Isoenergy Surface Research Articles

Billiard with slipping by an arbitrary rational angle

The class of billiards in a disc with slipping along the boundary circle by an angle commensurable with $\pi$ is considered. For such billiards it is shown that an isoenergy surface of the system is homeomorphic to a lens space $L(q,p)$ with parameters satisfying $0 < p <q$. The set of pairs $(q, p)$ such that there exists a billiard in a disc realizing the corresponding lens space $L(q,p)$ is described in terms of solutions of a linear Diophantine equation in two variables. This result also holds for planar billiards with slipping in simply connected domains with smooth boundary, that is, it is not confined to the integrable case. Bibliography: 30 titles.

Anisotropic and valley-resolved beam-splitter based on a tilted Dirac system

We investigate theoretically valley-resolved lateral shift of electrons traversing an n–p–n junction bulit on a typical tilted Dirac system (8-Pmmn borophene). A gauge-invariant formula on Goos–Hänchen (GH) shift of transmitted beams is derived, which holds for any anisotropic isoenergy surface. The tilt term brings valley dependence of relative position between the isoenergy surface in n region and that in the p region. Consequently, valley double refraction can occur at the n–p interface. The exiting positions of two valley-polarized beams depend on the incident angle and energy of incident beam and barrier parameters. Their spatial distance D can be enhanced to be ten to a hundred times larger than the barrier width. Due to tilting-induced high anisotropy of the isoenergy surface, D depends strongly on the barrier orientation. It is always zero when the junction is along the tilt direction of Dirac cones. Thus GH effect of transmitted beams in tilted Dirac systems can be utilized to design anisotropic and valley-resolved beam-splitter.

Longitudinal magnetoresistance of uniaxially deformed n-type silicon

The study of Galvano-magnetic effects (as well as tensoeffects) in silicon under extreme conditions allows not only to identify the mechanisms of these effects but also to identify the possibility of creating gaussmeters, infrared detectors, sensitive strain gauges, amplifiers and generators of a wide frequency range. The reliability of the mechanism of negative magnetoresistance was verified using uniaxial elastic deformation of the studied crystals. Uniaxial deformation excludes interline transitions of electrons, as a result of which negative magnetoresistance disappears with an increase in uniaxial pressure. When cubic symmetry is violated, anisotropic phenomena occur in such crystals. The multipath of the isoenergetic surface of the bottom of the silicon conduction band causes anisotropies of the effective mass and relaxation time, which are associated with the features of the transfer phenomenon. In particular, magnetoresistance (piezoresistance), which is the most sensitive to the anisotropy of the iso energy surface. The influence of the latter on magnetoresistance is most clearly revealed in the region of strong magnetic fields, where the magnetoresistance is saturated since the longitudinal magnetoresistance is entirely due to the anisotropy of electron mobility.

Longitudinal magnetoresistance of uniaxially deformed n-type silicon

The study of Galvano-magnetic effects (as well as tensoeffects) in silicon under extreme conditions allows not only to identify the mechanisms of these effects but also to identify the possibility of creating gaussmeters, infrared detectors, sensitive strain gauges, amplifiers and generators of a wide frequency range. The reliability of the mechanism of negative magnetoresistance was verified using uniaxial elastic deformation of the studied crystals. Uniaxial deformation excludes interline transitions of electrons, as a result of which negative magnetoresistance disappears with an increase in uniaxial pressure. When cubic symmetry is violated, anisotropic phenomena occur in such crystals. The multipath of the isoenergetic surface of the bottom of the silicon conduction band causes anisotropies of the effective mass and relaxation time, which are associated with the features of the transfer phenomenon. In particular, magnetoresistance (piezoresistance), which is the most sensitive to the anisotropy of the iso energy surface. The influence of the latter on magnetoresistance is most clearly revealed in the region of strong magnetic fields, where the magnetoresistance is saturated since the longitudinal magnetoresistance is entirely due to the anisotropy of electron mobility.

Realizing integrable Hamiltonian systems by means of billiard books

Fomenko’s conjecture that the topology of the Liouville foliations associated with integrable smooth or analytic Hamiltonian systems can be realized by means of integrable billiard systems is discussed. An algorithm of Vedyushkina and Kharcheva’s realizing 3-atoms by billiard books, which has been simplified significantly by formulating it in terms of f f -graphs, is presented. Note that, using another algorithm, Vedyushkina and Kharcheva have also realized an arbitrary type of the base of the Liouville foliation on the whole 3-dimensional isoenergy surface. This algorithm is illustrated graphically by an example where the invariant of the well-known Joukowsky system (the Euler case with a gyrostat) is realized for a certain energy range. It turns out that the entire Liouville foliation, rather than just the class of its base, is realized there; that is, the billiard and mechanical systems turn out to be Liouville equivalent. Results due to Vedyushkina and Kibkalo on constructing billiards with arbitrary values of numerical invariants are also presented. For billiard books without potential that possess a certain property, the existence of a Fomenko–Zieschang invariant is shown; it is also proved that they belong to the class of topologically stable systems. Finally, an example is presented when the addition of a Hooke potential to a planar billiard produces a splitting nondegenerate 4-singularity of rank 1.

Influence of quantum electron transport and surface scattering of charge carriers on the conductivity of nanolayer

We developed a theoretical model of an electron conductivity of a nanolayer in the view of quantum theory of transport phenomena. The layer thickness can be comparable to or less than the de Broglie wavelength of charge carriers. We suppose an isoenergy surface has a form of an ellipsoid of revolution. We derived analytical expressions for integral conductivity tensor components as functions of the non-dimensional layer thickness, chemical potential, ellipticity parameter and surface roughness parameters. The dependences of longitudinal and transverse conductivity tensor components on the above parameters were analyzed. We compared the results obtained for a metal and semiconductor. The comparison between theoretical calculations and experimental data for bismuth and silicon films was made.

Non-monotonic thickness dependent and anisotropic in-plane thermal transport in layered titanium trisulphide

Due to the potential applications in many fields such as thermoelectric, transistor, and electrode materials, a novel two-dimensional layered material TiS3 has attracted tremendous attention recently. In this article, the thickness dependent and anisotropic in-plane thermal transport behavior is explored by first-principles calculation with Boltzmann transport equation. The calculation results show that when the thickness is smaller than phonon confinement size (PCS), the phonon confinement effect is dominant and the in-plane thermal conductivity has a negative dependence on thickness due to the increase of phonon–phonon scattering; when the thickness is larger than PCS, the confinement effect is weak and the dependence becomes positive due to the decrease of phonon–boundary scattering. Through further analysis, we find this non-monotonic dependence or phonon confinement phenomenon is mainly caused by the low-frequency phonons. The calculation results also show strong thermal anisotropy within the basal plane. The coupling strength, bond length, phonon group velocity, dispersiveness of optical phonon branches, iso-energy surface and iso-surface of electron density distribution are analyzed to explain the anisotropy. Based on the 1D diatomic chain model, a unit cell having similar atomic mass and strong intracell bond can enhance the dispersiveness of optical phonon branches. Those findings here can give physical insights into the thickness dependent and anisotropic in-plane thermal transport in layered materials.

Topological type of isoenergy surfaces of billiard books

The homeomorphism class of the isoenergy surface of a billiard book, of low complexity and not necessarily integrable, is determined using methods of low-dimensional topology. In particular, a series of billiard books is constructed that realize isoenergy 3-surfaces homeomorphic to the connected sum of a number of lens spaces and direct products . The Fomenko-Zieschang invariants, which classify Liouville foliations on isoenergy surfaces up to fibrewise homeomorphisms — that is, up to Liouville equivalence of the corresponding integrable Hamiltonian systems — are calculated for several integrable billiards of this type. Bibliography: 14 titles.

The influence of isoenergy surface anisotropy and surface scattering kinetics on the conductivity of a thin metal layer

A kinetic theory of the conductivity of a thin metal layer in a longitudinal alternative electric field is constructed. We assume the layer thickness is much greater than the electron de Broglie wavelength and less than the skin layer depth. Therefore, the skin effect is neglected and electron energy spectrum quantization is not considered. The Soffer model is used as the boundary conditions for the Boltzmann equation. We suppose the roughness parameters of the upper and lower layer surface have different values. The Fermi surface is an ellipsoid of revolution, the main axis of which lies in the layer plane. The dependences of conductivity tensor components on the layer thickness, electric field frequency, Fermi surface anisotropy parameter, and surface roughness parameters are analyzed. The results are compared with the ones performed within the framework of diffuse-mirror boundary conditions and with experimental data.

Calculation of the conductivity of a thin conductive layer taking into account Soffer boundary conditions and isoenergy surface anisotropy of conductor

The task about the conductivity of a thin conductive layer in a longitudinal alternative electric field is solved. The relationship between layer thickness and charge carrier mean free path is arbitrary. The Soffer model is used as boundary conditions for Boltzmann equation. The isoenergy surface of a semiconductor is an ellipsoid of revolution. Analytical expressions are obtained for conductivity tensor components as functions of layer thickness, isoenergy surface anisotropy parameter, electric field frequency, surface roughness parameters, and chemical potential. The limiting cases of a degenerate and non-degenerate electron gas are considered. The dependences of conductivity tensor components on the above parameters are analyzed. The results obtained for cases of a degenerate and non-degenerate electron gas are compared. A comparative analysis of results is made with calculations in the view of diffuse-mirror boundary conditions and with known experimental data.

Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra so(3, 1)

In this paper, we study the topology of the Liouville foliation of an analogue of the Kovalevskaya integrable case on the Lie algebra so(3,1). The Fomenko-Zieschang invariants (i.e., marked molecules) of a given foliation on each regular isoenergy surface were calculated.Bibliography: 17 titles.

Unified description of the electronic structure of M2AC nanolamellar carbides

The nanolamellar ${M}_{2}A\mathrm{C}$ or carbon-based ``211 MAX'' phases, where $M$ is an early transition metal, $A$ belongs to groups 13--15, and C is carbon, can be described by rigid band models. The same band model applies to all possible $A$ elements belonging to a given group of the periodic table. Changing $M$ for a given $A$ is then equivalent to shifting the Fermi energy ${E}_{F}$ through a band structure common to all phases in the group. This is shown by comparing predictions of density functional theory (DFT) to angle-resolved photoemission spectroscopy (ARPES) measurements. In particular, the Fermi surface of a given Al-based MAX phase can be obtained with an acceptable degree of accuracy by simply selecting the appropriate ARPES isoenergy surface of another Al-based phase. In ${\mathrm{V}}_{2}\mathrm{AlC}$, and in addition to conventional metal energy bands, both DFT and ARPES show the existence of a gapped nodal line located around 0.2 eV below ${E}_{F}$ or complex crossing points at ${E}_{F}$ with Dirac-like features in some directions. Application of the rigid band model suggests that these topological features as well as others, also predicted by DFT, can be positioned at or close to ${E}_{F}$ by an appropriate choice of $M$ and $A$, or by using an appropriate combination of various $M$ and $A$ elements.

Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra so(4)

This paper is concerned with the topology of the Liouville foliation in the analogue of the Kovalevskaya integrable case on the Lie algebra . The Fomenko-Zieschang invariants (that is, marked molecules) for this foliation are calculated on each nonsingular iso-energy surface. A detailed description of the resulting stratification of the three- dimensional space of parameters of the iso-energy surfaces is given. Bibliography: 23 titles.

Anisotropic thermal transport property of defect-free GaN

Non-equilibrium molecular dynamics (MD) simulation is performed to calculate the thermal conductivity of defect-free GaN along three high-symmetry directions. It is found that the thermal conductivity along [001] direction is about 25% higher than that along [100] or [120] direction. The calculated phonon dispersion relation and iso-energy surface from lattice dynamics show that the difference of the sound speeds among the three high-symmetry directions is quite small for the same mode. However, the variation of phonon irradiation with direction is qualitatively consistent with that of the calculated thermal conductivity. Our results indicate that the anisotropic thermal conductivity may partly result from the phonons in the low-symmetry region of the first Brillouin zone due to phonon focus effects, even though the elastic properties along the three high-symmetry directions are nearly isotropic. Thus, the phonon irradiation is able to better describe the property of thermal conductivity as compared to the commonly used phonon dispersion relation. The present investigations uncover the physical origin of the anisotropic thermal conductivity in defect-free GaN, which would provide an important guide for optimizing the thermal management of GaN-based device.

Nonlinear optical response induced by non-Abelian Berry curvature in time-reversal-invariant insulators

We propose a general framework of nonlinear optics induced by non-Abelian Berry curvature in time-reversal-invariant (TRI) insulators. We find that the third-order response of a TRI insulator under optical and terahertz light fields is directly related to the integration of the non-Abelian Berry curvature over the Brillouin zone. We apply the result to insulators with rotational symmetry near the band edge. Under resonant excitations, the optical susceptibility is proportional to the flux of the Berry curvature through the iso-energy surface, which is equal to the Chern number of the surface times $2\pi$. For the III-V compound semiconductors, microscopic calculations based on the six-band model give a third-order susceptibility with the Chern number of the iso-energy surface equal to three.

Spatial scales and temporal scales in the theory of magnetohydrodynamic turbulence

In the study of incompressible magnetohydrodynamic (MHD) turbulence the correlation lengths along the directions parallel and perpendicular to the local mean magnetic field may be defined by iso-energy surfaces derived from second order structure functions. The correlation time of the turbulence may be defined in a similar manner by means of a temporal second order structure function. Moreover, there is a natural correspondence between the lengthscales of fluctuations with a given energy and the timescale of fluctuations of the same energy so that each iso-energy surface is associated with a unique pair of parallel and perpendicular correlation lengths and a unique correlation time. In the case when the magnetic Prandtl number is unity, Prm≡ν/η=1, it is shown that the correlation time τ associated with an iso-energy contour of energy E is equal to the energy cascade time of the turbulence in the sense that E/τ∼ε, where ε is the energy cascade rate. For balanced MHD turbulence, turbulence with vanishing cross-helicity, both the Goldreich and Sridhar theory and Boldyrev’s theory are shown to have this same property. An attempt is made to generalize these ideas to imbalanced MHD turbulence, turbulence with nonvanishing cross-helicity. It is shown that if the normalized cross-helicity σc is a constant in the inertial range, independent of wavenumber, then one obtains a model in which the energy cascade times of the two Elsasser fields are equal throughout the inertial range, that is, τ+/τ−=1. This result appears to be contradicted by numerical simulations and, therefore, the generalization to imbalanced turbulence discussed here requires further development; this is an important unsolved problem.

An algorithm for the uniform sampling of iso‐energy surfaces and for the calculation of microcanonical averages

In this article an algorithm is proposed to efficiently perform the uniform sampling of an iso-energy surface corresponding to a fixed potential energy U of a molecular system, and for calculating averages of certain quantities over microstates having this energy (microcanonical averages). The developed sampling technique is based upon the combination of a recently proposed method for performing constant potential energy molecular dynamics simulations [Rapallo, A. J Chem Phys 2004, 121, 4033] with well-established thermostatting techniques used in the framework of standard molecular dynamics simulations, such as the Andersen thermostat, and the Nose-Hoover chain thermostat. The proposed strategy leads to very accurate and drift-free potential energy conservation during the whole sampling process, and, very important, specially when dealing with high-dimensional or complicated potential functions, it does not require the calculation of the potential energy function hessian. The technique proved to be very reliable for sampling both low- and high-dimensional surfaces.

New type of topological electronic transition in metals with a change in the Fermi energy

A new type of electronic topological transition due to an abrupt change in the differential-geometric characteristics of the Fermi surface, whose topology remains unchanged, at some critical energy εd is predicted. This type of electronic topological transition differs from known electronic topological transitions due to a change in the topology of the Fermi surface in that the isoenergy surface ε(p)=εd does not contain any singular points where the velocity of the electrons vanishes. It is shown that a feature common to both types of electronic topological transitions associated with an abrupt change in the topology of the Fermi surface and its differential-geometric characteristics with no change in topology is that a qualitative change occurs in the spherical image of the Fermi surface as a result of the abrupt change in the number of pre-images of a point under a spherical mapping.

Formulation of a tail electron hydrodynamic model based on Monte Carlo results

In this paper, the Monte Carlo method is applied to uniform and n/sup +/-n/sup -/-n/sup +/ structures of silicon to study the behavior of tail electrons and to develop a new set of hydrodynamic equations based on tail electron statistics. Each term in these equations is calibrated under both nonhomogeneous and homogeneous electric fields. Terms associated with surface integral of the carriers and carrier momentum over an iso energy surface: (/spl Escr/=/spl Escr//sub th/) are introduced. The new tail electron hydrodynamic model yields the density (n/sub 2/) and the average energy (w/sub 2/) of tail electrons and is shown to predict hot electron effects accurately.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On discreteness of the spectrum of some operator sheaves associated with a periodic Schr�dinger equation

Three-dimensional periodic Schroedinger operators with potentials that are square integrable on the unit cell (single-electron model of a crystal) are considered. A description is given of the class of rational curves that do not have more than a finite number of common points with any isoenergy surface (in particular, the Fermi surface) of an arbitrary operator of the considered form. A consequence of a theorem proved in the paper is the absence on the isoenergy surfaces of elements of planes, cones, and cylinders with straight generators, and all possible paraboloids and hyperboloids. Another interesting consequence is the following assertion: The topological dimension of an isoenergy manifold does not exceed two, which justifies the use of the word surface. The results generalize the assertion of Thomas's theorem on the absence on isoenergy surfaces of straight edges.