Large Spin Hall Conductivity and Excellent Hydrogen Evolution Reaction Activity in Unconventional PtTe1.75 Monolayer

Two-dimensional (2D) materials have gained lots of attention due to the potential applications. In this work, we propose that based on first-principles calculations, the (2 × 2) patterned PtTe2 monolayer with kagome lattice formed by the well-ordered Te vacancy (PtTe1.75) hosts large and tunable spin Hall conductivity (SHC) and excellent hydrogen evolution reaction (HER) activity. The unconventional nature relies on the A1 @ 1b band representation of the highest valence band without spin–orbit coupling (SOC). The large SHC comes from the Rashba SOC in the noncentrosymmetric structure induced by the Te vacancy. Even though it has a metallic SOC band structure, the ℤ2 invariant is well defined because of the existence of the direct bandgap and is computed to be nontrivial. The calculated SHC is as large as 1.25 × 103 ℏe (Ω cm)−1 at the Fermi level (EF). By tuning the chemical potential from EF − 0.3 to EF + 0.3 eV, it varies rapidly and monotonically from −1.2 × 103 to 3.1 ×103ℏeΩ cm−1. In addition, we also find that the Te vacancy in the patterned monolayer can induce excellent HER activity. Our results not only offer a new idea to search 2D materials with large SHC, i.e., by introducing inversion–symmetry breaking vacancies in large SOC systems, but also provide a feasible system with tunable SHC (by applying gate voltage) and excellent HER activity.


SUPPLEMENTARY MATERIAL
A. Lattice parameters of the patterned PtTe2 monolayer with a Te vacancy The pristine PtTe 2 system crystallizes in the CdI 2 -type trigonal (1T) structure (SG P3m1) is a layered material stacking along the z axis. Recently, the monolayer structure with kagome lattice formed by one Te vacancy in the 2×2 supercell has been successfully grown 18 . The patterned PtTe 2 monolayer contains two Te layers, with 4 Te atoms in the bottom layer while 3 Te atoms in the top layer, as shown in Fig. 2(a). The corresponding lattice parameters are a = b = 8.1846Å, α = β = 120 • , and the thickness of the vacuum layer along z axis was set to 30 − d 0Å , with d 0 (= 2.7253Å) denoting the distance between the bottom Te layer and the top Te layer. To characterize the topological properties in the patterned PtTe 2 monolayer, the weak topological invariants are calculated by the 1D Wilson loop method. Taking Te: 5s 2 5p 4 and Pt: 5d 9 6s 1 orbitals into consideration, there are N e = 82 valance electrons, resulting in N e valence bands. From Figs. S1(a-d), the calculated weak topological invariants for 78, 80, 82 and 84 (corresponds to N e − 4, N e − 2, N e and N e + 2) occupied bands are Z 2 = 1, Z 2 = 1, Z 2 = 1 and Z 2 = 0, respectively. Thus, the patterned PtTe 2 monolayer with N e valence bands is a 2D TI with Z 2 = 1.

C. Topological surface states of the patterned PtTe2 monolayer
Exotic topological surface states serve as significant fingerprints to identify various topological phases. Based on the tight-binding (TB) model constructed with the maximally localised Wannier functions and surface Green function methods 59-61 , we have calculated the corresponding surface states to identify the nature of 2D TI. As shown in Fig.S2(a,b), we have chosen Pt-d and Te-p orbitals as the projected bases of the wannier-based TB model, which can reproduce the DFT band structures perfectly. As expected, the helical edge states corresponds to the 2D TI can be found in the (01) edge states, as shown in Fig. S3(b) and Fig. S3(d). There are two different mechanisms of the releasing hydrogen in HER. One is achieved by combining a solvated proton with an adsorbed H atom H + + e − + H * → H 2 , known as Heyrovsky reaction, while the other one is achieved by combining two adsorbed H atoms 2H * → H 2 , i.e., the Tafel reaction. As shown in Fig. S4, the energy barrier in the Tafel reaction is assessed with a smaller value (0.87 eV) by the climbing-image nudged elastic band (CI-NEB) approach 62 , which is 0.57 eV lower than that in the Heyrovsky reaction.

E. Screening adsorption sites and the correction of Gibbs free energy
In order to screen multiple thermodynamically stable adsorption sites, we uniformly generated 100 hydrogen adsorption structures with the {x,y} coordinates of the H atoms dispersing uniformly in the quarter of the 2×2 PtTe 1.75 supercell. After optimizing the z coordinate of the adsorbed H atoms, we choose five most thermodynamically stable structures. After the fully structural optimization, we find that four of them host the same ultimate configuration (denoted as PtTe 1.75 -I) with the lowest energy. While the other ultimate configuration (denoted as PtTe 1.75 -II) is metastable with a 6.14 meV/atom higher energy than the PtTe 1.75 -I phase.
The change of Gibbs free energy induced by hydrogen adsorption (∆G H * ) can be defined as 55 where E is internal Energy, ZPE is zero-point energy, C p dT is the correction of enthalpy, T is temperature, while S denotes entropy. For the adsorption structure (H*), we only pay attention to the vibration contribution of the adsorbed H atom, while the correction energy of the whole PtTe 1.75 slab can be considered as unchanged before and after the absorbed process. In order to take the effect of structural size on ∆G H * into consideration, we have chosen the most thermodynamically stable PtTe1.75 adsorption structures with 1×1, 2×2, 3×3 and 4×4 supercell. As shown in Table S2, we can find that the effect of the size is almost negligible.

F. Band structures vs. dopping
Since the electronic properties and SHC of the patterned PtTe 2 monolayer are sensitive to the E F , we can adopt different dopings at the Te-vacancy position to tune E F effectively. As shown in Figs. S5(a-c), we can find that both the introduced Pb doping and Tl doping behave as electron dopings, which increase the E F significantly. The decreased magnitude of the E F are estimated to be 0.482232 eV and 0.256149 eV, respectively.