Exciton in phosphorene: Strain, impurity, thickness and heterostructure
EExciton in phosphorene: Strain, impurity, thickness and heterostructure
Srilatha Arra, ∗ Rohit Babar, ∗ and Mukul Kabir
2, 3, † Department of Chemistry, Indian Institute of Science Education and Research, Pune 411008, India Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India Centre for Energy Science, Indian Institute of Science Education and Research, Pune 411008, India (Dated: October 30, 2018)Reduced electron screening in two-dimension plays a fundamental role in determining excitonproperties, which dictates optoelectronic and photonic device performances. Considering the ex-plicit electron-hole interaction within the GW − Bethe-Salpeter formalism, we first study the exci-tonic properties of pristine phosphorene and investigate the effects of strain and impurity coverage.The calculations reveal strongly bound exciton in these systems with anisotropic spatial delocaliza-tion. Further, we present a simplified hydrogenic model with anisotropic exciton mass and effectiveelectron screening as parameters, and the corresponding results are in excellent agreement with thepresent GW − BSE calculations. The simplified model is then used to investigate exciton renormal-ization in few-layer and heterostructure phosphorene. The changes in carrier effective mass alongwith increasing electron screening renormalizes the exciton binding in these systems. We establishthat the present model, where the parameters are calculated within computationally less expen-sive first-principles calculations, can predict exciton properties with excellent accuracy for largertwo-dimensional systems, where the many-body GW − BSE calculations are impossible.
I. INTRODUCTION
Excitonic properties of two-dimensional (2D) materi-als are markedly different from their bulk counterpartsdue to the fundamental difference in electron screening,and has attracted much attention conceptually in re-cent times. [1–3] Many 2D materials interact stronglywith light, and the concurrently generated electron-holepairs interact strongly due to reduced screening. [4–9]For example, while exciton binding is small, 84 meV, inbulk MoS , [10] due to reduced screening in monolayerMoS , it is measured to be in 220–570 meV range. [6–8] Moreover, unlike the Frenkel excitons, the excitonsin 2D materials can be delocalized in space and extendover 1 nm. [5, 9, 11] Further, the widely varying excitonbinding has been reported in the different class of ma-terials and has important implications in device applica-tions. For applications in solar cells, photodetectors, andcatalytic devices, exciton with weak binding leading toeasy dissociation is desirable. In contrast, the materialswith strong exciton binding are ideal to study plausibleexciton-polariton condensate and for applications such aspolariton lasing. [12]Phosphorene has attracted a lot of attention due toits many plausible technological applications, [13–17] andhas become an interesting proving ground for many-body physics.[18–20] The Dirac semimetal state has beenexperimentally realized in few-layer phosphorene underadatom absorption. [19] We have recently reported anintrinsic, robust and high-temperature Kondo state in de-fected phosphorene doped with a transition-metal impu-rity. [20] In the present context, phosphorene has unique ∗ These authors contributed equally † Corresponding author: [email protected] optical properties that are mainly determined by thequasiparticle band structure and screening. Originatingfrom a puckered honeycomb network, the electronic bandstructure is highly asymmetric in phosphorene and conse-quently, the optical properties are also found to be highlyanisotropic. [1, 14, 15, 21] The quasiparticle and opticalgaps of single-layer phosphorene (SLP) are experimen-tally measured to be 2.2 ± ± ± GW method produces the correct quasiparticle energies. [23]In this approach, the electron self-energies are expressedin terms of Green’s function G and screened Coulombinteraction W . Further, the optical properties of semi-conductors and insulators are strongly affected by inter-acting electron-hole pairs, which is described through theBethe-Salpeter equation (BSE). [24–26] These theoreti-cal descriptions lead to an excellent agreement with theexperimental results. [22]Here, we study the quasiparticle and optical proper-ties of SLP and its derivatives within the GW and BSEformalisms, and also investigate the effect of strain. Re-sults on the pristine SLP are in excellent agreement withthose obtained from the transmission and photolumi-nescence spectroscopies. [21, 27] The optical absorptionand the corresponding excitons are found to be highlyanisotropic. Due to the 2D confinement of photogener-ated electron-hole pairs, the exciton binding is exception-ally strong, which is in accordance with the experimental a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t measurements. [21]While a rigorous treatment of electron-hole interactionwithin the BSE formalism provides an excellent descrip-tion of exciton binding, [9, 11, 22, 28, 29] it is computa-tionally very expensive and thus restricted to the systemswith small size. In this context, we describe a hydrogeniceffective exciton mass model in which the parameters, theeffective carrier masses and the static dielectric constant,are calculated within the conventional density functionaltheory (DFT) calculations. [30, 31] The results of thissimplified model are in excellent agreement with thosecalculated within the BSE formalism for the pristine andstrained SLP along with the SLPs with 1 ML impuritycoverage. Further, the anisotropic hydrogenic model isextended for larger systems with low impurity coveragesthat are earlier predicted to be good candidate materialsfor water redox reactions. [17]The quantum confinement of exciton should be ex-traordinarily affected by the varied thickness of the 2Dmaterial, which alters electron screening. This pictureis well captured within the present model through therenormalization of exciton binding in few-layer phospho-rene. Furthermore, the practical applications of phos-phorene are limited due to its fast degradation in ambi-ent conditions resulting in a severe alteration in the cor-responding electronic properties. [32–34] Thus to avoiddegradation, phosphorene is encapsulated with a cappinglayer and substrate, and the devices restore the intrinsiccarrier mobility of phosphorene. [13, 21, 32, 35–37] Inthis regard, we also investigate the electronic and exci-ton properties in SLP encapsulated with atomically thinhexagonal boron-nitride (h-BN), which is often used toprotect phosphorene from degradation. [35–37] II. METHODOLOGY
The structural optimizations were carried out withinthe conventional DFT formalism [38, 39] and the elec-trons are treated within the projector augmented wavemethod. [40] The Kohn-Sham orbitals were expandedin plane-wave basis with 400 eV energy cutoff. Theexchange-correlation energy was described with Perdew-Burke-Ernzerhof (PBE) functional. [41] The Brillouin-zone was sampled using Γ-centred 17 × × k -grid. [42] Complete structural optimization wascarried until the forces exerted on each atom are lessthan 0.01 eV/˚A threshold. In phosphorene, three elec-trons participate in the covalent σ -bonding with threeneighboring P atoms, whereas the remaining two elec-trons occupy a lone pair orbital. Thus, for phosphoreneand its derivatives such as P X (X=O and S), we con-sidered van der Waals interaction through the non-localcorrelation functional optB88-vdW during the structuraloptimization, [43, 44] while for the larger systems in few-layer and heterostructure phosphorene we used the D3functional with zero damping. [45] The obtained latticeparameters for SLP are a = 4.58 and b = 3.32 ˚A along the armchair and zigzag directions, respectively, are con-sistent with the previous reports and experimental blackphosphorus (Supplemental Material). [27, 32, 46–49]The optimized structures with the PBE exchange-correlation functional are then used for the subsequent GW -BSE calculations. The quasiparticle (QP) pictureis investigated within the partially self-consistent GW approach by iterating the one-electron energies in theGreen’s function G . [23, 50] Two self-consistent updatesfor the Green’s function ( G W ) are found to be suffi-cient to converge the QP band gap. The convergenceof QP gap as a function of unoccupied bands was foundto be much faster for phosphorene, [1] and we find that158 such bands to be sufficient in this regard. Further,the electron-hole interactions are incorporated within theBSE formalism, which provides the optical gap and thecorresponding exciton binding energy. [22, 25, 26]The optical properties are calculated using frequencydependent complex dielectric tensor, ε ( ω ) = ε (cid:48) ( ω ) + iε (cid:48)(cid:48) ( ω ). The imaginary part ε (cid:48)(cid:48) ( ω ) of the linear dielec-tric tensor is calculated in the long-wavelength q → limit, [51] ε (cid:48)(cid:48) αβ ( ω ) = 4 π e Ω lim q → q (cid:88) c,v, k w k δ ( (cid:15) c k − (cid:15) v k − ω ) × (cid:104) u c k + e α q | u v k (cid:105)(cid:104) u c k + e β q | u v k (cid:105) ∗ , (1)where Ω is the volume of the primitive cell, ω k are k -point weights, and the factor 2 inside the summationaccounts for the spin degeneracy. The (cid:15) c k ( (cid:15) v k ) are k -dependent conduction (valence) band energies, u c k ,v k arecell periodic part of the pseudo-wave-function, and e α,β are unit vectors along the Cartesian directions. The realpart ε (cid:48) ( ω ) is calculated using Kramers-Kronig transfor-mation, and the absorption co-efficient is calculated asΛ αα ( ω ) = ωc [ | ε αα ( ω ) | − ε (cid:48) αα ( ω )] .Within the BSE formalism, an exciton state | S (cid:105) canbe written as, [26] | S (cid:105) = (cid:88) k hole (cid:88) v elec (cid:88) c A Svc k | vc k (cid:105) ; (2)where | vc k (cid:105) = ˆ a † v k ˆ b † c k + q | (cid:105) with | (cid:105) being the groundstate, and ˆ a † (ˆ b † ) is hole (electron) creation operator. q is the momentum of the absorbed photon, and A Svc k areelectron-hole amplitudes. The corresponding excitationenergies E S are determined via BSE, [52] (cid:16) (cid:15) QP c k + q − (cid:15) QP v k (cid:17) A Svc k + (cid:88) v (cid:48) c (cid:48) k (cid:48) A Sv (cid:48) c (cid:48) k (cid:48) (cid:104) vc k | K eh | v (cid:48) c (cid:48) k (cid:48) (cid:105) = E S A Svc k , (3)where (cid:15) QP are quasiparticle energies, and K eh is theelectron-hole interaction. The imaginary part of the di-electric function ε (cid:48)(cid:48) ( ω ) is calculated by the optical tran-sition matrix element of the excitations.Such a rigorous treatment within the many-body the-ory coupled with BSE scheme provides an excellent de-scription of QP and optical gaps; and exciton bind-ing, which all compare well with the experimental re-sults. [9, 11, 22, 28, 29] However, such treatment is re-stricted to the systems with small size due to its excep-tional computational cost. Thus, one needs to developsimplified models to investigate excitons in realistic sys-tems with appreciable accuracy. For three-dimensionalmaterials, the simplistic Mott-Wannier model predictsthe exciton binding energy as, E = ( µ/ε ) R ∞ , where R ∞ is the Rydberg constant. The excitonic effective mass µ and the static dielectric constant ε can easily be cal-culated within the standard electronic structure calcula-tions. [53]In contrast, the excitonic properties of two-dimensionalmaterials are fundamentally different from their 3Dcounterpart, and cannot be described within the Mott-Wannier approach. In 2D materials, excitons are stronglyconfined and the dielectric screening is considerably re-duced. [4–9] There have been recent efforts to develop ex-citonic models for 2D materials, however, the significanteffort has been devoted to the isotropic materials suchas transition metal dichalcogenides. [3, 30, 31, 54, 55]Here, we present a generalized scheme appropriate forthe anisotropic electronic materials such as phosphoreneand its various derivatives. [30, 31]For 2D semiconducting systems, the effective excitonHamiltonian can be written as, H x = − (cid:126) ∇ r µ + V ( r ) , (4)where µ − = m − e + m − h is the exciton reduced mass, and r is the electron-hole separation. Following Keldysh, thenon-locally screened electron-hole interaction is describedby, [56] V ( r ) = − e ε + ε ) ε r (cid:20) H (cid:18) rr (cid:19) − Y (cid:18) rr (cid:19)(cid:21) , (5)where ε and ε are the dielectric constant of the up-per and lower media; and ε is the vacuum permittiv-ity. H and Y are Struve and Bessel functions. Thescreening length r is related to the 2D polarizability χ as r = 2 πχ , [57] where χ is calculated usingthe static dielectric constant ε of the concerned 2D ma-terial, ε ( L v ) = 1 + 4 πχ /L v , where L v is the transversevaccuum size. The ε is calculated from the real part ofcomplex dielectric tensor ε ( ω ) at zero frequency. Notethat the interaction V at large separation r >> r fol-low the 1 /r Coulomb interaction, whereas at the r << r limit, the interaction reduces to a weaker log( r ) depen-dence.The variational excitonic wave function, for ananisotropic electronic material such as phosphorene with m xe (cid:54) = m ye and m xh (cid:54) = m yh , is written as, ψ ( x, y ) = 2 (cid:115) πλa x exp (cid:104) − (cid:8) ( x/a x ) + ( y/a y ) (cid:9) (cid:105) , (6) where a y = λa x is the anisotropic exciton extension alongthe x (armchair) and y (zigzag) directions, respectively,and treated as variational parameters. Using this formof excitonic wave function ψ ( x, y ), the expectation valueof the kinetic energy is calculated to be, E k ( λ, a x ) = (cid:126) a x (cid:20) µ x + 1 λ µ y (cid:21) . (7)Here µ x and µ y are the reduced exciton mass along the x and y directions, respectively. The corresponding po-tential energy is given by, [30] E p ( λ, a x ) = (cid:90) (cid:90) V ( x, y ) | ψ ( x, y ) | dxdy. (8)The variational exciton binding energy, E ( λ, a x ) = E k ( λ, a x ) + E p ( λ, a x ), is minimized with respect to thevariational parameters λ and a x to obtain the excitonbinding energy, and (anisotropic) exciton extension. Theparameters in the above model, effective electron andhole masses in different crystallographic directions andthe static dielectric constant are then calculated from thefirst-principles calculations. In principle, these parame-ters can be calculated using any level of approximationto the exchange-correlation functional, and here we haveused the Heyd-Scuseria-Ernzerhof (HSE06) hybrid func-tional. [58, 59] III. RESULTS AND DISCUSSION
First, we discuss the optical and excitonic properties ofpristine phosphorene and investigate the effect of uniax-ial strain along the different crystallographic directions.Here, we compare the computationally expensive GW -BSE results for these systems with the simplified modelcalculations. Once we demonstrate an excellent agree-ment between these methods, we extend our investiga-tion within the simplified model for the realistically largesystems, where GW -BSE calculations are practically im-possible. A. Pristine phosphorene
We start with the electronic and optical properties ofsingle-layer pristine phosphorene [Figure 1(a)] within the GW -BSE approach. Due to the long-range Coulomb in-teractions, these properties of 2D materials are stronglyinfluenced by the vertical separation L z between the pe-riodic images. [1, 60] As it was shown earlier that the QPband gap converges as 1 /L z , we extrapolate the gap to L z → ∞ limit, which we found to be 2.14 eV [Figure 1(b)and Table I]. This extrapolated E QP g is in excellent agree-ment with the previous theoretical, [1] and with the twoexperimental results that are available to date. [21, 27]The high-resolution transmission spectroscopy predicted FIG. 1. (a) The top and side view of single-layer phosphoreneis shown indicating the armchair and zigzag crystallographicdirections. (b) The QP and optical gaps are calculated withvaried vertical separation L z between the layers and extrapo-lated to L z → ∞ . The results are in excellent agreement withthe experimental results. [14, 21, 27] Absorption coefficientcalculated without and with the electron-hole interaction for L z =30 ˚A. Λ( ω ) shows strong absorption anisotropy betweenthe light polarization along the (c) armchair and (d) zigzagdirections. The vertical lines indicate the band edges corre-sponding to the first absorption peak. The difference betweenthe QP and optical gaps indicates strongly bound exciton. a transport gap of 2.05 eV, [27] while the photolumi-nescence excitation spectroscopy suggests a QP gap of2.2 ± G W resultswith earlier G W calculations, [61, 62] we argue that theself-consistent correction to the self-energy (Σ = iGW ),through the Green’s function update, is essential to pre-dict the QP gap correctly. It is important to note herethat the optical gap converges much faster than the QPgap with varying L z [Figure 1(b)].The absorption coefficient Λ( ω ) calculated withoutand with the electron-hole interaction shows stronganisotropy for the light polarizations along the armchairand zigzag directions [Figures 1(c)-(d)]. The L z → ∞ extrapolated optical gap of 1.40 eV [Figure 1(b) andTable I] along the armchair direction is in excellentagreement with the experimental range of 1.30 – 1.45eV. [14, 21] The large exciton binding energy of 0.74 eV indicates strongly bound exciton in SLP, which is in ex-cellent agreement with the previous GW -BSE calcula-tions [1, 62] and the experimental prediction of 0.9 ± E ( k ) = (cid:126) k / m ∗ . The estimated m ∗ e is muchsmaller 0.16 m e along the armchair direction than thesame along the zigzag direction with 1.40 m e , where m e is the electron rest mass. The qualitative picture is thesame for the hole, as m ∗ h along the armchair direction ismuch lighter 0.12 m e compared to 4.69 m e along the zigzagdirection. Thus, the effective exciton mass along thearmchair direction is much lighter compared to the samealong the zigzag direction ( µ x << µ y ). These results arein good agreement with the previous results. [46, 63] Theother parameter of the model, static dielectric constantis calculated from the real part of the complex dielec-tric function at zero frequency. The average of static ε along the armchair and zigzag directions is used to cal-culate the 2D polarizability χ (Table I). Using theseparameters, the effective exciton mass model predicts anexciton binding energy of 0.79 eV, which is in excellentagreement with the present GW -BSE prediction and theexperimental estimation (Table I).The spatial distribution of exciton state is found to beanisotropic and extended along the armchair direction( a x =12.26 and a y =4.90 ˚A), with spatial anisotropy sat-isfying the relation, λ = a y /a x ∼ ( µ x /µ y ) / , that wasanalytically predicted earlier. [30] In an agreement, suchelliptic spatial structure of bound hole has recently beenobserved in black phosphorus through STM tomographicimaging. [64] B. Effect of uniaxial strain
The effect of uniaxial strain on the electronic structureof SLP has been studied earlier within the conventionalexchange-correlation functional. [20, 63, 65, 66] However,the considerations of self-energy correction and electron-hole interaction are scarce in this regard. Here, we inves-tigate the QP and optical gaps; and the corresponding ex-citon binding for SLP under ε a/zs = ±
5% uniaxial strain(Table I), while the strained lattice is relaxed along thetransverse direction. We earlier reported that the strainenergy along the zigzag direction is much higher com-pared to the same along the armchair direction and thus,straining the SLP along the armchair direction is com-paratively easier. [20] Within − (cid:54) ε a/zs (cid:54) +5% strain TABLE I. With varied L z , the quasiparticle E QP g and optical E opt g gaps are calculated within the G W and G W -BSEapproaches, respectively, which are subsequently interpolated to L z → ∞ . The parameters for the simplified exciton model,the effective carrier masses m ∗ e and m ∗ h along the armchair and zigzag directions; and the average static dielectric constantare calculated within the HSE06 hybrid functional. The corresponding 2D polarizability χ is also tabulated. The uniaxialstrain severely affects the many-body interaction in phosphorene, and thus the E QP g , E opt g , and E x are altered. The effectof high impurity coverage is also investigated. The exciton binding energies E x calculated within the hydrogenic excitonmodel compares excellently with those from the accurate and computationally expensive GW -BSE formalism, and availableexperimental results.System E QP g E opt g E x (eV) armchair ( x ) zigzag ( y ) ε χ (eV) (eV) BSE model m ∗ e /m e m ∗ h /m e m ∗ e /m e m ∗ h /m e (˚A)SLP 2.05 [27] 1.45 [14](experiment) 2.2 ± ± L z → ∞ ) 2.14 1.40 0.74 0.79 0.16 0.12 1.40 4.69 2.60 3.55SLP, ε as = −
5% 1.78 1.03 0.75 0.72 0.14 0.12 1.35 3.10 2.78 3.95SLP, ε as =+5% 2.38 1.51 0.87 0.80 0.13 0.12 1.45 3.28 2.51 3.36SLP, ε zs = −
5% 1.89 1.12 0.77 0.78 0.23 0.13 1.39 2.64 2.70 3.77SLP, ε zs =+5% 2.20 1.41 0.79 0.78 0.20 0.14 1.39 2.76 2.71 3.79P O 3.22 2.31 0.91 0.81 0.76 0.24 0.17 3.42 2.48 3.17P S 2.74 1.87 0.87 0.67 0.63 0.24 0.80 0.40 3.19 4.63 range, the L z → ∞ interpolated gaps E QP g and E opt g decrease with uniaxial compressive strain, while they in-crease with tensile strain (Table I). Clearly, the many-body interaction is comparatively more affected by theuniaxial strain along the armchair direction. Further,the compressive strain triggers a stronger renormaliza-tion than the tensile strain of equal magnitude. Theabsorption anisotropy in different crystallographic direc-tions remains intact [Figure 2(a)]. Within the appliedstrain range, the first absorption peak appears within1 − ±
10 % for the strain ε as = ±
5% along the armchair direction, the change iseven more significant, ∓
30% for applied strain along thezigzag direction, ε zs = ± ε zs =5%,we observe a competing absorption peak just below itsQP gap. Such strain dependent E QP g and E opt g may ex-plain the variation observed in photoluminescence peakon different substrates. [14, 21]The excitons remain strongly bound under appliedstrain, while the G W -BSE calculated E x varies be-tween 700–900 meV within the investigated strain range.The subsequent excited states reflect a sensitive depen-dence on the strain dependent dielectric screening [Fig-ure 2(b)]. Thus, strain engineering in phosphorene leadsto wide photoluminescence energy, enhanced absorption,and multiple exciton formation.Next, we investigate the effective carrier masses witha varied uniaxial strain within the HSE06 functional. Inagreement with an earlier report, [63] the effective elec- tron and hole masses are severely affected by strain (Ta-ble I), however their qualitative anisotropic nature re-mains intact with µ x << µ y . These m ∗ e and m ∗ h are usedto estimate E x for the strained SLP within the hydro-genic model, which are in excellent agreement with themore accurate GW -BSE results (Table I). Further, thespatial anisotropy of exciton λ ∼ ( µ x /µ y ) / remain sim-ilar under strain as in pristine SLP (Supplemental Mate-rial). [49] These results essentially validate the applica-bility of the present hydrogenic model for the anisotropic2D phosphorene. C. Effect of impurity coverage
The presence of lone-pair electrons in phosphorenemakes it reactive and can easily absorb impurities witha strong binding energy resulting from the P to impuritychange transfer. We have earlier discussed O, S and Nchemisorption with varied impurity coverage and it wasconcluded that 0.25–0.5 monolayer (ML) O/S coveragesbecome conducive to both water redox reactions. [17]While the exciton binding energy plays an important rolein efficient charge separation and in turn affects the per-formance of a catalytic device, the SLP derivatives withsuch low impurity coverages are very difficult to investi-gate within the GW -BSE formalism. Thus at first, weinvestigate the derivatives with high 1 ML O/S cover-ages, P O and P S that can be represented by a smallcell, and compare the GW -BSE results with the effectivemass model.The SLPs with 1 ML O and S coverage are found tobe indirect gap semiconductors, and both QP and opticalgaps are severely altered (Table I). While for the 1 MLO-covered SLP the extrapolated E QP g and E opt g increase FIG. 2. (a) The effects of strain on the absorbance and dipole oscillator strength calculated including the electron-holeinteraction within the GW -BSE formalism ( L z = 30 ˚A). The vertical lines indicate the relative dipole oscillator strengths andare significantly affected by the applied uniaxial strain. (b) The corresponding exciton spectra for the pristine and uniaxiallystrained SLP. Strain-dependent dielectric screening influences the spectra. The E = 0 eV refers to the quasiparticle gap,and excitons generated from the transitions with non-vanishing dipole oscillator strengths are shown. The relative oscillatorstrengths are shown in (a). Absorption coefficient calculated without and with the electron-hole interaction ( L z = 30 ˚A) forthe light polarization along (c) armchair and (d) zigzag directions, shown for 1 ML O coverage. The absorption edges arecalculated from the corresponding E vs ( E Λ) / plot for this indirect gap semiconductor. The absorption anisotropy along thearmchair and zigzag directions is greatly reduced due to monolayer impurity coverage. The insets show the side and top viewsof SLP with 1ML O coverage. upon impurity coverage, the picture is reversed for the 1ML S-covered SLP. The absorption edge, calculated byconsidering the electron-hole interaction, lies at a muchlower energy than the same calculated without the in-teraction (Figure 2). This observation indicates stronglybound excitons in these derivatives similar to the pris-tine SLP (Table I). The qualitative anisotropic featurein the carrier effective masses are intact but are severelyaltered in all crystallographic directions. [49] However,the anisotropy in effective exciton mass disappears with λ ∼
1, and thus the corresponding exciton extension be-comes isotropic with 1 ML coverage (Supplemental Mate-rial). [49] Furthermore, the E x calculated within the GW -BSE formalism are in excellent agreement with thosecalculated using the hydrogenic model (Table I), whichimplies the applicability of this simplistic model to in-vestigate exciton in such phosphorene derivatives withimpurity coverage.The SLPs with sub-monolayer coverages are both ther-modynamically and kinetically stable. Further, the bandedges align with the redox potentials for water splittingreactions for SLPs with 0.33 − E x within thehydrogenic model is found to be 0.81 and 0.85 eV (0.85and 0.86 eV) for 0.33 and 0.5ML oxygen (sulfur) cover-ages. Such high exciton binding makes charge separationdifficult in optoelectronic and catalytic devices. Thus, al- though the conduction and valence bands in these deriva-tives align with the redox potentials, the catalytic activ-ity is expected to be negatively impacted due to highexciton binding. On the other hand, the high absorp-tion coefficient [Figure 2] and robust exciton with highbinding energy are desirable features for light emittingdevice applications. In this regard, the impurity cov-ered phosphorene can extend such application to green-light emission.[67, 68] The anisotropy in exciton exten-sion is in agreement with the analytical estimation of λ ∼ ( µ x /µ y ) / , which monotonically decreases with in-creasing coverage. [49] D. Exciton renormalization in few-layer andheterostructure phosphorene
While the layer-dependent evolution of gaps has beenstudied in few-layer phosphorene, [46, 62, 69, 70] it is in-triguing to investigate the layer-dependent exciton bind-ing. The effective screening increases with layer thick-ness and in addition, the carrier masses along differentcrystallographic directions are also expected to be modi-fied. The evolution in lattice parameters with increasinglayer thickness agrees well with the previous predictionand converges to the bulk values (Supplemental Mate-rial). [46, 49] Further, the bandgap in few-layer phos-phorene decreases with thickness and converges to thevalue for black phosphorus [Figure 3(b)]. The calculatedbulk gap of 0.26 eV agrees reasonably well with the ex-
FIG. 3. (a) Calculated χ = ( ε − L v / π increases withlayer thickness N (cid:96) indicating an increase in electron screening.(b) The renormalization of bandgap E g and exciton binding E x in few-layer phosphorene show a power-law dependencewith the layer thickness and both quantities converge veryslowly to the corresponding bulk value. The variation in ex-citon binding is largely determined by the effective hole mass m ∗ h along the zigzag direction, in addition to the change ineffective electron screening. perimental measurement of 0.33 eV. [71] Considering thefact that the gaps are underestimated within the HSE06functional, it is imperative to investigate its qualitativedependence on thickness. A power-law fit E g = aN − α(cid:96) + c ,with N (cid:96) being the number of layers, indicates that E g de-cays much slower as α = 0 .
83 than the usual quantumconfinement with α = 2. While α < α in few-layer phosphorene is much smallercompared to MoS , where α = 1 .
10. [72]The effective electron screening increases with thick-ness N (cid:96) , and the static dielectric constant ε in few-layerphosphorene increases with the number of layers. Theanisotropic ε in black phosphorus ( ε x = 14.75, ε y = 10.87, ε z = 8.68) are consistent with the experimental measure-ment of 16.5, 13 and 8.3 along the armchair, zigzag andperpendicular directions, respectively. [73] The depen-dence of χ , calculated using the average of anisotropicstatic ε , indicates increasing screening with thickness[Figure 3(a)].Effective carrier masses are mostly unaffected in few-layer phosphorene except for the hole mass along thezigzag direction, which exhibits a strong layer depen-dence (Supplemental Material). [49] The calculated m ∗ h along this direction decreases sharply with thickness N (cid:96) ,which is in agreement with a previous prediction. [46]To confirm the evolution of effective carrier masses with N (cid:96) , we calculated the same for the bulk black phospho-rus, which compares well with the experimental values(Supplemental Material). [49, 74]Consequently, the renormalization of exciton bindingin few-layer phosphorene [Figure 3(a)] is dictated by thestrong dependence of effective hole mass along the zigzagdirection and the increase in effective screening with layerthickness. This result is in contrast to the assumptionthat the exciton binding and thus its variation is inde- FIG. 4. (a)-(b) The top and side view of the h-BN/SLP/h-BNvan der Waals encapsule. The structure is optimized using thePBE+D3-vdW functional with zero damping. The encapsu-lation induces a compressive (tensile) strain of 5.2% (1.3%) inthe phosphorene layer along the armchair (zigzag) direction.(c) The valence band maxima and conduction band minimaoriginate from the SLP and thus indicate a type I band align-ment. While the apparent band structure looks similar to thepristine SLP, the corresponding carrier effective masses areseverely altered, which in turn modifies the exciton charac-ter. pendent of carrier effective mass. [3, 75] The large excitonbinding sharply decreases from 0.72 eV in 1L to 0.48 eVin 2L and 0.31 eV in 5L phosphorene, which is still muchhigher than the corresponding bulk value. We estimatedthe Mott-Wannier exciton binding in black phosphorus E = ( µ/ε ) R ∞ to be about 11 meV, which is muchsmaller than k B T at room temperature, and also muchsmaller than the same in bulk transition-metal dichalco-genides. [10] A similar power-law fit with α = 0 .
53 in-dicates a much slower dependence of E x on thicknessthan for E g . Further, the exciton extension in few-layer phosphorene remains anisotropic while the degreeof anisotropy decreases monotonically with thickness as λ increases (Supplemental Material). [49]Black phosphorous is often encapsulated to avoiddegradation and restore the intrinsic electronic proper-ties. [13, 21, 32] The hexagonal boron nitride (h-BN)was demonstrated to be a good candidate material forthis purpose. [35–37] Thus, it would be worth investi-gating that how the electronic structure and effective di-electric screening are affected due to the substrate andcapping. In this regard, we investigate the h-BN/SLPand h-BN/SLP/h-BN heterostructures [Figure 4(a) and(b)]. To minimize the lattice strain in the phosphorenelayer, we used a 1 × × E x is determined by the effec-tive mass and 2D polarizability. Moreover, the spatialanisotropic structure is directly related to the carrier ef-fective masses, λ ∼ ( µ x /µ y ) / . Thus, in addition tothe dielectric environment, any change in the intrinsicelectronic structure in phosphorene due to the substrateand capping layer is important to consider for a correctdescription of the exciton. Indeed, the effective carriermass in the phosphorene layer is affected by h-BN, [49]which along with the electron screening from the h-BNlayer reduces the exciton binding. The exciton binding isrenormalized to 0.55 eV for the SLP/h-BN heterostruc-ture, while the presence of a second h-BN layer in theh-BN/SLP/h-BN encapsulation does not alter the exci-ton binding further (0.53 eV). The excitons in these het-erostructures are generated in the phosphorene layer, andno interlayer exciton is possible [Figure 4(c)]. Further,the spatial anisotropy of excitons in these heterostruc-tures is reduced. [49] Similar renormalization of excitonbinding was predicted earlier for Al O /phosphorene/h-BN encapsulation, and phosphorene on SiO or PDMSsubstrates. [75, 78, 79] IV. SUMMARY
We have studied the quasiparticle and optical proper-ties of phosphorene and investigated the role of uniax-ial strain, and impurity coverages. The excitonic prop-erties described within a tractable anisotropic hydro-genic model exhibit excellent agreements with those cal-culated by explicitly considering the electron-hole inter-action within the GW -BSE formalism. In contrast tothe previous assumption, the exciton binding stronglydepends on effective carrier masses, which further deter-mines the anisotropic spatial extension of excitons. Sim-ilar to the pristine SLP, exciton in strained and impuritycovered phosphorene remain strongly bound. However,due to the change in the corresponding carrier masses,the anisotropy in spatial extension is severely altered.Owing to a severe alteration in the effective hole massalong the zigzag direction and increase in the electronscreening, the exciton binding is greatly renormalized infew-layer phosphorene. In contrast, the exciton bind-ing is relatively less affected in the h-BN and phospho-rene heterostructures. The robust large exciton bindingenergy and tunable photoluminescence in encapsulatedand impurity covered phosphorene derivatives raise theirprospective applications in light-emitting devices. Theresults indicate that the present model will be applicableto other phosphorene based superstructures and othertwo-dimensional anisotropic materials. ACKNOWLEDGMENTS
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2, 3, † Department of Chemistry, Indian Institute of Science Education and Research, Pune 411008, India Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India Centre for Energy Science, Indian Institute of Science Education and Research, Pune 411008, India (Dated: October 29, 2018)[1] A. Brown and S. Rundqvist, Acta Cryst. , 684 (1965).[2] L. Cartz, S. R. Srinivasa, R. J. Riedner, J. D. Jorgensen,and T. G. Worlton, J. Chem. Phys. , 1718 (1979).[3] Y. Akahama, S. Endo, and S.-i. Narita, J. Phys. Soc. Jpn. , 2148 (1983).[4] J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, Nat. Commun. , 4475 (2014).[5] A. Morita, H. Asahina, C. Kaneta, and T. Sasaki, in Proc.17th Int. Conf. Phys. Semicond. , edited by J. D. Chadi andW. A. Harrison (1985) pp. 1320–1324. ∗ These authors contributed equally † Corresponding author: [email protected] TABLE S1. Evolution of lattice parameters with layer thickness in few-layer phosphorene. Results are in good agreement withthe experimental lattice parameters of bulk black phosphorus (BP). Lattice parameters slightly differ within different treatmentof van der Waals interaction. For the vdW-D3 correction we used the zero damping method. The optimized structures withoptB88 and D3 vdW corrections have been used to calculate the HSE06 bandstructure.Lattice parameter (˚A) HSE06 gap (eV) a (armchair) b (zigzag)1L vdW-optB88 4.58 3.32 1.54vdW-D3 4.59 3.30 1.532L vdW-D3 4.51 3.30 1.033L vdW-D3 4.49 3.30 0.784L vdW-D3 4.47 3.30 0.645L vdW-D3 4.45 3.30 0.51Bulk BP vdW-D3 a =4.42, b =3.31, c =10.67 0.26vdW-optB88 a =4.48, b =3.34, c =10.72 0.26Experiment [1, 2] a =4.37, b =3.31, c =10.47 0.33 [3]TABLE S2. The effect of uniaxial strain along the armchair and zigzag directions in pristine SLP. The bandgap decreases(increases) with compressive (tensile) strain in the studied strain range of ε a/z λ is in excellent agreement with the analytical prediction of ( µ x /µ y ) / , and largely remains unaltered under strain.System HSE06 Gap(eV) λ ( µ x /µ y ) / a x (˚A) a y (˚A)SLP 1.54 0.40 0.40 12.26 4.90SLP, ε a = −
5% 1.30 0.41 0.41 13.20 5.40SLP, ε a =+5% 1.73 0.39 0.39 12.59 4.91SLP, ε z = −
5% 1.39 0.45 0.45 11.62 5.19SLP, ε z =+5% 1.65 0.44 0.44 11.65 5.17TABLE S3. Effective carrier masses for SLP with O/S impurity coverages and χ calculated within HSE06 functional. Theseparameters are used with the hydrogenic model to calculate the exciton binding energy E x and the corresponding excitonextension anisotropy λ .System armchair zigzag ε (avg) χ (˚A) E x (eV) λm ∗ e /m e m ∗ h /m e m ∗ e /m e m ∗ h /m e χ for few-layer phosphorene calculated within HSE06 functional. These parametersare used with the hydrogenic model to calculate the exciton binding energy E x and the corresponding exciton extensionanisotropy λ . Exciton binding for the bulk BP is calculated within the Mott-Wannier model.System vdW armchair zigzag ε (avg) χ (˚A) E x (eV) λm ∗ e /m e m ∗ h /m e m ∗ e /m e m ∗ h /m e
1L optB88 0.16 0.12 1.40 4.69 2.60 3.55 0.793 0.401L D3 0.12 0.10 1.38 5.34 2.69 3.75 0.723 0.372L D3 0.12 0.11 1.45 1.95 4.59 6.45 0.478 0.423L D3 0.12 0.11 1.43 1.76 5.98 8.00 0.402 0.464L D3 0.13 0.12 1.42 1.25 6.86 9.27 0.359 0.495L D3 0.13 0.12 1.41 1.03 7.51 10.71 0.313 0.57BP D3 0.12 0.10 1.39 0.82 11.43 − − TABLE S5. Calculated carrier effective masses calculated for bulk BP within the HSE06 hybrid functional. Present resultscompare well with an earlier theoretical calculation and experimental estimation. a , armchair b , zigzag c , perpendicular m ∗ e /m e m ∗ h /m e m ∗ e /m e m ∗ h /m e m ∗ e /m e m ∗ h /m e Bulk BP Present 0.12 0.10 1.39 0.82 0.17 0.30Ref. [4] 0.12 0.11 1.15 0.71 0.15 0.30Experiment [5] 0.082 0.078 1.027 0.648 0.128 0.28TABLE S6. Lattice parameter and the corresponding HSE06 gap for the h-BN/SLP heterostructure and the h-BN/SLP/h-BNencapsulate. The gap is mostly unaffected, while the slight change in gap is attributed to the strain induced in the SLP layer inthe heterostructures. Note that, while the SLP layer in both the heterostructures have similar lattice parameters, they producedifferent bandgaps. This is largely due to the change in the distance between the two-half layers of SLP (Figure S1).System Lattice parameters (A) Strain in phosphorene layer Bandgap (eV) a (armchair) b (zigzag) ε a ε z Pristine SLP 4.59 9.90 − − − − − − − − B N P (a) (b)
FIG. S1. (a) SLP/h-BN and (b) h-BN/SLP/h-BN heterostructures. The vertical separation between the layers changes slightlyin these heterostructure. While the strain induced in the SLP layer for both these heterostructures are very similar, the distancebetween the two half-layers of phosphorene differs substantially by about 0.06 ˚A, which in turn is responsible for very differentbadgap.TABLE S7. Carrier effective masses, χ , exciton binding E x and spatial anisotropy λ for phoshorene heterostructures withh-BN. Heterostructure a , armchair b , zigzag χ (˚A) E x (eV) λm ∗ e /m e m ∗ h /m e m ∗ e /m e m ∗ h /m ee