Matrix Elements Of Coulomb Interaction Research Articles

Exciton-Exciton Interactions in Van der Waals Heterobilayers

Exciton-exciton interactions are key to understanding nonlinear optical and transport phenomena in van der Waals heterobilayers, which emerged as versatile platforms to study correlated electronic states. We present a combined theory-experiment study of excitonic many-body effects based on first-principle band structures and Coulomb interaction matrix elements. Key to our approach is the explicit treatment of the fermionic substructure of excitons and dynamical screening effects for density-induced energy renormalization and dissipation. We demonstrate that dipolar blueshifts are almost perfectly compensated by many-body effects, mainly by screening-induced self-energy corrections. Moreover, we identify a crossover between attractive and repulsive behavior at elevated exciton densities. Theoretical findings are supported by experimental studies of spectrally narrow, mobile interlayer excitons in atomically reconstructed, h-BN-encapsulated MoSe2/WSe2 heterobilayers. Both theory and experiment show energy renormalization on a scale of a few meV even for high injection densities in the vicinity of the Mott transition. Our results revise the established picture of dipolar repulsion dominating exciton-exciton interactions in van der Waals heterostructures and open up opportunities for their external design. Published by the American Physical Society 2024

Downfolding from ab initio to interacting model Hamiltonians: comprehensive analysis and benchmarking of the DFT+cRPA approach

Model Hamiltonians are regularly derived from first principles to describe correlated matter. However, the standard methods for this contain a number of largely unexplored approximations. For a strongly correlated impurity model system, here we carefully compare a standard downfolding technique with the best possible ground-truth estimates for charge-neutral excited-state energies and wave functions using state-of-the-art first-principles many-body wave function approaches. To this end, we use the vanadocene molecule and analyze all downfolding aspects, including the Hamiltonian form, target basis, double-counting correction, and Coulomb interaction screening models. We find that the choice of target-space basis functions emerges as a key factor for the quality of the downfolded results, while orbital-dependent double-counting corrections diminish the quality. Background screening of the Coulomb interaction matrix elements primarily affects crystal-field excitations. Our benchmark uncovers the relative importance of each downfolding step and offers insights into the potential accuracy of minimal downfolded model Hamiltonians.

Calculations of shake-up satellites intensities in photoelectron spectra by generalized configuration interaction method

Satellites i.e. two-hole-one-particle states appear due to many-electron correlations in photoelectron spectra in addition to main lines corresponding to one-hole states. The correct assignment of these states is important for understanding the photoionization processes and for materials analysis by photoelectron spectroscopy. In the present work two different existing methods, which use overlap or Coulomb interaction matrix elements are considered and a new method is proposed. In this technique many-electron effects of satellites origin and interactions between satellite states are accounted for simultaneously by solving secular matrix. The method is applied to satellite states 2p −23p(4p), 2p −23s, 2p −23d of Ne photoionization for which contradiction between two interpretations existed. The results of calculations unambiguously support one the interpretations.

Microscopic theory of exciton-exciton annihilation in two-dimensional semiconductors

Auger-like exciton-exciton annihilation (EEA) is considered the key fundamental limitation to quantum yield in devices based on excitons in two-dimensional (2d) materials. Since it is challenging to experimentally disentangle EEA from competing processes, guidance of a quantitative theory is highly desirable. The very nature of EEA requires a material-realistic description that is not available to date. We present a many-body theory of EEA based on first-principle band structures and Coulomb interaction matrix elements that goes beyond an effective bosonic picture. Applying our theory to monolayer MoS$_2$ encapsulated in hexagonal BN, we obtain an EEA coefficient in the order of $10^{-3}$ cm$^{2}$s$^{-1}$ at room temperature, suggesting that carrier losses are often dominated by other processes, such as defect-assisted scattering. Our studies open a perspective to quantify the efficiency of intrinsic EEA processes in various 2d materials in the focus of modern materials research.

Representation of the Coulomb Matrix Elements by Means of Appell Hypergeometric Function F2

Exact analytical representation for the Coulomb matrix elements by means of Appell’s double series F2 is derived. The finite sum obtained for the Appell function F2 allows us to evaluate explicitly the matrix elements of the two-body Coulomb interaction in the lowest Landau level. An application requiring the matrix elements of Coulomb potential in quantum Hall effect regime is presented.

RCCPAC: A parallel relativistic coupled-cluster program for closed-shell and one-valence atoms and ions in FORTRAN

We report the development of a parallel FORTRAN code, RCCPAC, to solve the relativistic coupled-cluster equations for closed-shell and one-valence atoms and ions. The parallelization is implemented through the use of message passing interface, which is suitable for distributed memory computers. The coupled-cluster equations are defined in terms of the reduced matrix elements, and solved iteratively using Jacobi method. The ground and excited states of coupled-cluster wave functions obtained from the code could be used to compute different properties of closed-shell and one-valence atom or ion. As an example we compute the ground state correlation energy, attachment energies, E1 reduced matrix elements and hyperfine structure constants. Program summaryProgram Title: RCCPACProgram Files doi:http://dx.doi.org/10.17632/34rwtn27nb.1Licensing provisions: MIT licenceProgramming language: FORTRAN 90Nature of problem: Compute the ground and excited state wave functions, correlation energy, attachment energies, and E1 transition amplitude and hyperfine structure constant of closed-shell and one-valence atoms or ions using relativistic coupled-cluster theory.Solution method: The basic input data required is an orbital basis set generated using the Dirac–Coulomb Hamiltonian. For the present case, closed-shell and one-valence systems, the orbitals are grouped into occupied, valence and virtual. The relativistic coupled-cluster equations of the single and double excitation cluster amplitudes, which form a set of coupled nonlinear equations, are defined in terms of reduced matrix elements of the residual Coulomb interaction. The equations are solved iteratively in parallel using Message Passing Interface (MPI) with the Jacobi method. However, to overcome the slow convergence of the method, we use Direct Inversion in the Iterated Sub-space (DIIS) to accelerate the convergence. For enhanced performance, the two-electron integrals and 6j- symbols are precalculated and stored. Furthermore, to optimize memory requirements, selected groups of integrals are computed and stored only by the thread which needs the integrals during computation.Restrictions: For efficient computations, the two-electron reduced Coulomb matrix elements consisting of three and four virtual states are stored in RAM. This limits the size of the basis set, as there should be sufficient space in RAM to store all the integrals.Unusual features: To avoid replication of data across the cores, and optimize the RAM use, the three particle and four particle two-electron Coulomb integrals are distributed. That is, following the loop structure in the driver, each core stores only the integrals it requires. With this feature there is enormous reduction in the RAM required to store the integrals.Additional comments: The code can be modified, with minimal changes, to compute properties other than electromagnetic transitions and hyperfine constants with appropriate modifications. The required modifications are addition of subroutine to compute the single-electron matrix element, and inclusion of calling sequence in the main driver subroutine.

Interband Coulomb coupling in narrow-gap semiconductor nanocrystals:k·ptheory

We derive the matrix elements of Coulomb interaction between states with different numbers of electrons and holes in a semiconductor nanocrystal within the eight-band $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ theory. These matrix elements are responsible for multiple exciton generation which may contribute to the enhancement of the efficiency of solar cells. Our calculations are performed within the multiband envelope function formalism based on the states resulting from diagonalization of the eight-band $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ Hamiltonian. We study in detail and compare two contributions to the interband Coulomb coupling: the mesoscopic one, which involves only the envelope functions and relies on band mixing, and the microscopic one, which relies on the Bloch parts of the wave functions and is nonzero even between single-band states. We show that these two contributions are of a similar order of magnitude. We also study the statistical distribution of the magnitudes of the interband Coulomb matrix elements and show that the overall coupling to remote states decays according to a power law favorable for the convergence of numerical computations.

Influence of excited carriers on the optical and electronic properties of MoS₂.

We study the ground-state and finite-density optical response of molybdenum disulfide by solving the semiconductor Bloch equations, using ab initio band structures and Coulomb interaction matrix elements. Spectra for excited carrier densities up to 10(13) cm(-2) reveal a redshift of the excitonic ground-state absorption, whereas higher excitonic lines are found to disappear successively due to Coulomb-induced band gap shrinkage of more than 500 meV and binding-energy reduction. Strain-induced band variations lead to a redshift of the lowest exciton line by ∼110 meV/% and change the direct transition to indirect while maintaining the magnitude of the optical response.

Modeling anisotropic plasmon excitations in self-assembled fullerenes

The plasmon excitations in Coulomb-coupled spherical two-dimensional electron gases (S2DEGs) reveal an interesting dependence on the displacement vector between the centers of the spheres with respect to the axis of quantization for the angular momentum quantum number L. Specifically, plasmon modes for a bundle of three S2DEGs have been obtained within the random-phase approximation (RPA). The inter-sphere Coulomb interaction matrix elements and their symmetry properties were also investigated in detail. The case of a bundle gives an adequate picture of the way in which the Coulomb interaction depends on the orbital angular momentum quantum number L and its projection M. We concluded that the interaction between the S2DEGs aligned at an angle of 45o with the axis of quantization is negligible compared to the interaction along and perpendicular to the quantization axis, which are themselves unequal to each other. Consequently, the plasmon excitation frequencies reveal an interesting orientational anisotropic coupling to an external electromagnetic field probing the charge density oscillations. Our result on the spatial correlation may be experimentally observable. In this connection, there have already been some experimental reports pointing to a similar effect in nanoparticles

Method of calculating self-capacitance in planar structural elements of a spacecraft

The paper presents a methodology for calculating distribution of potentials based on the triangulation method of plane approximation, surface method and analytical determination of the matrix elements of Coulomb interaction. The results obtained demonstrate good agreement with the experiments on the electrization process in small spacecraft AIST.

Plasmons in a free-standing nanorod with a two-dimensional parabolic quantum well caused by surface states

The collective charge density excitations in a free-standing nanorod with a two-dimensional parabolic quantum well are investigated within the framework of Bohm—Pine's random-phase approximation in the two-subband model. The new simplified analytical expressions of the Coulomb interaction matrix elements and dielectric functions are derived and numerically discussed. In addition, the electron density and temperature dependences of dispersion features are also investigated. We find that in the two-dimensional parabolic quantum well, the intrasubband upper branch is coupled with the intersubband mode, which is quite different from other quasi-one-dimensional systems like a cylindrical quantum wire with an infinite rectangular potential. In addition, we also find that higher temperature results in the intersubband mode (with an energy of 12 meV (∼ 3 THz)) becoming totally damped, which agrees well with the experimental results of Raman scattering in the literature. These interesting properties may provide useful references to the design of free-standing nanorod based devices.

Theory of fine structure of correlated exciton states in self-assembled semiconductor quantum dots in a magnetic field

A theory of the fine structure of correlated exciton states in self-assembled parabolic semiconductor quantum dots in a magnetic field perpendicular to the quantum dot plane is presented. The correlated exciton wave function is expanded in configurations consisting of products of electron and heavy-hole 2D harmonic oscillator states (HO) in a magnetic field and the electron spin ${S}_{z}=\ifmmode\pm\else\textpm\fi{}1/2$ and a heavy-hole spin ${\ensuremath{\tau}}_{z}=\ifmmode\pm\else\textpm\fi{}3/2$ states. Analytical expressions for the short- and long-range electron-hole exchange Coulomb interaction matrix elements are derived in the HO and spin basis for arbitrary magnetic field. This allows the incorporation of short- and long-range electron-hole exchange, direct electron-hole interaction, and quantum dot anisotropy in the exact diagonalization of the exciton Hamiltonian. The fine structure of ground and excited correlated exciton states as a function of a number of confined shells, quantum dot anisotropy, and magnetic field is obtained using exact diagonalization of the many-body Hamiltonian. The effects of correlations are shown to significantly affect the energy splitting of the two bright exciton states.

Fine structure of exciton levels in carbon nanotubes: A semianalytical approach

We propose a new approach toward excitons in carbon nanotubes whereby the matrix elements of the electron-hole Coulomb interaction are expanded into a series over the nanotube's one-dimensional reciprocal lattice vectors. We show that only a few terms of this expansion give a nonvanishing contribution to the Coulomb matrix elements. The proposed approach allows one to single out Fourier components of the Coulomb potential responsible for the intervalley coupling and formation of the exciton fine structure for each particular nanotube chirality.

Effective Coulomb interactions in solids under pressure

Correlated materials are extremely sensitive to external stimuli, such as temperature or pressure. Describing the electronic properties of such systems often requires applying many-body techniques to effective low energy problems in the spirit of the Hubbard model or extensions thereof. While the effect of pressure on structures and bands has been investigated extensively within density-functional-based methods, the pressure dependence of electron-electron interactions has so far received little attention. As a step toward ab initio pressure studies for realistic systems within a setup of maximally localized Wannier functions and the constrained random-phase approximation, we examine in this paper the paradigmatic pressure dependence of Coulomb interactions. While compression commonly causes the ``extension'' of Wannier functions, and thus transfer elements, to grow, we find the---seemingly counterintuitive---tendency that the bare Coulomb interaction increases under compression as well. We reconcile these behaviors by appealing to a semianalytical tight-binding model. We moreover argue that, for this model, the requirement of maximal Wannier localization is equivalent to maximizing the Coulomb interaction matrix elements. We then apply the above first-principles techniques to fcc hydrogen under pressure. While we find our comprehension of the bare Coulomb interaction confirmed, the induced changes in screening strengths lead to an effective one-band model with a Hubbard interaction that is nonmonotonous under pressure.

Bounds to coulomb interaction integrals

Inequalities are derived for matrix elements of the two-electron Coulomb interaction from Holder's and Sobolev's inequalities. They offer upper and lower bounds. Lower bounds from Dirichlet's principle can be provided with error limits.

Separable representations of the Coulomb interaction

AbstractError limits are developed for some separable approximations to the representation of the Coulomb interaction matrix elements. They are directed towards the application of Slater atomic orbital basis sets in large‐scale molecular calculations. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.

Homogeneous broadening in quantum dots due to Auger scattering with wetting layer carriers

The homogeneous broadening in semiconductor quantum dot (QD) lasers and optical amplifiers is studied theoretically. Based on a model for the electronic states of the coupled QD--wetting layer (WL) system, Coulomb interaction matrix elements are calculated, including both screening and exchange interaction. The homogeneous broadening due to various Auger processes, involving scattering of carriers between WL states and confined QD states, is calculated. The effects of the orthogonalization of WL states, QD confinement, QD density, and carrier density on the homogeneous broadening are studied systematically. We found that such WL-assisted Auger scattering is very efficient with subpicosecond dephasing times, and it is the dominant channel for the homogeneous broadening at high carrier density. Good agreement is achieved when comparing our theoretical results and recent experimental data.

Coherent rotations of a single spin-based qubit in a single quantum dot at fixed Zeeman energy

Coherent rotations of single spin-based qubits may be accomplished electrically at fixed Zeeman energy with a qubit defined solely within a single electrostatically-defined quantum dot; the $g$-factor and the external magnetic field are kept constant. All that is required to be varied are the voltages on metallic gates which effectively change the shape of the elliptic quantum dot. The pseudospin-1/2 qubit is constructed from the two-dimensional $S=1/2$, $S_z=-1/2$ subspace of three interacting electrons in a two-dimensional potential well. Rotations are created by altering the direction of the pseudomagnetic field through changes in the shape of the confinement potential. By deriving an exact analytic solution to the long-range Coulomb interaction matrix elements, we calculate explicitly the range of magnitudes and directions the pseudomagnetic field can take. Numerical estimates are given for {GaAs}.

ABSORPTION SPECTRUM OF AN EXCITON IN AN InxGa1 - xAs/GaAs QUANTUM DOT IN THE PRESENCE OF A MAGNETIC FIELD

We calculate the absorption spectrum of a single exciton, confined in an In x Ga 1 - x As/GaAs disk-shaped quantum dot, as a function of the strength of an applied magnetic field. Exciton eigenstates are obtained by numerical diagonalization of the Hamiltonian, within the effective mass approximation. A correction in the expression of the Coulomb interaction matrix elements is verified.