Electronic structures of solids made of C₂₀ clusters
C ₂₀团簇固体的电子结构

Discovery of the fullerene structure of carbon cluster C₆₀¹ had triggered a new area of research on modern materials in both the theoretical and experimental aspects. Because of the excellent properties of these fullerenes, they have broad range of applications such as in high-Tc superconductivity, modern microelectronics, bio-sensors, drug delivery, bio-film resistant surfaces, nano-bearings, mechanical reinforcement for membranes, artificial molecular motors, and polymer composites.^2–8 Due to the interesting properties of fullerenes and their derivatives, search for new fullerenes sparked since the discovery of buckminister fullerene, C₆₀. The motivation to study the small carbon clusters, C₂₀, is aroused because of its potential applications in novel functional single molecular devices or nanometer electronics.⁹ The study on the C₂₀ molecules and C₂₀ solids could help understand the applications of the C₂₀ molecules and solids. Some theoretical investigations on the C₂₀ molecules and solids have been performed. For example, Ronald et al.¹⁰ investigated the quantum transport through paradigmatic NEMS devices consisting of chains of C₂₀ cages. Otani et al.¹¹ reported that conductance is quite sensitive to the number of C₂₀ molecules between electrodes. Yamamoto et al.¹² investigated inelastic electronic transport properties of C₂₀ molecule under low bias voltage assisted by molecular vibrations. Yi Peng et al.¹³ studied quantum transport properties of C₂₀ fullerenes and observed NDR behavior, and Ouyang et al.¹⁴ investigated nano electric switch device based on C₂₀ molecule.
碳簇C ₆₀ ¹富勒烯结构的发现，在理论和实验方面引发了现代材料研究的新领域。由于这些富勒烯的优异性能，它们具有广泛的应用，例如高温超导、现代微电子学、生物传感器、药物输送、生物膜抗性表面、纳米轴承、膜的机械增强、人造分子电机和聚合物复合材料。 ^2–8由于富勒烯及其衍生物的有趣特性，自从发现巴克米斯特富勒烯 C ₆₀以来，人们开始寻找新的富勒烯。研究小碳簇C _{20 的}动机是由于其在新型功能单分子器件或纳米电子器件中的潜在应用。 ⁹对C ₂₀分子和C ₂₀固体的研究有助于了解C ₂₀分子和固体的应用。对C ₂₀分子和固体进行了一些理论研究。例如，罗纳德等人。 ¹⁰研究了通过由 C ₂₀笼链组成的典型 NEMS 设备的量子传输。大谷等人。 ¹¹报道电导对电极之间的C ₂₀分子的数量非常敏感。山本等人。^图12研究了分子振动辅助下低偏压下C ₂₀分子的非弹性电子输运特性。易鹏等.¹³研究了C ₂₀富勒烯的量子输运性质并观察了NDR行为，欧阳等人。 ¹⁴研究了基于C ₂₀分子的纳米电开关器件。

In addition to above mentioned applications, a prominent application of C₂₀ fullerene molecule is in the field of high temperature T_c superconductivity due to its larger electron phonon coupling than that of C₆₀ fullerene^15,16 because the vibronic coupling becomes larger as the cluster size decreases.¹⁷ As a result, it was proposed that the condensed form of the smallest fullerene C₂₀^18–20 could be the best potential candidate for high-Tc superconductors.^15,21 However, unlike the C₆₀, C₂₀ is highly reactive due to its pentagonal configuration, so it could be polymerized spontaneously as in the case of silicon clatherate Si₄₆ which consists of Si₂₀ fullerenes.²² It is well-known that molecular C₆₀ cluster can be condensed to a solid with either a face-centered cubic^23–25 or a hexagonal close-packed (hcp) phase.²⁶ This might suggest that the condensed phase of C₂₀ could also be expected when they are condensed in the same way, and, this is what we are trying to simulate.
除了上述应用之外，C ₂₀富勒烯分子的一个突出应用是在高温T _c超导领域，因为其电子声子耦合比 C ₆₀富勒烯^15,16更大，因为电子振动耦合随着团簇变得更大尺寸减小。 ¹⁷因此，有人提出，最小的富勒烯 C ₂₀ ^18-20的凝聚形式可能是高温超导体的最佳潜在候选者。 ^15,21然而，与C ₆₀不同，C ₂₀由于其五边形构型而具有高反应性，因此它可以自发聚合，如由Si ₂₀富勒烯组成的硅包合物Si ₄₆的情况。 ²²众所周知，分子 C ₆₀簇可以凝聚成具有面心立方^23-25或六方密堆积 ( hcp ) 相的固体。 ²⁶这可能表明，当 C ₂₀以相同方式凝结时，也可以预期出现凝结相，这就是我们试图模拟的。

In 2000, Prinzbach et al.²⁷ provided experimental evidence that C₂₀ could be synthesized from dodoecahedrane, C₂₀H₂₀, in the gas phase. They reported experimental evidences for the existence of three different structures of carbon clusters of C₂₀, i.e., cage, bowl, and ring. These clusters, cage, bowl, and ring, were also experimentally identified by anion photoelectron spectroscopy. Above experiment along with theoretical calculations^28,29 reveals the existence of single stable C₂₀ clusters but it doesn’t unfold the stability of C₂₀ clusters-assembled solid.³⁰ So far, there are some attempts to experimentally produce C₂₀ based solid.^31–33 Kurokawa et al.³¹ have made experimentally one-dimensional polymeric chain of C₂₀ fullerenes on Ag(111) and Au(111) surfaces. It was clearly observed in this experiment that C₂₀ molecules align in one dimension in the form of chain. Wang et al.³² and Iqbal et al.³³ reported the successful syntheses of three dimensional solid phases of C₂₀ fullerenes. Wang et al.³² used the ion beam irradiation method to produce a hexagonal close-packed crystal of C₂₀, while Iqbal et al.³³ used the laser ablation to deposit C₂₀ fullerene molecules into a face-centered cubic phase with the fullerene molecules interconnected by two additional carbon atoms per unit cell in the interstitial tetrahedral sites. However, in all the above cases polymeric solids were formed, therefore, no pure cluster-assembled solid from C₂₀ has been realized yet.
2000 年，普林茨巴赫等人。^图27提供了C ₂₀可以在气相中由十二面体C ₂₀ H ₂₀合成的实验证据。他们报告了存在三种不同结构的C ₂₀碳簇，即笼式、碗式和环式的实验证据。这些簇、笼、碗和环也通过阴离子光电子能谱进行了实验鉴定。上述实验和理论计算^28,29揭示了单个稳定C ₂₀团簇的存在，但没有揭示C ₂₀团簇组装固体的稳定性。 ³⁰迄今为止，已有一些尝试通过实验生产 C ₂₀基固体。 ^31-33黑川等人。 ³¹已通过实验在Ag(111) 和Au(111) 表面上制备了C ₂₀富勒烯的一维聚合链。在该实验中清楚地观察到C ₂₀分子以链的形式在一维排列。王等人。 ³²和伊克巴尔等人。 ³³报道了C ₂₀富勒烯三维固相的成功合成。王等人。 ³²采用离子束辐照方法制备了C ₂₀六方密排晶体，而Iqbal等人。^图33使用激光烧蚀将C ₂₀富勒烯分子沉积到面心立方相中，其中富勒烯分子通过间隙四面体位点中的每个晶胞的两个额外碳原子互连。然而，在所有上述情况下都形成了聚合物固体，因此，尚未实现来自C ₂₀的纯簇组装固体。

In this paper, we have simulated five different crystal structures assembled from C₂₀, by the first-principles calculations. We have found that a C₂₀ cluster-assembled solid can be formed in 1D when C₂₀ molecules are orientated to be head to head. Moreover, we have got very interesting results from the optimized diamond structure of C₂₀ clusters, where two C₂₀ fullerenes are connected and form a (C₂₀)₂ dimer which are then condensed by van der Waals forces, forming a cluster-assembled solid. The band structures along with DOS calculations have suggested that 1D head to head and 3D-diamond structure of C₂₀ cluster-assembled solids are semiconductors, while, 2D-graphene and 3D-fcc structures are metallic rather than insulating.
在本文中，我们通过第一性原理计算模拟了由C ₂₀组装而成的五种不同的晶体结构。我们发现，当C ₂₀分子头对头定向时，可以在一维中形成C ₂₀簇组装固体。此外，我们从优化的C ₂₀簇金刚石结构中得到了非常有趣的结果，其中两个C ₂₀富勒烯连接并形成(C ₂₀ ) ₂二聚体，然后通过范德华力凝聚，形成簇组装固体。能带结构以及DOS计算表明，C ₂₀簇组装固体的1D头对头结构和3D金刚石结构是半导体，而2D石墨烯和3D-fcc结构是金属而不是绝缘的。

Our calculations have been performed by employing Vienna ab initio simulation package (VASP),^34,35 which is based on density function theory, the plane wave basis and the projector augmented wave (PAW) representation.³⁶ The exchange-correlation potentials have been approximated by the generalized gradient-corrected function (GGA) in the form of Perdew-Burke-Ernzerhof functional (PBE).³⁷ We have also performed van der Waals (vdW) correction calculation by employing PBE-D2 method for the diamond structure. Wave functions are expanded by plane waves with a cut-off of plane wave kinetic energy up to 400 eV in all the calculations. Brillouin-zone integrations are performed by special k-point sampling of Monkhorst-Pack scheme³⁸ with a $\mathrm{\Gamma }$-centered $60\times 1\times 1,\mathrm{\hspace{0.17em}15}\times 15\times 1,\text{and}\mathrm{\hspace{0.17em}10}\times 10\times 10$ grids for 1D, 2D and 3D structures relaxations. All the crystal geometries were fully relaxed until the force upon each atom is less than 0.02 eV/Å.
我们的计算是通过采用维也纳从头算仿真软件包 (VASP) ^34,35进行的，该软件包基于密度函数理论、平面波基础和投影增强波 (PAW) 表示。 ³⁶交换相关势已通过 Perdew-Burke-Ernzerhof 泛函 (PBE) 形式的广义梯度校正函数 (GGA) 进行了近似。 ³⁷我们还采用 PBE-D2 方法对金刚石结构进行了范德华 (vdW) 校正计算。所有计算中波函数均通过平面波展开，平面波动能截止值高达 400 eV。布里渊区积分通过 Monkhorst-Pack 方案³⁸的特殊 k 点采样执行， $\mathrm{\Gamma }$ 以-为中心 $60\times 1\times 1,\mathrm{\hspace{0.17em}15}\times 15\times 1,\text{and}\mathrm{\hspace{0.17em}10}\times 10\times 10$ 用于 1D、2D 和 3D 结构松弛的网格。所有晶体几何形状都完全松弛，直到每个原子上的力小于 0.02 eV/Å。

We present our results by considering five cluster-assembled solids made out of C₂₀ fullerene molecules. The input atomic coordinates of a C₂₀ molecule were initially taken from the computation chemistry database.³⁹ The carbon atoms in the C₂₀ fullerene were allowed to relax without constrains, until the interatomic forces were all lower than 0.02 eV/Å. The coordinates of C₂₀ obtained from our DFT calculations are in good agreement with those obtained by Saito et al.⁴⁰ These coordinates are then used for all the further DFT calculations for the C₂₀ cluster-assembled solids in this paper. The cohesive energy of the ground state of a C₂₀ cluster is found to be -8.087 eV/atom, which is consistent with the value by Gotthard et al.⁴¹ The molecular structure of an isolated C₂₀ fullerene consists of 12 pentagons with equal size on the surface of a sphere and without hexagon, having a bond length of 1.450 Å. The molecular diameter of a C₂₀ is then about 3.89 Å, agreeing with those obtained by Dolgonos⁴² and Hussain.⁴³ 
我们通过考虑由 C ₂₀富勒烯分子制成的五种簇组装固体来展示我们的结果。 C ₂₀分子的输入原子坐标最初取自计算化学数据库。 ³⁹ C ₂₀富勒烯中的碳原子可以不受约束地松弛，直到原子间作用力全部低于 0.02 eV/Å。从我们的 DFT 计算中获得的 C ₂₀坐标与 Saito等人获得的坐标非常一致。 ⁴⁰这些坐标随后用于本文中 C ₂₀簇组装固体的所有进一步 DFT 计算。发现C ₂₀团簇基态内聚能为-8.087 eV/atom，这与Gotthard等人的数值一致。 ⁴¹孤立的C ₂₀富勒烯的分子结构由12个球体表面大小相等的五边形组成，不含六边形，键长为1.450 Å。 C ₂₀的分子直径约为3.89 Å，与Dolgonos ⁴²和Hussain 获得的结果一致。 ⁴³

We have considered two cases for the C₂₀ cluster-assembled solids in one-dimension, that is, the head to head and the ordinary cases. In the case of head to head, two neighboring C₂₀’s are rotated such that each C₂₀ cluster has a carbon atom lying on the x-axis so that the nearest carbon atoms from neighboring C₂₀’s are in head to head arrangement. In the ordinary case, the orientations of C₂₀’s are in general. The C₂₀ cluster-assembled solids thus formed are then calculated. The insets in Fig. 1(a)–(b) show the variation in cohesive energy as a function of inter-cluster distance of the C₂₀ cluster-assembled solids in one-dimension, for both the 1D head to head and general cases. We have found energy minimum at lattice constants (distances between the centers of two nearest C20’s) of 7.5 Å and 6.09 Å for the head to head and the ordinary cases, respectively. The inter-fullerene bond length (i.e., C-C nearest distance between two C₂₀’s) are then 3.609 Å for the head to head case and 1.548 Å for the ordinary case. Judged from the C-C nearest distances between two C₂₀’s, it shows that a molecular crystal (i.e., cluster-assembled solid) was formed for the head to head configuration, however, C₂₀ clusters were inter-linked by covalent bonds (noted that bond length in C₂₀ is 1.450 Å) in the ordinary case, where C₂₀ fullerenes are coalesced like 1D polymeric chain of C₂₀ molecules. Kurokawa et al.³¹ have made experimentally one-dimensional polymeric chain of C₂₀ fullerenes on Ag(111) and Au(111) surfaces after deposition of carbon using an arc plasma gun (APG). It was clearly observed in this experiment that C₂₀ molecules align in one dimension in the form of a chain in the same manner as we have observed in our DFT calculations. However, the formation of molecular crystal in 1D head to head case can also be judged clearly from the charge density distributions. The charge density plots in Fig. 1(a)–(b) showed clearly that a molecular crystal was formed for the head to head structure while polymeric chain of C₂₀ molecules was form for the ordinary structure. We have obtained a very small cohesive energy of 0.015 eV/C₂₀ (see Table I) for the 1D head to head configuration, where C₂₀ fullerene molecules are condensed by a rather weak van der Waals forces. The reason that a molecular crystal can be formed in 1D is that the repulsive force between two head to head carbon atoms of C₂₀’s can be established to compete against all the attractive forces, i.e., repulsive force between two C atoms can be very large (due to Pauli exclusion principle). However, in the ordinary case, there is no head to head atom, so that repulsive forces between C atoms can only be established when two C20’s are close, leading to the coalescence of the C₂₀ clusters.
我们考虑了一维C ₂₀团簇组装固体的两种情况，即头对头情况和普通情况。在头对头的情况下，两个相邻的C ₂₀旋转，使得每个C ₂₀簇具有位于x轴上的碳原子，使得距相邻C ₂₀最近的碳原子处于头对头排列。在一般情况下，C ₂₀的取向是一般的。然后计算由此形成的C ₂₀簇组装固体。图1(a) – (b)中的插图显示了一维 C ₂₀簇组装固体的内聚能随簇间距离的变化，无论是一维头对头还是一般情况。我们发现头对头和普通情况下的晶格常数（两个最近的 C20 中心之间的距离）分别为 7.5 Å 和 6.09 Å 时能量最小。那么，对于头对头情况，富勒烯间键长（即，两个C ₂₀之间的CC最近距离）为3.609 Å，对于普通情况为1.548 Å。从两个C ₂₀之间的CC最近距离判断，表明形成了头对头构型的分子晶体（即簇组装固体），然而，C ₂₀簇通过共价键相互连接（注意到在通常情况下，C ₂₀中的键长为 1.450 Å，其中 C ₂₀富勒烯像 C ₂₀分子的一维聚合链一样聚结。黑川等人。^{31 等人}在使用电弧等离子体枪（APG）沉积碳后，在 Ag（111）和 Au（111）表面上实验性地制备了 C ₂₀富勒烯的一维聚合物链。在此实验中清楚地观察到，C ₂₀分子以链的形式在一维排列，其方式与我们在 DFT 计算中观察到的方式相同。然而，一维头对头情况下分子晶体的形成也可以从电荷密度分布中清楚地判断。图1（a） - （b）中的电荷密度图清楚地表明，头对头结构形成分子晶体，而普通结构形成C ₂₀分子的聚合链。对于一维头对头构型，我们获得了0.015 eV/C ₂₀的非常小的内聚能（参见表I ），其中C ₂₀富勒烯分子通过相当弱的范德华力凝聚。分子晶体之所以能在一维形成，是因为两个头对头的C ₂₀碳原子之间可以建立排斥力来对抗所有的吸引力，即两个C原子之间的排斥力可以非常大。大（由于泡利不相容原理）。然而，在通常情况下，不存在头对头的原子，因此只有当两个C20靠近时，C原子之间才会产生排斥力，导致C ₂₀团簇聚结。

Band structures, along with the electronic density of states for head to head configuration are presented in Fig. 1(c), (d), respectively. The band structures show that 1D head to head structure is semiconducting, consistent with its character of being a molecular solid. Dispersions of bands are small for those far below the Fermi level, while for those close to the Fermi level the dispersions are significant.
能带结构以及头对头配置的电子态密度分别如图1(c)、(d)所示。能带结构表明一维头对头结构是半导体，与其分子固体的特性一致。对于远低于费米能级的能带，色散很小，而对于接近费米能级的能带，色散很大。

A two-dimensional cluster-assembled solid made out of C₂₀ fullerene molecules has been studied, where C₂₀’s are arranged in a graphene structure. The optimized structure is shown in Fig. 2(a). It is found that there is strong bonding between the nearest neighbor C₂₀ fullerene molecules. The inter-molecular bond length is found to be 1.54 Å which shows strong covalent bonding between the C₂₀ fullerenes (noted that bond length in C₂₀ is 1.450 Å). Formation of inter-cage bonds results in a slight change in intra-cage bonds around the connection sites between C₂₀’s. A bond in C₂₀ fullerene is broken and cage opens up, enabling the formation of C-C bonds with the nearest fullerene molecules, although C₂₀ cage still retains its shape in general. Each fullerene cage is connected with three neighboring fullerenes. In order to describe the distortion of C₂₀ quantitatively, we have defined the following geometric distortion parameter:
研究了由C ₂₀富勒烯分子制成的二维簇组装固体，其中C ₂₀排列成石墨烯结构。优化后的结构如图2(a)所示。发现最近邻的C ₂₀富勒烯分子之间存在强键合。发现分子间键长为1.54 Å，这显示了C ₂₀富勒烯之间的强共价键合（注意C ₂₀中的键长为1.450 Å）。笼间键的形成导致C ₂₀之间的连接位点周围笼内键的轻微变化。 C ₂₀富勒烯中的键断裂并且笼打开，从而能够与最近的富勒烯分子形成CC键，尽管C ₂₀笼总体上仍保持其形状。每个富勒烯笼与三个相邻的富勒烯连接。为了定量描述C ₂₀的畸变，我们定义了以下几何畸变参数：

where (x_i, y_i, z_i) are the coordinates of the i-th atom of an independent C₂₀ molecule (an ideal cluster), while $(x_i^{\prime },y_i^{\prime },z_i^{\prime })$ are the coordinates of the i-th atom of a C₂₀ in the cluster-assembled solid. Such a parameter will describe the distortion of C₂₀ clearly. For 2D-graphene structure, the distortion parameter $\alpha$ is 0.1647. The larger value of this parameter indicates a larger distortion of C₂₀ in the formation of a solid made out of C₂₀.
其中(x , y , z )是独立 C ₂₀分子（理想簇）的第 i 个原子的坐标，而 $(x_i^{\prime },y_i^{\prime },z_i^{\prime })$ 是簇组装固体中 C ₂₀第 i 个原子的坐标。这样的参数可以清楚地描述C ₂₀的畸变。对于二维石墨烯结构，畸变参数 $\alpha$ 是0.1647。该参数值越大，表明在由C ₂₀制成的固体形成过程中C ₂₀的变形越大。

Total cohesive energy per C₂₀ of 2D solid phase is 6.271 eV as shown in Table I. The total charge density plot is also shown in Fig. 2(b). From both the cohesive energies and the charge density plot, we conclude that the molecular solid of C₂₀ can not be formed in the 2D graphene structure. The energy bands and the density of states (DOS) for the cluster-assembled solid made out of C₂₀ molecules in the graphene structure are shown in Fig. 2(c), (d), respectively. Since there are strong interactions between the neighboring molecules, a reasonable dispersion in energy bands has been observed, as presented in Fig. 2(c). The band structures and DOS profile of this 2D C₂₀ solid phase indicate a semiconducting property. This can be explained by the geometric structure of the connecting pattern of the optimized 2D C₂₀’s. For example, from Fig. 2(a), it shows that all the C atoms in this structure are three-coordinated, including the atoms at the joints which resemble the bonds in the C₂₀ cluster, leading to all the electrons in the system are bonded. Since there is no movable electron in this structure, it can exhibit a semiconducting character, although it is not a molecular solid.
2D固相的每C ₂₀的总内聚能是6.271eV，如表I中所示。总电荷密度图也如图2(b)所示。从内聚能和电荷密度图来看，我们得出结论，在二维石墨烯结构中不能形成C ₂₀分子固体。由石墨烯结构中的C ₂₀分子制成的簇组装固体的能带和态密度(DOS)分别如图2(c)、(d)所示。由于相邻分子之间存在强烈的相互作用，因此观察到了能带的合理分散，如图2(c)所示。该 2D C ₂₀固相的能带结构和 DOS 分布表明其具有半导体特性。这可以通过优化的2D C ₂₀的连接图案的几何结构来解释。例如，从图2(a)可以看出，该结构中的所有C原子都是三配位的，包括连接处的原子类似于C ₂₀团簇中的键，导致系统中的所有电子被绑定。由于该结构中没有可移动的电子，因此尽管它不是分子固体，但它可以表现出半导体特性。

In the 3D cases, we have placed the C₂₀ fullerenes at both the fcc crystalline lattice sites (named as ‘C₂₀ fcc structure’) and the diamond lattice sites (named as ‘C₂₀ diamond structure’) initially. First, the C₂₀ fcc structure is discussed here. It shows that C₂₀ fullerenes were interconnected together by covalent bonds and formed a 3D polymeric network of C₂₀ fullerene units. In general, fullerene cages were only slightly distorted although they were well interconnected. In the C₂₀ fcc solid, each C₂₀ has 12 chemical bonds formed between the C₂₀ and its nearest neighbor C₂₀ clusters, since each C₂₀ fullerene cage has 12 nearest neighbor C₂₀ clusters. In this optimized C₂₀ fcc structure, the intra-fullerene bonds were slightly elongated and the inter-C₂₀ fullerenes link due to polymerization, however no bond was broken in each fullerene unit, i.e., shapes of the fullerenes remained intact. Inter-fullerene bond is around 1.539 Å which is typically in a covalent nature, noted that bond length in C₂₀ is 1.450 Å. Since the inter-fullerene bonds were longer than those of intra-fullerenes ones, this indicates that strength of inter-fullerene bond was weaker than that of intra-fullerene. The geometric distortion parameter $\alpha$ is 0.0308.
在3D情况下，我们最初将C ₂₀富勒烯放置在fcc晶格位点（命名为“C ₂₀ fcc结构”）和金刚石晶格位点（命名为“C ₂₀金刚石结构”）处。首先，这里讨论 C ₂₀ fcc结构。它表明C ₂₀富勒烯通过共价键互连在一起并形成C ₂₀富勒烯单元的3D聚合物网络。一般来说，富勒烯笼虽然相互连接良好，但仅轻微变形。在C ₂₀ fcc固体中，每个C ₂₀具有在C ₂₀与其最邻近的C ₂₀簇之间形成的12个化学键，因为每个C ₂₀富勒烯笼具有12个最邻近的C ₂₀簇。在该优化的C ₂₀ fcc结构中，富勒烯内的键稍微伸长，并且C ₂₀富勒烯间的键由于聚合而连接，但是每个富勒烯单元中没有键断裂，即富勒烯的形状保持完整。富勒烯间键约为1.539 Å，其通常具有共价性质，注意到C ₂₀中的键长为1.450 Å。由于富勒烯间键比富勒烯内键长，这表明富勒烯间键的强度弱于富勒烯内键的强度。几何畸变参数 $\alpha$ 是 0.0308。

The cohesive energy of C₂₀ fcc solid is 8.346 eV/atom (as shown in Table I) which is larger than that of an isolated C₂₀ cage (8.087 eV/atom), indicating that the fcc solid made of C₂₀ is energetically stable. The cohesive energy between C₂₀ clusters in the fcc C₂₀ solid is 5.74 eV/C₂₀, i.e. 0.287 eV/atom, which is larger than that of fcc C₆₀ solid which is only 1.6 eV/C₆₀, i.e., 0.03 eV/atom.⁴⁴ The cohesive energy between C₂₀ clusters is also larger than that of the simple cubic B₈₀ solid which is 4.9 eV/B₈₀, i.e., 0.061 eV/atom and fcc B₈₀ solid which is 18.2 eV/B₈₀, i.e., 0.23 eV/atom).⁴⁵ It should be noted that, unlike C₂₀ fcc solid, C₆₀ fcc solid is a molecular solid condensed by van der Waals forces.⁴⁶ Hence the solid made of C₂₀ clusters in the fcc structure is not a molecular solid, since C₂₀’s are condensed to form a polymeric continuous network of C₂₀ fullerenes inter-connected. The metallic behavior of C₂₀ based fcc solid is similar to that of fcc B₈₀ solid.⁴⁵ In order to show the contribution of orbitals, we have plotted the DOS profile along with PDOS for 3D-fcc C₂₀ solid, shown in Fig. 3. Since there is no electronic state around the Fermi level for semiconducting system, we discuss here only the 3D-fcc system. From Fig. 3, we can see that near Fermi level only p-orbitals and s-orbitals take part in the interactions while the major contribution springs from p-orbitals of carbon atoms.
C ₂₀ fcc固体的内聚能为8.346 eV/atom（如表1所示），大于孤立的C ₂₀笼的内聚能（8.087 eV/atom），表明由C ₂₀制成的fcc固体是能量稳定的。 fcc C ₂₀固体中C ₂₀团簇之间的内聚能为5.74 eV/C ₂₀ ，即0.287 eV/atom，这比fcc C ₆₀固体的内聚能只有1.6 eV/C ₆₀ ，即0.03 eV/atom还要大。原子。 ⁴⁴ C ₂₀簇之间的内聚能也大于简单立方 B ₈₀固体的内聚能，为 4.9 eV/B ₈₀ ，即 0.061 eV/atom 和fcc B ₈₀固体，为 18.2 eV/B ₈₀ ，即 0.23 eV/原子）。 ⁴⁵需要注意的是，与C ₂₀ fcc固体不同，C ₆₀ fcc固体是通过范德华力凝聚的分子固体。 ⁴⁶因此，由fcc结构中的 C ₂₀簇组成的固体不是分子固体，因为 C ₂₀缩合形成相互连接的 C ₂₀富勒烯的聚合连续网络。 C ₂₀基面心立方固体的金属行为与面心立方B ₈₀固体相似。 ⁴⁵为了显示轨道的贡献，我们绘制了 3D-fcc C ₂₀固体的 DOS 剖面和 PDOS，如图3所示。 由于半导体系统的费米能级周围不存在电子态，因此我们这里仅讨论3D-fcc系统。从图3中我们可以看出，在费米能级附近，只有p轨道和s轨道参与相互作用，而主要贡献来自于碳原子的p轨道。

Finally, C₂₀ diamond structure is discussed here. That is, we consider a diamond crystalline structure made of C₂₀ fullerenes. Due to the importance of the 3D-diamond structure, we have also performed the first-principles calculations by taking into account of the van der Waals interactions. The full relaxation on this diamond structure leads to the formation of (C₂₀)₂ fullerene dimers in the unit cell, as shown in Fig. 4(a). The charge density distribution is also plotted in Fig. 4(b). Results show that the two C₂₀ fullerenes are bounded to form a C₂₀ dimer by strong covalent bond, although retaining their individual identity in general. The binding energies between the two isolated C₂₀ fullerenes of an independent (C₂₀)₂ dimer are 3.74 eV/C₂₀ (without vdW) and 3.751 eV/C₂₀ (with vdW), respectively, which is around the energy of a typical C-C bond (greater than 3 eV). So, formation of the (C₂₀)₂ dimer is energetically favorable. The distance between two nearest C-C atoms of a single (C₂₀)₂ dimer is 1.516 Å, while the inter-dimer distance, i.e., the nearest C-C atoms between two (C₂₀)₂ fullerene dimers, is 3.35 Å, which is almost twice of the distance between two single C₂₀ clusters in a dimer. The large distance between two (C₂₀)₂ fullerene dimers indicates a van der Waals interactions between the (C₂₀)₂ dimers, therefore, leading to the formation of a molecular solid (i.e., a cluster-assembled solid) consisting of (C₂₀)₂ dimers. It should be emphasized that this molecular solid consists of (C₂₀)₂ dimers rather than C₂₀ clusters, that is, the periodically repeating unit is the (C₂₀)₂ dimer instead of an individual C₂₀. A (C₂₀)₂ fullerene dimer is formed without breaking any bond in forming adjacent neighboring C₂₀ cages. The dimer length is defined as end to end axial distance between two cages which is 10.286 Å. The intra-fullerene bonds in each fullerene unit around inter-C₂₀ fullerene link are relatively long (around 1.56 Å) but not broken. The geometric distortion parameter $\alpha$ is 0.02685.
最后，这里讨论 C ₂₀金刚石结构。也就是说，我们考虑由C ₂₀富勒烯制成的金刚石晶体结构。由于 3D 金刚石结构的重要性，我们还考虑了范德华相互作用来进行第一原理计算。该金刚石结构的完全弛豫导致晶胞中形成(C ₂₀ ) ₂富勒烯二聚体，如图4(a)所示。电荷密度分布也绘制在图4(b)中。结果表明，两个C ₂₀富勒烯通过强共价键结合形成C ₂₀二聚体，尽管总体上保留了它们各自的特性。独立的 (C ₂₀ ) ₂二聚体的两个分离的 C ₂₀富勒烯之间的结合能分别为 3.74 eV/C ₂₀ （无 vdW）和 3.751 eV/C ₂₀ （有 vdW），这大约是典型的 C 20 富勒烯的能量。 CC 键（大于 3 eV）。因此，(C ₂₀ ) ₂二聚体的形成在能量上是有利的。单个(C ₂₀ ) ₂二聚体的两个最近的CC原子之间的距离为1.516 Å，而二聚体间距离，即两个(C ₂₀ ) ₂富勒烯二聚体之间最近的CC原子之间的距离为3.35 Å，这几乎是二聚体中两个单个 C ₂₀簇之间距离的两倍。两个 (C ₂₀ ) ₂富勒烯二聚体之间的大距离表明 (C ₂₀ ) ₂二聚体之间存在范德华相互作用，因此导致分子固体的形成（即，一种簇组装固体），由 (C ₂₀ ) ₂二聚体组成。需要强调的是，该分子固体由(C ₂₀ ) ₂二聚体而不是C ₂₀簇组成，即周期性重复单元是(C ₂₀ ) ₂二聚体而不是单个C ₂₀ 。形成(C ₂₀ ) ₂富勒烯二聚体，而不会破坏形成相邻C ₂₀笼中的任何键。二聚体长度定义为两个笼之间端到端的轴向距离，为 10.286 Å。每个富勒烯单元中围绕C ₂₀富勒烯间连接的富勒烯内键相对较长（约1.56 Å），但未断裂。几何畸变参数 $\alpha$ 是 0.02685。

On the other hand, the cohesive energy between (C₂₀)₂ dimers is calculated to be 0.93 eV/(C₂₀)₂ (without vdW) and 1.171 eV/(C₂₀)₂ (with vdW), that is, 0.023 eV/atom (without vdW) and 0.029 eV/atom (with vdW), which is much less than the binding energy of a typical C-C bond energy in a C₂₀ cluster (more than 3 eV). Compared with the binding energies between C₆₀ clusters in the fcc C₆₀ solid (1.6 eV/C60 i.e 0.026 eV/atom),⁴⁴ the molecular solids of hcp and fcc C₅₀Cl₁₀ (0.9 eV and 0.3 eV, respectively),⁴⁶ the hexagonal monolayer phase of C₆₄F₄ (0.48 eV per C₆₄F₄ molecule⁴⁷ and K₃C₆₀ (0.18 eV/molecule),^48,49 it seems that our case is almost equally stable as that of fcc C60 molecular solid and more stable than other molecular solids such as hcp and fcc C₅₀Cl_10, C₆₄F₄, and K₃C₆₀. Calculations indicate that there is no chemical bond between (C₂₀)₂ molecules which is similar to fcc C₆₀ solid⁴⁴ and other molecular solids.^46–49 This feature is clearly evident from the electronic charge density contour map as shown in Fig. 4(b). Furthermore, the small value of the binding energy between two (C₂₀)₂ dimers indicates a rather weak van der Waals interactions between the (C₂₀)₂ dimers, reminding that the binding energies between two C₂₀ fullerenes in the dimer are 3.74 eV/C₂₀ (without vdW) and 3.751 eV/C₂₀ (with vdW), respectively as shown in Table I.
另一方面，(C ₂₀ ) ₂二聚体之间的内聚能经计算为0.93 eV/(C ₂₀ ) ₂ (无vdW)和1.171 eV/(C ₂₀ ) ₂ (有vdW)，即0.023 eV /atom（无vdW）和0.029 eV/atom（有vdW），这远小于C ₂₀簇中典型CC键能的结合能（大于3 eV）。与fcc C ₆₀固体中 C ₆₀团簇之间的结合能（1.6 eV/C60，即 0.026 eV/原子）相比， ⁴⁴ hcp 和 fcc C ₅₀ Cl ₁₀分子固体（分别为 0.9 eV 和 0.3 eV）， ⁴⁶ C ₆₄ F ₄的六方单层相（每个 C ₆₄ F ₄分子 0.48 eV ⁴⁷和 K ₃ C ₆₀ （0.18 eV/分子）， ^48,49看来我们的情况几乎与 fcc C60 分子固体同样稳定比其他分子固体如hcp和fcc C ₅₀ Cl _10、 C ₆₄ F ₄和K ₃ C ₆₀更稳定。计算表明(C ₂₀ ) ₂分子之间不存在与fcc C ₆₀类似的化学键。固体⁴⁴和其他分子固体^46-49从电子电荷密度等高线图中可以清楚地看出这一特征，如图4(b)所示。 此外，两个(C ₂₀ ) ₂二聚体之间的结合能值较小，表明(C ₂₀ ) ₂二聚体之间的范德华相互作用相当弱，这提醒二聚体中两个C ₂₀富勒烯之间的结合能为3.74 eV /C ₂₀ (无vdW)和3.751 eV/C ₂₀ (有vdW)，分别如表I所示。

Electronic band structures and density of states of (C₂₀)₂-dimered diamond structure are shown in Fig. 4(c)–(d), respectively. We can notice multiple flat bands in the electronic band structure, the reason for these flat bands is that the interaction between the C₂₀ cluster fullerene is very weak (weak van der Waals). The band gap without van der Waals interaction is 0.415 eV, while it reduces to 0.291 eV when van der Waals interaction is included. The reduction of the bandgap implies an increase of interaction between the (C₂₀)₂ dimers, when van der Waals interaction is included. The condensed phase of (C₂₀)₂ dimers is found to be a semiconductor with a fundamental gap to be only 0.291 eV, which is much smaller than the gap value of fcc C₆₀ solid (1.5 eV).⁴⁴ 
(C ₂₀ ) ₂二聚体金刚石结构的电子能带结构和态密度分别如图4(c) - (d)所示。我们可以注意到电子能带结构中有多个平带，这些平带的原因是C ₂₀团簇富勒烯之间的相互作用非常弱（弱范德华）。没有范德华相互作用时的带隙为 0.415 eV，而当包括范德华相互作用时，带隙减小到 0.291 eV。当包括范德华相互作用时，带隙的减小意味着(C ₂₀ ) ₂二聚体之间的相互作用增加。发现(C ₂₀ ) ₂二聚体的凝聚相是半导体，基本能隙仅为0.291 eV，远小于fcc C ₆₀固体的能隙值(1.5 eV)。 ⁴⁴

In summary, first-principles calculations have been performed to investigate the structural and electronic properties of 1D, 2D, and 3D lattices with C₂₀ clusters are used as building block of the solid. We found that 1D molecular solid could be formed when C₂₀ clusters were head to head. The investigation on the 2D graphene structure made of C₂₀ showed that, in this 2D case, C₂₀ units were distorted and interlinked by covalent bonds. In the three-dimensional case, the optimized diamond structure composed of C₂₀ molecules yielded dimered C₂₀’s, i.e., (C₂₀)₂, which were then condensed by weak van der Waals interaction between (C₂₀)₂ dimers. Hence the molecular solid consists of (C₂₀)₂ dimers rather than C₂₀ clusters were formed. This study will help to understand the properties and applications of C₂₀ clusters in the field of new nanometer electronics.
总之，我们进行了第一性原理计算，以研究使用 C ₂₀簇作为固体构建块的 1D、2D 和 3D 晶格的结构和电子特性。我们发现当C ₂₀簇头对头时可以形成一维分子固体。对由C ₂₀制成的2D石墨烯结构的研究表明，在这种2D情况下，C ₂₀单元扭曲并通过共价键互连。在三维情况下，由C ₂₀分子组成的优化金刚石结构产生二聚体C ₂₀ ，即(C ₂₀ ) ₂ ，然后通过(C ₂₀ ) ₂二聚体之间的弱范德华相互作用进行缩合。因此形成了由(C ₂₀ ) ₂二聚体而不是C ₂₀簇组成的分子固体。该研究将有助于了解C ₂₀团簇的性质及其在新型纳米电子学领域的应用。

This work is supported by the National Key R&D Program of China under grant No. 2016YFA0202601 and 2016YFB0901502, the National Natural Science Foundation of China under Grant Nos. 21233004.
该工作得到了国家重点研发计划（批准号：2016YFA0202601和2016YFB0901502）和国家自然科学基金（批准号：21233004）的支持。