Measuring the Electron–Phonon Interaction in Two-Dimensional Superconductors with He-Atom Scattering

Abstract

Helium-atom scattering (HAS) spectroscopy from conducting surfaces has been shown to provide direct information on the electron–phonon interaction, more specifically the mass-enhancement factor λ from the temperature dependence of the Debye–Waller exponent, and the mode-selected electron–phonon coupling constants λ_Qν from the inelastic HAS intensities from individual surface phonons. The recent applications of the method to superconducting ultra-thin films, quasi-1D high-index surfaces, and layered transition-metal and topological pnictogen chalcogenides are briefly reviewed.

1. Introduction

Helium-atom scattering (HAS) from a conducting surface can exchange energy and momentum with the lattice vibrations of the surface exclusively via the surface charge-density oscillations produced by the atomic motion, i.e., via electron–phonon (e–ph) interaction. Although this mechanism has been suggested since the early days of HAS spectroscopy [1], as a consequence of the discovery by HAS of the ubiquitous anomalous longitudinal surface resonance at metal surfaces [2,3,4,5], only more recent theoretical studies based on density functional perturbation theory (DFPT) [6,7] proved that the inelastic HAS intensities from surface phonons are directly proportional to their specific e–ph coupling constants λ_Qν, inaugurating what has been termed as mode-lambda spectroscopy.

An interesting aspect of the e–ph mechanism is that subsurface phonons that produce a modulation of the surface electron density can also be detected by HAS, the detection depth being equal to the range of the e–ph interaction. Thus, a surface probe such as He atoms with incident energies in the range of tens of meV that only tickle the surface where the electron density is ~10⁻⁴ a.u. [8], can even measure the dispersion curves of phonons propagating at several atomic planes beneath the surface (quantum sonar effect), e.g., at the interface of ultra-thin metal films with the substrate, provided the e–ph coupling is sufficiently strong [7,9].

A natural consequence of this mechanism is that the thermal mean-square distortion of the surface charge-density profile, providing the Debye–Waller attenuation of the specular intensity with increasing temperature, is proportional to the mean-square phonon displacement via the total e–ph coupling, represented by the mass-enhancement factor λ. This permits the direct derivation of λ from the temperature dependence of HAS reflectivity for any conducting surface, as demonstrated in a recent series of papers devoted to metal surfaces [10], ultra-thin metal films [11], layered transition-metal chalcogenides [12,13], topological semimetals [14,15] and one-dimensional metals [16], multidimensional materials [17] and graphene [18]. The very high surface sensitivity of HAS for surface dynamic corrugations of the order of 10⁻² a.u. also permits the detection of surface charge-density waves, undetectable by other current surface probes, together with the associated charge density wave (CDW) excitations like phasons and amplitons [14,19,20].

This ability of HAS spectroscopy, including its comparatively high resolution in energy and parallel momentum (see Chap. 9 of Ref. [1]), makes it a choice tool to investigate various aspects of the e–ph interaction in two-dimensional (2D) superconductors. After a theoretical summary (Section 2), with the relevant equations whose detailed derivation is found in refs. [7,10,11,15,17], a few examples for different classes of 2D superconducting materials are discussed in the following sections.

2. The Electron–Phonon Theory of He-Atom Scattering from Conducting Surfaces

The repulsive part of the interaction between a He atom and a conducting surface is described to a good approximation by the Esbjerg–Nørskov (EN) potential [21]

$$V(r)=An(r),$$

(1)

where n(r) is the surface electron density, A = 364 eV a₀³ the EN constant, and a₀ the Bohr’s radius [22]. Consider in Figure 1 an inelastic scattering process of a He atom from an initial state |i> of energy E_i and wave vector k_i = (K_i, k_iz) into a final state <f| of energy E_f and wave vector k_f = (K_f, k_fz), where K_i and K_f are the components parallel to the surface and k_iz, k_fz the respective normal components. The inelastic scattering probability P(k_i, k_f) for one–phonon creation processes in the standard distorted wave Born approximation takes the form (up to a constant factor) [23]:

$$P\left({\mathbf{k}}_i,{\mathbf{k}}_f\right)\propto f\left(\Delta E\right)N\left(E_F\right){\sum }_{\mathbf{Q}\nu } {\lambda }_{\mathbf{Q}\nu }\delta \left(\Delta E-{\epsilon }_{\mathbf{Q}\nu }\right),$$

(2)

where N(E_F) is the electron density of states (DOS) at the Fermi level, ΔE = E_f − E_i the energy gain, ε_Qν the phonon energy of wave vector Q = K_f − K_i and branch index ν, and λ_Qν the mode-selected e–ph coupling constants (mode lambdas) [24,25]. The coefficient

$$f\left(\Delta E\right)\equiv \frac{k_f}{\left|k_{iz}\right|} \Delta E\left[1+n_{BE}\left(\Delta E\right)\right]A^2{|⟨f|{\psi }_{Kn}^{*}\left(r\right){\psi }_{K+Q{n}^{\prime }}\left(r\right)|i⟩|}^2 ,$$

(3)

with n_BE(ΔE) the Bose-Einstein phonon occupation number, is proportional to the square matrix element between final and initial He-atom wave functions of the electron density matrix, connecting two Fermi level electronic states of bands n and n’ and wave vectors K and K + Q, respectively. Please note that the He-atom wave-functions decay very rapidly as they penetrate into the surface electron density near the classical turning point, far away from the first surface atomic plane, where the wave function ψ_Kn(r) decays asymptotically as e^−κz. In the Wentzel-Kramers-Brillouin (WKB) approxi-mation, κ = (2m*ϕ)^1/2ħ where ϕ is the surface work function and m* the effective mass of the surface free electrons at the Fermi level. Thus, the square matrix element in Equation (3) can be taken as independent of the electronic band indices and of ΔE, the latter being much smaller than ϕ. Since at sufficiently high temperature ΔE[1 + n_BE(ΔE)] ≅ k_BT, the coefficient f(ΔE) is also approximately constant, and the inelastic HAS intensity for a phonon (Q, ν) is just proportional to the mode-selected e–ph coupling constant λ_Qν. This conclusion enables inelastic HAS spectra to indicate which phonons are strongly coupled to electrons, and which are less so. As compared to other methods which permit an extraction of the Eliashberg function, the information obtained from HAS spectroscopy is particularly relevant for 2D Bardeen-Cooper-Schrieffer superconductors for providing, besides the frequency, the parallel momentum Q and branch index ν of phonons that are mostly involved in pairing.

Another important piece of information is obtained from the temperature dependence of HAS reflectivity. Reflectivity, expressed by the ratio I(T)/I_i of the specular peak intensity to the intensity of the incident beam, is attenuated with increasing surface temperature T due to lattice vibrations that modulate the profile of the surface electron density. At conducting surfaces this modulation also contains the contribution of thermal elementary excitations of the electronic Fermi sea, which is however negligible at the usual surface temperatures of HAS experiments. Thus, only phonon excitations, in a first approximation, are responsible for reflectivity attenuation, and this occurs via e–ph interaction. The attenuation of the specular peak intensity is expressed by the Debye–Waller (DW) factor

$$I(T)=I_0{e}^{-2W\left(k_f,k_i;T\right)},$$

(4)

where I₀ is the rigid surface intensity, and the so-called DW exponent for HAS from insulators, as well as for other probes directly scattered by the atom cores like X-rays and neutrons, is simply related to the mean-square displacement of surface atoms by

$$2W\left(k_f,k_i;T\right)=-\ln [I(T)/I_0]= <{[(k_f-k_i)\cdot u]}^2{>}_T.$$

(5)

For a conducting surface the vibrational displacement u must be replaced by the corresponding modulation of the surface profile at the He-surface classical turning point, proportional to the mean-square phonon displacement. Since the latter includes the contribution of all phonons weighted by the respective mode lambdas, it was shown that at sufficiently high temperature, where <u²>_T grows linearly with T [10,11] according to

$$2W(k_f,k_i,T)\cong N(E_F)\frac{{ħ}^2{(\Delta k_z)}^2}{2m^{*}\varphi }{k}_{B}T \lambda .$$

(6)

For a two-dimensional electron gas (2DEG) the Fermi level DOS referred to the surface unit cell of area a_c, is given by N(E_F) = m*a_c/πħ². A multilayer system as considered in the following example is conveniently viewed as a stack of n_s 2DEGs, provided n_s does not exceed the number n_sat of layers encompassed by the e–ph interaction range. With these definitions, the mass-enhancement factor λ (also denoted λ_HAS when obtained from HAS DW data) can be written as

$$\lambda =\frac{\pi }{2n_s}\alpha , \alpha \equiv -\frac{4\varphi }{a_c{(\Delta k_z)}^2}\frac{\partial \ln I(T)}{ k_B\partial T}, n_s\le n_{sat}.$$

(7)

3. Mode-Selected Electron–Phonon Coupling from HAS: Lead Ultra-thin Films

The early HAS studies of surface phonons in Pb(111) ultra-thin films (3–8, 10 and 50 monolayers (ML)) grown on a Cu(111) substrate [26,27,28] showed something surprising: the number of dispersion curves detected by HAS increases with the number of layers, well beyond the few dispersion curves observed in the semi-infinite crystal. Due to the large acoustic mismatch between Pb and Cu (acoustic impedance ratio Z_Pb/Z_Cu = 0.56), the observed phonons are confined within the film. Actually the number of modes resolved by HAS reaches a maximum at about 8 ML, then it decreases to the four typical surface phonon branches of the face-centered-cubic (111) metal surface localized on the surface atomic bilayer, with prevalent shear-vertical (SV) or longitudinal (L) polarization. This suggests n_sat ≅ 8 for Pb(111). Figure 2a,b illustrates the HAS dispersion curves for 5 ML- and 7 ML-Pb(111)/Cu(111) (filled circles) as compared to the theoretical dispersion curves calculated with DFPT calculations [7]: substantially all branches with prevalent SV polarization are detected. The calculated eigenvectors indicate that the highest mode (ε₂) is localized on the surface bilayer, whose interlayer distance is quite contracted, whereas the second highest mode (ε₁) is localized near the film–substrate interface, a few layers beneath the surface. This offers a clear demonstration of the quantum sonar effect.

The inelastic HAS spectra measured in the 90° planar scattering geometry along the scan curves at incident angles of 37° and 39.5° for 5 ML (Figure 2a), and 35.5° and 37.5° for 7 ML (Figure 2b) are displayed in Figure 2c,e, respectively. The spectra of the mode-lambdas λ_Qν calculated with DFPT for all the theoretical quasi-SV modes intersected by the scan curves (Figure 2d,f) compare very well with the experimental inelastic spectra, thus demonstrating the approximate proportionality predicted by Equation (2).

The calculated λ_Qν for the different modes (ν = α₁, α₂, …, ε₁, ε₂) are plotted as functions of Q in Figure 3 for both 5 ML and 7 ML. It is rather instructive that the maxima of λ_Qν occur for the Q = 0 acoustic modes α₁. These modes have a small but finite frequency at Q = 0 (Figure 2a,b), because of the film–substrate interaction, and consist of an almost rigid translation of the whole film against the substrate. Since the wave penetration into the substrate is quite small, due to the large acoustic mismatch, the displacement field gradient, measuring the local stress, is concentrated at the interface, and therefore a large contribution to the e–ph coupling comes in this case from the interface electrons.

Measurements of the temperature-dependent DW exponent during the layer-by-layer deposition of Pb films on Cu(111) permit the obtaining of λ as a function of the number of layers via Equation (7) (Figure 4a) [27]. Please note that due to oscillating surface relaxation where the interlayer distance oscillates so as to form a sequence of atomic bilayers near the surface, the reflectivity peaks actually occur at n = 1 (wetting layer), 3, 4.5, 6.5, 8…ML. As appears in Figure 4a, λ < 0.8 for a single Pb wetting layer and for 3 ML, then rises to values above 1 for n ≥ 4 ML, exhibiting quantum-size oscillations [26]. This behavior corresponds quite well to what has been reported from scanning-tunneling microscopy/spectroscopy (STM/STS) for the superconducting T_c of Pb ultra-thin films grown on semiconductor surfaces (Figure 4b) [29,30,31,32,33,34].

For 3 to 8 ML (Figure 4b) the HAS data reproduce quite well the results of DFPT calculations [7]. For thicknesses of 15 to 24 ML, the HAS data in Figure 4a show quantum-size oscillations that look different from those of the corresponding angle-resolved photoemission spectroscopy (ARPES) data [35]. It should be noted that ARPES values for λ are derived from the deformation of an electronic band induced by the e–ph coupling near the Fermi surface, and therefore refer to that specific electronic band, whereas the HAS values from the DW exponent are averages over all possible virtual phonon-induced electronic transitions around the Fermi level, as considered in the definition of λ.

4. A Quasi-1D Metal: Bi(114)

Although bulk crystalline bismuth at normal pressure is not superconducting above 0.53 mK [36,37], nanostructured [38] and amorphous bismuth [37] are reported to become superconductors with critical temperatures above 1 K. It is, therefore, interesting to investigate the e–ph interaction in Bi nanostructures by means of HAS. Of particular interest is the (114) surface of bismuth (Figure 5a). Bi(114) exhibits, in its stable reconstructed (1 × 2) form, resulting from a missing-row transition, the features of a one-dimensional (1D) topological metal consisting of conducting atomic rows 28.4 Å apart [15,38]. Below a critical temperature of ≈240 K, a CDW arises along the rows with a critical exponent of 1/3 due to a Peierls dimerization along the chains [16], leading to a (2 × 2) reconstruction with an oblique unit cell in real (Figure 5a) and reciprocal space (Figure 5b). The electronic structure in the (1 × 2) phase, calculated by Wells et al. [39], is characterized at the Fermi level by carrier pockets at the zone center $\overline{\Gamma }$ and the zone-boundary points $\overline{\mathrm{{\rm X}}}$ (Figure 5c). The $\overline{\Gamma }$-$\overline{\mathrm{{\rm X}}}$ intervalley e–ph coupling explains the (2 × 2) reconstruction.

The HAS diffraction spectra of Bi(114) along the row direction, reported in [15,16], show the temperature evolution of the CDW peaks, the most prominent being that at (±3/2,0); an example, measured at 113 K, is shown in Figure 5d. The best fit of the order parameter, proportional to ${[I_{\mathrm{CDW}}\left(T\right)/I_{\mathrm{CDW}}\left(0\right)]}^{1/2}$ for the CDW peak (inset of Figure 5d), yields a CDW critical temperature of 242 ± 7 K and a critical exponent β = 0.32 ± 0.03, consistent with the universal exponent of 1/3 as predicted in the presence of fluctuations [40,41], and in agreement with the critical exponent for other CDWs in layered chalcogenides and quasi-1D systems [17,42,43,44,45].

The e–ph coupling constant λ can be derived from the temperature dependence of the DW exponent, Equation (7), with the experimental parameters taken from Ref. [16]: a_c = 129 Ų, (Δk_z)² = 54.24 Å⁻², ϕ = 4.23 eV (for Bi), −∂lnI(T)/k_B∂T = 104.3 eV⁻¹ and n_s = 1 (assuming that only the surface atomic rows contribute). This gives λ = 0.40, to be compared with the HAS value 0.57 for Bi(111) [10]. A slightly larger value for Bi(114), λ = 0.45, is obtained when treated as a 1D free-electron gas system, as discussed in Ref. [17]. This value is probably more reliable, in consideration of the quasi-1D metallic nature of Bi(114)(1 × 2), although the difference is well within the uncertainty of the present method.

5. Superconducting Layered Chalcogenides

The large family of layered chalcogenides includes several superconductors. Although their superconductivity generally occurs at fairly low temperatures and under special conditions (proximity, pressure, high doping, low dimensionality), their great interest is related to the frequent coexistence/competition with CDWs and the interplay of electron-electron and e–ph interactions.

The high sensitivity and resolution of HAS permit detection of CDWs where other conventional surface probes fail, and measurement of the dispersion curves of their collective excitations [14,16,19]. The additional possibility to directly estimate the e–ph interaction makes HAS an excellent tool to investigate superconducting (SC) layered chalcogenides.

5.1. 1T-TaS₂(001)

Tantalum di-chalcogenides as superconductors exemplify the special conditions mentioned above. The SC critical temperature of 1T-TaS₂(001), originally reported by Wilson et al. to be about 0.5 K [46], was later determined to be about 1.5 K at normal pressure [47]. Pressure was shown by Sipos et al. [48] to produce a large increase of T_c up to 5 K at 5 GPa and then to keep it about constant at larger values up to 25 GPa. A raise of T_c can also be obtained, as recently demonstrated [49,50], by reducing the sample thickness to a few layers, and even in a monolayer an increase of T_c up to 3.61 K is induced by the formation of structural defects [51]. An alternate stacking of the 1T and 2H layers of TaS₂ to form the so-called 4Hb polytype was also shown to raise T_c to 2.7 K [52]. The coexistence of the CDW with superconductivity in 1T-TaS₂ has been investigated by Sipos et al. [48] and more recently by Ritschel et al. [53]. Wagner et al. [54] reported on the interesting observation that a Cu doping of 2H-TaS₂ raises the SC T_c from 0.5 K of the undoped material to a maximum of 4.5 K for a 4% doping, while dramatically reducing the CDW coherence length by a factor ten.

The complex CDW structure of 1T-TaS₂, as described by Wilson et al. [46,55], Coleman et al. [56], and more recently by Yu et al. [57], has been studied with HAS by Cantini et al. [58] and by Heimlich et al. [59,60]. The slopes of the HAS specular intensity as a function of temperature, shown in Figure 6a [59] for increasing temperature in the low-T commensurate CDW (C-CDW), intermediate-T nearly commensurate CDW (NC-CDW) and high-T incommensurate CDW (I-CDW) regions, permit to obtain from Equation 7 the respective e–ph coupling constant. With the input data k_iz² = 14.58 Å⁻² [60], ϕ = 5.2 eV [61], a_c = 9.43 Ų [62] and n_s = 2, from a Thomas-Fermi screening length not exceeding 1 nm [57], the observed specular intensity slopes give λ(C-CDW) = 0.61 ± 0.06, λ(NC-CDW) = 0.91 ± 0.09, and λ(I-CDW) = 0.61 ± 0.10 for the three phases. The data for decreasing temperature, presented in [52], give for the I-CDW and NC-CDW comparable values within the experimental error. For the low temperature C-CDW phase, the McMillan formula yields, with μ^∗ = 0.20 [63] and a Debye temperature Θ_D of 249 K [46,64,65], a superconducting critical temperature T_c = 1.2 K, in fair agreement with the value at normal pressure of 1.5 K reported by Nagata et al. [47].

For the high-pressure (5 GPa) NC-CDW phase, Rossnagel derives from the McMillan formula and the input data T_c = 5 K, μ^∗ = 0.20 and Θ_D = 249 K, an e–ph interaction constant λ = 0.85 [63], also in agreement with the present value of λ for the NC-CDW phase, although HAS experiments are performed in vacuum conditions. Density functional theory calculations by Amy Liu [66] also predict a dramatic effect of pressure, with λ varying from 2.09 at 5 GPa (T_c = 16 K) to 0.69 at 30 GPa (T_c = 5.9 K). Please note that the Debye temperature, here corresponding to a phonon energy of 21.6 meV, is adopted in the McMillan formula as an average energy of phonons contributing to λ, and may have little to do with that derived from the specific heat, according to Debye theory. Moreover, this theory has been conceived for monoatomic crystals with only three acoustic phonon branches, whereas for the additional flat optical branches of polyatomic crystals the Einstein model would be more appropriate. The simple fit of the Debye theory to polyatomic crystals with high-energy optical phonon branches leads to an important temperature dependence of Θ_D. Since in general T_c << Θ_D, it is implicit that Θ_D should be conveniently taken in its low-T limit. In the present three-atomic case, the Debye phonon energy ε_D = k_BΘ_D = 21.6 meV falls just above the acoustic region but well below the optical phonon region (27–40 meV), according to pseudo-charge model calculations for the C-CDW phase in its (√13 × √13)R13.9° superstructure (1T-TaS₂) [67]. This is indicative of a relevant contribution of acoustic phonons to the e–ph interaction, consistently with the fact that HAS measurements of the surface phonon dispersion curves of 1T-TaS₂ show the Rayleigh wave branch with several umklapp replicas corresponding to CDW reciprocal lattice vectors [68].

5.2. 2H-TaSe₂(0001)

HAS reflectivity data for 2H-TaSe₂ (Figure 6b) show a decrease for increasing temperature similar to that of 1T-TaS₂, with imperceptible discontinuities at the CDW transition temperatures. The HAS intensities of the CDW diffraction peaks [59,60] show instead clear features at the transition temperatures, from the C-CDW to the striped-incommensurate (SI-CDW) phase (92 K), from the SI-CDW to the I-CDW phase (113 K) and from the I-CDW to the normal phase (122 K), in agreement with previous McWhan et al. X-ray data [69]. Similar agreement is found for the transition temperatures during cooling, with hysteresis at the I-CDW → C-CDW transition (88 K). The decay slopes of the specular intensity for increasing temperature (Figure 6b) in the C-CDW linear region (70–90 K) and in the (SI + I) – CDW region (90–122 K) are about the same, within the experimental uncertainty. With the input data k_iz² = 68.45 Å⁻² [59,60], ϕ = 5.5 eV [70], a_c = 10.22 Ų [71] and n_s = 2 (from a spherically averaged coherence length of ≈14 Å [72], vs. the unit cell thickness including two Ta layers of 12.70 Å), a single value for λ = 0.57 ± 0.08 is obtained. The data for decreasing temperature displayed in [59] show a smaller slope in the I-CDW region (122–88 K), giving λ = 0.30 ± 0.06, but approximately the same slope in the C-CDW region (<88 K). The factor of two difference in λ found in the incommensurate phase between heating and cooling data may reflect the structural dependence on the thermal history, as suggested by the fact that the SI phase is observed when heating and not when cooling the sample. In any case the superconducting temperature resulting from McMillan’s formula, with a Debye temperature of 140 K [73], μ^∗ = 0.20, and λ = 0.57 for the C-CDW phase is equal to 410 mK. The C-CDW values of λ and the corresponding T_c are larger than those found in the literature, e.g., λ = 0.49 reported by Rossnagel [64], 0.39 by Bhoi et al. [73], 0.4 by Luo et al. [74], and correspondingly rather small values of T_c, such as 200 mK by Kumakura et al. [75] and 130 mK by Yokota et al. [76].

As in the case of 1T-TaS₂ the interplay between CDW and superconductivity is an interesting issue, as discussed in recent works by Lian et al. [77], Wu et al. [78], Liu et al. (for 1T-TaSe_2−xTe_x) [79], etc. Similarly, a large rise of T_c up to above 2 K has been demonstrated as an effect of doping [80] or ion gating (intercalation) [78], as well as of film thickness reduction down to 1 ML. For example, Lian et al. have shown that a value of λ = 0.4, valid for bulk 2H-TaSe₂ and giving a T_c of 100 mK, rises to λ = 0.74 in a single monolayer, with a T_c = 2.2 K [78]. Thus, the larger values of λ observed with HAS clearly refer to the surface, more precisely to the surface bilayer (n_s = 2).

The conjecture, originally formulated for graphite, that in layered crystals with weak interlayer van der Waals forces the surface phonon dispersion curves should be almost coincident with those of the bulk, proved invalid for layered chalcogenides (Figure 7) [1]. A specific role of e–ph interaction at the surface of 2H-TaSe₂, and in general of transition-metal chalcogenides, has been surmised since the early HAS measurements of the surface phonon dispersion curves [43,81,82]. An intriguing HAS observation in 2H-TaSe₂ was a temperature-dependent Kohn anomaly in the Rayleigh wave (RW ≡ S₁) dispersion curve at about ½$\overline{\Gamma }\overline{\mathrm{{\rm M}}}$ [82], whereas the bulk longitudinal acoustic (Σ₁) mode, as measured with inelastic neutron spectroscopy [70], shows a T-dependent anomaly at ²/₃$\overline{\Gamma }\overline{\mathrm{{\rm M}}}$(Figure 7a). Moreover, the maximum softening observed in the bulk Σ₁ mode occurs at the normal-to-CDW transition (≈120 K), whereas the deepest S₁ Kohn anomaly occurs at ≈110 K. Actually the HAS intensities of the CDW diffraction peaks [43] exhibit hysteresis loops around the I↔SI transition temperature (113 K), which would associate in some way the SI phase to the surface region, within the penetration depth of the RW at ½$\overline{\Gamma }\overline{\mathrm{{\rm M}}}$. As mentioned above, HAS data on the temperature dependence of the CDW diffraction peaks also provided the order-parameter critical exponent for the CDW transition at 122 K equal to the universal β = 1/3 [42], as generally found for CDW transitions driven by e–ph interaction in this class of materials.

The shift of the surface phonon anomaly to a wave vector different from that for the bulk phonon anomaly may be ascribed to a different nesting of the Fermi level surface electronic states with respect to that for the bulk states. A similar behavior is found with HAS in 2H-NbSe₂ (Figure 7b), where the dip in the dispersion curve of the bulk Σ₁ phonon also falls at ≈0.76 Å⁻¹ while the dips of both S₁ and S₃ surface phonons branches, as measured by HAS, are shifted to a smaller wave vector, ≈0.65 Å⁻¹. It is unfortunate that this HAS data, originally measured in 1986 by Brusdeylins and Toennies and reproduced in [83], have not been extended to lower temperatures, where, as found with inelastic X-ray spectroscopy (IXS) [84,85], the S₁ frequency at ≈0.65 Å⁻¹ vanishes at about 33 K. The Weber et al. analysis based on state-of-the-art ab-initio calculations indicate the wave vector dependence of the e–ph coupling as the driving mechanism for CDW formation in 2H-NbSe₂ [85]. Considering the superior resolution of inelastic HAS spectroscopy in the meV range with respect to IXS, and its ability to obtain direct information on e–ph coupling constants, revisiting with HAS the CDW phases and transitions in this class of 2D superconductors would be highly desirable. Recent examples in this direction are the HAS measurements of λ in 2H-MoS₂ [12], 1T-PtTe₂ [13] and 1T-PdTe₂ [86].

5.3. Pnictogen Chalcogenides

Layered pnictogen chalcogenides, with a bulk electronic band gap and a cone of topological Dirac states localized at the surface and crossing the Fermi level, are topological insulators (TI) unsuitable to be superconductors at any temperature. They can however become so by intercalating donor atoms [87,88,89], doping [90], by interfacing with conventional superconductors via the proximity effect [91,92,93,94,95,96], or under pressure [97,98,99]. Although critical temperatures of induced superconductivity are in the best cases in the range of a few Kelvin, the involvement of surface topologically protected states confers a particular stability against external disturbances. This would make pnictogen chalcogenides and their heterostructures particularly suitable for device applications. Moreover, TI-superconductor heterostructures have provided a quantum playground for some fundamental advances, e.g., the evidence of Majorana fermions in condensed matter with promising applications to quantum computing (for brief accounts see, e.g., [100,101,102]).

A successful option was looking for Majorana fermions in the vortices of a 2D chiral topological superconductor [103,104,105,106,107,108,109]. Xu et al. [110], were able to detect with STM/STS a Majorana mode in a heterostructure made of the topological insulator Bi₂Te₃ and the superconductor NbSe₂. The spin-selected Andreev reflection effect [111] was exploited by Sun et al. [112] in a similar 2D heterostructure where five quintuple layers of the topological insulator Bi₂Se₃ are interfaced to a NbSe₂ superconductor, which enabled unveiling Majorana zero modes by probing the vortex core states with spin-polarized-STM/STS. An important issue is the coexistence of superconductivity with the topological order [113,114,115]. The recent HAS observation of diffraction features attributed to long-period CDWs in Bi₂Se₃ [19] raises another coexistence issue, as mentioned above for layered transition-metal chalcogenides, with the corresponding question about the origin and role of e–ph interaction in topological pnictogen chalcogenides.

HAS reflectivity measurements on Bi₂Se₃(111), Bi₂Te₃(111), Bi₂Te₂Se(111) [15,116,117] and Sb₂Te₃(111) [118] have permitted obtaining λ from the temperature dependence of the DW exponent and Equation (7) (λ_HAS in Figure 8b). Their respective values are 0.23, 0.19, 0.14 and 0.08, all with an experimental uncertainty of around 10%. The large range of values as compared to the similarity of these compounds can be understood by considering their electronic structure at the Fermi level, as resulting from the ARPES data for the bismuth chalcogenides used in HAS experiments [15,19,116] and from ab-initio calculations for Sb₂Te₃(111) [118] (Figure 8a,c). The substantial differences in the sets of topological states concurring to the e–ph interaction can qualitatively account for the values of λ and indicate which Fermi level transitions contribute most.

In Bi₂Se₃(111) the Fermi level cuts the Dirac cone arising from the Dirac point, as well as pairs of topological quantum-well states and the bottom of the conduction band; in Bi₂Te₃(111) the Dirac cone and apparently only the conduction-band bottom; in Bi₂Te₂Se(111) only the Dirac states, in Sb₂Te₃(111), where the Dirac point is above the Fermi level and there are altogether six hole pockets (three spin-up and three spin-down, as indicated by arrows in Figure 8c), there is an intervalley e–ph coupling, in addition to the contribution from Dirac holes. It appears that the Dirac states alone contribute little to the total e–ph interaction, and that larger contributions come from the surface quantum-well states and the bottom of the conduction band at the surface. The observation that λ_HAS increases when quantum-well and conduction-band states are involved agrees with the result of DFPT calculations of λ as a function of the Fermi level position recently reported by Heid et al. [119]. The above values of λ_HAS are also in agreement with the corresponding values from other sources: for Bi₂Se₃(111) λ = 0.17 [120], 0.25 [121], 0.26 [122]; for Bi₂Te₃(111) λ = 0.19 [120], for Bi₂Te₂Se(111) λ = 0.12 [15]. No other value of λ to compare with λ_HAS is available, to our knowledge, for Sb₂Te₃(111), though it is noted in [123] that in Sb₂Te₃ the e–ph interaction from ab-initio calculations is weaker than in Bi₂Te₃, as actually found here.

6. Conclusions

In this brief review HAS from conducting surfaces has been shown to provide quantitative information on the electron–phonon (e–ph) coupling constants in some classes of 2D superconductors. Although the temperature dependence of HAS reflectivity, expressed via the DW exponent, allows for the determination of the global e–ph interaction through the mass-enhancement factor λ, inelastic HAS intensities from single phonons turn out to be proportional to the mode-selected e–ph coupling constants λ_Qν (mode-lambda spectroscopy). The latter possibility has been illustrated by the study of Pb ultra-thin films of 5 and 7 monolayers, while the derivation of λ from the temperature dependence of the DW exponent has been discussed for Pb films 1 to 24 ML thick, as well as for a high-index, quasi-1D bismuth surface and for superconducting layered chalcogenides. Concerning transition-metal dichalcogenides, an important aspect that HAS spectroscopy can help elucidate is whether and where the coexistence of CDWs and superconductivity is competitive or cooperative. Pnictogen chalcogenides, where topological superconductivity can be induced by pressure, intercalation, or proximity to superconductors, offer possible scenarios of coexistence with CDWs, after the recent observation of CDWs in some of these materials. The values of λ obtained by HAS in n-type Bi chalcogenides permit the appreciation of the role of surface quantum well and conduction-band states with respect to that of Dirac states.

In view of these potentialities it would be desirable to investigate by HAS various classes of high-T_c superconductors, in particular cuprates. No HAS study addressing e–ph interaction has been carried out so far in these materials. E–ph interaction in perovskites has been shown in the 1980s by Heinz Bilz and his school to receive an important contribution from the non-linear polarizability of oxygen ions [124], and to be therefore responsible for ferroelectricity [125,126,127,128,129] and other structural phase transitions [130,131], with possible implications for high-T_c superconductivity in perovskites [131]. Given the layered structure of these materials and the good quality of their surfaces, high-resolution HAS studies of their complex phonon structure and of mode-selected e–ph coupling constants would certainly help to further advance the understanding of high-T_c superconductivity. The simultaneous information on dynamics and structure of the surface electron density that can be obtained with HAS would also be instrumental in investigating certain aspects of high-T_c materials, e.g., phase separation [132], heterogeneity [133,134] and the superstripes landscape [135,136,137] made of multiscale puddles of CDWs from 3 nm to hundreds of nm [138] controlled by doping and elastic strain [139]. The nanoscale CDW texturing has been observed in 1T-TiSe₂ [140] where by Ti self-doping the Fermi level is tuned near a Lifshitz transition in correlated multi-band systems as was predicted theoretically [141]. Moreover, it has been confirmed that the CDW texture in Sr doped Bi₂Se₃ [142] can be manipulated by uniaxial strain to control superconductivity.

Inelastic HAS spectroscopy, besides measuring single surface phonon frequencies and the related λ_Qν, allows for the observation of low-energy collective electronic excitations like phasons, amplitons and acoustic surface plasmons (ASP) [1,19,20,143] due to the direct “mechanical” interaction (Pauli repulsion) of He atoms with the surface electron density. When the ASP dispersion curve (linear in the long-wavelength limit [144]) enters the surface-projected phonon density, there are avoided crossings with the surface phonon dispersion curves and a robust renormalization of ASP phase velocity, all driven by e–ph interaction. Actually, a strong ASP-phonon interaction is at the basis of a recent prediction by Shvonski et al. [145] of polaron plasmons as possible hybrid excitations at topological metal surfaces, reviving the old concept by Lemmens and Devreese [146] of collective excitations of the polaron gas. In view of the importance that has been attributed to polaron pairing (bipolarons) in the early days of high-T_c superconductivity [147,148,149,150], the strong e–ph coupling in cuprates has been confirmed experimentally by the anomalous doping dependent isotope effect: (i) on the superconductivity critical temperature [151], which shows the proximity of the chemical potential to a Lifshitz transition at 1/8 doping [152], (ii) on the polaronic CDW ordering temperature T* [153] and (iii) the kink in the electronic dispersion from angle-resolved photoemission spectra [154]. Therefore, the study of exotic collective polaron excitations in topological superconductors by HAS could have many surprises in store.