Rayleigh Surface Phonons Research Articles

Electron-phonon interaction on the surface of a three-dimensional topological insulator

We analyze the role of a Rayleigh surface phonon mode in the electron-phonon interaction at the surface of a 3D topological insulator. A strong renormalization of the phonon dispersion, leading eventually to the disappearance of Rayleigh phonons, is ruled out in the ideal case of a continuum long-wavelength limit, which is only justified if the surface is clean and defect free. The absence of backscattering for Dirac electrons at the Fermi surface is partly responsible for the reduced influence of the electron-phonon interaction. A pole in the dielectric response function due to the Rayleigh phonon dispersion could drive the electron-electron interaction attractive at low frequencies. However, the average pairing interaction within the weak coupling approach is found to be too small to induce a surface superconducting instability.

Surface spin waves of fcc cobalt films on Cu(100): High-resolution spectra and comparison to theory

With the help of a recently developed electron energy-loss spectrometer we have studied the surface spin waves on an eight-monolayer cobalt film deposited on Cu(100) surfaces with unprecedented energy resolution. Standing waves of the bulk of the film are discovered in the region of small wave vectors ${q}_{\ensuremath{\parallel}}l0.35$ \AA{}${}^{\ensuremath{-}1}$. The dispersion of surface spin waves is isotropic even for large ${q}_{\ensuremath{\parallel}}$. Because of the high-energy resolution and the complete characterization of the electron optical properties of the spectrometer reliable data for the linewidth of the surface spin waves are obtained. As a byproduct the dispersion of the Rayleigh surface phonon was measured. Data are compared to theoretical spin wave spectra extracted from calculations of the transverse spin susceptibility based on an ab initio electronic structure that incorporates both the metallic substrate and the magnetic film. The calculation takes fully into account the itinerant nature of the electrons responsible for the magnetic moments. The agreement between theoretical and experimental spin wave energies and linewidths is remarkably good.

Electron–phonon interaction on an Al(001) surface

We report an ab initio study of the electron–phonon (e–ph) interaction and its contributionto the lifetime broadening of excited hole (electron) surface states on Al(001). Thecalculations based on density-functional theory were carried out using a linear responseapproach in the plane-wave pseudopotential representation. The obtained results show thatboth the electron–phonon coupling and the linewidth experience a weak variation with theenergy and momentum position of a hole (electron) surface state in the energy band. Ananalysis of different contributions to the e–ph coupling reveals that bulk phonon modesturn out to be more involved in the scattering processes of excited electrons and holes thansurface phonon modes. It is also shown that the role of the e–ph coupling in the broadeningof the Rayleigh surface phonon mode is insignificant compared to anharmoniceffects.

Brillouin light scattering from porous silicon films and multilayers

AbstractPorous silicon films and multilayers were studied using Brillouin spectroscopy. Acoustic wave velocities and elastic constants were determined for films with porosity ∼0.60 formed from p–, p+ and n+ crystalline silicon. The velocities and elastic constants depend on the pore/crystallite geometry and morphology. Porous silicon multilayers were fabricated from (100) p+ crystalline silicon. For a multilayer with 250 nm layer thickness and layer porosity sequence of 0.45/0.65/0.45/0.65/0.45, the Rayleigh surface phonon velocity was found to be 2850 m/s. The origin(s) of other peaks in the multilayer spectra are unknown but it is unlikely that they are surface modes because their shift(s) are independent of incident angle. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Anharmonicity on Al(100) and Al(111) surfaces

We present a molecular-dynamics analysis of the stable nonmelting (100) and (111) surfaces of Al. A many-body potential derived from first-principles calculations is used. The molecular-dynamics method includes anharmonic effects of all orders. We study static and dynamical properties of the surface. An expansion of the (111) surface and a contraction of the (100) surface results from the calculations. At low temperature, the vertical mean-square vibrational amplitude is larger than the in-plane component, while at higher temperature the in-plane component approaches the vertical one. Both components are at least twice as large as the bulk value. The interactions due to the surface decay very rapidly going into the crystal from the surface, as indicated by the analysis of the Debye-Waller factor. The evaluated linewidths for the Rayleigh surface phonon reproduce quite well the temperature dependence of the He-surface scattering data. The experimental behavior of the energy shifts, as a function of the scattering momentum transfer, presents a minimum inside the Brillouin zone, which is also found by our calculations. The surface energy shifts are about $30%$ larger than the bulk ones at the same temperature. The effect of surface anharmonicity is much larger for the static properties than for the dynamical properties.

Effects of isotropic overlayer on the focusing of surface phonons

Based on the geometrical acoustic approximation, we study theoretically the phonon focusing at the (100) face of silicon with a thin oxide overlayer. The oxide layer is assumed to be isotropic elastically and the focusing factors of both the Rayleigh-surface and pseudosurface phonons are calculated for several values of k (wave number) \ifmmode\times\else\texttimes\fi{} h (thickness of the oxide layer). As kh increases we find that the strong focusing region of Rayleigh-surface phonons in between two caustic directions becomes narrow and moves toward the [110] direction. Also the directions around the [110] axis, where the pseudosurface phonons are allowed to exist, are diminished and eventually their spatial intensity distribution disappears.

Experimental evidence of the transformation of the Rayleigh surface phonon in CaF2/GaAs(111) heterostructures of the accelerating type

The spectrum of surface acoustic phonons in films of CaF2 on GaAs(111) has been investigated by means of Brillouin light scattering. We report the first experimental observation of the disappearance of the Rayleigh phonon with increasing film thickness in these structures of the fast-on-slow or accelerating type. For propagation along <121> at the disappearance of the Rayleigh surface phonon, a quasi-resonance in the continuum, whose spectral position tends to that of the Rayleigh phonon of CaF2(111) for larger value of film thickness, is detected in the spectrum. Because of this behaviour we identify this quasi-resonance as the pre-Rayleigh mode of the film. When propagation is along (110) the role of the pre-Rayleigh mode of the film is played by the pseudosurface acoustic mode. The theoretical discussion is performed by comparing the Brillouin spectra with the surface projected phonon density of states for shear vertical phonons numerically calculated for CaF2/GaAs(111) heterostructures.

Comparison of semiempirical and ab initio pseudopotential calculations of the lattice dynamics of aluminum surfaces with He-atom-scattering time-of-flight spectra.

We report a detailed comparison of two theoretical methods for the study of the surface lattice dynamics of sp-bonded aluminum and compare the results with He-atom time-of-flight spectra for the (001), (110), and (111) surfaces of this metal. One of the theoretical methods is a semiempirical Born--von Karman scheme in which the interatomic force constants are related to central pairwise forces and also to ``angle-dependent'' interactions between triplets of atoms. The other method is an ab initio scheme in which the interatomic force constants are calculated from the electron-density response function and second-order pseudopotential perturbation theory. Both schemes require the inclusion of long-range central forces (up to ten nearest neighbors) in order to obtain converged results. In addition, the semiempirical approach must include three-body forces up to second-nearest neighbors to allow for the breakdown of the Cauchy relation, which in the ab initio method arises as a direct consequence of the volume dependence of the effective pair potential. The semiempirical approach reproduces accurately the surface phonon-dispersion curves calculated with the ab initio scheme. The leading surface force constants obtained in both methods have similar values, which provides justification for the physical significance of the semiempirical force constants. Both methods reproduce, with great accuracy, the experimental dispersion relations for the Rayleigh surface phonons, except for the case of the (110) surface. Assuming that the He atom couples only to the ion cores, the reflection coefficients calculated in both methods reproduce well the intensities of the weak resonances observed in the He time-of-flight spectra below the longitudinal edge. These structures can be explained in terms of Fano resonances.

Brillouin light scattering from the Rayleigh surface phonon of thermally annealed a-C:H films

Brillouin light scattering experiments are performed on amorphous diamond-like carbon films (a-C:H). Inelastic scattering from the Rayleigh surface phonon is found. Young's modulus, the shear modulus, bulk modulus, and microhardness of a-C:H films annealed at various temperatures are evaluated combining light scattering and indentation techniques.

Rayleigh mode in amorphous hydrogenated carbon films.

Brillouin-scattering experiments are performed on amorphous hydrogenated carbon (a-C:H) films grown by plasma deposition on silicon substrates at various bias voltages (-50 to -1400 V) and annealed after deposition at a bias of -400 V between 100 and 1100 \ifmmode^\circ\else\textdegree\fi{}C. Scattering from Rayleigh surface phonons is observed. The results show that the Reyleigh mode frequencies are between those of diamond and graphite. With increasing substrate bias voltage the Rayleigh mode frequency first increases to a maximum value at -200 V and then decreases monotonically. Upon annealing, the Rayleigh phonon frequency decreases drastically for temperatures above 500 \ifmmode^\circ\else\textdegree\fi{}C with minor changes below 400 \ifmmode^\circ\else\textdegree\fi{}C. These results correspond to the results of the ${\mathit{sp}}^{3}$-to-${\mathit{sp}}^{2}$ bonding ratio, indicating that the frequency variation of the Rayleigh phonon is due to phase transformations of a-C:H. The coupling between the relaxation mechanism and the phonon also confirms the phase transitions. Based on the hydrogen concentration, the ${\mathit{sp}}^{3}$-to-${\mathit{sp}}^{2}$ bonding ratio, and the phonon frequency, the correlated densities and elastic constants of annealed a-C:H films are calculated.

Surface lattice dynamics of Cu(111).

We present and analyze off-specular electron-energy-loss data which explore surface vibrations of Cu(111), along the line from \ensuremath{\Gamma}\ifmmode\bar\else\textasciimacron\fi{} to M\ifmmode\bar\else\textasciimacron\fi{} in the two-dimensional surface Brillouin zone. The dispersion relation of the Rayleigh surface phonon and the longitudinal resonance found in inelastic-He-scattering studies have been measured out to M\ifmmode\bar\else\textasciimacron\fi{}, thus extending the earlier He work. In addition, we obtain the frequency of a (high-frequency) surface phonon which exists at and near M\ifmmode\bar\else\textasciimacron\fi{}, in a gap in the projected bulk-phonon density of states. The data are analyzed within a simple model of surface lattice dynamics. We have also carried out extensive calculations of the energy variation of the various features observed in the electron-loss spectrum, using multiple-scattering theory. Very good agreement between theory and experiment is obtained within a model where the intraplanar force constant between surface atoms is softened by a modest amount, 15%. In our view, the results presented here raise questions about recent suggestions that a very large (70%) softening of the intraplanar surface force constant is present in the Cu(111) surface.

Lattice dynamics of rare-gas overlayers on smooth surfaces.

We discuss the lattice dynamics of rare-gas overlayers on metal surfaces, with attention to monolayers, bilayers, and trilayers on the Ag(111) surface. For this system, the adsorbate overlayers are incommensurate with the metal substrate. Our interest is in the role of the interaction between the motions of the atoms in the adlayers and those of substrate atoms. In certain regions of the two-dimensional Brillouin zone, the adlayer phonon dispersion curves (calculated with the substrate atoms held in fixed position) overlap the bulk phonon bands of the semi-infinite substrate. A consequence is that when excited, the adlayer phonon may decay by emitting phonons into the substrate. They thus become leaky modes. Also, the adlayer modes may hybridize with the Rayleigh surface phonon on the substrate, to produce anomalous dispersion near those wave vectors where crossover occurs. For Ar, Kr, and Xe overlayers on models of Ag(111), we explore these phenomena through calculations of the phonon spectral density functions probed in inelastic He-surface scattering experiments.

Observation of surface optical phonons in NaF(001) by inelastic He-atom scattering.

Previous time-of-flight experiments at 20-meV He beam energies, which provided the dispersion curve of Rayleigh surface phonons, have been extended to 90-meV energies on NaF(001) along $〈100〉$. Despite a large multiphonon background from the Rayleigh phonons it has been possible to detect small peaks due to optical surface phonons. The results provide the first direct experimental evidence in the dispersion curves of microscopic surface optical modes.

Anomalously long lifetimes of high-energy surface acoustic phonons

The lifetime of Rayleigh surface phonons as opposed to decay via anharmonic three-phonon processes is calculated in the regime $\ensuremath{\hbar}{\ensuremath{\omega}}_{R}\ensuremath{\gg}{k}_{B}T$. The surface phonon decays predominantly by combining with a bulk transverse phonon to create another bulk transverse phonon. The energymomentum conservation of the processes imposes the condition that those bulk phonons which may interact with the surface phonons should possess frequencies of the same order of ${\ensuremath{\omega}}_{R}$. However, such high-frequency phonons are rarely excited thermally under the condition considered here, leading to an anomalously long lifetime of the surface phonons in proportion to ${\ensuremath{\omega}}_{R}^{\ensuremath{-}4.5}{T}^{\ensuremath{-}0.5}\mathrm{exp}(\frac{A\ensuremath{\hbar}{\ensuremath{\omega}}_{R}}{{k}_{B}T})$, where $A$ is a constant of the order of 0.1. We also estimate numerically the magnitude of the lifetime.

On the Debye-Waller factor in atom-surface scattering

Abstract A theory for the Debye-Waller factor in atom-surface scattering is presented, to lowest order in the phonon contributions. Multiple-scattering effects as well as the cross-correlated surface atom displacements are included. The theory accounts for experimental data without the necessity of introducing the Armand effect, which is due to the finite size of the incident atom. The work presented here implies that the Kirchhoff approximation fails when the energy of the incident particle is in the energy range of the phonon spectrum. The results of the calculation are presented in the high-temperature limit, and it is observed that the Rayleigh surface phonons contribute three-quarters of the Debye–Waller factor, while the bulk phonons account for the rest. This result is interesting because the calculation of the former contribution is simpler than that, of the latter.

Measurement of Surface Phonon Dispersion Curves of Alkali Halide Single Crystals by Time of Flight Spectroscopy of He Atom Beams

The development of highly expanded He nozzle beams with velocity half widths less than 1% has made it possible to observe the annihilation and creation of single surface phonons in time of flight spectra of the scattered atoms. Measurements have been carried out for a wide range of incident angles at a beam energy of about 20 meV for the following crystals and directions: LiF ⟨100⟩, LiF ⟨110⟩, NaF ⟨100⟩, KCl ⟨100⟩, NaCl ⟨100⟩, and MgO ⟨100⟩. In all cases the spectra show up to 5 sharp well resolved maxima. From the final angle and energy of each peak the parallel surface momentum component and frequency of the phonon is determined. The results show a predominant coupling with Rayleigh surface phonons with only a weak coupling with bulk phonons. Rayleigh dispersion curves have been measured out to the zone boundary for all crystals except LiF ⟨110⟩ and MgO. There is also some evidence for excitation of modes of higher frequencies. Except for LiF ⟨100⟩ at the zone boundary the results compare well with theoretical calculations. Benedek has been able to fit the new LiF He atom and previous neutron dispersion curves with a revised breathing shell model calculation.

Measurement of the Dispersion Relation for Rayleigh Surface Phonons of LiF(001) by Inelastic Scattering of He Atoms

Time-of-flight spectra of low-energy (\ensuremath{\approx}20 me V) He scattered from a LiF(001) crystal along the $〈100〉$ direction reveal up to five sharp inelastic-scattering maxima. Their scattering angle and flight time determine the frequencies and wave numbers of surface phonons over the entire Brillouin zone. The results agree well with calculations of Rayleigh modes by de Wette and Benedek except near the zone boundary. Further unidentified features are also observed.

Rotationally Inelastic Molecule-Surface Scattering

Time-of-flight measurements have been performed on the inelastic scattering of room-temperature beams of ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$ molecules from a LiF (100) surface. The results show discrete inelastic peaks corresponding to energy losses approximately equal to the molecular rotational excitation energy but shifting with scattering angle. This indicates that rotationally inelastic scattering is dissipative rather than diffractive; i.e., it includes phonon excitations in the solid. The phonons involved can be identified as single Ray-leigh surface phonons.