Enhanced electron–phonon coupling at metal surfaces

Abstract

Recent advances in experimental techniques and theoretical capabilities associated with the study of surfaces show promise for producing in unprecedented detail a picture of electron–phonon coupling. These investigations on surfaces of relatively simple metals can be the platform for understanding functionality in complex materials associated with the coupling between charge and the lattice. In this article, we present an introduction to electron–phonon coupling, especially in systems with reduced dimensionality, and the recent experimental and theoretical achievements. Then, we try to anticipate the exciting future created by advances in surface physics.

Introduction

Electron–phonon interaction in metals is a very old subject. Göran Grimvall in the preface to his classic book on this subject pointed out that 1000 papers had been published in this subject field between 1960 and 1980 [1]. What can be new and exciting in such an old and mature field? The answer is surfaces and surface science techniques. Recent advances in experimental techniques coupled with ever-growing theoretical capabilities promise to create a renaissance in this subject. High-resolution angular-resolved photoemission is producing direct images of the distortion of the two-dimensional surface state bands near the Fermi energy because of electron–phonon coupling (EPC) [2], [3], [4], [5], [6]. First-principles calculations of the EPC for surface states are appearing in the literature [7], [8], [9] which not only explain the origin of the EPC-induced band distortions but also produce exquisite pictures of the Eliashberg function α²F(ω) [1]. The Eliashberg function is the product of a coupling constant times the phonon density of states and is at the heart of any theory of EPC [1]. A recent theoretical advance that will be described in this paper will allow experimentalists to extract the Eliashberg function directly from the high-resolution photoemission data [6]. These developments mark the beginning of a new era or a renaissance in the elucidation of many-body effects in reduced dimensionality.

Before continuing, it is important to back up and describe exactly what is implied by the Born-Oppenheimer approximation or the adiabatic approximation because EPC is a breakdown of the adiabatic approximation [1]. The total Hamiltonian for electrons and ions contains terms for the kinetic energy of the electrons and ions as well as the e–e and ion–ion Coulomb potential energy. The potential energy U_ion−e $\text{(}\text{r}{}_{\mathrm{i}}\text{,}\text{R}{}_{\mathrm{j}}\text{)}$ between the ZN electrons located at $\text{r}{}_{\mathrm{i}}$ and the N ions located at $\text{R}{}_{\mathrm{j}}$ couples the ions and the electrons. In principle, we would like to find the solution for the Schrödinger equation ψ_tot for this Hamiltonian, but it is impossible, and assumptions have to be made. The first assumption is that ψ_tot can be written as a product of two wave functions

$$\text{ψ}{}_{\mathrm{<mtext>tot</mtext>}}\text{(}\text{r}{}_{\mathrm{i}}\text{R}{}_{\mathrm{j}}\text{)=ψ}{}_{\mathrm{<mtext>e</mtext>}}\text{(}\text{r}{}_{\mathrm{i}}\text{,}\text{R}{}_{\mathrm{j}}\text{)ψ}{}_{\mathrm{<mtext>ion</mtext>}}\text{(}\text{R}{}_{\mathrm{j}}\text{).}$$

The second assumption is the adiabatic approximation where it is assumed that since the ions move so slowly compared with the electrons that the electrons follow their motion adiabatically. In an adiabatic motion, an electron does not make transitions from one state to another; instead, the electronic state $\text{ψ}{}_{\mathrm{<mtext>e</mtext>}}\text{(}\text{r}{}_{\mathrm{i}}\text{,}\text{R}{}_{\mathrm{j}}\text{)}$ is the ground state for any configuration of the ions. At T=0, the ground state of the electronic system can be calculated with the equilibrium positions of the ions. This static lattice configuration is what will be used for the dispersion of the two-dimensional surface states, $\text{ε}{}_0\text{(}\text{k}\text{)}$, in the absence of any EPC.

Now, it is clear what must be considered in the nonadiabatic regime necessary to explain EPC––the coupling of the excitation spectra of the electrons and the lattice. The static electronic band structure represented by $\text{ε}{}_0\text{(}\text{k}\text{)}$ will be distorted by coupling to the phonon modes of the lattice. The screening of the electrons by the lattice is represented by the self-energy function $\text{Σ(}\text{k}\text{,E)}$ where the quasi-particle band dispersion with EPC will be given by

$$\text{ε(}\text{k}\text{)=ε}{}_0\text{(}\text{k}\text{)+}\text{Re}\text{Σ(}\text{k}\text{,E).}$$

The imaginary part of the self-energy is related to the lifetime τ of the excited electronic states,

$$\text{τ}{}^{\mathrm{-1}}\text{=2}\text{Im}\text{Σ(}\text{k}\text{,ET).}$$

In this case, the temperature dependence of the self-energy will give the temperature-dependent linewidth.

A simple representation for the two independent systems is shown in Fig. 1 where (a) is the electron dispersion for a simple free-electron metal showing the Fermi energy (E_F) and Fermi wave vector (k_F) and (b) shows the phonon dispersion for a monatomic lattice. It is clear from the previous discussion that what is required when the systems are coupled is the excitation spectrum. For EPC to be important, the spectrum of electronic excitations must be degenerate, or nearly so, with the elementary lattice excitation (phonons) in the system [Fig. 1(b)]. Metals, by definition, satisfy this condition because there are always low-energy excitations at the Fermi energy. Fig. 2 shows the allowed excitation energy as a function of the momentum transfer for (a) a one-dimensional metal and (b) a higher dimensional metal. Fig. 1(b) is the lattice excitation spectra as a function of momentum. It is essential to understand the scale in these two figures. The temperature of the Fermi energy electron in Al is 10⁵ K, while the Debye temperature of the lattice vibrations is only 394 K. Therefore, the distortion of the electronic bands due to EPC will occur only within a narrow energy window around the Fermi energy [see Fig. 3(a)] defined by the maximum energy in the phonon density of states. This statement is not true for the EPC contribution to the linewidth (Eq. (3)). Excitations from deep in the band will display a temperature-dependent linewidth because EPC is involved in the decay process [6], [10], [11], [12], [13], [14], [15], [16], [17].

Fig. 3(a) displays the distortion to the electronic band dispersion near the Fermi energy anticipated from EPC. It is easy to see that one consequence is that the EPC causes a mass enhancement in the electronic band at the Fermi energy. The mass enhancement factor λ is defined by [1]

$$\text{λ=(m}{}^{\mathrm{*}}\text{−m}{}_0\text{)/m}{}_0$$

where m^* is the electron effective mass at the Fermi energy and m₀ is the effective mass in a frozen lattice, i.e., without EPC. If the low-energy lattice excitations (phonons) distort the electronic bands, then the low-energy electronic excitation close to the Fermi energy should distort the phonon dispersion curves shown in Fig. 1(b). In 1959, Kohn pointed out that the 2k_F excitations shown in Fig. 1(a) could lead, via the screened ion–ion interaction, to a dip in the phonon dispersion curves at this value of q [19]. These dips are represented in Fig. 3(b) and are known as Kohn anomalies.

A Kohn anomaly occurs in the phonon dispersion curves when there is a single vector connecting extremal sections of the Fermi surface (nesting). Two sections of the Fermi surface need to be parallel over some extended range in k-space. This is much easier to achieve as the dimensionality is reduced. Saturated H adsorption on W(1 1 0) or Mo(1 1 0) is probably the best known and studied example of Kohn anomalies and Fermi surface nesting at surfaces [5], [15], [20], [21], [22], [23], [24] and will be described subsequently. In one-dimension, a monatomic chain is unstable because of Fermi surface nesting. Fig. 1, Fig. 2 show that there is one and only one vector that connects two points on the Fermi surface, i.e., perfect nesting. This leads to a singularity in the one-dimensional Lindhard response function [25] and “giant Kohn anomalies” in the phonon dispersion at 2k_F. In the one-dimensional system, k_F is one half of the zone boundary at π/a (a lattice spacing); therefore the giant Kohn anomaly occurs at the zone boundary. If the phonon frequency at 2k_F=π/a goes imaginary, the monatomic chain will reconstruct. The new periodicity will be 2π/q (soft phonon) = 2π/2k_F=2π/(πa)=2a. The pairing of the atoms creates a new unit cell with the zone boundary at π/2a, a gap in the electronic states at the Fermi energy (metal–insulator transition), and an optical phonon branch. Dimensionality of the system is important when discussing EPC.

As promised, the H/W(1 1 0) system will now be described. It offers a beautiful example of all the concepts discussed above. Adsorption of H onto a clean W(1 1 0) surface shifts the surface states continuously [20], and at saturation, a Kohn anomaly is observed in the surface phonon dispersion using He atom scattering (HAS) [21]. The HAS data in Fig. 4(a) show a very deep dip in the phonon energy at q=0.93 Å⁻¹. This has to be the signature of 2k_F Fermi surface nesting, but the early photoemission Fermi contour measurements for the hydrogen-covered surface did not show this nesting vector [20]. Subsequent calculations of the Fermi contours showed segments that were appropriately nested [22]. These calculated surface states [22] are shown on the plot of the Fermi contours in Fig. 4(b). This surface state, labeled 1, is connected by a spanning vector of magnitude 0.96, in close agreement with the experimental value. The calculated phonon dispersion is shown as the solid lines in Fig. 4(a), displaying dips, but not as large as seen with HAS [23]. On the other hand, inelastic electron scattering data (EELS) [24] shown in Fig. 4(a) are almost identical to the calculations. There are two explanations in the literature for the discrepancy between the HAS and EELS data. The first, and obviously the one most compatible with EPC, is that in this region of energy and momentum there is mixing between the excitations of the electrons and the lattice [22]. The deep dip seen by HAS is primarily electronic in origin due to Fermi surface nesting, while the shallow branch seen by ELS is primarily lattice vibrations [22]. The second explanation [23] is that at saturation coverage the hydrogen forms a disordered layer and the mode is a collective plasmonlike mode associated with the diffusive motion.

The recent photoemission measurements of the Fermi contours for this system displayed in Fig. 4(b) expand upon these results in an important and interesting way––enhanced spin-orbit coupling at a surface [5], [15], [26]. The spin-orbit interaction is enhanced because of the broken symmetry at the surface [27]. The Fermi contour implicated in the calculations as being responsible for the anomaly was found to be significantly split into two bands, labeled 1 and 2 in this figure. The two states are observed to be 100% spin polarized and to support an unusual spin structure in k-space [26]. The observed phonon anomaly is now understood to be associated with nesting between contours 1 and 2, which are coupled by a vector of magnitude 0.87 Å⁻¹. As shown in this figure, nesting between each individual contour and its image opposite the $\text{Σ}$ line does not match the position of the observed anomaly as well. Moreover, the initially proposed [5] and recently measured [26] spin structure does not allow coupling between each contour and its image because these states have opposite spin, and the electron-phonon interaction is spin-independent. In this simple surface system, we find all the ingredients of strong coupling between spin, charge, and lattice degrees of freedom.

How strong are these couplings? An indication is offered in Fig. 4(c), which displays the surface state 1 dispersion revealing the distortion due to EPC near the Fermi energy, and by Fig. 4(d), which shows ReΣ(ε) extracted from these data [15]. The authors measured a mass enhancement factor of λ=1.4 for surface state #1 near the Fermi energy––an anomalously high value indicative of strong coupling indeed. Moreover, they assigned the coupling to the high-energy (161-meV) hydrogen symmetric stretching mode, and the energy-coupling parameter pair is notably anomalous [24]. Even though this system has been studied for many years and both the data and theory are consistent, there are still important unresolved questions, especially pertaining to the origin of the EPC. The EPC, as displayed by the ReΣ in Fig. 4(d), cannot be explained with any simple phonon model, and it is clear from looking at these data and the arrow showing the position of the H stretching mode that the eventual answer will be more complicated than having a single vibrational mode coupling to the electrons [5], [15].

Before leaving this section, it is important to talk briefly about line shapes and linewidths. Fig. 5(a) displays the temperature dependence of the photoemission linewidth from the bottom of the Be(0 0 0 1) surface state (2.78-eV binding energy) [12]. The increase in width as the temperature is increased is a consequence of the temperature dependence in ImΣ(T) expressed in Eq. (3). The high-temperature limit gives the following equation [12]

$$\text{dΔ}\text{E/}\text{d}\text{T=2πλk}{}_{\mathrm{<mtext>B</mtext>}}\text{,}$$

where λ is the mass enhancement factor given in Eq. (4) and ΔE(T) is the width of the peaks in the photoemission spectra. Using this equation, the data shown in Fig. 5(a) yield a value of λ=1.06 [11]. The H and D vibrational modes [shown in Fig. 5(b)] are measured using infrared spectroscopy [28], exhibiting another fundamental line shape effect due to EPC. The H–W mode at 1270 cm⁻¹ is assigned to the overtone of the wagging-mode. The Fano line shape is a direct result of the EPC of this mode to the continuum of electronic excitations [29].