⇦Chinese Physics Letters, 2018, Vol. 35, No. 1, Article code 013301 Width-Dependent Optical Properties for Zigzag-Edge Silicene Nanoribbons * Hai-Rui Bao(鲍海瑞)1, Wen-Hu Liao(廖文虎)1,2**, Xin-Cheng Zhang(张新成)1, Min Zuo(左敏)1 Affiliations 1College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou 416000 2Key Laboratory of Mineral Cleaner Production and Exploit of Green Functional Materials in Hunan Province, Jishou University, Jishou 416000 Received 25 September 2017 *Supported by the National Natural Science Foundation of China under Grant Nos 11664010 and 11264013, the Hunan Provincial Natural Science Foundation of China under Grant Nos 2017JJ2217 and 12JJ4003, the Scientific Research Fund of Hunan Provincial Education Department of China under Grant No 14B148, the Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, the Innovation Project for Postgraduate of Hunan Province under Grant No CX2015B549, and the Research Program of Jishou University under Grant Nos 15JDY026 and Jdy16021.
**Corresponding author. Email: whliao2007@aliyun.com Citation Text: Bao H R, Liao W H, Zhang X C and Zuo M 2018 Chin. Phys. Lett. 35 013301 Abstract We study theoretically the optical response for perfect zigzag-edge silicene nanoribbons with $N$ silicon atoms of the A and B sublattices ($N$-ZSiNRs) under the irradiation of an external electromagnetic field at low temperatures. The 8- and 16-ZSiNRs are demonstrated to exhibit a broad energy regime of absorption coefficient, refractive index, extinction coefficient, and reflectivity from infrared to ultraviolet, utilizing the dipole-transition theorem for semiconductors. The optical spectra for 8- and 16-ZSiNRs may be classified into two types of the transitions, one between valence and conduction subbands with the same parity, and the other among the edge state and bulk state subbands. With the increase of the ribbon width, the optical spectra for ZSiNRs are proved to exhibit red shift and blue shift at the lower and higher energy regimes, respectively. The obtained novel features are believed to be of significance in designs of silicene-based optoelectronic devices. DOI:10.1088/0256-307X/35/1/013301 PACS:33.20.Ea, 42.25.Bs, 07.57.Hm © 2018 Chinese Physics Society Article Text Many fascinating properties for two-dimensional graphene-like Dirac materials[1-5] have been investigated in the past few years. As one of the Dirac materials, silicene has been successfully synthesized[6-9] on silver substrates. Silicene is more compatible with the silicon-based electronic technology than graphene, and possesses a controllable electronic band structure from its buckled geometry.[10] Great interest has been attracted in design and fabrication of future nanoscale optoelectronic devices based on silicene[11-13] and its nanostructures.[14-18] In contrast to graphene, a strong intrinsic spin–orbit coupling[10,11] plays a crucial role in the low-energy electronic properties of silicene. Several significant phenomena, such as the electrically induced topological phase transition,[19,20] the quantum (anomalous) Hall effect,[21,22] the spin and valley polarized transport,[23,24] and giant tunneling magnetoresistance[25] have been observed in silicene. Similar to graphene nanoribbons, there are two basic edge shapes for silicene nanoribbons (SiNRs), armchair- and zigzag-edge ones. As is well known,[14-17] the armchair-edge silicene nanoribbons exhibit a semiconducting or metallic behavior depending on its ribbon width, while all of the ZSiNRs are metallic due to the gapless edge state at Fermi level. Recently, Kang et al.[26] have investigated the symmetry-dependent transport properties and magnetoresistance in ZSiNRs using the first-principles calculation. Moreover, the electric fields engineered band gap[27] of the ZSiNRs have been manifested. Many exceptional features ranging from the interplay between edge and bulk states,[28] and the spin and valley transports in silicene junctions[29] to thermoelectric effects[30] in ZSiNRs have been reported. On the other hand, the spin-valley optical selection rule and strong circular dichroism[31] in silicene have been proposed. Tabert et al.[32,33] have investigated the magneto-optical behavior to circularly polarized light of the buckled silicene. Based on the ab initio calculation, Arzate et al.[34] studied theoretically the optical absorption spectrum for one-dimensional silicon nanotubes. Furthermore, the optical response in different energy ranges, such as infrared and terahertz frequency regimes of the buckled silicene,[35-37] other than the obtained optical spectra for armchair-edge silicene nanoribbons, have been demonstrated recently.[38] However, to date there have been rare theoretical investigations[39,40] on the interesting optical response from infrared to visible for ZSiNRs, and these significant issues are addressed in the present work. In this Letter, based on the tight-binding approximation and dipole-transition theorem for semiconductors,[11,41-43] we examine the electronic structure (not shown here for simplicity) and optical spectra for 8- and 16-ZSiNRs, under the irradiation of an external electromagnetic field. The observed optical spectra (such as in absorption coefficient, refractive index, extinction coefficient, and reflectivity) in the range of infrared to ultraviolet may be induced from the transitions between the valence and conduction subbands with the same parity, as well as the ones between the edge state and bulk state subbands. With the increase of the ribbon width, the optical spectra of ZSiNRs are demonstrated to red shift at the lower energy regime and blue shift in the higher frequency range. The obtained results may be of importance in understanding the electronic structure for zigzag-edge silicene nanoribbons and shed some light on design of silicene-based optoelectronic devices.[31-38]
cpl-35-1-013301-fig1.png
Fig. 1. (Color online) Sketch of an $N$-ZSiNR consisting of sublattices A and B (as labeled by the red and yellow balls). Each unit cell denoted by the (blue) dashed-line rectangle contains $N$ numbers of A and B sublattices. Two additional hard-walls are imposed on both edges.
As illustrated in Fig. 1 for zigzag-edge silicene nanoribbons, the system consists of two types of sublattices A and B labeled by the red and yellow balls, respectively. The unit cell (see the (blue) dashed-line rectangle) contains $N$ numbers of A and B sublattices. Two additional hard-walls (as denoted by the parallel black dashed-line) are imposed on both edges. The plane-wave basis is along with the $x$-direction according to the translational invariance. Within the framework of the hard-wall boundary condition, the tight-binding energy spectrum[38,44,45] for $N$-ZSiNR is $$\begin{align} E=\,&s\Big\{t^2\Big[1+4\cos^2\Big(\frac{k_xa}{2}\Big) +4\cos\Big(\frac{k_xa}{2}\Big)\cos\kappa\Big]\\ &+\frac{16\lambda_{\rm SO}^2}{27}\sin^2\Big(\frac{k_xa}{2}\Big)\Big[\cos\Big(\frac{k_xa}{2}\Big) -\cos\kappa\Big]^2\Big\}^{1/2},~~ \tag {1} \end{align} $$ where $s=\pm$ represents the conduction (C) and valence (V) bands, respectively, the nearest-neighbor hopping with the transfer energy $t=1.60$ eV, $k_x$ is the longitudinal wave vector, $a=3.86$ Å is the lattice constant, the effective spin–orbit coupling $\lambda_{\rm SO}=3.9$ meV, and the transversal wave vector $\kappa$ (in units of $1/a$) can be numerically[44] determined by $\sin(N\kappa)+2\cos(k_xa/2)\sin[(N+1)\kappa]=0$. The second term under the radical in Eq. (1) can be ignored since the spin–orbit coupling is much smaller than that of the nearest-neighbor hopping energy.[38] Furthermore, when the longitudinal wave vector $k_x$ is located at the interval of $2\arccos[N/2(N+1)] < |k_xa| < \pi$, the band spectrum[45] for edge state can be written as $E_{\rm ES}=\pm t\sinh\tau/[\sinh(N+1)\tau]$, where the wave vector $\tau$ (in units of $1/a$) can be obtained from $\sinh(N\tau)=2\cos(k_xa/2)\sinh[(N+1)\tau]$. One notes that the edge states for ZSiNRs are localized on the zigzag edge[44-47] and decay exponentially into the center. The system wave function can be written as $$ |\psi\rangle=C_{\rm A}|\psi\rangle_{\rm A}+C_{\rm B}|\psi\rangle_{\rm B},~~ \tag {2} $$ where $C_{\rm A}$ and $C_{\rm B}$ are the normalized coefficients, and $|C_{\rm A}|^2+|C_{\rm B}|^2=1$. The wave functions for A and B sublattices are $|\psi\rangle_{\rm A/B}=N^{-1}_{\rm A/B}\sum_{j,x_{A_j/B_j}}e^{ik_xx_{A_j/B_j}} \phi_{A_j/B_j}|A_j/B_j\rangle$, where $N^{-1}_{\rm A/B}$ is the normalized constants, $x_{A_j/B_j}$ is the $x$-component of position vector at site $j$, $\phi_{A_j/B_j}$ is the wave function in the confined $y$-direction, and $|A_j/B_j\rangle$ is the $p_z$ orbit wave functions of the silicon atom, respectively, for A and B sublattices at site $j$. The bulk state wave functions[44] along the confined direction for A and B sublattices can be obtained as $(-s\sin[(N+1-j)\kappa],\sin(j\kappa))^{\rm T}$, while the edge state ones will be $e^{i\pi j}(s\sinh[(N+1-j)\tau],\sinh(j\tau))^{\rm T}$ if $k_x$ is located at the interval of $2\arccos[N/2(N+1)] < |k_xa| < \pi$. Furthermore, the imaginary part of the dielectric function[37,38] for $N$-ZSiNRs irradiated by an external electromagnetic field reads $$\begin{align} {\rm Im}\,\varepsilon({\it \Omega})=\,&\frac{8\pi^2 e^2}{3m_{\rm e}^2N_xa{\it \Omega}^2}\sum_{k_x,c,v}[f(E_{k_x,c})-f(E_{k_x,v})]\\ &\times |M(k_x)|^2\delta(E_{k_x,c}-E_{k_x,v}-\hbar{\it \Omega}),~~ \tag {3} \end{align} $$ where $e$ and $m_{\rm e}$ denote the charge and mass of electron, respectively, $N_x$ is the number of unit cells along the longitudinal direction, $f(E_{k_x,c/v})$ is the Fermi–Dirac distribution function, $\hbar{\it \Omega}$ is the photon energy, and the dipole-transition matrix element $M_{\rm cv}(k_x)$ among the bulk state subbands reads $$\begin{alignat}{1} &M_{\rm cv}(k_x)\\ \equiv\,&\langle\psi_{c,k_x,\kappa}|\hat{p}_x|\psi_{v,k_x,\kappa'}\rangle =\frac{2m_{\rm e}at}{(2N+1)\hbar}\sin\Big(\frac{k_xa}{2}\Big)\\ &\times\Big[\frac{\sin[N(\kappa-\kappa')/2]\cos[(N+1)(\kappa+\kappa')/2]} {\sin[(\kappa-\kappa')/2]}\\ &-\frac{\sin[N(\kappa+\kappa')/2]\cos[(N+1)(\kappa-\kappa')/2]} {\sin[(\kappa+\kappa')/2]}\Big],~~ \tag {4} \end{alignat} $$ while that between the edge state and bulk state subbands should be $$\begin{alignat}{1} &M_{\rm cES}(k_x)\\ \equiv \,&\langle \psi_{c,k_x,\kappa}|\hat{p}_x|\psi_{\rm ES,k_x,\tau}\rangle=i\frac{2m_eat} {(2N+1)\hbar}\sin\Big(\frac{k_xa}{2}\Big)\\ &\times\Big[\frac{\sin[N(\kappa-i\tau)/2]\cos[(N+1)(\kappa+i\tau)/2]} {\sin[(\kappa-i\tau)/2]}\\ &+\frac{\sin[N(\kappa+i\tau)/2]\cos[(N+1)(\kappa-i\tau)/2]} {\sin[(\kappa+i\tau)/2]}\Big].~~ \tag {5} \end{alignat} $$ Furthermore, the real part of the dielectric function can be calculated from the Kramers–Krönig relation as $$ {\rm Re}\,\varepsilon({\it \Omega})=1+\frac{2}{\pi}\wp{\int}^\infty_0d{\it \Omega}' \frac{{\it \Omega}'{\rm Im}{\varepsilon}({\it \Omega}')}{{\it \Omega}'^2-{\it \Omega}^2},~~ \tag {6} $$ where the integral principal value $\wp$ should be $$ \wp{\int}^\infty_0=\lim_{\eta\rightarrow0^+}{\int}^{{\it \Omega}-\eta}_0 +\lim_{\eta\rightarrow0^+}{\int}^\infty_{{\it \Omega}+\eta}.~~ \tag {7} $$ In addition, the absorption coefficient can be obtained from the dielectric function as $$\begin{alignat}{1} \!\!\!\!\!\!\!\!I({\it \Omega})=\sqrt{2}{\it \Omega}[\sqrt{{\rm Re}\,\varepsilon({\it \Omega})^2+{\rm Im}\,\varepsilon({\it \Omega})^2}-{\rm Re}\,\varepsilon({\it \Omega})]^{1/2}.~~ \tag {8} \end{alignat} $$ One further obtains the refractive index and extinction coefficient as $n({\it \Omega})=[\sqrt{{\rm Re}\,\varepsilon({\it \Omega})^2+{\rm Im}\,\varepsilon({\it \Omega})^2}/2 +{\rm Re}\,\varepsilon({\it \Omega})/2]^{1/2}$ and $k({\it \Omega})=[\sqrt{{\rm Re}\,\varepsilon({\it \Omega})^2+{\rm Im}\,\varepsilon({\it \Omega})^2}/2 -{\rm Re}\,\varepsilon({\it \Omega})/2]^{1/2}$, respectively. Finally, the system reflectivity is given by $$ R({\it \Omega})=\frac{[n({\it \Omega})-1]^2+k({\it \Omega})^2}{[n({\it \Omega})+1]^2+k({\it \Omega})^2}.~~ \tag {9} $$
cpl-35-1-013301-fig2.png
Fig. 2. (Color online) Optical absorption spectrum $I({\it \Omega})$ for (a) 8- and (b) 16-ZSiNRs as a function of the photon energy $\hbar{\it \Omega}$ (in units of eV), as denoted by the (black) solid and (red) dashed lines, respectively.
The calculated optical absorption spectra $I({\it \Omega})$ (in arbitrary units) for 8- and 16-ZSiNRs are shown in Fig. 2. In the presence of an external electromagnetic field, a peak in $I({\it \Omega})$ indicates an absorption photon with energy $\hbar{\it \Omega}$ allowed by the optical transition from valence to conduction subbands. As is shown, the absorption spectra for 8- and 16-ZSiNRs are presented in the energy range from 0 to 3.20 eV. The 8-ZSiNR shows the edge state induced absorption peaks at 0.60, 1.12, and 1.46 eV [see Fig. 2(a)], respectively, from $V_{\rm ES/5}\rightarrow C_{\rm 5/ES}$, $V_{\rm ES/6}\rightarrow C_{\rm 6/ES}$, and $V_{\rm ES/7}\rightarrow C_{\rm 7/ES}$. In consideration of the momentum conservation,[41,46] the bulk-state-induced optical transitions may be only induced from the same indexed subbands. The observed sharp peaks (located at 1.20, 2.24, and 2.92 eV) in $I({\it \Omega})$, should be attributed to $V_7\rightarrow C_7$, $V_6\rightarrow C_6$, and $V_5\rightarrow C_5$, respectively. Furthermore, relatively richer peaks for 16-ZSiNR are shown by the (red) dashed line in Fig. 2(b) because of the many more optical transitions in this case. It is found that the absorption peaks at 0.30, 0.87, 1.12, 1.32, 1.47 and 1.56 eV for 16-ZSiNR can be identified to the transitions $V_{{\rm ES}/l}\rightarrow C_{l/{\rm ES}}$ with $l=9$–13 and 15. The edge state and/or bulk state induced peak near 0.60 eV may be from the transitions $V_{\rm ES/14}\rightarrow C_{\rm 14/ES}$ and/or $V_{15}\rightarrow C_{15}$. Additionally, the remaining peaks are induced from the transitions between the occupied (valence) and unoccupied (conduction) subbands with the same parity. It should be pointed out that the optical absorption spectra for ZSiNRs are manifested to be red-shifted at lower energy regime and a weaker blue-shifted in the higher energy range, with the increase of the ribbon width.
cpl-35-1-013301-fig3.png
Fig. 3. (Color online) Refractive index $n({\it \Omega})$ for (a) 8- and (b) 16-ZSiNRs as a function of the photon energy $\hbar{\it \Omega}$ (in units of eV), with the other parameters being identical to those in Fig. 2.
Figure 3 demonstrates the refractive index $n({\it \Omega})$ as a function of the photon energy $\hbar{\it \Omega}$ (in units of eV). As shown by the (black) solid line in Fig. 3(a), the refractive index at the vicinity of 0.60, 1.12, and 1.46 eV can be attributed to the transitions from/to the edge states for 8-ZSiNR. One notes that the remaining refractions are induced from the resonances between bulk state valence and conduction subbands with the same parity. The normal and anomalous dispersion of the refractive index for 8-ZSiNR appear in the vicinity of the transition energies (i.e., peak positions), as can be explained by the Kramers–Krönig relation with respect to the extinction coefficient. Obviously, the resonance structure of the refractive index exhibits a red shift at lower energy regime for the 16-ZSiNR case [see Fig. 3(b)]. One notices that the optical response is similar to the 8-ZSiNR case, since the optical selection rules for ZSiNRs under the momentum conservation do not alter with the ribbon width. As is expected, the presented refractive index for 8- and 16-ZSiNRs are very different from the optical spectra for armchair silicene nanoribbons.[38] On the other hand, the variation of the system extinction coefficient for 8- and 16-ZSiNRs with the photon energy $\hbar{\it \Omega}$ (in units of eV) are shown in Fig. 4. As demonstrated by the (black) solid and (red) dashed lines in Figs. 4(a) and 4(b), all the bulk/edge induced peaks for 8 and 16-ZSiNRs correspond to the same energy positions as the absorption spectra [see Figs. 2(a) and 2(b)] since $I({\it \Omega})\propto{\it \Omega} k({\it \Omega})$. However, the extinction coefficient for 16-ZSiNR is somewhat different from the case of 8-ZSiNR, especially the numbers and energies of the resonance structures, similar to the situation of graphene nanoribbons.[41,46] It seems that the red shift at lower frequency regime of the extinction coefficient is more sensitive than the blue shift in higher energy region with the increase of the ribbon width. In addition, one notes that the edge states (make the zigzag-edge silicene nanoribbons metallic) are of particular importance in the optical response at lower frequency regime. It is indicated that[37,38,41,46] the higher the photon energy, the weaker the extinction coefficient to the electromagnetic field.
cpl-35-1-013301-fig4.png
Fig. 4. (Color online) Extinction coefficient $k({\it \Omega})$ (in arbitrary units) for (a) 8- and (b) 16-ZSiNRs as a function of the photon energy $\hbar{\it \Omega}$ (in units of eV), with the other parameters being identical to those in Fig. 2.
cpl-35-1-013301-fig5.png
Fig. 5. (Color online) Reflectivity $R({\it \Omega})$ for (a) 8- and (b) 16-ZSiNRs as a function of the photon energy $\hbar{\it \Omega}$ (in units of eV), with the other parameters being identical to those in Fig. 2.
Finally, the system reflectivity $R({\it \Omega})$ as a function of photon energy $\hbar{\it \Omega}$ (in units of eV) are illustrated in Fig. 5. As shown by the (black) solid line in Fig. 5(a) for the 8-ZSiNR case, these reflectivity spectra in the energy range from 0 to 3.2 eV may be assigned to the transitions between the same indexed bulk state valence and conduction subbands other than the ones from/to edge state subbands. A set of reflectivity peaks for 16-ZSiNR at the same energy positions of above absorption spectra, refractive index, and extinction coefficient are induced by the transitions $V_{{\rm ES}/l}\rightarrow C_{l/{\rm ES}}$ with $l=9$–13 and 15, as indicated by the (red) dashed line in Fig. 5(b). It is worthwhile to point out that the reflectivity for 16-ZSiNR is originated from the mentioned resonances (i.e., the transitions from bulk/edge to and/or bulk/edge states subbands). As is expected, with the increase of the ribbon width, the system reflectivity has been observed to red shift at lower energy range and blue shift in the higher energy region, as can be understood from the electronic structure in Fig. 2. Note that the reflectivity for 8- and 16-ZSiNRs shows a higher amplitude near the resonance energies range from infrared to visible since it is combined with the refractive index and extinction coefficient. One also notices that the reflectivity structure for 16-ZSiNR is more closed in the lower frequency regime than that for the case of 8-ZSiNR. In summary, within the framework of the tight-binding approximation and dipole-transition theorem for semiconductors, we have investigated the absorption coefficient, refractive index, extinction coefficient, and reflectivity of the intrinsic 8- and 16-zigzag-edge silicene nanoribbons, under the irradiation of an external electromagnetic field. It has been demonstrated that the 8- and 16-ZSiNRs show broad optical spectra at the infrared and ultraviolet frequency regimes. The optical spectra for 8- and 16-ZSiNRs are observed to the transitions between valence and conduction subbands with the same parity, as well as the ones between edge state and bulk state subbands. However, with the increase of the ribbon width, the optical spectra for zigzag-edge silicene nanoribbons are red-shifted at the lower energy regime, while present a weaker blue shift in the higher energy range. The obtained results may be of importance in comprehension of the electronic structure for silicene nanoribbons and shed some light on the design of the nanoscale optoelectronic devices based on silicene. 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