Study on optical absorption of silicene nanodisks under strain modulation

Using the tight-binding Hamiltonian approach, we theoretically study the position of energy level structures and optical absorption peaks by applying stress forces laterally or longitudinally of triangular zigzag silicene nanodisks (TZSN). We consider the effect of strain on the structure. We found that by applying a lateral force similar to that applied to the stretched structure (the force applied to the compressed structure), all absorption peaks can be observed to move to the left or to the right as a whole in the optical absorption spectrum, and this shift is more pronounced at the main absorption peaks, indicating the possibility of TZSN as a strain sensor, and we also considered which size of TZSN would be more suitable as a sensor. On the other hand, it is found that the compressive strain can effectively reduce the light absorption intensity when the incident energy is less than 1.4 eV (0.89 µm). The work described in the system can be used as new material is applied to the sensor and imaging equipment, etc.


Introduction
Silicene is a two-dimensional, single-atom silicon film with a honeycomb structure similar to graphene [1].Similar to graphene, it has extremely rich physical properties [2], such as K and K' valley low energy dynamics.It can also be described using Dirac theory Silicene has massless fermions at a point in the Brillouin region.When spin-orbit interactions are considered, a gap as small as 1.5 meV is created.It is worth mentioning that the structure of silicene has small out-of-plane buckling due to the silicene itself.Thus, the two sublattices of silicene (atoms A and B) lie on two parallel planes with separation distances of about 0.46 Å [3].
Quantum nanodisks, sometimes referred to as artificial atoms, are low-dimensional finite semiconductor nanostructures that can bind excitons in all directions in space.The dots are several nanometres across and several layers thick vertically.Thus, in such small quantum nanodiscs, edge effects could easily occur.So far, quantum nanodisks have been studied extensively in low-dimensional systems, such as graphene nanobands [4][5], phosphorene [6], and transition metal dichalcogenide monolayer [7].
Previously, Ezawa et al. analyzed the electrical properties of energy levels near the Fermi level in a triangular zigzag silicene quantum nanodisc.This study also discusses the topological phase transition characteristics of the system when an electric field is applied [8].Qi et al. studied that the strain of triangular graphene quantum dots can significantly regulate the energy level splitting and optical absorption of the dots.When tensile and compressive strains are applied, a linear shift of the optical absorption peak to the left or right can be observed in the spectrum, respectively [9].Meanwhile, recently, the results of Peng et al. show that gaps appear in edge states with the increase in the quantum size.Both the upper and lower levels and the Fermi level are separated by spin-orbital coupling [10].However, to the best of our knowledge, the electrical and optical absorption properties of triangular zigzag silicene quantum nanodisks under uniaxial strain have not been explored.Therefore, the optical absorption characteristics of triangular zigzag silicene quantum nanodiscs under uniaxial strain are investigated.

Theoretical formalism
The object of our study is a triangular silicene quantum dot, which is depicted in Figure 1.It is well known that the energy spectra of triangular quantum dots are significantly rearranged relative to those of hexagons.The triangular disk has more size-controlled zero energy levels than the hexagonal disk, meaning that the larger the size is, the more zero energy levels are.And the size is related to the total number of atoms in the nanodisk.These zero energy levels will allow the electrons to move around and bring in an extra degenerate shell.Previous studies have also shown that degenerate shells can control the optical properties of armchair edges in silicene quantum dots with zigzag edges.Its edge atom number is represented by edge N and the total silicon atom number is described as In order to study the silicon nanodiscs under uniaxial strain, we adopt a four-strip model, which can be applied to the basic properties of graphene-like materials with low bending.The Hamiltonian of the system can be written as where j i, ( ) where e , e m , c and 0 ε represent electron charge, electron mass, speed of light in vacuum, and dielectric constant in vacuum, respectively.Subscript i denotes all occupied and subscript j denotes all unoccupied molecular orbitals.α p is the component of the momentum operator that is different from the position operator in the α direction and ( )

Ψ
is the strength of the oscillator with energy f i, ε .Under normal circumstances, the Fermi energy F ε in TZSN is zero, but it can be by changing the size of the external grid voltage to adjust.When uniaxial strain is applied to TZSN, its energy level structure decreases in degeneracy and splits like spectral lines, so the energy required to move from one energy level to another changes and so does the light absorption spectrum.We will give some results in the analysis below.

Results and discussion
In general, in graphene and graphene-like structures, there are two types of edge structures, a zigzag structure and an armchair structure.And the armchair structure is much more stable than the zigzag structure.So we are looking at zigzag triangular silicene nanodisks.We found that the uniaxial strain of x axis applied in the system should be changed more effectively than the uniaxial strain of y axis to regulate the range of energy spectrum, so we mainly focus on the uniaxial strain of x axis.We should note that due to the characteristics of the silicene structure, the stress should not be too large, because excessive profits will cause the structure to collapse, thus losing the original characteristics of the structure, and the significance of our study will be lost.In this case, we only change the strain from -0.09 to 0.09 regardless of the transverse stress or longitudinal stress., while η represents the uniaxial strain intensity.The uniaxial strain is applied in the x direction.In contrast, the uniaxial strains in (b) and (d) are applied in the y direction.The results show that when we apply strain from zero to 0.06 along the transverse, both the main absorption peak of the spectrum measured along the transverse and the main absorption peak of the spectrum measured along the longitudinal will shift to the left, which is often referred to as redshift.And when the compression strain changes from 0 to 0.09, the main absorption peaks y σ and y σ move to the right, which is often said as the blue shift.In addition, the number of peaks changes during stretching or compression.
It is also to be considered that when the stress is applied uniformly along the vertical direction, the change of this small and uniform force will bring about not only small changes in the structure but also changes in the transverse and longitudinal light absorption peaks.However, compared with the stress applied along the y direction, the absorption peaks in the y direction only shift slightly from left to right.So we will only consider the case of adding strain in the zigzag direction in the following discussion, as shown in Figures 2(b) and 2(d).The red shift (blue shift) occurs because as the tension (compression) applied to TZSN increases, the energy levels will split regardless of which direction this stress is in.Therefore, the energy required for the transition between the beginning energy level of the transition and the end energy level of the transition increases or decreases. .Therefore, by analyzing the above results, we can propose a strain sensor based on uniaxial strain.By detecting the position of light absorption peaks in the system, we can determine the magnitude and type of stress (tension or compression).In addition, in order to show the system's sensitivity to stress size, we plot the frequency location of the first significant peak relative to the strain when applied along the direction of the armchair.As can be seen from Figure 3(c), when the strain is -0.09-0.06, the peak displacement of changes linearly with the strain.Thus, one can easily measure the strain based on the peak displacement detected.For N=6, the curve is less linear because the redshift and blueshift change less linearly as the strain changes.Combined with Figures 3(a), 3(b) and 3(c), it can also be seen that when the tensile strain is higher than 5%, the red shift and blue shift of quantum dots of all sizes become blurred.This is because the increase of tensile strain leads to the increase of peak splitting and absorption peaks.Therefore, the sensitivity of the sensor will be affected under the condition of high strain.Meanwhile, the peak size x Through the analysis of Figure 3, it can be seen that with the increase of strain, two significant effects occur.First, the absorption peak becomes fuzzy when the tensile strain reaches 5%.This effect is due to the fact that ZBs in silicene are no longer uniformly located near the Fermi level.The transition energy from FB to ZB is different from that from ZB to UB.Second, with the change of the above absorption peaks, when edge N = 8 or 10, the absorption coefficient of light absorption decreases by about 60% when the tensile strain increases to 9%.This will have an impact on current optical devices.

Conclusions
To conclude, we investigate tight binding calculations of TZSN for optical properties.We consider uniaxial strains which can reduce the degeneracy of energy levels.The degeneracy of the energy level decreases when a small force is applied in the direction of a zigzag or armchair and causes it to stretch or compress evenly.Therefore, this type of strain can significantly regulate the energy level structure and light absorption of silicene nanodiscs.When the tensile strain and compressive strain increase, the red shift and blue shift of the optical absorption peak can be observed in the spectra respectively.Therefore, a silicon nanodisk is proposed as a sensor to detect the magnitude and direction of strain, which can determine the direction of the magnitude of the applied stress by locating the light absorption peak at the position shown by the spectrum.On the other hand, the optical absorption coefficient also changes significantly through the application of strain, and the highest circumstance can double the optical absorption coefficient.This result can guide the research and development of some optical devices.

Figure 1 .
Figure 1.Structure of a TZSN.s i δ (i= 1, 2, 3) indicates the lattice vectors.The black and white circles distinguish between the two types of atoms in the upper and lower layers of the phenyl-like ring lattice.In this figure, 6 = edge N neighbor transition and the second nearest neighbor transition, respectively, ( ) † is is c c is the creation or annihilation of electrons at lattice i, and the spin at this lattice point is s.σ represents the Pauli matrix.The t in the first term denotes the overlap integral also called the nearest neighbor hopping strength.In the second term, λ so is the coefficient term of intrinsic spin orbit coupling.1 = ij ν indicates that if the next next-nearest neighbor transition is clockwise, otherwise ij ν has a value of -1.The third term represents the Rashba spin interaction with the coupling constant R λ .And finally, the lattice constant of silicene is taken as the general value, a 0 = 3.86 Å.

Figure 2 .
Figure 2. (a) and (c) plot the optical absorption spectra in the x and y directions respectively when 6 = edge N, while η represents the uniaxial strain intensity.The uniaxial strain is applied in the x direction.In contrast, the uniaxial strains in (b) and (d) are applied in the y

Figure 2
Figure 2 plots light absorption spectra of TZSN with

Figure 3 .
Figure 3. Optical absorption spectra of TZSN for