A convenient parameter to characterize the e-ph coupling strength is the mass-enhancement λ (Grimvall, 1981). Since the e-ph coupling describes the interaction between the electronic system and the lattice dynamics (phonons), experimental studies at finite temperatures can either concentrate on the electronic or the phononic system. The former is mostly available via ARPES by examining the renormalization of the electron energy dispersion in the vicinity of the Fermi surface as a result of e-ph interactions. The latter can be carried out using HAS, which on the contrary studies the renormalisation of the surface phonon dispersion due to e-ph interactions. As shown in recent works (Tamtögl et al., 2017; Benedek et al., 2018; Benedek et al., 2020c; Manson et al., 2022), the temperature dependence of the Debye-Waller (DW) exponent plotted in Figure 2 permits to extract, for a conducting surface, the mass-enhancement parameter λ. The Debye-Waller factor takes into account the thermal attenuation of the elastic helium intensity due to atomic motion. The attenuated intensity I(T _S ) with respect to I ₀, the intensity at rest (T _S = 0 K) in the absence of zero-point motion is given as I T S = I 0 e − 2 W k i , k f , T S , (1) with k _i and k _f being the wave vectors before and after scattering, respectively. Since a He beam with energies in the meV region is scattered on a conducting surface by the surface free electron density, the exchange of energy with the phonon gas occurs via the phonon-induced modulation of the surface electron gas, that is, via the e-ph interaction (Benedek et al., 2020c; Tamtögl et al., 2021a; Manson et al., 2022). Therefore, the DW-factor, originating from the integrated action of all phonons weighted by their respective Bose factors, turns out to be directly proportional, under reasonable approximations, to the mass-enhancement factor λ (Manson et al., 2022).

The relation between λ and the DW-exponent is given by the equations: λ = π 2 n s α , α ≡ ϕ A c k i z 2 ∂ ⁡ ln ⁡ I T S k B ∂ T S , (2) where ϕ is the work function, A _c the unit cell area, I(T _S ) the He-beam specular intensity, T _S the surface temperature, k _iz the normal component of the incident wavevector and n _s the number of conducting layers which contribute to the phonon-induced modulation of the surface charge density. The latter is estimated to be n _s = 2λ _TF/c ₀, where λ _TF is the Thomas–Fermi screening length characterizing the surface band-bending region, c ₀ the thickness of the cleaved layer, and the factor two considers two metallic sheets per layer.

The advantage of HAS is that it directly provides a surface sensitive measure of the mode-averaged λ. On the contrary, with ARPES a dependence of λ on the initial-state electron energy has been reported (Kumar et al., 2022) and it remains difficult to distinguish between surface and bulk states which is why often time-resolved methods are used (Sobota et al., 2023). Similarly, Raman measurements provide access to the e-ph interaction of optical modes, but again, the method is rather bulk sensitive (Shojaei et al., 2021).

Since thermal helium atoms have a wavelength comparable to inter-atomic distances the elastic scattering of a helium beam from a periodic surface gives rise to a diffraction pattern. The distances between the diffraction peaks can then be used to calculate the surface lattice constant and in a second step the in-plane linear thermal expansion coefficient. Figure 3 shows several recorded angular diffraction scans (ϑ-scans) obtained by rotating the sample in the scattering plane. Conversion to parallel momentum transfer ΔK follows from Δ K = | Δ K | = | K f − K i | = | k i | sin ϑ f − sin ϑ i , (3) with k _i being the incident wave vector and ϑ _i and ϑ _f the incident and final angles with respect to the surface normal, respectively. The scans in Figure 3 were taken along the high symmetry Γ M ̄ orientation keeping the surface at temperatures of T _S = 113 K (left panel) and 296 K (right panel), respectively. The intensity of each specular peak was used for normalization. All scans show clearly a very pronounced specular peak (ϑ _i = ϑ _f ) and two first order diffraction peaks.

The small side peak next to the specular peak for the scan with E _i = 10.4 meV occurs due to the presence of the atom-surface potential, which will be further described in Section 3.4. In addition to diffraction peaks, the angular scan at E _i = 12.6 meV and T _S = 113 K (red curve in the left panel of Figure 3) shows a very sharp satellite peak next to the (1,0)-diffraction peak at ΔK = 2.04 Å. At this temperature, the crystal is in the C-CDW phase, which is characterized by the star-of-David clusters covering the entire lattice forming a commensurate ( 13 × 13 ) R13.54° superstructure. Interestingly, while we performed diffraction scans at various nozzle temperatures, the satellite peak is only visible for E _i = 12.6 meV and secondly, we are able to resolve the peak even though it should be rotated with respect to the scanning direction as illustrated in Figure 1D. The latter could be due to a slight azimuthal misalignment of the sample, although we note that Brusdeylins et al. (1989) also observed the peaks arising from the corresponding superstructure, not only in the C-CDW phase but also in the NC-CDW phase at T _S = 340 K.

For TlBiTe₂ on the other hand, due to the difficulty associated with the cleavage and the reported likelihood of a patchy termination, we are only able to resolve the specular and a small diffraction peak on one side as shown in the inset of Figure 2B. In general, the intensity scattered into the specular direction is typically by a factor of 10 higher for 1T-TaS₂. As described above, this is sufficient for a determination of the e-ph coupling strength of TlBiTe₂ however, it does not permit us to perform any detailed analysis of the TlBiTe₂ surface structure and electronic corrugation.

Nevertheless, since a measurement of the angular spread in the specular peak provides an estimate of the surface quality (Farías and Rieder, 1998) and following the approach described in Tamtögl et al. (2017) we can use the measured full-width at half maximum (FWHM) of the specular peak to compare surface order and more specifically the terrace width for both samples.

The peak broadening is proportional to the average domain size, also known as the surface coherence length. As the measured specular width Δθ _exp is a convolution of the angular broadening of the apparatus Δθ _app and the broadening form the domain size Δθ _w , it follows: Δ θ e x p 2 = Δ θ w 2 + Δ θ app 2 . The coherence length can then be determined using: l c = 5.54 Δ θ w k i cos ϑ f (4) with k _i and ϑ _f as defined beforehand (Farías and Rieder, 1998; Tamtögl et al., 2017).

For 1T-TaS₂, the measured FWHM is typically about 0.04° in ϑ _i -scans, which corresponds to a terrace width of ≈ 500  Å. For TlBiTe₂ the FWHM is about 0.05 Å⁻¹ in momentum space (inset of Figure 2B) or 0.07° in ϑ _i giving rise to a terrace width of ≈ 200  Å. While STM studies report a Tl island size of the order of 1 − 2 nm and thus much smaller than our result, observed terraces in STM measurements exhibit also widths of about 20 nm (Kuroda et al., 2013; Pielmeier et al., 2015). As the absence of the Tl islands in room temperature STM measurements was attributed to increased mobility of the Tl atoms (Pielmeier et al., 2015), we note that addressing these details and differences would require additional measurements and a more in-depth study.

In summary, we conclude, that as already anticipated, TlBiTe₂ is “rougher” and exhibits smaller domain sizes compared to 1T-TaS₂. However, neither of the surface qualities is comparable to some of the binary TIs, where the angular broadening of the specular peak had been reported to be mainly limited by the angular broadening of the apparatus (Tamtögl et al., 2017; Ruckhofer et al., 2019).

Its unique surface sensitivity makes He diffraction an ideal method to determine the lattice constant of solely the surface layer without any contribution from the underlying layers. From our angular diffraction scans we can thus calculate the surface lattice constant of 1T-TaS₂ and in a second step the linear thermal expansion coefficient. Following the two-dimensional Laue condition for a hexagonal lattice, the surface lattice constant can be calculated. For T _S = 113 K, we recorded 14 angular scans in Γ M ̄ direction at different incident beam energies resulting in an average lattice constant of a = (3.35 ± 0.03) Å. Similarly, we used eight scans in Γ M ̄ direction for the lattice calculation at T _S = 296 K. From the position of the first order diffraction peaks with respect to the specular peak we thus obtain a surface lattice constant of a = (3.37 ± 0.03) Å at room temperature, which is an increase of roughly 0.6%.

Both values are in good agreement with other experimental studies (scanning tunneling microscopy, LEED, X-ray powder diffraction, surface-enhanced Raman spectroscopy (SERS), and theoretical studies using density functional theory) of the 1T-TaS₂ in-plane lattice parameter (Wilson et al., 1975; Givens and Fredericks, 1977; Sanders et al., 2016; Kratochvilova et al., 2017; Bao et al., 2022).

With these values we can further calculate the in-plane (linear) thermal expansion coefficient (TEC) defined as α _‖ = 1/a ₀ ⋅Δa/ΔT, where a ₀ is the lattice parameter at T _S = 113 K, Δa the difference between the lattice constant values at different temperatures and ΔT the temperature difference. We obtain a value of α ‖ = 33 ± 12 ⋅ 1 0 − 6 K − 1 .

Our value is larger compared to the value that Givens and Fredericks obtained in their X-ray diffraction (XRD) study. As we have only considered two different surface temperatures our result should be treated carefully even though it seems unlikely to be caused by such a large uncertainty and might well be a consequence of the different probing techniques. Table 1 provides a short overview of the thermal expansion coefficients of other transition metal dichalcogenides, which experimental technique had been used, and for which temperature range. Previous studies of TMdCs reported values for the TEC in the range of 5–25 ⋅ 10^–6 K⁻¹, thus, comparable but slightly smaller than our value (Anemone et al., 2022), which may point toward a particularly weak vdW coupling between the layers.

As every polarisable object in nature is subjected to the vdW force a fundamental understanding of the force is crucial for any quantitative description and theoretical treatment of molecular adsorption or surface reaction processes and is also of paramount importance for a better design and control of nanoscale devices. Beyond the applications of weak interactions in devices and materials, there are several experimental techniques analyzing nano-structured materials or explicitly using effects occurring from these weak interactions including atomic force microscopy and matter-wave scattering experiments which are probing dispersion forces (Fiedler et al., 2023).

However, in the case of topological nontrivial materials, HAS is the only experimental technique used to date to provide experimental information about the vdW interaction with a topological insulator (Tamtögl et al., 2021b). The latter stresses the need for experimental measurements even more so as, e.g., it is expected that peculiar effects such as the topological magnetoelectric effect (Dziom et al., 2017) cause an unconventional contribution to the vdW potential (Martín-Ruiz and Urrutia, 2018). Before we illustrate the experimental determination of the atom-surface vdW potential of 1T-TaS₂ we will first describe HAS measurements of the step-height distribution in the following.

The surfaces of layered materials such as 1T-TaS₂ are not perfectly flat, rather they are characterized by steps and parallel terraces, e.g., due to defects and the sample preparation process. The step heights and distributions of such periodically modulated surfaces can be calculated and resolved from interference effects in HAS (Farías and Rieder, 1998; Mayrhofer-Reinhartshuber et al., 2013a). As a result of the combination of the He beam being scattered from different terraces, constructive and destructive interference occurs. In a so-called drift measurement, the specular intensity is monitored while the kinetic energy of the incident He beam is changed. In the experiment this can be achieved by changing the nozzle temperature. By modulating the kinetic energy, the phase shift Δϕ for the He beam emerging from different terraces is varied. For the specific case of ϑ _i = ϑ _f (specular peak) the phase difference of two adjacent terrace levels is given by (Mayrhofer-Reinhartshuber et al., 2013a) φ k i = 2 h k i ⁡ cos ϑ i = h Δ k z , (5) where h is step height and Δk _z is the change of the wave vector component perpendicular to the surface. For constructive interference, the condition of in-phase scattering Δϕ = 2πn with an integer value of n must be fulfilled, whereas a half-integer value n leads to destructive interference due to anti-phase scattering. As a result, maxima and minima can be observed in the drift spectrum Figure 4A, where the normalized specular intensity is plotted against the incident wave vector k _i . The measured specular intensity as a function of incident wave vector k _i upon varying the nozzle temperature is plotted for the sample aligned along Γ M ̄ and the crystal being held at room temperature.

As the incident wave vector k _i is varied by changing the nozzle temperature T _N , the He beam intensity scales via I ∝ 1 / T N and the signal has to be corrected for this factor which has been done in Figure 4A. In addition, the intensity decreases also with increasing k _i according to the Debye-Waller attenuation (Ruckhofer et al., 2019). Such a drift spectrum can now be used to determine the step height of the current 1T-TaS₂ sample.

Following a simple theoretical model to describe the periodic intensity modulation which assumes a coherent overlap of plane waves, emerging from different terrace levels, the specular intensity can be calculated with (Tölkes et al., 1997; Mayrhofer-Reinhartshuber et al., 2013a) I k i = I 0 e − 2 W ∑ j = 0 ∞ a j e − i j φ k i 2 . (6)

In Equation 6, I ₀ is the intensity for a flat surface without steps, φ(k _i ) the phase shift (see Eq. 5) and a _j the visible fraction of the terrace level j. Eq. 6 has been fitted to the measured drift spectrum in the wave vector range 4.54 Å⁻¹ < k _i < 5.41 Å⁻¹ and the best fit result is shown in Figure 4 as a blue line with the inset indicating the distribution of the visible terrace fractions. We obtain a surface step height of h = (6.1 ± 0.1) Å, which is in excellent agreement with the distance between two adjacent TaS₂ layers (Bovet et al., 2003; Li et al., 2019; Kahraman et al., 2020).

As mentioned above, HAS provides direct experimental information about the vdW interaction between an atom and the surface and thus invaluable details about London dispersion forces of vdW layered materials. As a first step to determine the three-dimensional corrugated atom-surface interaction potential of 1T-TaS₂, we follow a further analysis of the drift spectrum. The simple model for interference effects from steps (see Eq. 6) neglects the occurrence of selective adsorption resonances (SARs). These phenomena occur when an incoming helium atom is bound temporarily on the surface due to the attraction of the He-surface interaction potential. A SAR follows from the kinetic condition that the energy of the incident energy E _i equals the kinetic energy of the atom on the surface plus the binding energy ϵ _n of the potential: E i = ℏ 2 k i 2 2 m = ℏ 2 2 m K i + G 2 + ϵ n K i , G . (7)

Such adsorption processes give rise to peaks and dips in the drift spectrum (Tamtögl et al., 2021b). Therefore, these peaks and dips can be used to obtain the bound state energies of the potential. As a first step we concentrate on the laterally averaged atom-surface potential. Each peak/dip at a specific incident wave vector k _i can be associated with a certain bound state energy ϵ _n and reciprocal lattice vector G pair which fulfills the condition given in Eq. (7). In the so-called free atom approximation, ϵ _n (K _i , G) is independent of K _i and G (Ruckhofer et al., 2019; Tamtögl et al., 2021b) and thus Figure 4B provides a direct measure of the bound state energies ϵ _n of the laterally averaged potential.

As a next step, we will then model the three-dimensional He-surface interaction potential, whereupon we use the corrugated Morse potential (CMP) due to its algebraic simplicity (Tamtögl et al., 2021b). It consists of two exponentials considering the repulsive and the attractive part of the potential as a function of the lateral position R on the surface and the distance z from the surface in the following form: V R , z = D 1 ν 0 e − 2 κ z − ξ R − 2 e − κ z . (8)

Here, κ is the stiffness and D the well depth of the potential, ξ(R) the corrugation function, which reflects the periodicity of the crystal surface, and ν ₀ is the surface average over e ^2κξ(R). The laterally averaged surface potential (i.e., without corrugation in the exponential) of Eq. 8 can be described analytically and the bound state energies are given by ϵ n = − D + ℏ ω n + 1 2 1 − n + 1 2 2 γ (9) with n being a positive integer, ω = κ 2 D / m the Debye-frequency and γ = 2D/ℏω. By fitting Eq. 9 to the ϵ _n -values obtained from the SAR analysis of the drift spectrum we can thus obtain the laterally averaged potential with the well depth and stiffness: D = 8.40 ± 0.15 m e V a n d κ = 0.45 ± 0.02 Å − 1 .

The resulting bound state energies and their uncertainties are displayed in Table 2 while in Figure 5A a plot of the laterally averaged potential is shown. The positions of the corresponding SARs in the drift spectrum are displayed as vertical lines in Figure 4B, where each color belongs to one specific bound state energy ϵ _n and the red dashed lines illustrate the threshold energies. The labels next to the lines denote the corresponding reciprocal lattice vector G. For completeness, it should be mentioned that two more analytical bound state energies ϵ ₇ and ϵ ₈ exist, but they are quite close to zero, that is, to the threshold condition and therefore not displayed in Figure 4B and Table 2. In Figure 5A, the horizontal colored lines represent the first seven analytically calculated bound state energies. We note that SAR measurements are typically obtained with a cooled sample since with increasing temperature inelastic channels give rise to a broadening of the linewidth and may also cause changes from maxima to minima and vice versa (Tamtögl et al., 2018; Tamtögl et al., 2021b). Here we use measurements taken at room temperature as the position of the resonances becomes clearer at that temperature. It may be a consequence of the different phases of the cooled sample and the increased likelihood of rest gas adsorption for the latter.

Our value of D is slightly smaller than the value D = 8.7 meV from Brusdeylins et al. whose value is based on intensity calculations (Brusdeylins et al., 1989) following the hard corrugated wall model together with the Beeby correction and thus does not directly consider scattering from a soft potential resulting in less accuracy compared to the elastic close-coupling method used in the previous works (Sanz and Miret-Artés, 2007; Mayrhofer-Reinhartshuber et al., 2013b). In general compared to other semiconductors, metals, and semimetals, 1T-TaS₂ exhibits one of the “deepest” atom-surface interaction potentials, only surpassed by the graphene/graphite potential (Benedek and Toennies, 2018). At the same time, the stiffness κ is quite close to the value found for the topological insulator Bi₂Se₃ (Ruckhofer et al., 2019; Tamtögl et al., 2021b). With the outermost layer for Bi₂Se₃ being Se, located in the same column of the periodic table just below sulphur, it seems reasonable that the He−TaS₂ potential exhibits a similar stiffness and supports a similar number of bound states.

Following the results of the laterally averaged atom-surface interaction potential we continue with the determination of the three-dimensional atom-surface interaction potential and the corresponding surface electronic corrugation (Tamtögl et al., 2021b; Allison et al., 2022). Following the CMP (Eq. 8), the corrugation ξ(R) for a hexagonal surface such as 1T-TaS₂, can be described by a two-parameter Fourier ansatz, ξ R = ξ x , y = ξ 0 cos 2 π a x − y 3 + cos 2 π a x + y 3 + cos 2 π a 2 y 3 + h . o . , (10) where x and y depict the coordinates in the surface plane and ξ ₀ determines the corrugation amplitude. The ansatz above is based on the sixfold symmetry of the topmost surface layer. As the He beam provides an average of the long-range order, we omit the star of David clusters in this description. The corrugation is then typically expressed in terms of the peak-to-peak value ξ _pp = max(ξ) − min(ξ). With the CMP (Eq. 8) we are able to calculate the diffraction intensities and compare those with the experimentally measured ones to obtain a value for the peak-to-peak corrugation ξ _pp.

A theoretical framework to describe the elastic scattering of He atoms from surfaces is the close-coupling (CC) formalism, which is more exact than the hard-wall approximation, but still computationally manageable (Sanz and Miret-Artés, 2007; Allison et al., 2022). The starting point here is the time-independent Schrödinger equation together with the CMP, which contains the Fourier series expansion (Eq. 10) in the exponentials, yielding a set of coupled differential equations for the diffracted waves. These equations are solved for using the CC algorithm, taking into account all open channels and 110 closed channels. The z − boundaries of the integration were set to [−6, 18]. A detailed theoretical treatment and the corresponding equations can be found in several references (Miret-Artés, 1995; Sanz and Miret-Artés, 2007; Mayrhofer-Reinhartshuber et al., 2013b; Kraus et al., 2015), while an implementation of the CC algorithm is available from https://repository.tugraz.at/records/cd0y0-xa478 under the GNU General Public License v3.0.

By comparing the elastic diffraction intensities calculated with the close-coupling algorithm we are able to determine the corrugation of the sample surface (Ruckhofer et al., 2019; Schmutzler et al., 2022). For the measured diffraction intensities we used the peak areas instead of the peak height to account for broadening due to the energy spread of the He atoms, angular resolution of the apparatus, and size effects of the crystal surface (Ruckhofer et al., 2019). For the CC calculations, the intensities have been corrected by the Debye-Waller factor, and the values for D and κ from above were kept constant, while the corrugation amplitude ξ ₀ (see Eq. 10) was varied between 0.05 and 0.90 Å⁻¹. The best corrugation value ξ ₀ is the one that minimises the deviation R between simulated and experimentally obtained diffraction intensities I G sim and I G exp given by R = 1 N ∑ G I G exp − I G sim 2 , (11) with N the number of experimentally measured diffraction scans. As for the calculation of the lattice constants we used in total 22 scans at various incident energies for the optimization process of Eq. 11 to find a global minimum of R as a function of ξ ₀. It results in a surface electronic peak-to-peak corrugation (see inset of Figure 5A) of ξ pp = 0.25 Å , which corresponds to 7.5% of the surface lattice constant a. Figure 5B shows a plot of the best-fit surface electronic corrugation according to Equation 10.

So far, there barely exist any other HAS studies of other TMdCs that determine the electronic corrugation. Anemone et al. studied the 1T-PtTe₂ surface and provided a rough estimation of the maximum corrugation amplitude with 0.33 Å (Anemone et al., 2020). It should be mentioned that their estimation is based on the HCW model, which makes a direct comparison difficult. Similarly, various low-index metal surfaces have been studied in atomic beam scattering experiments and revealed a significantly smaller corrugation (Benedek and Toennies, 2018). On the other hand, in comparison to the binary topological insulators and single-element semimetals, which have been studied recently with HAS (Tamtögl et al., 2021b) and analyzed in a similar manner, TaS₂ exhibits a similar surface electronic corrugation.