Periodic Electric Potential Research Articles

On the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential

We study the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential $V\in L^2_{\mathrm{loc}}(\mathbb R^2;\mathbb R)$ assuming that the magnetic flux of the homogeneous magnetic field $B>0$ satisfies the condition $(2\pi)^{-1}Bv(K)=Q^{-1}$, $Q\in \mathbb N $, where $v(K)$ is the area of the unit cell $K$ of the period lattice of the potential $V$. For arbitrary periodic potentials $V\in L^2_{\mathrm {loc}}(\mathbb R^2;\mathbb R)$ with zero mean $V_0=0$ we show that the spectrum has no eigenvalues different from Landau levels. For periodic potentials $V\in L^2_{\mathrm{loc}}(\mathbb R^2;\mathbb R)\setminus C^{\infty}(\mathbb R^2;\mathbb R)$ we also show that the spectrum is absolutely continuous. Bibliography: 23 titles.

Modifications of single walled carbon nanotubes by ion-induced plasma

Ion beam technology is used efficiently for tailoring the properties of various nanomaterials. Here, we focus on the structural modifications of carbon nanotubes by highly energetic heavy ions of MeV–GeV kinetic energy. The irradiation with high ion fluence induces significant topographic change along the irradiated nanotubes. We introduce a hydrodynamic fluid model for explaining the observed results by investigating the ion-induced plasma. The calculations shows that the evolution of the periodic electric potential coincides qualitatively with the observed broken nanostructured segments along the irradiated nanotube. Moreover, the increase in ion fluence leads to an increase in the magnitude of the normalized electric potential. The dependence of the electric potential on the ion kinetic energy is discussed.

Beyond Diophantine Wannier diagrams: Gap labelling for Bloch–Landau Hamiltonians

It is well known that, given a 2d purely magnetic Landau Hamiltonian with a constant magnetic field b which generates a magnetic flux \varphi per unit area, then any spectral island \sigma_b consisting of M infinitely degenerate Landau levels carries an integrated density of states \mathcal{I}_b=M \varphi . Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any 2d Bloch–Landau operator H_b which also has a bounded \mathbb{Z}^2 -periodic electric potential. Assume that H_b has a spectral island \sigma_b which remains isolated from the rest of the spectrum as long as \varphi lies in a compact interval [\varphi_1,\varphi_2] . Then \mathcal{I}_b=c_0+c_1\varphi on such intervals, where the constant c_0\in\mathbb{Q} while c_1\in \mathbb{Z} . The integer c_1 is the Chern marker of the spectral projection onto the spectral island \sigma_b . This result also implies that the Fermi projection on \sigma_b , albeit continuous in b in the strong topology, is nowhere continuous in the norm topology if either c_1\ne0 or c_1=0 and \varphi is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.

Spectrum of the Landau Hamiltonian with a Periodic Electric Potential

We define a class of periodic electric potentials for which the spectrum of the two-dimensional Schrodinger operator is absolutely continuous in the case of a homogeneous magnetic field B with a rational flux η = (2π)−1Bυ(K), where υ(K) is the area of an elementary cell K in the lattice of potential periods. Using properties of functions in this class, we prove that in the space of periodic electric potentials in Lloc2(ℝ2) with a given period lattice and identified with L2(K), there exists a second-category set (in the sense of Baire) such that for any electric potential in this set and any homogeneous magnetic field with a rational flow η, the spectrum of the two-dimensional Schrodinger operator is absolutely continuous.

On the spectrum of a relativistic Landau Hamiltonian with a periodic electric potential

On the spectrum of a relativistic Landau Hamiltonian with a periodic electric potential

Enhancement of Seebeck coefficient in graphene superlattices by electron filtering technique

We show theoretically that the Seebeck coefficient and the thermoelectric figure of merit can be increased by using electron filtering technique in graphene superlattice based thermoelectric devices. The average Seebeck coefficient for graphene-based thermoelectric devices is proportional to the integral of the distribution of Seebeck coefficient versus energy of electrons. The low energy electrons in the distribution curve are found to reduce the average Seebeck coefficient as their contribution is negative. We show that, with electron energy filtering technique using multiple graphene superlattice heterostructures, the low energy electrons can be filtered out and the Seebeck coefficient can be increased. The multiple graphene superlattice heterostructures can be formed by graphene superlattices with different periodic electric potentials applied above the superlattice. The overall electronic band gap of the multiple heterostructures is dependent upon the individual band gap of the graphene superlattices and can be tuned by varying the periodic electric potentials. The overall electronic band gap of the multiple heterostructures has to be properly chosen such that, the low energy electrons which cause negative Seebeck distribution in single graphene superlattice thermoelectric devices fall within the overall band gap formed by the multiple heterostructures. Although the electrical conductance is decreased in this technique reducing the thermoelectric figure of merit, the overall figure of merit is increased due to huge increase in Seebeck coefficient and its square dependency upon the Seebeck coefficient. This is an easy technique to make graphene superlattice based thermoelectric devices more efficient and has the potential to significantly improve the technology of energy harvesting and sensors.

Effective numerical method of spectral analysis of quantum graphs

We present in the paper an effective numerical method for the determination of the spectra of periodic metric graphs equipped by Schrödinger operators with real-valued periodic electric potentials as Hamiltonians and with Kirchhoff and Neumann conditions at the vertices. Our method is based on the spectral parameter power series method, which leads to a series representation of the dispersion equation, which is suitable for both analytical and numerical calculations. Several important examples demonstrate the effectiveness of our method for some periodic graphs of interest that possess potentials usually found in quantum mechanics.

Absolute continuity of the periodic Schrödinger operator in transversal geometry

We show that the spectrum of a Schrödinger operator on \mathbb R^n, n ≥ 3 , with a periodic smooth Riemannian metric, whose conformal multiple has a product structure with one Euclidean direction, and with a periodic electric potential in L^{n=2}_{\mathrm {loc}} (\mathbb R^n) , is purely absolutely continuous. Previous results in the case of a general metric are obtained in [12], see also [9], under the assumption that the metric, as well as the potential, are reflection symmetric.

Commensurability Oscillations of Composite Fermions Induced by the Periodic Potential of a Wigner Crystal.

When the kinetic energy of a collection of interacting two-dimensional (2D) electrons is quenched at very high magnetic fields so that the Coulomb repulsion dominates, the electrons are expected to condense into an ordered array, forming a quantum Wigner crystal (WC). Although this exotic state has long been suspected in high-mobility 2D electron systems at very low Landau level fillings (ν≪1), its direct observation has been elusive. Here we present a new technique and experimental results directly probing the magnetic-field-induced WC. We measure the magnetoresistance of a bilayer electron system where one layer has a very low density and is in the WC regime (ν≪1), while the other ("probe") layer is near ν=1/2 and hosts a sea of composite fermions (CFs). The data exhibit commensurability oscillations in the magnetoresistance of the CF layer, induced by the periodic potential of WC electrons in the other layer, and provide a unique, direct glimpse at the symmetry of the WC, its lattice constant, and melting. They also demonstrate a striking example of how one can probe an exotic many-body state of 2D electrons using equally exotic quasiparticles of another many-body state.

An overview of periodic elliptic operators

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schrodinger operator with periodic electric potential −∆ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases. Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.

Electric and magnetic superlattices in trilayer graphene

The properties of one dimensional Kronig–Penney type of periodic electric and vector potential on ABC-trilayer graphene superlattices are investigated. The energy spectra obtained with periodic vector potentials shows the emergence of extra Dirac points in the energy spectrum with finite energies. For identical barrier and well widths, the original as well as the extra Dirac points are located in the ky=0 plane. An asymmetry between the barrier and well widths causes a shift in the extra Dirac points away from the ky=0 plane. Extra Dirac points having same electron hole crossing energy as that of the original Dirac point as well as finite energy Dirac points are generated in the energy spectrum when periodic electric potential is applied to the system. By applying electric and vector potential together, the symmetry of the energy spectrum about the Fermi level is broken. A tunable band gap is induced in the energy spectrum by applying both electric and vector potential simultaneously with different barrier and well widths.

Effects of Periodic Electric Potential and Electrolyte Recirculation on Electrochemical Remediation of Contaminant Mixtures in Clayey Soils

This paper investigates the operational factors that affect an integrated electrochemical remediation (IECR) process for the degradation and removal of both organic and heavy metal contaminants in low permeability soils. IECR aims to degrade the organic contaminants through chemical oxidation process and at the same time remove the heavy metals through electromigration. Hydrogen peroxide is used as an oxidant which is transported into the soil by the electroosmotic flow. The natural iron in the soil acts as a catalyst for the decomposition of hydrogen peroxide in a Fenton-like process. Kaolin soil was used as model low permeability soil. Soil was polluted with phenanthrene and nickel at concentrations of 500 mg/kg each. Experiments were carried out at a constant voltage gradient of 1 VDC/cm using 10 % H2O2 as a flushing solution at the anode. The use of periodic voltage gradient and the recirculation of the cathodic solution to the anode favored the electroosmotic flow and the contaminant removal. Phenanthrene was eliminated from the soil in an average value of 30 %. Though electroosmotic flow was registered in all the experiments, negligible amount of phenanthrene was detected in the electrode solutions, confirming that the removal of phenanthrene is due to chemical oxidation. Nickel was solubilized and transported towards the cathode, but it was accumulated in the close vicinity of the cathode due to the high pH. Overall, this study showed that IECR has potential to remediate soils contaminated by both organic compounds and heavy metals.

Characteristics of Transient Electroosmotic Flow in Microchannels With Complex-Wavy Surface and Periodic Time-Varying Electric Field

A numerical investigation is performed into the flow characteristics of the electroosmotic flow induced within a microchannel with a complex-wavy surface by a time-varying periodic electric field. The simulations focus specifically on the effects of the Strouhal number of the periodic electric potential, the amplitude of the periodic electric potential, the amplitude of the complex-wavy surface, and the waveform geometry. The results show that under steady-time periodic conditions, the flow pattern induced within the microchannel varies over the course of the oscillation period. In particular, it is shown that a flow recirculation structure is generated in the trough region of the wavy surface as the applied electric field falls to zero if the amplitude of the wavy surface exceeds a certain threshold value. In addition, it is shown that the phases of the electric field and electroosmotic velocity near the wall surface are almost identical. However, a phase shift exists between the electric field and the bulk flow velocity in the central region of the channel; particularly at larger values of the Strouhal number. Finally, it is shown that the velocity profile near the wavy surface is more sensitive to changes in the waveform geometry than that in the center of the channel. Overall, the simulation results presented in the study provide a useful source of reference for the development of new microfluidic systems incorporating microchannels with complex-wavy surfaces.

Pressure boundary condition of the lattice Boltzmann method for fully developed periodic flows

The general periodic boundary condition for the lattice Boltzmann method has been modified to incorporate the pressure difference for fully developed periodic flows. The results demonstrated that, unlike other existing pressure boundary treatments, the proposed procedure/treatment does not generate nonphysical inlet and outlet flow disturbances while preserving the system periodicity. This method is readily applicable to a range of lattice Boltzmann simulations for systems with periodic electric potential and temperature fields.

Asymptotic of the density of states for the Schrödinger operator with periodic electric potential

Asymptotic of the density of states for the Schrödinger operator with periodic electric potential

Temperature dependence of nonlinear absorption in InP doping superlattices

Absorptive optical nonlinearity in InP doping superlattices grown by vapor-phase epitaxy is investigated in a 2–300 K temperature range. The absorption spectra have tails below InP band-gap energy, caused by the Franz–Keldysh effect due to the internal electric field. The absorption tail is almost temperature independent. This means that the periodic electric potential in the doping superlattice, responsible for the nonlinearity, is unchanged across this temperature range. Nonlinear absorption is evaluated by the pump-probe method. The large reduction in absorption coefficient is obtained even with a weak excitation light intensity. This enhancement in nonlinearity is due to prolongation of the excited-carrier lifetime. Nonlinear absorption increases as temperature decreases. This is due to the temperature dependence of the carrier lifetime.

Decoupling of Bloch Bands in the Presence of Homogeneous Fields

Wave functions which are valid if a charge moves in a superposition of a periodic electric potential and a uniform magnetic field are constructed. The wave functions are not themselves solutions of the Schroedinger equation, but yield the traditional effective Hamiltonian for this problem. Contrary to the electric field case the mainfold of states linked by the band index does not form a Bloch band; the reason is that the cellular transforms of the Bloch-like functions are modified by the Peierls phase. At present, the derivation of these results is in closed form, but justifiable only to all powers of the magnetic field.'' This was also the case for the previous electric derivation. The limitation may not be genuine. The existence of closed Bloch bands in the presence of a homogeneous electric field is proved; the case of free electrons is given as an example. The results for the magnetic field should be independent of the power series method used for their jnstification. The procedure is extended to crossed electric and magnetic fields. (auth)