金属间化合物Pt7Sb投影Berry相位与析氢催化关联的第一性原理计算 (2024)

金属间化合物Pt₇Sb投影Berry相位与析氢催化关联的第一性原理计算

周彦余¹^,², 李江旭¹, 刘晨¹^,², 赖俊文¹^,², 高强¹, 马会¹^,², 孙岩^,¹^,², 陈星秋¹^,²

1 中国科学院金属研究所沈阳材料科学国家研究中心沈阳 110016

2 中国科学技术大学材料科学与工程学院沈阳 110016

First-Principles Study of Projected Berry Phase and Hydrogen Evolution Catalysis in Pt₇Sb

ZHOU Yanyu¹^,², LI Jiangxu¹, LIU Chen¹^,², LAI Junwen¹^,², GAO Qiang¹, MA Hui¹^,², SUN Yan^,¹^,², CHEN Xingqiu¹^,²

1 Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China

2 School of Materials Science and Engineering, University of Science and Technology of China, Shenyang 110016, China

Abstract

With the increase of global energy consumption and related environment pollution, new types of renewable clean energy resources and carriers are desirable. Given its high gravimetric energy density and combustion product (i.e., water), molecular hydrogen has attracted considerable attention. Obtaining molecular hydrogen from water splitting is the ideal strategy because inputs and outputs are carbon-free clean matter. In achieving this process, a suitable and highly efficient catalyst is a crucial parameter. Novel metal Pt is an excellent catalyst with high efficiency and chemical stability. However, owing to its high cost and insufficient reserves on Earth, the wide application of Pt in catalysis is strongly limited. Correspondingly, the design of a highly efficient hydrogen evolution reaction (HER) catalyst with low Pt loading is an important task for electrochemical water splitting in the field of renewable energy resources. Understanding the hidden mechanism is essential for the guiding principle of such a design. In this study, an excellent HER catalyst in cubic Pt₇Sb is proposed, in which Gibbs free energy for hydrogen adsorption ( $Δ G_{H^{*}}$ ) is smaller than that from Pt. Thus, together with its good chemical stability, a better HER catalytic activity with reduced Pt loading can be obtained. Based on the analysis of electronic structures, a good agreement between the two descriptors of $Δ G_{H^{*}}$ and the projected Berry phase (PBP) is revealed. Considering that the PBP is purely decided by the bulk state, such an agreement indicates a strong relationship between the good catalytic performance and the topological nature of the intrinsic electronic structure. This work provides an excellent HER catalytic candidate with reduced Pt loading and a good example to show the role of the intrinsic topological nature in catalysts.

Keywords：hydrogen evolution reaction;catalyst;projected Berry phase;first-principles calculation

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本文引用格式

周彦余, 李江旭, 刘晨, 赖俊文, 高强, 马会, 孙岩, 陈星秋. 金属间化合物Pt₇Sb投影Berry相位与析氢催化关联的第一性原理计算[J]. 金属学报, 2024, 60(6): 837-847 DOI:10.11900/0412.1961.2022.00129

ZHOU Yanyu, LI Jiangxu, LIU Chen, LAI Junwen, GAO Qiang, MA Hui, SUN Yan, CHEN Xingqiu. First-Principles Study of Projected Berry Phase and Hydrogen Evolution Catalysis in Pt₇Sb[J]. Acta Metallurgica Sinica, 2024, 60(6): 837-847 DOI:10.11900/0412.1961.2022.00129

在过去的几十年里，随着全球能源消耗的快速增长和化石燃料带来的相关环境污染问题，人们致力于研究新型可再生能源和清洁能源载体。在所有替代品中，H₂因其高能量密度、可持续性和无碳燃烧副产物^[1~4]而成为最有希望的候选之一。目前，大部分H₂是由化石燃料的蒸汽重整产生^[5]，其中的主要副产物CO₂是不可避免的。因此，要使H₂成为一种实用的无碳燃料，还需要无碳资源。电化学分解H₂O为H₂和O₂是最理想的方法，在整个循环中，起始点和H₂燃烧产物都是H₂O。由于H₂O的电解需要较大的正Gibbs能量，因此高效、低成本的析氢反应(HER)催化剂是实现氢经济的关键。Pt具有低的反应活化能、Tafel斜率和过电位，被认为是最有效的HER电催化剂^[6~8]。但是，由于铂基催化剂的高成本和低储量，以降低Pt负载量为目标的设计和优化以及对相应机理的理解引起催化界的广泛关注^[9~12]。

近年来，学者们主要利用吸附自由能和金属-氢键强度的计算来寻找性能优异的催化剂，它们都与催化剂的功函数、d带中心或d带形状等简单的物理参数有关^[13]。科研人员^[14~16]发现高效催化剂Pt、Au等是Z₂拓扑金属，这表明拓扑能带结构与催化之间可能存在某种关系。近10年来，随着对拓扑材料中一些高效催化剂的研究，发现拓扑非平庸材料具有许多有趣的性质，如非平庸拓扑表面态、高载流子迁移率、手性Fermi子(Fermion)、超长的Fermi弧等。拓扑材料中受拓扑保护的表面态^[17~19]和由线性能带交叉引起的高迁移率^[20~23]在理论和实验上被证明可以有效调整表面吸附行为和异质催化反应中的电子转移，是高效催化剂的良好候选材料。虽然HER的化学反应发生在表面上，但表面本身强烈依赖于体带结构。于是，Xu等^[24]提取了一个纯粹的本征物理参数，即投影Berry相位，它只依赖于体电子结构，且与HER的催化效率之间存在线性关系，可作为预测和设计HER催化剂的新型描述符。

根据寻找低Pt负载量高效催化剂的指导原则，本工作以Pt₇Sb和Pt₇Cu为研究对象，利用基于密度泛函理论(DFT)的第一性原理计算方法，计算了氢吸附Gibbs自由能( $Δ G_{H^{*}}$ )以确定催化活性，同时基于对催化反应中电子拓扑作用的理解^[21,25]，计算了体投影Berry相位^[24]，将表面Gibbs自由能得到的火山图与从体带结构得到的投影Berry相位进行比较，从而深入理解Pt₇Sb的优异催化性能与本征拓扑性质之间的关系。

1 理论方法和模型

本工作采用从头计算模拟软件包VASP^[26,27]用于所有能量最小化计算，并使用经过充分优化的fcc Pt₇Sb获得表面模型。所有DFT计算均采用广义梯度近似(GGA)下的Revised-Perdew-Burke-Ernzerhof (RPBE)交换关联泛函来完成^[28]。采用高斯展宽(Gaussian smearing)方法^[29]描述总能量，宽度设置为0.05 eV。计算过程中，分别将Pt_5d⁹6s¹、Sb_5s²5p³和H_1s¹轨道上的电子视为价电子。采用共轭梯度(CG)算法将离子弛豫到基态。收敛计算期间的平面波截断能、能量标准和力的阈值分别设置为400 eV、10^-6 eV和0.1 eV/nm。对于Pt₇Sb的体和平板模型，Brillouin区k点采样分别采用8 × 8 × 8和8 × 8 × 1网格的Monkhorst-Pack方案计算，所有平板模型的真空高度均为1.5 nm。选择具有不同终端的Pt₇Sb平板模型，并考虑晶体结构的晶格对称性，如图1所示，A和B分别表示Pt-Sb混合终端和纯Pt终端的表面。所考虑的平板模型的表面能可通过下列公式计算^[30~33]，表面能( $σ$ )的广泛定义为：

$σ = \frac{1}{2 A} (E_{s l a b} - N_{P t} μ_{P t} - N_{S b} μ_{S b} + P V - T S)$ (1)

式中，E_slab是5层的Pt终端或混合终端平板模型弛豫后的总能量，A是表面积，µ_Pt和µ_Sb分别是Pt₇Sb体结构中Pt和Sb的化学势， $N_{P t}$ 和 $N_{S b}$ 是平板模型中相应的原子数目，P是压强，V是体积，T是温度，S是熵。在0 K和常压下，PV和TS项可忽略不计。结合Pt₇Sb体化学势( $μ_{P t_{7} S b}$ (bulk))，式(1)可改写为：

$σ = \frac{1}{2 A} [E_{s l a b} - \frac{1}{7} N_{P t} μ_{P t_{7} S b} (b u l k) + (\frac{1}{7} N_{P t} - N_{S b}) μ_{S b}]$ (2)

图1

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图1具有不同晶格方向和终端的Pt₇Sb平板模型和相应的表面能

Fig.1Pt₇Sb slab models with the different lattice orientations and terminations (a) and the corresponding surface energy variations with relative chemical potential ( $μ_{S b} - μ_{S b} (b u l k)$ ) (b) (The surface energies of six situations with lattice orientation along with three high symmetrical directions of (100), (110), and (111), respectively; A and B indicate different types of terminations, with A representing Pt-Sb mixed termination and B representing pure Pt termination; $μ_{S b}$ is the chemical potential of Sb in the Pt₇Sb bulk structure and $μ_{S b} (b u l k)$ is the chemical potential of Sb in the monomer)

Pt₇Sb的形成能(ΔH_f⁰)的定义如下：

$Δ H_{f}^{0} = μ_{P t_{7} S b} (b u l k) - 7 μ_{P t} (b u l k) - μ_{S b} (b u l k)$ (3)

当假设表面和体处于平衡状态时，体的化学势( $μ_{P t_{7} S b}$ (bulk))、0 K下的ΔH_f⁰和元素在其单质中的化学势(µ_Pt(bulk)，µ_Sb(bulk))由上式关联。此外，化合物中每种元素的化学势必小于在其单质中的化学势：

$μ_{P t} \leq μ_{P t} (b u l k)$ (4)

$μ_{S b} \leq μ_{S b} (b u l k)$ (5)

同时，利用式(3)~(5)可得Sb的相对化学势( $μ_{S b} - μ_{S b} (b u l k)$ )的变化范围：

$Δ H_{f}^{0} \leq μ_{S b} - μ_{S b} (b u l k) \leq 0$ (6)

氢吸附Gibbs自由能( $Δ G_{H^{*}}$ )的计算公式为^[34~36]：

$Δ G_{H^{*}} = Δ E_{H} + Δ E_{Z P E} - T Δ S_{H}$ (7)

式中，ΔE_H是H的吸附能，ΔE_ZPE和ΔS_H分别是吸附状态和气相之间的零点振动能差和熵差。T设定为298 K。吸附状态下的振动熵很小，相对于1/2H₂可以忽略，所以 $Δ S_{H} = - 1 / 2 S_{H_{2}}$ ，其中 $S_{H_{2}}$ 使用标准状态下H₂的熵值。因此，此公式中-TΔS_H这项被定义为-0.2 eV^[37]。除此之外，H的吸附能可通过以下公式计算^[34~36]：

$Δ E_{H} = E_{s l a b + H^{*}} - E_{s l a b} - \frac{1}{2} E_{H_{2}}$ (8)

式中， $E_{s l a b + H^{*}}$ 是催化剂表面氢吸附状态总能量， $E_{H_{2}}$ 是气相中H₂的能量。火山图模型中pH = 0时的交换电流密度i₀作为 $Δ G_{H^{*}}$ 的函数，计算公式如下^[37]：

$i_{0} = - e k_{0} \frac{1}{1 + e x p (- \frac{|Δ G_{H^{*}}|}{k_{B} T})}$ (9)

式中，k₀是速率常数(设定为200 s^-1·site^-1)，e是元电荷，k_B是Boltzmann常数。

对于投影Berry相位的计算，方法参考文献[24]。

$F_{x y} (k) = 2 I m \sum_{E_{n} < E_{o c c}} \sum_{m \neq n} \frac{〈n k | v_{x}^{'} | m k〉〈m k | v_{y} | n k〉}{{(E_{n k} - E_{m k})}^{2}}$ (10)

$γ = |\int_{k} F_{x y} (k) d k|$ (11)

式中， $v_{x}^{'} = \{p v_{x}\}$ ， $p = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})$ ， $v_{i}$ ( $i = x, y$ )表示速度算符； $| n k〉$ 是Hamilton量H在k点和能带n的特征向量，特征值为 $E_{n k}$ 。类似于普通的Berry曲率，把 $F_{x y} (k)$ 叫做投影Berry曲率， $γ$ 是投影Berry相位。

2 计算结果及讨论

2.1 晶体结构与表面能

化合物Pt₇Sb与Pt共属同一空间群Fm $\bar{3}$ m，空间群编号为225。Pt₇Sb和Pt的晶体结构如图2a和b所示。可以看出，Pt₇Sb单胞比Pt单胞含有更多的原子。与Pt相比，Pt₇Sb的Pt负载量更低。首先对Pt₇Sb进行了晶体结构弛豫，得到晶格常数a = b = c = 0.8089 nm，与之前实验^[38]报道的a = b = c = 0.7948 nm相近。

图2

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图2Pt₇Sb和Pt的晶体结构图，Pt₇Sb(111)表面不同吸附位点示意图及Pt₇Cu、Pt、Pt₇Sb的能带结构

Fig.2Crystal structure diagrams of Pt₇Sb and Pt, schematic of different adsorption sites on Pt₇Sb (111) surface, and the band structures of Pt₇Cu, Pt, and Pt₇Sb
(a, b) crystal structures of cubic Pt₇Sb (a) and Pt (b)
(c) (111) surface of Pt₇Sb with different surface adsorption sites labeled by crosses (The most stable site locates at the face-cubic center, as highlighted by the red cross)
(d-f) bulk band structures of Pt₇Cu (d), Pt (e), and Pt₇Sb (f), respectively (E is the energy, E_f is the Fermi level, and the horizontal coordinate represents the point of high symmetry in the inverse space)

为了进行对比分析，本工作还计算了与Pt₇Sb结构完全相同的Pt₇Cu。基于Pt₇Sb与Pt₇Cu原胞的晶体结构，计算了含有相同原子数量Pt的超胞的电子能带结构，并与Pt₇Sb和Pt₇Cu的电子能带结构进行了比较。如图2d~f所示，由于Pt₇Cu和Pt₇Sb可以被看作是在Pt超胞的基础上由Cu或Sb对Pt原子的有序替代掺杂而成，他们的能带结构必然会因为Cu或Sb轨道的引入在Pt的基础上发生变化。将图1f的Pt₇Sb能带与图2e的Pt能带结构相比，Sb的s、p轨道均分布在远离Fermi能级的区域，且主要分布在深能处，由于没有d轨道的存在，使Pt元素整体能带相较于Fermi能级下移。虽然Pt₇Sb和Pt具有相同的空间群，Sb原子对Pt原子的替代仍然会降低体系的对称性，使得Pt超胞中原来处于简并状态的能带在Pt₇Sb中发生劈裂。这种劈裂使得Fermi能处的一些带宽变窄，尤其是Γ点附近，如图2f所示。与Pt₇Sb类似，在Pt₇Cu中，由于对称性的降低，能带发生了劈裂。由于Cu含有3d电子，与Pt₇Sb相比，Pt₇Cu的电子能带结构更接近与Pt，如图2d所示。

一般来说，表面能对材料的稳定性和化学性质有很大的影响，表面能越小，相应的表面就越容易形成。为了得到稳定的表面，沿着Pt₇Sb的(100)、(110)和(111) 3个高对称方向构建平板模型，分别计算了其表面能。每个晶格方向主要有2种不同的情况，Pt-Sb混合终端(A)和纯Pt终端(B)，详见图1。基于3种不同的晶格方向和2种不同的终端，按照式(2)和(6)计算了每种情况的表面能。

图1b给出了这6个表面的表面能与Sb的相对化学势的函数关系，其中正或负的斜率分别表示富Pt或富Sb的表面更稳定。它们都表现为随化学势变化的直线，并且上下排列，由此很容易确定Pt-Sb混合终端的(111)表面是Pt₇Sb中最稳定的情况。作为参考，还计算了Pt₇Cu的表面能。按照类似的方法，可以得出，对于Pt₇Cu，在3个高对称方向中，纯Pt终端的(111)表面是最稳定的情况，如图3所示。对比计算Pt₇Sb和Pt₇Cu最稳定表面原子位置(即原子的Z坐标)的变化，发现Pt₇Sb (111)B表面的2种Pt变化分别为-0.004689和-0.004691 nm，Sb变化为0.012883 nm，而Pt₇Cu(111) A表面的2种Pt原子的变化分别为-0.001291 nm和0.002426 nm。同一表面原子位置的变化有正有负，这表明在所考虑的表面上发生了某些重构。并且值得注意的是，Cu和Pt的价电子均为s、d电子，而Sb原子的价电子为s、p电子，Pt-Sb混合表面的Pt_s轨道与Sb_p轨道之间可能存在一定的耦合，这不同于Pt₇Cu的Pt_s轨道与次外层表面的Cu_d轨道之间的耦合，从而导致表面重构，间接影响表面能使其稳定表面产生差异。基于(111)取向的平板模型，分别分析Pt₇Sb、Pt₇Cu和Pt的相应表面HER催化性能。

图3

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图3具有不同晶格方向和终端的Pt₇Cu平板模型和相应的表面能

Fig.3Pt₇Cu slab models with the different lattice orientations and terminations (a) and the corresponding surface energy variations with relative chemical potential ( $μ_{C u} - μ_{C u}$ (bulk)) (b) (The surface energies of six situations with lattice orientations along the three high symmetrical directions of (100), (110), and (111), respectively; A and B indicate the different types of terminations, with A representing Pt-Cu mixed termination and B representing pure Pt termination; $μ_{C u}$ is the chemical potential of Cu in the Pt₇Cu bulk structure and $μ_{C u} (b u l k)$ is the chemical potential of Cu in the monomer)

2.2 HER性能

催化剂表面的氢吸附Gibbs自由能是理解催化剂活性的一个有效描述符^[39]。HER机制涉及3个步骤^[40]。在第一步的Volmer反应中，一个电子转移到一个质子上，在催化剂表面形成氢原子H^*。接下来的步骤可以是Heyrovsky反应或Tafel反应(或2者的组合)，2个H^*相遇，产生H₂。因此，根据Sabatier原理^[41]，当催化剂与氢的键合不强也不弱时，吸附和解吸步骤都相对容易，对应的是高效的催化反应。基于这种理解，人们提出了 $Δ G_{H^{*}}$ 和交换电流密度之间的火山图^[37]，在相应的催化剂中绝对值小的Gibbs自由能往往伴随着高活性。

为了确定H原子在催化剂表面吸附时的相对稳定位置，利用式(7)和(8)及根据上述表面能计算得到的可吸附表面，对Pt₇Sb、Pt₇Cu和Pt进行了不同位点的吸附能计算。如图2c所示，Pt₇Sb的给定表面有9个吸附位点，分别是2个顶位、3个桥位和4个空心位点(2个hcp位和2个fcc位)。计算各个位点的吸附自由能，当 $Δ G_{H^{*}}$ 绝对值越大时即要使H脱附所需的能量越大，则该吸附构型越稳定。比较吸附自由能的大小发现，H原子在Pt-Pt-Pt空心(fcc)位点上的构型是Pt₇Sb能量最稳定的构型，在图2c中标记为红色交叉。得到Pt₇Sb的氢吸附自由能约为-0.0357 eV。而H原子在Pt₇Cu表面的Pt-top位点更稳定，其吸附自由能为-0.0390 eV。计算了具有相同H覆盖率的Pt平板模型的吸附自由能约为-0.0943 eV，与文献[37]中-0.09 eV的结果一致。因此，Pt₇Sb或Pt₇Cu的 $Δ G_{H^{*}}$ 值均小于Pt，且Pt₇Sb的值更小，意味着Pt₇Sb的HER性能更好，如图4a所示。

图4

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图4Pt₇Sb、Pt₇Cu和Pt的台阶图及其与文献^[20,37,42]对比所得火山图

Fig.4Step diagrams of Pt₇Sb, Pt₇Cu, and Pt; and the volcano plot (exchange current density i₀vs the free energy for hydrogen adsorption ( $Δ G_{H^{*}}$ )) obtained by comparison with literatures

(a) calculated free energy diagram for hydrogen evolution reaction (HER) at a potential U = 0 V relative to the standard hydrogen electrode at pH = 0 (The free energy of H⁺ + e^- is defined as the same as that of 1/2H₂ at standard conditions)

(b) volcano plot for the HER of Pt₇Sb and Pt₇Cu in comparison with various pure metals (The experimental^[37] and calculated^[42] data of Pt, Pd, Co, Ag, and Cu), topological Weyl semimetals (The calculated data of NbP, NbAs, and TaAs^[20]), and other candidates (the theoretical data of TaS₂(2H) and TaS₂(1T)^[20])

此外，与其他报道^[20,37,42]的典型催化剂进行了更为详细的比较，包括过渡族金属、拓扑Weyl半金属和过渡族金属硫化物，见图4b。可以看到，Pt₇Sb在火山图中处于一个特殊的位置，具有所有选定材料中最大的交换电流密度，满足了比Pt更低的成本及更好的HER催化活性，有望成为出色的催化剂候选材料。

2.3 电子结构

为了进一步理解Pt₇Sb的良好HER性能，详细分析了其电子结构。以Pt₇Sb为例，将Pt₇Sb与元素形式的Sb和Pt体态密度(DOS)进行了比较。从图5a可以看出，Pt₇Sb的DOS的形状接近于Pt，但与Sb非常不同。Pt₇Sb可以被看作是在Pt超胞的基础上由Sb掺杂而成，并且Pt_5d轨道主导了Fermi能级附近的态。同样，Pt₇Cu Fermi能级附近的态也是由Pt_5d轨道主导的，但Cu_3d轨道也做出了少量贡献，如图5b所示。这与前面关于电子能带结构的分析一致。基于这种理解，分析了表面有效区域(顶层与H相互作用的Pt原子)投影态密度(PDOS)的变化。图6a和b显示了Pt₇Sb吸附H原子前后PDOS的变化。虽然Pt_5d轨道在H₂吸附前后对Fermi能级附近的态的贡献占主导地位，但H₂吸附后顶部Pt_5d态的PDOS明显下降。这种变化意味着电荷从Pt₇Sb的表面转移到了H，同时形成一个弱化学键。图6c和d显示了Pt吸附H前后PDOS的变化，情况与Pt₇Sb类似。对于Pt₇Cu来说，Pt_5d轨道的PDOS在氢吸附后下降得更多，接近一个电子，如图6e和f所示，这是因为H在Pt₇Cu表面的top位与其成键，仅有一个Pt原子与H发生相互作用，将电荷转移给H并形成了化学键。

图5

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图5Pt₇Sb、Pt₇Cu、Pt体结构电子态密度及差分电荷密度分析图

Fig.5Bulk structure electronic densities of states and charge density difference analyses of Pt, Pt₇Sb, and Pt₇Cu (PDOS—projected electronic densities of states)

(a) PDOS for bulk Pt₇Sb, fcc Pt, and Sb, respectively

(b) PDOS for bulk Pt₇Cu, fcc Pt, and Cu, respectively

(c-e) top and side views of electron charge density differences for the hydrogen adsorption on Pt (c), Pt₇Sb (d), and Pt₇Cu (e) surfaces, respectively (Color code: Pt—grey; Sb—orange; H—red. The charge accumulation and depletion are depicted by the yellow and light blue regions, respectively. The isosurface levels are set to ± 0.002)

图6

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图6Pt₇Sb、Pt₇Cu、Pt投影电子态密度图

Fig.6PDOS for Pt₇Sb (a, b), Pt (c, d), Pt₇Cu (e, f) in the cases before (a, c, e) and after (b, d, f) hydrogen evolution

进一步，计算了Pt、Pt₇Sb及Pt₇Cu表面吸附H原子引起的实空间电荷密度差(Δρ)。Δρ通过 $Δ ρ = ρ^{s y s} - ρ^{s u r} - ρ^{H}$ 计算，其中 $ρ^{s y s}$ 、 $ρ^{s u r}$ 和 $ρ^{H}$ 分别为催化表面、纯表面和H原子的电荷密度。从图5c、d和e的比较可知，Pt、Pt₇Sb与Pt₇Cu的电荷转移均主要发生在平板模型的顶部原子层和吸附物H之间，与上述DOS分析完全一致。揭示了在这些表面催化反应中，吸附的H原子和催化剂的顶部原子层之间很容易发生原子相互作用。由于Pt₇Sb和Pt都具有空心的吸附位点，吸附的H和3个相邻Pt原子周围均发生了明显的电荷累积和耗散，H和表面Pt原子之间形成了一个空间。图5c和d中的电荷累积(黄色)表明，表面Pt原子与H之间存在着弱化学键。同时，由于H和Pt之间的电荷重排，H和Pt原子外的电子密度被耗散(蓝色)。而Pt₇Cu的吸附位点在top位，电荷的累积与耗散发生在H与特定Pt原子之间，有较明显的电荷累积，如图5e所示，形成了一个化学键。与Pt相比，Pt₇Sb的平板模型中有效区域内Pt原子之间的电荷耗散量及H原子上的电荷量累积量均略有减少，这表明平板模型和H之间的结合强度降低，H更易脱附，这与自由能绝对值的计算结果一致。

2.4 投影Berry相位

最近，有研究^[24]提出了一个新的描述符——投影Berry相位。由于它只取决于本征的体能带结构，不受溶液pH值影响，具有相对稳定和普适性，并显示出与交换电流密度之间近乎线性的关系，投影Berry相位已成为了一个连接本征拓扑状态和催化性能的工具。为了解其对HER催化性能的本征贡献，本工作计算了Pt₇Sb的投影Berry相位。由图7a投影Berry相位随能量的变化曲线可以看出，Pt₇Sb Berry投影相位的峰值处于-0.5 eV附近，其绝对值可以达到0.09，与Pt的值比较接近。在Fermi能处，Pt₇Sb Berry投影相位的绝对值可以达到0.065。在催化材料中，这是一个很大的数值，位于2个著名的高效HER催化剂Pd和Pt之间，如图7g所示，这也暗示Pt₇Sb具有很高的催化效率。这也可以从Pt₇Sb氢吸附Gibbs自由能的计算看出，如图4b所示。由于Pt和Pd的 $Δ G_{H^{*}}$ 几乎相互重叠(图4b)，Pt₇Sb的PBP和氢吸附Gibbs自由能之间的顺序差异可以在数值精度范围内理解。Pt₇Sb的PBP和 $Δ G_{H^{*}}$ 都几乎达到了已知材料上限值。

图7

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图7Pt₇Sb和Pt₇Cu的投影Berry曲率和投影Berry相位

Fig.7Projected Berry phases (γ) and Berry curvatures (F_xy ) projected of Pt₇Sb and Pt₇Cu
(a, d) the curves of the projected Berry phase (PBP) of Pt₇Sb (a) and Pt₇Cu (d) with energy
(b, c, e, f) electronic band structures (b, e) and local distributions of PBP (c, f) of Pt₇Sb (b, c) and Pt₇Cu (e, f)
(g) the theoretical results (red star) of PBP vs exchange current density of Pt₇Sb in comparison with various pure metals (the experimental data (solid blue points) of Pt^[43], Pd^[44], Rh^[45], Ir, Au, Cu^[39], and Ag) and compounds (the theoretical data^[24] of Pt₇Cu (purple hollow dots)) with space group Fm $\bar{3}$ m

由于Pt₇Sb与Pt₇Cu具有相同的晶体结构，都相当于在Pt的基础上做其他元素的掺杂。相比于Pt₇Sb，Pt₇Cu的投影Berry相位更接近于元素Pt，因此Pt₇Sb与Pt₇Cu之间的对比有助于对Pt₇Sb中的强投影Berry相位的理解。如图7d所示，Pt₇Cu的投影Berry相位在Fermi能处有0.095左右，几乎等于Pt的值。与Pt₇Sb不同之处在于投影Berry相位的峰值正好位于Fermi能级处，这与元素Pt比较类似。Pt₇Cu的高效率催化性能也被表面氢吸附Gibbs自由能和直接的实验测量所验证^[24]。在反应电流密度为10 mA/cm²的情况下，Pt₇Cu的转换频率比商用20% Pt/C高出近5倍^[24]。从2个材料的对比可以看出，不同元素掺杂的效果在细节上虽然不同，但是都能保持高的投影Berry曲率和小的氢吸附Gibbs自由能。

为了揭示Pt₇Sb和Pt₇Cu中强投影Berry相位的机制，本工作计算了这2种化合物沿着高对称方向的分布，并和相应的电子能带结构作了对比。如图7b和c所示，在高对称方向上，Pt₇Sb投影Berry曲率局部分布有3个明显的峰值，分别处于L-Γ、Γ-K和X-Γ之间。结合电子结构可以看出，这3个极值对应的能带细节非常类似，都是不同能带间的线性反交叉，这种能带结构一个显著的特点是能够带来不同能带间的强耦合作用。这种效果在Pt₇Cu中表现得更明显，尤其是Γ点附近。如图7e和f所示，由于自旋轨道耦合的作用，Γ点附近的能带简并被自旋轨道耦合作用打开局部带隙，导致了Fermi能附近能带的强烈耦合，并伴随着投影Berry曲率一个宽峰的出现。Pt₇Sb和Pt₇Cu中的强投影Berry相位主要是由这种自旋轨道耦合作用导致的退简并所导致。此外，这种能带结构经常伴随着拓扑电子态电子，因此Pt₇Sb和Pt₇Cu中的强投影Berry相位和其伴随的高效催化效率与其拓扑性存在着关联作用，属于材料的本征属性。

3 结论

从获得低Pt负载的良好催化剂的目标出发，找到了含Pt化合物Pt₇Sb和Pt₇Cu，并从理论上研究了其HER催化性能。计算结果显示，其 $Δ G_{H^{*}}$ 几近为零，且与H之间存在较弱的电荷转移，其中Pt₇Sb的性能更优。此外， $Δ G_{H^{*}}$ 和投影Berry相位这2个描述符之间的良好一致性意味着其良好的催化性能是由拓扑引起的。优异的催化活性使Pt₇Sb有希望替代Pt成为高效HER催化剂，而催化与拓扑之间的潜在联系也为寻找经济高效的HER电催化剂提供了更多的思路。

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The vital role of electrocatalysts in determining the efficiency of renewable energy conversion inspired the uncovering of the relation between the catalytic efficiency and electronic structure, in which the volcano-type plot based on adsorption energies and d-band model has achieved great success. At the same time, catalysts with nontrivial topological electronic structures have received considerable attention because of their robust topological surface states and high-mobility electrons, which favor the electrons transfer processes in the heterogeneous catalysis reactions. Under the guidance of this theory, excellent catalysts were reported among topological materials. Inspired by the current development of catalyst and topological materials, we tried to extract a pure intrinsic physical parameter, projected Berry phase (PBP), that only depends on the bulk electronic structure. Applying this parameter to the well-known nonmagnetic transition-metal electrocatalysts, we found a linear relationship between PBP and catalytic efficiency of hydrogen evolution reaction (HER) after considering the symmetry constraint. This can be used as a descriptor for the prediction and designing of promising catalysts for HER, which is realized experimentally in PtCu nanostructures. This work illustrates the importance of the pure bulk band structure effect on electrochemical activities and implies an effective way to understand the mechanism of HER catalysts.Copyright © 2020 American Chemical Society.

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