Parabolic States Research Articles

The electronic transport properties of parabolic topological edge states

Abstract The broadly studied topological edge states in two-dimensional materials are usually of a linear dispersion relation. However, the transport of non-Dirac topological edge states is rarely studied. Here, we report the finding of topological edge states with a parabolic dispersion. The topological nature of the states is supported by spatial wave function, Chern number calculation and the quantized linear conductance. The parabolic dispersion aspect of the edge states can be verified via a double-barrier junction which acts as a Fabry–Perot interferometer. For the parabolic dispersion edges states, the Fermi energy E dependent resonance peaks spacing δ shows a δ ∼ E relation. In contrast, for the linear dispersion edges states, δ is a constant. Finally, the transport properties of parabolic topological edge states under Anderson disorder are studied. It is found that under a strong disorder, the quantized conductance is sensitive to the Fermi energy, but unaffected by the sample length.

Atomic Data for Calculation of the Intensities of Stark Components of Excited Hydrogen Atoms in Fusion Plasmas

Motional Stark effect (MSE) spectroscopy represents a unique diagnostic tool capable of determining the magnitude of the magnetic field and its direction in the core of fusion plasmas. The primary excitation channel for fast hydrogen atoms in injected neutral beams, with energy in the range of 25–1000 keV, is due to collisions with protons and impurity ions (e.g., He 2 + and heavier impurities). As a result of such excitation, at the particle density of 10 13 –10 14 cm − 3 , the line intensities of the Stark multiplets do not follow statistical expectations (i.e., the populations of fine-structure levels within the same principal quantum number n are not proportional to their statistical weights). Hence, any realistic modeling of MSE spectra has to include the relevant collisional atomic data. In this paper we provide a general expression for the excitation cross sections in parabolic states within n = 3 for an arbitrary orientation between the direction of the motion-induced electric field and the proton-atom collisional axis. The calculations make use of the density matrix obtained with the atomic orbital close coupling method and the method can be applied to other collisional systems (e.g., He 2 + , Be 4 + , C 6 + , etc.). The resulting cross sections are given as simple fits that can be directly applied to spectral modeling. For illustration we note that the asymmetry detected in the first classical cathode ray experiments between the red- and blue-shifted spectral components can be quantitatively studied using the proposed approach.

Reinvestigation of the giant Rashba-split states on Bi-covered Si(111)

We study the electronic and spin structures of the giant Rashba-split surface states of the Bi/Si(111)-($\sqrt{3} \times \sqrt{3}$)R30 trimer phase by means of spin- and angle-resolved photoelectron spectroscopy (spin-ARPES). Supported by tight-binding calculations of the surface state dispersion and spin orientation, our findings show that the spin experiences a vortex-like structure around the $\bar{\Gamma}$-point of the surface Brillouin zone - in accordance with the standard Rashba model. Moreover, we find no evidence of a spin vortex around the $\bar{\mathrm{K}}$-point in the hexagonal Brillouin zone, and thus no peculiar Rashba split around this point, something that has been suggested by previous works. Rather the opposite, our results show that the spin structure around $\bar{\mathrm{K}}$ can be fully understood by taking into account the symmetry of the Brillouin zone and the intersection of spin vortices centered around the $\bar{\Gamma}$-points in neighboring Brillouin zones. As a result, the spin structure is consistently explained within the standard framework of the Rashba model although the spin-polarized surface states experience a more complex dispersion compared to free-electron like parabolic states.

Unexpected orbital magnetism in Bi-rich Bi2Se3 nanoplatelets

The discovery of two-dimensional electron gas states with giant Rashba spin splitting (RSS) in electron-doped three-dimensional topological insulators (TIs) uncovered new fascinating physics and raised hopes for novel spintronic devices. Significant challenges, including synthetic constraints and control of the magnetic properties, must be addressed before any breakthroughs are possible. Here, we show how RSS in Bi-rich Bi2Se3 nanoplatelets is responsible for the appearance of remarkable orbital magnetic properties, as observed using magnetization and conduction electron spin resonance experiments and confirmed by theoretical simulations. In view of the strong spin-orbit coupling (SOC) and the proximity to the TI surface states, this discovery enlightens fundamental aspects of SOC-based functionalities of TI materials with aims for future applications. Plate-like bismuth nanocrystals can give physicists enriched mechanisms to control spin through unexpected, surface-localized magnetism. Topological insulators have crystal structures that make them non-conductive everywhere except for the quantized states on their surfaces. Hae Jin Kim's group at KBSI and co-workers (PI, “Demokritos”, and UoI) has now used a liquid-phase, solvothermal synthesis to turn bismuth selenide (Bi2Se3) topological insulators into nanometre-thin platelets with unique hexagonal shapes. The team discovered that sandwiching bismuth atoms between the nanohexagons disrupts the normal coupling interactions between electron spin and orbital motion at topological insulators surface states. This splitting generates so-called Rashba states that produce a special orbital-based type of magnetism, where spin states can be flipped with help from a weak magnetic field. Further spin engineering of this system is possible by altering the degree of bismuth intercalation. Bi-layer intercalation in Bi2Se3 nanoplatelets gives rise to an intriguing crystal structure comprised of randomly stacked Bi2Se3 and Bi2Se2. Detailed conduction electron spin resonance (CESR) and AC/DC magnetization studies prove that controlling Bi intercalation results in fine tuning the two-dimensional electron gas parabolic Rashba states, which enables the appearance of extraordinary orbital magnetism, through the coupling of the spin and orbital degrees of freedom. The methodology presented herein provides a unique and simple way for efficient spin engineering, with important potential applications.

Electron-hole asymmetry, Dirac fermions, and quantum magnetoresistance inBaMnBi2

We report two-dimensional quantum transport and Dirac fermions in BaMnBi2 single crystals. BaMnBi2 is a layered bad metal with highly anisotropic conductivity and magnetic order below 290 K. Magnetotransport properties, nonzero Berry phase, small cyclotronmass, and the first-principles band structure calculations indicate the presence of Dirac fermions in Bi square nets. Quantum oscillations in the Hall channel suggest the presence of both electron and hole pockets, whereas Dirac and parabolic states coexist at the Fermi level.

Comparison of effective rate coefficients for high energy charge-exchange with measurements of the Rydberg series of Ar16+ at the tokamak TEXTOR

The charge-exchange (CX) rate coefficients for highly ionized impurity ions play a crucial role in fusion plasma diagnostics. However, till today a substantial difference exists in data for the nl-resolved cross-sections based on the different approximations underlying the classical trajectory Monte Carlo (CTMC) calculations either based on the standard initial momentum distribution of target electron orbits (pCTMC, as in the CX data provided by Whyte et al (1998 Phys. Plasmas 5 3694) and Schultz et al (2010 J. Phys. B: At. Mol. Opt. Phys. 43 144002) or based on the alternate initial radial distribution of orbits (rCTMC, as in the calculations of Errea et al (2006 J. Phys. B: At. Mol. Opt. Phys. 39 L91). In this paper, results of new pCTMC and rCTMC calculations for CX in 16.7, 25, and 50 keV/u Ar17+ + H(1s), H(2s), and H(2p) are compared against X-ray line measurements performed at the tokamak TEXTOR. The Rydberg series (1snp–1s2) and the Kα-spectrum (1s2l–1s2) of He-like argon were measured directly in the beam-line of a 16.7–50 keV/u hydrogen injector. The intensities of the spectral lines are compared to the effective CX rate coefficients utilizing both sets of cross sections. While both data sets show good agreement with respect to the observed impact on the Kα transition, only the pCTMC data allow a consistent description of the CX ‘resonance’ observed on the Rydberg lines around n ≈ 8, 9. Similar to the case of low energy ion–atom interactions reported from different tokamaks, the observed influence of CX is separable into contributions from beam particles in the ground and excited states, respectively. It is shown, that the number of beam excited states nh contributing to the CX signal, where nh is the principal quantum number, is limited to nh ≲ 10, confirming the results of recent collisional-radiative models of beam atoms in parabolic states.

Electric dipole moments of He atoms excited to the 1s5l (l ⩾ 2) states by He+-ion impact at intermediate energies

The post-collisional 1s5l (l ⩾ 2) states of He atoms after He+-ion impact (10 keV–28 keV) have been investigated using anticrossing spectroscopy. In particular, the intensity of the spectral line λ(1s5l 3D-1s2p 3P) = 402.6 nm emitted by the impact-excited He atoms was measured as a function of an axial electric field (which varied from −30 kV cm−1 to +30 kV cm−1). By fitting the theoretical intensity functions to the measured ones, the post-collisional states of the atoms and their electric dipole moments were determined. The results indicate that for projectile energies below 20 keV, the electric dipole moments are small; however, for energies above 20 keV, mainly the parabolic Stark states with maximal electric dipole moments are excited. We conclude that in the upper section of the energy range investigated here, the Paul-trap promotion is the dominant excitation mechanism for He+–He collisions.

Non-statistical population distributions for hydrogen beams in fusion plasmas

Most atomic models for neutral hydrogen beams in fusion plasmas assume a statistical (Boltzmann) distribution of populations for excited states with the same principal quantum number n. Here we analyze population distributions for the excited magnetic sublevels of a beam under typical conditions of existing and future fusion devices. The collisional–radiative model NOMAD based on completely m-resolved parabolic states up to n = 10 is used to study this problem. The model utilizes new proton-impact excitation data calculated with the atomic-orbital close-coupling method and the Glauber approximation and takes into account electric-field-induced ionization from highly excited states. Our simulations show that the statistical assumption for a specific n is generally not valid for typical fusion conditions due to radiative processes and strong field ionization. The deviation increases considerably for higher beam energies and stronger magnetic fields. The calculated line intensities of σ and π components and beam-emission parameters are discussed in detail.

DNN-state identification of 2D distributed parameter systems

There are many examples in science and engineering which are reduced to a set of partial differential equations (PDEs) through a process of mathematical modelling. Nevertheless there exist many sources of uncertainties around the aforementioned mathematical representation. Moreover, to find exact solutions of those PDEs is not a trivial task especially if the PDE is described in two or more dimensions. It is well known that neural networks can approximate a large set of continuous functions defined on a compact set to an arbitrary accuracy. In this article, a strategy based on the differential neural network (DNN) for the non-parametric identification of a mathematical model described by a class of two-dimensional (2D) PDEs is proposed. The adaptive laws for weights ensure the ‘practical stability’ of the DNN-trajectories to the parabolic 2D-PDE states. To verify the qualitative behaviour of the suggested methodology, here a non-parametric modelling problem for a distributed parameter plant is analysed.

Neural Identification of 3D Distributed Parameter Systems

AbstractDifferential Neural Networks (DNN) state identification of 3D Distributed Parameters Systems is studied in this paper. A lot of examples in science and engineering of systems described mathematically by partial differential equations (PDE's), posses the disadvantage of having many sources of uncertainties around their mathematical representation. Moreover to find the exact solutions of those PDE's is not a trivial task especially if the PDE is described in two or more dimensions. It is well known that neural networks can approximate a large set of continuous functions defined on a compact set to an arbitrary accuracy. A strategy based on DNN for the non parametric identification of a mathematical model described by a class of three dimensional (3D) PDE is proposed. The adaptive laws for weights ensure the “practical stability” of the DNN trajectories to the parabolic 3D-PDE states. To verify the qualitative behavior of the suggested methodology, here a non parametric modeling problem for a distributed parameter plant is analyzed.

Consistency of atomic data for the interpretation of beam emission spectra

Several collisional–radiative (CR) models (Anderson et al 2000 Plasma Phys. Control. Fusion 42 781–806, Hutchinson 2002 Plasma Phys. Control. Fusion 44 71–82, Marchuk et al 2008 Rev. Sci. Instrum. 79 10F532) have been developed to calculate the attenuation and the population of excited states of hydrogen or deuterium beams injected into tokamak plasmas. The datasets generated by these CR models are needed for the modelling of beam ion deposition and (excited) beam densities in current experiments, and the reliability of these data will be crucial to obtain helium ash densities on ITER combining charge exchange and beam emission spectroscopy. Good agreement between the different CR models for the neutral beam (NB) is found, if corrections to the fundamental cross sections are taken into account. First the Hα and Hβ beam emission spectra from JET are compared with the expected intensities. Second, the line ratios within the Stark multiplet are compared with the predictions of a sublevel resolved model. The measured intensity of the full multiplet is ≈30% lower than expected on the basis of beam attenuation codes and the updated beam emission rates, but apart from the atomic data this could also be due to the characterization of the NB path and line of sight integration and the absolute calibration of the optics. The modelled n = 3 to n = 4 population agrees very well with the ratio of the measured Hα to Hβ beam emission intensities. Good agreement is found as well between the NB power fractions measured with beam emission in plasma and on the JET Neutral Beam Test Bed. The Stark line ratios and σ/π intensity ratio deviate from a statistical distribution, in agreement with the CR model in parabolic states from Marchuk et al (2010 J. Phys. B: At. Mol. Opt. Phys. 43 011002).

Non-statistical populations of magnetic sublevels of hydrogen beam atoms in fusion plasmas

Emission of spectral lines from the excited states of neutral beam atoms in fusion plasmas is the basis of beam emission spectroscopy and motional Stark effect diagnostics. It is well known that the measured intensities of Stark multiplet lines between the n=3 and 2 excited states of hydrogen systematically deviate from the full statistical model. Here we study the H α Stark component intensities using a newly developed collisional-radiative model that is based on the description of magnetic sublevels in a parabolic basis. Proton excitation cross-sections between the parabolic states are calculated in the Glauber (eikonal) approximation. The model incorporates various collisional processes between the hydrogen beam atoms and plasma particles (protons, impurity ions and electrons) as well as the radiative processes. The simulated line component ratios σ 1 / σ 0 , π 2 / π 3 and π 4 / π 3 are found to agree with the measured ratios from JET plasmas within the experimental uncertainties.

A transition from local equilibrium to paraequilibrium kinetics for ferrite growth in Fe–C–Mn: A possible role of interfacial segregation

Ferrite growth kinetics was measured under highly controlled decarburization conditions in an Fe–0.94%Mn–0.57%C (mass%) alloy. The observed growth kinetics closely followed the predictions of the local equilibrium non-partitioning (LE-NP) model over the temperature range 725–775 °C. A transition from LE-NP to paraequilibrium (PE) kinetics was observed, for the first time, as the temperature was increased from 775 °C to 825 °C. Long-lived parabolic kinetic states that are intermediate between LE-NP and PE were observed at these intermediate temperatures. Above 825 °C, ferrite growth appeared to follow the predictions of the PE model up to the T 0 temperature at which ferrite growth stopped. These novel results are attributed to the segregation of Mn to the moving interface. It is thought that this process has the effect of decreasing the Mn concentration on the austenite side of the interface and, consequently, extending the PE state.

An accurate quantum expression of the z-dipole matrix element between nearby Rydberg parabolic states and the correspondence principle

We give an exact quantum formula for the z-component of the dipole matrix element between parabolic states of a hydrogen atom in terms of the Jacobi polynomials. The formula extends the range of numerical computation to larger values of the parabolic quantum numbers for which computation from the standard textbook formula, which is in terms of the hypergeometric functions, is defined. We obtain an accurate quantum expression of the z-dipole matrix element in terms of the ordinary Bessel functions for transition between nearby Rydberg parabolic states. We derive for the first time the formula of the z-dipole matrix element of the correspondence principle method directly from the quantum expression, and in the process of derivation, clarify the nature of classical–quantum correspondence. The expressions obtained in this work solve the problem of computation of the z-dipole matrix element of hydrogen to a large extent.

Anticrossing spectroscopy of He atoms excited by 30–300 keV He+ impact

By measuring the anticrossing spectra of the 1s4ℓ and 1s5ℓ He I levels, we investigate single-electron excitation in 30–300 keV He+–He collisions. The post-collisional He I states are highly coherent superpositions of spherical states with large electric dipole moments. These superposition states vary gradually with the impact energy from purely parabolic states at 30 keV to superpositions of ℓ ⩽ 2 states at 300 keV. The results indicate that for the whole energy range single-electron excitation should be considered as a process mainly localized in the saddle region of the two-centre Coulomb potential of the (He+)2 molecular core. A theoretical model based on the symmetry of the collision system is discussed.

Quantum localization in the three-dimensional kicked Rydberg atom

We study the three-dimensional (3D) unidirectionally kicked Rydberg atom. For parabolic initial states elongated in the direction of the kicks we show that the ionization of the quantum system is suppressed as compared to the classical counterpart and that the quantum wave function is localized along all degrees of freedom, whereas the classical system is globally diffusive. We discuss the connection to the previously studied one-dimensional (1D) model of the kicked Rydberg atom and verify that the 1D model is a good approximation to the 3D quantum case in the limiting case of the most elongated initial states. We further study the quantum phase-space distribution (Husimi distribution) of the eigenstates of the period-one time-evolution (Floquet) operator and show that the eigenstates are localized in phase space. For the most elongated parabolic initial state, we are able to identify the unstable periodic orbits around which Floquet states localize. We discuss the possibility of observing quantum localization in high Rydberg states in $ng100.$

Cascade feeding of the 1s3d $ \mathsf {^3}$ D level of He I from high-l states excited by proton impact

The 1 s 3 d 3D level of atomic helium cannot be excited directly by proton impact, but it is strongly populated by cascade feeding from 1snl states with l ≥ 3, if 10÷15 keV protons are used for excitation. This cascade feeding process can be analysed in detail by investigating the intensity of the spectral line at λ(1 s 3 d 3 D- 1 s 2 p 3 P ) = 588 nm as a function of an electric field directed parallel and antiparallel to the proton beam. In this paper the intensity functions are calculated assuming collisional excitation of parabolic singlet states | 1 s ; n , n 1 , n 2 , m 〉 with large electric dipole moments. The theoretical results are compared with the experimental intensity function published earlier. Excellent agreement is obtained if the excitation cross-sections σ n of the parabolic states scale as n-3.

Direct comparison of electric field measurement byfluorescence and optogalvanic Stark spectroscopies

Electric field in the cathode fall region of a 2.2 Torr heliumdc discharge with 0.8 mA cm-2 current density has beenmeasured from the Stark splittings of 11 1P (unperturbed)Rydberg transition by simultaneous optogalvanic (OG) andfluorescence techniques. The LIF Stark spectra showed narrowerlinewidths compared to the OG spectra for higher parabolicquantum number states; LIF data also showed broader linewidthsfor higher parabolic quantum number states compared to the linesin the centre of the manifold. However, the electric fieldsobtained from the two methods were in agreement with each otherwithin three per cent for all the measured locations in thesheath. A comparison of the OG spectrum with the off-axis LIF datashows a combination of signal gain in the high-field regionalong with the Stark intensity distribution permits the line-of-sight-averaged OG spectrum to obtain accurate field values in adischarge with modest radial field gradients.

The correspondence principle and quantum dipole matrix element between parabolic Rydberg states

We derive a quantum formula of the dipole matrix element for parabolic Rydberg states in terms of the Bessel functions without appealing to any quasiclassical methods. We exploit this quantum formula to explain the discrepancy between the other quasi-classical expressions available in the literature in terms of the Airy functions. We use the quantum formula to examine critically the region of validity of the corresponding formula derived by Born using the classical electron orbit and correspondence principle.

Form factor for transition between parabolic Rydberg states

An analytical formula of the form factor for transition between parabolic Rydberg states is derived directly from the full quantum formula. This is done by expressing, for the first time, the quantum form factor for transition between arbitrary parabolic states in terms of the Jacobi polynomials. Usefulness of this formulation in deducing various properties of the form factor is also discussed.