Integer Number Of Electrons Research Articles

Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice

Moir\'e superlattices of van der Waals materials, such as twisted graphene and transitional metal dichalcogenides, have recently emerged as a fascinating platform to study strongly correlated states in two dimensions, thanks to the strong electron interaction in the moir\'e minibands. In most systems, the correlated states appear when the moir\'e lattice is filled by integer number of electrons per moir\'e unit cell. Recently, correlated states at fractional fillings of 1/3 and 2/3 holes per moir\'e unit cell has been reported in the WS2/WSe2 heterobilayer, hinting the long range nature of the electron interaction. In this work, employing a scanning microwave impedance microscopy technique that is sensitive to local electrical properties, we observe a series of correlated insulating states at fractional fillings of the moir\'e minibands on both electron- and hole-doped sides in angle-aligned WS2/WSe2 hetero-bilayers, with certain states persisting at temperatures up to 120 K. Monte Carlo simulations reveal that these insulating states correspond to ordering of electrons in the moir\'e lattice with a periodicity much larger than the moir\'e unit cell, indicating a surprisingly strong and long-range interaction beyond the nearest neighbors. Our findings usher in unprecedented opportunities in the study of strongly correlated states in two dimensions.

МАГНИТНЫЙ ПОТОК И ЕГО КВАНТОВАНИЕ

It is noted that the elementary electric charge is equal to e , from which fact the well-known principle of quantization of the electric charge definitely follows, namely, the electric charge is quantized and a quantum is the charge of the electron e (or positron). Or any change in the charge is equal to an integer number of electrons (or positrons). The formally identical transformation of the principle of quantization of the electric charge allows formulating the principle of quantization of the magnetic flux, namely, the quantity inverse to the magnetic flux is quantized and a quantum is the inverse of the quantum of the F. London magnetic flux. Or any change in the reciprocal of the magnetic flux is equal to an integer number of quantities inverse to the quantum of the F. London magnetic flux. The reciprocal of the magnetic flux quantum is equal to the sum of two quantities inverse to the quantum of the F. London magnetic flux. The validity of the principle of quantization of the magnetic flux with respect to the hydrogen atom is illustrated by Theorem 1: quantization of the energy of the hydrogen atom is a consequence of the principle of quantization of the magnetic flux. Theorem 2 is proved: the magnetic flux of a hydrogen atom in the ground state is equal to the quantum of the F. London magnetic flux. Theorem 3 is proved: the quantum of the magnetic flux is not the minimum possible for a nonzero magnetic flux. It is established that the quantum of the magnetic flux is not a quantum in the sense of a portion (like the F. London quantum). A quantum is the inverse of the quantum of the F. London magnetic flux. It is established that the magnitude of the magnetic flux quantum is not the minimum possible for a nonzero magnetic flux. It is established that the magnetic flux of a hydrogen atom in the ground state is equal to the quantum of the F. London magnetic flux. A discrete set of energies of the hydrogen atom is noted to be a consequence of the solution of the Schrödinger equation, which, in turn, is phenomenological. The reverse discourse used in the proof of Theorem 1 can show that the Schrödinger equation is a consequence of the principle of quantization of the magnetic flux.

Computational study of the structures and photoelectron spectra of 12 azabenzenes

The molecular structures of 12 azabenzenes have been optimized with the Gaussian09 package at the level of coupled cluster singles and doubles with the basis set cc-pVTZ. The optimized geometry of each is used in the ADF13 program for the calculation of the vertical ionization energies of all the electrons. For both outer-shell and inner-shell valence electrons, the 2009 method of ΔPBE0(SAOP) is used, whereas the 1999 method of ΔPW86PW91 + Crel is employed for the core electrons. For degenerate orbitals, the alternative method chosen is to use localized orbitals, keeping the integer number of electrons, while giving up proper symmetry, rather than to keep symmetry with fractional electrons. The success of the computed results of valence ionization potentials of pyridine and the diazabenzenes gives confidence for the predicted values for the higher azabenzenes. The calculated results for core-electron binding energies provide incentive to experimentalists to measure them with X-ray photoelectron spectrometers and (or) synchrotron facilities.

Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers

We show that moiré bands of twisted homobilayers can be topologically nontrivial, and illustrate the tendency by studying valence band states in ±K valleys of twisted bilayer transition metal dichalcogenides, in particular, bilayer MoTe_{2}. Because of the large spin-orbit splitting at the monolayer valence band maxima, the low energy valence states of the twisted bilayer MoTe_{2} at the +K (-K) valley can be described using a two-band model with a layer-pseudospin magnetic field Δ(r) that has the moiré period. We show that Δ(r) has a topologically nontrivial skyrmion lattice texture in real space, and that the topmost moiré valence bands provide a realization of the Kane-Mele quantum spin-Hall model, i.e., the two-dimensional time-reversal-invariant topological insulator. Because the bands narrow at small twist angles, a rich set of broken symmetry insulating states can occur at integer numbers of electrons per moiré cell.

Quantized charge pump of massive Dirac electrons

We study a new scheme to realize a quantized two-parameter charge pump based on massive Dirac electrons. It is shown that the two time-dependent and out-of-phase staggered potentials introduced in graphene can pump out an integer number of electrons in a pumping cycle as long as the Fermi energy resides in the effective energy gap opened by pumping potentials. The dependence of the pumped charge per mode on the pumping phase or the dynamic phase exhibits a binary alternation from $+e$ to $\ensuremath{-}e$. This quantization has a topological origin and can be accounted for by adiabatic evolution of the topologically protected interfacial state forming between the two pumping sources.

Communication: Two types of flat-planes conditions in density functional theory.

Using results from atomic spectroscopy, we show that there are two types of flat-planes conditions. The first type of flat-planes condition occurs when the energy as a function of the number of electrons of each spin, Nα and Nβ, has a derivative discontinuity on a line segment where the number of electrons, Nα + Nβ, is an integer. The second type of flat-planes condition occurs when the energy has a derivative discontinuity on a line segment where the spin polarization, Nα - Nβ, is an integer, but does not have a discontinuity associated with an integer number of electrons. Type 2 flat planes are rare-we observed just 15 type 2 flat-planes conditions out of the 4884 cases we tested-but their mere existence has implications for the design of exchange-correlation energy density functionals. To facilitate the development of functionals that have the correct behavior with respect to both fractional number of electrons and fractional spin polarization, we present a dataset for the chromium atom and its ions that can be used to test new functionals.

Interpolation of property-values between electron numbers is inconsistent with ensemble averaging.

In this work we explore the physical foundations of models that study the variation of the ground state energy with respect to the number of electrons (E vs. N models), in terms of general grand-canonical (GC) ensemble formulations. In particular, we focus on E vs. N models that interpolate the energy between states with integer number of electrons. We show that if the interpolation of the energy corresponds to a GC ensemble, it is not differentiable. Conversely, if the interpolation is smooth, then it cannot be formulated as any GC ensemble. This proves that interpolation of electronic properties between integer electron numbers is inconsistent with any form of ensemble averaging. This emphasizes the role of derivative discontinuities and the critical role of a subsystem's surroundings in determining its properties.

The Redox Processes in Lithium Batteries : Beyond the Oxidation State Formalism

In general, electrochemistry is based on electron exchange between different species (ions, atoms, molecules,grains…), in a liquid or solid environment, and this electron exchange is usually written in using the concept of oxidation state of the different elements constituting the species. Redox processes are thus introduced, where oxidation and reduction for a specie correspond to the loss or a gain of electrons, respectively. This concept is generally well adapted to electrochemistry because the oxidation state of an element corresponds to the number of electrons which is lost or gained in a chemical bond, supposed to be purely ionic, to form a cation or an anion, respectively. The electronic structure of an ion or a molecule is described by discrete energy levels which are occupied by a small and integer number of electrons. So, during an electrochemical process, electrons are added or removed and this electronic transfer can be explained by using quite simple concept. The problem is different in a solid, The problem is different in a solid for which the only way to describe the electronic structure is to use the band structure model: atomic orbitals of all elements are more or less mixed together to form the bands. Consequently, the number of valence electrons is very large and the ionic model is very often unsuitable. In this presentation we will show through different examples that use of the oxidation state can be misleading in order to explain the true behavior of a solid electrode material in lithium batteries. We will propose a more general description. First, we will focus on X-ray Absorption Spectroscopy (XAS), one of the most powerful tools to analyze precisely the state of charge of an element. It has been showed many times, on chalcogenides (1) and oxides (2), that this technique combines very nicely with the calculated electronic structure of a solid. It will be evidenced that in a redox process all elements of a material are modified by the electron transfer. Secondly, we will show how the oxidation state concept could lead the researchers to exclude a priori some reactions because they would put some elements in unusual low or high oxidation states (3). Finally, we will show recent results obtained on Li2MnO3 and HE-NMC materials (4-6). From them, it appears clearly that the oxidation and reduction processes in solid are complex phenomenon. They generally impact all elements which have to be considered as possible redox centers. The general model we propose is based on solid electronic band structure for solid, considered as an electron reservoir which can be emptied (oxidation) or filled (reduction). The reservoir is built a priori by the orbitals of all elements. The fact that an electrochemical reaction occurs, becomes reversible, or leads to degradation of the material, has to be discussed in terms of energetic stability of a given atomic structure. This can be related to the shape and the filling of the electronic reservoir. References 1 – Z.Y. Wu, G. Ouvrard, S. Lemaux, P. Moreau, P. Gressier, F. Lemoigno and J. ROUXEL, Phys. Rev. Lett. 77 (1996) 2101-2104. 2- F. Boucher, N. Bourgeon, K. Delbé, P. Moreau, D. Guyomard and G. Ouvrard, Journal of Physics and Chemistry of Solids, 67 (2006) 1238. 3 – N. Tran, L. Croguennec, M. Ménétrier, F. Weill, Ph. Biensan, C. Jordy, and C. Delmas, Chem. Mater. 20 (2008) 4815–4825 4 – H. Koga, L. Croguennec, M. Ménétrier, P. Mannessiez, F. Weill, C. Delmas, and S. Belin J. Phys. Chem. C 118 (2014) 5700−5709 5 – Yukinori Koyamaa, Isao Tanakaa, Miki Nagaob, Ryoji Kanno Journal of Power Sources 189 (2009) 798–801 6 – A. Pradon, C. La Fontaine, S. Belin, P. E. Petit, P. Moreau,, L. Lajaunie, E. Dumont, E. Elkaim, C. Tessier, G. Ouvrard, M.T. Caldes, Submitted to Chemistry of Materials

On the stability of some analytically solvable maximum probability domains

The stability of some maximum probability domains (MPDs) determined for a set of analytically solvable models is examined. MPDs are regions of the physical space that maximize the probability of finding an exact, integer number of electrons and have been shown to provide vivid images of cores, lone and bonding pairs, etc. They are obtained by numerical maximization techniques based on the use of the shape derivative, which has to vanish at any stationary point. Here, we give the first steps to use the shape Hessian to gain information about the true nature of the numerical solutions. As a general conclusion, electron correlation seems to decrease the multiplicity of solutions found at the Hartree–Fock level, the remaining MPDs remaining closer to what it is expected from chemical insight.

Uma abordagem diferente na estatística do experimento Millikan

A experiência clássica de Millikan para a determinação da carga do elétron é muito interessante do ponto de vista de estudos estatísticos de medidas experimentais. Em uma experiência didática de laboratório de física, é sugerido dividir os valores da carga elétrica obtida pelo valor da carga do elétron e, desta forma, estabelecer experimentalmente que as gotas de óleo carregam números inteiros de elétrons. Este trabalho apresenta um algoritmo estatístico novo para tratamento das medidas e para o cálculo do melhor valor da carga do elétron em um experimento Millikan.

Curvature and Frontier Orbital Energies in Density Functional Theory.

Perdew et al. discovered two different properties of exact Kohn-Sham density functional theory (DFT): (i) The exact total energy versus particle number is a series of linear segments between integer electron points. (ii) Across an integer number of electrons, the exchange-correlation potential "jumps" by a constant, known as the derivative discontinuity (DD). Here we show analytically that in both the original and the generalized Kohn-Sham formulation of DFT the two properties are two sides of the same coin. The absence of a DD dictates deviation from piecewise linearity, but the latter, appearing as curvature, can be used to correct for the former, thereby restoring the physical meaning of orbital energies. A simple correction scheme for any semilocal and hybrid functional, even Hartree-Fock theory, is shown to be effective on a set of small molecules, suggesting a practical correction for the infamous DFT gap problem. We show that optimally tuned range-separated hybrid functionals can inherently minimize both DD and curvature, thus requiring no correction, and that this can be used as a sound theoretical basis for novel tuning strategies.

Toward Modeling Charge-Defect Reactions at the Atomistic Level

AbstractDefect reactions involving charged species are commonplace in nuclear fuels fabrication and burn-up. Even the simplest of these fuels, uranium dioxide (UO2), typically involves the nominal charge states of +3, +4, and +5 or +6 in U and -1 and -2 states in O. Simulations that attempt to model evolutionary processes in the fuels require tracking changes among these charge states. At the atomistic level, modeling defect reactions poses a particularly vexing problem. Typical potential energy surfaces do not have this type of physical phenomena built into them. Those models that do attempt to model charge-defect reactions do not have especially strong physical bases for the models. For instance, most do not obey established limits of charge behavior at dissociation or lack internal consistency. This work presents substantial generalizations to earlier work of Perdew et al. No matter the size of the system, total system hamiltonians can be decomposed into subsystem or site hamiltonians and coulombic interactions. Site hamiltonians can be evaluated in a spectral representation, once an integer number of electrons are assigned. For both pair and individual site hamiltonians a dilemma emerges in that many sites are better understood as possessing a fractional charge. The dilemma is how to weight the site integer-charge states in a physically consistent manner. One approach to solving the dilemma results in two distinct charge-dependent energy contributions emerge, arising from intra- and inter-subsystem charge transfer. Further analysis results in a model of the intra-subsystem charge-transfer that can accommodate the mixed valence states of either U or O in nuclear fuels. Mixed valence properties add complications to the model that originate in the phenomenological fact that it typically requires different amounts of energy to increase or decrease charge. As a result of the inherent complexity one has the option of using multiple charges, a concept with strong ties to shell models, or modeling parameters not directly related to charge as functions of charge. This latter approach is illustrated by invoking a minimization principle that does preserve the important dissociation limits of Perdew et al., in order to complete the model.

Electron number probability distributions for correlated wave functions

Efficient formulas for computing the probability of finding exactly an integer number of electrons in an arbitrarily chosen volume are only known for single-determinant wave functions [E. Cances et al., Theor. Chem. Acc. 111, 373 (2004)]. In this article, an algebraic method is presented that extends these formulas to the case of multideterminant wave functions and any number of disjoint volumes. The derived expressions are applied to compute the probabilities within the atomic domains derived from the space partitioning based on the quantum theory of atoms in molecules. Results for a series of test molecules are presented, paying particular attention to the effects of electron correlation and of some numerical approximations on the computed probabilities.

Exact study of charge-spin separation, pairing fluctuations, and pseudogaps in four-site Hubbard clusters

An exact study of charge-spin separation, pairing fluctuations and pseudogaps is carried out by combining the analytical eigenvalues of the four-site Hubbard clusters with the grand canonical and canonical ensemble approaches in a multidimensional parameter space of temperature (T), magnetic field (h), on-site interaction (U) and chemical potential. Our results, near the average number of electrons <N>=3, strongly suggest the existence of a critical parameter U_{c}(T) for the localization of electrons and a particle-hole binding (positive) gap at U>U_{c}(T), with a zero temperature quantum critical point, U_{c}(0)=4.584. For U<U_{c}(T), particle-particle pair binding is found with a (positive) pairing gap. The ground state degeneracy is lifted at U>U_c(T) and the cluster becomes a Mott-Hubbard like insulator due to the presence of energy gaps at all (allowed) integer numbers of electrons. In contrast, for U< U_c(T), we find an electron pair binding instability at finite temperature near <N>=3, which manifests a possible pairing mechanism, a precursor to superconductivity in small clusters. In addition, the resulting phase diagram consisting of charge and spin pseudogaps, antiferromagnetic correlations, hole pairing with competing hole-rich (<N>=2), hole-poor (<N>=4) and magnetic (<N>=3) regions in the ensemble of clusters near 1/8 filling closely resembles the phase diagrams and inhomogeneous phase separation recently found in the family of doped high T_c cuprates.

Acoustic Transport of Electrons in Parallel Quantum Wires

Over the last few years we have developed a new method to control single-electrons by isolating and moving them through a submicron width channel formed in a GaAs/AlGaAs heterostructure using a surface acoustic wave. The acoustic wave acts to push electrons through the depleted submicron channel in packets each containing an integer number of electrons. Our primary motivation for studying this system has been to develop a new standard of dc current for metrological purposes, but our recent focus has widened to investigate the possibility of single-photon emission. Here we show new experimental results which demonstrate acoustoelectric current flow in adjacent 1D wires. These results have relevance both to the use of the system in a single-photon emission scheme, as well as in the creation of a proposed acoustoelectric quantum computer.

Resonance approximation and charge loading and unloading in adiabatic quantum pumping

Quantum pumping through mesoscopic quantum dots is known to be enhanced by resonant transmission. The pumped charge is close to an integer number of electrons when the pumping contour surrounds a resonance, but the transmission remains small on the contour. For noninteracting electrons, we give a quantitative account of the detailed exchange of electrons between the dot and the leads (to the electron reservoirs) during a pumping cycle. Near isolated distinct resonances, we use approximate Breit-Wigner expressions for the dot's Green function to discuss the loading/unloading picture of the pumping: the fractional charge exchanged between the dot and each lead through a single resonance point is related to the relative couplings of the dot and the leads at this resonance. If each resonance point along the pumping contour is dominated by the coupling to a single lead (which also implies a very small transmission), then the crossing of each such resonance results in a single electron exchange between the dot and that lead, ending up with a net quantized charge. When the resonance approximation is valid, the fractional charges can also be extracted from the peaks of the transmissions between the various leads.

The non-integer cyclovoltammetric electron-exchange numbers of reversible redox reactions of adsorbates: theoretical considerations

Two types of non-integer electron-exchange numbers from uniform and reversible surface-redox reactions without side reactions have been distinguished. The first being the apparent number, napp, of the apparent faradaic charge corresponding to cyclovoltammetric peak areas above the interpolated baseline, and the second the thermodynamically defined surface-redox valency, n', of Nernstian slopes of cyclovoltammetric peak potentials depending on different solution pH. An analytical expression has been derived for napp based on a simplified capacitive equivalent circuit and for n' using the potential-dependent free adsorption energies of the reactants involved. It should be pointed out that the different experimental values of napp and n' refer to the same integer number of electrons per molecule oxidized or reduced.

Proposal for a quantum Hall pump

A device is proposed that is similar in spirit to the electron turnstile except that it operates within a quantum Hall fluid. In the integer quantum Hall regime, this device pumps an integer number of electrons per cycle. In the fractional regime, it pumps an integer number of fractionally charged quasiparticles per cycle. It is proposed that such a device can make an accurate measurement of the charge of the quantum Hall effect quasiparticles.

Relationship between the highest occupied Kohn-Sham orbital eigenvalue and ionization energy

Recent arguments for and against the equivalence of the highest occupied orbital eigenvalue of the Kohn-Sham theory and ionization energy are discussed. It is shown that for physically realistic systems with a nonintegral number of electrons, which are described by the thermal average of two systems, each with an integer number of electrons, an equivalent Kohn-Sham system exists. This is done by writing explicit expressions for the exchange-correlation potential constructed to give the mixed-state density, and then relating it to the mixed-state exchange-correlation energy functional by employing the virial theorem sum rule of Levy and Perdew [Phys. Rev. A 32, 2010 (1985)]. Further, the functional derivative of the mixed-state exchange-correlation energy functional is obtained in terms of this potential. This is then used to show, without recourse to Janak's theorem [Phys. Rev. B 18, 7165 (1978)], that ${\ensuremath{\varepsilon}}_{\mathrm{max}}(N)=\ensuremath{-}I(Z),$ where Z is an integer and $(Z\ensuremath{-}1)lNlZ.$ Thus the original arguments about the equivalence of the highest occupied Kohn-Sham orbital eigenenergy and the ionization energy which were based on Janak's theorem are valid, and the two quantities are equal.

A challenge for density functionals: Self-interaction error increases for systems with a noninteger number of electrons

The difficulty of widely used density functionals in describing the dissociation behavior of some homonuclear and heteronuclear diatomic radicals is analyzed. It is shown that the self-interaction error of these functionals accounts for the problem—it is much larger for a system with a noninteger number of electrons than a system with an integer number of electrons. We find the condition for the erroneous dissociation behavior described by approximate density functionals: when the ionization energy of one dissociation partner differs from the electron affinity of the other partner by a small amount, the self-interaction error will lead to wrong dissociation limit. Systems with a noninteger number of electrons and hence the large amount of self-interaction error in approximate density functionals arise also in the transition states of some chemical reactions and in some charge-transfer complexes. In the course of analysis, we derive a scaling relation necessary for an exchange-correlation functional to be self-interaction free.