The influence of disorder on the exciton spectra in two-dimensional structures

Quantum chaotic features of the spin-orbit coupled excitons in disordered two-dimensional insulators

We study the synergy between disorder (phenomenologically modeled by the introduction of Riesz fractional derivative in the corresponding Schrödinger equation) and spin-orbit coupling (SOC) on the exciton spectra in two-dimensional (2D) semiconductor structures. We demonstrate that the joint impact of "fractionality" and SOC considerably modifies the spectrum of corresponding "ordinary" (i.e., without fractional derivatives) hydrogenic problem, leading to the non-Poissonian statistics of the adjacent level distance distribution. The latter fact is strong evidence of the possible emergence of quantum chaotic features in the system. Using analytical and numerical arguments, we discuss the possibilities to control the above chaotic features using the synergy of SOC, Coulomb interaction, and "fractionality," characterized by the Lévy index μ.

Screened Coulomb interaction in insulators with strong disorder

We study the effect of disorder on the excitons in a semiconductor with screened Coulomb interaction. Examples are polymeric semiconductors and/or van der Waals structures. In the screened hydrogenic problem, we consider the disorder phenomenologically using the so-called fractional Scrödinger equation. Our main finding is that joint action of screening and disorder either destroys the exciton (strong screening) or enhances the bounding of electron and hole in an exciton, leading to its collapse in the extreme case. Latter effects may also be related to the quantum manifestations of chaotic exciton behavior in the above semiconductor structures. Hence, they should be considered in device applications, where the interplay between dielectric screening and disorder is important. Our theoretical results permit one to predict the various excitonic properties in semiconductor samples with different degrees of disorder and Coulomb interaction screenings.

Fractional quantum oscillator and disorder in the vibrational spectra

We study the role of disorder in the vibration spectra of molecules and atoms in solids. This disorder may be described phenomenologically by a fractional generalization of ordinary quantum-mechanical oscillator problem. To be specific, this is accomplished by the introduction of a so-called fractional Laplacian (Riesz fractional derivative) to the Scrödinger equation with three-dimensional (3D) quadratic potential. To solve the obtained 3D spectral problem, we pass to the momentum space, where the problem simplifies greatly as fractional Laplacian becomes simply \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^\mu $$\end{document}kμ, k is a modulus of the momentum vector and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}μ is Lévy index, characterizing the degree of disorder. In this case, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow 0$$\end{document}μ→0 corresponds to the strongest disorder, while \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow 2$$\end{document}μ→2 to the weakest so that the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =2$$\end{document}μ=2 corresponds to “ordinary” (i.e. that without fractional derivatives) 3D quantum harmonic oscillator. Combining analytical (variational) and numerical methods, we have shown that in the fractional (disordered) 3D oscillator problem, the famous orbital momentum degeneracy is lifted so that its energy starts to depend on orbital quantum number l. These features can have a strong impact on the physical properties of many solids, ranging from multiferroics to oxide heterostructures, which, in turn, are usable in modern microelectronic devices.

1D solitons in cubic-quintic fractional nonlinear Schrödinger model

We examine the properties of a soliton solution of the fractional Schrö dinger equation with cubic-quintic nonlinearity. Using analytical (variational) and numerical arguments, we have shown that the substitution of the ordinary Laplacian in the Schrödinger equation by its fractional counterpart with Lévy index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α permits to stabilize the soliton texture in the wide range of its parameters (nonlinearity coefficients and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α) values. Our studies of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (N)$$\end{document}ω(N) dependence (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document}ω is soliton frequency and N its norm) permit to acquire the regions of existence and stability of the fractional soliton solution. For that we use famous Vakhitov-Kolokolov (VK) criterion. The variational results are confirmed by numerical solution of a one-dimensional cubic-quintic nonlinear Schrödinger equation. Direct numerical simulations of the linear stability problem of soliton texture gives the same soliton stability boundary as within variational method. Thus we confirm that simple variational approach combined with VK criterion gives reliable information about soliton structure and stability in our model. Our results may be relevant to both optical solitons and Bose-Einstein condensates in cold atomic gases.

Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity

We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under "fractional" we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index [Formula: see text]. We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Laplacian (corresponding to Lévy index [Formula: see text]), the soliton is unstable, even infinitesimal difference [Formula: see text] from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of [Formula: see text] dependence ([Formula: see text] is soliton frequency and N its mass) show (within the famous Vakhitov-Kolokolov criterion) the stability of our soliton texture in the fractional [Formula: see text] case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at [Formula: see text], which is in accord with existing literature data. These results may be relevant to both Bose-Einstein condensates in cold atomic gases and optical solitons in the disordered media.

The influence of Coulomb interaction screening on the excitons in disordered two-dimensional insulators

We study the joint effect of disorder and Coulomb interaction screening on the exciton spectra in two-dimensional (2D) structures. These can be van der Waals structures or heterostructures of organic (polymeric) semiconductors as well as inorganic substances like transition metal dichalcogenides. We consider 2D screened hydrogenic problem with Rytova-Keldysh interaction by means of so-called fractional Scrödinger equation. Our main finding is that above synergy between screening and disorder either destroys the exciton (strong screening) or promote the creation of a bound state, leading to its collapse in the extreme case. Our second finding is energy levels crossing, i.e. the degeneracy (with respect to index [Formula: see text]) of the exciton eigenenergies at certain discrete value of screening radius. Latter effects may also be related to the quantum manifestations of chaotic exciton behavior in above 2D semiconductor structures. Hence, they should be considered in device applications, where the interplay between dielectric screening and disorder is important.

Lévy distributions and disorder in excitonic spectra

We study analytically the spectrum of excitons in disordered semiconductors like transition metal dichalcogenides, which are important for photovoltaic and spintronic applications. We show that ambient disorder exerts a strong influence on the exciton spectra. For example, in such a case, the well-known degeneracy of the hydrogenic problem (related to Runge-Lenz vector conservation) is lifted so that the exciton energy starts to depend on both the principal quantum number n and orbital l. We model the disorder phenomenologically substituting the ordinary Laplacian in the corresponding Schrödinger equation by the fractional one with Lévy index μ, characterizing the degree of disorder. Our variational treatment (corroborated by numerical results) shows that an exciton exists for 1 < μ≤ 2. The case μ = 2 corresponds to the "ordered" hydrogenic problem, while in the opposite case μ = 1 the exciton collapses. The exciton spectrum is dominated by the sample areas with moderate disorder. Our theory permits controlled predictions to be made of the excitonic properties in semiconductor samples with different degrees of disorder.