Strong polaronic effect in a superatomic two-dimensional semiconductor

Assembling nanoscale building blocks into functional materials has been a longstanding goal in nanoscience. Prominent examples include crystalline solids assembled from nanoparticles (NPs) via van der Waals interactions or DNA hybridization.^1,2 Both approaches are limited by intrinsic disorder resulting from variations in NP size, shape, surface chemistry, and inter-NP interaction and by weak inter-NP electronic coupling due to intervening capping ligands. The recent development of superatomic crystals overcomes both limitations as the building blocks are atomically precise clusters and inter-cluster interaction can be sufficiently strong from intimate van der Waals, electrostatic, or covalent interactions.^3–5 The replacement of atoms in conventional solids by superatomic units offers the enticing prospect of combining multifunctionality and tunability.

Here, we investigate the polaronic properties of a model superatomic semiconductor, Re₆Se₈Cl₂. In this solid, the [Re₆Se₈] clusters are covalently linked in-plane through Re–Se bonds to form two-dimensional (2D) sheets terminated with Cl atoms; the 2D sheets are stacked to form a layered van der Waals structure [Fig. 1(a)].⁶ This semiconductor has an indirect bandgap of 1.58 eV.⁷ Its hierarchical atomic structure translates into a hierarchical phonon landscape, including zero-dimensional (0D) cluster breathing modes, 2D inter-cluster twisting modes, and three-dimensional (3D) inter-sheet acoustic modes, with the 2D modes found to strongly couple to electronic transitions.⁸ Interestingly, a recent theoretical analysis⁹ of a related 3D cluster solid PbMo₆S₈ suggests that strong coupling of electrons to the inter-cluster twisting modes is critical to superconductivity,^10,11 and superconductivity was observed in electron-doped 2D layered Re₆Se₈Cl₂ crystals.¹² Besides superconductivity, electron–phonon coupling can affect conventional carrier properties via polaron formation. Density functional theory (DFT) calculation shows that the valence bandwidth of Re₆Se₈Cl₂ is 300–400 meV.⁷ This is smaller than those in typical 2D atomic semiconductors such as MoSe₂, making electron–phonon coupling more important to carrier properties in the superatomic crystal.

In this work, we apply angle-resolved photoemission spectroscopy (ARPES) to determine the valence band dispersion in Re₆Se₈Cl₂. ARPES is particularly powerful in probing electron–phonon coupling, as it directly measures the spectral function, which, in the commonly accepted “sudden approximation,” describes the quasiparticle of an electron interacting with phonons and other electrons.¹³ Our ARPES measurements show that the valence bandwidth (W) is a strong function of temperature; it is 120 ± 30 meV at 70 K and decreases by one order of magnitude to W ∼ 10 ± 20 meV as temperature is increased to 300 K. This band narrowing with increasing temperature is consistent with strong electron–phonon coupling, as described by the Holstein polaron model.¹⁴ For comparison, we show that the valence bandwidth in the traditional atomic 2D semiconductor, MoSe₂, is independent of temperature (100–300 K).

Re₆Se₈Cl₂ single crystals were synthesized by chemical vapor transport following reported procedures, as detailed elsewhere.⁷ Figure 1(b) shows an optical image of a cleaved single crystal. The nearly square but anisotropic structure in the ab plane is confirmed by indexing the crystal edges.⁷ Figure 1(c) illustrates the reciprocal lattice (blue) and the first Brillouin zone (BZ, black) in the a–b plane. We performed ARPES experiment using a setup that features an extreme ultraviolet (EUV) laser source (21.6 eV, 35 fs, 10 kHz) and a hemispherical analyzer with angular (θ) resolution.¹⁵ We carried out ARPES measurement along the Γ–Y direction [Fig. 1(d)]. The kinetic energy of photoelectrons as a function of the parallel momentum k_// directly probes the hole dispersion in the valence band.

Figure 2(a) shows room temperature ARPES of Re₆Se₈Cl₂ along the Γ–Y direction; the corresponding energy distribution curve (EDC) at the Γ point is shown in Fig. 2(b). The observations of the valence band edge (VBE) and midgap defect states are in agreement with previous photoemission and scanning tunneling spectroscopy measurements.⁷ The large number (16) of atoms in the unit cell of the Re₆Se₈Cl₂ superatomic solid results in a dense manifold of valence bands.⁷ In contrast to atomic solids such as MoSe₂ (see Fig. 3), the individual valence bands of Re₆Se₈Cl₂ are not resolved. The most surprising and important finding from Fig. 2(a) is that the experimental dispersion over the momentum range (Γ → Y) is nearly flat (±20 meV). For comparison, DFT calculation for a rigid lattice gives a valence band dispersion of ∼300 meV in the Γ → Y direction.⁷ 

The valence band dispersion is a strong function of temperature. Figure 2(c) shows the ARPES of Re₆Se₈Cl₂ along the Γ–Y direction near the VBE at three temperatures: T = 70 K (bottom), 108 K (middle), and 300 K (top). It is obvious that the valence band dispersion increases as temperature decreases. If we approximate Re₆Se₈Cl₂ in the ab plane as consisting of a square lattice and use the tight-binding model, we can describe the band dispersion E in the Γ–Y direction as E = E₀ + W/2 · cos(k_∥ · b) at k_⊥ = 0; E₀ is a constant offset, and b is the lattice constant. Fits to this approximation [red dashed curves in Fig. 2(c)] yield valence bandwidths of W = 120 ± 30 meV, 80 ± 25 meV, and 10 ± 20 meV at T = 70 K, 108 K, and 300 K, respectively. Similar results on the temperature dependent valence bandwidth can be obtained by comparing the VBE at the Γ point (k_// = 0 Å⁻¹) and the Y point (k_// = 0.6 Å⁻¹), as shown by the EDCs in Fig. 2(d).

The observed band narrowing with increasing temperature is a signature of strong electron–phonon coupling, as is implicit in the Holstein polaron theory.¹⁴ The nearly flat valence band at 300 K suggests that the photo-hole forms a small polaron. Theoretical analysis of the experimentally observed band narrowing with temperature in molecular crystals^16–18 has been described in the Holstein–Peierls model.^19,20 In the simple case when one considers local electron–phonon coupling (Holstein model) and assumes no energy or polarization dispersion of phonon modes, W is given by¹⁹ 

where N_λ = [exp(⁠$ℏ$ω_λ/k_BT) − 1]⁻¹ is the phonon occupation number for phonon mode λ, $ℏ$ω_λ is the phonon energy, g_λmm and g_λnn are the electron-lattice coupling constants for on-site and nearest neighbor, respectively, and W₀ is the asymptotic bandwidth at 0 K. Lacking a detailed knowledge of the coupling of each individual phonon to the hole, we fit the temperature dependent bandwidth using the effective phonon occupation number (N_eff) and coupling strength (g_eff),

with N_eff = [exp(⁠$ℏ$ω_eff/k_BT) − 1]⁻¹ given by the effective phonon frequency ω_eff. A recent study based on coherent phonon spectroscopy has suggested that the coupling of electrons to the inter-cluster twisting mode at $ℏ$ω = 10.8 meV is particularly strong, while those to other modes are much weaker,⁸ and we take this as the effective phonon mode. By fitting the experimental data in Fig. 3(a) (solid circles) to Eq. (2), we obtain W₀ = 280 ± 50 meV and g_eff = 1.2 ± 0.2. This asymptotic W₀ value is in excellent agreement with the DFT valence bandwidth for a rigid Re₆Se₈Cl₂ crystal without electron–phonon coupling.⁷ 

The decrease in electronic bandwidth with temperature is related to the well-known phenomenon of decrease in semiconductor bandgap, and both can be traced to the decrease in exchange integral. The decrease in semiconductor bandgap with temperature has been described by the semi-empirical Varshni equation^20,21 and attributed also to electron–phonon coupling.²² Note that thermal expansion increases the bond length and decreases the exchange integral, but this effect is much smaller than that of electron–phonon coupling. The thermal expansion coefficient of Re₆Se₈Cl₂ is not known but is close to ∼9 × 10⁻⁶ K⁻¹, which is related to the super-atom solid of PbMo₆S₈.²³ This corresponds to a thermal expansion of only ∼0.2% in the temperature range of 70–300 K. Note that the Re₆Se₈Cl₂ crystal remains in the same phase in this temperature range and there are no structural changes.^12,23,24

To put the observed band narrowing of Re₆Se₈Cl₂ in perspective, we compare these results in Fig. 3(a) to the temperature dependent bandwidths in a monolayer atomic solid, MoSe₂, and a layered van der Waals molecular semiconductor, pentacene. The two MoSe₂ data points (open circles) are from ARPES spectra [Figs. 3(b) and 3(c)] measured on a mm-size MoSe₂ single crystal monolayer exfoliated onto a SiO₂/Si substrate by a gold tape method.²⁵ The spectra in Figs. 3(b) and 3(c) are in excellent agreement with the previous ARPES measurement on monolayer MoSe₂.²⁶ As temperature is decreased from 300 K to 106 K, we observe a rigid downward shift in the band structure due to an increase in surface workfunction, but no changes in band dispersion or valence bandwidth. We conclude that the polaronic effect is negligible in the 2D atomic solid of MoSe₂.

For crystalline pentacene, we reproduce the ARPES results of Nakayama et al.¹⁸ as open triangles in Fig. 3(a). Hatch et al. reported that an effective phonon energy of $ℏ$ω = 11 meV, corresponding to an intermolecular vibration mode in pentacene crystal, is strongly coupled to the hole.¹⁷ Fitting to Eq. (2) using this effective phonon energy yields W₀ = 240 ± 40 meV and g_eff = 0.4 ± 0.1. While the asymptotic bandwidth in the molecular solid of pentacene is close to that in the superatomic solid Re₆Se₈Cl₂, the effective hole–phonon coupling constant in the former is three times smaller than that in the latter. The larger g_eff in Re₆Se₈Cl₂ than that in pentacene may be attributed to the covalent nature of inter-cluster interaction in the former.

In summary, we have quantified the temperature dependent valence bandwidth in the layered Re₆Se₈Cl₂ superatomic semiconductor using ARPES. We observe a band-narrowing effect with increasing temperature. The bandwidth is W = 120 ± 30 meV at 70 K and decreases by one order of magnitude to W ∼ 10 ± 20 meV as the temperature is increased to 300 K. This observation is consistent with a dominant polaronic effect. Fitting the temperature-dependent bandwidth to the Holstein polaron model gives an asymptotic low temperature valence bandwidth of 280 ± 50 meV, in excellent agreement with the DFT bandwidth for the rigid Re₆Se₈Cl₂ structure. A comparison to the well-known layered molecular semiconductors, pentacene, and the layered atomic semiconductor, MoSe₂, suggests that the Re₆Se₈Cl₂ superatomic semiconductor is closer to the former in terms of polaronic characters.