First-Principles Study of the Electronic, Mechanical, and Optical Properties of Schwarzites
Schwarzites 电子、力学和光学性质的第一性原理研究

¹,........................................

¹ Key Laboratory of materials and surface technology (Ministry of Education), School of Materials Science and Engineering, Xihua University, Chengdu, 610039, Sichuan, China
¹ 材料与表面技术教育部重点实验室，西华大学材料科学与工程学院，成都，610039，四川，中国

² Institute of Fundamental and Frontier Sciences & School of Resources and Environment, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, China
² 电子科技大学基础与前沿科学研究院 & 资源与环境学院，成都 611731，四川，中国

Abstract
摘要

The distinctive three-dimensional mesh configuration and porous nature of Schwarzites render them a valuable subject for investigation within the discipline of materials science. A first principles approach was used in the investigation of the carbon allotropes known as "Schwarzites," with an analysis conducted on three of its structures. The structures were subject to systematic investigation with a view to ascertaining their electronic, mechanical, and optical properties. The D8-0 structure exhibits an indirect band gap of 2.87 eV, while the D8-2 structure displays an indirect band gap of 1.12 eV. In comparison, the P8-1 structure has a direct band gap of 0.91 eV. The D8-0 structure has been found to exhibit the highest elastic modulus (2713.5 GPa) among the structures examined, as well as displaying anisotropy. In regard to tensile strength, the D8-0 structure exhibits the greatest strength in the [100], [010], and [001] directions (81 GPa) and the least in the [110] direction (44 GPa). In the D8-2 and P8-1 models, plastic deformation occurs with slip from a tensile strain of 0.0 to 0.7, as evidenced by the deformation of the material under tensile stress. Additionally, in section 5, we calculated the optical properties of the three materials, including absorption coefficient, extinction coefficient, refractive index, and reflectivity.
Schwarzites 独特的三维网格结构和多孔性质使其成为材料科学领域中一个有价值的研究对象。本文采用第一性原理方法，对碳同素异形体“Schwarzites”进行了研究，并对其三种结构进行了分析。对这些结构进行了系统的研究，以确定它们的电子、机械和光学性质。D8-0 结构表现出 2.87 eV 的间接带隙，而 D8-2 结构表现出 1.12 eV 的间接带隙。相比之下，P8-1 结构具有 0.91 eV 的直接带隙。研究发现，在所检查的结构中，D8-0 结构表现出最高的弹性模量（2713.5 GPa），并且表现出各向异性。在抗拉强度方面，D8-0 结构在[100]、[010]和[001]方向上表现出最大的强度（81 GPa），而在[110]方向上表现出最小的强度（44 GPa）。在 D8-2 和 P8-1 模型中，如材料在拉应力下的变形所证明的那样，塑性变形发生在 0.0 到 0.7 的拉伸应变滑动时。 此外，在第 5 节中，我们计算了三种材料的光学性质，包括吸收系数、消光系数、折射率和反射率。

Keywords: Optical properties; Band structure ; Mechanical Properties ; Schwarzites .
关键词：光学性质；能带结构；机械性能；Schwarzites。

Introduction
导言

Among the elements comprising the periodic table, carbon is arguably the most versatile and indispensable for sustaining human civilization. Due to its distinctive electronic orbitals in the ground state, it forms three types of hybridization (sp1, sp2, sp3), which gives rise to a multitude of allotropes observed in natural systems, including amorphous carbon, graphite, and diamonds. In 1985, Harold Kroto and Richard Errett Smalley used a high-energy laser beam to knock carbon atoms off graphite, creating the first fullerene[1]. In 1991, Lijima discovered tubular products while studying fullerenes using the arc discharge method, opening the door to research on one-dimensional materials—carbon nanotubes[2]. In 2004, Andre Geim and Konstantin Novoselov isolated graphene from graphite using mechanical exfoliation[3, 4]. Following fullerenes, carbon nanotubes, and graphene, in 2010, a team led by Academician Li Yuliang synthesized graphdiyne, pioneering a new paradigm for carbon materials[5, 6]. In 1991, Mackay and Terrones[7] introduced the concept of negative curvature in the context of periodic graphite structures, similar in shape to triply periodic minimal surfaces (TPMS). In other words, schwarzites are composed of TPMS decorated with carbon atoms. These sponge-like structures, called carbon-based schwarzites, exhibit negative curvature due to the incorporation of eight-membered and seven-membered rings in the hexagonal rings[9]. Carbon-based schwarzites are allotrope structural materials of carbon and are porous materials. Schwarzites use the concept of negative curvature in the context of periodic graphite structures, and their topology is similar to triply periodic minimal surfaces (TPMS), meaning that schwarzites are composed of TPMS decorated with carbon atoms. Their structural characteristics are very interesting, and many studies have shown that carbon-based schwarzites have high stability[11], even more so than fullerenes with positive Gaussian curvature. This has attracted the interest of many researchers.
在构成元素周期表的众多元素中，碳无疑是最通用且对维持人类文明不可或缺的元素。由于其在基态下独特的电子轨道，碳可以形成三种杂化方式（sp1、sp2、sp3），从而产生自然系统中观察到的多种同素异形体，包括无定形碳、石墨和钻石。1985 年，Harold Kroto 和 Richard Errett Smalley 使用高能激光束将碳原子从石墨上击落，创造了第一个富勒烯[1]。1991 年，Iijima 在使用弧放电法研究富勒烯时发现了管状产物，开启了一维材料——碳纳米管的研究之门[2]。2004 年，Andre Geim 和 Konstantin Novoselov 使用机械剥离法从石墨中分离出石墨烯[3, 4]。继富勒烯、碳纳米管和石墨烯之后，2010 年，李玉良院士带领的团队合成了石墨炔，开创了碳材料的新范例[5, 6]。1991 年，Mackay 和 Terrones[7]在周期性石墨结构中引入了负曲率的概念，其形状类似于三重周期极小曲面（TPMS）。 换句话说，黑碳是由装饰有碳原子的 TPMS 组成的。这些被称为碳基黑碳的海绵状结构，由于在六边形环中加入了八元环和七元环[9]，表现出负曲率。碳基黑碳是碳的同素异形体结构材料，是多孔材料。黑碳在周期性石墨结构的背景下使用了负曲率的概念，它们的拓扑结构与三重周期性最小曲面（TPMS）相似，这意味着黑碳是由装饰有碳原子的 TPMS 组成的。它们的结构特性非常有趣，许多研究表明，碳基黑碳具有很高的稳定性[11]，甚至超过了具有正高斯曲率的富勒烯。这引起了许多研究人员的兴趣。

Schwarzites have unique structures, excellent stability, and mechanical properties[16]. The electronic properties of schwarzites[17] are determined by their negative Gaussian curvature. The interest in these new materials not only stems from their scientific beauty but also from their unique mechanical properties[12], electrical properties, magnetism[13], and good storage properties[14]. These properties make carbon-based schwarzite materials promising for applications in supercapacitors, battery electrodes, catalysis, gas storage, and separation[15].
Schwarzites 具有独特的结构、出色的稳定性和机械性能[16]。Schwarzites 的电子特性[17]由其负高斯曲率决定。人们对这些新型材料的兴趣不仅源于它们的科学之美，还源于它们独特的机械性能[12]、电气性能、磁性[13]和良好的存储性能[14]。这些特性使碳基 Schwarzite 材料在超级电容器、电池电极、催化、气体存储和分离方面具有广阔的应用前景[15]。

Therefore, many researchers have conducted extensive studies on methods to synthesize schwarzites. Pun S H's group[18] found that negatively curved carbon allotropes have great potential applications but have not yet been synthesized. They found that synthesizing negatively curved nanographenes could yield fragments of negatively curved carbon allotropes, which could be used as templates or monomer units for synthesizing schwarzites and ring-shaped carbon nanotubes. They discovered that some negatively curved nanographenes exhibit semiconductor properties, leading them to speculate that similar synthesis methods for negatively curved nanographenes could be used to synthesize new negatively curved carbon allotropes. They suggested that this approach would expand the research field of carbon nanomaterials.
因此，许多研究人员对合成黑碳的方法进行了广泛的研究。Pun S H 的研究小组[18]发现，负曲率碳同素异形体具有巨大的潜在应用价值，但尚未被合成出来。他们发现，合成负曲率纳米石墨烯可以产生负曲率碳同素异形体的碎片，这些碎片可以用作合成黑碳和环状碳纳米管的模板或单体单元。他们发现一些负曲率纳米石墨烯表现出半导体特性，这使他们推测，类似的负曲率纳米石墨烯合成方法可以用于合成新的负曲率碳同素异形体。他们认为，这种方法将扩展碳纳米材料的研究领域。

Efrem Braun's group[19] demonstrated that carbon precursors could be obtained on zeolite templates using chemical vapor deposition, which, after removing the template, yielded zeolite-templated carbons (ZTCs). ZTCs are unique porous framework materials with a three-dimensional pore network contained within the polycyclic aromatic walls of atomic membranes. The goal of these materials is to develop carbon-based frameworks with ordered, uniform microporosity, as opposed to randomly porous activated carbon. They developed a theoretical framework for generating ZTC models from zeolite structures, establishing a relationship between ZTCs and schwarzite materials with negative Gaussian curvature. Their results suggest that schwarzites should no longer be considered purely hypothetical materials and can be generated using zeolite templates. Selecting the correct zeolite allows for the tuning of schwarzite structures to achieve desired properties.
Efrem Braun 的研究小组 [19] 证明，可以使用化学气相沉积在沸石模板上获得碳前驱体，去除模板后，即可得到沸石模板碳（ZTC）。ZTC 是一种独特的多孔骨架材料，具有包含在原子膜多环芳烃壁内的三维孔网络。这些材料的目标是开发具有有序、均匀微孔的碳基骨架，而不是随机多孔的活性炭。他们开发了一个理论框架，用于从沸石结构生成 ZTC 模型，从而建立了 ZTC 与具有负高斯曲率的施瓦茨材料之间的关系。他们的结果表明，施瓦茨材料不应再被视为纯粹的假设材料，并且可以使用沸石模板生成。选择正确的沸石可以调整施瓦茨结构，以实现所需的性能。

碳同素异形体种类多，从石墨-石墨烯-富勒烯-石墨炔等等，-汞黝矿碳结构-因其结构的独特性，研究其特性是很有意思的。

Models and Computational Methods
模型和计算方法

To systematically reveal the impact of surface topology on the mechanical properties of triply periodic sp²-carbon foams, three different schwarzites were studied: D8-0, D8-2-2, and P8-1, which belong to the G, P, and Iwp48 families[20]. Due to the presence of large carbon polygons covalently bonded by sp² carbon atoms exceeding six carbon atoms, these carbon schwarzites exhibit surfaces with negative Gaussian curvature. It is noteworthy that the structural characteristics and complexity of carbon schwarzites are mainly determined by the number of non-hexagonal carbon rings in the nanostructures. Similar to mathematical complexity, the structures also exhibit varying degrees of complexity and curvature. To generate atomic models of carbon schwarzites, the Gauss-Bonnet theorem and Euler's law were applied[21].
为了系统地揭示表面拓扑结构对三周期 sp 碳泡沫力学性能的影响，我们研究了三种不同的石墨烯负曲面结构：D8-0、D8-2-2 和 P8-1，它们分别属于 G、P 和 Iwp48 族[20]。由于存在由超过六个碳原子的 sp 碳原子共价键合的大碳多边形，这些碳石墨烯负曲面结构表现出具有负高斯曲率的表面。值得注意的是，碳石墨烯负曲面结构的结构特征和复杂性主要取决于纳米结构中非六边形碳环的数量。与数学复杂性类似，这些结构也表现出不同程度的复杂性和曲率。为了生成碳石墨烯负曲面结构的原子模型，我们应用了高斯-博内定理和欧拉定律[21]。

（1）

Where N4, N5, N7, and N8 are the numbers of four-membered, five-membered, seven-membered, and eight-membered carbon rings in the structure, respectively, and "g" is the genus of the original structure. For example, the G family with g = 3 would have 12 eight-membered rings in the original unit. The calculations in this document use first-principles calculation software, which obtains physical constants and structures of the entire system by self-consistently solving the Schrödinger equation. The first-principles calculation method is based on density functional theory and has become an important research method in materials science, chemistry, and physics. Due to the unique semiconductor properties of these three structures, their electronic and mechanical properties were studied.
其中，N4、N5、N7 和 N8 分别是结构中四元碳环、五元碳环、七元碳环和八元碳环的数量，“g”是原始结构的亏格。例如，亏格 g = 3 的 G 族在原始单元中将有 12 个八元环。本文档中的计算使用第一性原理计算软件，通过自洽地求解薛定谔方程来获得整个系统的物理常数和结构。第一性原理计算方法基于密度泛函理论，已成为材料科学、化学和物理学的重要研究方法。由于这三种结构独特的半导体特性，因此研究了它们的电子和机械性能。

Figure 1 a, b, and c are respectively D8-0, D8-2, and P8-1 model structure diagrams.
图 1 a、b 和 c 分别是 D8-0、D8-2 和 P8-1 模型结构图。

D8-0 is a simple cubic lattice constant of 6.02, adjacent angles of 90 degrees, and the cell contains 24 atoms. D8-2 has a face-centered cubic lattice constant of 12.2 and a cell containing 72 atoms. P8-1 is a simple cube with a lattice constant of 12.8 and a cell containing 96 atoms. ig.1 a, b, and c are respectively D8-0, D8-2, and P8-1 model structure diagrams. D8-0 is a simple cubic lattice constant of 6.02, adjacent angles of 90 degrees, and the cell contains 24 atoms. D8-2 has a face-centered cubic lattice constant of 12.2 and a cell containing 72 atoms. P8-1 is a simple cube with a lattice constant of 12.8 and a cell containing 96 atoms.
D8-0 是一个简单的立方晶格，晶格常数为 6.02，相邻角度为 90 度，晶胞包含 24 个原子。D8-2 具有面心立方晶格，晶格常数为 12.2，晶胞包含 72 个原子。P8-1 是一个简单的立方体，晶格常数为 12.8，晶胞包含 96 个原子。图 1 a、b 和 c 分别是 D8-0、D8-2 和 P8-1 的模型结构图。D8-0 是一个简单的立方晶格，晶格常数为 6.02，相邻角度为 90 度，晶胞包含 24 个原子。D8-2 具有面心立方晶格，晶格常数为 12.2，晶胞包含 72 个原子。P8-1 是一个简单的立方体，晶格常数为 12.8，晶胞包含 96 个原子。

Electronic Structure Calculations
电子结构计算

In this section, all structure optimizations and electronic property calculations were performed using the VASP software package, employing the projected augmented wave pseudopotential (PAW). The structures were optimized using the Perdew-Burke-Ernzerhof form of the generalized gradient approximation (GGA-PBE), with self-consistent accuracies of eV for energy and 0.05 eV/Å for forces, and a plane-wave cutoff energy of 520 eV. K-points were sampled using the Monkhorst-Pack method. We introduced and discussed the calculated band structures and DOS of the three aforementioned schwarzite models. We listed the calculated electronic structures, including the band gaps for insulators or DOS at the Fermi level (N(EF)) for metals. For insulators, zero energy is set at the top of the valence band (VB); for metals, zero energy is set at the Fermi level.
在本节中，所有结构优化和电子性质计算均使用 VASP 软件包进行，该软件包采用投影缀加波赝势（PAW）。结构优化采用广义梯度近似（GGA-PBE）的 Perdew-Burke-Ernzerhof 形式，能量的自洽精度为 eV，力的自洽精度为 0.05 eV/Å，平面波截断能为 520 eV。K 点采用 Monkhorst-Pack 方法进行采样。我们介绍并讨论了上述三种黑锌矿模型的计算能带结构和 DOS。我们列出了计算的电子结构，包括绝缘体的带隙或金属在费米能级的 DOS（N(EF)）。对于绝缘体，零能量设置在价带（VB）的顶部；对于金属，零能量设置在费米能级。

Figure 2 Band structures and density of states. D8-0, D8-2 and P8-1 black zinc ore. (a)(b)(c)Where a and b are indirect bandgap semiconductors, and c are direct bandgap semiconductors.
图 2 能带结构和态密度。D8-0、D8-2 和 P8-1 黑锌矿。(a)(b)(c)其中 a 和 b 为间接带隙半导体，c 为直接带隙半导体。

This is the OAS-constructed polybenzene model or the previously mentioned D8-0 surface. The sc unit cell contains only 24 atoms and can be considered to have four hexagonal benzene rings on the faces of a tetrahedron. Figure 3 shows the band structures and DOS of this model. The conduction band minimum is at M, the valence band maximum is at Γ, and the indirect band gap is 2.87 eV. The D8-2 model has an indirect band gap with a width of 1.12 eV, with the valence band maximum at Γ and the conduction band minimum at L. P8-1 has a direct band gap, with both the valence band maximum and conduction band minimum at T, and a band gap width of 0.91 eV. All three structures exhibit semiconductor properties, with the electronic properties of P8-1 being superior to the other two materials based on the band structure. Therefore, the mechanical properties of these three structures will be analyzed to study their mechanical properties for practical applications.
这是 OAS 构建的聚苯模型或前面提到的 D8-0 表面。该 sc 晶胞仅包含 24 个原子，可以认为在四面体的面上具有四个六边形苯环。图 3 显示了该模型的能带结构和 DOS。导带最小值位于 M 点，价带最大值位于Γ点，间接带隙为 2.87 eV。D8-2 模型的间接带隙宽度为 1.12 eV，价带最大值位于Γ点，导带最小值位于 L 点。P8-1 具有直接带隙，价带最大值和导带最小值均位于 T 点，带隙宽度为 0.91 eV。所有这三种结构都表现出半导体特性，基于能带结构，P8-1 的电子特性优于其他两种材料。因此，将分析这三种结构的机械性能，以研究其在实际应用中的机械性能。

Mechanical Properties Calculations
机械性能计算

Carbon schwarzites have unique nanoporous structures composed of sp2-carbon atoms, which are expected to exhibit high mechanical performance. In this section, the elastic constants and uniaxial tensile mechanical properties of three carbon schwarzites are studied and compared. Given the corresponding values of cubic elastic constants and compliance tensors, the average values of Young's modulus and Poisson's ratio are calculated. These values refer to the macroscopic effective values for each quantity. Equations (3) and (4) provide the Voigt average values of Young's modulus and Poisson's ratio[22]. These values represent the upper limit of Young's modulus and the lower limit of Poisson's ratio. Equations (5) and (6) give the Reuss average values for these quantities, representing the lower limit of Young's modulus and the upper limit of Poisson's ratio. Equation (7) provides the Voigt-Reuss-Hill average values for Young's modulus, while Equation (9) gives the Voigt-Reuss-Hill average values for Poisson's ratio, which requires the calculation of the Voigt-Reuss-Hill average values of the shear modulus given in Equation (8). These equations are described by Toonder, Dommelen, and Baaijens[22]. The table below calculates the bulk modulus, Young's modulus, shear modulus, and Poisson's ratio for the three models.
碳黑具有独特的由 sp2 碳原子组成的纳米多孔结构，有望表现出优异的机械性能。本节研究并比较了三种碳黑的弹性常数和单轴拉伸力学性能。给定立方弹性常数和柔度张量的对应值，计算杨氏模量和泊松比的平均值。这些值指的是每个量的宏观有效值。公式（3）和（4）提供了杨氏模量和泊松比的 Voigt 平均值[22]。这些值代表杨氏模量的上限和泊松比的下限。公式（5）和（6）给出了这些量的 Reuss 平均值，代表杨氏模量的下限和泊松比的上限。公式（7）提供了杨氏模量的 Voigt-Reuss-Hill 平均值，而公式（9）给出了泊松比的 Voigt-Reuss-Hill 平均值，这需要计算公式（8）中给出的剪切模量的 Voigt-Reuss-Hill 平均值。 这些方程由 Toonder、Dommelen 和 Baaijens[22]描述。下表计算了三种模型的体积模量、杨氏模量、剪切模量和泊松比。

;; (2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

The elastic constants of the three structures were calculated using ELATE (Elastic Tensor Analysis) open-source software[23], which is both an open-source Python module for manipulating elastic tensors (available at https://github.com/fxcoudert/elate) and a standalone online application for general analysis of elastic tensors (available at http://progs.coudert.name/elate). The bulk modulus, Young's modulus, shear modulus, and Poisson's ratio for the three structures were obtained. Table 1 compares the bulk modulus, Young's modulus, shear modulus, and Poisson's ratio for the three structures. Among them, the maximum Young's modulus for D8-0 is 3914.5 Gpa, and the minimum is 1814.2 Gpa. For D8-2, the maximum Young's modulus is 1450.3 Gpa, and the minimum is 760.14 Gpa. For P8-1, the maximum Young's modulus is 1775.7 Gpa, and the minimum is 779.9 Gpa. Additionally, the spatial dependence of Young's modulus for the three models was compared, showing that D8-0 has the highest anisotropy.
三种结构的弹性常数使用 ELATE（弹性张量分析）开源软件[23]计算，该软件既是用于操作弹性张量的开源 Python 模块（可在 https://github.com/fxcoudert/elate 获取），也是用于弹性张量通用分析的独立在线应用程序（可在 http://progs.coudert.name/elate 获取）。获得了三种结构的体积模量、杨氏模量、剪切模量和泊松比。表 1 比较了三种结构的体积模量、杨氏模量、剪切模量和泊松比。其中，D8-0 的最大杨氏模量为 3914.5 Gpa，最小值为 1814.2 Gpa。对于 D8-2，最大杨氏模量为 1450.3 Gpa，最小值为 760.14 Gpa。对于 P8-1，最大杨氏模量为 1775.7 Gpa，最小值为 779.9 Gpa。此外，比较了三种模型的杨氏模量空间依赖性，表明 D8-0 具有最高的各向异性。

Table1: The volume modulus, Young's modulus, Shear modulus, Poisson's ratio, and Vickers Hardness table of all the structures studied in this paper. Listed in Gpa perpendicular to the cube plane (in the direction [100]) or as the macro average. Poisson's ratio (ν) is also perpendicular to the cube plane and is given as the macroscopic average.
表 1：本文研究的所有结构的体积模量、杨氏模量、剪切模量、泊松比和维氏硬度表。以 Gpa 为单位，垂直于立方体平面（沿 [100] 方向）或作为宏观平均值列出。泊松比 (ν) 也垂直于立方体平面，并以宏观平均值给出。

Figure 3 D8-0 (a), D8-2 (b), and P8-1 (c) show the spatial dependence of Young's modulus of the three structures, respectively.
图 3 D8-0 (a)、D8-2 (b) 和 P8-1 (c) 分别显示了三种结构的杨氏模量的空间依赖性。

The simulated uniaxial tensile stress-strain curves of the constructed carbon schwarzites are shown. Although different carbon schwarzites display surfaces with negative Gaussian curvature, they exhibit significant nonlinear uniaxial tensile responses. The tensile mechanical response of the D8-0 model shows an initial linear relationship, followed by a nonlinear increase to the final uniaxial tensile strength, reaching the maximum mechanical load. When the tensile strain further exceeds the ultimate tensile strength, the loading stress suddenly drops sharply, indicating failure due to the breakage of covalent C-C bonds. However, the nonlinear characteristic of loading stress is closely related to the type of carbon schwarzite[Surface-topology-controlled]. In the D8-0 model, the maximum tensile strength in the <100>, <010>, and <001> planes reaches 80 Gpa, and the tensile strain curves in these three planes completely overlap, while in the <110> plane, the maximum tensile stress reaches 44 Gpa. The <110> plane undergoes four stages of change: lattice elongation near the 0.0 to 0.15 strain interval, molecular chain segment activity from 0.15 to 0.18 strain, crystallization region fragmentation from 0.18 to 0.4 strain, and molecular chain orientation change from 0.4 to 0.5 strain, indicating high plasticity. This analysis shows that the structure has good ductility in the <100>, <010>, and <001> planes. In the D8-2 and P8-1 models, sliding and plastic deformation occur from 0.0 to 0.7 strain. The mechanical performance of D8-0 is superior to the other two structures.
构建的碳黑晶的模拟单轴拉伸应力-应变曲线如图所示。虽然不同的碳黑晶显示出具有负高斯曲率的表面，但它们表现出显著的非线性单轴拉伸响应。D8-0 模型的拉伸力学响应显示出初始的线性关系，随后非线性增加到最终的单轴拉伸强度，达到最大机械载荷。当拉伸应变进一步超过极限拉伸强度时，加载应力突然急剧下降，表明由于共价 C-C 键的断裂而失效。然而，加载应力的非线性特征与碳黑晶的类型密切相关[表面拓扑控制]。在 D8-0 模型中，<100>、<010>和<001>平面上的最大拉伸强度达到 80 Gpa，并且这三个平面上的拉伸应变曲线完全重叠，而在<110>平面上，最大拉伸应力达到 44 Gpa。 <110> 晶面经历了四个阶段的变化：0.0 到 0.15 应变区间附近的晶格伸长，0.15 到 0.18 应变区间的分子链段活动，0.18 到 0.4 应变区间的结晶区碎片化，以及 0.4 到 0.5 应变区间的分子链取向变化，表明其具有高塑性。该分析表明，该结构在<100>、<010>和<001>晶面具有良好的延展性。在 D8-2 和 P8-1 模型中，滑动和塑性变形发生在 0.0 到 0.7 应变之间。D8-0 的机械性能优于其他两种结构。

Figure 4 (a), (b), and (c) are the tensile stress-strain curves of D8-0, D8-2, and P8-1 structures, respectively.
图 4（a）、（b）和（c）分别是 D8-0、D8-2 和 P8-1 结构的拉伸应力-应变曲线。

Optical Properties Calculation
光学性质计算

When light is incident on the surface of an object, its optical properties may change to some extent due to interactions influenced by variations in the frequency of the incident light. These changes are primarily reflected in aspects such as refractive index, reflectivity, interference, diffraction, light absorption, and dispersion[24]. When light passes through a material, the oscillation of the photon's electric field causes transitions of electrons within the medium, which need to be represented by a complex dielectric function.
当光线照射到物体表面时，由于受到入射光频率变化影响的相互作用，其光学性质可能会在一定程度上发生改变。这些变化主要体现在折射率、反射率、干涉、衍射、光吸收和色散等方面[24]。当光穿过材料时，光子电场的振荡会引起介质内电子的跃迁，这需要用复介电函数来表示。

(10)

The same symbol represents the frequency of the incident light. represents the imaginary part of the complex dielectric function, which is related to interband transitions. is the real part of the dielectric function. These can be obtained using the Kramers-Kronig relations, as shown below[25-28].
相同的符号 表示入射光的频率。 表示复介电函数的虚部，它与带间跃迁有关。 是介电函数的实部。这些可以使用克拉默斯-克朗尼关系获得，如下所示[25-28]。

(11) (12)

Where p is the Cauchy principal value, and η is the complex shift parameter. In summary, we can obtain frequency-dependent linear spectra; for instance, the frequency-dependent linear optical spectra such as the refractive index n(), extinction coefficient k(), absorption coefficient () , energy loss function L(), and reflectivity R() can be calculated using the following equations [29].
其中 p 是柯西主值，η 是复位移参数。总之，我们可以获得频率相关的线性光谱；例如，可以使用以下公式计算频率相关的线性光学光谱，如折射率 n( )、消光系数 k( )、吸收系数 ( ) 、能量损失函数 L( ) 和反射率 R( ) [29]。

(13) (14)

(15) (16)

In Figure 5, we present the linear optical spectra of D8-0, D8-2, and P8-1 determined by solving the Bethe-Salpeter Equation (BSE) near the top of the approximation. It can be observed that D8-0's absorption coefficient only becomes significant after 3.0 eV. D8-2's absorption coefficient starts to become significant after 1.3 eV and reaches a maximum at 20 eV. P8-1's absorption coefficient becomes significant around 2 eV and peaks at 18 eV. This is due to all three materials having indirect bandgaps, resulting in lower absorption coefficients. As shown in Figure 6, the reflectance of D8-0 peaks at 0.43 at 4.1 eV, the reflectance of D8-2 peaks at 0.2 at 2.8 eV, and the reflectance of P8-1 peaks at 0.34 at 1.8 eV. Additionally, the extinction coefficients and refractive indices of the three materials were calculated. Figure 7 displays the extinction coefficients and refractive indices of the three materials. The refractive index of D8-0 reaches a maximum of 3.8 at 3.6 eV, the refractive index of D8-2 reaches a maximum of 2.3 at 2.3 eV, and the refractive index of P8-0 reaches a maximum of 3.1 at 1.3 eV.
在图 5 中，我们展示了通过求解近似顶部的 Bethe-Salpeter 方程（BSE）得到的 D8-0、D8-2 和 P8-1 的线性光学光谱。可以看出，D8-0 的吸收系数仅在 3.0 eV 后才变得显著。D8-2 的吸收系数在 1.3 eV 后开始变得显著，并在 20 eV 处达到最大值。P8-1 的吸收系数在 2 eV 左右变得显著，并在 18 eV 处达到峰值。这是由于这三种材料都具有间接带隙，导致较低的吸收系数。如图 6 所示，D8-0 的反射率在 4.1 eV 处达到峰值 0.43，D8-2 的反射率在 2.8 eV 处达到峰值 0.2，P8-1 的反射率在 1.8 eV 处达到峰值 0.34。此外，还计算了这三种材料的消光系数和折射率。图 7 显示了这三种材料的消光系数和折射率。D8-0 的折射率在 3.6 eV 处达到最大值 3.8，D8-2 的折射率在 2.3 eV 处达到最大值 2.3，P8-0 的折射率在 1.3 eV 处达到最大值 3.1。

Figure 5 (a) The absorption coefficients of the three structures. The black, red, and blue lines represent the absorption coefficients for structures D8-0, D8-1, and P8-1, respectively. (b) The refractive coefficients of the three structures. The black, red, and blue lines correspond to structures D8-0, D8-1, and P8-1, respectively.
图 5 (a) 三种结构的吸收系数。黑色、红色和蓝色线条分别代表结构 D8-0、D8-1 和 P8-1 的吸收系数。(b) 三种结构的折射系数。黑色、红色和蓝色线条分别对应于结构 D8-0、D8-1 和 P8-1。

Fig. 6.(a) Refractive index and extinction coefficient of D8-0, (b) Refractive index and extinction coefficient of D8-2, (c) Refractive index and extinction coefficient of P8-1.n is the refractive index and k is the extinction coefficient.
图 6.(a) D8-0 的折射率和消光系数，(b) D8-2 的折射率和消光系数，(c) P8-1 的折射率和消光系数。n 为折射率，k 为消光系数。

Conclusion
结论

In summary, we optimized the geometries of three types of schwarzites and computed their related physical properties using DFT-based first-principles calculations. We analyzed the electronic, mechanical, and optical properties of the three schwarzites and reached the following conclusions. Firstly, the study of the electronic structure revealed that D8-0 has an indirect bandgap of 2.8 eV, D8-2 has an indirect bandgap of 1.12 eV, and P8-1 has a direct bandgap of 0.91 eV. Secondly, the study of the tensile properties of the three schwarzites showed that D8-0's elastic constants meet the criteria for mechanical stability, and analyses of the elastic modulus and Poisson's ratio indicate that D8-0 is a ductile material. The study of optical properties revealed that D8-0 has the highest peak in the optical absorption coefficient and provided the extinction coefficient, refractive index, and reflectance for all three materials.
总之，我们优化了三种 Schwarzite 的几何结构，并使用基于 DFT 的第一性原理计算了它们相关的物理性质。我们分析了这三种 Schwarzite 的电子、力学和光学性质，并得出以下结论。首先，对电子结构的研究表明，D8-0 具有 2.8 eV 的间接带隙，D8-2 具有 1.12 eV 的间接带隙，P8-1 具有 0.91 eV 的直接带隙。其次，对三种 Schwarzite 拉伸性能的研究表明，D8-0 的弹性常数满足力学稳定性的标准，并且对弹性模量和泊松比的分析表明 D8-0 是一种延性材料。对光学性质的研究表明，D8-0 在光吸收系数中具有最高的峰值，并提供了所有三种材料的消光系数、折射率和反射率。

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