Magneto-electronic and optical properties of zigzag silicene nanoribbons

Physica E 87 (2017) 178–185

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Physica E journal homepage: www.elsevier.com/locate/physe

Magneto-electronic and optical properties of zigzag silicene nanoribbons

MARK

Feng-Lin Shyu Department of Physics, R.O.C. Military Academy, Kaohsiung 830, Taiwan, ROC

A R T I C L E I N F O

A BS T RAC T

Keywords: Silicene nanoribbon Tight-binding model Magnetic field Electric field Electronic property Optical property

The tight-binding model including the spin-orbit coupling (SOC) is used to study electronic and optical properties of zigzag silicene nanoribbons (ZSiNRs) in magnetic and electric fields. The SOC affects the lowenergy bands and induces new selection rules leading to richer optical spectra. Except an increase in bandgaps, perpendicular magnetic field further exhibits spin-polarized Landau levels, in which electron's probability density of band-edge states distributes like a standing-wave. Landau levels could enhance the DOS and increases absorption frequency and strength. Perpendicular electric field (Fz) increases bandgap and thus absorption frequency, but it does not change band symmetry, edge-states, and selection rules. Moreover, Fz enhances the split of spin-polarized states inducing more absorption peaks. Parallel electric field (Fx) leads to an overlap between conduction and valence bands and destroys band symmetry and Landau levels. Consequently, Fx exhibits new selection rules and enriches absorption spectra.

1. Introduction Layered-graphene systems including monolayer graphene, fewlayer graphite films, and 3D graphites, by regulating their stacking conditions, show the intriguing physical properties, such as quantum Hall effect (QHE) [1] and field-modulated electronic properties [2–4]. Moreover, monolayer graphene has been experimentally proven as potential candidates for photodetectors [5,6]. Although monolayer graphene displays extremely high electrical conductivity and electron mobility, the more applications are limited due to its zero bandgap. Except various stacking conditions, another way to overcome this problem is to induce lateral quantum confinement, i.e., by shaping the 2D graphene into 1D graphene nanoribbons (GNRs) in the sub-nanometer regime. The reduction of dimensionality would cause strong dependences of electronic properties and quantum spin Hall effect on ribbon's width and boundary geometry [7]. What is more, energy dispersions of GNRs could be modulated by electric field leading to a rich change in optical, magnetic, and electrical properties [8–11]. Apart from electric field, band structures of GNRs in the presence of the perpendicular magnetic field exhibit the oscillating Landau subbands and Landau levels (LLs) [12] that further affects optical and transport properties [13,14]. Following the discovery of graphene, 2D silicene, purely composed of Si atoms, with nano-scale thickness and hexagonal symmetry also has been synthesized on silver substrate [15]. However, the larger ionic size of silicon atoms causes a sp3 hybridization. The mixture of atomic orbitals makes the 2D lattice of silicene a low-buckled structure such

that sites on the A and B sublattices sit in different vertical planes with a separation about 0.47 Å. The buckled structure breaks inversion symmetry about the silicene plane resulting in a significant spin-orbit coupling (SOC) [16] which could bring about the quantum spin Hall effect (QSHE) with a nontrivial topological property [17–19]. The SOC in silicene opens up a spin-orbit bandgap (≈7.9 mev) and changes linear bands around Dirac point into parabolic bands giving a mass to Dirac electrons. Additionally, applied perpendicular electric field could control the effective mass further inducing a phase transition from a quantum spin Hall state to band insulator [20]. After the synthesization of silicene, 1D silicene nanoribbons (SiNRs), considered as cutting a monolayer silicene, have also been discovered in experiments [21–23]. In contrast to GNRs, the interplay between the quantum confinement and the SOC in SiNRs reveals richer physical properties including quantum spin Hall effect [24], thermoelectric effect [25], magnetic properties [26], spin-polarized transport [27,28]. Moreover, thermoelectric measurements in SiNRs demonstrated that thermoelectric power factor was enhanced by the strong quantum confinement [29]. Electronic structures of SiNRs, except the strong geometry dependence, also could be tuned by applying electric field [27,28,30], deformation [31], and doping [32]. Since both SiNRs and GNRs have similar plane geometry, for SiNRs, magnetic field also provides an effective confinement making electron to undergo a cyclotron motion. At sufficiently large B-field strength, the cyclotron radius of electron motion becomes smaller than SiNR's width, promoting the formation of Landau levels. Different from GNRs, the interplay between the SOC and magnetic field effect in

E-mail address: fl[email protected]. http://dx.doi.org/10.1016/j.physe.2016.12.005 Received 19 October 2016; Received in revised form 23 November 2016; Accepted 13 December 2016 Available online 18 December 2016 1386-9477/ © 2016 Elsevier B.V. All rights reserved.

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Ui = Fz l + Fx xi (Ui = −Fz l + Fx xi ) is for sublattice A (B). Electric fields Fz and Fx (unit eV/Å) are, respectively, perpendicular to the ribbon plane and along the ribbon width; xi represents the x coordinate of the ith Si atom. In the presence of electric fields, the Bloch function including the spin polarization could be written as

SiNRs could break the spin-degeneracy of LLs to exhibit spin-polarized LLs. LLs with partial flat bands could enhance the density of states and absorption strength that will be beneficial to the development in nanodevices based on spintronics. On the other hand, for GNRs, in the presence of magnetic field, the sensitivity of bandgap modulations due to extra parallel electric field is further increased [33]. Therefore, under the existence of LLs in SiNRs, effects of applied perpendicular (Fz) and parallel (Fx) electric fields on LLs deserve to be further studied. The enhancement of spin-polarized states and the collapse of LLs by applied electric fields will significantly affect electronic and optical properties including optical spectra and selection rules. In this work, we study mainly magneto-electronic and optical properties of zigzag silicene nanoribbons within the pz orbital tightbinding model including the spin-orbit coupling. Applied perpendicular magnetic field could affects electronic properties of ZSiNRs such as energy dispersions, bandgaps, and the DOS. For sufficiently strong magnetic field, Landau levels are produced and their partial flat bands could be significantly reflected by the DOS and optical absorption. Then, applied perpendicular and parallel electric fields are further used to observe a change in Landau levels and optical absorption. Our study shows that the spin-orbit coupling changes energy dispersions of partial flat bands near the Fermi energy and further induces new selection rules resulting in richer optical spectra. Landau levels could be exhibited with increasing B-field which make significant changes in the DOS and optical spectra of ZSiNRs. The field strength constructing Landau levels is effectively reduced by increasing nanoribbon's width. Perpendicular electric field does not change the band symmetry, edgestates, and optical selection rules; however, it enhances a split of spinpolarized states in LLs and induces more absorption peaks. On the other hand, parallel electric field could cause an overlap between conduction and valence bands making ZSiNR a metal; additionally, it destroys the band symmetry and Landau levels. The drastically fieldmodulated energy dispersions and induced new selection rules by parallel electric field would enrich optical spectra.

Nx

Ψ c, v (F , k y , J ) =

〈ij〉α

iα

−i

2λR 3

∑ 〈〈ij〉〉αβ

(Ciα+Cjα ) + i

λSO 3 3

z μij Ciα+(→ σ × dlij )αβ Cjβ +

∑

|An↑,↓ 〉 =

1 N

↑,↓ ↑,↓ ↑,↓ ∑ exp{i [k·R↑,↓ n + Gn ]}|ϕA (r − R n )〉.

(3)

Rn

The Peierls phase is expressed as Gn↑,↓ =

2π ϕ0

r

∫R↑,↓ A·d l , where ϕ0 = n

hc e

is

the flux quantum. The similar calculation is also applied for Since the chosen vector potential simply depends on the x coordinate, it does not break the periodic condition along yl direction. Thus the wave vector ky is still a good quantum number and the Hamiltonian matrix remains a 4Nx × 4Nx Hermitian matrix. By diagonalizing the Hamiltonian matrix, we could obtain energy dispersion relation (E c, v (B, F , k y, J )) and wave function (Ψ c, v (B, F , k y, J )). The superscript c(v) corresponds to the unoccupied π * band (the occupied π band). J = 1, 2, …, 4Nx serves as the subband index of conduction (valence) bands according to the minimum energy spacing between the subband and the Fermi energy EF. J does not represent a good quantum number which is just used to identify the optical excitation channels. For narrower SiNRs (Nx < 40), the first-principle calculation has shown that the bond-length of SiNRs has a nearly uniform distribution except a sudden decrease at the ribbon's edges [34,35]. According to the calculations in Nx-dependent bandgaps, the nearestneighbor transfer integral at edges will be modified with the value γ0edge = −1.15 ev . Since the wider zigzag SiNR (Nx > 40) is considered in this study, such an edge-effect could be ignored. In the absence of external field, the band structure of ZSiNR without the SOC is similar to that of zigzag graphene nanoribbon (ZGNR). The band structure with spin degeneracy is symmetric about the zone center k y = 1 and contains parabolic bands with band-edge states at k y = 2/3 (or k y = 4/3). Specially, the J c = 1 and J v = 1 subbands are dispersionless and degenerate at the Fermi energy EF = 0 which are demonstrated by the black solid line in the inset of Fig. 1(a). Those two partial flat bands are composed of localized edge states between 2/3 ≤ k y ≤ 1 (or 1 ≤ k y ≤ 4/3). While the SOC is included, it affects energy dispersion for subbands close to the Fermi energy instead of subbands with higher energies. For example, the J c = 1 and J v = 1 subbands no longer merge together and are changed from partial flat bands into two linear bands intersecting at k y = 1 state, as shown in the red solid line of the inset of Fig. 1(a). That could be reflected in the lowfrequency optical properties, as discussed in following section. While the perpendicular magnetic field with the magnitude B=20 T is applied, Fig. 1(b) shows the band symmetry about the zone center and the Fermi energy keeps unchanged. For parabolic subbands with J c (v ) > 1, their band-edge states are still at k y = 2/3 (or k y = 4/3). However, the spin degeneracy is broken due to magnetic field such that J c (v ) subband at B=0 will be split into a spin-up (Juc (v ) ) and spindown (Jdc (v ) ) states. The splitting energy between spin-up and spindown states is very tiny with the value about 0.5 mev and gradually decreases with increasing subband index J c (v ). For other heavier groupIV elements such as Ge and Sn, the splitting energy will be more easily observed due to the stronger SOC. It is noticed that the linear bands with J c (v ) = 1, owing to applied magnetic field, are changed into two parabolic bands and open a tiny bandgap about Eg = 0.22 mev .

|Bn↑,↓ 〉.

z νij Ciα+σαβ Cjβ

〈〈ij〉〉αβ

∑ Ui Ciα+Ciα, iα

periodic

When SiNRs are subjected to a uniform perpendicular magnetic field B = Bzl , an extra Peierls phase characterized by the vector potential A = Bxyl is introduced into the tight-binding function. The tight-binding function |An↑,↓ 〉 in Eq. (2) is thus changed into

The zigzag silicene nanoribbon (ZSiNR) with Nx zigzag dimer lines along the x-axis contains two sublattices composed of Si atoms at A and B sites. The number of Si atom in a primitive unit cell is 2Nx. The lattice constant is a=3.897 Å and two sublattices are separated with the distances 2l=0.466 Å due to the buckled structure. The lattice along the y-axis has the periodic distance Ic = 3b (b=2.25 Å) such that the first Brillouin zone could be defined by k y Ic ≤ 2π . For convenience, the unit of wave vector ky is chosen as π / Ic , i.e., 0 ≤ k y ≤ 2 . The π band could be calculated in terms of the pz-orbital nearest-neighbor tight-binding Hamiltonian, which could be written as

∑ ϵi Ciα+Ciα + γ0 ∑

(2)

n =1

where |An↑,↓ 〉 (|Bn↑,↓ 〉〉 is the tight-binding function related to atom An↑,↓ (Bn↑,↓) with spin-up ( ↑ ) or spin-down ( ↓ ) state.

2. Electronic properties of zigzag silicene nanoribbon under external fields

H=

∑ an↑ |An↑ 〉 + an↓ |An↓ 〉 + bn↑ |Bn↑ 〉 + bn↓ |Bn↓ 〉,

(1)

Ciα+

is a (Ciα ) creation (annihilation) operator of an electron with where spin-polarization α at the i site. ϵi is the on-site energy and γ0 = −1.03 ev is the nearest-neighbor transfer integral [18]. The third term λSO = 3.9 mev is the spin-orbit coupling (SOC) and the intrinsic Rashba SOC (RSOC) is λR = 0.7 mev in the fourth term [2]. The summation indices 〈ij〉 and 〈〈ij〉〉 include the nearest and next-nearest neighbor hopping sites, respectively. → σ = (σx , σy, σz ) is the vector of Pauli matrix and νij =+ 1(−1) indicates the anticlockwise (clockwise) next nearest-neighbor interactions about the +zl direction. The unit vector dlij connects different sites i and j with the same sublattice; μij =+ 1 or μij = −1 is involved in the sublattice A or B. Regarding the electrostatic potential energy, in the last term 179

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Fig. 2. (a)–(d) are the distribution of electron's probability density for J c = 1 and J c = 2 subbands at ky = 2/3 state and different spins for B=0. (e)–(h) are similar to (a)–(d) but for B=60 T.

subband (Fig. 2(c) and (d)), the distributions of PDWF oscillate like sinusoidal functions. As for higher energy subbands (J c > 2 ), variations of the PDWF with atomic positions are similar to those of J c = 2 but having higher oscillating frequencies. The above suggested features of probability density distribution are mainly conducted by quantum confinement due to finite nanoribbon's width. While magnetic field is applied, the interference induced by magnetic phase will compete with the effect of quantum confinement. With increasing field strength, the localized probability density are gradually formed. At B=60 T, in Fig. 2(e) and (f), the spin-degenerate state J c = 1 is split into two spin-polarized states with a tiny splitting energy ΔE ∼ 0.5 mev . For the spin-up state J c = 1u , the probability density is only dominated by B-site electrons and localized near the nanoribbon's boundary. As for the spin-down state J c = 1d , A-site electrons play the main role for the PDWF, which is localized at the nanoribbon's center. However, the PDWF of B-site electrons does not completely vanish and shows a tail (blue solid line) near nanoribbon's boundary, which seems to be an incomplete standing wave. With increasing field strength, it would become an exact standing wave. For higher energy J c = 2 subband, magnetic field exhibits the first Landau level (LL1), as shown in Fig. 1(d). In fact, LL1 has two spin-polarized states denoted by J c = 2d and J c = 2u with a tiny separated energy (≈0.02 mev ). Except the partial dispersionless of subbands, the PDWAs of LLs, in Fig. 2(g) and (h), reveal an exact standing wave and strong localization. Such a profound effect will be expected to exhibit special signatures in the density of states and optical properties. For wider ZSiNR, e.g., Nx = 100 , the needed magnetic field to create LL1 is about B=20 T; moreover, LLs, with increasing magnetic field, would be extended to higher energy subbands. From the point of view of experimental measurements, it is much easier, for wide ZSiNRs, to verify the special properties of Landau levels. Since magnetic-fieldmodulated electronic (or optical) property is similar for Nx = 60 and Nx = 100 , we choose the smaller Nx as a model study for saving computing time.

Fig. 1. The low-energy band structures of 60-ZSiNR at (a) B=0, (b) B=20 T, (c) B=40 T, and B=60 T. The inset in (a) is low-energy band structure with and without the SOC. The inset in (c) is low-energy band structure around the Fermi energy at different Bs. The inset in (d) is the B-dependent bandgap.

Meanwhile, the two split band-edge states are away from k y = 1. Bandgap and the separation of two band-edge states are enhanced with increasing magnetic field, as shown in the inset of Fig. 1(c). Furthermore, the inset of Fig. 1(d) reveals that bandgap increases linearly with increasing B that may be considered as the B-dependence of threshold absorption frequency. Except the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) states, J c (v ) > 1 subbnads with higher energy are gradually modulated as B increases such as the lifted band energy and flattened band-edge states. For sufficiently large field strength (B=60 T), the flattened subbands will exhibit Landau levels, as shown in Fig. 1(d). Except the partial flat bands, the characteristics of LLs could be further illustrated form the changes in wave functions discussed as follows. The above demonstrates the effect of magnetic field on electronic properties including energy dispersion, band energy and bandgap shift. It is noticeable that the increasing magnetic field also makes wave functions become strong localized states. We select conduction subbands with J c = 1 and J c = 2 at k y = 2/3 state for B=0 and B=60 T to depict the changes in wave functions. The analyzed results could be applied for other subbands. In Fig. 2, Au(Bu) represents A(B)-site spinup state and A(B)-site spin-down state is denoted by Ad(Bd). Since wave functions are complex, we calculate the distribution of probability density of wave functions (PDWF) along the direction of finite width which is denoted by |Ψ c|2 . The spin-degenerate state J c = 1 could be divided into spin-up (J c = 1u ) and spin-down (J c = 1d ) states, seen in Fig. 2(a) and (b). Fig. 2(a) displays that the PDFW of Au state slowly decays from the boundary, whereas Bu state shows similar behavior from another boundary. The similar condition, due to the spindegeneracy, also occurs for spin-down state in Fig. 2(b). For J c = 2 180

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about k y = 1 and the partial flat bands in LLs are destroyed by Fx = 0.0001. The vanishing LLs could be interpreted as that Fx destroys the cyclotron motion of electrons on the nanoribbon's plane. Meanwhile, the HOMO and LUMO states gradually become a mixed state containing partial flat bands and parabolic bands with increasing Fx. Also noted that there exists an overlap between conduction and valence bands revealing a semiconductor-metal transition (SMT). Such a band-overlap is enhanced by increasing Fx, as shown in Fig. 3(d) for Fx = 0.0005. In order to conveniently specify the optical transitions, the subbands indices are thus changed by Js (J's) for the lower (upper) part. Except the SMT, partial flat bands also shift toward higher energy and LLs (J′ = 2(3) and J = 2(3)) start to be significantly distorted with increasing Fx. Band-edge states shift toward opposite ky directions for J = J ′ subbands. As a result, the strength of direct optical transitions may be reduced exhibiting weaker optical absorption, as discussed in the following section. Field-modulated electronic properties such as dispersion relation, Landau level, band-splitting, and bandgap are directly reflected by special structures of the density of states (DOS). The numbers of special structures are related to the excitation channels; in other words, the frequency and strength of optical excitations could be determined by their peak positions and heights. The density of states is defined as

D (B , F , ω ) =

2Ic Nx

∑∫ h, J

1stBZ

dk y

δ , 2π [ω − E h (B, F , J , k y )]2 + δ 2

(4)

where h = c, v represents conduction or valence bands, and the phenomenological broadening parameter is δ = 5 × 10−4 ev . Without external fields, Fig. 4(a) shows that special structures of the DOS are symmetric about EF = 0 and each peak respectively comes from J=1 to J=6 subbands. Bandgap is the energy distances of the two peaks closest to EF between them the DOS vanishes, i.e., ZSiNRs are metallic. Except the first peak of the DOS, for state energy ω > EF , it diverges in the

Fig. 3. The low-energy band structures of 60-ZSiNR at B=60 T and (a) Fz = 0.1, (b) Fz = 0.3 , (c) Fx = 0.0001, (d) Fx = 0.0005. The inset in (b) shows the details of split LL1.

In addition to magnetic field, applied electric field is also a convenient method to observe a change in electronic properties [30]. In this study, the electric field does not play the main role to investigate field-modulated physical properties. We are mainly interested in how electric field, under strong magnetic field B=60 T, further modulates electronic properties such as energy dispersion, destruction and split of Landau level, and bandgap. Electric fields along two directions are considered; Fz is perpendicular to the ribbon's plane and Fx is along the ribbon's finite width. For a 60-ZSiNR, its height and width are, respectively, 0.466 Å and 200 Å. Thus, the electric potential differences along the perpendicular direction and between two boundaries are about 0.04466 ev for Fz = 0.1 and 0.02 ev for Fx = 0.0001, respectively. At Fz = 0.1, Fig. 3(a) shows that energy dispersions are similar to those of Fz = 0 case, as compared with Fig. 1(d). However, subbands closest to EF = 0 have significant change. The splitting energy between spin-up and spin-down states increases from 0.6 mev to 8 mev due to electric field Fz = 0.1. Moreover, bandgap also grows from Eg = 0.66 mev to Eg = 41 mev . As for Landau levels, there is a tiny splitting energy about 1.2 mev just for LL1. While electric field increasingly reaches to Fz = 0.3, in Fig. 3(b), bandgap is enlarged to Eg = 134 mev . Additionally, the splitting energy of LL1 clearly grows to 3.4 mev (the inset of Fig. 3(b)); meanwhile, the band-splitting of LL2 is also found. Obviously, increasing Fz mainly enlarges bandgap and enhances band-splitting rather than change energy dispersions. From the above, Fz does not change band symmetry, band-edge states and energy dispersions. The main reason is that the interaction between Si atoms, for low-buckled SiNRs, is principally contributed by plane bonding. In contrast, parallel electric field Fx will strongly affect electronic structures of ZSiNRs including energy dispersion, destruction of LLs, band symmetry, bandgap, and band-edge states. Comparison between Figs. 1(d) and 3(c) shows that the band symmetry

Fig. 4. The density of states of 60-ZSiNR at different magnetic and electric fields. The three insets in (b)–(d) are the detailed DOS around ω = 0 .

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transitions firstly must satisfy Δk y = 0 , when ZSiNRs are excited by the EM field with the yl polarization. The spectral function could be written as

form 1/ ω − E c (k ye ) from k ye = 2/3 (or k ye = 4/3) band-edge state. The peak height of the DOS slightly increases with increasing energy. It may be predicted that strong optical absorption occurs at higher energy. It is noticeable that the first peak with the largest height does not certainly imply an occurrence of the strongest optical absorption, since optical spectra are profoundly associated with the optical selection rule and velocity matrix element. While applied magnetic field increases from B=20 T to B=60 T, the DOS could significantly reveal changes in B-dependent electronic properties, as shown form Fig. 4(b) to (d). The detailed changes of the DOS around the Fermi energy could be found in the three insets of Fig. 4 by reducing the phenomenological broadening parameter to δ = 10−4 ev in Eq. (4). At B=20 T, the first pair of peaks near ω = 0 is from electronic states with wave vector close to k y = 1 and implies a tiny bandgap. The second pair of peaks with energy ω = ± 3.3 × 10−3 ev is contributed from spin-degenerate states around k y = 2/3 (or k y = 4/3) due to the spin-orbit coupling. As magnetic field increases, for the first pair of peaks the widened energy spacing implies larger bandgap and the increasing sharpness of peaks reveals a change in effective mass. As for the second pair of peaks, at B > 20 T the doubled peak numbers are attributed to two spin-polarized states induced by strong magnetic field. The result may imply an increase in the low-frequency excitation channel. For the higher energy region, each peak gradually shifts far away from ω = 0 as B increases. Most importantly, at B=60 T (Fig. 4(d)) the peak height of the first (LL1) and second (LL2) peaks in the DOS is significantly enhanced due to the created Landau levels in Fig. 1(d). Therefore, it could be predicted that the frequency and strength of optical spectra will grow with increasing magnetic field. Following the exhibition of LLs created by B=60 T, applied electric field Fz is irrelevant to the planar cyclotron motion of electrons such that it could not destroy the LLs. The HOMO and LUMO states are away from each other causing an increase in bandgap. However, Fz = 0.1 causes an electric potential difference between non-coplanar A-site and B-site atoms such that the peak sharpness of the DOS at LL1 starts distorted slightly resulting in two spin-polarized states, as shown in Fig. 4(e). For stronger field strength Fz = 0.3, Fig. 4(f) shows the splitting spin-polarized states are extended form LL1 to higher energy LL2 subbands. The result might increase transition channels and enrich the optical spectra. Fx, as above-mentioned, destroys the band symmetry about the zone center but the DOS is remains symmetric about ω = 0 . Moreover, the band-overlap at ω = 0 induced by Fx could also be seen in the DOS. For example, Fig. 4(g) shows that at Fx = 0.0001 and near ω = 0 , there are three special structures for ω > 0 (or ω < 0 ); they simultaneously include both J=1 and J′ = 1 subbands. In addition, Fx also makes LLs become parabolic bands resulting in the weaker DOS. The most significant peak structures in the DOS are from the nearly partial flat bands which gradually shift to higher energy subbands with increasing Fx. While field strength reaches to Fx = 0.0005 shown in Fig. 3(d), except the destruction of band symmetry, energies of band-edge states at k y ≈ 2/3 and k y ≈ 4/3 are no longer equal. That results in peaksplitting structures in the DOS, e.g., two pairs of peaks at ω ≈ 0.133 eV and ω ≈ 0.2 ev in Fig. 4(h). Such a peak-splitting in the DOS may increase optical transition channels of ZSiNRs. On the other hand, the band-overlap induced by Fx makes the DOS around ω = 0 nonvanishing. Both electrons and holes thus make a contribution to the low-temperature thermal properties, such as electronic specific heat, thermal conductivity, and thermal power. Therefore, Fx could be used to change physical properties at low-temperature.

A (B , F , ω ) ∝

2Ic Nx

∑ ∫ 1stBZ

c, v , J , J ′

dk y 2π

|〈Ψ c (B, F , J ′, k y )|

⎤ ⎡ 1 ⎥, × Im ⎢ ⎣ ω − ωvc (B, F , J , J ′, k y ) + iδ ⎦

py me

|Ψ v (B, F , J , k y )〉|2

(5)

where ωcv (B, F , J , J ′, k y ) = E c (B, F , J ′, k y ) − E v (B, F , J , k y ) is the interband transition energy and δ (=0.002 ev) is the broadening parameter due to various de-excitation mechanism. The velocity matrix py element 〈Ψ c (B, F , J ′, k y )| m |Ψ v (B, F , J , k y )〉 is calculated within the e gradient approximation [36]. At zero temperature, the inter-π-band excitation (π → π *) is the only excitation channel. The significant absorption peaks in the spectral function directly reveal the available channels which are induced by inter-π-band optical excitations. At an excitation energy ωex, which corresponds to optical transitions from states near the band-edge, the number of such excitations becomes infinite due to the divergent DOS. Based on symmetric properties of velocity matrix element in Eq. (5), there could exist selection rules to make certain transition channels allowable. In addition, low-frequency optical absorptions are mainly addressed, since effects of the weak field on physical properties are easily verified by experimental measurements. The number of optical excitations is thus finite. In the previous study [30], for ASiNRs the spectral function, without external field, satisfies the selection rule ΔJ = 0 which is defined as ΔJ = J c − J v . Here, the notation ΔJ both include excitation and deexcitation. Thus, the optical excitation energy could be generally written as ω J vJ c = E (B, F , J c, k y ) − E (B, F , J v, k y ) demonstrating the transition channel J v → J c . The optical selection rule follows symmetric characteristics of nano-structures, i.e., profoundly related to nanoribbon's geometry. In Fig. 5(a), in the absence of the SOC, absorption peaks (black solid line) based on peak height could be classified into principal peaks (ωnP) and subpeaks (ω1J c ). The selection rule corresponding to principal peaks must basically satisfy ΔJ = odd ; however, ΔJ for one absorption peak, at higher absorption frequency, could include more different odd numbers. For example, ω1P (J v → J c: 1 → 2 ) and ω2P (J v → J c: 2 → 3) satisfy ΔJ = 1. ω3P has ΔJ = 1, 3 ΔJ = 1, 3, 5 (J v → J c: 3 → 4, 2 → 5), ω4P has 5P v c (J → J : 4 → 5, 3 → 6, 2 → 7), ω corresponds to ΔJ = 1, 3, 5, 7 (J v → J c: 5 → 6, 4 → 7, 3 → 8, 2 → 9), and so on. Additionally, there exists one subpeak between two adjacent principal peaks, which simply comes from the optical transition J v = 1 → J c (J c = 4, 6, 8, …). With increasing excitation frequency, the selection rule of a subpeak sequentially satisfies ΔJ = 3, ΔJ = 5, ΔJ = 7, and so on. In summary, the selection rule (ΔJ = odd ) of ZSiNRs significantly differs from that of ASiNRs (ΔJ = 0 ). This above-mentioned selection rules could be understood via the following analysis. In the absence of the SOC, the 4Nx × 4Nx hermitian Hamiltonian matrix in Eq. (1) is reduced into a 2Nx × 2Nx real symmetric matrix for ZSiNRs. The Bloch function could be written as Nx |Ψ h〉 = ∑n =1 a nh |An 〉 + bnh |Bn 〉(h = c, v ) . By diagonalizating the Hamiltonian matrix of 60-ZSiNR at band-edge state k y = 2/3, we find the amplitudes of the conduction and valence bands follow the relations: (1) c v anc (J c ) = (−1) J b Ncx +1− n (J c ) and (2) anv (J v ) = (−1) J +1b Nvx +1− n (J v ). Then, we further use the gradient approximation to calculate the velocity matrix element in Eq. (5) which plays a major role in determining the selection rule. The calculated velocity matrix element is proportional to c v Nx [1 + (−1) J + J +1] ∑n =1 anc (J c ) bnv (J v ) implying the selection rule ΔJ = odd . The same method could be used to obtain the selection rule ΔJ = 0 for armchair silicene nanoribbons [30]. While the SOC is considered, it could induce new absorption peaks and construct new selection rules due to the symmetry-breaking on ribbon's plane by introducing Pauli matrices σx and σy. As suggested in the inset of Fig. 1(a), the SOC makes conduction and valence bands at

3. Field-modulated optical properties of zigzag silicene nanoribbons Since applied external fields considered in this study does not destroy the nanoribbon's period along the yl direction so that wave vector ky is still a good quantum number. The selection rule in optical 182

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as B increases, as shown in the right inset of Fig. 5(b). However, the peak height is getting lower with increasing B. The reason could be seen by the B-dependent DOS in the three insets of Fig. 4. They reveal that stronger magnetic field just makes a split to edge-state at k y = 2/3 and slightly affects its energy dispersion and energy spacing between conduction and valence bands. Next, the dependence of optical spectra with higher frequency (ω > 0.1 ev ) on magnetic field are addressed, as shown in Fig. 5(b)–(d). Since applied magnetic field is perpendicular to ribbon's plane, the above-mentioned selection rules keep the same. That is to say the allowed transition channels of each absorption peak could be explained by the above discussions. The changes in optical spectra are mainly attributed to band shift and energy dispersion (Landau levels). At weaker magnetic field B=20 T, a blue-shift for the principal peaks is more significant than that for subpeaks and next-subpeaks. But there are still one subpeak and one next-subpeak between consecutive two principal peaks. These result could be predicted in the DOS calculation (in Fig. 4). Except the band-shift, B-modulated energy dispersions could be observed by means of changes in absorption peaks. For example, Fig. 5(b) shows that the peak height gradually vanishes with increasing frequency for next-subpeaks but not for subpeaks. On the other hand, the peak height of ω1P starts to be slightly higher than that of ω2P, which reveals J c (v ) = 2 subbands are more flattened at B=20 T than B=0. Such a result predicts that a Landau level is gradually formed with increasing B. While magnetic field reaches to B=40 T, there are three lower peaks between ω1P and ω2P that is not from a new transition channel but just from a band-shift. As field strength increases to B=60 T, two LLs at subbands J c, v = 2 and J c, v = 3 (LL1 and LL2) are created and then exhibits the stronger DOS at lower frequency, as shown in Figs. 1(d) and 4(d). The lower-frequency principal peaks are profoundly related to the LLs such that the peak height of principal peaks drastically grows for ω1P, but it slowly reduces from ω3P. The behavior of principal peak strength versus frequency at B=60 T quite differs from that at B=0 case and the first principal peak dramatically changes that could be considered as an evidence of Binduced LLs in optical measurements. The above studies demonstrate that perpendicular magnetic field could significantly affect the frequency and strength of optical absorption. Then, whether applied electric field, after forming LLs, could further modulate optical spectra is our following issue, especially for the exhibition of new transition channel. Applied perpendicular electric field Fz does not destroy plane symmetry such that the optical selection rules are the same as those at Fz = 0 . Except the band-shift, Fz also plays an important role to enhance the destruction of spin-degeneracy resulting in two significant spin-polarized states. As a result, absorption frequency is not only modulated but spectral function becomes richer, i.e., exhibiting more absorption peaks. At Fz = 0.1, the significant spinspilt states at k y = 2/3 induces two low-frequency peak structures at ω = 0.043 ev and ω = 0.0575 ev , as shown in the inset of Fig. 6(b). While field strength increasingly reaches to Fz = 0.3, the above-mentioned two absorption peaks shift to higher frequency (ω = 0.136 ev and ω = 0.151 ev ) but their energy spacing remains the same. The main reason is the two spin-polarized states come from the same J=1 subband at zero field. In addition to low-frequency optical absorption, the LL-splitting induced by Fz also pronouncedly change the spectral function at higher frequency region (ω > 0.15 ev ). As Fz increases, except a blue-shift in principal peaks, the peak-splitting starts to occur from the first peak extending to higher-order peaks. At Fz = 0.1, it makes LL1 slightly split (seen in Fig. 4(e)), thus the two-peak structure mainly form the split J=1 subbands. For stronger field strength Fz = 0.3, peak-splitting of the first principal peak is more significant due to the stronger LL1-splitting (seen in Figs. 3(b) and 4(f)). Fx, as above-mentioned, more easily modulates band structures and makes 60-ZSiNR metallic such that optical selection rules become complicated. Except the inter-band transitions, the intra-band transitions could also be included due to applied Fx. Fig. 6(c), at weak field

Fig. 5. (a) is the spectral function of 60-ZSiNR with and without the SOC at B=0, the details near ω = 0 are shown in the inset. (b)–(d) are the spectral functions for different Bs and the details of low-frequency regions are shown in the two insets.

k y = 2/3 and k y = 4/3 split that exhibits a new low-frequency absorption peak with excitation energy ω ≈ 0.01 ev (red solid line), shown in the inset of Fig. 5(a). It satisfies the selection rule ΔJ = 0 and is from the optical transition J v = 1 → J c = 1. Besides, the SOC induces an extra next-subpeak, except a subpeak, between two adjacent principal peaks. The next-subpeaks are from the optical transitions J v = 1 → J c = 3, 5, 7, … corresponding to the selection rules ΔJ = 2, 4, 6 ,... Therefore, the SOC could construct a new selection rule ΔJ = even , which quite differs from ΔJ = odd for the SOC=0 case. Though the peak height of next-subpeaks is relatively weaker than that of principal peaks and subpeaks, it is still observable and enhanced by the stronger SOC for heavier group-IV elements such as Ge, Sn, etc. As a result, the extra next-subpeaks could be considered as a signature of the SOC in measured optical spectra. Except the intrinsic SOC, optical absorption could be modulated by applied perpendicular magnetic field. As mentioned previously in Fig. 1(c) and (d), energy dispersions around the Fermi energy are changed and bandgaps gradually grow as magnetic field increases that could be reflected in low-frequency optical spectra. The left inset of Fig. 5(b) shows that magnetic field initiates a threshold optical absorption at very low frequency (ωth) due to the bandgap. For B=20 T, threshold optical absorption occurs at ωth ≈ 0.25 mev , then it linearly increases to ωth ≈ 0.75 mev for B=60 T. Moreover, the strength of spectral function rises due to the more flattened band-edge state with increasing B. For 60-ZSiNR, both ωth and absorption strength are relatively much smaller and weaker than those in higher frequency regions such that they are not easily measured in experiments. For wider ZSiNRs (e.g., Nx = 150 ), the needed magnetic field for effectively modulating energy dispersion and bandgap is largely reduced. Thus the experimental measurements of the B-dependent threshold optical absorption will be valid. As for another low-frequency optical absorption at B=0, its absorption frequency ω ≈ 0.01 ev is almost unchanged 183

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and (3) J = 1(J = 3) → J ′ = 3(J ′ = 1). Therefore, the significant bandmodulations induced by stronger Fx exhibit richer optical spectra and more complicated selection rules. 4. Conclusion We study low-energy electronic and optical properties of ZSiNRs by using tight-binding model including the spin-orbit coupling in the presence of magnetic and electric fields. The spin-orbit coupling opens the partial flat band and further induces new optical absorptions and selection rules leading to richer optical spectra. The increasing magnetic field causes an increase in bandgap and constructs spinpolarized Landau levels; in which at band-edges electron's probability density distributes like a standing-wave. Landau levels could enhance the DOS and increases absorption frequency and strength. While extra perpendicular electric field is applied, it does not change the band symmetry, edge-states, and optical selection rules. With increasing Fz, bandgap and absorption frequency gradually grows. More importantly, it enhances a split of spin-polarized states in LLs and induces more absorption peaks that will be beneficial to develop nano-devices based on spintronics. However, parallel electric field leads to an overlap between conduction and valence bands that makes ZSiNRs metallic. As Fx increases, drastic modulations of band structures causes the destruction of band symmetry and Landau levels. As a result, optical excitations including both intra- and inter-band transitions construct new selection rules leading to richer absorption spectra. The Landau levels induced by magnetic field could be further verified by measured absorption strength and frequency in experiments. By the modulations of applied electric fields, the optical devices made by SiNRs with diversified optical spectra are expected to have more extensive applications.

Fig. 6. (a)–(b) are the spectral functions of 60-ZSiNR at B=60 T for different Fzs. (c)–(d) are similar to (a)–(b) but for different Fxs. The three insets show the detailed spectral functions of low-frequency regions.

Acknowledgements strength Fx = 0.0001, shows that the four principal peaks from ω1P to ω4P are similar to those without Fx, except the weaker ω1P, as compared with Fig. 5(d). Moreover, the selection rules remains unchanged. However, there are some differences in optical spectra including (1) the low-frequency excitation increases from ω = 0.01 eV to ω = 0.018 ev (see the insets in Figs. 5(b) and 6(c)) and (2) the subpeaks and next-subpeaks between two adjacent principal peaks almost disappear. The former is due to the modulation of Fx on edgestates k y = 2/3 and k y = 4/3 at J=1 and J′ = 1 subbands. The latter could be attributed to the destruction of partial flat bands (J=1 and J′ = 1) and Landau levels (J=2 and J′ = 2 ). For sufficiently large field strength Fx = 0.0005, optical properties are significantly affected including absorption frequency and strength and even the selection rules, as shown in Fig. 6(d). The stronger band-overlap and distortion at edge-states k y ≈ 2/3 and k y ≈ 4/3 with J=1 and J′ = 1 make a shoulder structure in lowfrequency spectra, shown in the inset of Fig. 6(d). It is further found the low-frequency optical absorption (ω < 0.1 ev ) becomes vanishing with increasing Fx. Furthermore, the first principal peak ω1P is split due to the destruction of band symmetry about k y = 1 induced by larger field Fx = 0.0005. We further analyze corresponding transition channels, the split ω1P is from the transitions J = 2 → J = 1 around k y = 2/3 state and J ′ = 1 → J ′ = 2 around k y = 4/3 states. Such a pure intra-band transition differs from the pure inter-band transition in above-mentioned cases for zero or weaker applied Fx. In contrast to ω1P, the second principal peak ω2P includes both inter-band ( J = 4 → J ′ = 1) and intra-band (J ′ = 1 → J ′ = 4 ) transitions. Similar selection rules also occur for other principal peaks such as ω3P, ω4P, ω5P, etc. In addition, subpeaks and next-subpeaks between two adjacent principal peaks are also from both inter- and intraband transitions. For example, the weak special structures between ω1P and ω2P mainly come from the following transition channels: (1) J = 2(J = 1) → J ′ = 1(J ′ = 2), (2) J = 3 → J = 1 and J ′ = 1 → J ′ = 3,

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