changes for $3d$ compounds $\mathrm{Co}(\mathrm{S},\mathrm{S}\mathrm{e}{)}_{2},$ $\mathrm{Lu}(\mathrm{C}\mathrm{o},\mathrm{A}\mathrm{l}{)}_{2},$ and $\mathrm{Lu}(\mathrm{C}\mathrm{o},\mathrm{G}\mathrm{a}{)}_{2}$ and also for itinerant $5f$ compound U(Co,Fe)Al are estimated and compared with those observed for MnFe(P,As) and $\mathrm{La}(\mathrm{F}\mathrm{e},\mathrm{S}\mathrm{i}{)}_{13}.$ It has been found generally that the giant magnetocaloric effect in itinerant-electron metamagnets is expected when the coefficient ${b}_{0}$ of ${M}^{4}$ in the Landau energy expansion with respect to the magnetization M is negative and large.

Numerical determination of vortices in superconductors: simulation of cooling

We use a gradient descent method to numerically calculate critical points of the Ginzburg–Landau energy functional for a two-dimensional domain, possibly including holes. By directly minimizing the functional we avoid the difficulty of treating the Ginzburg–Landau equations with their associated nonlinear boundary conditions. The descent method is made efficient by the use of Sobolev gradients. In order to find the minimum-energy critical point we simulate a cooling process in which we compute a sequence of critical points, each associated with a slightly lower temperature. The solution at each temperature value serves as a good initial estimate for the next value. We present test results that demonstrate the effectiveness of the method.

A product-estimate for Ginzburg–Landau and corollaries

We prove a new inequality for the Jacobian (or vorticity) associated to the Ginzburg–Landau energy in any dimension. It allows to retrieve existing lower bounds on the energy, to extend them to the case of unbounded vorticity, and to get a few other corollaries. It also provides a new estimate on the time-variation for time-dependent families, which has applications for the study of Ginzburg–Landau dynamics.

A Landau free energy for diblock copolymers with compressibility difference between blocks

A new Landau free energy is derived for diblock copolymers of incompatible pairs based on the recently developed compressible random-phase approximation analysis. Finite compressibility of each block is generally allowed. The inhomogeneity of each block density and free volume is analyzed in the weak segregation regime. Free volume inhomogeneity fluctuates in two ways: One represents compressibility difference between blocks and the other stands for the screening of unfavorable cross-contacts. It is shown from the Landau energy that a continuous transition, observed in a symmetric block copolymer either incompressible or with no compressibility difference, disappears provided that one block is more compressible. Microphase transitions and their pressure response of commonly used diblock copolymers are calculated and compared with experimental results. A Flory-type interaction parameter χcRPA, which is suggested from the effective second-order vertex function in the free energy, is shown to be useful, owing to its compressible nature in understanding the phase behavior of various copolymers.

Critical behavior from Landau theory in nanothermodynamic equilibrium

Two modern thermostatistical techniques are used to treat classical Landau-like theories. A Ginzburg–Landau model is simplified using Kadanoff blocks that fluctuate independently, and the Landau energy is lowered using a nanothermodynamic ensemble with no constraints. Advantages of these techniques include eliminating explicit surface terms between real-space blocks, and minimizing the free energy to give the true thermal equilibrium. The theory yields non-uniform clustering, non-exponential relaxation, and non-classical critical scaling, similar to behavior found near the liquid-glass and ferromagnetic transitions.

Two-dimensional hydrogen atom in a strong magnetic field

We study the Schrödinger equation of the hydrogen atom in the (arbitrarily) strong magnetic field in two dimensions, which is an integrable and separable system. The energy spectrum is very interesting as it has infinitely many accumulation points located at the values of the Landau energy levels of a free electron in the uniform magnetic field. In the polar coordinates, the canonical (not kinetic!) angular momentum has a precise eigenvalue and we have the one-dimensional radial Schrödinger equation, which is an ordinary second-order differential equation whose analytical exact solution is unknown. We describe the qualitative properties of the energy spectrum, and we propose a semi-analytical method to numerically calculate the eigenenergies (the representation matrix of the Hamiltonian in the Landau basis is analytically calculated and is exactly known). Also, we use a number of useful analytical approximation methods, such as semiclassical approximations, the perturbation method, the variational method and the Taylor power expansion of the potential around the minimum to estimate the ground-state energy and the higher levels.

A product estimate for Ginzburg–Landau and application to the gradient-flow

We prove a new inequality for the Jacobian (or vorticity) associated to the Ginzburg–Landau energy in any dimension, and give static and dynamical corollaries. We then present a method to prove convergence of gradient-flows of families of energies which Gamma-converge to a limiting energy, which we apply to establish, thanks to the previous dynamical estimate, the limiting dynamical law of a finite number of vortices for the heat-flow of Ginzburg–Landau in dimension 2, with and without magnetic field. To cite this article: E. Sandier, S. Serfaty, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

On Cahn—Hilliard systems with elasticity

Elastic effects can have a pronounced effect on the phase-separation process in solids. The classical Ginzburg—Landau energy can be modified to account for such elastic interactions. The evolution of the system is then governed by diffusion equations for the concentrations of the alloy components and by a quasi-static equilibrium for the mechanical part. The resulting system of equations is elliptic-parabolic and can be understood as a generalization of the Cahn—Hilliard equation. In this paper we give a derivation of the system and prove an existence and uniqueness result for it.

Limits of a Ginzburg–Landau model with codimension-one defects

We derive a nondimensional Ginzburg–Landau energy functional that does not use temperature-dependent scaling quantities. Using the machinery of gamma-convergence, the asymptotic limits of the functional are computed in the presence of thin defects centered around co-dimension 1 manifolds. We classify the defects into three groups depending on the L1-norm of the defect potential. We show that each group has its own distinguished asymptotic limit as the thickness of the defect converges to zero.

Vortices inp-Wave Superconductivity

In the theory of p-wave superconductivity, the Ginzburg--Landau energy functionals with multicomponent order parameters were employed. Here we find a minimizer of a reduced form of the p-wave Ginzburg--Landau free energy with two-component order parameters. The minimizer has distinct degree-one (or minus one) vortices in each component. We also derive a system of ordinary differential equations as the motion equations of vortices in the approximated gradient flow for p-wave superconductivity.

A Mathematical Representation of Quantizing the Motion of Relativistic Electrons in a Magnetic Field

It is shown that the quadratic component of the kinetic energy of continuous longitudinal motion of relativistic electrons in the external magnetic field is varied continuously between 0 and 2(2mec2μBH) within each Landau energy level, undergoing an abrupt change at the boundaries of the levels. This results in the fact that in the quantum limit of a superstrong magnetic field where all electrons are at the zero Landau level, the maximum quadratic component of the kinetic energy of free longitudinal electron motion along the direction of the magnetic field is twice as high as the maximum quadratic component of the kinetic energy of its bound transverse motion.

Quasilinear elliptic system arising in a three-dimensional type-II superconductor for infinite κ

We study a quasilinear elliptic system arising in a three-dimensional superconductor Ω. This model is formally derived from the Ginzburg–Landau energy at κ=+∞ for a Meissner solution. If the tangential trace of the magnetic field H is given on ∂Ω, we prove the existence of a unique solution for small data, and nonexistence for large data. On the other hand we prove that the current J = curl H is such that J 2 is maximum on the boundary ∂Ω.