Introduction

Global environmental problems and energy crisis are two of the most serious problems facing humanity1,2,3. Therefore, hydrogen energy, as a clean and sustainable energy, has received unprecedented attention4,5. In contrast with steam methane reforming6, which also emits a mass of CO2, water decomposition7,8 is a cleaner and easier way to produce hydrogen. However, the energy consumption required for water electrolysis is very high, making the development of high-efficiency and stable catalysts particularly necessary. The hydrogen evolution reaction (HER) is precisely the half-reaction that produces hydrogen in water electrolysis, and the design of high-activity catalysts for HER indeed enhances the activity of water splitting and effectively reduces energy consumption9,10. In general, regulating the adsorption behavior of intermediates such as ⋆H can promote the electron transfer directly and be used to improve catalytic activity. At present, the volcano plot11 which correlates the electrocatalytic efficiency with the Gibbs free energy ΔG has been widely used to reveal the activity of HER catalysts. The Gibbs free energy of hydrogen adsorption on the surface of an efficient HER catalyst is close to zero, i.e., at the peak of the volcano plot. The well-known high-activity HER catalysts Pt and Pd both follow this rule, but their high prices prompt people to seek alternatives.

Besides single-atom catalysts12,13, heterostructures14, and alloying15 can reduce the cost, topological materials, including topological insulators (TIs)16,17,18 and topological semimetals (TSMs)19,20 are also excellent candidates for HER catalysts21,22,23,24,25. TI is insulated internally, while its surface is conductive. The electronic properties of TSMs are similar to those of TIs, except that its bulk band structures have band crossing points near Fermi level, which makes it behave as a semimetal internally. According to the degeneracy of the band crossing points known as topological points, TSMs can be divided into four categories: Weyl TSMs26,27,28,29,30,31,32,33, Dirac TSMs34,35, nodal-line TSMs36, and high-fold degenerate TSMs37,38. Therefore, they all have nontrivial topological surface states with high-mobility electrons moving along unidirectional conduction channels. The topological surface states are protected by bulk crystal symmetries and robust against backscattering. The high-mobility electrons and stability are exactly what HER catalysis requires, and the crucial role of topological surface states in facilitating surface electrochemical reactions has been demonstrated by serving as an effective electron bath39. Recently, nonmagnetic Weyl TSMs TaAs family of materials40, nodal-line TSMs TiSi family of materials41, Dirac TSM PtSn442, ferromagnetic Weyl TSM Co3Sn2S243, type-II Weyl TSMs W/MoTe244,45, and several high-fold degenerate TSMs with chiral structures, such as PtGa/Al46, CoSi47, NiSi48, and TcSi49, are predicted as high-activity HER topological catalysts theoretically. However, only a few topological catalysts have been verified in experiment, and Pt remains the most efficient catalyst for applications. Among the limited number of topological catalysts, the high-fold degenerate TSMs with chiral structures show remarkable potential in catalysis. Due to the lack of mirror and inversion symmetry, the nontrivial topological points with nonzero Chern number in TSMs with chiral structures locate at different energy, making the energy window of nontrivial states larger. According to the experimental study of PtGa and PtAl46, which have better HER catalytic performance than Pt, this large energy window of nontrivial states plays an important role in enhancing the catalytic activity. Therefore, it is expected to discover more efficient topological catalysts in high-fold degenerate TSMs with chiral structures.

In this work, in order to explore high-activity HER topological catalysts through high-throughput screening founded on the first-principles calculations, based on the database of high-fold degenerate TSMs with chiral structures that we have established in previous work50, we focus on TSMs with the same point group T as CoSi family of materials, only a few of which have been theoretically and experimentally predicted as excellent HER catalysts51,52. We have not only verified the high catalytic performance of PtGa and PtAl in experiment, but also predicted that the catalytic activity of PtPbTe and Pd3Pb2S2 is not inferior to them. In addition, a large number of topological catalysts with good catalytic performance, such as PtBiTe, NiPS, and Pd3Bi2S2, have also been discovered. Totally 16 high-fold degenerate topological catalysts, whose Gibbs free energies ∣ΔG∣ are smaller than the calculated ∣ΔG∣ of Pt (i.e., 0.156 eV), are explored. More importantly, in the same compound, the surface with topological surface states can provide higher catalytic activity than the surface without topological surface states, which directly verifies the positive impact of topological surface states with high-mobility electrons on HER catalysis. Thus, this work not only provides more promising topological catalysts for HER catalysis but also proposes an effective method for developing high-activity catalysts.

Results and discussion

Among the 146 nonmagnetic high-fold degenerate TSMs with chiral structures discovered in our previous work50, we focus on 82 TSMs having the same point group T with CoSi family of materials to investigate the impact of topological states on topological catalysis and explore more high-activity HER topological catalysts. Since the adsorption calculations on the surface of supercell require a lot of computational resources, the TSMs having too many atoms (more than 20 atoms per unit cell) are not considered for the time being. Therefore, we obtain 47 high-fold degenerate TSM candidates, including 42 TSMs with space group P213 (No. 198) and 5 TSMs with space group I213 (No. 199), whose lattice constants are shown in Supplementary Tables S1–2 and Supplementary Figs. S1–3.

According to the crystal structures, we classify them into three types: the binary compounds with space group P213, the ternary compounds with space group P213, and the compounds with space group I213. The CoSi family of materials is all in the first type. To better exhibit the crystal structures, as shown in Fig. 1, PtGa, PtPbTe, and Pd3Pb2S2, which respectively have 8, 12, and 14 atoms per unit cell, are chosen as representatives of these three types. Since all the compounds belong to the same point group T (or 23 in international symbols), they manifest three kinds of symmetry operations: three-fold rotation symmetry with [111] axis, two-fold screw rotation symmetry with [100], [010] and [001] axes, and three-fold screw rotation symmetry with [\(1\overline{1}\overline{1}\)], [\(\overline{1}1\overline{1}\)] and [\(\overline{1}\overline{1}1\)] axes. The high-fold degenerate points located at the high-symmetry k-points are marked in the band structures in Figs. 1i–k. For compounds with space group P213 like PtGa and PtPbTe, two four-fold degenerate points \({H}_{PG/PPT}^{\Gamma }\) and \({H}_{PG/PPT}^{M}\) with Chern number −4 and ±2 locate at Γ and M points, respectively, while the Chern number of six-fold degenerate point \({H}_{PG/PPT}^{R}\) at R point is +4. The compounds with space group I213 can also host three-fold degenerate point \({H}_{PPS}^{P}\) with Chern number +2 at P point and two four-fold degenerate points \({H}_{PPS}^{\Gamma }\) and \({H}_{PPS}^{H}\) with Chern number  ± 4 at Γ and H, respectively.

Fig. 1: Crystal and band structures of the high-fold degenerate topological semimetals (HDTMs).
Fig. 1: Crystal and band structures of the high-fold degenerate topological semimetals (HDTMs).The alternative text for this image may have been generated using AI.
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a–c Lattice structure of binary HDTMs with space group P213 (No. 198), ternary HDTMs with space groups P213 (No. 198) and HDTMs with space group I213 (No. 199), respectively. d–f Side view of the lattice structure in a–c. g, h Brillouin zone of space group P213 and I213. i–k Band structure of PtGa with space group P213, PtPbTe with space group P213, and Pd3Pb2S2 with space group I213, respectively.

For HER catalysis, the adsorption of H atoms on the surface of a solid plays a decisive role in the reaction. In order to construct the adsorption surface, we test all the possible surfaces of PtGa, PtPbTe, and Pd3Pb2S2 in Supplementary Fig. S4. The differences in the surface energy are not significant for each compound, but comparatively speaking, the (001) surface possessing topological surface states is most stable. Therefore, we mainly focus on the HER on (001) adsorption surface when discussing the catalytic performance of the high-fold degenerate TSMs candidates.

The distribution of adsorption sites has a significant impact on the efficiency, selectivity, and performance of catalytic reactions in the adsorption process. Accordingly, we construct slab models with vacuum space along the [001] direction for each compound, and consider all possible adsorption configurations to identify the optimal adsorption sites for H atoms, which is shown in Figs. 2a–c. After the atoms on surface and adsorbed H have fully relaxed, the most stable adsorption sites for three representatives are shown in Figs. 2d–f, while the other high-fold degenerate TSM candidates can be found in Supplementary Figs. S8–S10. There are totally 36 TSM candidates possessing top adsorption sites, most of which are located at the top of transition metals due to the effect of d orbits, such as PtGa shown in Fig. S14. Meanwhile, the bridge sites exist in 11 TSM candidates, and the adsorbed H shifts towards top site slightly in most of them. As a whole, the top site is most stable for the compounds with T point group. If the candidate contains Pt, the top site of Pt is the optimal choice for adsorption, such as in representatives PtGa and PtPbTe. Besides, the top site of S is another choice when Pt is not contained, such as in representative Pd3Pb2S2.

Fig. 2: Adsorption site for H atom and differential charge density after adsorption.
Fig. 2: Adsorption site for H atom and differential charge density after adsorption.The alternative text for this image may have been generated using AI.
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a–c The potential adsorption sites in PtGa, PtPbTe, and Pd3Pb2S2, respectively. t/b/h represents top/bridge/hollow site. d–f The most stable adsorption sites in PtGa, PtPbTe, and Pd3Pb2S2. Only one layer of atoms is shown in the top views. g–i The differential charge density of PtGa, PtPbTe, and Pd3Pb2S2 when H atom is adsorbed. The yellow/blue surfaces represent the increasing/decreasing of density of electrons. All red balls are H atoms.

After determining the optimal adsorption site, the differential charge density of adsorption of H atoms is calculated to determine the transfer of electrons. As shown in Figs. 2g, h, the representatives PtGa and PtPbTe, in which H atom is adsorbed at the top of the transition metal Pt, the electrons transfer from Pt to H evidently. A few atoms in the next layer are also getting involved in the process of adsorption, which affects the direction of bonding. Similarly, all TSM candidates transfer the electrons from atoms on surface to adsorbate H atom except CuSeX (X=S or Te) with space group P213 and A3B2S2 (A=Ni or Pd, B=Bi or Pb) with space group I213, which is shown in Supplementary Figs. S11–13. For example, in the representative Pd3Pb2S2, H is adsorbed at the top of nonmetallic element S, the electronic distribution around H and S atoms changes complicatedly as depicted in Fig. 2i. The charge density increases near H and S atoms, and at the same time, the distribution of charge surrounding H atom is narrowed down. The transition metal Pd on surface also contributes electrons during the transferring process, although H is not adsorbed at the top of Pd atoms directly. Other A3B2S2 compounds with space group I213 have the same characteristics with Pd3Pb2S2. Therefore, for all of the high-fold degenerate TSMs candidates, adsorbate H atoms obtain electrons from atoms on surface, and transition metals also play an important role if they are included.

Subsequently, based on the adsorption site that is most stable in energy, the catalytic efficiency of HER for all the TSM candidates is characterized by the adsorption energy and displayed in the volcano plots of three TSM types in Fig. 3. The hydrogen adsorption Gibbs free energy ΔG for each TSM candidate is calculated by Eq. 2. Firstly, for binary TSM candidates with space group P213 (i.e., typical high-fold degenerate TSMs CoSi family of materials), there are seven compounds TcSi, PtGa, PtAl, CoGe, CoSi, RhSn and NiSi showing better catalytic performance than Pt as illustrated in Fig. 3a. If considering the deviation of Gibbs free energy between calculation in this work and experiment in Ref. 11, the catalytic performance of PtGa, PtAl, CoGe and CoSi are also superior to that of Pt. Among them, PtGa and PtAl have already been verified to have lower overpotential, lower Tafel slope, higher turnover frequency (TOF), and higher HER operational stability than Pt in experiment46. Besides PtGa and PtAl, TcSi49, CoSi47, and NiSi48 have also been theoretically proposed to have good HER catalytic activity. All these results show that our high-throughput calculation on HER topological catalysts is reliable enough to explore high-activity catalysts. Secondly, among the ternary TSM candidates with space groups P213, which have been experimentally verified as high-fold degenerate TSMs53, 7 compounds display better catalytic performance than Pt in calculation, while 6 compounds are demonstrated to have better performance than Pt in experiment as shown in Fig. 3b. Except for NiPS, they all contain precious metals Pt that play a unique role in catalysis. Finally, in the candidates with space group I213 predicted theoretically as high-fold degenerate TSMs, both Pd3Pb2S2 and Pd3Bi2S2 can exhibit excellent performance in HER according to the volcano plot in Fig. 3c. It is evident that the representatives PtGa, PtPbTe, and Pd3Pb2S2 located at the peak of the volcano plots of three TSM types are the best choices for catalysis. By comparing these three compounds and Pt in experiment and theory, illustrated in the reaction coordinate diagram (Fig. 3d), the Gibbs free energy of PtGa and PtPbTe have little difference and is the smallest among all the TSM candidates. Overall, we determine that 16 high-fold degenerate TSMs, of which 11 compounds contain precious metals, achieve high catalytic efficiency comparable to the traditional high-efficiency catalyst Pt by using high-throughput calculation.

Fig. 3: Volcano plots.
Fig. 3: Volcano plots.The alternative text for this image may have been generated using AI.
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a Volcano plot of binary HDTMs with space group P213. The red solid dots represent the results of this work, and the blue ones are the results of other previous work46,47,48,49. b Volcano plot of ternary HDTMs with space group P213. The red hollow dots show the results in this work. c Volcano plot of HDTMs with space group I213. The blue hollow dots show the results in this work. In all the figures above, the black solid and hollow dots represent experimental results and calculated Pt in this work, respectively. d Reaction coordinate diagram of three representatives, PtGa, PtPbTe, and Pd3Pb2S2, which are the best candidates of catalyst for each kind of HDTMs.

For all the TSM candidates with Pt element except PtMg, the optimal adsorption sites are same, i.e., the top site of Pt. Meanwhile, they all represent good HER performance with ∣ΔG∣ ≤ 0.163 eV near the peak of the volcano plot. Thus, the Pt element in TSM candidates plays an indispensable role in the surface adsorption of hydrogen. Compared to Pt, the topological surface states of topological catalysts containing Pt can provide high-mobility electrons, enhancing the catalytic efficiency. The topological surface state dispersion and Fermi arc of (001) surface are analyzed systematically for the representatives PtGa and PtPbTe in Figs. 4a–d. The four-fold degenerate point at M and the six-fold degenerate point at R are projected into the high symmetry k-points \(\overline{M}\) and \(\overline{\Gamma }\) of the surface Brillouin zone, respectively. Although most of the topological surface states are merged into the bulk state, especially in PtPbTe, four surface states connecting the four-fold degenerate point \({H}_{PG/PPT}^{\Gamma }\) with Chern number −4 can be observed. For PtGa, the surface states mainly provided by Pt elements on surface, as shown in Supplementary Fig. S14, extend from the degenerate point about 0.1 eV below Fermi level to higher energy above Fermi level. The large energy window of topological states and extremely long Fermi arcs enable surface adsorption to adapt to greater changes in energy and momentum. Besides Pt, precious metal Pd also appears in some topological catalysts. For example, in Pd3B2S2 (B=Bi or Pb), belonging to the third type of TSM with space group I213, the Pd atoms located in the top two atomic layers contribute to the H adsorption according to the differential charge density in Supplementary Fig. S13. Consistent with topological catalysts containing Pt element, when a compound containing Pd element hosts topological surface state correlated with Pd as shown in Figs. 4e, f, the catalytic performance is slightly improved compared with the traditional high-efficiency catalyst Pd.

Fig. 4: The (001) topological surface states (TSS).
Fig. 4: The (001) topological surface states (TSS).The alternative text for this image may have been generated using AI.
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a Surface dispersion of binary HDTM PtGa. b Topological Fermi arcs of binary HDTM PtGa with energy fixed at the energy E1 of four-fold degenerate fermion \({H}_{PG}^{\Gamma }\). c Surface dispersion of ternary HDTM PtPbTe. d Topological Fermi arcs of ternary HDTM PtPbTe with energy fixed at the energy E2 of four-fold degenerate fermion \({H}_{PPT}^{\Gamma }\). e Surface dispersion of HDTMs Pd3Pb2S2. f Topological Fermi arcs of HDTMs Pd3Pb2S2 with energy fixed at the energy E3 of four-fold degenerate fermion \({H}_{PPS}^{\Gamma }\). The green, blue, and red dots are four-fold, six-fold, and three-fold degenerate fermion, respectively.

In order to delve into the effect of topological surface state in the high-efficiency topological catalysts, the surfaces with and without topological surface states for CoSi unrelated to the precious metals, are established when H is adsorbed on the catalyst. It has been verified in experiment that the Fermi arcs can emerge in (001) surface, but not in (111) surface in high-degenerate TSM CoSi54. In the [111] direction, similar to topological catalyst PtGa in Supplementary Fig. S5, there are three possible surfaces with Co-layer, Si-layer, or CoSi-layer. The optimal adsorption sites in (111) surfaces conform the symmetry of crystal well, such as Co-layer with the hole site of three Co atoms in Figs. 5a, b. The absolute value of Gibbs free energy ∣ΔG∣ of (111) surface with Co-layer, Si-layer, and CoSi-layer is respectively calculated to be −0.874 eV, 0.208 eV, and −0.243 eV, and larger than ∣ΔG∣ of (001) surface as illustrated in Fig. 5c. Thereafter, the topological surface states in (001) and (111) surface of CoSi are also calculated in Figs. 5d, e. The topological Fermi arcs of (001) surface are observed to connect high-fold degenerate points clearly, whereas no topological surface state exists in (111) surface because two high-fold degenerate points with opposite Chern number projected into same point \(\overline{\Gamma }\). The fact that catalytic performance of the surface with topological surface state is better than that of the surface without nontrivial state in CoSi indicates that the topological surface states in high-fold degenerate topological catalysts indeed optimize the catalytic performance in HER.

Fig. 5: The (111) surface of CoSi.
Fig. 5: The (111) surface of CoSi.The alternative text for this image may have been generated using AI.
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a, b The side and top views of stable adsorption site on (111) surface with Co-layer. c Reaction coordinate diagram when H is adsorbed on (001) surface and (111) surface with Co-layer, Si-layer, and CoSi-layer. d, e The constant energy surfaces of (001) and (111) surfaces, respectively, in case the energy is fixed at the energy of the high-fold degenerate point.

Considering the stability of topological materials and the high-mobility electrons provided by topological surface states, topological catalysts have an advantage over other kinds of catalysts in electrochemical reactions. Referring to the outstanding HER catalytic performance of high-fold degenerate TSMs with chiral structures, which have extremely long topological surface states in k-space, we have discovered 16 high-fold degenerate topological HER catalysts whose catalytic activity is not inferior to that of traditional high-efficiency catalyst Pt. Among them, only 5 compounds have been predicted theoretically, and PtGa, PtAl, and CoSi have been verified experimentally. Based on the study of 11 topological catalysts containing precious metals Pt, Pd, or Rh, the topological surface states mainly formed by the d orbits of the precious metals can enhance catalytic efficiency. Almost all the high-fold degenerate TSMs containing Pt element in our database exhibit excellent HER catalytic performance, indicating that high-efficiency HER catalyst design can be achieved by synthesizing TSMs using monometallic catalysts. Meanwhile, 5 topological catalysts without precious metals, namely CoSi, CoGe, TcSi, NiSi, and NiPS, can also exhibit excellent HER catalytic performance, making it possible to greatly reduce the cost of catalysts. Most importantly, by comparing (001) surface and (111) surface with and without topological surface states in CoSi, the adsorption of H atoms on the surface with topological surface states exhibits smaller absolute value of Gibbs free energy. The significance of topological surface states providing high-mobility electrons is intuitively verified to improve catalytic efficiency. Therefore, this work not only significantly expands the number of topological HER catalysts and promotes their industrial application, but also theoretically validates the importance of topological surface states in topological catalysis, which also potentially leads to efficient topological catalysts in crucial electrocatalytic reactions beyond HER, such as the oxygen evolution reaction (OER) or CO2 reduction.

Methods

The density functional theory (DFT) calculations have been conducted in the Vienna Ab Initio Simulation Package (VASP)55 with the Perdew–Burke–Ernzerhof (PBE)56 exchange-correlation functional. The cut-off energy for plane-wave basis is 400 eV. The Monkhorst-Pack k-mesh for bulk and supercell calculations in self-consistent process are 8 × 8 × 8 and 8 × 8 × 2, respectively. All the lattice constants are fully relaxed. The spin-orbit coupling (SOC) is considered from self-consistent field calculations throughout the work. In order to investigate the topological surface states of high-fold degenerate TSMs, the maximally localized Wannier functions (MLWF)57 are used to generate the tight-binding Hamiltonian based on the orbits near Fermi level. Afterwards, the Green’s function method58,59 is used to calculate the surface states by establishing a half-infinite boundary model. The topological invariants are calculated by using the WannierTools package60.

For the surface modeling, a slab with (1 × 1 × 3) unit cell for ternary compounds with space group P213 and (1 × 1 × 4) unit cell for binary compounds with space group P213 and I213 is constructed. A 12 Å vacuum slab in the direction perpendicular to the surface for adsorbing is applied to avoid the periodic interaction. The atomic positions of the top layer of supercells are fully relaxed until the force on each atom is less than 10−3 eV/Å. Thus, according to the adsorption process M+H++e− → M-H, the adsorption energy could be calculated by

$$\Delta {\text{E}}={\text{E}}_{{\rm{M}}-{\rm{H}}}-{\text{E}}_{{\rm{s}}{\rm{c}}}-\frac{1}{2}{\text{E}}_{{{\rm{H}}}_{2}}$$
(1)

where EM − H, Esc, and \({E}_{{{\rm{H}}}_{2}}\) represent the total energy of adsorption system M-H including TSM supercell and adsorbate ⋆H, TSM supercell M without adsorbate, and isolated adsorbate molecule H2, respectively. Here, the energy of adsorbate atom ⋆H is approximated as half of \({E}_{{{\rm{H}}}_{2}}\). Furthermore, the Gibbs free energies are calculated by

$$\Delta {\text{G}}=\Delta {\text{E}}+\Delta {\rm{Z}}{\rm{P}}{\rm{E}}-{\text{T}}\Delta {\text{S}}=\Delta {\text{E}}+0.24\,{\rm{e}}{\rm{V}}$$
(2)

where ΔE is the adsorption energy, ΔZPE is the difference of zero-point energy between adsorbed and gas phase, ΔS is the entropy of adsorption of H, which is taken as half of the entropy of hydrogen at standard conditions, and T is the room temperature. The value of ΔZPE is calculated to be 0.04 eV for H/Cu(111)11, and then applied to all compounds in this work since the vibrational frequencies have been found to depend much less on the metal than the bond strength61.