† Corresponding author. E-mail:
Project supported by the Major Program of Aerospace Advanced Manufacturing Technology Research Foundation from NSFC and CASC, China (Grant No. U1537204), the National Key Research and Development Program of China (Grant No. 2017YFA0206301), and the National Natural Science Foundation of China (Grant No. 51702146).
Unusual quadratic dispersion of flexural vibrational mode and red-shift of Raman shift of in-plane mode with increasing layer-number are quite common and interesting in low-dimensional materials, but their physical origins still remain open questions. Combining ab initio density functional theory calculations with the empirical force-constant model, we study the lattice dynamics of two typical two-dimensional (2D) systems, few-layer h-BN and indium iodide (InI). We found that the unusual quadratic dispersion of flexural mode frequency on wave vector may be comprehended based on the competition between atomic interactions of different neighbors. Long-range interaction plays an essential role in determining the dynamic stability of the 2D systems. The frequency red-shift of in-plane Raman-active mode from monolayer to bulk arises mainly from the reduced long-range interaction due to the increasing screening effect.
Two-dimensional crystals have been attracting tremendous attention in the last few decades due to its great potential in fundamental research and multitude of applications. The quantum confinement due to dimension reduction gives rise to rich and novel physical properties which cannot be obtainable in their three-dimensional (3D) counterparts. It is very essential to pinpoint and understand the novel properties before it can be fully explored for diverse applications. So far there have been some phenomena which have not been fully understood and still await for a definitive answer, for example, the unusual quadratic phonon dispersion of flexural mode near the Brillouine zone center,[1–7] the counterintuitive frequency red-shift of in-plane mode with increasing layer thickness.[8–11]
The quadratic phonon dispersion behavior (ω ∼ q2) of the flexural (out-of-plane) acoustic mode against the linear dispersion (ω ∼ q) of the other acoustic phonon modes in 2D crystals have been observed widely ever since debut of graphene.[1] Due to the lack of a good understanding on the novel dispersion relation, there is a controversy on the contribution of the flexural mode to lattice thermal conductivity.[2–5] Liu et al.[6] studied such a quadratic dispersion relation of quasi-two-dimensional structures based on continuum elasticity theory, but the microscopic mechanism is still unknown. Jiang et al. studied microscopically the effect of rotation symmetry on the nonlinear dispersion of the flexural mode.[1,7] However, this analysis may be only limited in graphene system, not applicable to other 2D systems.
Another issue we are concerned with here is the unconventional frequency red-shift of in-plane Raman mode with increasing number of layers. It was firstly observed experimentally in MoS2 by Lee et al.[8] and also in h-BN crystals by Gorbachev et al.[9] Molina–Sanchez et al.[10] theoretically calculated the long-range screening as a function of layer thickness of MoS2 and WS2 crystals, and ascribed the thickness-reduced frequency of the in-plane
This study will focus on the lattice dynamics of two typical 2D systems, few-layer h-BN and indium iodide (InI), to get an insight of the two aforementioned phenomena by combining ab initio density functional calculations with the force-constant model. We found that the unusual quadratic dispersion of vibrational frequency on wave vector may be comprehended based on the competition between atomic interactions of different neighbors. Long-range interaction plays an essential role in determining the dynamic stability of the 2D systems. The frequency red-shift of in-plane Raman-active mode from monolayer to bulk is mainly due to the reduced long-range interaction by the increasing screening effect.
We used ab initio density functional theory as implemented in the Quantum Espresso package.[12] A periodic boundary condition with monolayer structures represented by a periodic array of slabs separated by a vacuum region(∼ 15 Å). We used the Perdew–Zunger exchange–correlation functional[13] within the local-density approximation (LDA) and norm-conserving pseudopotentials.[14] The Brillouin zone of the primitive unit cell of the 2D structures was sampled by 13 × 13 × 1 k-points.[15] The kinetic energy cutoff for wave functions was set at 150 Ry (1 Ry = 13.6056923 eV) and 10−10 Ry for a total energy difference between subsequent self-consistency iterations as the criterion for reaching self-consistency. All geometries are optimized using the conjugate gradient method,[16] until none of the residual Hellmann–Feynman forces exceeds 4 × 10−5 Ry/Bohr.
The phonon dispersion relation of InI and h-BN was calculated based on two methods, one is density functional perturbation theory (DFPT)[17] and the other the force constant model. The non-resonant Raman intensities were calculated based on Placzek approximation as introduced by Lazzeri and Mauri.[18]
The inter-atomic long-range interaction of polar materials includes the non-analytic term in the lattice dynamical matrices, which comes from the dipole–dipole interactions as an effect of vibration-induced polarization. Such a non-analytical behavior is available through the knowledge of the Born effective charges tensor Z* and the electronic dielectric permittivity tensor ɛ∞. The non-analytical, direction-dependent term at the vicinity of BZ center can be expressed by the following equation:[19,20]
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The force constant model is also used to calculate the phonon dispersion relation. Within this model, interatomic interactions up to the 4th neighbor are considered. The force constant tensor
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The atomic structures of monolayer and bulk indium iodide (InI) and hexagonal BN (h-BN) are shown in top view and side view, respectively, in Fig.
To study their lattice dynamics, it is essential to calculate the phonon dispersion relation. Figure
To have a better view of such an unusual behavior, we show in Fig.
![]() | Table 1. Screening effect of h-BN as a function of layer number. ψlr and ψsr are long-range and short-range force constants, respectively. Z* is Born effective charge. . |
The linear N dependence of Raman intensity observed in many 2D materials can be well understood from the quantized optical absorption[24,25] (∼ Nπα with α = e2/hc ≈ 1/137, N as the number of layers). The thickness-induced frequency redshift of in-plane mode has also been observed in other 2D systems such as MoS2, WS2, and MoTe2.[8,10,11] It is still in argument and not clear whether this is due to enhanced screening effect on the long-range interaction[8,10] or by Davydov splitting.[11] From our calculation results, we found that, unlike the split frequency values which go both up and down with increasing layer number in MoTe2, the split frequency values all go down, which excludes the origin from the Davydov splitting.
To unveil the possible origin due to the screening effect, we obtained both long-range and short-range lattice force constants, as well as dielectric constant as a function of layer number N of h-BN from ab initio calculations, as shown in Table
Now we have a look at the possible origin behind the quadratic dispersion of out-of-plane acoustic (ZA) mode in h-BN. In Fig.
Monolayer InI was theoretically predicted to be stable by Wang et al.[23] It has two atomic layers of alternating indium and iodine atoms with the In atom extruding slightly (≈ 0.50 Å) out of the layer plane. Few-layer and bulk phase have orthorhombic structure of Cmcm symmetry (D2h point group). Here we studied bi-layer, tri-layer, and bulk besides monolayer structure. Phonon dispersion relation for both monolayer and bulk is shown in Fig.
Two typical Raman active vibrational modes are studied here, one is the layer breathing Ag mode having the highest frequency value and the other the shearing Bg mode having the lowest frequency value, as highlighted in green and blue dots, respectively, in Fig.
The N dependence of both Ag and Bg modes of InI is studied in Fig.
![]() | Table 2. Screening effect of InI as a function of layer number. ψlr and ψsr are long-range and short-range force constants, respectively. Z* is Born effective charge. . |
In contrast, the N dependence of the lowest Bg shearing mode in Figs.
We further studied the source of quadratic dispersion of ZA mode frequency of InI monolayer. As shown in Fig.
In summary, we have studied the N dependent lattice dynamical properties of indium iodide (InI) and h-BN crystals, by combining ab initio density functional theory calculations with the empirical force-constant model. Unconventional thickness-induced frequency red-shift of in-plane vibrational modes (E2g in h-BN and Bg in InI) has been found from our calculations, and the theoretical results agree quite well with the experimental results available. This behavior can be well comprehended based on the dielectric screening effect, which is enhanced with increasing layer number and therefore can suppress long-range interaction responsible for those in-plane Raman active modes. Meanwhile, unconventional quadratic dispersion of the out-of-plane flexural ZA modes appears due to the sufficient competition between interaction among different nearest neighbors.
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