The crystal structure of Cs2⁢𝑀⁢𝑋6 ( 𝑀 = Ti, Zr; 𝑋=I,Br) VODP can be visualized by removing every other 𝑀-site cation from the Cs⁢𝑀⁢𝑋3 perovskite structure, thus the name of vacancy-ordered double perovskite [38,49], as shown in Fig. 1(a) in which the dotted red circles indicate the ordered vacancies. We mentioned that VODPs are a class of well-characterized, stable complex halides with the general composition 𝐴2⁢𝑀⁢𝑋6 ( 𝐴 = Cs) [37–39]. The [𝑀⁢𝑋6]2− octahedral clusters in VODP Cs2⁢𝑀⁢𝑋6 do not form covalent bonds with one another because they are separated by the evenly distributed 𝑀-site vacancies. The band structures of these four VODPs are quite similar (Fig. S1 in Supplemental Material [50]), and replacing Ti (I) with Zr (Br) in Cs2⁢𝑀⁢𝑋6 increases the fundamental band gap [37]. Both Cs2⁢Ti⁢𝑋6 and Cs2⁢Zr⁢𝑋6 have an indirect band gap between the valence band maximum at Γ and the conduction band minimum at 𝑋. The low-energy conduction bands are mainly derived from the 𝑀-site 𝑑 states, whereas the highest valence bands are mostly contributed by the halogen element 𝑋. Under the influence of the octahedral crystal field, the 𝑑-derived conduction bands split into a doublet and a triplet group; the low-energy triplet bands are fairly flat with an extremely narrow band width (0.29, 0.25, 0.44, and 0.37 eV for Cs2⁢TiI6, Cs2⁢TiBr6, Cs2⁢ZrI6, and Cs2⁢ZrBr6, respectively). The top valence band is virtually dispersionless from Γ to 𝑋, and the direct gap of Cs2⁢TiI6 at Γ is only 40 meV larger than the indirect gap between Γ and 𝑋.
晶体结构 Cs2⁢𝑀⁢𝑋6 （ 𝑀 = 钛、锆； 𝑋=I,Br ) VODP 可以通过删除所有其他内容来可视化 𝑀 -位点阳离子来自 Cs⁢𝑀⁢𝑋3 钙钛矿结构，因此被称为空位有序双钙钛矿[38, 49] ，如图1（a）所示，其中红色虚线圆圈表示有序空位。我们提到过 VODP 是一类特征良好、稳定的复杂卤化物，其一般组成为 𝐴2⁢𝑀⁢𝑋6 （ 𝐴 =铯） [37–39] 。这 [𝑀⁢𝑋6]2− VODP 中的八面体簇 Cs2⁢𝑀⁢𝑋6 彼此不形成共价键，因为它们被均匀分布的 𝑀 - 网站空缺。这四种VODP的能带结构非常相似（补充材料[50]中的图S1），并且用Zr（Br）代替Ti（I） Cs2⁢𝑀⁢𝑋6 增加基本带隙[37] 。两个都 Cs2⁢Ti⁢𝑋6 和 Cs2⁢Zr⁢𝑋6 在价带最大值之间有一个间接带隙 Γ 和导带最小值 𝑋 。低能导带主要来源于 𝑀 -地点 𝑑 态，而最高价带主要由卤素元素贡献 𝑋 。 在八面体晶体场的影响下， 𝑑 -衍生的导带分裂成双线态和三线态组；低能三重态能带相当平坦，带宽极窄（0.29、0.25、0.44 和 0.37 eV Cs2⁢TiI6, Cs2⁢TiBr6, Cs2⁢ZrI6 ， 和 Cs2⁢ZrBr6 ， 分别）。顶部价带实际上是无色散的 Γ 到 𝑋 ，以及直接差距 Cs2⁢TiI6 在 Γ 仅比之间的间接间隙大40 meV Γ 和 𝑋 。

FIG. 1. 如图。 1.

In order to facilitate later discussion and to better illustrate the roles of Cs+ and [𝑀⁢𝑋6]2− in defining the frontier states, we compare the DFT band structure of Cs2⁢TiI6 with those of two fictitious systems, namely, a periodic system of [TiI6]2− without the Cs+ ions (with the same lattice constant as Cs2⁢TiI6) and an isolated [TiI6]2− octahedron, as shown in Fig. 1(b). The DFT band structures are calculated using the Perdew-Burke-Ernzerhof (PBE) functional within the quantum espresso package (after Refs. [51–54]; for details of the calculations, see the Supplemental Material [50]). The nearly identical frontier band structures of Cs2⁢TiI6 and [TiI6]2− suggest that Cs+ ions merely function as electron donors and spacers without significantly affecting the band dispersion of the system. The role of Cs+ here is similar to that of CH3⁢NH+3 in CH3⁢NH3⁢PbI3 [55]. The highest-occupied molecular orbital (HOMO) of [TiI6]2− is mostly composed of the iodine 5⁢𝑝 orbitals, while the titanium 3⁢𝑑 orbitals dominate the lowest-unoccupied molecular orbital (LUMO). These [TiI6]2− derived LUMO and HOMO states interact to form the lowest conduction bands and highest valence bands of Cs2⁢TiI6, respectively. The narrow widths of the frontier energy bands in Cs2⁢TiI6 further suggest that [TiI6]2− behaves like a polyatomic superanion [56], which interacts weakly with itself. In this manner, Cs2⁢TiI6 crystal can be understood as an assembly of [TiI6]2− units and Cs+ spacers. As we will discuss below, such a unique structure is responsible for extremely strong excitonic effects in VODPs.
为了方便后面的讨论以及更好的说明其作用 Cs+ 和 [𝑀⁢𝑋6]2− 在定义前沿态时，我们比较了 DFT 能带结构 Cs2⁢TiI6 与两个虚构的系统，即周期系统 [TiI6]2− 没有 Cs+ 离子（具有相同的晶格常数 Cs2⁢TiI6 ）和一个孤立的 [TiI6]2− 八面体，如图1(b)所示。 DFT 能带结构是使用量子浓缩咖啡包中的 Perdew-Burke-Ernzerhof (PBE) 函数计算的（参考文献[51–54]之后；有关计算的详细信息，请参阅补充材料[50] ）。几乎相同的前沿能带结构 Cs2⁢TiI6 和 [TiI6]2− 建议 Cs+ 离子仅充当电子供体和间隔物，而不会显着影响系统的能带色散。的作用 Cs+ 这里类似于 CH3⁢NH+3 在 CH3⁢NH3⁢PbI3 [55] 。最高占据分子轨道（HOMO） [TiI6]2− 主要由碘组成 5⁢𝑝 轨道，而钛 3⁢𝑑 轨道主导最低未占据分子轨道（LUMO）。这些 [TiI6]2− 衍生的 LUMO 和 HOMO 态相互作用形成最低导带和最高价带 Cs2⁢TiI6 ， 分别。 前沿能带的宽度较窄 Cs2⁢TiI6 进一步建议 [TiI6]2− 其行为类似于多原子超阴离子[56] ，其自身相互作用较弱。以这种方式， Cs2⁢TiI6 晶体可以理解为一个集合体 [TiI6]2− 单位和 Cs+ 垫片。正如我们将在下面讨论的，这种独特的结构导致了 VODP 中极强的激子效应。

The QP band structure of Cs2⁢TiI6 is calculated using the berkeleygw package [43–48] as shown in Fig. 1(c). The indirect QP band gap of Cs2⁢TiI6 is 1.79 eV, which is slightly smaller than the direct band gap at Γ (1.83 eV). The calculated QP band gap within the 𝐺0⁢𝑊0 approximation is significantly larger than the measured optical gap of 1.02 eV [49]. It is noteworthy that our calculated QP gap is smaller than that reported in a previous work [37] by 0.54 eV. This discrepancy mainly stems from the use of different cutoff parameters and treatment of the frequency-dependent dielectric function [57–60]. Indeed, a small cutoff of dielectric matrices used in 𝐺𝑊 calculations can lead to an overestimation of the quasiparticle band gap by as much as 0.5 eV (Fig. S2 in Supplemental Material [50]). We also compare the QP band gaps of all four VODPs in Table I and the DFT-PBE band structures in Fig. S1 of Supplemental Material [50].
QP 能带结构 Cs2⁢TiI6 使用berkeleygw包[43-48]计算，如图1(c)所示。间接 QP 带隙为 Cs2⁢TiI6 为 1.79 eV，略小于 处的直接带隙 Γ （1.83 eV）。计算出的 QP 带隙在 𝐺0⁢𝑊0 近似值明显大于测量的光学间隙 1.02 eV [49] 。值得注意的是，我们计算的 QP 间隙比之前的工作[37]中报告的小 0.54 eV。这种差异主要源于使用不同的截止参数和对频率相关介电函数的处理[57-60] 。事实上，介电矩阵的小截止用于 𝐺𝑊 计算可能导致准粒子带隙高估多达 0.5 eV（补充材料中的图 S2 [50] ）。我们还比较了表中所有四种 VODP 的 QP 带隙和补充材料[50]图 S1 中的 DFT-PBE 能带结构。

A comparison of the band structures calculated with and without spin-orbit coupling (SOC) (Fig. S1 of Supplemental Material [50]) reveals strong relativistic effects on the top valence bands, and the gap at the Γ point of Cs2⁢TiI6 is reduced by about 130 meV due to the SOC effects. Figure 1(d) highlights the impacts of the SOC effects on the frontier electronic states at the Γ and 𝑋 points of Cs2⁢TiI6. If the SOC effects are neglected, the lowest-energy direct transitions at the Γ and 𝑋 points ( Γ15′→Γ25′ and 𝑋4→𝑋3) are dipole forbidden, while the second lowest-energy direct transition at the Γ point ( Γ15→Γ25′) is dipole allowed [61]. After the SOC effects are considered, the triply degenerate states split into a quadruplet and a doublet (including Kramer's degeneracy). The dipole-allowed optical transitions are indicated by green arrows in Fig. 1(d). Due to the SOC splitting, the energy of the dipole-allowed transition Γ−8→Γ+8 is smaller than that of Γ15→Γ25′ by about 0.31 eV. In the following, we show that this trend also holds when the excitonic effects are considered.
对使用和不使用自旋轨道耦合 (SOC) 计算的能带结构进行比较（补充材料[50]的图 S1）揭示了顶部价带和顶部价带处的带隙具有很强的相对论效应。 Γ 的点 Cs2⁢TiI6 由于 SOC 效应，降低了约 130 meV。图1(d)突出显示了 SOC 效应对前沿电子态的影响 Γ 和 𝑋 点的 Cs2⁢TiI6 。如果忽略 SOC 效应，则最低能量直接跃迁 Γ 和 𝑋 点（ Γ15′→Γ25′ 和 𝑋4→𝑋3 ）是偶极禁戒，而第二低能量的直接跃迁在 Γ 观点 （ Γ15→Γ25′ ) 是偶极子允许的[61] 。考虑 SOC 效应后，三重简并态分裂为四重态和二重态（包括克莱默简并态）。偶极子允许的光学跃迁由图1(d)中的绿色箭头表示。由于 SOC 分裂，偶极子允许跃迁的能量 Γ−8→Γ+8 小于 Γ15→Γ25′ 约 0.31 eV。在下文中，我们表明，当考虑激子效应时，这种趋势也成立。

Figure 2(a) shows the imaginary part of the frequency-dependent dielectric constant of Cs2⁢TiI6, calculated with and without including the e-h interaction. Our 𝐺⁢𝑊+BSE calculations reveal several prominent excitonic absorption peaks far below the quasiparticle band gap. In fact, the lowest-energy exciton locates at 0.85 eV, indicating a giant binding energy of nearly 1 eV and the formation of Frenkel excitons. The lowest-energy exciton is a dark exciton with a negligible optical dipole moment, while the lowest-energy bright exciton locates at 1.04 eV, agreeing well with the measured optical gap of 1.02 eV [49]. We have carefully checked the convergence of the calculated exciton binding energies and absorption spectra and found that the results converge quickly with respect to the density of the 𝑘-point sampling (as shown in Table S1 and Fig. S3 of Supplementary Material [50]). In contrast, binding energies of Wannier excitons in bulk and 2D systems often require extremely dense 𝑘 grids to achieve proper convergence [62,63]. As we will show later, the e-h amplitudes of the low-energy excitons spread across a large part of the Brillouin zone (BZ), which explains the rapid convergence behavior of the calculated exciton binding energy.
图2(a)显示了随频率变化的介电常数的虚部 Cs2⁢TiI6 ，在包含和不包含eh交互作用的情况下进行计算。我们的 𝐺⁢𝑊+BSE 计算揭示了远低于准粒子带隙的几个突出的激子吸收峰。事实上，最低能量的激子位于0.85 eV，表明具有接近1 eV的巨大结合能，并且形成了弗兰克尔激子。最低能量的激子是光学偶极矩可以忽略不计的暗激子，而最低能量的亮激子位于1.04 eV，与测量的1.02 eV光学间隙非常吻合[49] 。我们仔细检查了计算的激子结合能和吸收光谱的收敛性，发现结果相对于激子的密度很快收敛。 𝑘 点采样（如补充材料[50]的表S1和图S3所示）。相比之下，块体和二维系统中万尼尔激子的结合能通常需要极其致密的 𝑘 网格以实现适当的收敛[62, 63] 。正如我们稍后将展示的，低能激子的eh振幅分布在布里渊区 (BZ) 的大部分区域，这解释了计算的激子结合能的快速收敛行为。

FIG. 2. 如图。 2.

The strong SOC effects on the calculated band structure, especially on the top valence states as shown in Fig. 1(c), result in significant changes in the calculated excitonic structure and optical absorption. To uncover the effects of SOC on the low-energy excitons, we compare the excitonic structures calculated with and without the SOC effects. When the SOC effects are neglected, an excitonic state can be either spin-0 (singlet) or spin-1 (triplet). The triplet excitons are always dark since spin is conserved in optical dipole transitions if the SOC effects are neglected. The excitation energies of the singlet and triplet excitons for Cs2⁢TiI6 are shown in Fig. 2(b), in which each vertical line corresponds to an excitonic state; these lines are color coded according to their brightness, i.e., the square of the dipole matrix elements. The lowest-energy triplet exciton is about 0.1 eV lower than the lowest-energy singlet one, which is also a dark exciton due to the orbital symmetry as discussed earlier. The lowest-energy bright exciton is located at around 1.34 eV and can be attributed to the Γ15→Γ25′ transitions shown in Fig. 1(d).
SOC对计算的能带结构的强烈影响，特别是如图1（c）所示的顶价态，导致计算的激子结构和光吸收的显着变化。为了揭示 SOC 对低能激子的影响，我们比较了有和没有 SOC 影响时计算的激子结构。当忽略 SOC 效应时，激子态可以是自旋 0（单线态）或自旋 1（三线态）。如果忽略 SOC 效应，三线态激子总是暗的，因为自旋在光学偶极跃迁中是守恒的。单线态和三线态激子的激发能 Cs2⁢TiI6 如图2(b)所示，其中每条垂直线对应于一个激子态；这些线根据其亮度（即偶极子矩阵元素的平方）进行颜色编码。最低能量的三线态激子比最低能量的单线态激子低约 0.1 eV，由于前面讨论的轨道对称性，单线态激子也是暗激子。最低能量的亮激子位于 1.34 eV 左右，可归因于 Γ15→Γ25′ 转变如图1(d)所示。

When the SOC effects are considered, the excitonic states are a mixture of spin singlets and spin triplets [45]. The lowest-energy exciton, which is mainly derived from the low-energy spin-triplet states, is dark, as shown in Fig. 2(c). Moreover, the SOC effects cause a significant redshift (from 1.34 to 1.04 eV) of the absorption edge, bringing theory in better agreement with experiment [49]. The exciton binding energy of Cs2⁢TiI6 ( 𝐸ex𝑏=0.98eV) is nearly twice that of monolayer 2H- MoSe2 ( 𝐸ex𝑏=0.55eV) [5], which has a comparable quasiparticle gap ( 𝐸𝑔= 2.1 eV). This finding is rather surprising, as it is well known that the excitonic effects in bulk semiconductors are often much weaker than those in 2D semiconductors [64–66] owing to the stronger screening effect in 3D solids.
当考虑 SOC 效应时，激子态是自旋单线态和自旋三线态的混合态[45] 。最低能量的激子主要源自低能自旋三重态，是暗色的，如图2(c)所示。此外，SOC效应导致吸收边发生显着的红移（从1.34到1.04 eV），使理论与实验更加一致[49] 。激子结合能 Cs2⁢TiI6 ( 𝐸ex𝑏=0.98eV ）几乎是单层2H-的两倍 MoSe2 ( 𝐸ex𝑏=0.55eV ) [5] ，具有可比较的准粒子能隙 ( 𝐸𝑔= 2.1 eV）。这一发现相当令人惊讶，因为众所周知，由于 3D 固体中的屏蔽效应更强，块体半导体中的激子效应通常比 2D 半导体中的激子效应弱得多[64-66] 。

The calculated energies of the lowest-energy bright excitons for the other three VODPs ( Cs2⁢TiBr6, Cs2⁢ZrI6, and Cs2⁢ZrBr6) also agree well with the experimental optical gaps, as summarized in Table I. The imaginary parts of the dielectric functions of these three VODPs are compared in Figs. S4–S6 of Supplemental Material [50]. In addition, we compared the calculated absorption spectra of Cs2⁢TiI6 with the experimental results [67], which achieved best match, as shown in Fig. S9 of Supplemental Material [50]. Similar to the case of Cs2⁢TiI6, the lowest-energy excitons of these three VODPs are also dark, and the corresponding exciton binding energies are extremely large. Perhaps the most striking finding is the binding energy of the lowest dark exciton in Cs2⁢ZrBr6, which reaches 1.69 eV.
计算出其他三个 VODP 的最低能量亮激子的能量（ Cs2⁢TiBr6, Cs2⁢ZrI6 ， 和 Cs2⁢ZrBr6 ）也与实验光学间隙非常吻合，如表所示。这三种 VODP 的介电函数的虚部在图 2 和 3 中进行了比较。补充材料[50]的 S4–S6。此外，我们还比较了计算的吸收光谱 Cs2⁢TiI6 与实验结果[67] ，达到最佳匹配，如补充材料[50]的图S9所示。类似的情况 Cs2⁢TiI6 ，这三种VODP的最低能量激子也是暗的，相应的激子结合能极大。也许最引人注目的发现是最低暗激子的结合能 Cs2⁢ZrBr6 ，达到 1.69 eV。

The abnormally large exciton binding energies in these moderate-gap VODPs deserve closer scrutiny. To gain a deeper insight into the low-energy excitons, we examine their wave functions in both the reciprocal and real spaces. Within the Tamm-Dancoff approximation [45,68], the exciton wave functions can be expanded as a linear combination of products of the electron and hole wave functions 𝜓𝑐⁢𝒌⁡(𝒓𝒆) and 𝜓𝑣⁢𝒌⁡(𝒓𝒉):
这些中等能隙 VODP 中异常大的激子结合能值得更仔细的研究。为了更深入地了解低能激子，我们研究了它们在倒易空间和实空间中的波函数。在 Tamm-Dancoff 近似[45, 68]中，激子波函数可以展开为电子和空穴波函数乘积的线性组合 𝜓𝑐⁢𝒌⁡(𝒓𝒆) 和 𝜓𝑣⁢𝒌⁡(𝒓𝒉) :

where 𝑆 indexes the excitonic state and 𝐴𝑆𝑣⁢𝑐⁢𝒌 are often called the e-h amplitudes, which give the weights of independent e-h pair states in an excitonic state 𝑆. We have analyzed the BZ distribution of the lowest-energy dark and bright excitons in VODPs by defining a 𝑘-dependent e-h amplitude |Ψ𝑆⁡(𝒌)|2=∑𝑣⁢𝑐|𝐴𝑆𝑣⁢𝑐⁢𝒌|2. The radii of the circles in Fig. 3(a) are proportional to |Ψ𝑆⁡(𝒌)|2. In contrast to halide perovskites, in which the Wannier-Mott excitons are highly localized in small regions of the BZ [69–71], the low-energy excitons in Cs2⁢TiI6 are far more extended in the BZ, as can be seen from Fig. 3(a). Therefore, the excitonic states can be accurately described even with a relatively coarse 𝑘 grid. To further illustrate the Frenkel nature of the excitons, we compare Ψ𝑆⁡(𝒌) with that of a hydrogenic model [72,73], Ψhy⁡(𝒌)=(2⁢𝑎0)32/𝜋⁢(1+𝑎20⁢𝑘2)2, as shown in the inset of Fig. 3(a). The calculated |Ψ𝑆⁡(𝒌)|2 of the lowest-energy exciton agrees reasonably with |Ψhy⁡(𝒌)|2 using very small Bohr radius 𝑎0=3.7a.u., suggesting a highly localized exciton in this system.
在哪里 𝑆 索引激子态和 𝐴𝑆𝑣⁢𝑐⁢𝒌 通常称为eh振幅，它给出激子态中独立eh对状态的权重 𝑆 。我们通过定义 VODP 中最低能量暗激子和亮激子的 BZ 分布进行了分析 𝑘 依赖的eh幅度 |Ψ𝑆⁡(𝒌)|2=∑𝑣⁢𝑐|𝐴𝑆𝑣⁢𝑐⁢𝒌|2 。图3(a)中圆的半径与 |Ψ𝑆⁡(𝒌)|2 。与卤化物钙钛矿相反，其中万尼尔-莫特激子高度集中在 BZ 的小区域中[69-71] ，低能激子 Cs2⁢TiI6 从图3(a)可以看出，BZ 中的边界更加扩展。因此，即使使用相对粗糙的参数，也可以准确地描述激子态。 𝑘 网格。为了进一步说明激子的弗兰克尔性质，我们比较 Ψ𝑆⁡(𝒌) 与氢模型[72, 73] ， Ψhy⁡(𝒌)=(2⁢𝑎0)32/𝜋⁢(1+𝑎20⁢𝑘2)2 ，如图3(a)的插图所示。计算出的 |Ψ𝑆⁡(𝒌)|2 最低能量激子的合理符合 |Ψhy⁡(𝒌)|2 使用非常小的玻尔半径 𝑎0=3.7a.u. ，表明该系统中存在高度局域化的激子。

FIG. 3. 如图。 3.

The localization of excitons can also be directly visualized in real space. To this end, we fix the hole position at an iodine atom, and plot the electron distribution of the lowest-energy dark and bright excitons for Cs2⁢TiI6 in Fig. 3(b), which shows that the electron is mainly localized within one [TiI6]2− octahedron. This conclusion does not change with different choices of the hole position, as shown in Fig. S7 of the Supplemental Material [50]. The fact that excitons are highly localized is in line with our previous analysis of the electronic structure, and the conclusion that Cs2⁢𝑀⁢𝑋6 can be considered as a 3D assembly of weakly coupled [𝑀⁢𝑋6]2− clusters, resulting in the formation of Frenkel excitons like those in molecular solids [74]. The electronic dielectric constants ɛ∞ of Cs2⁢𝑀⁢𝑋6 range from 2.97 to 4.46 (Table I), which are very small compared to typical semiconductors such as Si or GaAs, but are comparable to other molecular solids such as naphthalene or anthracene [75] (Table S2 in Supplemental Material [50]). The relatively weak dielectric screening of Cs2⁢𝑀⁢𝑋6 also contributes to an overall strong e-h interaction, thus the exceptionally large exciton binding energies.
激子的局域化也可以在真实空间中直接可视化。为此，我们固定了碘原子上的空穴位置，并绘制了最低能量暗激子和亮激子的电子分布： Cs2⁢TiI6 在图3(b)中，表明电子主要局域于一个区域内。 [TiI6]2− 八面体。这一结论不会随着孔位置选择的不同而改变，如补充材料[50]的图S7所示。激子高度局域化这一事实与我们之前对电子结构的分析一致，得出的结论是 Cs2⁢𝑀⁢𝑋6 可以被视为弱耦合的 3D 组装 [𝑀⁢𝑋6]2− 团簇，导致形成像分子固体中那样的弗伦克尔激子[74] 。电子介电常数 ɛ∞ 的 Cs2⁢𝑀⁢𝑋6 范围从 2.97 到 4.46（表），与 Si 或 GaAs 等典型半导体相比非常小，但与萘或蒽等其他分子固体相当[75] （补充材料[50]中的表 S2）。相对较弱的介电屏蔽 Cs2⁢𝑀⁢𝑋6 也有助于整体强的电子相互作用，从而产生异常大的激子结合能。

Finally, we would like to address possible exciton-phonon coupling effects on the calculated exciton binding energy. As recently shown by Filip et al. [76], exciton-phonon coupling can considerably renormalize the exciton binding energies in ionic materials. We estimate the correction ( Δ⁢𝐸𝑏) to the exciton binding energy due to phonon-screening effects using a simplified formula [76]:
最后，我们想解决激子-声子耦合对计算的激子结合能的可能影响。正如 Filip等人最近所表明的。 [76] ，激子-声子耦合可以显着重整离子材料中的激子结合能。我们估计修正值（ Δ⁢𝐸𝑏 ）使用简化公式将由于声子屏蔽效应引起的激子结合能[76] ：

where 𝜀∞ is the high-frequency dielectric constant, 𝜀0 is the static dielectric constant, and 𝜔LO is the frequency of the dominant longitudinal optical (LO) phonon, which is usually the highest LO phonon. The phonon dispersions [77,78] of four VODPs are shown in Fig. S8 of the Supplemental Material [50]. We show in Table I the estimated Δ⁢𝐸𝑏, which are less than 50 meV for all four systems studied. Therefore, while these corrections are not negligible, they do not significantly affect our conclusion.
在哪里 𝜀∞ 是高频介电常数， 𝜀0 是静态介电常数，并且 𝜔LO 是主要纵向光学 (LO) 声子的频率，通常是最高的 LO 声子。补充材料[50]的图 S8 显示了四种 VODP 的声子色散[77, 78] 。我们在表中显示了估计 Δ⁢𝐸𝑏 ，所研究的所有四个系统均小于 50 meV。因此，虽然这些修正不可忽略，但它们并不会显着影响我们的结论。