MoS2 多层电子和光学特性对层间耦合和范霍夫奇异性的依赖性

其中 ℏω 等于 E ck − E vk , S k 表示由 E 表示的恒定能量表面 ck − E vk =ℏω =常数。状态的联合密度J 简历 (ω ) 与从价带到导带的跃迁以及 J 中的大峰有关 简历 (ω ) 将起源于 ∇k (E ck − E vk ) ≈ 0. k 中的点 -空间其中∇k (E ck − E vk ) =0 称为临界点或范霍夫奇点 (VHS)，而 E ck − E vk 被称为临界点能量 [26, 27]。临界点∇k E ck =∇k E vk =0 通常只出现在高对称点，而临界点 ∇k E ck =∇k E vk ≠ 0 可能出现在布里渊区的任何一般点 [27, 45]。在二维情况下，存在三种类型的临界点，即P 0（最低点），P 1（鞍点）和 P 2（最高点）。在点 P 0 或 P 2、JDOS中出现阶跃函数奇点，而在鞍点P 1，JDOS 被描述为对数奇点 [27]。更详细地说，E c (k x , k 是 ) − E v (k x , k 是 ) 可以在关于临界点的泰勒级数中展开。将展开限制为二次项，由于奇点的性质不出现线性项，则有

因此，出现了三种类型的临界点。对于 P 0, (b x> 0, b 是> 0)，对于 P 1, (b x> 0, b 是 <0) 或 (b x <0, b 是> 0)，对于 P 2, (b x <0, b 是 <0)。在本文中，对于 MoS2 多层膜，只有 P 0个临界点。

图 6a 给出了 E 的 MoS2 多层介电函数的虚部，\( {\varepsilon}_2^{xx}\left(\omega \right) \) ||x.我们发现了一个有趣的现象，介电函数的虚部\( {\varepsilon}_2^{xx}\left(\omega \right) \) 具有高原，并且不同层MoS2 的高原在范围内几乎相等1.75 eV~2.19 eV。从阈值能量到 1.75 eV，\( {\varepsilon}_2^{xx}\left(\omega \right) \) 对于不同的 MoS2 多层有很大不同。不同层的平台的阈值和结束能量不同，尤其是单层\( {\varepsilon}_2^{xx}\left(\omega \right) \) 平台的能量范围明显宽于那些其他多层。单层 MoS2 介电函数的阈值能量等于其 1.64 eV 的直接带隙。然而，双层介电函数的阈值能量不是1.17 eV的间接带隙，而是价带和导带之间1.62 eV的直接带隙的最小值。这是因为我们只研究了具有相同电子波矢量的价带和导带之间的跃迁，这些跃迁被归类为直接光学跃迁 [36, 47]。随着 MoS2 层数增加到 4，我们发现多层 MoS2 系统的 \( {\varepsilon}_2^{xx}\left(\omega \right) \) 几乎与体相无法区分。因此，我们在此仅详细讨论单层、双层和三层以及块状 MoS2 的平台。单层、双层、三层和块状 MoS2 的 \( {\varepsilon}_2^{xx}\left(\omega \right) \) 平台分别以 2.57 eV、2.28 eV、2.21 eV 和 2.19 eV 结束。为了更准确地解释这一点，图 7 中显示了单层、双层、三层和块状 MoS2 的 JDOS。从图 7 中，平台也显示在 JDOS 中。单层、双层和三层 JDOS 的平台分别以 2.57 eV、2.28 eV、2.21 eV 结束，与它们的 \( {\varepsilon}_2^{xx}\left(\omega \right ) \)对于块体 MoS2，JDOS 的平台期结束于 2.09 eV，略小于介电函数中的 2.19 eV \( {\varepsilon}_2^{xx}\left(\omega \right) \).

一 介电函数的虚部，b 介电函数的实部，c 吸收系数和 d 不同数量的 MoS2 层的反射率光谱。 c 中的插图 还展示了实验数据 [46]

单层、双层、三层和块状 MoS2 的联合态密度

为了准确分析电子跃迁并详细分析介电函数 \( {\varepsilon}_2^{xx}\left(\omega \right) \)，直接能隙 ΔE(NC − NV)，在单层、双层、三层和块状 MoS2 的导带和价带如图 8 所示。符号 NC 和 NV 表示导带和价带的序数。因此，NC =1、2 和 3 表示材料的最低、第二低和第三低的未占用带。另一方面，NV =9、18 和 27（取决于晶胞中的电子数）分别表示单层、双层和三层 MoS2 的最高占据带。对于单层，在 0 ~ 2.57 eV 范围内，发现电子跃迁仅从最高占据带 NV =9 到最低未占据带 NC =1。从图 8a 中，出现最小值出现在高对称点K 和 JDOS 的阈值（图 7a）出现在 1.64 eV，这实际上是单层 MoS2 的直接带隙。在高对称点K附近，ΔE(NC =1 − NV =9)的曲线类似于单层MoS2的抛物线。因此，∇k (E ck − E vk ) =0 在 K 点，表示高对称点 K 处的临界点。在二维结构中，该临界点属于 P 0 型奇点 [27]，因此它导致了 JDOS 中的一个步骤。因此，在临界点能量 1.64 eV 处发现了 JDOS 平台的阈值能量。 JDOS 平台的结束能量接近 2.57 eV，这是由于两个 P 的出现 B1 点处的 0 型奇点 (k =(0.00, 0.16, 0.00)) 和点 B2 (k =(- 0.10, 0.20, 0.00))。 ΔE(NC =1 − NV =9) 曲线在两个临界点 B1 和 B2 附近的斜率非常小，这导致 JDOS 的快速增加（见方程（5））。 JDOS 的这些长平台的主要临界点列于表 2 中，包括类型、位置、过渡带和直接能隙 ΔE(NC − NV)。此外，我们发现 ∇k E ck =∇k E vk =0 发生在高对称点 K 处，其中价带和导带的斜率是水平的。而 ∇k E ck =∇k E vk ≠ 0 发生在 B1 和 B2 点，这意味着两个波段的斜率是平行的。同时，对单层的能带结构和直接能隙（见图8a）的分析表明，当直接能隙ΔE低于2.65 eV时，只有NV =9和NC =1之间的跃迁对JDOS有贡献；当 ΔE 大于 2.65 eV 时，NV =9 到 NC =2 的转变也开始对 JDOS 做出贡献；而当 ΔE 达到 2.86 eV 以上时，NV =9 到 NC =3 的转变对 JDOS 有影响。应该指出的是，对于大于 2.65 eV 的能量，图 8a 中的许多波段将有助于 JDOS。单层 MoS2 的 JDOS 在 1.64 ~ 2.57 eV 范围内呈现平稳状态，并且表达式 |Mvc| ² /ω ² 发现在这个范围内很小。根据方程。 (1)和(5)，介电函数的虚部\( {\varepsilon}_2^{xx}\left(\omega \right) \) 主要由JDOS和过渡矩阵元素决定，这给与 JDOS 相比，介电函数的虚部 \( {\varepsilon}_2^{xx}\left(\omega \right) \) 具有类似的平台。

a 导带和价带之间的直接能隙ΔE(NC − NV) 单层，b 双层，c 三层和d 块状 MoS2。 一 –d 单层、双层、三层和块状 MoS2 的带间跃迁分别有 3、6、12 和 6 个临界点

对于双层 MoS2，在 0 ~ 2.28 eV（JDOS 平台的终点）范围内，电子跃迁有助于 NV =17, 18 到 NC =1, 2。ΔE(NC − NV) 中的最小能量位于在 K 点，间隙为 1.62 eV。在图 8b 中，类似于单层 MoS2，双层 MoS2 在 K 点保持两条向上的抛物线曲线（来自 ΔE(NC =1 − NV =18) 和 ΔE(NC =2 − NV =18)）。因此，有两个 P 0 型奇点 (∇k (E ck − E vk ) =0) 在 K 点，导致 JDOS 中的一个步骤。 The critical point energies are both 1.62 eV, this is because that the conduction bands (NC =1 and NC =2) are degenerate at point K (as shown in Fig. 3b), which results in the same direct energy gap between transitions of NV =18 to NC =1 and NV =18 to NC =2. From Fig. 8b, as the direct energy gap is increased to 1.69 eV, two new parabolas (which come from ΔE(NC = 1 − NV = 17) and ΔE(NC = 2 − NV = 17)) appear and two new singularities emerge again at K point in the direct energy gap graph, leading to a new step in JDOS for bilayer MoS2 (see Fig. 7b). As a result, the JDOS of the bilayer MoS2 has two steps around the threshold of long plateau (see inset in Fig. 7b). Two parabolas (in Fig. 8b) contribute to the first step and four parabolas contribute to the second step in JDOS. It means that the value of the second step is roughly the double of the first one. As the ΔE reaches to 2.28eV, two new singularities appear at Γ point (where interband transitions come from Γ(NV =18→NC =1) and Γ(NV =18→NC =2)), which have great contribution to the JDOS and bring the end to the plateau. Our calculations demonstrate that ∇k E ck  = ∇k E vk  = 0 are satisfied not only at high symmetry point K, but also at high symmetry point Γ. Similar to the case of monolayer, we found that the term of |Mvc| ² /ω ² is a slowly varying function in the energy range of bilayer JDOS plateau; hence, \( {\varepsilon}_2^{xx}\left(\omega \right) \) of bilayer have a similar plateau in the energy range.

For trilayer MoS2, in the region of 0 ~ 2.21 eV, the JDOS are contributed from electronic transitions of NV =25, 26, and 27 to NC =1, 2, and 3. As shown in Fig. 8c, trilayer MoS2 have nine singularities at three different energies (ΔE =1.61 eV, 1.66 eV, and 1.72 eV, respectively) at the K point. Figure 3c depicts that the conduction bands (NC =1, 2, 3) are three-hold degenerate at point K; this means that there are three singularities at each critical point energy. According to our previous analysis, the JDOS and \( {\varepsilon}_2^{xx}\left(\omega \right) \) of trilayer MoS2 will show three steps near the thresholds of the long plateaus, the endpoints of the long plateaus of trilayer JDOS, and \( {\varepsilon}_2^{xx}\left(\omega \right) \) are then owing to the appearance of three singularities at Γ point with ΔE =2.21 eV (see Fig. 7c), which come from the interband transitions of Γ(NV =27→NC =1, 2, 3).

For bulk MoS2, the thresholds of \( {\varepsilon}_2^{xx}\left(\omega \right) \) and JDOS are also located at K point, with the smallest ΔE(NC − NV) equals to 1.59 eV. Nevertheless, there is no obvious step appeared in the thresholds of plateaus for both the \( {\varepsilon}_2^{xx}\left(\omega \right) \) and JDOS (see Fig. 6a and Fig. 7d). Based on the previous analysis, the number of steps in the monolayer, bilayer, and trilayer MoS2 are 1, 2, and 3, respectively. As the number of MoS2 layers increases, the number of steps also increases in the vicinity of the threshold energy. Thus, in the bulk MoS2, the JDOS curve is composed of numerous small steps around the threshold energy of the long plateau, and finally the small steps disappear near the threshold energy since the width of the small steps decreases. In the region of 0 ~ 2.09 eV, the electron transitions of bulk MoS2 are contributed to NV =17, 18 to NC =1, 2. The 2.09 eV is the endpoint of JDOS plateau of bulk MoS2, which is attributed to two singularities, i.e., the interband transitions of Γ(NV =18→NC =1) as well as Γ(NV =18→NC =2), as presented in Fig. 8d. However, the plateau endpoint of the imaginary part of dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) is 2.19 eV, which is greater than the counterpart of JDOS (e.g., 2.09 eV). Checked the transition matrix elements, it verified that some transitions are forbidden by the selection rule in the range of 2.09 eV to 2.19 eV. Therefore, the imaginary part of the dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) is nearly invariable in the range of 2.09 ~ 2.19 eV. As a result, the plateau of \( {\varepsilon}_2^{xx}\left(\omega \right) \) of bulk MoS2 is then 1.59 ~ 2.19 eV.

It has been shown that these thresholds of the JDOS plateaus are determined by singularities at the K point for all of the studied materials, see Fig. 8. The endpoint energy of the monolayer JDOS plateau is determined by two critical points at B1 and B2 (Fig. 8a). Nevertheless, the endpoint energies of bilayer, trilayer, and bulk JDOS plateaus are all dependent on the critical points at Γ(Fig. 8b–d). The interlayer coupling near point Γ is significantly larger than the near point K for all the systems of multilayer MoS2. The smallest direct energy gap decreases and the interlayer coupling increases as the number of layers grow. With the layer number increases, a very small decrease of direct energy gap at point K and a more significant decrease of direct energy gap at point Γ can be observed, as a result, a faint red shift in the threshold energy and a bright red shift in the end of both JDOS and \( {\varepsilon}_2^{xx}\left(\omega \right) \) plateaus can also be found. For monolayer MoS2, the smallest ΔE(NC − NV) at point Γ is 2.75 eV which is larger than that at point B1 (or point B2) with a value around 2.57 eV. When it goes to multilayer and bulk MoS2, the strong interlayer coupling near point Γ makes the smallest ΔE(NC − NV) at Γ less than those at point B1 (or point B2). Hence, monolayer owns the longest plateau of JDOS, which is between 1.64 eV and 2.57 eV. The shortest plateau of JDOS (from 1.59 eV to 2.09 eV) is shown in the bulk.

As the energy is increased to the value larger than the endpoint of long platform of the dielectric function, a peak A can be found at the position around 2.8 eV, for almost all the studied materials (Fig. 6a). The width of peak A for monolayer is narrower compared with those of multilayer MoS2; however, the intensity of peak A for monolayer is found to be a little stronger than multilayers. The differences between the imaginary parts of dielectric function for the monolayer and multilayer MoS2 are evident, on the other hand, the differences are small for multilayer MoS2.

In order to explore the detailed optical spectra of MoS2 multilayers, the real parts of the dielectric function ε 1(ω ), the absorption coefficients α (ω ), and the reflectivity spectra R (ω ) are presented in Fig. 6b–d. Our calculated data of bulk MoS2 for the real and imaginary parts of the dielectric function, ε 1(ω ) 和 ε 2(ω ), the absorption coefficient α (ω ) and the reflectivity R (ω ) agree well with the experimental data, except for the excitonic features near the band edge [48,49,50]. The calculated values of , which is called the static dielectric constant, for MoS2 multilayers and bulk can be found in Table 1. From Table 1, the calculated values of \( {\varepsilon}_1^{xx}(0) \) for multilayers and bulk MoS2 are all around 15.5, which is very close to the experimental value of 15.0 for bulk MoS2 [50]. The values of \( {\varepsilon}_1^{xx}(0) \) increase with the increasing number of MoS2 layers. For monolayer MoS2, a clear peak B of \( {\varepsilon}_1^{xx}\left(\omega \right) \) appears about 2.54 eV. Peak B of monolayer is clearly more significant than multilayers, and they are all similar for multilayer MoS2. As the layer number increases, the sharp structures (peak B) also move left slightly. In Fig. 6c, we also observe the emergence of long plateaus in the absorption coefficients, and absorption coefficients are around 1.5 × 10 ⁵ cm ⁻¹ at the long plateaus. There are also small steps around the thresholds for the absorption coefficients, which are consistent to those of the imaginary parts of dielectric function. With the layer number increases, the threshold energy of absorption coefficient decreases, while the number of small steps increases at the starting point of the plateau. For monolayer and multilayer MoS2, strong absorption peaks emerge at visible light range (1.65–3.26 eV), and the monolayer MoS2 own a highest absorption coefficient of 1.3 × 10 ⁶ cm ⁻¹ . The theoretical absorption coefficients for different number of MoS2 layers are compared with confocal absorption spectral imaging of MoS2 (the inset) [46], as shown in Fig. 6c. For monolayer and multilayer MoS2, a large peak of α (ω ) can be found at the position around 2.8 eV for both the calculation and experiment [46, 51]. Furthermore, a smoothly increase of α (ω ) can be found between 2.2 and 2.8 eV for both the theoretical and experimental curves. Therefore, from Fig. 6c, the calculated absorption coefficients of MoS2 multilayers show fairly good agreement with the experimental data [46], except for the excitonic peaks. The reflectivity spectra are given in Fig. 6d. The reflectivity spectra of MoS2 multilayers are all about 0.35–0.36 when energy is zero, which means that MoS2 system can reflect about 35 to 36% of the incident light. In the region of visible light, the maximum reflectivity of monolayer MoS2 is 64%, while the maxima of multilayer and bulk MoS2 are all about 58%. Because of the behaviors discussed, MoS2 monolayer and multilayers are being considered for photovoltaic applications.