Surface Phonons

Abstract

The existence of surface lattice vibrations, surface phonons, is explained by means of the model of a linear chain of atoms. The extension to two- and three-dimensional systems as surfaces on bulk crystals consisting of more than one atomic species leads to Rayleigh waves and optical surface phonons. Hereby Fuchs–Kliewer phonons, their coupling to electronic surface plasmons and their detection by HREELS play a major role. Atom and molecular beam scattering are presented as essential investigation tools for surface phonons.

Appendices

Panel XI: Atom and Molecular Beam Scattering

Atoms and molecules such as He, Ne, H₂, D₂, impinging on a solid surface as neutral particles with a low energy (typically <20 eV), cannot penetrate into the solid. Scattering experiments with neutral particle beams therefore provide a probe that yields information exclusively about the outermost atomic layer of a surface. Such experiments have now become an important source of information in surface physics. Both elastic and inelastic scattering can be studied. A schematic overview of the various scattering phenomena is given in Fig. XI.1. Since He atoms, for example, with a kinetic energy of 20 meV have a de Broglie wavelength of 1 Å, scattering phenomena must be described in the wave picture (Sect. 4.1). A particle approaching the surface interacts with the surface atoms through a typical interatomic or intermolecular potential V(r _∥,z), r _∥ being a vector parallel to the surface, and z the coordinate normal to the surface. V(z) consists of an attractive and a repulsive part (as in chemical bonding). The scattering from a two-dimensional periodic lattice of atoms (surface) is dominated by the specular quasi-elastic peak (intensity I ₀₀) and elastic Bragg diffraction (intensity I _hk ) in well-defined directions (as in LEED, Sect. 4.2). This elastic scattering is adequately described in the rigid-lattice approximation with only an intensity correction for inelastic effects, provided by the temperature-dependent Debye–Waller factor. An incident atom or molecule can lose so much energy that it is trapped at the surface or “selectively adsorbed”. This trapping of atoms in bound states on the surface can strongly modify the scattered intensities at specific angles and energies.

Fig. XI.1

Schematic diagram showing the different collision processes that can occur in the non-reactive scattering of a light atom with the de Broglie wavelength comparable to the lattice dimensions. Since the lattice vibrational amplitudes are small, phonon inelastic scattering is expected to be weak relative to elastic diffraction (specular beam I ₀₀ and Bragg diffraction beams I _hk ). Additionally, high energy losses can lead to selective adsorption of impinging atoms in the attractive part of the surface atom potential V(z) [XI.1, XI.2]

Inelastic scattering comes into play due to the fact that the crystal is in reality not rigid: the atoms vibrate about their average positions. The incident particle can therefore transfer part of its kinetic energy to the dynamic modes of the vibrating surface, the surface phonons. Similarly, it can gain energy via the annihilation of a surface phonon.

The mathematical description of the scattering is analogous to that of electron-surface scattering (Sect. 4.1). The most general interaction potential V(r) between the incident particle and the crystal surface (4.1) which enters the formula for the scattering cross section (4.17) is conveniently written as a function of r _∥, a coordinate parallel to the surface, the coordinate z normal to the surface, and s _n (t) the vibrational coordinate of the nth surface atom:

$$ V\bigl[\boldsymbol{r}_{\parallel}, z, \boldsymbol{s}_{\boldsymbol{n}}(t) \bigr] = V(\boldsymbol{r}_{\parallel}, z)|_{\boldsymbol{s}_{\boldsymbol{n}} = 0} + \sum_{\boldsymbol{n}}(\boldsymbol{\nabla}V)\cdot \boldsymbol{s}_{\boldsymbol{n}}(t) + \cdots. $$

(XI.1)

The first term in the potential expansion is the corrugated elastic potential, which can be determined by fitting the intensities of the elastic diffraction peaks using model potentials. Elastic scattering thus yields information about the topology of the surface and about details of the interatomic potentials. The second- and higher-order terms, which couple to the vibrations s _n (t) of the surface atoms, are responsible for inelastic scattering. An understanding of these coupling terms is fundamental for an interpretation of such phenomena as sticking coefficients (Sect. 9.5) and energy transfer between surface atoms and incident particles.

Before presenting some detailed examples of the application of atom scattering, the experimental set-up will be discussed briefly.

The experimental apparatus consists of a source of monoenergetic molecules or atoms which are directed as a beam towards the surface under investigation; the back-scattered distribution is recorded by a detector. Both sample and detector can be rotated around a common axis in the surface plane to allow the detection of higher diffraction orders under different angles. Since neutral particles are used, neither electric nor magnetic fields can be used as focussing or dispersive elements. A schematic diagram of a typical experimental set-up is shown in Fig. XI.2. An important feature is the nozzle beam source producing the monochromatic rare-gas beam. The beam of Ne or He atoms is produced in a high-pressure expansion source. In the expansion of the gas from a source pressure of about 2 atm through a thinwalled orifice (diameter ≈5⋅10⁻² mm) to a beam pressure of about 10⁻⁴ Torr, the random translational energy is converted into a forward mean velocity of the beam. Thus the magnitude of the random velocity component which determines the velocity spread Δv is reduced relative to the most probable velocity v. In the apparatus shown in Fig. XI.2 the resultant Δv/v is about 10 %. With improved nozzle beam sources Δv/v values on the order of 1 % are achieved. Toennies [XI.4] used a He source cooled down to 80 K. The beam is expanded from a pressure of 200 atm through a 5 μm hole into vacuum. To improve the forward velocity distribution further, the beam passes a skimmer after expansion. This funnel-shaped tube skims off atoms with insufficiently forward-directed velocity. During the expansion, chaotic thermal motion is converted into a concerted forward motion of the atoms and as a result of enthalpy conservation, the temperature in the moving gas is drastically reduced; behind a distance of about 20 mm to ≈10⁻² K. This corresponds to a relative velocity spread of less than 1 %. With modern nozzle-beam sources, primary energies from 6 meV up to 15 eV can be produced. He atoms with de Broglie wavelengths of 1 Å have an energy of about 20 meV. In Fig. XI.2 the primary beam is modulated by a chopper and phase-sensitive detection is employed using a lock-in amplifier. This technique allows detection of the modulated scattered beam against a relatively high background pressure. Either standard ion gauges or more sophisticated mass spectrometers are employed as detectors.

Fig. XI.2

Schematic diagram of a typical low-energy molecular beam scattering apparatus [XI.3]

In the following, some examples are presented of the different applications of atom and molecular scattering based on the processes of Fig. XI.1. Since He atoms are essentially scattered on the almost structureless “electron sea”, far from the uppermost surface lattice plane, an ideal well-ordered, close-packed metal surface gives rise to virtually no interesting scattering phenomena. But deviations from ideality, such as steps, defects or adsorbates, can affect the elastically scattered intensity in reflection direction, i.e. the specular beam intensity I ₀₀ quite dramatically. Figure XI.3 shows the intensity variation of the specular beam of He atoms reflected from Pt(111) surfaces with differing distributions of steps and terraces. For the measurement the angle of incidence θ _i(=θ _r) is varied over a small range and the backscattered intensity I ₀₀ is recorded. For this purpose, of course, an experimental set-up with detection under variable observation direction is necessary. According to the differing terrace width (average values around 300 Å and above 3000 Å) an interference pattern or an essentially monotonic variation is observed in the angular region considered. The oscillations are explained in terms of constructive and destructive interferences of the He wave function reflected from (111) terraces which are separated by monatomic steps. The average terrace width, i.e., the step density, determines similarly as in an optical grid slit width and distance, the phase differences of the evading He waves under certain observation directions. The oscillation period of curve (b) allows an estimation of the step atom density of about 1 %. Better preparation techniques lead to step atom densities lower than 0.1 % which then give rise to a higher total reflection intensity, and the interference oscillations are absent (curve (a)). The technique is thus useful for characterizing the degree of ideality of a clean surface after preparation.

Fig. XI.3

Relative specular intensity I ₀₀ (referred to primary beam intensity) of a low-energy He atom beam (energy E=63 meV) versus angle of incidence θ _i (=θ _r reflection angle) for two Pt(111) surfaces with differing average terrace widths [XI.5]

Elastic He atom scattering, i.e. diffraction, can provide information about the structural properties of a surface. In contrast to electron scattering in LEED (Panel VIII: Chap. 4), where the electrons penetrate several Ångstroms into the solid, only the outermost envelope of the electron density about the surface is probed by the He atoms. This makes the technique relatively insensitive to clean, well-ordered, densely-packed metal surfaces; but ordered adsorbate atoms or molecules whose electron density protrudes significantly from the surface, give rise to stronger scattering intensities in certain Bragg spots. This is shown for the example of a well-ordered oxygen layer with p(2×2) superstructure on Pt(111) in Fig. XI.4. For the clean Pt surface the ($\bar{1},\bar{1}$) Bragg spot has ten times less intensity than on the oxygen covered surface. The diffraction spots ($\bar{1}/2, \bar{1}$/2) and ($\bar{3}/2, \bar{3}$/2) due to the oxygen superlattice occur with much higher intensity. Adsorbate effects are thus clearly distinguished from substrate spots and the interpretation problems sometimes encountered for adsorbate LEED patterns (substrate vs. adsorbate superstructure) do not exist. The method of atom and molecule diffraction is therefore complementary to LEED because of its extreme sensitivity to the outermost atomic layer. In the inelastic scattering regime, atom and molecule scattering from surfaces also provides interesting advantages over other scattering techniques because of its high energy resolution. Because the possible energy and wave vector transfer are well matched throughout the whole Brillouin zone, surface-phonon dispersion branches (Chap. 5) can be measured with extremely high accuracy. Figure XI.5 shows inelastic He beam spectra measured with different angles of incidence θ _i on Cu(110). The detection direction is chosen for wave-vector transfers along $\overline{\varGamma Y}$. The experimental resolution readily allows the determination of peak half-widths below 1 meV. Thus information about broadening due to phonon coupling etc. can also be derived from the experimental data. This is by no means possible from electron scattering data (HREELS, Panel X: Chap. 4), where the best energy resolution is on the order of 1 meV. Surface phonon dispersion curves derived from spectra such as those of Fig. XI.5 are given in Fig. XI.6, but here along the $\overline{\varGamma X}$ direction of the Cu(110) surface Brillouin zone. The data denoted by R correspond to the Rayleigh surface waves (Chap. 4).

Fig. XI.4

He-beam polar diffraction patterns in the [112] direction from the clean (bottom) and p(2×2)O/Pt(111) oxygen covered surface (top). The primary He energy E _He is 17.3 meV and the sample temperature 300 K [XI.6]

Fig. XI.5

Inelastic He scattering spectra taken along the $\overline{\varGamma X}$ direction of the surface Brillouin zone on Cu(110). The primary He beam energy is 18.3 meV [XI.7]

Fig. XI.6

Surface phonon dispersion curves as obtained by inelastic He scattering (primary energy E _He=18.3 eV) along the $\overline{\varGamma Y}$ direction of the surface Brillouin zone on Cu(110). The reduced wave vector ξ is defined by $\xi= k/k_{\mathrm{BZ}}(\overline{X})$ with as the Brillouin-zone dimension in the $\overline{X}$ direction [XI.7]

References

• XI.1. J.P. Toennies, Phonon interactions in atom scattering from surfaces, in Dynamics of Gas-Surface Interactions, ed. by G. Benedek, U. Valbusa. Springer Ser. Chem. Phys., vol. 21 (Springer, Berlin, 1982), p. 208

• XI.2. E. Hulpke (ed.), Helium Atom Scattering from Surfaces. Springer Ser. Surf. Sci., vol. 27 (Springer, Berlin, 1992)

• XI.3. S. Yamamoto, R.E. Stickney, J. Chem. Phys. 53, 1594 (1970)

• XI.4. J.P. Toennies, Physica Scripta T 1, 89 (1982)

• XI.5. B. Poelsema, G. Comsa, Scattering of Thermal Energy Atoms from Disordered Surfaces. Springer Tracts Mod. Phys., vol. 115 (Springer, Berlin, 1989)

• XI.6. K. Kern, R. David, R.L. Palmer, G. Comsa, Phys. Rev. Lett. 56, 2064 (1986)

• XI.7. P. Zeppenfeld, K. Kern, R. David, K. Kuhnke, G. Comsa, Phys. Rev. B 38, 12329 (1988)

Problems

Problem 5.1

Information can be transmitted through a solid by bulk sound waves or by Rayleigh surface waves. What phonons provide a faster transmittance velocity? Discuss the problems which arise for the signal propagation by means of short pulses when long wavelengths λ≫a (lattice parameter) and short wavelengths λ≈a are used.

Problem 5.2

The dielectric response of an infrared active, n-doped semiconductor is described in the IR spectral region by a dielectric function ϵ (5.66) which contains an oscillator contribution due to TO phonons (5.50) and a Drude-type contribution (5.60) due to free electrons in the conduction band. Calculate the surface loss function $\operatorname{Im}\{-1/[\epsilon(\omega) - 1]\}$ and discuss the loss spectrum expected in an HREELS experiment as a function of carrier concentration. Flat-band situation is assumed at the surface.

Problem 5.3

Surface phonon polaritons (Fuchs–Kliewer phonons) are excited on a clean GaAs(110) surface in an HREELS experiment with a primary energy of 5 eV. Calculate from the corresponding loss peak at 36.2 meV the exponential decay length of the polarisation field of the surface phonons. Discuss the consequence for an HREELS measurement which is performed on a GaAs film which is thinner than the calculated decay length.

Problem 5.4

Calculate the frequency of a surface phonon on the (100) surface of an fcc crystal at the Brillouin-zone boundary in the [110] direction. Only central forces between next neighbour atoms are assumed. The surface phonon should have odd symmetry with respect to the mirror plane defined by the phonon wave vector q and the surface normal.

Why is the calculation so simple?

Does a second surface phonon exist on this surface which is localized on the first atomic monolayer?