Quasiperiodic ordering in thick Sn layer on $i$-Al-Pd-Mn: A possible quasicrystalline clathrate

Realization of an elemental solid-state quasicrystal has remained a distant dream so far in spite of extensive work in this direction for almost two decades. Here, we report the discovery of quasiperiodic ordering in a thick layer of elemental Sn grown on icosahedral ($i$)-Al-Pd-Mn. The STM images and the LEED patterns of the Sn layer show specific structural signatures that portray quasiperiodicity but are distinct from the substrate. Photoemission spectroscopy reveals the existence of the pseudogap around the Fermi energy up to the maximal Sn thickness. The structure of the Sn layer is modeled as a novel form of quasicrystalline clathrate on the basis of the following: Firstly, from ab-initio theory, the energy of bulk Sn clathrate quasicrystal is lower than the high temperature metallic $\beta$-Sn phase, but higher than the low temperature $\alpha$-Sn phase. A comparative study of the free slab energetics shows that surface energy favors clathrate over $\alpha$-Sn up to about 4 nm layer thickness, and matches $\beta$-Sn for narrow window of slab thickness of 2-3 nm. Secondly, the bulk clathrate exhibits gap opening near Fermi energy, while the free slab form exhibits a pronouced pseudogap, which explains the pseudogap observed in photoemission. Thirdly, the STM images exhibit good agreement with clathrate model. We establish the adlayer-substrate compatibility based on very similar (within 1%) the cage-cage separation in the Sn clathrate and the pseudo-Mackay cluster-cluster separation on the $i$-Al-Pd-Mn surface. Furthermore, the nucleation centers of the Sn adlayer on the substrate are identified and these are shown to be a valid part of the Sn clathrate structure. Thus, based on both experiment and theory, we propose that 4 nm thick Sn adlayer deposited on 5-fold surface of $i$-Al-Pd-Mn substrate is in fact a metastable realization of elemental, clathrate family quasicrystal.


I. INTRODUCTION
The mathematical concept of aperiodic ordering and the discovery of quasicrystals brought paradigm shift in crystallography [1,2]. Quasiperiodicity has been observed in naturally occurring minerals [3], different inter metallic compounds [4], binary nanoparticle super lattices [5] colloidal quasicrystal [6], oxide thin film [7] and large molecular assemblies [8,9]. A recent exciting finding is the nontrivial topological property of quasiperiodic systems [10,11]. The technological importance of quasicrystals arise from their low frictional coefficient, heat insulation and photonic band gap [12,13].
However, the chemical complexity of quasicrystals discovered so far hinders their applicability and basic understanding of their unusual properties. An elemental quasicrystal would be best suited for this purpose, but a single-element bulk quasicrystal has not been yet observed. Efforts in this direction for over a decade have involved the use of quasicrystalline substrates as a template to grow quasiperiodic elemental layers [14][15][16][17][18][19][20][21][22][23][24][25][26]. Pseudomorphic growth of one monolayer (ML) quasiperiodic Sn has been reported in the past on decagonal (d)-Al-Ni-Co [15] and icosahedral (i)-Al-Cu-Fe [16]. A density functional theory (DFT) study predicted pseudomorphic growth of quasiperiodic Sn on i-Al-Pd-Mn [17]. In most of the studies in the past, only the first wetting layer has been reported to exhibit five-fold quasiperiodicity with a structure similar to the substrate. Pb/i-Ag-In-Yb showed five-fold growth isostructural with the substrate and pentagonal motifs with maximum height of about 0.7 nm were identified [25]. It may be noted that although the thickness was substantial, five-fold quasiperiodicity was not observed in 25 ML Cu/i-Al-Pd-Mn [27] or in 1.8 nm thick Ag on GaAs [28], rather aperiodic modulation of row separation were observed only in one direction.
It is interesting to note that elements from group XIV of the periodic table have specific propensity to form pentagonal or even icosahedral structures. Although Sn exhibits a diamond type structure (α-Sn, space group Fd3m) at low temperature and a tetragonal structure (β-Sn, space group I4 1 /amd) above 286 K, it is well known that Sn can form sp 3bonded clathrate structures whose canonical constituents are four kinds of cages that are bounded exclusively by pentagonal and hexagonal faces. Out of the four canonical cages, the most abundant one in common clathrate crystals is the dodecahedral cage with icosahedral symmetry, bounded exclusively by 20 pentagonal faces, in contrast to purely hexagonal 3 diamond structure (see Fig.5 in Section III.D). Intriguingly, in their computer simulation Engel et al. [29] observed spontaneous formation of three icosahedral phases, and the "loosely packed" phase can be regarded as an imperfect realization of the icosahedral quasicrystal clathrate. Last but not the least, the dodecahedron cage found in clathrates is identical with the second shell of the so called mini-Bergman clusters, playing fundamental role in i-Al-Pd-Mn structure.
In this work, we observed that Sn deposited on i-Al-Pd-Mn surface retains quasiperiodic ordering up to maximal achievable coverage leading to nearly 4 nm thickness, indicating the existence of the first intrinsic -although apparently metastable -monoatomic realization of a quasicrystal. We model it as a quasicrystal structure belonging to the clathrate family strictly maintaining tetrahedral coordination consistent with sp 3 bonding. We justify plausibility of the model by energetic considerations within the DFT framework, based on the structural comparison with experimentally observed motifs, and from the character of electronic density of states (DOS) near Fermi energy.

II. METHODS
The single grain fivefold i-Al-Pd-Mn quasicrystal surface was prepared by repeated cycles of Ar + ions sputtering and annealing to about 900-930 K for 2−2.5 h. The cycles were repeated until sharp fivefold low energy electron diffraction (LEED) pattern was observed.
Sn (99.99% purity) was evaporated from a water cooled Knudsen evaporation cell [30]. The evaporation cell was operated in the temperature range of 1073−1163 K at a pressure of 2×10 −10 mbar during the deposition. For the thick layer growth, we used an effective deposition rate of 0.06 to 0.4 ML/min in repeated cycles of 1 min deposition followed by 1 min waiting time. Sn deposition was carried out with the substrate close to room temperature (RT≈ 300-330 K) and at 150 K. The temperature of the substrate was measured using a K-type thermocouple.
The STM measurements were carried out at a base pressure of 2×10 −11 mbar using a variable temperature STM from Omicron GmbH. STM measurements were performed at room temperature (RT) and 80 K in the constant current mode using a tungsten tip that was cleaned by field emission, sputtering and voltage pulse method. The tip was biased and the sample was kept at the ground potential. It may be noted that same kind of tiles 4 and motifs are observed for different bias voltages ranging from -2.6 to 2.8 V for different coverages, which demonstrate that the features are topography related. The zero in the vertical scale of a STM image is set at the bearing height, which is the most dominant height value, based on a height distribution histogram. LEED was performed using 4-grid rear view optics from OCI Vacuum Microengineering.
The ultraviolet photoemission spectroscopy (UPS) measurements at 150 K were performed using R4000 electron energy analyzer with wide angle lens using 21.2 eV photon energy incident at 45 • with respect to (w.r.t) the surface normal and measured in normal emission, with an analyzer acceptance angle of ±15 • in transmission mode and a resolution of 50 meV. The Fermi edge was measured on a metallic sample in electrical contact with the specimen. X-ray photoelectron spectroscopy (XPS) measurements were performed with 1253.6 eV (MgKα) photon energy with a resolution of 0.8 eV using Phoibos 100 electron energy analyzer. The thickness of the Sn adlayer has been determined by STM and also by XPS from the intensities of Sn 3d and Al 2p core level peaks using the well known relations  [32] by performing an iterative solution of the Kohn-Sham equations of DFT within a plane wave basis. We use projector augmented wave potentials [33] in the PW91 generalized gradient approximation [34]. For both bulk and free slab structures, cohesive energies are converged to less than 1 meV/atom with basis set containing plane waves with a kinetic energy up to E cut−off = 150 eV. The self-consistency iteration were stopped when total energies are converged to within 10 −6 eV. All the structures, bulk models as well as slabs, were fully relaxed without constraints, and including the cell parameters. k-point meshes have been converged to satisfy for each mesh dimension α N kα × c α ∼100, where c α are edges of the periodic cell. We begin our discussion by a high resolution STM image of 1 ML Sn/i-Al-Pd-Mn in Fig. 1(a). It is clearly quasiperiodic, as demonstrated by its Fourier transform (FT) that exhibits two sets of spots of tenfold symmetry highlighted by yellow circles [ Fig. 1(b)], whose distance from the center (white lines) is in the ratio of τ , where τ is the golden mean (≈ 1.618). The most prevalent tile in the Sn layer is a pentagon with a central dark region Material (SM) [35]. Hexagonal tiles, although rare, are also observed with length scale similar to the P -tiles [ Fig. 1(d)]. The different motifs formed by the congregation of such tiles are highlighted by circles in Fig. 1(a). A distinctive motif of the Sn layer that is not observed for i-Al-Pd-Mn is a circular congregation having common sides that we refer to as wheel motif (yellow dashed circles), while half-circular or incomplete wheel motif is named as crown motif (white dashed circle). An expanded view of a crown motif, seen in the bottom right corner of Fig. 1(a), is shown in Fig. 1(e). It shows five P tiles (C1 to C5) clearly not in the same plane forming a half circle. Noteworthy is that the height profiles ( Fig. S1(b) of SM [35]) along the sides of these tiles show the maximum puckering to be about 0.06 nm, which is sizeable compared to the thickness of the 1st layer (0.26 nm). The puckered nature of the Sn layer can also be quantified by the root mean square roughness (S q ), which turns out to be 0.04 nm, which is twice that of i-Al-Pd-Mn (S q = 0.02 nm). A hexagonal tile centered motif in red circle in the top left side of Fig. 1(a) also indicates that the tiles are not in the same plane. It is noteworthy that even in the submonolayer regime, for example at 0.6 ML, as the Sn adatoms form islands, the crown motif is observed [white dashed circle in Fig. 1(f)].
In contrast to the P -tiles that are uniform, pentagons with bright vertices with side length of about 1.1 nm are occasionally observed [white circles in Fig. 1(a)]. These are different from the white flowers of i-Al-Pd-Mn because here a central dark region is observed.
The LEED of the Sn monolayer is distinctly different from the substrate: the latter recorded with beam energy (E p ) of 81 eV exhibits two sets of ten spots, inner and outer [ Fig. 1(g)], in agreement with literature [36,37]. A blue pentagon connects the intense 7 (10000) spots of the inner set, while (00110) and (11000) spots on the outer set are also indicated by circles. As Sn coverage increases, the intensities of (00010) spots increase (white dashed pentagon) whereas in contrast the (10000) intensities decrease (blue pentagon) [ Fig. 1(h)]. Furthermore, a set of five innermost spots appears forming a smaller pentagon  [35] show that all the islands representing the second layer are ≈0.2 nm thick indicating layered growth. This is expected [38] because the surface energy of Sn be assigned to the 1st (floor of the pit P), 2nd (floor of the pit Q), and 3rd layer (at edges of pits P and Q), while the main peak corresponds to the 4th layer. The height profiles along lines gh and mn in Fig. 2(b) also show this, the transition from 3rd to 4th layer is shown by a change in slope (pink arrow). In fact, the pink height profile along gh shows formation of three Sn layers of nearly equal heights. This is incompatible with the quasiperiodic step heights of the i-Al-Pd-Mn substrate, hence confirming the growth of four Sn layers.
A high resolution STM image for the 4th layer in Fig. 2f show the wheel, crown (dashed circles), and the triplet motifs (red circle) that are similar to the thinner layers. island growth after the 1st quasiperiodic layer [16] possibly because in that work the deposition was done at much higher temperatures (573-623 K) and large deposition rate of 15 ML/min, where activated diffusion of Sn across terraces might have resulted in clustering and island growth. In contrast, our deposition temperature for growing thicker layers is 150 K and the deposition rate is much lower (see section II).

C. Thick Sn layer studied by STM and LEED
In order to obtain quasiperiodic Sn of even larger thickness, the deposition was done at 150 K for twice the time compared to the previous (4th layer) deposition and the layer was cooled down to 80 K. In this case, the STM image in Fig. 3(a) shows dome-like structures of uniform shape with a circular base (highlighted by dashed circles). Interestingly, an expanded view of a dome in Fig. 3  In order to find the thickness of this layer, we note that the peak of the height histogram at z= 0 nm (red arrow in Fig. 3f) shows that about 50% of the area (shaded with blue color) has a thickness (or height) of ≥2 nm w.r.t. the minimum at z≈ -2 nm (black arrow). The minimum region is identified to be around a in Fig. S6(b) from the height profile abc shown in  Fig. 3(a), by the same argument discussed above, is ≥4 nm, since these are at a height of 12 3 nm from the base (Fig. 3(c)), and the base can be taken to be ≥1 nm in thickness. The appearance of domes and undulations [ Fig. 3(a,d)] is indicative of transformation towards quasiperiodic ordering in the vertical direction, which could be expected as the Sn-adlayer becomes thicker. The growth transforms from puckered but primarily two dimensional to quasiperiodic growth in all three directions similar to a bulk quasicrystal. We find that the thick film is not stable when warmed up to RT, as is evident from 40-50% decrease of the Sn Auger signal, which is possibly caused by the Sn atoms diffusing out to the sides of the substrate. Diffusion of Sn from the back of Al-Ni-Co substrate to the front with increasing temperature has been reported in literature [15]. The function of the substrate is to suppress initial formation of stable crystalline forms of Sn i.e. the α and β-Sn.
In the Sections III.D through III.I, we combine the geometrical concepts and DFT calculations, while in Section III.J, we discuss how Sn white flower (SnWF) cluster facilitates nucleation of the intrinsic Sn structure. We show that the intrinsic Sn structure is a metastable clathrate quasicrystal, and that in a narrow window of covering widths of 3-4 nm, an unsupported pentagonal clathrate structure is more stable compared to α-Sn. In the α-Sn structure, atoms are tetrahedrally coordinated and satisfy the sp 3 type bonding scheme. But there exists a whole family of alternative structures -clathrates -satisfying the same tetrahedral bonding scheme (Fig.5), achieving that goal by grouping atoms around point centers into empty "cages". Indeed "empty" clathrates are experimentally achievable metastable states of Ge [45] or Si [46], and there is a broad family of clathrate structures that are stabilized by addition of large "host" atoms captured inside the cages. The clathrate structures are modular in the sense that the constituting polyhedral cages can be nontrivially recomposed into variety of periodic arrangements.
Two of the known clathrate structures commonly designated as type II and III exhibit special relationship, illustrated in Fig. 6(a,b)  or disordered.
We note that the cage-cage linkage length in Sn clathrates of ≈12.6Å almost exactly coincides (to within 1%) with the pseudo-Mackay cluster-cluster separations in i-Al-Pd-Mn of 12.55Å (τ -times the intercluster linkages of 7.7Å between "mini-Bergman" or between "pseudo-Mackay icosahedra" clusters). In the clathrates, the prevalent dodecahedral cages have icosahedral symmetry, and sections through the cage normal to a 5-fold axis will have pentagonal symmetry. Since the symmetry of the LEED patterns [ Fig. 4, Fig. S5] is determined by the surface layer, we predict that in general their symmetry will be 5-fold.  [51], whose strictly quasiperiodic version would be based on Stampflis square-triangle inflation rule [52].
The possibility of icosahedral FK structure whose dual would be perfect icosahedral clathrate is not clear. The only locally icosahedral and strictly tetrahedrally close-packed structure known is, to our knowledge, the "1/1" icosahedral approximant with prototypical natural realization in Al-Mg-Zn system, discovered by Bergman et al. [53] In their milestone paper, Henley and Elser [54] described this structure as a simple decoration of Penrose rhombohedra, and proposed that generalization of that decoration scheme eventually leads to FK quasicrystal enjoying global icosahedral symmetry. But there is a bottleneck: the decoration rule requires that all oblate rhombuses in the 3D Penrose tiling are paired along their short body diagonals, but such tiling of 3D golden rhombuses is not known at present. An interesting aspect of the bulk icosahedral clathrate is its possible genuine entropic advantage over layered quasicrystals.
Relaxing the strict sp 3 -bonding requirement (that translates into tetrahedral bonding rule) brings interesting clue between spontaneously formed icosahedral quasicrystals in toy mono-particle system interacting via oscillating pair potentials [29] and real tin-based systems. As is clear from the phase diagram outlined in Fig. 3    mediate between sp 3 -bonding type and metallic, which is reflected by the close competition of α and β tin variants, such "imperfect" quasicrystal ordering seems entirely plausible.

F. Cohesive energies and diffraction patterns of bulk Sn-clathrate models
While intermetallic clathrates are usually stabilized by 15% content of large guest atom that fits into large clathrate cages (and do not fit inside α-Sn structure), even empty clathrate structures are energetically competitive. Fully relaxed cohesive DFT energies of these structures relative to the α-Sn structure ground state are shown in Table I. The smallness of these energy differences (28-48 meV/atom range) as well as the fact that the decagonal approximant energy is intermediate between energies of type II and III prototype clathrates, are the first argument supporting the plausibility of the clathrate structure for Sn. We expect that the leading term in energetics of any clathrate structure will depend on the fraction of each type of the cage in the structure. Since the decagonal clathrate approximant (row 3 in  formed by completing all cages with next higher fractional coordinate z.
To represent the slab of the decagonal clathrate, we chose periodic approximant from Finally, when the coverage exceeds 0.7 atoms/Å 2 or slab thickness ≥ 3 nm i.e approaching the bulk limit, the most stable structure is α-Sn.
H. Photoemission spectroscopy and density of states from the clathrate model In Fig. 8(a,b), Sn 3d 5/2 and Al 2p XPS core level spectra that appear at 485 eV and 73.1 eV, respectively, show no change in their binding energies with Sn coverage. This is a significant observation because core-level binding energies are sensitive to the transfer of electronic charge from the adsorbate to the substrate, which obviously can be ruled out in this case. Absence of core level shift or change in the shape also excludes the possibility of any alloying or intermixing of Sn with the substrate. This observation also implies a propensity for condensed island growth [18] as also shown in Fig. 2(a), indicating the dominance of adsorbate-adsorbate interaction. The Sn 3d 5/2 core level binding energy (485 eV) is similar to that of α-Sn [56], indicating the similarity of the quasiperiodic Sn layer with that of α-Sn, thus reaffirming the clathrate model where sp 3 hybridization dominates. In Fig. 8(c), the intensity variation of the Sn 3d and Al 2p XPS core-level spectra exhibits exponential trend as a function of thickness indicating a layered growth, as also shown by STM. Fig. 8(d) depicts the valence band of Sn/i-Al-Pd-Mn, where the prominent peak at 4 eV binding energy arises from Pd 4d-like states. It decreases in intensity with Sn deposition, but its binding energy remains unchanged lending further support to the conclusions drawn from the core-level spectra [ Fig. 8(a,b)].
The stability of quasicrystals has been related to the existence of a pseudogap in the electronic DOS around the Fermi level (E F ) originating from the interaction between the with the specimen. However, for thicker layers, the pseudogap becomes shallower compared to the substrate, which is possibly related to the increase in disorder, as also shown by the relative weakening of the LEED spots, as discussed earlier in Section III.C.
The DOS for relaxed bulk and slab Sn clathrate is shown in Fig. 8(f). It is interesting to note that the bulk Sn quasiperiodic clathrate is semiconducting with a band gap of about 0.4 eV, in contrast to bulk i-Al-Pd-Mn that shows a pseudogap [60]. In contrast to bulk, the DOS for the slab of Sn quasiperiodic clathrate shows a pronounced pseudogap with the minimum at 0.3 eV (red arrow), the states near E F having almost equal contributions from the Sn p and Sn d states, while the contribution from the s states is marginal. The pseudogap is deeper compared to i-Al-Pd-Mn, where the DOS was calculated for MS and M slabs [60].
Thus, the Sn clathrate model explains the interesting observation from photoemission that the pseudogap is deeper for the Sn monolayer compared to the substrate i-Al-Pd-Mn. It is worth to note that the inter atomic bonding in Sn is different from Pb, where a wider pseudogap was reported from scanning tunneling spectroscopy, but it was later ascribed to the splitting of Pb 6p band due to large spin-orbit coupling [19,24,25]. Pb does not have s − p hybridization since the s and p bands are clearly separated, and thus the bonding is mediated by p orbitals only. Here, the deepening of the pseudogap around E F results from formation of the puckered clathrate structure with high covalency and sp 3 bonding between the Sn atoms. The clathrate structures perfectly support the sp 3 bonding scheme, with exclusively tetrahedral local environments like in α-Sn structure -but while in the latter the empty spaces are condensed into flat interlayer area, in clathrates they are isolated in four types of approximately spherical cages (Fig. 5).

I. The relaxed clathrate model and the STM motifs
The structure of the most optimal surface of the Sn clathrate obtained by relaxing the atom positions in our DFT calculation is shown in Fig. 9(a). We find that it exhibits The relaxed clathrate structure is in good agreement with STM and the motifs are also reproduced. For example, in Fig. 9(b), a part of the clathrate structure enclosed by green dashed rectangle is overlaid on a STM image from 3 nm thick Sn layer that includes a crown motif (encircled by pink dashed circle) after τ 3 inflation and the agreement is found to be satisfying. τ -inflation is well known in quasicrystals due to their self-similar nature, for example, in i-Al-Pd-Mn, the fundamental intercluster linkage of 7.75Å is τ inflated along the 2-fold direction, while τ 3 inflation is observed along 5-fold or 3-fold directions. τ 3 inflation has also been reported in a binary quasicrystal Yb-Cd with the formation of cluster of clusters [61]. Fig. 9(c) provides another example, where a STM image containing a hexagonal tile centered motif (encircled by black dashed circle) observed for 1 ML Sn is reproduced by a hexagon and the pentagons surrounding it from the violet dashed rectangle in Fig. 9(a) with τ 2 inflation. Thus, the relaxed clathrate model is able to reproduce the STM images [ Fig. 9(b, c)], which reconfirms the validity of this structural model.

J. Nucleation and compatibility of the Sn clathrate structure with i-Al-Pd-Mn
In this section, in order to find the genesis of the Sn clathrate structure and establish its compatibility with the substrate i-Al-Pd-Mn, we have studied the sub-monolayer coverages of Sn by STM to identify the nucleation centers. We find occurrence of pentagonal clusters that look like flowers with five bright petals at 0.2 ML coverage (encircled by orange dashes in [ Fig. 10(a)], a larger area image is shown in Fig. S8[35]). These clusters are formed on specific areas of the 5-fold surface recognized in STM images as white flowers [63], and so we call them as the Sn white flower (SnWF) clusters. As shown in Fig. S8 by orange circles, the SnWF's are oriented in the same direction and are mostly isolated. However, we are able to identify regions, for example, inside the red rectangle in Fig. 10(a) that has two SnWF's that are nearest neighbors.
Similar five-fold clusters have also been observed for Pb [21] and Bi [20], which on the i- STM image this motif is seen as the white flower mentioned above [63]. If the center of the pMC is deeper below the surface plane, this part of the surface corresponds to another characteristic pattern known as the dark star [63]. In the STM image of the bare surface shown in Fig. S1(c) [35], motifs of the white flower and the dark star can be well recognized.

Al
At the submonolayer coverage, the SnWF cluster formed on the white flowers consists of 10 adatoms [24], as shown by the red atoms in Fig. 10(c). The calculated STM image of the SnWF is shown in Fig. 10(b), where shape is similar to that observed in the experiment

IV. CONCLUSIONS
Our work demonstrates formation of a new quasicrystalline structure in a thick Sn layer up to a thickness of atleast 4 nm. Based on multiple evidences from both experiment and theory, we propose a quasiperiodic clathrate structural model for the Sn layer. Sn retains quasicrystallinity up to a thickness that is highest reported so far for any element, and this is almost in the realm of bulk-like growth, where the influence of the substrate potential is negligible. The STM motifs (crown, wheel and hexagonal tile centered motif in Fig. 1) and the LEED IV curves (Fig. S3 of SM [35]) of the Sn monolayer are different from i-Al-Pd-Mn that is described by the Penrose P1 tiling. The motifs of the thick Sn layer are similar to the motifs of the first layer. This shows a structural difference between the Sn layer and the substrate, indicating that Sn grows with its intrinsic quasiperiodic structure and the substrate is unable to force a pseudomorphic growth. The role of the substrate is solely to suppress initial formation of any stable crystalline forms of Sn. Photoemission spectroscopy shows that the pseudogap around the Fermi level is deeper for the first Sn monolayer compared to i-Al-Pd-Mn and exists for the thick layer, indicating its stability.
Using density functional theory, a comparative study of the free slab energies shows that surface energy favors clathrate over α-tin up to about 4 nm layer thickness, and matches β-