Large Spin Hall Conductivity and Excellent Hydrogen Evolution Reaction Activity in Unconventional PtTe<sub>1.75</sub> Monolayer

Large Spin Hall Conductivity and Excellent Hydrogen Evolution Reaction Activity in Unconventional PtTe_1.75 Monolayer

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Dexi Shao¹^,², Junze Deng¹^,³, Haohao Sheng¹^,³, Ruihan Zhang¹^,³, Hongming Weng¹^,³, Zhong Fang¹^,³, Xing-Qiu Chen⁴^,⁵, Yan Sun⁴^,⁵^,^*, Zhijun Wang¹^,³^,^*

Research. Vol 6 Article ID 0042

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Research. Vol 6 Article ID 0042

• Research Article •

Large Spin Hall Conductivity and Excellent Hydrogen Evolution Reaction Activity in Unconventional PtTe_1.75 Monolayer

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Dexi Shao¹^,², Junze Deng¹^,³, Haohao Sheng¹^,³, Ruihan Zhang¹^,³, Hongming Weng¹^,³, Zhong Fang¹^,³, Xing-Qiu Chen⁴^,⁵, Yan Sun⁴^,⁵^,^*, Zhijun Wang¹^,³^,^*

Affiliations

¹Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.

²Department of Physics, Hangzhou Normal University, Hangzhou 311121, China.

³University of Chinese Academy of Sciences, Beijing 100049, China.

⁴Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Science, Shenyang 110016, Liaoning, China.

⁵School of Materials Science and Engineering, University of Science and Technology of China, Hefei, China.

Published: 2023-02-20 doi: 10.34133/research.0042

Outline

Abstract

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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 (PtTe_1.75) 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 $ℏ e$ (Ω cm)⁻¹ at the Fermi level (E_F). By tuning the chemical potential from E_F − 0.3 to E_F + 0.3 eV, it varies rapidly and monotonically from −1.2 × 10³ to 3.1 $× 103 ℏ e Ω cm − 1$ . 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.

Cite this Article

Dexi Shao, Junze Deng, Haohao Sheng, Ruihan Zhang, Hongming Weng, Zhong Fang, Xing-Qiu Chen, Yan Sun, Zhijun Wang. Large Spin Hall Conductivity and Excellent Hydrogen Evolution Reaction Activity in Unconventional PtTe_1.75 Monolayer[J]. Research, 2023 , 6 (2) : 0042 - . DOI: 10.34133/research.0042

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Introduction

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In the past decade, many topological semimetals with various quasiparticle dispersions and fascinating properties have been proposed [1–5]. The layered noble transition metal dichalcogenide PtTe₂ is extraordinary with heavily tilted type-II Dirac fermion [6]. It hosts unique properties, such as topological nontrivial band structure [6,7], ultrahigh electrical conductivity [8,9], and robustness of the remaining semimetal phase even down to just 2 triatomic layers [10,11]. Soon after, many PtTe₂ derivatives have been proposed, including the monolayer, multilayer, doping, vacancy, heterojunction structures, and so on. For example, the Ir-doped PtTe₂ (i.e., Pt_{1 − x}Ir_xTe₂) has realized the Fermi level (E_F) tunability and superconductivity, which opens up a new route for the investigation of Dirac physics and topological superconductivity [12–14]. More recently, PtTe₂-based broadband photodetectors and image sensors have been fabricated, demonstrating tremendous potential application value in various photoelectric devices [15–17]. Very recently, the patterned monolayer with kagome lattice formed by one Te vacancy in a 2 × 2 supercell has been grown successfully [18], whose band topology and potential properties are unknown. The study of PtTe₂ derivatives can not only reveal novel condensed matter physics but also facilitate the versatile development in device physics.

In this work, we theoretically propose that the recently synthesized patterned PtTe₂ monolayer with the Te vacancy (i.e., PtTe_1.75) hosts large spin Hall conductivity (SHC) because of the Rashba spin–orbit coupling (SOC), where the Te vacancy breaks inversion symmetry (

I

). The momentum offset and strength of the Rashba SOC are estimated, k₀ = 0.12 Å⁻¹ and α_R = 0.8112 eV Å. The momentum offset k₀ is very large and comparable with the largest one reported in the Bi/Ag(111) surface alloy in literature [19], which induces visible Rashba band splitting. Using the Kubo formula approach at the clean limit, we find the Rashba SOC will induce large SHC, as large as 1.25

× 103 ℏ ℏ e Ω cm − 1

at E_F. Furthermore, the SHC changes rapidly and monotonically as the chemical potential evolving in a wide range (E − E_F ∈ [−0.3,0.3] eV), which benefits to the potential applications in spintronics. In the end, the variation of the Gibbs free energy for hydrogen adsorption progress is considered, which indicates that the PtTe_1.75 monolayer exhibit an excellent hydrogen evolution reaction (HER) activity.

Crystal Structure and Methodology

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The pristine PtTe₂ crystallizes in the CdI ₂-type trigonal (1T) structure with

P 3 ¯ m 1

space group (SG). It hosts the layered structure stacking along the z axis and can be easily tuned by strain, which indicates that it can be grown under various substrates. The patterned monolayer with kagome lattice formed by one Te vacancy in the 2 × 2 supercell has been successfully grown on the Pt(111) surface [18]. As shown in Fig. 1A, the patterned PtTe₂ monolayer contains 2 Te layers: 4 Te atoms (blue balls) in the bottom layer and 3 Te atoms (green balls; with one vacancy schematized by “×” at 1b Wyckoff site) in the top layer. The distance between the bottom and top layers is d₀ = 2.7253 Å. The Te vacancy breaks

I

, resulting in a noncentrosymmetrical structure with the p3m1 layer group (LG 69; corresponding to SG P3m1 excluding translational symmetry along the z axis). Thus, the Rashba SOC-induced band splitting is inevitable. The lattice parameters and atomic positions are listed in Table S1 of the Supplementary Materials (SM).

We performed the first-principles calculations based on the density functional theory (DFT) using projector augmented wave method [20,21] implemented in the Vienna ab initio simulation package (VASP) [22,23]. The generalized gradient approximation with exchange–correlation functional of Perdew, Burke, and Ernzerhof for the exchange–correlation functional [24] was adopted. The kinetic energy cutoff was set to 500 eV for the plane wave bases. The thickness of the vacuum layer along z axis was set to >20 Å. The BZ was sampled by Γ-centered Monkhorst–Pack method with a 12 × 12 × 1 k-mesh for the 2-dimensional (2D) periodic boundary conditions in the self-consistent process. The Wilson loop technique [25] was used to calculate the ℤ₂ topological invariant. In addition, the electronic structures near E_F were doubly checked by the full-potential local-orbital code [26] and fully consistent with those from VASP. To compute SHC, a Wannier-based tight-binding (TB) model under bases of the Te-p and Pt-d orbitals is extracted from the DFT calculations.

Calculations and Results

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Electronic band structures

The band structures of PtTe_1.75 monolayer without and with SOC are presented in Fig. 1C and D, respectively. Comparing them, we notice that the band dispersions change dramatically. Each band splits into two nondegenerate bands in Fig. 1D. It is the Te vacancy in the monolayer that breaks

I

(and

TI

), inducing the visible Rashba band splitting. From the orbital-resolved band structures in Fig. 1C, we find that there exist visible band hybridizations between Te-p_x,y and Te-p_z orbitals around E_F. Using IRVSP [27], the irreducible representations of the high-symmetry k points are calculated and labeled in Fig. 2A. Accordingly, the band representation (BR) analyses indicate that the 2 conduction bands belong to E @ 1b BR, while the highest valence band belongs to A1 @ 1b BR, suggesting the unconventional nature of the obstructed atomic limit [28–31].

SOC often plays important roles to engineering topological states, such as quantum spin Hall effect in graphene [32,33] and Ta₂M₃Te₅ (M = Pd,Ni) compounds [34,35], 3D large-SOC-gap topological insulator in Bi₂Se₃ and NaCaBi families [36,37], and so on. In PtTe_1.75, once including SOC, the Te-p_z dominated band around Γ splits because of the Rashba SOC induced by the Te vacancy, as shown in Fig. 1D. To get more insights in the nontrivial band topology and Rashba SOC band splitting, we have explored how the band structure evolves with the increasing strength of SOC (denoted by λ) gradually in Fig. 2. We notice that the nontrivial band topology for N_e − 4 occupied bands is due to the SOC (can be infinitesimal)-induced bandgap at Γ without involving band inversion [38,39]. In addition, the nontrivial topologies for N_e − 2 and N_e occupied bands are due to a gap closing and reopening process as varing λ. Taking N_e − 2 occupied bands as an example, the critical Weyl band crossing between the (N_e − 2)th band and the (N_e − 1)th band appears on the M–K line with λ = 0.8, as highlighted by a red dashed ring in Fig. 2E. Similarly, the critical Weyl point (WP) between the N_eth band and the (N_e + 1)th band appears on the K–Γ line with λ = 0.98, as the right inset shown in Fig. 2F. However, it becomes topologically trivial for N_e + 2 occupied bands because there are 2 nontrivial gap openings around both Γ and M.

Because of the existence of R_3z and M₁₀₀ symmetries, the critical WPs abovementioned appear in sextuplet in the first BZ, as shown in Fig. 3A and B. Similar with WPs in 3D Weyl semimetals [40–43], these critical WPs also conform to the codimensional analysis. This can be deduced as follows. First, both the M–K (k_y = π) and K–Γ (k_y = 0) lines are M₁₀₀ ∗ T invariant. In the 2-band k · p Hamiltonian depicting the Weyl band crossing, the combined antiunitary symmetry with [TM₁₀₀]² = 1 will reduce the number of the independent σ matrices in the k · p Hamiltonian to 2. Second, the k_x value in both the TM₁₀₀ invariant lines and the SOC strength λ are 2 tunable parameters to search a WP. Thus, the number of the independent σ matrices in the k · p Hamiltonian equals to the number of the tunable parameters, which indicates that a WP is stable in the 2D parameter space {k_x, λ}. In other words, a topological phase transition can happen by tuning λ in the M₁₀₀ ∗ T invariant lines. Through the gap closing and reopening process in the evolution, it becomes topologically nontrivial for N_e (N_e − 2) occupied bands. As a result, we can expect the existence of the helical edge states of the patterned PtTe₂ monolayer. The edge spectra are presented in Fig. S3B and D (in Section C of the SM).

Rashba SOC at Γ

Because the Te vacancy breaks

I

in the patterned PtTe₂ monolayer, the Rashba SOC band splitting will appear inevitably. As the projected band structures shown in Fig. 3C, the Te-p_z dominated parabolic bands splits clearly near Γ. The splitting bands near Γ can be well fitted by

E = ℏ ℏ k ± k 0 2 2 M

with M = 2.02659 m_e (m_e denoting the free electronic mass) and k₀ = 0.12 Å⁻¹ (the momentum offset), as the 2 blue parabolas shown in Fig. 3C. The coupling strength of the Rashba SOC can be derived as

α R = 2 E R k 0 = ℏ ℏ 2 k 0 M = 0.81

eVÅ. The estimated k₀ is super large in Fig. 3C, as large as the the Bi/Ag(111) surface alloy [19].

Large SHC effect

To explore the intrinsic SHC in the patterned PtTe₂ monolayer, the Wannier-based TB model under bases of the Te-p and Pt-d orbitals is extracted from the DFT calculations. As shown in Fig. S2A and B, the fitted Wannier-based TB bands can reproduce the DFT bands perfectly. On the basis of this Wannier-based TB model, we have employed the Kubo formula approach at the clean limit [44–48] to calculate the SHC of the patterned PtTe₂ monolayer,

σ α β γ = e ℏ ∑ n ‍ ∫ BZ ‍ d k 2 π 2 f n k Ω αβ; n γ k, Ω αβ; n γ k = 2 i ℏ 2 ∑ m ≠ n u k n | J ̂ α γ | u k m u k m | v ̂ β | u k n ε k n − ε k m 2,

(1)

where

J ̂ α γ = 12 v ̂ α, s ̂ γ

is the spin current operator, with

s ̂

denoting the spin operator,

v ̂ α = ∂ H ̂ ℏ ℏ ∂ k α

denoting the velocity operator, and α, β, γ = {x, y, z}. f_n(k) is the Fermi–Dirac distribution.

| u k n 〉

and

ε k n

are the eigenvectors and eigenvalues of the TB Hamiltonian, respectively. The distributions of

O N k ≡ ∑ n = 1 N ‍ Ω yz; n x k

for N = N_e − 1 and N_e occupied bands are presented in Fig. 3D and E, respectively. As the calculated SHC as a function of the chemical potential shown in Fig. 3G, one can find that the calculated SHC is as large as 1.25

× 103 ℏ ℏ e Ω cm − 1

at E = E_F. The corresponding distribution at E = E_F is presented in Fig. 3F, which indicates that the large contribution of the SHC at K comes from the SOC band splitting. In addition, the SHC changes rapidly and monotonically in a wide energy window ranging from E_F − 0.3 eV to E_F + 0.3 eV. At E − E_F = −0.3 eV, the SHC changes the sign and becomes

− 1.2 × 103 ℏ e Ω cm − 1

, while at E − E_F = 0.3 eV, the SHC nearly triples and becomes 3.1

× 103 ℏ ℏ e Ω cm − 1

. In general, the chemical potential can be tuned by applying gate voltage or introducing chemical doping at the vacancy. As shown in Fig. S5A to C, we can find that the absorption of Tl/Pb at the vacancy behaves as electron dopings, which will increase the E_F with negligible changes in the band structure. We think our results will be benefitial to the potential applications in spintronics.

Excellent HER activity

According to the new principle for active catalytic sites [30,49,50], the obstructed bulk states in the patterned monolayer (which can be seen as the limit of obstructed surface states) may bring measured catalytic activity. By exposing undercoordinated atoms as the active sites, vacancy engineering is an important strategy to optimize the HER performance of the basal planes in 2D materials [51,52]. As the acidic HER of the PtTe_1.75 is schematized in Fig. 4A, protons (H⁺) in solution generate adsorbed H atoms (H^∗) as intermediate and then the H atoms on the catalyst surface are desorbed to produce hydrogen (H₂), which can be formulized as

H + + e − + ∗ → H ∗ → 12 H 2 + ∗ .

(2)

Here “*” denotes some site on the surface, i.e., a “*” by itself denotes a free site, while H* denotes a hydrogen atom absorbed on the surface. Te vacancy-induced states near E_F give PtTe_1.75 monolayer larger electrical conductivity than pristine PtTe₂ monolayer, which will effectively facilitate electron transfer for HER. We used a 2 × 2 PtTe_1.75 supercell to simulate the basal plane. Compared with the fully coordinated Te atoms, H atoms are more likely to be adsorbed near the undercoordinated Pt atoms, just as the most stable and metastable structures shown in Fig. 4B and C. Details of screening stable adsorption sites can be found in the SM. It is well known that the change of Gibbs free energy induced by hydrogen adsorption (ΔG_H^∗) is an important descriptor of HER activity [53–55], and an ideal catalyst for HER should host a near-zero ΔG_H^∗, which can effectively maintain the balance between adsorption and desorption steps [54]. As shown in Fig. 4D, unlike the pristine PtTe₂ monolayer with a large positive ΔG_H^∗ due to its extremely inert basal plane, the PtTe_1.75 monolayer hosts an optimal ΔG_H^∗ (0.08 eV), which is slightly superior to the benchmark material Pt (ΔG_H^∗ = −0.09 eV) [53]. Details of the free energy correction can be found in the SM. Here, we noted that the effect of the size on the ΔG_H^∗ is negligible, which can be deduced from Table S2. Thus, the active Pt sites induced by Te vacancy greatly optimize hydrogen adsorption in the intermediate, which will significantly improve HER performance [56]. According to Nørskov et al. [53], the theoretical exchange current density (i₀) as a function of ΔG_H^∗ is calculated. As shown in Fig. 4E, the PtTe_1.75 monolayer approaches the volcanic peak from the right with i₀ = 0.68 mA cm⁻², which is comparable to commercial Pt/C catalyst (i₀ = 1.2 mA cm⁻²) [57]. In addition, as shown in Fig. S4A and B, the energy pathways and corresponding energy barriers of the (a) Heyrovsky and (b) Tafel reactions to release hydrogen in HER are exhibited, from which we can find that the Tafel reactions to release hydrogen is preferred. Therefore, Te vacancy can greatly stimulate the catalytic activity of PtTe₂ basal plane and produce excellent HER performance.

Discussion

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We find that the PtTe_1.75 not only hosts the unique band structure with 3 lower-energy bands belonging to (A1 + E) @ 1b BRs at an empty site but also exhibits large and tunable SHC and excellent HER performance. First, we have calculated the time reversal invariant ℤ₂, which indicates the 2D TI nature in the patterned PtTe₂ monolayer. We demonstrate that the topological phase can be deduced by a gap closing and reopening process with the evolution of the strength of SOC from λ = 0 to λ = 1.0. The critical phase transition occurs at λ = 0.98, which gives a sextuplet of critical WPs. Second, the Te vacancy breaks

I

and induces Rashba SOC band splitting. The estimated momentum offset is super large with k₀ = 0.12 Å⁻¹. Third, we find that the SHC is as large as 1.25

× 103 ℏ ℏ e Ω cm − 1

at E_F. Furthermore, the SHC varies quickly and almost monotonically from −1.2 to 3.1

× 103 ℏ ℏ e Ω cm − 1

, indicating that the SHC in the patterned PtTe₂ monolayer can be conveniently tuned for various applications. Last, we also find the Te vacancy in the patterned monolayer can induce excellent HER activity. These 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 for the potential application in spintronics and HER catalysts.

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Year 2023 volume 6 Issue 2

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doi: 10.34133/research.0042

Receive Date：2022-09-06
Online Date：2026-10-27
Published：2023-02-20

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History

Received：2022-09-06
Accepted：2022-12-16

Affiliations

¹Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.

²Department of Physics, Hangzhou Normal University, Hangzhou 311121, China.

³University of Chinese Academy of Sciences, Beijing 100049, China.

⁴Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Science, Shenyang 110016, Liaoning, China.

⁵School of Materials Science and Engineering, University of Science and Technology of China, Hefei, China.

Corresponding:

^* Address correspondence to: sunyan@imr.ac.cn (Y.S.)

wzj@iphy.cn.cn (Z.W.)

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表12种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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Fig. 1. (Color online) (A)Crystal structure of the patterned PtTe₂ monolayer (PtTe_1.75). The pristine PtTe₂ monolayer system in the kagome lattice contains 2 Te layers, with 4 Te atoms in both the bottom layer and the top layer of the (2 × 2) supercell. In addition, the PtTe_1.75 system comes from the patterned PtTe₂ monolayer with a well-ordered Te vacancy (schematized by “×” at the 1b Wyckoff site) at the top layer of the (2 × 2) supercell. Thus, there are 3 Te atoms (denoted by green ball) occupying the top layer, while there are 4 Te atoms (denoted by blue ball) occupying the bottom layer in the PtTe_1.75 system. (B) The corresponding 2D bulk BZ and 1-dimensional projected BZ orthogonal to the (01) edge. Band structures of the PtTe_1.75 system (C) without and (D) with SOC. The light green, blue, red, and gray zones in (D) indicate that there exist direct bandgaps between the corresponding adjacent bands. Thus, the time reversal ℤ₂ can be defined and calculated to be 1, 1, 1, and 0 with (N_e − 4), (N_e − 2), N_e, and (N_e + 2) occupied bands.

Fig. 2. (Color online) Band structures of PtTe_1.75 monolayer with the strength of SOC (A) λ = 0 (without SOC), (B) λ = 0.2, (C) λ = 0.4, (D) λ = 0.6, (E) λ = 0.8, and (F) λ = 1.0 (with actual SOC). For the case without SOC shown in (A), the 2 conduction bands schematized by 2 blue lines belong to E @ 1b BR, while the highest valence band schematized by the red line belongs to A1 @ 1b BR. There exists a WP along the M–K line below E_F when λ = 0.8. Bands near E_F undergoes a gap closing and reopening progress when the strength of SOC evolves from 0.0 to 1.0, which gives a topological nontrivial bandgap with SOC (λ = 1.0). The critical transition occurs at λ = 0.98 (right inset in the λ = 1.0 panel), which gives another WP along the K–Γ line.

Fig. 3. (Color online) The 6 symmetry-related (R_3z and M₁₀₀) WPs formed by band crossings (A) between the (N_e − 2)th band and the (N_e − 1)th band with λ = 0.8 as well as (B) the N_eth band and the (N_e + 1)th band with λ = 0.98. (C) The Te-p_z projected band structures with SOC, and the 2 blue lines depict the parabolically asymptotic behavior of the Rashba SOC induced splitting bands near the Γ point. The distribution of (D) $O N e − 1 k$ and (E) $O N e k$ in the 2D BZ. (F) The distribution of $σ xy z$ at E_F. (G) The calculated SHC vs. the chemical potential (ranging from E_F − 0.5 eV to E_F + 0.5 eV).

Fig. 4. (Color online) (A) Schematic diagram of HER process on PtTe_1.75 monolayer. Top and side views of (B) the most stable (PtTe_1.75-I) and (C) metastable (PtTe_1.75-II) structures after H atom adsorption. The red ball denotes the absorbed H atom. (D) Variation of the Gibbs free energy for hydrogen adsorption (ΔG_H^∗) to different compounds. (E) Volcano plot depicting the relationship between exchange current density (i₀) and ΔG_H^∗, in which cases of Pt(111) [53], Rh [53], Pt/C [57], PtTe₂ with ordered trigonal Te vacancies (PtTe₂-V_Te) [51], MoS_1.76 [52], and 2H-1T phase boundaries of MoS₂ (MoS₂-PBs) [58] are also included for comparison.

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