Two-dimensional Brillouin Zone Research Articles

Numerical integration of exchange energy in the two-dimensional Brillouin zone

A method is described for performing accurate numerical integration of the electronicexchange energy over polygonal regions in the two-dimensional Brillouin zone. It isillustrated by application to Bloch states constructed from Gaussian-type orbitals.

Intensity of the resonance Raman excitation spectra of single-wall carbon nanotubes

The electron-phonon matrix elements are calculated for the radial breathing mode (RBM) and the $G$-band $A$ symmetry mode of single-wall carbon nanotubes. The RBM intensity decreases with increasing nanotube diameter and chiral angle. The RBM intensity at van Hove singular $k$ points is larger outside the two-dimensional Brillouin zone around the $K$ point than inside the Brillouin zone. For the $G$ band $A$ symmetry mode, the matrix element shows that all semiconducting nanotubes have nonzero LO mode intensity, and the LO mode generally has a larger intensity than the TO mode, while the ratio of the intensity of the LO mode to that of the TO mode decreases with increasing chiral angle. In particular, zigzag nanotubes have zero intensity for the TO mode, and armchair nanotubes have zero intensity for the LO mode. Using the matrix elements thus obtained, the resonance Raman excitation profiles are calculated for nanotube samples under different broadening factor $\ensuremath{\gamma}$ regimes. For semiconducting nanotubes, the excitation profiles for the RBM are consistent with experiments. For metallic nanotubes, a quantum interference effect in the Raman intensity is found for both the RBM and LO modes. For the RBM and LO modes, different kinds of excitation profiles are discussed for nanotube samples in the large and small $\ensuremath{\gamma}$ regimes by considering the electron-phonon matrix element and the trigonal warping effect. For nanotube samples in the large $\ensuremath{\gamma}$ regime, a shift in the energy of the peak in the RBM intensity relative to the corresponding peak in the joint density of states is found.

Ultrathin antiferromagnetic films on a ferromagnetic substrate: a first-principles study ofMn on Fe(001)

The electronic and magnetic structures of Mn films with thicknesses of up to 12 monolayerson Fe(001) are investigated by means of first-principles calculations, with a focus on thethicker films. The Mn films show an in-layer antiferromagnetic structure and coupleantiferromagnetically to the Fe substrate. The magnetic moments at the Mn surface and atthe Mn–Fe interface are significantly enhanced, as compared to the respective bulk values.Several surface states are found at the centre of the two-dimensional Brillouin zone, some ofwhich might explain recent experimental scanning tunnelling spectroscopy results.

First-principles calculation of the electronic structure and EELS spectra at the graphene/Ni(111) interface

A spin-polarized first-principles calculation of the atomic and electronic structure of the graphene/Ni(111) interface is presented. Different structural models have been considered, which differ in the positions of the carbon atoms with respect to the nickel topmost layer. The most probable structure, which has the lowest energy, has been determined. The distance between the floating carbon layer and the nickel surface is found smaller than the distance between graphene sheets in bulk graphite, in accordance with experimental measurements. The electronic structure of the graphene layer is strongly modified by interaction with the substrate and the magnetic moment of the surface nickel atoms is lowered in the presence of the graphene layer. Several interface states have been identified in different parts of the interface two-dimensional Brillouin zone. Their influence on the electron energy loss spectra has been evaluated.

Specific Features of Low-Energy Electron Scattering by Thin Films of Cubic Crystals

The specific features of the electron states in thin films of cubic crystals are considered in the energy range above the vacuum zero of the crystal potential. It is demonstrated that bands of bound states embedded in the energy continuum can exist along particular directions of the two-dimensional Brillouin zone. These bands can substantially affect the intensity of low-energy electron scattering.

Optical absorption of graphite and single-wall carbon nanotubes

A review of the electronic dipole transitions in graphite and single-wall carbon nanotubes is presented. Because of its singular electronic structure, the optical absorption matrix element as a function of wave vector has a node in the two-dimensional Brillouin zone of graphite, which depends linearly on the optical polarization direction. In the case of the single-wall carbon nanotubes, the dipole selection rule and the van Hove singularity in the joint density of states will give a characteristic behavior, which is observed by luminescence and resonance Raman spectroscopy.

Evidence from surface phonons for the (2 × 1) reconstruction of the (101̅4) surface of calcite from computer simulation

A new force field for modeling calcium carbonate has been derived that corrects deficiencies of previous models. The model correctly reproduces the structure of the gas phase species, as predicted from ab initio calculations, as well as the bulk structure and properties of calcite and aragonite. With this new model, a (2 × 1) reconstruction is predicted to occur for the dominant (101̄4) surface of calcite, involving rotation of half of the surface carbonate anions. This reconstruction matches the results of low energy electron diffraction measured in vacuo and provides the first independent verification of this observation, as well as yielding the atomic detail of the nature of the reconstruction. While there is only a small exothermic energy associated with the formation of the supercell, the presence of an imaginary phonon mode at (½0) in the two-dimensional Brillouin zone for a single surface cell verifies the existence of the reconstruction.

Inhomogeneous optical absorption around theKpoint in graphite and carbon nanotubes

The optical absorption spectra of \ensuremath{\pi} electrons are calculated for graphite and carbon nanotubes. Particular attention is paid to the processes contributing to the optical absorption as a function of the electron wave vector k and light polarization direction. The optical absorption amplitude around the K point in the Brillouin zone has a node in the two-dimensional Brillouin zone of graphite. The formula for the absorption scattering matrix around the K point is given analytically by expanding the matrix element into a Taylor series. The chirality dependence of the absorption matrix element of a single-wall carbon nanotube is presented.

Electronic configuration of the c(2 × 2)MnCutwo-dimensional alloy in layered structures supported on Cu(100)

The c(2 × 2)MnCusurface alloy on Cu(100) can be considered as a purely two-dimensional magneticsystem where the Mn atoms exhibit a large corrugation closely related to their highspin moment. In this paper we investigate the influence of the atomic environmenton the electronic and magnetic properties of the two-dimensional alloyed layer,extending our study to the less known multilayered system made of MnCutwo-dimensional alloy layers embedded in a Cu crystal. The analysis is based onangle-resolved photoelectron spectroscopy measurements and calculations usingthe Green function matching method, which allows us to treat exactly theprojection of the three-dimensional lattice on the c(2 × 2)plane. A complete study of the valence band is performed along the two-dimensionalBrillouin zone in a wide energy range. We show that the presence of Mn results inan important redistribution of the spin-polarized electronic states of theneighbouring Cu atoms. This redistribution is not accompanied by a netcharge transfer between different atoms, and also the spin moment of Curemains small. Most of the new features induced by Mn in the surfacealloy are also present in the multilayered system, evidencing that theyare specific to the two-dimensional alloyed layer and not surface effects.

Improved triangle method for two-dimensional Brillouin zone integrations to determine physical properties

An improvement of the linear triangle method for two-dimensional Brillouin zone integrations is presented. A simple correction formula for this improvement is given and applied to several systems --- a three layer Fe(001) slab, a three layer slab of Si(001), and three layers of ${\mathrm{MgB}}_{2}(001)$---to investigate its validity and efficiency. This ``improved triangle method'' is seen to give better convergence behavior of the total energy, atomic force, and magnetic moment with respect to the number of k-points than do the normal triangle method and the special k-point method of Monkhorst and Pack.

Phase analysis of quantum well states in metallic multilayers: Periodicity and thickness dependence of the oscillatory magnetic coupling

A phase accumulation approach has been used to model Cu metallic quantum well (MQW) states in the fcc $M/\mathrm{C}\mathrm{u}/\mathrm{f}\mathrm{c}\mathrm{c}$ $M(100)$ systems, where $M=\mathrm{Fe},$ Co, or Ni. The electronic states at ${k}_{\ensuremath{\parallel}}=0.00{\AA{}}^{\ensuremath{-}1}$ and at ${k}_{\ensuremath{\parallel}}=0.98{\AA{}}^{\ensuremath{-}1}$ along the $\overline{\ensuremath{\Gamma}X}$ direction\char22{}the locations of the belly and the neck of the Cu Fermi surface, respectively\char22{}are considered. MQW states crossing the Fermi energy ${(E}_{F})$ at these points in the two-dimensional Brillouin zone are associated with the long- and short-period oscillatory magnetic coupling observed in these systems. The model predicts that states cross ${E}_{F}$ at the belly with a period of 5.7 Cu ML and a neck with a period of 2.6 Cu ML, in excellent agreement with the observed crossings in photoemission and inverse photoemission, as well as the oscillation periods seen in magnetic measurements. The fact that the period with which MQW states cross the Cu Fermi energy is solely a property of the spacer layer follows naturally from application of the model. Furthermore, the ${E}_{F}$ crossing of a given state near the neck is found to occur at successively greater thicknesses when the ferromagnetic (FM) layer is changed from Fe to Co to Ni. This systematic shift is consistent with what is seen in magnetic measurements and can be directly related to the energy of projected minority spin band gaps in the FM layer. The model gives an intuitive picture of how the minority spin band structure of the FM layer affects the electronic states of the nonmagnetic layer, showing why the thickness shift of a Fermi energy crossing depends on the location of projected band gaps of the FM layer.

Surface and interface states in theCu/V(110)structure

We present ab initio calculations of the electronic structure of copper multilayers on V(110) surface. The calculations were based on density-functional theory and the real-space linear muffin-tin orbital method in the atomic-sphere approximation was used. Surface and interface states were investigated using the Green-function based transfer-matrix method. The results shown a variety of electronic states along the main symmetry lines of the two-dimensional Brillouin zone, and the presence of surface and interface states was verified. Also, the existence of a binding state about 1.8 eV below the Fermi level is in good agreement with recent photoemission experiments, such state can be interpreted as a quantum-well state.

Influence of the substrate electronic structure on metallic quantum well state dispersions in ultrathin metal films

We have studied ultrathin Cu films grown on three related ferromagnetic (FM) layers to clarify the role of the substrate in determining the two-dimensional dispersion of metallic quantum well (MQW) states in the overlayer. The dispersions with parallel momentum of Cu MQW states above the Fermi level E F were measured in the Cu/fccFe/Cu(1 0 0) and Cu/fccCo/Cu(1 0 0) systems using inverse photoemission, and below E F in the Cu/fccNi/Cu(1 0 0) system using angle resolved photoemission spectroscopy. This study focussed on the ΓX direction of the two-dimensional Brillouin zone. For states away from the projected band gaps of the FM layer, the identity of the FM layer (i.e., Fe or Co) has little influence on Cu MQW states and their dispersions closely resemble those of a free standing Cu film. In contrast, all three systems exhibit nondispersing MQW states when the Cu bands overlap the minority spin projected band gaps of the FM metal. A phase accumulation approach gives a very simple explanation of this behavior, showing that the flat dispersion occurs because the phase shift upon reflection from the FM layer has a strong energy dependence in the projected gap.

The Role of Next-Nearest Neighbours for the Existence Conditions of Subsurface Spin Waves in Magnetic Films

Spin-wave energies and respective band structures throughout the two-dimensional Brillouin zone are investigated in magnetic (cubic) thin films for the surface orientations sc(110), bcc(110) and fcc(110). We apply the Heisenberg localized spin model assuming exchange nearest (NN) and next-nearest (NNN) neighbour interactions and elucidate the role of the geometrical disposition of the NN and NNN neighbourhoods for the emergence of surface (SSW) and subsurface spin waves (SSSW). The necessary condition for the emergence of SSSW is found to require the NNN bonds to be cut at the surface obliquely thereto.

The Role of Next-Nearest Neighbours for the Existence Conditions of Subsurface Spin Waves in Magnetic Films

Spin-wave energies and respective band structures throughout the two-dimensional Brillouin zone are investigated in magnetic (cubic) thin films for the surface orientations sc(110), bcc(110) and fcc(110). We apply the Heisenberg localized spin model assuming exchange nearest (NN) and next-nearest (NNN) neighbour interactions and elucidate the role of the geometrical disposition of the NN and NNN neighbourhoods for the emergence of surface (SSW) and subsurface spin waves (SSSW). The necessary condition for the emergence of SSSW is found to require the NNN bonds to be cut at the surface obliquely thereto.

Photoemission study of the bulk and surface electronic structure of Bi(111)

Angle-resolved energy distributions for photoelectrons emitted from the Bi(111) single crystal face are presented at 16.85 eV photon energy. The variation of peak-energy positions as a function of the electron emission angle shows four main dispersing features located between the Fermi level and 5 eV. The bulk electronic structure is found in qualitative agreement with the pseudopotential calculation performed by Golin along the TMU and TQW symmetry lines of the Brillouin zone. The experimental features located at around 0.3 and 3 eV below the Fermi level are sampled in the two-dimensional Brillouin zone along the and symmetry directions (dispersion about 1 eV) and are assigned to electronic surface states.

First principles calculation of the interface state in fcc-Fe/Cu(111)

The interfacial properties of fcc-Fe/Cu(111) superlattices are investigated by the first principles all-electron linearized augmented plane wave method in the local spin-density functional approximation. For the paramagnetic phase an interface state located at 2.61 eV below the Fermi level at the K ̄ point of the two-dimensional Brillouin zone is obtained and it develops extending along the K ̄ − Γ ̄ direction. This is consistent with the experimental observation.

Far-infrared ellipsometric study of the spectral gap in thec-axis conductivity ofY1−xCaxBa2Cu3O7−δcrystals

The temperature (T) and doping dependence of the spectral gap in the far-infrared c-axis conductivity of ${\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ has been studied by ellipsometry. For underdoped crystals a T-independent spectral gap is observed which persists almost unchanged into the normal state. The normal-state gap disappears at optimum doping whereas a conventional T dependence of the spectral gap is found for overdoped samples. The analogy to the results of angle-resolved photoemission suggests that the c-axis conductivity may be dominated by the carriers at the X point of the two-dimensional Brillouin zone.

Magnetic interlayer coupling and interaction between interface states in a quantum-well system

A first-principles Green's function technique has been used to investigate the magnetic interlayer coupling and the electronic structure responsible for the magnetic interaction in a bcc system. We discuss the close relation between the interlayer coupling, the bulk Fermi surface of the spacer material, and the dispersion of the quantum-well states in the two-dimensional Brillouin zone. As a function of spacer thickness, we find an oscillatory long-range energy splitting of Fe/Cu interface states which is shown to be an example of a long-range oscillatory interface interaction transmitted by the spin-polarized quantum-well states of the Cu spacer. These interacting interface states may provide a signature of the magnetic coupling in multilayers.

Prediction of bias-voltage-dependent corrugation reversal for STM images of bcc (110) surfaces: W(110), Ta(110), and Fe(110)

We predict a bias-voltage dependent corrugation reversal for scanning tunneling microscopy (STM) images with atomic resolution of bcc-(110) transition metal surfaces: Atoms which appear usually on STM images of metal surfaces as protrusions, may appear on these images anticorrugated, e.g., as hollow sites and vice versa hollow sites may appear as atoms. This makes the absolute determination of atom sites by STM unreliable. We investigate the image-reversal in detail for the W(110) surface and explain its origin on the basis of the electronic structure. We found, the image is determined by a competition between surface resonance states with ${d}_{\mathrm{xz}}$ and ${d}_{{z}^{2}}$ character contributing to a direct image of atomic sites and surface-state bands around the $\overline{S}$ point of the two-dimensional Brillouin zone with $\mathrm{pd}$ bonding character and bonding charge between the surface atoms. The surface states contribute to anticorrugated images. Finally the image depends on the bias voltage. For W(110) and positive bias voltages the surface resonances dominate over the surface states leading to a direct image of atomic sites. For negative voltages below a critical value of $\ensuremath{-}0.4$ V calculations show a reversed image, i.e., anticorrugation. The critical bias voltage depends slightly on the tip-sample separation. For bias voltages around the critical value corrugating and anticorrugating contributions to the STM image compensate each other, the corrugation amplitudes become extremely small, the atomic resolution disappears, which is consistent with the experimental difficulties in achieving atomic resolution on W(110) for negative voltages and stripelike images are predicted. For positive bias voltages we found a good agreement between the theoretical results and our measured STM images. The competition between surface resonances and surface states is a quite general mechanism and anticorrugation is expected to occur on (110) surfaces of other bcc transition-metals [i.e., Nb(110), Mo(110), Ta(110)]. We demonstrated this explicitly for Ta(110), anticorrugation occurs practically over the entire bias-voltage range available by a STM and an image reversal from an anticorrugated to a corrugated image is predicted for 1.3 V. For magnetic surfaces the image reversal may occur twice, once for majority and once for minority states. For Fe(110) we show that the minority spin channel controls the STM image. We predict a direct image for majority states and an anticorrugated image for minority states below a bias voltage of 0.7 V, and we predict only one image reversal at about 0.4 V for an ordinary non-spin-polarized STM. Employing the full-potential linearized augmented plane wave method in film geometry, the electronic structure is determined by first principles calculations within the framework of the density functional theory in the local (spin) density approximation. The STM analysis is carried out within the s-orbital tip-model of Tersoff and Hamann. An efficient analysis of the corrugation amplitude in terms of two-dimensional star coefficients of the vacuum density of states is presented. The tip-sample distance dependent ${\mathbf{k}}_{\ensuremath{\parallel}}$-point selection to the tunneling current is analyzed. The enhancement of the corrugation amplitude due to ${p}_{z}$- and ${d}_{{z}^{2}}$-type tip-orbitals are determined. We show that the enhancement factors calculated are close to the analytical factors given by Chen.