As seen in Fig. 1, both 1T and 3R have octahedral coordination for each unit layer but differ in layer stacking, i.e., AA (Fig. 1b) and ABC (Fig. 1c), respectively. Based on the calculated formation energies shown in Table 1, 1T is the stable structure for bulk PtX 2 , while 3R is a metastable phase. With regards to our proposed metastable 3R structure, we based our initial assumptions on the experimental study20 in which the authors were able to find PtSe 2 structures that are not 1T. From this result, we examined known bulk structures of TMDs—1T, 2H, and 3R combined with octahedral and trigonal prismatic coordination, and we found that octahedral 1T is indeed the most stable for bulk PtX 2 structure. Surprisingly, we found that octahedral 3R has a comparable stability relative to 1T based on the E F , indicating the possibility of synthesis. Also, we define for this study the system band gap which is the difference in energy between the conduction band minimum (CBM) and the valence band maximum (VBM). Using this definition, we further define a semi-metal as a material with a negative band gap.

Here, we discuss the calculated properties of the bulk PtX 2 structures. We start with 1T-PtS 2 , which is the only semiconductor among the three 1T bulk structures. Our calculated lattice constants and band gap (Table 1) are a = 3.591 Å, c = 4.877 Å, and E G = 0.25 eV which is in good agreement with the experimental lattice parameters (a = 3.543 Å, c = 5.039 Å) and band gap (E G = 0.25 eV), respectively.25,26 Furthermore, 1T-PtSe 2 and 1T-PtTe 2 are found to be semi-metallic30,31 with calculated (experimental25) lattice parameters of a = 3.785 Å, c = 4.969 Å and a = 4.069 Å, c = 5.236 Å (a = 3.728 Å, c = 5.081 Å and a = 4.026 Å, c = 5.221 Å), respectively. However, we note that previous studies30,31 show that bulk 1T-PtTe 2 is metallic, in contrast with our results, which we attribute the difference due to the inclusion of spin-orbit coupling (SOC) in our calculations.

Scrutinizing the bulk 1T band structures, we see that the system band gap originates from the top of the VBM at Γ point up to the conduction band at K point for all the chalcogens. With regards to the calculated band structures in 3R, we see that the band of bulk 3R-PtS 2 is still indirect but the gap is now 0.93 eV, which is higher as compared to its 1T counterpart. In addition, bulk 3R-PtSe 2 now has an indirect semi-conducting gap of 0.12 eV. Among the three bulk 3R-PtX 2 , 3R-PtTe 2 remained to be semi-metallic. In Fig. 2, we compare the band structures of bulk 3R with bulk 1T. We see that the system band gap still originates from the VBM at Γ point, but this time, the CBM is now at the middle of Γ and K points. The difference in the electronic band structures of bulk 1T and 3R is due to the breaking of crystal symmetry which affects the high-symmetry points of the BZ.

Next, we explore the thickness dependent properties. The bulk lattice constants (see Table 1) are used to construct the layered structures starting from monolayer up to 10L. Crystal structure relaxations were performed again before determining the formation energies and electronic properties. To understand the effect of the thickness to the electronic structures of layered PtX 2 TMDs, we explored the structural and electronic properties of PtX 2 with respect to increasing the number of layers. We note that for the case of thin film structures, we opted to only show the band structures for 1T-PtX 2 because, to the best of our knowledge, only this structure has been synthesized experimentally.

We now analyze the band structures of each chalcogen starting off with PtS 2 . As shown in Fig. 3, the band gap increases as the number of layers decreases, with E G = 0.25 eV in bulk, and E G = 1.68 eV in monolayer. However, we only observe indirect band gaps as the number of layers decrease,32 which is unlike MoS 2 and WS 2 . But looking closely at the band structures, we see that the VBM moved from Γ point at 10L to the in-between of M and Γ points at the monolayer with the apparent adjustment occurring between the 3L and 4L. We see that at 4L, the VBM is still at the Γ point, but upon decreasing to 3L, the VBM shifted its position. Still, the CBM remains in the middle of M and Γ points.

The evolution of electronic band structure of thin film PtSe 2 from monolayer to 10L is shown in Fig. 4. Our results show that 1T-PtSe 2 is an indirect band semiconductor only at its mono- and bi-layer with gaps of 1.18 eV and 0.21 eV, respectively, while the structures from 3L to bulk are semi-metallic. Just like in the case of PtS 2 , we see that in PtSe 2 from 10L to 4L, the VBM is located at Γ point while the CBM is located in-between the M and Γ points. The change also occurs in 3L where both the VBM and CBM, respectively, are located in-between M and Γ points.29 The difference between PtS 2 and PtSe 2 is that the VBM of PtSe 2 goes back to the Γ point at monolayer, as opposed to monolayer PtS 2 in which the VBM is still in-between M and Γ points. With this, we observe that there is minimal to no shift of indirect-to-direct band gap as the thickness decrease for the PtSe 2 structure.

Among the chalcogens incorporated in this study, Te combined with Pt has the smallest monolayer band gap of 0.40 eV, while the bilayer up to its bulk structure is semi-metallic (see Table 2). Looking closely at band structures of 1T-PtTe 2 with varying thickness in Fig. 5, we are still able to observe the decreasing system band gap with increasing number of layers from bilayer to 10 layers. From 3L to 10L, the VBM is located at the Γ point and the CBM is in-between the M and Γ points. However, in bilayer, the VBM and CBM, respectively, are now both located in-between the M and Γ points. Proceeding to the monolayer, we see another transition of the VBM going back to Γ point, but this time, the structure is now a semiconductor with an indirect band gap of 0.40 eV. Although PtX 2 are an indirect bandgap semiconductors, a transition from indirect to direct bandgap can be tuned by strain.14,33,34

In all 1T-PtX 2 (X = S, Se, and Te) TMDs investigated in this study, we were able to observe an increasing system band gap with decreasing number of layers from bulk down to monolayer structures. However, unlike other TMDs like MX 2 (M = Mo and W; X = S, Se, and Te) which are direct bandgap semiconductors at monolayer,2 we were not able to observe a shift from indirect-to-direct band gap as the number layers decrease from bulk to monolayer structures. A possible reason for this uniqueness is due to the difference in crystal structure – MX 2 has the 2H structure while PtX 2 has the 1T structure. This dissimilarity in crystal symmetries greatly contributes on how the electronic properties will behave. Nevertheless, we are still able to observe the inverse relationship between the band gap and the number of layers. This phenomenon is governed by factors such as quantum confinement effect18 and interlayer interaction through van der Waals interaction.43

Moreover, upon further inspection of the band structures, we were able to observe a flat band dispersion near the Fermi level of unstrained (0.0%) 4L 1T-PtS 2 as shown in Fig. 6b. When a band in a dispersion curve is flat, this indicates that a lot of allowed states occupy almost the same energy levels. This results to high or diverging density of states (DOS) which is the characteristic feature of vHs.39,44 Moreover, previous studies45,46 on graphene and phosphorene have shown that by inducing strain, one can tune the presence and/or location of the vHs relative to the Fermi level, which is indicated by saddle points, resulting to high density of states. With this in mind, we plotted the band energy contour and band structure with DOS for monolayer to 4L PtS 2 (Fig. 6). We immediately see the presence of vHs near the Fermi level in monolayer along Γ-K (Fig. 6a), as confirmed in DOS indicated by the red arrow (Fig. 6b). We also see the possible vHs in bilayer and 4L along Γ-M (highlighted by the red arrows).

Since the vHs in 4L is closer to the Fermi level than bilayer, we applied biaxial strain (from −4 to 4%) to the former and plotted the band energy contours (Fig. 7a) and band structures with DOS (Fig. 7b) to investigate the effect of strain. We found a substantial enhancement of vHs in the −2.0% strain with respect to the unstrained case, as indicated by the higher DOS intensity in the former. Our results show that, indeed, the vHs can be tuned by controlling the thickness and straining the material.