Madelung Energy Research Articles

Classic ionic crystals and quantal Wigner electron solids: Role of electron correlation

AbstractMotivated by the pioneering studies of Löwdin on ionic crystals like NaCl and CsCl, and on electron correlation, recent work on the polarizability of the anion in such materials, and its relation to the free‐space polarizability, will first be reported. The essential role of electron correlation in treating negative ions will be emphasized. The Madelung energy, crucial for the stability of such classic ionic crystals, is then pointed out, within the framework of the jellium model of strongly interacting electrons, to lead to quantal Wigner electron crystals. Work will be summarized first on such an electron solid in three dimensions and in particular its melting curve will be discussed. Attention will then be focused on the freezing of the 2‐D liquid in a GaAs–AlGaAs heterojunction in a transverse magnetic field. Finally, it is stressed that electron–electron repulsion plus localization due to the applied magnetic field order the low‐density electrons into such a magnetically induced Wigner electron solid. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Lattice energies and structural distortions in Pb(ZrxTi1-x)O3 solid solutions

The changes in Madelung and non-electrostatic energies ofPb(ZrxTi1-x)O3 (PZT) solid solutions in the high-symmetry(Pm3m)phase have been calculated using the heats of formationfrom the oxides. The non-electrostatic contribution (ΔEN) decreaseswith decreasing xand becomes negative for compositions x ≤ 0.35,corresponding to perovskite tolerance factors t ≥ 1.Correlation of the strong increase in tetragonal distortion inthe ferroelectric phase with more exothermic values of ΔENsuggests a softening of short-range repulsions for Ti-rich compositions. The influenceof complex solid solution behaviour on the character of Pm3m ↔ P 4mmtransition is investigated by high-temperature specific heat and unit-cellparameter measurements. The implications of the thermochemical results withrespect to mean-field theoretical descriptions of lattice instabilities and phaseboundaries in the PZT system are briefly discussed.

Monte Carlo simulation of phase separation inKxC60

Kinetic Monte-Carlo methods have been used to simulate phase transformations in ${\mathrm{K}}_{x}{\mathrm{C}}_{60}$ layers. It has been found that the phase composition is controlled by a balance between the Madelung energy and the energy of the electronic-shells interaction. A density-functional calibrated tight-binding method is used to calculate electronic effects. Our calculations point out a phase separation in the fulleride layer into K-rich $({\mathrm{K}}_{3}{\mathrm{C}}_{60})$ and K-depleted areas at room temperature. The two-phase system is formed as a fine mixture which is shown to be stable against aggregation. The average diameter of ${\mathrm{K}}_{3}{\mathrm{C}}_{60}$ particles is about 20 to 40 nm and depends on the K content. This phase separation explains nanostructuring effects observed in electrochemically doped potassium fulleride layers. The size of the simulated ${\mathrm{K}}_{3}{\mathrm{C}}_{60}$ particles correlates with the size of experimentally observed clusters. Since ${\mathrm{K}}_{3}{\mathrm{C}}_{60}$ particles are metallic, the system considered can serve as an array of nanoelectrodes.

Screened Coulomb interactions in metallic alloys. I. Universal screening in the atomic-sphere approximation

We have used the locally self-consistent Green's function (LSGF) method in supercell calculations to establish the distribution of the net charges assigned to the atomic spheres of the alloy components in metallic alloys with different compositions and degrees of order. This allows us to determine the Madelung potential energy of a random alloy in the single-site mean field approximation which makes the conventional single-site density-functional- theory coherent potential approximation (SS-DFT-CPA) method practically identical to the supercell LSGF method with a single-site local interaction zone that yields an exact solution of the DFT problem. We demonstrate that the basic mechanism which governs the charge distribution is the screening of the net charges of the alloy components that makes the direct Coulomb interactions short-ranged. In the atomic sphere approximation, this screening appears to be almost independent of the alloy composition, lattice spacing, and crystal structure. A formalism which allows a consistent treatment of the screened Coulomb interactions within the single-site mean-filed approximation is outlined. We also derive the contribution of the screened Coulomb interactions to the S2 formalism and the generalized perturbation method.

Critical points of metals of three main groups and selected transition metals

The liquid–gas critical points of metals of the first three main groups and those of selected transition metals have been estimated by scaling with the ionization potential and valence. The critical density is shown to be attached to the percolation threshold of the classically accessible spheres. Corrections to the classically accessible radius are made allowing for deviation from the Coulomb potential. The scaling laws follow from a van der Waals-like equation of states of overlapping virtual atoms with partially free valence electrons. The attraction of virtual atoms is represented by a renormalized Madelung energy with corrections due to finite numbers of neighboring atoms and screening electrons in atomic cells. With given corrections, a set of the critical parameters of metals obtained by scaling becomes more reliable than any semi-empirical estimates which neglect the electron structure.

Coulombic Insight into Charge Ordering in Yb4As3and Sm3Te4

Charge ordering in Yb 4 As 3 and Sm 3 Te 4 , complementary f -electron compounds with the same crystalline structure, is investigated on the basis of a Coulomb lattice gas model; the Coulomb interactions between ions are fully taken into account within a classical treatment. The Madelung energy of the system at finite temperatures is calculated with a Monte Carlo (MC) method simulating the Boltzmann distribution of multivalent cations on equivalent lattice sites. The unit cell for the MC simulations is extended to contain 32 molecular formulas in it. The model is successful in distinguishing the two mixed-valence compounds with different characteristics of charge ordering. Yb 4 As 3 shows a phase transition (Verwey transition) due to charge ordering while Sm 3 Te 4 continuously freezes into a glassy state on cooling.

Interactions between gallium and nitrogen dopants in ZnO films grown by radical-source molecular-beam epitaxy

It has been recently predicted that the co-doping of an acceptor (nitrogen) and a donor (aluminum, gallium, indium) in a 2:1 ratio will dope ZnO p-type due to a reduction in the Madelung energy making the nitrogen acceptor energy level more shallow. We have been growing gallium and nitrogen co-doped ZnO films by radical-source molecular-beam epitaxy by use of oxygen and nitrogen radicals supplied via rf radical source cells. Diode-like current–voltage characteristics and donor acceptor pair-like photoluminescence emission were observed for a Ga and N doped ZnO film grown on an undoped ZnO buffer layer. However, Hall measurements revealed that the conductivity was n-type. Formation of a non-ZnO phase in the sample was confirmed by secondary ion mass spectroscopy and x-ray diffraction measurements. Zn and Zn+O secondary ion intensities fell sharply by two orders of magnitude in going from the undoped ZnO layer to the highly co-doped ZnO. X-ray diffraction measurements indicated the formation of ZnGa2O4.

Growth of N-doped and Ga+N-codoped ZnO films by radical source molecular beam epitaxy

It has been theoretically predicted that the codoping of an acceptor (nitrogen) and a donor (aluminium, gallium, indium) in a 2 : 1 ratio will enable the fabrication of low resistivity p-type ZnO due to reduction of the Madelung energy making the nitrogen acceptor energy level shallower. We have been growing N-doped and Ga+N codoped ZnO films by radical-source molecular beam epitaxy. Nitrogen and oxygen radicals were separately supplied via radio-frequency radical cells. Nitrogen did not incorporate in ZnO films at the high substrate temperature of 600°C. Lower substrate temperatures were necessary for nitrogen incorporation. From the fact that the nitrogen concentration also depended on the Zn fluxes, it is concluded that Zn-rich conditions are indispensable for nitrogen doping. The presence of Ga was observed to enhance nitrogen incorporation, one of the predicted effects of codoping. However, high Ga concentrations were found to lead to formation of the additional phase of ZnGa 2O 4 in ZnO films.

Charge ordering in Eu4As3

The one-dimensional magnetic chain model clarified the unusual properties of Yb4As3. However, there are still unexplained problems. The Madelung energy calculation pointed out that the actual arrangement of Yb3+ and Yb2+ in the charge-ordered state, which forms the one-dimensional Yb3+ chains, is not a lowest energy state. To examine the role of the orbital moment and the exchange interaction, which are candidates dominating the charge ordering, we prepared a single crystal of Eu4As3, in which only spin moment is dominant. The measurements of X-ray diffraction, transport properties and susceptibility revealed that the charge ordering in Eu4As3 is very similar to that in Yb4As3, however, an appearance of ferromagnetism in Eu4As3 is a striking contrast to Yb4As3. These results may indicate that the charge-ordering formation is subject to neither orbital moment nor exchange interaction.

Crucial effects of intramolecular charge distribution on the neutral-ionic transition of tetrathiafulvalene–p-chloranil

We calculated the Madelung energies of both the ground state and excited states in tetrathiafulvalene - p-chloranil (TTF-CA) by taking into account intramolecular charge distribution. The distribution is found to be significant in the neutral-ionic (NI) transition. In the ionic phase, the Madelung energy depends more strongly on the intermolecular distance perpendicular to the $\pi$-stacking chains than on that along the chains. This indicates that simple single-chain models neglecting interchain electrostatic coupling are not adequate. The gain of the Madelung energy due to dimerization is concluded to be small compared with the other structural changes. We also calculated the formation energy of excited state domains, which appear in the initial process of the phase transition. A one-dimensional excited domain has the smallest energy among the possible domains with the same number of molecules, and the energy per molecule is considerably reduced in increasing the domain size. It is consistent with the experimental suggestion that a large number of excitations were generated by only one photon.

Chemical Stability of Binary Coulomb Compounds

Chemical stability of binary Coulomb compounds is studied through extensive calculation of their Madelung energies. A remarkable finding is that the compounds with disordered ionic arrangements in a certain range of the composition are stable against chemical separation beyond the charge ratio around 6; clearly the entropic effect due to finite temperature promotes the mixing of ions of different species. For the ordered AB3-type compounds, there exists a crystalline structure which is more chemically bounded than that of Dyson (1971). Relative stability o the ordered and disordered compounds is also discussed.

Chemical Stability of Binary Coulomb Compounds

Chemical stability of binary Coulomb compounds is studied through extensive calculation of their Madelung energies. A remarkable finding is that the compounds with disordered ionic arrangements in a certain range of the composition are stable against chemical separation beyond the charge ratio around 6; clearly the entropic effect due to finite temperature promotes the mixing of ions of different species. For the ordered AB 3 -type compounds, there exists a crystalline structure which is more chemically bounded than that of Dyson (1971). Relative stability of the ordered and disordered compounds is also discussed.

Electronic States of C60 Ions vs Electron Interaction and Energetics of A3C60 Fullerides

The charge dependencies of energies E(Q of free correlated C60 cations and anions are calculated accounting for the full configuration interaction (CI) within the corresponding active spaces and the icosahedral C60 symmetry. For Hubbard and Coulomb-like electron-electron (e-e) potentials, opposite Q-sequences of ground-state anion energies E(Q) are found, predicted energies are consistent with experimental data only when the Coulomb long-range interactions are involved. Basing on the calculated E(Q) and the Madelung energy evaluation, the energetics of A3C60, crystals (A=Li, Na, K, Rb) in different charge states is considered with varied screenings of e-e potentials inside and outside fullerene cages. The screening areas are specified, in which a partial metal-fullerene electron transfer is more preferable than complete.

Electrostatically driven charge-ordering in magnetite below the Verwey temperature

This work describes the stability of different charge models for magnetite, Fe 3O 4, below the Verwey temperature (T V). Thirty-seven configurations allowed by the Anderson condition are analysed by means of Madelung calculations, including models proposed by Verwey, Mizoguchi and Zuo. A new approach based on summation over cubic cells is applied that ensured rapid convergence of lattice sums. The lowest Madelung energy has been found for the model in which coupled chains of Fe 2+ and Fe 3+ ions form structures parallel to the c-axis. These structures are similar to double helices having a period of twice the cubic lattice constant. Thus it has been shown, contrary to previous studies, that the Coulomb energy alone is sufficient to stabilize the charge order involving doubling of the c-axis.

On the Madelung energy of spherical Coulomb clusters

We perform large-scale molecular dynamics (MD) simulations of Coulomb clusters in a spherical trapping field. The fast multipole method have been employed to simulate clusters of up to 5×10 4 particles while previous investigations are limited to clusters of less than 5000 particles. It is found that the Madelung energy of bcc clusters with reconstructed surfaces is substantially smaller than the extrapolated values based on the previous results for smaller multilayered icosahedral clusters.

Madelung Energy and Charge Ordering inβ-Phase Vanadate Bronzes

β-phase vanadate bronzes β-A0.33V2O5 (A=Na, Li, Ag, Ca, Sr, etc.) have recently attracted much attention because of their conjectured metal-insulator (MI) phase transition associated with charge ordering (CO).2-5) The transition temperature TCO ranges from 90K for A=Ag to 180K for A=Li.1-3) Because three crystallographically inequivalent V sites are involved in the structure and drastic change in the valence state of V ions has been observed at the phase transition, 5) the question one has to ask first is: where are the electrons at T > TCO and at which sites do the electrons localize below TCO? The band structure calculations will provide useful information on this issue but they are unavailable at present. In this Short Note, we therefore adopt the ionic model and calculate the Madelung energy of the system for various electron configurations to gain some insight into the stable electron configurations. Below, we will show that the V1 position (zigzag chain sites of VO6 octahedra) is the lowest in energy for the electrons to be located, the V2 position (ladder sites of VO6 octahedra) is the next lowest, and the V3 position (zigzag chain sites of VO5 pyramids) is the highest in energy. Implication of the results will be discussed. The crystal structure of the β-phase compounds has a characteristic V2O5 framework consisting of three kinds of infinite chain along the crystallographic b-axis (see Fig. 1 of refs. 2 and 3). The V1 sites form a zigzag chain of edge-shared VO6 octahedra, and the V2 sites form a two-leg ladder of corner-shared VO6 octahedra. The V1 chain and V2 ladder share the edges of the VO6 octahedra. The V3 sites form a zigzag chain of edgeshared VO5 pyramids, which bridges the V1 chain and V2 ladder. The average valence of V ions β-A1/3V2O5 is V(5−1/6)+ when A=monovalent cations (i.e., there is one d-electron per six V-ions), and V(5−1/3)+ when A=divalent cations (i.e., is one d-electron per three V-ions). Quasi-one-dimensional (1D) metallic conduction along the b-axis at T > Tc and the MI plus CO phase transition at TCO have been confirmed in the single crystal of β-Na0.33V2O5. 2, 5) At low temperatures (T 50K), the antiferromagnetic (AF) long-range order occurs in A=Na and Ag, while in A=Ca and Sr, the

Physics and control of valence states in ZnO by codoping method

We investigate unipolarity in ZnO based on ab initio electronic band structure calculations. We find that p-type doping using Li, N or As species causes an increase in Madelung energy, n-type doping using B, Al, Ga, In or F species gives rise to a decrease in Madelung energy. In order to solve the unipolarity, to fabricate p-type ZnO, we propose a codoping method using acceptors and reactive donors simultaneously. A codopant pair including Ga-reactive donors and N-acceptors is eminently suitable for use in the codoping. For Li-acceptors, F species is a good candidate for reactive donors. For As-acceptors, Ga species is a preferable codopant.

The electron-phonon interaction renormalized by strong correlations: The way to HTS

Tunneling and optic spectroscopies give evidence for the strong electron-phonon interaction ( EPI) in HTS materials. Strong electronic correlations ( SEC), which renormalize the EPI in HTS oxides, together with the pronounced Madelung energy give rise to the forward scattering peak in the EPI (FSP - EPI) for small hole doping δ ⪡ 1. The FSP - EPI mechanism explains relevant properties of the normal and superconducting state in HTS oxides.

The Madelung Function of Elpasolites

Abstract Madelung Energy, Coulomb Interaction, Elpasolite For binary ionic compounds the Madelung energy can be calculated using the Madelung constant, the ionic charges and the next neighbour distance of the ions only. For ternary and higher systems the well-known standard formula cannot be applied, because the ionic charges and distances combine in different ways. In the present work we calculate the Madelung energy of the quaternary elpasolite structure by straightforward computer summation and propose a Madelung function which describes the Madelung energy in dependence on the lattice constant and the free structural parameter. This function has been applied to several chemical compounds of the elpasolite type to determine the individual Madelung energy.

Unipolarity of ZnO with a wide-band gap and its solution using codoping method

We have investigated the electronic structures of p- or n-type-doped ZnO based on ab initio electronic band structure calculations. We find unipolarity in ZnO; p-type doping using Li or N increases the Madelung energy while n-type doping using Al, Ga, In or F species decreases the Madelung energy. We find a very weak repulsive interaction between Li acceptors in Li-doped ZnO (ZnO : Li) with a remarkable increase in the Madelung energy, in contrast with the case of ZnO : N. For ZnO : (2Li, F), total energy calculations show that the formation of the complex with Li Zn–F O–Li Zn which occupy the nearest-neighbor sites is energetically favorable with a decrease in the Madelung energy, which produces low-resistivity p-type ZnO crystals.