The Electron-Phonon Interaction of Low-Dimensional and Multi-Dimensional Materials from He Atom Scattering

Atom scattering is becoming recognized as a sensitive probe of the electron-phonon interaction parameter $\lambda$ at metal and metal-overlayer surfaces. Here, the theory is developed linking $\lambda$ to the thermal attenuation of atom scattering spectra (in particular, the Debye-Waller factor), to conducting materials of different dimensions, from quasi-one dimensional systems such as W(110):H(1$\times$1) and Bi(114), to quasi-two dimensional layered chalcogenides and high-dimensional surfaces such as quasicrystalline 2ML-Ba(0001)/Cu(001) and d-AlNiCo(00001). Values of $\lambda$ obtained using He atoms compare favorably with known values for the bulk materials. The corresponding analysis indicates in addition the number of layers contributing to the electron-phonon interaction that is measured in an atom surface collision.


I. INTRODUCTION
Electron-phonon (e-ph) interaction at conducting surfaces together with its dimensionality are of great importance both at fundamental and technological levels. 1,2 Very recently, this interaction has been shown to play a relevant role in topological semimetal surfaces such the quasi-one-dimensional charge density wave system Bi(114) and the layered pnictogen chalcogenides. 3 The e-ph coupling in these materials for individual phonons λ Q,ν (where Q denotes the surface parallel wave vector and ν the branch number), and its average λ, the well-known mass-enhancement factor 4-6 can be directly measured with supersonic He-atom scattering (HAS). [7][8][9][10][11] With this experimental technique, subsurface phonons were detected on multilayer metallic structures 7,10 exploring the fairly long range of the e-ph interaction, e.g., spanning as many as 10 atomic layers in Pb films 7,8 (known as the quantum sonar effect). Under reasonable approximations, from the thermal attenuation of the diffraction peaks ruled by the so-called Debye-Waller (DW) factor as well as the interaction range through the number of layers n sat , λ HAS values can actually be extracted which agree fairly well with previous values for the bulk or obtained from other surface techniques. 3,9,10 This quantity n sat indicates the number of layers above which the measured λ becomes thicknessindependent.
In this work we focus on the specific role of dimensionality in the e-ph mass-enhancement factor as derived from HAS. 3 In particular, we extract λ HAS values from HAS data for different classes of conducting surfaces characterized by nearly free-electron gases of growing dimensions, from the quasi-1D systems such as W(110):(1×1)H and Bi(114), and the quasi-2D layered chalcogenides, to quasicrystalline surfaces such as the dodecagonal 2ML-Ba(0001)/Cu(001) and decagonal d-AlNiCo(00001), which can be regarded as behaving like periodic 4D and 5D materials, respectively.

A. The new Debye-Waller factor
As is well known, the DW factor describes the thermal attenuation, due to the surface atomic motion, of the elastic scattered intensity I(T ) observed at temperature T with respect to the elastic intensity of the corresponding rigid surface I 0 . It is usually written as where the factor, exp{−2W (k f , k i , T )}, depends explicitly on the final (k f ) and incident (k i ) wave vectors of the scattered atom. When zero point motion vibrations can be neglected, which holds for T comparable to or larger than the surface Debye temperature of the material, 2W (T ) is approximately linear in T .
When the incident atom directly interacts with the surface, the DW exponent is simply expressed by 2W (k f , k i , T ) = (∆k · u) 2 T , ∆k = k f − k i being the scattering vector, u the phonon displacement experienced by the projectile atom upon collision, and · · · T means a thermal average. For energies typically well below 100 meV, incident atoms are exclusively scattered by the surface free-electron density, a fewÅ away from the first atomic layer. Thus, the exchange of energy with the phonon gas mainly occurs via the phononinduced modulation of the surface free-electron gas or the so-called e-ph interaction. Under reasonable approximations, it has been recently shown that the DW exponent is proportional to λ and in the simplest case reads as 9 where N (E F ) is the electron density of states at the Fermi energy E F , m and m * e are the projectile atomic mass and the electron effective mass, respectively, φ is the work function and k B is the Boltzmann constant. Eq. (2) is written here specifically for the specular diffraction peak for which E iz = E i cos 2 (θ i ) = 2 k 2 iz /2m is the incident energy associated with the motion normal to the surface for the given incident angle θ i . For application to non-specular diffraction peaks or to other elastic features, Eq. (2) needs to be adjusted to account for the correct scattering vector appropriate to the scattering configuration, i.e., Using the simple expression of Eq. (2), a previous analysis of the thermal attenuation of the specular peaks of He atom scattering from several simple metals extracted values of λ HAS . 9 These values must be regarded as values of the electron-phonon constant relevant to the region near the surface.
An important observation concerning the DW factor 2W (k f , k i , T ) is that it is rigorously proportional to the temperature, subject to the condition that T is large compared to the Debye temperature (which in practice means that T should be larger than about half the Debye temperature). This results because the average phonon mean-square displacement satisfies similar conditions as long as the crystal obeys the harmonic approximation. Thus, plots of 2W (k f , k i , T ) versus T (usually called DW plots) have linear slopes at large T , although at small values of T where zero point motion becomes effective the curve saturates to a constant value.
There have been extensive He atom scattering measurements of successive layers of alkali metals grown on various metal substrates, and similar studies of multiple Pb layers grown on Cu(111). 10 These systems are of interest for studies of the e-ph constants, for example to see how λ HAS varies as a function of numbers of monolayers. The case of multiple Pb monolayers is of particular interest because thin films of Pb on Si and GaAs are known to remain superconductors down to one monolayer. 7,[12][13][14][15] An interesting first observation of the thermal attenuation of specular He atom scattering for layer-by-layer growth is that, for a given system, the slope of the DW plots increased linearly with layer number n for the first few layers up to a saturation number n sat which was typically about five layers. 10 This behavior suggests that each layer of these simple metals contributes similarly and additively to the Fermi level density of states appearing in Eq. (2), a property which is also indicated by theoretical calculations for multi-layer alkali films. 16 For the surface of bulk simple metals, using for the Fermi level density of states that of a three-dimensional (3D) free electron gas, N (E F ) = 3Zm * e / 2 k 2 F , where k F is the Fermi wave vector and Z the number of free electrons per atom, was a satisfactory approximation. 10 However, the linearly increasing slope observed for ultrathin films with n < n sat suggests that it is appropriate to attribute to each metal layer an independent contribution of the 2D free electron density of states (DOS), which implies that N (E F ) = nm * e a c /π 2 for n ≤ n sat where a c is the area of the surface unit cell. Combining this last expression (2D form) for the Fermi level density of states, and recognizing that ln[I(T )/I 0 ] = −2W , the following expression for λ from Eq. (2) is obtained where T 1 and T 2 are any two temperatures in the linear region of the DW plot. If n sat is known one can use the 2D expression for n > n sat by setting n = n sat (this is what has been done for all layered crystals); alternatively, the 3D expression can be used. Interestingly, Eq. (3) is written in a form in which the factor of 1/n is cancelled out by the linear increase in n of ln[I(T 1 )/I(T 2 )] resulting in a value of λ HAS that is essentially independent of the monolayer number n, even for n > n sat where the 3D Fermi density of states is applied.
Analysis of the data for thermal attenuation of the specular He atom diffraction peak for up to 10 layers of the alkali metals Li, Na, K, Rb and Cs deposited on various metal substrates produced values for λ HAS that are in quite reasonable agreement with known tabulated values for bulk crystals, and similarly for up to 25 ML of Pb deposited on Cu(111). 10 The success of the theory expressed in Eqs. (2) and (3) where d being the dimension, and Γ the Riemann gamma-function, which for integer d has values given by For d = 2, γ 2 = 2π, and d = 3, γ 3 = 2π 2 , the usual two-dimensional electron gas (2DEG) and three-dimensional electron gas (3DEG) expressions, namely N = m * e k F /π 2 2 , respectively, are readily obtained with k F = (2m * e E F ) 1/2 / . As discussed elsewhere, 10 the 3DEG of a thick anisotropic degenerate semiconductor slab can be viewed as a stack of a number n = n sat of 2DEGs. This yields a definition of n sat as where k F ⊥ is the Fermi wave vector normal to the surface and c * has the meaning of the e-ph interaction range normal to the surface or the maximum depth beneath the surface from where phonon displacements can modulate the surface charge density. In this way, the 2D expression for the e-ph coupling constant for a thick layer crystal as given in Ref.
For a general d-dimensional free-electron system (for any d, even fractional), one finds where r d 0 is the unit cell hyper-volume. Similarly, in the case of a measurement of the dependence on the HAS specular reflectivity as a function of the incident wave vector at constant T , the expression for λ (d) .
The factor of k η i multiplying the intensity is a correction for the energy dependence of the incident beam flux. The standard theoretical treatment of a jet beam nozzle expansion flow shows that the beam energy varies inversely as the square root of the stagnation temperature.
The correction factor is then simply k i , or η = 1, 18 although in some cases different behaviors on incident energy have been measured. Further discussion on the dependence of the incident beam on stagnation temperature and pressure has been reported by Palau et al. 19 When dealing with layered semimetal surfaces, the free electron gas is protected by an anion surface layer leading to an essentially hard-wall potential plus a more or less deep attractive van der Waals potential, k 2 iz should be corrected due to the presence of the attractive well before being repelled by the hard wall (Beeby correction 20 ). This implies that both the incident and final normal momenta should be replaced by k 2 z −→ k 2 z + 2mD/ 2 , where D is the attractive potential depth. Usually, the incident energy E i is generally much larger than D, so that this correction can be neglected. With regard to Eq. (9), notice that the Beeby correction cancels out in the differential in the denominator, but retains a minor effect through the term in the numerator.

III. ONE DIMENSIONAL ELECTRON GAS
The high sensitivity of the HAS technique permits the detection of weak surface charge density waves (CDW) that are difficult to detect with other surface techniques. Recently, it has been shown that from the temperature dependence of the CDW diffraction peaks information about the e-ph interaction sustaining the CDW transition is possible. 3 There is an instability below a critical temperature T c generally induced by the e-ph coupling according to the theory developed by Fröhlich-Peierls 21,22 or the Kelly-Falicov multivalley mechanism. [23][24][25] In the latter case, the phonon-induced transitions between narrow pockets (nests) literally realize what is meant as perfect nesting. The occurrence of a CDW instability below T c yields additional T -dependent diffraction peaks in the elastic scattering angular distribution at parallel wave vector transfers ∆K equal or close to the nesting vectors Q c (e.g., Q c = 2k F for the 1D Peierls mechanism).
When examining the thermal attenuation for a given diffraction peak intensity due to the DW factor, the corresponding wave vector ∆K transfer parallel to the surface equals to either a G-vector of the normal surface lattice (∆K = G), or to a CDW wave vector Q c . In this case, Eq. (8) can also be applied to diffraction peaks by simply replacing 4k 2 iz by ∆k 2 z + ∆K 2 . Usually in HAS experiments, the condition ∆K 2 << ∆k 2 z holds, therefore the T -dependence of the diffraction and specular peaks leads to a λ HAS value which is independent of the diffraction channel. In Eq. (1), the temperature dependence of I(T ) comes from thermal vibrations. However, this is no longer completely true when considering the diffraction from a surface CDW which forms below T c from a Fermi surface instability.
Clearly, the temperature-dependent population of electron states near the Fermi level follows Fermi statistics. Here, I 0 has an implicit dependence on T , which is generally negligible with respect to that of W (T ), except near T c ; in this case, its square root √ I 0 can be considered as an order parameter, 26,27 and vanishes when T −→ T c as (1 − T /T c ) β , where β is the orderparameter critical exponent (typically β = 1/3). [28][29][30] In the following, 1DEG examples are shown that, away from the critical region, a CDW diffraction peak may be used to extract λ HAS .

A. Bi(114)
The case of Bi(114), a topological surface exhibiting properties of a 1D free electron gas, as well as a CDW, has been discussed in an earlier letter. 3 As shown in that letter, the weak 2D character of this surface, characterized by a long period (28.4Å) in the X Y direction, is responsible for the charge density wave, observed with HAS below T c ≈ 280 K, via the Kelly-Falicov multivalley X -X nesting mechanism. The pronounced 1D metal character of Bi(114) 31 is confirmed by the reliable value λ where γ 2 /γ 1 = 2 and a c = r 2 0 / √ 2 is the unit cell area for the CDW lattice. This gives HAS /λ (1D) HAS = 2 3/2 /π = 0.90 , which means that also the 2D expression provides a reasonable approximation for W(110):H(1×1). Since Eqs. (8) and (9) have been derived for an isotropic nDEG, the deviation of the above ratio from unity essentially measures the deviation from isotropy of the actual 2DEG. Genuine 2DEG systems are now considered in the next Section.

IV. 2D TOPOLOGICAL MATERIALS. CHALCOGENIDES
The cases of recently measured topological Bi pnictogens Bi 2 Se 3 , Bi 2 Te 3 and Bi 2 Te 2 Se have been considered elsewhere. 3 In addition, several other chalcogenide crystal surfaces that have been investigated with He atom scattering are listed in Table I. In all but one of these systems it was the specular thermal attenuation that was measured and two examples, for 1T-TaS 2 and 2H-TaSe 2 , are shown in Fig. 2. Although the experiments on the chalcogenides with n s given by The quantity n s plays a role similar to n sat in the case of layer-by-layer growth, i.e., it is the number of layers whose electronic states at the Fermi level concur to give the surface charge density probed by the He atoms, and N (E F ) is their total density of states; thus F . This implies that n s represents the number of layers of the compound that are contributing to the value of λ HAS as measured in a He atom collision with the surface. Eqs. (11) and (12) suggest that the question of determining the density of states N (E F ) appropriate to the reflection of atomic He is cast into the problem of determining a single parameter, namely the small number of layers n s of the surface that contribute.
In semimetals and degenerate semiconductors, as well as in normal semiconductors with a degenerate accumulation layer at the surface, the density of states N (E F ) is associated with the presence of surface states and quantum-wells states confined within the band bending, i.e., within the Thomas-Fermi (TF) screening length λ T F . 61 In transition-metal layered compounds the limited electron mobility in the normal direction, resulting in a large effective mass anisotropy, makes λ T F comparatively short, typically spanning about two triple layers.
Thus n s ≈ 2 appears to be an appropriate value for this class of materials (see Table I).
Pnictogen chalcogenides, whose e-ph coupling strength as derived from HAS measurements have been presented and discussed elsewhere, 3 are characterized by screening lengths an order of magnitude larger than in transition metal chalcogenides, due to their quintuple layer structure with a more pronounced 3D character. As a consequence, their λ HAS is seen to receive a far larger contribution from the surface quantum-well states than from the topological Dirac states.  Table I, with reasonable choices for λ T F determined using parameters taken from the literature, results for λ HAS are in fairly good agreement with values of λ determined from other known sources, which may be either bulk measurements or calculations. The entry for PtTe 2 differs from the others in that the elastic peak measured was not the specular one but the off-specular diffuse elastic peak measured at the constant temperature of 100 K. In this case, the incident angle was varied, keeping the source-todetector angle fixed. Thus, Eq. (3) was modified to account for the correct wave vector difference, i.e., k 2 iz → ∆k 2 /4 = (k f − k i ) 2 /4, and the factor α changes to reflect the fact that the intensities were evaluated at different incident angles, but at constant T .

V. QUASICRYSTALLINE SURFACES
Quasicrystals (QC) are characterized by a long-range orientational order 62-64 but no periodicity. They can be viewed as projections onto the ordinary space of a periodic lattice in a space nD of higher dimension. 65 Certain structural and dynamical properties of the QC may be more conveniently described in the corresponding nD periodic lattice. This representation is adopted here to derive the e-ph coupling constant for two QC metallic structures, a dodecagonal bilayer and a 3D decagonal QC, respectively, represented by periodic 4D and 5D lattices.

A. 2ML-Ba(0001)/Cu(001)
The LEED pattern for a barium bilayer grown on Cu(001) at the growth stage denoted as III by Bortholmei et al. 66 exhibits a dodecagonal QC structure as shown in Fig. 3a,b).
The structure is a superposition of two hexagonal 2D lattices with a 30 • twist. A twist of any angle other than integer multiples of π/3 breaks the hexagonal periodicity of a single layer while keeping a long-range orientational order. The present structure can be generated by the projection of a 4D {3, 3, 4, 3} honeycomb lattice (actually a 4D-bcc cubic lattice) 65 .
The DW exponent measured from the HAS reflectivity I(T ) as a function of temperature at a given incident wave vector k i ≈ 6.8Å −1 as shown in Fig. 3c), 66   . As it appears from this cut of SX-ARPES data, k F = 0.9Å −1 seems to be a reasonable choice (Fig. 5c)).
Then, with φ = 4.8 eV from He * (2 3 S, 1s2s) de-excitation spectroscopy, 75  HAS for the (00001) surface indicates an appreciable anisotropy, which favors the quasicrystalline plane. These values can be compared with the mass-enhancement factor extracted from the e-ph enhancement of thermoelectric power (TP) measurements in Y-AlNiCo, the monoclinic 3D approximant of decagonal d-AlNiCo. 76 At room temperature the fit in the a * direction (approximating [10000] in the decagonal sample) gives λ T P ∼ 0.2, a value which appears, however, to increase at lower temperature. 76 There is significant anisotropy, with λ T P for the quasicrystalline plane about five times larger than along the tenfold axis. Moreover, λ T P exhibits a large increase for decreasing T , though the present value λ It may be argued that an ideal free-electron model does not contain information on the lattice periodicity or quasi-periodicity. With this in mind, the 5D d-AlNiCo can also be treated as a 3D system with a Fermi surface from the highly-dispersed Al sp-band (k 0 F = 1.57Å −1 ). 74 In this case, with the same φ and r 0 , one finds λ (3D) HAS = 0.23 ± 0.05 (Bragg) and 0.15 (anti-Bragg) for the d=AlNiCo(00001) surface, and 0.22 ± 0.06 for the (00110).
It appears that, within the experimental uncertainties, the effects of dimensionality are in this case rather modest. On the other hand, it is easily seen that treating d-AlNiCo(00001) as a 2D system, with the response restricted to the surface bilayer (n sat = 2), the ratio λ (2D) HAS /λ (4D) HAS = (r 0 k F ) 2 /4πn sat is for this case equal to 1.16. Thus, in this case, like for 2ML-Ba/Cu(001), the 2D treatment yields only a slightly larger e-ph coupling constant than that obtained from the treatment with the more appropriate dimensions.

VI. CONCLUSIONS
In this paper, we have analyzed the thermal dependence of the Debye-Waller factor measured in the scattering of atoms from a selection of complex surface systems in order to extract values of the electron-phonon coupling constant λ. The analysis is based on a theory originally developed for obtaining λ for metal surfaces, but which here is adapted to the more complicated cases of layered chalcogenide semiconductors and systems that can be considered as having different dimensions such as 1D CDWs and quasicrystals. The original theory demonstrates that the argument of the Debye-Waller factor 2W (k f , k i , T ) is, to a good approximation, proportional to λ. The current analysis shows that, with suitable interpretation of the theory, values of λ can be obtained from the surfaces of these more complex systems. For all of these systems, the values of λ HAS obtained from atom-surface scattering experiments compare favorably with established values for the bulk materials as published in the literature.