Optical absorption properties of few-layer phosphorene
OOptical absorption properties of few-layer phosphorene
Zahra Torbatian and Reza Asgari
1, 2, ∗ School of Nano Science, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran
We investigate the optical absorption and transmission of few-layer phosphorene in the frameworkof ab initio density functional simulations and many-body perturbation theory at the level of randomphase approximation. In bilayer phosphorene, the optical transition of the valence band to theconduction band appears along the armchair direction at about 0.72 eV, while it is absent along thezigzag direction. This phenomenon is consistent with experimental observations. The angle-resolvedoptical absorption in few-layer phosphorene shows that it is transparent when illuminated by neargrazing incidence of light. Also, there is a general trend of an increase in the absorption by increasingthe number of layers. Our results show that the bilayer phosphorene exhibits greater absorbancecompared to that of bilayer graphene in the ultraviolet region. Moreover, the maximal peak in thecalculated absorption of bilayer MoS is in the visible region, while bilayer graphene and phosphoreneare transparent. Besides, the collective electronic excitations of few-layer phosphorene are explored.An optical mode (in-phase mode) that follows a low-energy √ q dependence for all structures, andanother which is a damped acoustic mode (out-of-phase mode) with linear dispersion for multilayerphosphorene are obtained. The anisotropy of the band structure of few-layer phosphorene along thearmchair and zigzag directions is manifested in the collective plasmon excitations. PACS numbers: 73.20.Mf, 71.10.Ca, 71.15.-m, 78.67.Wj
I. INTRODUCTION
Advanced two-dimensional (2D) crystalline layered ma-terials have received great attention during the pastdecade. The most well-known 2D crystalline materialsare graphene [1, 2] and molybdenum disulfide [3]. Thegapless nature of graphene and a low carrier mobility oftransition metal dichalcogenides (TMDCs) have limitedtheir potential applications in technology and electronicdevices [4]. Phosphorene, one or a few layers of black phos-phorous [5], is another 2D semiconductor that has latelybecome the focus of investigations owing to its high car-rier mobilities and a tunable band gap [6, 7]. High chargemobility on the order of cm /Vs has been observed inmonolayer phosphorene at low temperatures [8]. Also, thenarrow gap of phosphorene (between zero-gap grapheneand large-gap TMDCs), makes it an ideal material fornear and mid-infrared optoelectronics and new types ofplasmonic devices [9].Similar to graphene and TMDCs, phosphorene is alsoa layered material that can be exfoliated to yield indi-vidual layers [10]. Monolayer phosphorene with a hexag-onal puckered lattice has a direct band gap and highanisotropic band structure. The dispersion relation inmonolayer phosphorene is highly anisotropic, which givesrise to the direction depending on mechanical, optical,electronic and transport properties [11–14]. Bilayer phos-phorene has attracted considerable interest given its po-tential application in nanoelectronics owing to its naturalband gap and high carrier mobility [15–17]. Bilayer phos-phorene has a smaller direct band gap than monolayerphosphorene, which offers more opportunity to tune the ∗ [email protected] semiconductor, metal and maybe even topological insula-tor [5]. More efforts have been devoted to the band struc-ture and electronic properties of few-layer phosphorene,however, its optical properties deserve special consider-ation. Most importantly, phosphorene exhibits stronglight-matter interactions in the visible and infrared pho-ton energies. The application of few-layer phosphorene innanoplasmonics and terahertz devices is highly promis-ing [18, 19].The aim of this paper is to explore the angle-resolvedoptical absorption and transmission of few-layer phospho-rene. To do so, we use a recently proposed theoreticalformulation [20]. In this theory, the current-current re-sponse tensor is calculated in the framework of ab ini-tio density-functional theory (DFT) within a many-bodyrandom-phase approximation (RPA), where the electro-magnetic interaction is mediated by the free-photon prop-agator. The tensorial character of the theory allows us toinvestigate the response to a transverse electric, s (TE),and transverse magnetic, p (TM), external electromag-netic field, separately. It is important to mention that thepresent theory doesn’t include quasiparticle correction ofthe DFT band structure such as an electron-hole boundstate or excitons in semiconducting two-dimensional crys-talline materials. Also, it is possible to explore the de-pendence of the optical properties of the system on theincident angle of the external electromagnetic field.For the sake of completeness, we calculate the collec-tive modes of few-layer phosphorene. We use the for-malism that has been presented in previous works [21–23] and calculate the density-density response functionwithin the DFT-RPA approach. Eventually, invoking thedensity-density response function, the collective modesare established by the zero of the real part of the macro-scopic dielectric function. a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov In this work, the optical absorption and transmissionin terms of the ab initio current-current response tensorare calculated for few-layer phosphorene. We show thatthe optical absorbance of the systems monotonically de-creases, as the incident angle of light increases and few-layer phosphorene is transparent when it is illuminatedby near grazing incidence of light. But the transmissionincreases, as the incident angle of light and it becomes al-most for near grazing incidence. In the low-energyzone, the absorption spectrum is red-shifted by increasingthe number of layers along the armchair direction, whileit changes slightly along the zigzag direction. In addition,the plasmon excitations show a highly anisotropic plas-mon dispersion owing to the strong crystal anisotropy infew-layer phosphorene. Finally, we make a brief compar-ison of the plasmon dispersion and optical absorption infew-layer phosphorene.This paper is organized as follows. In Sec. IIA, wepresent the methodology used for calculating the ground-state properties of the system. In Sec. IIB, we presentthe optical quantities such as the optical absorption andtransmission which are needed for the study in this work.In Sec. III we present and discuss in details our numeri-cal results for the optical absorption spectra for pristinefew-layer phosphorene. Last but not least, we wrap upour main results in Sec. IV.
II. THEORY AND COMPUTATIONALMETHODSA. Structure and ground-stat calculations
In order to investigate the structural and optical prop-erties of few-layer phosphorene, we use ab initio simu-lations, based on the density functional theory as im-plemented in the QUANTUM ESPRESSO [24] code.Our DFT calculations are carried out using the Perdew-Burke-Ernzerh exchange-correlation functional [25] cou-pled with the DFT-van der Waals method. The Kohn-Sham orbitals are expanded in a plane wave basis set witha cutoff energy which is Ry and a Monkhorst-Pack k -point mesh of × × is used. The energy convergencecriteria for the electronic and ionic iterations are set tobe − and − eV, respectively. In order to minimizethe interaction between layers, a vacuum space of Å isapplied.In single layer phosphorene, each phosphorus atom co-valently bonds with three adjacent atoms, forming a sp hybridized puckered honeycomb structure. Therefore, itcontains two atomic layers and two kinds of bonds with . and . Å bond lengths for in-plane and inter-plane p − p connections, respectively. In the case of monolayerphosphorene, the calculated lattice constants are foundto be a = 4 . and b = 3 . Å along the armchair andzigzag directions, respectively, which are in good agree-ment with those predicted by other theoretical work [26].Bilayer phosphoren has been predicted to exist in three
AA AB AC x y (armchair) ( z i g za g ) (a)(b) Figure 1. (Color online) Three stacking structures of bilayerphosphorene. (a) Top view of AA, AB and AC stacking (b)Side view of AA, AB and AC stacking. Our numerical resultsof the ground-state energy show that the AB-stacked struc-ture is energetically the most preferred for bilayer phospho-rene. different stacking, namely AA, AB and AC structures(Fig.1). For the AA stacking, the top layer is directlystacked on the bottom layer. The AB stacking can beviewed as shifting the bottom layer of the AA stackingby half of the cell along either the a or b directions. Forthe AC stacking, the top and bottom layers are mirrorimages of each other. Regarding the trilayer phosphorene,three possible stacking types (AAB, ABA, and ACA) areconsidered.The ground-state energy calculations, stemming fromour numerical calculations, show that the AB-stackedand ABA-stacked structures are energetically the mostpreferred for bilayer and trilayer phosphorene, respec-tively. For bilayer phosphorene with the AB-stackedstructure, the interlayer separation between the closestphosphorene atoms is . Å while the in-plane latticeconstants remain the same. Throughout this study, weperform calculations on the most stable AB-stacked formof the bilayer and ABA-stacked form of trilayer phospho-rene systems.
B. Current-current response tensor
In this section, we consider independent electronswhich live in a local crystal potential obtained by DFTand interact with the electromagnetic field described bythe vector potential, A ( r , t ) . We pursue the proceduregiven in [20] and thus the screened current-current re-sponse tensor can be calculated by solving the Dysonequation Π = Π + Π ⊗ D ⊗ Π (1)where Π and D are the noninteracting current-currentresponse tensor and free-photon propagator, respectively. -3-2-101 X Γ Y -3-2-101 E n e r gy ( e V ) X Γ Y -3-2-101 X Γ Y k y . Γ k x k y . Γ k x k x k y k x k y . Γ . Γ VB k x k y CB . Γ k x k y Γ . (a) (b) (c)(d) (e) (f) Figure 2. (Color online) The calculated band structures for(a) monolayer (b) bilayer and (c) trilayer phosphorene struc-tures along the X − Γ − Y direction. The band gap for mono-layer phosphorene is 0.98 eV and it decreases to 0.6 and 0.52eV for bilayer and trilayer phosphorene, respectively. The na-ture of the band gap remains direct for trilayer to single-layerblack phosphorous. Isofrequency contour plots of the energiesaround the conduction band (CB) minimum and the valenceband (VB) maximum in the k-space are illustrated for the(d) monolayer, (e) bilayer and (f) trilayer phosphorene for E F = 0 . eV and E F = − . eV, with a step of . eV.The shape of the Fermi surfaces in the systems is almost anelliptic shape especially at low charge density. Our system contains slabs repeated periodically and Π µν is a periodic function along the z direction. Also, thephoton propagator has a long-range character that leadsto the interactions between adjacent crystal slabs. There-fore, the screened Π µν contains the effects of the couplingbetween supercells. One of the best idea to solve thisproblem is to suppose that supercells are not repeatedperiodically along the z direction and the system hasjust one crystal slab, which is restricted to the region − L/ < z < L/ . It denotes Π µν contains just one termin the z direction and photon propagator D µν couplesonly to the charge or current fluctuations in the region − L/ < z < L/ , even if it propagates interaction allover the space.With this well-known method for 2D materials, al-though the periodicity is broken along the z direction,it remains in the x − y plane. Therefore, we can performthe Fourier transform of the Dyson equation (Eq. 1) inthe x − y plane and results are given by Π G (cid:107) ,G (cid:48)(cid:107) ( q , ω, z, z (cid:48) ) = Π G (cid:107) ,G (cid:48)(cid:107) ( q , ω, z, z (cid:48) )+ (cid:88) G (cid:107) ,G (cid:107) (cid:90) L/ − L/ dz dz Π G (cid:107) ,G (cid:107) ( q , ω, z, z ) × D G (cid:107) ,G (cid:107) ( q , ω, z , z )Π G (cid:107) ,G (cid:48)(cid:107) ( q , ω, z , z (cid:48) ) (2)where q is the momentum transfer vector parallel to the x − y plane and G (cid:107) = ( G x , G y ) are 2D reciprocal lat-tice vectors. Also, the Fourier transformed free-photonpropagator is given by [27] D G (cid:107) ,G (cid:48)(cid:107) ( q , ω, z, z (cid:48) ) = D ( q + G (cid:107) , ω, z, z (cid:48) ) δ G (cid:107) ,G (cid:48)(cid:107) (3)where D ( q , ω, z, z (cid:48) ) = − πcω δ ( z − z (cid:48) ) z · z (cid:48) + 2 πicβ { e s . e s + e p . e p } e iβ | z − z (cid:48) | (4)The directions of s (TE) and p (TM) polarized fieldsdescribe by e s = q × z and e p = cω [ − βsgn ( z − z (cid:48) ) q + q z ] ,where c is the velocity of light, β = (cid:112) ω /c − | q | andthe q is the unit vector in the q direction.Since the integration in Eq. 2 is performed for − L/ S × L is the normalized volume and f n ( k ) is theFermi-Dirac distribution at temperature T . Notice thatwe define the three-dimensional vector r = ( ρ, z ) and thewave function φ n k is expanded over the plane waves withcoefficients which are obtained by solving the Kohn-Shamequations self-consistently.We can use a useful relation between the optical absorp-tion and current-current response tensor that was alreadythoroughly discussed in [28]. Now, we want to exploit thenormalized absorption power per unit area which is givenby A s,p = 4 πω S s,p ( q , ω ) , (9)where the dynamical spectral function is S s,p ( q , ω ) = (cid:61) m (cid:88) µ,ν e s,pµ e s,pν (cid:88) G z ,G (cid:48) z I + G z Π µ,ν,G z G (cid:48) z ( q , ω ) I + G (cid:48) z (10)in which, the form factors are I ± G z = 2 √ L sin[( β ± G z ) L/ β ± G z (11)By using the energy conservation law, the transmittedelectromagnetic energy flux can be calculated by T s,p = 1 − R s,p − A s,p (12)where R s,p is the reflected energy flux and given by R s,p = | R s,p | (13)Here, the amplitude of the reflected s and p waves are R s = 2 πicβ D xx ( q , ω ) ,R p = 2 πiω [ D yy ( q , ω ) cos θ − D zz ( q , ω ) sin θ tan θ ] , (14) where θ = sin − ( β/q ) is the angle between wave vector q and the normal vector of the system surface. The surfaceelectromagnetic field propagator is defined by D µν ( q , ω ) = (cid:88) G z G (cid:48) z I + G z Π µν,G z ,G (cid:48) z ( q , ω ) I − G (cid:48) z (15)At the end of this section, it is worth mentioning thatthe dielectric tensor and the conductivity tensor can beobtained with the current-current response. The first oneis defined as [29] ε µν ( ω ) = δ µν + 4 πω Π µν, ( q = 0 , ω ) (16)which describes the response of phosphorene to an ex-ternal homogeneous electrical field. With knowledge ofthe optical dielectric function, the refractive index of thesystem can be obtained. Moreover, the real part of thedielectric function describes the imaginary part of theoptical conductivity, however, the imaginary part of thedielectric function describes the absorption of light.Since ε µν ( ω ) = δ µν + (4 π/ω ) iσ ( ω ) , the longitudinalconductivity is given by (cid:60) eσ µµ ( ω ) = 1 ω (cid:61) m Π µµ, ( q = 0 , ω ) (17)which corresponds to the experimentally measurable op-tical conductivity [20]. The optical longitudinal conduc-tivity includes the interband transitions and the contri-bution of the intraband transitions, which leads to thefact that the Drude-like term is no longer relevant in thisstudy since the momentum relaxation time is assumed tobe infinite. This approximation is valid at low tempera-ture and for a clean sample where defect, impurity, andphonon scattering mechanisms are ignorable. III. NUMERICAL RESULTS ANDDISCUSSIONS In this section, we present our numerical results forfew-layer black phosphorus, based on first principles sim-ulations within the DFT-RPA approach at zero temper-ature. The band structure of few-layer phosphorene isplotted in Fig. 2. Clearly, the band characteristic of theAB bilayer and ABA trilayer phosphorene are similar tothat of the monolayer one, except that in the bilayer andtrilayer, energy level splitting occurs due to the interlayerinteractions. Our DFT calculations predict that the bandgap of monolayer phosphorene is direct and equal to 0.98eV and intriguingly, it decreases to 0.6 and 0.52 eV for bi-layer and trilayer phosphorene, respectively. The natureof the band gap remains direct for trilayer to single-layerblack phosphorous irrespective of the nature of stackingand it is more promising in terms of applying tunnel field-effect transistor devices [30].In the 2D case, the isofrequency profiles are obtainedby horizontally cutting the dispersion surface. Also, theisofrequency contour plots of the energies around the con-duction band minimum (CBM) and the valence bandmaximum (VBM) in the k − space of few-layer phospho-rene are illustrated in Figs. 2(d) - 2(f). Most importantly,the shape of the Fermi surfaces in the systems is almostan elliptic shape, especially at low charge density. Theelliptic shape of the Fermi surfaces, particularly in thehole-doped case demonstrates the anisotropic band en-ergy dispersion of the few-layer phosphorene along the Γ X and Γ Y directions. It turns out that, in the hole-doped cases, the anisotropic band energy dispersion de-creases by increasing the number of the layers.It is well-known that DFT underestimates the bandgap of semiconductors and the advance Green’s functionmethod can provide improved predictions. The Green’sfunction gap of monolayer phosphorene calculated byTran et al . [16] is about 2.0 eV. A recent Monte Carlostudy of monolayer phosphorene [31], using infinite peri-odic superlattices as well as finite clusters, predicted thatthe band gap is 2.4 eV. A. Optical absorption and transmission of bilayerphosphorene We explore the optical absorption calculated by usingEqs. 9-11 with the current-current response tensor Π µν (Eq. 7) . Having calculated the structure of few-layerphosphorene, we can obtain the Kohn-Sham wave func-tions φ m k ( r ) and energies E m ( k ) which are invoked tocalculate the current-current response, Π µν . In the sum-mation over k in Eq. 7, we use × × k -point meshsampling and band summation ( m , n ) is performed over and bands for bilayer and trilayer phosphorene, re-spectively. The damping parameter, η is meV in allfigures, unless we specifically define this value otherwise.The real and imaginary parts of the ab initio DFT di-electric function (Eq. 16) are shown in Fig 3. The dielec-tric function is calculated by invoking the Kohn-Shamwave functions and they change by changing the struc-tures and inter-atomic interactions. Therefore, the dielec-tric function depends on the material density and theinterlayer distance.We ought to note that at the onset of transparency atthe plasmon frequency we have (cid:60) eε xx/yy ( ω ) = 0 . Theimaginary part of the DFT dielectric function shows theexcitations in the system. The (cid:61) mε xx ( ω ) displays consid-erable structure before decreasing to become nearly zeroat (cid:126) ω ≈ eV. This leads us to imagine that below 2 eV, (cid:61) mε xx ( ω ) is due to the topmost occupied electron lev-els. Electrons above 10 eV play no significant part in theoptical spectrum. As shown in Fig. 3, the peaks in theimaginary part of the dielectric function are located inthe energy range between and eV, where the absorp-tions are maximum, which corresponds to the drop ofthe real part of the dielectric functions. The values of theextremum positions for both imaginary and real parts ofthe dielectric function are equal as they are related by the Kramers-Kroning relations. It is worth to mention thatthe finite values of (cid:61) mε ( ω ) around ω = 0 are a numericalartifact and it should be zero. This artifact is related tothe finite value of η , and from /ω factor in Eq. 16 andit can be reduced by decreasing of η [32]. In the insetof Fig. 3(b), we illustrate the imaginary part of ε yy ( ω ) for different values of η . The role of η in this problem iscompletely clear.There is an important f -sum rule for the dielectric func-tion which is used in the analysis of the absorption spec-tral. A general f -sum rule for the imaginary part of thedielectric function says that (cid:90) ∞ ω (cid:61) mε αβ dω = δ αβ N π e V m (18)where N = 20 is the number of electrons, V is the unitcell volume and m is the electron bare mass. We validateour numerical DFT-RPA results by considering 200 bandstructures in each system and find that the theory givessatisfactory results by 6%.The optical absorption and reflection of the p − and s -polarized normal incidence ( θ = 0 ) are shown in Fig.4 as a function of photon energy ( (cid:126) ω ) for bilayer phos-phorene. The optical absorption illustrates a maximumaround energy between − eV where the most profoundphoton-induced absorption of electrons between the men-tioned bands occurs. The optical transition of the valenceband to the conduction band appears along the armchairdirection at about . eV, while it is zero between and . eV along the zigzag direction. This phenomenon orig-inates from the contribution of (cid:61) mε xx ( ω ) in this regionand consistent with experimental observations reportedin Ref. [18]. These results demonstrate a strong lineardichroism in bilayer phosphorene, where the position ofthe lowest energy absorption peaks for the armchair andzigzag directions differ significantly.It can be found from the symmetry of the wavefunc-tions that dipole operator connects the VBM and CBMstates for the s -polarization, allowing the direct-band gapprocess, however, this is symmetry-forbidden for the p -polarization and transition occurs between the VBM andCBM states elsewhere in the Brillouin zone [6]. There-fore, we expect that phosphorene would be a suitablematerial for applications associated with liquid-crystaldisplays and optical quantum computers [33, 34].It should be noted that the reflection part of the inci-dent light is negligible as shown in Fig. 4(b). Therefore,the main part of the optical properties goes to the trans-mission and absorption parts.We also analyze the optical absorption and transmis-sion of bilayer phosphorene as functions of the incidentangle θ and photon energy (cid:126) ω along the armchair direc-tion (Fig. 5). It is observed that the optical absorbancemonotonically decreases, as the incident angle of lightincreases, however, the transmission increases. In partic-ular, Fig. 5 shows that bilayer phosphorene is transparentwhen it is illuminated by a nearly grazing incidence light.It is intriguing that the peaks in the absorption spectra h _ ω (eV) I m ε ( ω ) -3036 R e ε ( ω ) (a)(b) η= 0.05 eV η= 0.02 eV η= 0.005 eV η= 0.001 eV Figure 3. (Color online) (a) The real and (b) imaginary partsof the dielectric tensor of bilayer phosphorene vs ω for ε xx (blue line) and ε yy (red line). Note that the interband tran-sitions are included in the dielectric tensor. The imaginarypart of the dielectric function illustrates the excitations inthe system. Note that, the peaks in the imaginary part of thedielectric functions correspond to the drops in the real partof the dielectric functions. The inset shows (cid:61) mε yy ( ω ) around ω = 0 for different values of η . The (cid:61) mε yy ( ω = 0) tends tozero by decreasing the value of η . correspond to the dips in the transmitted spectra andthe peaks show that the reflected light is negligible. Sim-ilar physical behaviors have been obtained for few-layerphsophorene and monolayer MoS [28].We would like to recall well-known dipole selec-tion rules, which determines whether transitions are al-loweded or forbidden based on the symmetry of the va-lence or conduction wave functions. The optical absorp-tion can be calculated in the dipole-transition approxima-tion as [35] α ( ω ) = (2 π ) (cid:126) ω (cid:88) νc k | < c k | ˆ e · p | ν k > | δ ( (cid:126) ω − E c k + E ν k ) (19)where p is the dipole matrix operator and ˆ e is the direc-tion of the polarization of the incident light. The dipoleselection rules allow transitions in which angular momen-tum between the valence and the conduction states differfrom unity. Since the parity of p is odd, two wavefunc-tions of the valence and the conduction have oppositeparities in the direction of ˆ e and the same parity in otherdirections. It is worth mentioning that in order to havea significant absorption at a particular energy, the jointdensity of states, V (cid:80) k νc δ ( (cid:126) ω + E c k − E ν k ) , must havea Van-Hove singularity for a given energy. Most impor-tantly, this one-particle picture of the transition process A b s o r p ti on [ % ] h _ ω (eV) R e f l ec ti on [ % ] Γ X Γ Y (a)(b) Figure 4. (Color online) Optical (a) absorption and (b) reflec-tion of pristine bilayer phosphorene of the s -polarized (alongthe armchair direction) and p -polarized (along the zigzag di-rection) for normal incident ( θ = 0 ). Noticeably, the opticaltransition of the valence band to the conduction band appearsalong the armchair direction at about . eV, while it is zeroup to . eV along the zigzag direction. is totally inadequate and does not come close to describ-ing the absorption spectra observed in experiments. Con-sequently, our analysis based on the many-body opticalabsorption is needed.Note that when states near the VBM or the CBM havemulticomponent characters, the spinors describing thesecomponents can pick up nonzero winding numbers andin such systems, the strength and required light polariza-tion of an excitonic optical transition are dictated by theoptical matrix element winding number. This winding-number physics, which mainly emerge in nanoribbonstructures, leads to novel exciton series and optical selec-tion rules [36]. In this work, we focus on only the sum-ruleselection rules presented in Eq. 19.For bilayer phosphorene, the percent contribution fromeach atomic orbital to the valence and the conductionwavefunctions at the special k -point listed in Tabel. I.The optical absorption edge along the armchair directionis related to an interband transition from the VBM tothe CBM and the allowed transitions are d x − y → p x , d z → p x and d zx → p z .For a zigzag polarization, on the other hand, the opti-cal absorption edge around . eV is from the VBM toCBM+1 (the next band higher in energy than the conduc-tion band) at a k point along the Γ − Y direction whichmainly originates from the interband transition. In this Table I. The PDOS of the wavefunctions that contribute to allowed transitions along the armchair and zigzag directions. Thepercentage contributions from atomic orbitals of the wavefunctions are determined.type direction k (X,Y) state s p z p x p y d z d xz d yz d x − y d xy bilayer armchair (0,0) CBM 6 42 17 0 28 4 0 1 0VBM 8 72 4 0 0 10 0 5 0bilayer zigzag (0,0.35) CBM+1 2 24 6 21 1 7 3 6 26VBM 12 70 1 10 0 1 0 1 5trilayer armchair (0,0) CBM 9 38 16 0 30 6 0 1 0VBM 7 77 3 0 0 5 0 5 0trilayer zigzag (0,0.38) CBM+1 5 7 19 24 3 6 3 6 26VBM 6 72 1 0 0 1 8 1 8 case, CBM+1 contains components of p y and d xy whichit causes allowed transitions such as p y → s , p y → d x − y and p z → d xz . A similar analysis of the optical transition is applicablefor the trilayer phosphorene based on the details given inTable I. I.The optical absorption for the normal incidence ( θ = 0 )of light of a few-layer phosphorene along the armchairand zigzag directions are compared in Fig. 6. It can benoticed that the optical absorption spectra of the bilayerand trilayer are generally similar to that of the monolayerphosphorene. It is also noticeable that there is a generaltrend of an increase in the absorption by increasing thenumber of layers. This result shows that light absorp-tivity can be improved by appropriately increasing thenumber of layers in few-layer phosphorene [30]. In thelow-energy zone, the absorption spectrum is red-shiftedby increasing the number of the layers along the armchairdirection, while it changes slightly with the addition ofphosphorene’s layers along the zigzag direction.This phenomenon originates from the decreasing of theenergy band gap with increasing the number of the layers.The optical absorption edges are found to start approxi-mately from the band gap. Therefore, the optical absorp-tion edge can be tuned by changing of layers in few-layerphosphoren.The optical conductivity of few-layer phosphorene iscalculated by using Eq. 17. It is illustrated in Fig. 7 for q = 0 , the homogeneous electrical field directed in the x direction. It is clear there is no new feature regarding theoptical conductivity in comparison with the absorption.As mentioned in Ref. [20], the unscreened ( Π ) becomesequal to the screened ( Π ) current-current response for q = 0 , and Π is nonzero only for G z = G z (cid:48) = 0 wherethe form factor eventually becomes a constant. Therefore,it can be concluded that the absorption becomes propor-tional to the optical conductivity.In Fig. 8, we compare the optical absorption of bi-layer phosphorene with those of graphene and molybde-num disulfide for η = 30 meV. As shown in the case ofgraphene, the absorption onset starts from eV which isdue to the gapless dipole active π to π ∗ interband tran- sitions near the K point of the Brillouin zone. However,it is nearly zero in the region between − . eV forbilayer phosphorene and − . eV for bilayer MoS . Infact, it shows semimetal nature in bilayer graphene andsemiconductor characteristic of bilayer phosphorene andbilayer MoS .Regarding bilayer graphene, in the infrared region, thespectral absorption per pristine graphene layer is a con-stant, πα = 2 . ( α is the fine-structure constant), ingood agreement with that obtained in a recent experi-ment and theoretical predictions [37, 38]. Obviously thisvalue is valid for perfect graphene flake and dependsstrongly on the damping constant η used in the calcula-tion. In the visible energy, the absorption monotonicallyincreases. The first absorption maximum, which appearsin the ultraviolet region at ω = 4 . eV, is a consequenceof the dipole active interband π to π ∗ transitions alongthe MM’ and M Γ directions of the first Brillouin zone, asdiscussed in details in Ref [21].Bilayer phosphorene exhibits greater absorbance com-pared to the optical absorption of AB bilayer graphene inthe ultraviolet region. The same results pertain to theirmonolayer attribute [39]. Therefore, these results revealthat phosphorene absorb light strongly and it is a promis-ing material to utilize in thin-film solar cells and photo-electric converters [17].The onset of optical absorption in AB bilayer MoS isabout . eV and slowly increasing to a plateau, whichcorresponds to a transition of the valence band to theconduction band around the K point. The intense peakis at 2.7 eV which is red-shifted compared to the intensepeak in the monolayer. It is worth mentioning that themaximal peak of the absorption of bilayer MoS is in thevisible region, while bilayer graphene and phosphoreneare transparent in this region.Finally, we use the formalism that was presented in pre- A b s o r p ti on [ % ] h _ ω (eV) T r a n s m i ss i on [ % ] (a)(b) Figure 5. (Color online) Angle-resolved absorption and trans-mission of bilayer phosphorene with incident angle θ = n ∆ θ for ∆ θ = 10 ◦ and n = 0 , , ..., . The optical absorbance mono-tonically decreases, as the incident angle of light increases,however, the transmission increases. Notice that phosphoreneis transparent when it is illuminated by near grazing incidencelight. vious works [21–23] and calculate the collective modesof few-layer phosphorene. To do so, the Fermi energyis set to be . eV above the edge of the conductionband of each phosphorene structure and our numericalresults are illustrated in Fig. 9. One mode is the opticalplasmon mode, which has its counterpart in single-layersamples with ω ( q → √ q . This mode corresponds to acollective excitation of the electron gas in which the carri-ers of both layers oscillate in-phase. The optical plasmonmodes increase by doping the system. Although both op-tical plasmon modes along the Γ X and Γ Y directionsfollow a low-energy √ q dependence [40], the plasmon inthe armchair direction has a higher density than thatthe zigzag and armchair directions make the anisotropicplasmon mode in both two directions [41]. Also, it can beseen that the optical plasmon mode of ω p (monolayer) ≥ ω p (bilayer) ≥ ω p (trilayer). It is related to the electronconcentration that those systems contain. Basically, with E F = 0 . eV, the electron concentration of monolayerphosphorene is greater than that in the bilayer and thusthe trilayer has the lowest electron concentration at giventhe Fermi energy. In addition to the optical mode, weobserve the existence of an additional mode in the exci-tation spectrum dispersion for bilayer and trilayer phos- A b s o r p ti on [ % ] h _ ω (eV) A b s o r p ti on [ % ] (a)(b) Figure 6. (Color online)Optical absorption spectra of few-layer phosphorene along the (a) armchair and (b) zigzag direc-tions. Light absorptivity improves by increasing the number ofthe layers in few-layer phosphorene. The inset shows the lowenergy part of optical absorption. It shows that the absorptionspectrum is red-shifted with increasing the number of layersalong the armchair direction, while it changes slightly withthe addition of phosphorene layers along the zigzag direction. phorene. In Fig. 9(b), the acoustic modes of bilayer andtrilayer phosphorene along the Γ X and Γ Y directions areshown. These modes correspond to a collective oscillationin which the carrier density in the two layers oscillatesout-of-phase. These modes with a low-energy nearly lin-ear dispersion are damped as they lie on the electron-holecontinuum region. IV. CONCLUSION In this work, we have analyzed the angle-resolved op-tical absorption and transmission of few-layer phospho-rene using the current-current response tensor calculatedin the framework of ab initio DFT calculations and themany-body random phase approximation. The opticaltransition of the valence band to the conduction bandappears along the armchair direction at about 0.72 eV,while it is zero between 0 and 2.5 eV along the zigzag di-rection in bilayer phosphorene. In few-layer phosphorene,it is observed that the optical absorbance monotonicallydecreases, as the incident angle of light increases, andis transparent when it is illuminated by near grazing in- σ ( ω ) σ ( ω ) h _ ω (eV) σ ( ω ) Figure 7. (Color online) The real part of the optical conductiv-ity of few-layer phosphorene, in units of e / (cid:126) , along the arm-chair direction. Notice that since Π ( q = 0 , ω ) = Π( q = 0 , ω ) ,the absorption becomes proportional to the optical conductiv-ity, in phosphorene structures. cidence of light. But the transmission increases, as theincident angle of light increases and it becomes almost for near grazing incidence. It can be noticed thatthe optical absorption spectra increases by increasing thenumber of layers. In the low-energy zone, the absorptionspectrum is red-shifted by increasing the number of lay-ers along the armchair direction, while it changes slightlyalong the zigzag direction. Also, we have compared theoptical absorption of bilayer phosphorene with those ofbilayer graphene and MoS . It is shown that the bilayerphosphorene exhibits greater absorbance than the opticalabsorption of bilayer graphene in the ultraviolet region.The maximal peak in the absorption of bilayer MoS is in the visible region, while bilayer graphene and phos-phorene are transparent in this region. Moreover, theanisotropy of the band structure of few-layer phosphorenealong the armchair and zigzag directions is manifested inthe collective plasmon excitations. Our results provide amicroscopic understanding of the electronic and opticalcharacteristics of few-layer phosphorene. A b s o r p ti on [ % ] h _ ω (eV) BL.GBL.MoS BL.P Figure 8. (Color online) The optical absorption of pristine bi-layer phosphorene, graphene and molybdenum disulfide alongarmchair direction for normal incidence ( θ = 0 ). Here, η = 30 meV and the value of the peak of the optical absorption de-pends on the damping constant η used in the calculation.It would be noticed that the bilayer phosphorene exhibitsgreater absorbance compared to the optical absorption of bi-layer graphene in the ultraviolet region. V. ACKNOWLEDGMENTS We thank H. Akbarzadeh for fruitful discussions andhis help. Z. T. would like to thank the Iran NationalScience Foundation for its support. This work is also sup-ported by the Iran Science Elites Federation. [1] Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim,Nature. , 201 (2005).[2] M. B. Lundeberg, Y. Gao, R. Asgari, C. Tan, B. VanDuppen, M. Autore, P. Alonso-Gonzalez, A. Woessner,K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M.Polini, F. H.L. Koppens Science 187 (2017).[3] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. 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