On the Optical Properties of Excitons in Buckled 2D Materials in an External Electric Field
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l On the Optical Properties of Excitons in Buckled 2D Materials in an ExternalElectric Field
Matthew N. Brunetti , , Oleg L. Berman , , and Roman Ya. Kezerashvili , Physics Department, New York City College of TechnologyThe City University of New York,300 Jay Street, Brooklyn NY, 11201, USA The Graduate School and University CenterThe City University of New York,New York, NY 10016, USA (Dated: July 3, 2018)We study the binding energies and optical properties of direct and indirect excitons in monolayersand double layer heterostructures of Xenes: silicene, germanene, and stanene. It is demonstratedthat an external electric field can be used to tune the eigenenergies and optical properties of excitonsby changing the effective mass of charge carriers. The Schr¨odinger equation with field-dependentexciton reduced mass is solved by using the Rytova-Keldysh (RK) potential for direct excitons,while both the RK and Coulomb potentials are used for indirect excitons. It is shown that forindirect excitons, the choice of interaction potential can cause huge differences in the eigenenergiesat large electric fields and significant differences even at small electric fields. Furthermore, ourcalculations show that the choice of material parameters has a significant effect on the bindingenergies and optical properties of direct and indirect excitons. These calculations contribute to therapidly growing body of research regarding the excitonic and optical properties of this new class oftwo dimensional semiconductors.
I. INTRODUCTION
Following the discovery of stable graphene monolayers in 2004 and the subsequent isolation of two-dimensional (2D)insulators such as hexagonal boron nitride ( h -BN) and 2D semiconductors such as transition metal dichalcogenides(TMDCs) , researchers have continually sought to discover new 2D materials with novel properties. One recentaddition to the 2D universe are the group XIV elements, namely silicon, germanium, and tin, whose 2D forms arereferred to as silicene, germanene, and stanene (or sometimes, tinene), respectively. A recent paper referred tobuckled 2D monolayers consisting of silicon, germanium, and tin as “Xenes”. For the sake of brevity, we shall adoptthe same convention when collectively referring to the behavior or properties of silicene, germanene, and stanene.Early theoretical studies were soon followed by the first experimental reports of silicene nanoribbons and 2Dsilicene sheets .These early studies of silicene revealed one of the most crucial differences between silicene, the heavier group XIVelements germanene and stanene, and graphene: silicene’s most stable form is not a perfectly flat sheet, but is insteadslightly buckled . Among other novel phenomena, this buckling allows one to tune the band gap of Xenes byapplying an external electric field perpendicular to the plane of the monolayer . The tunable band gap of Xenesgives researchers, among other things, extraordinary in situ control over the binding energies and optical propertiesof excitons in these materials.In general, excitons in 2D materials are interesting because of their potential for large binding energies, strong opticalabsorption, and unique collective properties. Indeed, excitons in TMDCs are characterized by relatively high bindingenergies and significant spin-orbit coupling . Bose-Einstein condensation and superfluidity of spatially indirectexcitons in TMDC/ h -BN heterostructures, formed by two TMDC monolayers separated by N h -BN monolayers,were also analyzed . A theoretical study of intraexcitonic optical transitions in TMDC/ h -BN heterostructureswas performed . Recently, experimental studies of the excited states of direct excitons in monolayer MoS andspatially indirect excitons in multi-layer MoSe single crystals have also been performed. A comprehensive reviewof excitons in TMDCs is given in Ref. 28. While there is an abundance of research regarding excitons in TMDCs,there is relatively little research on excitons in buckled 2D materials.Experimental studies of intraexcitonic optical transitions have been performed in Cu O , and GaAs/GaAlAssemiconductor coupled quantum wells . Recently, similar experiments have been performed on direct excitons inmonolayer TMDCs , but there are not yet any comparable studies of the 2D Xenes. In this paper we perform atheoretical study of the binding energies and optical properties of direct and indirect excitons in buckled 2D materialsunder the effect of an external electric field.The objective of this paper is to study the exceptional tunability, via application of an external electric field, ofthe properties of excitons in Xenes. We demonstrate this by explicitly calculating the binding energies and opticalproperties of excitons in the case of (i) direct excitons in Xene monolayers and (ii) spatially indirect excitons inheterostructures formed by two Xene monolayers separated by N monolayers of h -BN, which we denote as X-BN-X.First, in the framework of the effective mass approximation, we consider the dependence of the exciton reducedmass µ as a function of the perpendicular external electric field, E ⊥ . This field-dependent mass is used when solvingthe Schr¨odinger equation for the eigenfunctions and eigenenergies of the direct or indirect exciton. This allows us tofurnish relevant optical quantities such as the oscillator strength and absorption coefficient. Second, we investigatethe dependence of the binding energies and optical properties of direct excitons in monolayer Xenes on the externalelectric field. For spatially indirect excitons in X-BN-X heterostructures, we study the dependence of these quantitieson the interlayer separation as well as on the external electric field.This Paper is organized in the following way. In Sec. II, we present a theoretical framework for excitons in buckled2D materials within the effective mass approach and consider their optical properties when the effective mass iselectric-field dependent. The binding energies and optical properties of direct excitons in monolayer Xenes and ofindirect excitons in X-BN-X heterostructures are presented in Secs. III and IV, respectively. A comparison betweendirect and indirect excitons is given in Sec. V. Our conclusions follow in Sec. VI. II. THEORETICAL FRAMEWORK OF 2D EXCITONS WITH ELECTRIC FIELD-DEPENDENT MASSA. Charge Carriers in buckled 2D materials
Monolayers of silicene, germanene, and low-buckled stanene can be pictured as graphene monolayers in which thetwo triangular sublattices are offset with respect to the plane of the monolayer by a particular distance, known asthe buckling constant or buckling factor. This offset between the two triangular sublattices gives rise to the intrinsicsensitivity of Xenes to an external electric field applied perpendicular to the plane of the monolayer. With no externalelectric field, the band structure of Xenes in the vicinity of the
K/K ′ points resembles graphene, though the intrinsicgaps of Xenes are significantly larger than that of graphene. The application of a perpendicular electric field creates apotential difference between the sublattices, causing a change in the band gap in the material, which in turn changesthe effective masses of the electrons and holes.The Hamiltonian in the vicinity of the K/K ′ points is given in Ref. 38 as:ˆ H = ~ v F ( ξk x ˆ τ x + k y ˆ τ y ) − ξ ∆ gap ˆ σ z ˆ τ z + ∆ z ˆ τ z , (1)where v F is the Fermi velocity, ˆ τ and ˆ σ are the pseudospin and real spin Pauli matrices, respectively, k x and k y arethe components of momentum in the xy -plane of the monolayer, relative to the K points, 2∆ gap is the intrinsic bandgap, ξ, σ = ± z = ed E ⊥ is the gap induced by the electric field, E ⊥ , acting along the z -axis, where d in the latter expression is the buckling constant. The first term in Eq. (1) isthe same as that of the low-energy Hamiltonian in graphene . The second term describes the spin-orbit coupling with an intrinsic band gap of 2∆ gap , while the last term describes the modification of the intrinsic band gap via anexternal electric field.Using Eq. (1), one may write the dispersion relation of charge carriers near the K/K ′ points as: E ( k ) = q ∆ ξσ + ~ v F k , (2)where ∆ ξσ = | ξσ ∆ gap − ed E ⊥ | (3)is the electric field-dependent band gap at k = 0. We note that when E ⊥ = 0, the spin-up and spin-down bands ofthe valence and conduction bands are degenerate. In other words, spin-orbit splitting only manifests itself at non-zeroexternal electric fields. At non-zero electric fields, both the valence and conduction bands split, into upper bands witha large gap (when ξ = − σ ), and lower bands with a small gap (when ξ = σ ). We call the excitons formed by chargecarriers from the large gap A excitons, and those formed by charge carriers in the small gap B excitons. When theexternal field reaches a critical value E c = ∆ gap / ( ed ), the lower bands form a Dirac cone at the K/K ′ points. Thevalues of these quantities are presented in Table I.In the vicinity of the K/K ′ points, the conduction and valence bands are parabolic. The effective mass of chargecarriers near the K/K ′ points can be written as m ∗ = ∆ ξσ /v F . The effective masses of electrons and holes are thesame due to the symmetry between the lowest conduction and highest valence bands, and can be written in terms ofthe external electric field as: m ∗ = | ξσ ∆ gap − ed E ⊥ | v F . (4) Material 2∆ gap (meV) d (˚A) v F ( × m/s) ǫ l [nm]Silicene (FS) 1.9 Silicene ( h -BN, Type I)
27 0.46 4.33 11.9 0.333Silicene ( h -BN, Type II)
38 0.46 5.06 11.9 0.333Germanene (FS) 33
16 0.45Stanene (FS) 101
24 0.5TABLE I: Parameters for buckled 2D materials: 2∆ gap is the total gap between the conduction and valence bands; d is thebuckling parameter; v F is the Fermi velocity; l is the monolayer thickness; ǫ is the dielectric constant of the bulk material. FSrefers the freestanding Xene monolayers. μ m ] ( a ) ( b ) E (cid:2) V / Å ] B, SiA, SiB, GeA, GeB, SnA, Sn
FIG. 1: Exciton reduced mass µ , in units of m , as a function of the external electric field, E ⊥ . (a) µ as a function of E ⊥ ,zoomed in to show the behavior at small values of E ⊥ . (b) µ as a function of E ⊥ across the full range of E ⊥ considered in thecalculations. The behavior of µ as a function of E ⊥ for A and B excitons in freestanding (FS) Xenes is shown in Fig. 1. Following ab initio calculations which determined that the crystal structure of silicene becomes unstable around 2.6 V/˚A, weconsider in our calculations electric fields up to 2.7 V/˚A. As one can see from Table I, silicene, which has the largest v F and the smallest d , has the smallest slope. Even at E ⊥ = 2 . m . On the other hand, in stanene, which has the smallest v F , the exciton reducedmass exceeds m at large fields.At small electric fields, germanene and especially stanene show significant differences between the reduced massesof the A and B excitons – this is due to their large intrinsic band gaps. Silicene, which has an intrinsic band gapon the order of a couple meV, exhibits very little difference between the reduced masses of A and B excitons, evenat relatively small electric fields. At large electric fields the difference between the A and B exciton reduced mass isnegligible in silicene and germanene. In all cases, the mass of the A exciton exceeds the mass of the B exciton. B. Effective mass approach for excitons in buckled 2D materials
In order to obtain the eigenfunctions and eigenenergies of an exciton in Xenes, we first write the Schr¨odingerequation for an interacting electron and hole: (cid:20) − ~ m e ∇ e + − ~ m h ∇ h + V ( r e , r h ) (cid:21) ψ ( r e , r h ) = Eψ ( r e , r h ) , (5)where e and h are the indices referring to the electron and hole, respectively, m e = m h = m ∗ are the masses of chargecarriers given by Eq. (4). Performing the standard procedure for the coordinate transformation to the center-of-mass, R CM = ( m e r e + m h r h ) / ( m e + m h ), and relative coordinates, r = r e − r h , one obtains an equation for the relativemotion of the electron and hole: (cid:20) − ~ µ ∇ + V ( r ) (cid:21) ψ ( r ) = Eψ ( r ) , (6)where µ = m e m h / ( m e + m h ) = m ∗ / r between the electron and hole can be written in cylindrical coordinates, r = ρ ˆ ρ + D ˆ z ,allowing us to treat the case of direct excitons in an Xene monolayer and spatially indirect excitons in X-BN-Xheterostructures on equal footing. For direct excitons, we set D = 0, and Eq. (6) becomes a purely 2D equation, with ρ representing the separation between the electron and hole sharing the same plane. Throughout this paper, we considerthe separation between two Xene monolayers in steps of calibrated thickness, l h − BN = 0 .
333 nm, corresponding tothe thickness of one h -BN monolayer. For spatially indirect excitons, the relative coordinate r = p ρ + D , where D = l + N l h − BN , l is the Xene monolayer thickness and N is the number of h -BN monolayers.The interaction between the electron and hole in 3D homogeneous dielectric environments is described by theCoulomb potential, but this interaction in 2D is modified and described by a potential which includes screeningeffects as a result of the reduced dimensionality. This potential was first considered in Ref. 46 and was independentlyderived in Ref. 47 – we refer to it as the Rytova-Keldysh (RK) potential. Therefore the interaction potential V ( r )between the electron and hole for direct excitons in an Xene monolayer is: V ( r ) = πke κρ (cid:20) H (cid:18) rρ (cid:19) − Y (cid:18) rρ (cid:19)(cid:21) (7)where ρ is the screening length, and H and Y are the Struve and Bessel functions of the second kind, respectively.In Eq. (7), κ = ( ǫ + ǫ ) / ǫ and ǫ are the dielectric constantseither (a) above and below the monolayer, in the case of direct excitons in an Xene monolayer, or (b) betweenand surrounding the Xene monolayers in the case of spatially indirect excitons in an X-BN-X heterostructure. Thescreening length ρ can be written as ρ = (2 πχ D ) / ( κ ), where χ D is the 2D polarizability, which in turn is givenby χ D = ( lǫ ) / ( ǫ + ǫ ), where ǫ is the bulk dielectric constant of the Xene monolayer.To better understand the importance of the screening effect in X-BN-X heterostructures, we perform calculationsusing both the RK and Coulomb potentials. For indirect excitons, the expressions for the interaction between theelectron and hole can be written as: V ( p ρ + D ) = πke κρ " H p ρ + D ρ ! − Y p ρ + D ρ ! , (8)for the RK potential, and V (cid:16)p ρ + D (cid:17) = ke κ ( ρ + D ) (9)for the Coulomb potential.Therefore, one can obtain the eigenfunctions and eigenenergies by solving Eq. (6) using the potential (7) for directexcitons, or for indirect excitons using either potential (8) or (9).Both the RK and Coulomb potentials have central symmetry, therefore the eigenstates of the system can be specifiedby a principal quantum number n = 1 , , , . . . and an angular momentum quantum number l = − n + 1 , − n +2 , . . . , , . . . , n − , n −
1. For the sake of brevity, we shall refer to the eigenstates of the exciton using the familiarnomenclature of the ideal 2D hydrogen atom, that is, 1 s refers to ( n, l ) = (1 , p would refer to ( n, l ) = (2 , ± . C. Optical Properties of Excitons in buckled 2D materials
Our approach to calculating the optical properties follows well-established methods for modeling optical transitionsin atom-like systems . This approach was used to describe the optical absorption by excitons in semiconductorcoupled quantum wells . We treat the exciton as a two-level system, modeling its polarization in response to anincident electromagnetic wave as a harmonic oscillator. The oscillator strength, f , of a particular optical transitionis given by, f = 2 µ ( E f − E i ) |h ψ f | x | ψ i i| ~ , (10)where E i and E f are the eigenenergies corresponding to the eigenfunctions ψ i and ψ f , and x represents the lin-ear polarization of the electric field of the incident electromagnetic wave. The dipole matrix element, |h ψ f | x | ψ i i| ,determines which transitions are allowed or forbidden. The allowed optical transitions are given by n f = n i and l f = l i ±
1. Hence, the allowed transitions from the ground state are those with n f = 2 , , . . . and l f = ±
1, i.e.the states 2 p, p, . . . . The oscillator strength is theoretically useful, as it is a dimensionless quantity which gives thestrength of a particular optical transition relative to all other possible transitions from the initial state ψ i .Experimentally studying the optical properties of a material generally involves observing how a sample absorbs,transmits, or reflects different wavelengths of electromagnetic waves. The intensity of an electromagnetic wave offrequency ω propagating a distance z through a medium is given by: I ( z ; ω ) = I e − α ( ω ) z (11)where I is the original intensity of the wave and α is the absorption coefficient, and is calculated as, α ( ω ) = (cid:18) ωω c πe ǫ √ ǫ h − BN µ nL X f (cid:19) (Γ / ω − ω ) + (Γ / ! , (12)where ω = ( E f − E i ) / ~ is the Bohr angular frequency of the optical transition, n is the 2D concentration of excitons, L X represents the thickness of the monolayer(s) which the electron and hole occupy, and Γ is the full width halfmaximum (FWHM) of the optical transition. We can deduce from Eq. (11) that the absorption coefficient is theinverse of the propagation distance z over which the intensity of the electromagnetic wave decreases by a factor 1 /e .Evaulating Eq. (12) for a single optical transition will yield a Lorentzian peak centered on ω with a FWHM of Γ.The absorption spectrum, obtained experimentally by measuring the absorption of a medium across a wide range ofincident frequencies ω , is represented theoretically by summing over Eq. (12) for all possible optical transitions in themedium (not limited to excitonic transitions). We focus on the maximal value of the absorption coefficient, obtainedwhen the incident electromagnetic wave is in resonance with a given optical transition, i.e. ω = ω . This maximalvalue is: α ( ω = ω ) = (cid:18) πe cǫ √ ǫ h − BN µ nL X f (cid:19) (cid:18) (cid:19) . (13)However, in 2D materials, where the thickness of a monolayer is a fixed value, the absorption coefficient is not themost efficient way to compare absorption properties across different materials. Recalling Eq. (11), one can considerthe absorption factor, A = 1 − I ( z = L X ; ω = ω ) /I = 1 − exp( − α ( ω = ω ) L X ): A = 1 − exp (cid:20) − (cid:18) πe n cǫ √ ǫ h − BN µ f (cid:19) (cid:18) (cid:19)(cid:21) , (14)which gives the fraction of the electromagnetic wave absorbed by a particular excitonic transition in direct excitonsin a single Xene monolayer or in spatially indirect excitons in an X-BN-X heterostructure. III. DIRECT EXCITONS IN XENE MONOLAYERS
Below we present the results of calculations for freestanding Xene monolayers as well as monolayer silicene on an h -BN substrate. The input parameters used in the calculations are given in Table I.While it is certainly instructive and informative to consider freestanding silicene, germanene, and stanene, it is alsoimportant to consider other scenarios which may be experimentally more practical, namely, the behavior of thesematerials when placed on different substrates. Hexagonal boron nitride is a promising substrate for silicene due toits atomically flat structure and relatively weak interactions with the silicene monolayer. Indeed, h -BN has beenidentified as an excellent substrate for other 2D materials such as graphene and TMDC monolayers . Theredoes, however, appear to be some disagreement regarding exactly how the weak interaction between the h -BN andsilicene affects the properties of the silicene, if at all.The authors of Ref. 44 performed ab initio calculations and found that the interaction between h -BN and siliceneleads to a rather significant modification of the material properties of silicene, increasing the band gap and decreasingthe Fermi velocity of silicene such that its material parameters more closely resemble those of freestanding germanene.The authors find that there are nine different stacking arrangements of silicene on h -BN, based on the slight latticemismatch between the two materials, and the variety of different rotation angles between the two lattices. All butthree of the nine different stacking arrangements result in a bandgap in silicene between 32 −
39 meV, and the otherthree arrangements lead to band gaps of 27, 28, and 29 meV. All but one of the lattice arrangements results in a Fermivelocity of at least 92% of v F in freestanding silicene, which the authors calculated to be 5 . × m/s. One latticearrangement results in a significantly lower value of the Fermi velocity, only 83% the magnitude of v F in freestandingsilicene. Interestingly, the authors find that the buckling parameter of silicene is not changed by the h -BN substrate,but remains constant at d = 0 .
46 ˚A, the same as for freestanding silicene.Fortunately, one lattice arrangement has both the largest bandgap and highest Fermi velocity, while a secondarrangement has both the smallest band gap and lowest Fermi velocity. This allows us to easily provide lower andupper bounds on the calculated properties of excitons in silicene on h -BN. These parameters are presented in Table Iand are taken from Ref. 44.Curiously, the authors of Ref. 63 also studied silicene on an h -BN substrate using ab-initio calculations, but arrivedat a completely different result compared to Ref. 44. They find that the buckling parameter of silicene is increasedfrom 0.46 to 0.54 ˚A, while they also find that the band gap and Fermi velocity remains largely unchanged comparedto freestanding silicene. For this reason, we did not perform a separate set of calculations corrsponding to these data,since the results would very closely resemble that of freestanding silicene. A. Eigenenergies of direct excitons in monolayer Xenes
The results of our calculations for the binding and optical excitation energies of direct excitons in monolayer Xenesare presented in Figs. 2 and 3, respectively. In Fig. 2 we compare the direct exciton binding energy of freestandingsilicene, germanene, and stanene, and encapsulated silicene. The direct exciton binding energies for FS Ge and FSSn are qualitatively similar to FS Si, but they are smaller than freestanding Si and larger than encapsulated Si.The freestanding monolayers exhibit by far the largest binding energies, due to the much weaker dielectric screeninginduced by the environment compared to silicene encapsulated by h -BN. The curves for FS Ge and FS Sn qualitativelyresemble that of FS Si, but FS Ge reaches a maximum binding energy of 725 meV, and the maximum binding energyfor FS Sn is roughly 525 meV, significantly smaller than for FS Si. The percent difference between the binding energyof FS Si and FS Ge at the largest electric field considered, E ⊥ = 2 . A and B excitons at electric fields up to about 1 V/˚A. These differences decreaseas the electric field increases. In FS Si, the difference between A and B excitons is always negligible.Overall, we find that FS Si exhibits the largest direct exciton binding energy, followed by FS Ge and then FS Sn,despite the fact that silicene has the lowest-mass charge carriers, while stanene has the highest mass charge carriers.Silicene has the lowest mass charge carriers because (a) it has the smallest intrinsic band gap, (b) it has the smallestbuckling parameter, so the external electric field induces the smallest change in its band gap, and (c) it has thelargest Fermi velocity, again implying that the charge carriers have intrinsically small mass. The opposite of points(a), (b), and (c) explain why stanene has the largest carrier masses. We infer that stanene has the smallest directexciton binding energy because it has by far the largest dielectric constant, and the largest monolayer thickness,and therefore, the screening length ρ is much larger than in silicene. This leads to significant dielectric screening,especially as carrier masses increase and the average exciton radius decreases. Direct excitons in h -BN-encapsulatedsilicene show the smallest binding energies, due to the aforementioned strong dielectric screening of the surroundingenvironment.Fig. 3 presents the optical transition energies and reveals an unexpected difference in behavior between freestandingXenes and encapsulated silicene. We use the data for FS Si as representative of the other two FS materials, withoutmaking the Figure visually cluttered. Since FS Si qualitatively resembles FS Ge and FS Sn, we will only show theresults for FS Si throughout the rest of the paper.At an electric field greater than 0 . − . s → p optical transition energy, that is, the transition energy does not change significantly as the electric fieldcontinues to increase. Furthermore, each of the three FS materials show the same rapid increase at small electric fields.In FS Si, we can see that the value of the transition energy at E ⊥ = 1 . E ⊥ = 2 . E max, s → p ≈
300 meV) at E ⊥ = 1 V/˚A. In FS Sn ( E max, s → p ≈
200 meV), the same happens at an electric field of only 0 . ■ FS Si ● FS Ge ▲ FS Sn ▼ Encapsulated, Type II ◆ Encapsulated, Type I ■ , ● , ▲ , ▼ , ◆ B Exciton □ , ○ , △ , ▽ , ◇ A Exciton B i nd i ng E ne r g y , E b [ m e V ] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ E ⟂ (cid:0) V / Å (cid:1) ■ FS Si ● Encapsulated, Type II ▲ Encapsulated, Type I ■ , ● , ▲ →
2p transition □ , ○ , △ →
3p transition T r an s i t i on E ne r g y [ m e V ] (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:13) (cid:14) (cid:15) (cid:16) (cid:17) (cid:18) (cid:19) (cid:20) (cid:21) (cid:22) (cid:23) (cid:24) (cid:25) (cid:26) (cid:27) (cid:28) (cid:29) (cid:30) □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ (cid:31) ! " ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ E ⟂ [ V < Å] FIG. 2: Direct exciton binding energies in freestanding silicene, germanene, and stanene, and in silicene encapsulated by h -BN.The solid symbols correspond to B excitons, while open symbols represent A excitons.FIG. 3: Dependence of the optical transition energy on the external electric field for the 1 s → p and 1 s → p transitions fordirect A excitons in freestanding and encapsulated silicene monolayers. that this saturation is due to the binding energy and the 2 p state eigenenergy increasing at roughly the same rateat high electric fields. The optical transition energy in the freestanding materials is therefore less tunable than thedirect exciton binding energy, since the transition energy for all three FS materials does not change significantly asthe electric field is increased. In contrast, encapsulated Si continues to show a linear increase in the 1 s → p opticaltransition energy even at high electric fields. It is also interesting to note that the transition energies of A and B excitons converge to nearly the same value at relatively small electric fields in FS Ge and Sn, even though Fig. 2demonstrates that the difference in binding energies of A and B excitons in these two materials remains noticableuntil the electric field becomes larger than the value at which the A and B transition energies converge. It would bevery interesting to probe these optical transitions experimentally to determine if both the A and B excitonic opticaltransitions may be induced by a single probe laser tuned to the common transition energy.Fig. 3 also shows the 1 s → p transition energies, which are consistently and significantly larger than the 1 s → p transition energies. Indeed, in all three FS materials, we see that the 1 s → p transition energy can be up to 50%larger than the 1 s → p transition, especially as the electric field approaches its maximum. In encapsulated Si, wefind that the difference is not so dramatic, but still on the order of 25% or greater.In addition, one can see from Figs. 2 and 3 that the dependence of the eigenenergies of direct excitons calculatedusing the RK potential on the electric field is non-linear, while the reduced mass of the exciton linearly depends onthe electric field according to Eq. (4). It is well known that the eigenenergies of direct 2D excitons calculated with theCoulomb potential are directly proportional to the exciton reduced mass. Therefore, in contrast to the RK potential,the eigenenergies of the exciton in the case of the Coulomb potential would depend linearly on the electric field. B. Optical properties of direct excitons
The results of calculations of the optical properties of direct excitons in monolayer Xenes are presented in Fig. 4.The oscillator strengths of the three freestanding materials quickly become saturated at a value of about 0 .
4, as shownin Fig. 4a. Furthermore, there is very little difference in f for a given material for A and B excitons. The oscillatorstrengths in encapsulated Si never quite reach saturation, and never come close to the same magnitude as that of thefreestanding materials.The oscillator strengths for the 1 s → p optical transition are also given in Fig. 4a. Surprisingly, the behaviorof f s → p as a function of the external electric field is qualitatively very similar to f s → p . We find that the valueof f s → p is roughly one-tenth the magnitude of the corresponding value of f s → p at a given electric field. Thisconsistent difference in magnitude of roughly a factor of 10 is somewhat surprising, considering the rather smallmagnitudes of f s → p at electric fields less than approximately 1 V/˚A. It was thought that perhaps this would meanthat f s → p would be of comparable magnitude to f s → p at small electric fields, however this is clearly not the case.On the other hand, in FS Si, f s → p quickly reaches a value of 0.04 at small electric fields, but we observe a veryslight decrease in the magnitude of f s → p as the electric field continues to increase beyond approximately 1 V/˚A.This slight decrease in f at electric fields greater than ≈ f s → p for indirectexcitons in X-BN-X heterostructures of FSE Sn, when the interlayer separation is large and the electric field is strong.The absorption coefficient and absorption factor for FS and encapsulated Si are shown in Figs. 4b and 4c, respec-tively. We observe that the freestanding materials should absorb significantly more light than encapsulated Si. Thisagain can likely be tied back to the difference in dielectric environment – recall from Eq. (13) the factor of √ ǫ h − BN in the denominator. For freestanding Xenes in a vacuum, this would translate to significantly stronger absorptionthan for h -BN encapsulated materials. It is also noteworthy that the absorption in encapsulated silicene becomessaturated by the electric field much more quickly, not exhibiting much change when the electric field is increasedbeyond E ⊥ = 1 V/˚A. On the other hand, FS Si exhbits a noticable change in its absorption through the entirerange of electric fields. In FS Ge, α and A lies roughly between the curves for FS Si and encapsulated Si. In FSSn, α converges towards encapsulated Si at large electric fields, while A remains larger than in encapsulated Si byapproximately one percentage point.Ultimately, we find that the minimum absorption factor, obtained at the maximum value of the electric field, isapproximately 3% for FS Si, 2% for FS Ge, 1.5% for FS Sn, and only about 1% for encapsulated silicene. For the sakeof comparison, at an electric field of E ⊥ = 1 V/˚A, the value of the absorption factor in FS Ge is approximately 5.5%(where B excitons absorb slightly more, and A excitons absorb slightly less), and in FS Sn, the corresponding valueis slightly less than 4% for B excitons, but slightly more than 3% for A excitons. As the electric field is increased,the difference between A and B excitons becomes much less significant. The FS materials show a stronger responsein their optical absorption as a function of electric field, suggesting that they are more tunable than encapsulated Si,which approaches its global minimum at an electric field of about 1.5 V/˚A. ■ FS Si ● Encapsulated, Type II ▲ Encapsulated, Type I ■ , ● , ▲ →
2p transition □ , ○ , △ →
3p transition ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ E ⟂ [ V /Å ] f ( a ) ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ × × × X × Y Electric Field, E ⟂ [ V /Å Z α [ m \ ^ ( b ) ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ _ ‘ a b c d e f g h i j k l m n o p q r s t u v w x y ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ E ⟂ [ V /Å ] ( c ) FIG. 4: Optical properties of direct excitons in Xene monolayers. The dependence of the (a) the oscillator strength, f , (b) theabsorption coefficient, α , and (c) the absorption factor, A , on the electric field, E ⊥ , for direct A excitons in FS and encapsulatedSi. In (a) and(c) both the 1 s → p and 1 s → p transitions are shown, while in (b), only the 1 s → p optical transition isshown. Surprisingly, the 1 s → p transition in freestanding Si is comparable to the 1 s → p transition in encapsulatedSi. On the other hand, the 1 s → p transition in encapsulated Si is quite strongly suppressed, barely surpassing 1%absorption at an electric field of on 0.1 V/˚A, and decreasing to only a small fraction of 1% absorption as the electricfield continues to increase. IV. PROPERTIES OF INDIRECT EXCITONS IN X-BN-X HETEROSTRUCTURES
In the following subsections, we study the dependence of the binding energy and optical properties of spatiallyindirect excitons on the external electric field, E ⊥ , as well as on the number of h -BN monolayers in the X-BN-Xheterostructure. We continue to perform calculations using the parameters corresponding to freestanding Si, Ge, andSn, even though it is of course unreasonable to expect the Xene monolayers to retain their freestanding parameterswhen placed in an X-BN-X heterostructure. In the following calculations, we now consider the dielectric environment κ = 4 .
89, unlike the case of direct excitons, where the truly freestanding materials were modeled to be surroundedby vacuum, i.e. κ = 1. To clearly denote the difference between calculations for direct excitons in freestandingmonolayers, and calculations using the freestanding parameters in an X-BN-X heterostructure, we will refer to thelatter as freestanding-encapsulated, or FSE. We shall present our results for indirect excitons in FSE materials in anX-BN-X heterostructure as a means of illustrating the importance of using physically accurate material parameterswhen calculating the properties of indirect excitons. A. Eigenenergies of spatially indirect excitons in an X-BN-X heterostructure
Fig. 5a shows the binding energies of indirect A excitons in FSE and encapsulated Si. Therefore, the larger intrinsicband gap and significantly smaller Fermi velocity of the encapsulated Si in turn leads to consistently larger bindingenergies than the FSE Si at all values of electric field and interlayer separation. Even at large interlayer separation,we see that there is a significant difference in the binding energy between FSE and encapsulated Si, and this differencebetween the binding energies increases significantly as the interlayer separation decreases. Therefore, the observeddifference in indirect exciton binding energy in Fig. 5a of greater than 10% at N = 5 is even more pronounced atsmaller interlayer separations. In Fig. 5b, it is shown that the binding energy of indirect A excitons in encapsulatedSi increases sharply as the electric field is increased up to about 1 V/˚A, but as the electric field continues to increase,the binding energy does not increase significantly. Increasing the interlayer separation from N = 1 to N = 5 reducesthe binding energy by about 33% at high electric fields.To understand the role of screening we perform calculations for the Coulomb and RK potentials. Fig. 6 provides acomparison of the value of the binding energy in Type II encapsulated Si using either the Coulomb or RK potentials.We find that the binding energies calculated with the Coulomb potential are always larger than those calculated usingthe RK potential. For one monolayer of h -BN, the percent difference in the binding energies for FSE Si range fromroughly 5% at small applied electric fields (5.35% at E ⊥ = 0 .
25 V/˚A) up to nearly 12% at the maximum calculatedelectric field (11.7% at E ⊥ = 2 .
75 V/˚A). For the same values of the electric field and interlayer separation, the percentdifference in the binding energies for FSE Ge is more prominent than in FSE Si, starting at 10.9% at E ⊥ = 0 .
25 V/˚A, ■ FS Si ● Encapsulated, Type I ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■● ● ● ● ● ● ● ● ● ● ● ● E z [ V / Å] B i nd i ng E ne r g y , E b [ m e V ] { a ) FIG. 5: (a) Indirect A exciton binding energy as a function of external electric field at N = 5 for FSE and encapsulated Si.(b) Dependence of the indirect A exciton binding energy in encapsulated Si on the interlayer separation, N and the externalelectric field, E ⊥ . Calculations are performed using the Rytova-Keldysh potential. and increasing up to 20.1% at E ⊥ = 2 .
75 V/˚A. The percent difference in FSE Sn is by far the most pronounced,beginning at 19.6% and increasing to 34.5% as the external electric field is increased.As one might expect, these differences in the binding energy decrease as the interlayer separation increases. This isdue to the fact that the RK potential converges towards the Coulomb potential at large distances. For an interlayerseparation of N = 5 in FSE Ge, the percent difference ranges between 3.7% and 5.8%. In FSE Sn, however, thepercent difference ranges from 7.2% to 12.5%, which is still rather significant. FIG. 6: Difference in binding energy for indirect A excitons calculated using Coulomb and Rytova-Keldysh potentials inencapsulated Type II Si, as a function of the interlayer distance, N , and the external electric field, E ⊥ .FIG. 7: Dependence of the 1 s → p optical transition energies of indirect A excitons in encapsulated Type II Si on the interlayerdistance, N , and external electric field, E ⊥ . Calculations are performed using the RK potential. The 1 s → p optical transition energies of indirect A excitons in Type II encapsulated Si are presented in Fig. 7.Curiously, at large N , the optical transition energy in encapsulated silicene begins to decrease slightly as the electricfield continues to increase. Furthermore, our calculations show again that the difference in the optical transitionenergy between the Coulomb and RK potentials is again quite significant, which reinforces the importance of using aphysically accurate interaction potential when calculating the properties of the indirect exciton eigensystem. Finallywe see that the optical transition energy in encapsulated Si is not, in fact, particularly tunable, since the transitionenergy plateaus at low electric field at all values of the interlayer separation, and it remains mostly constant, thoughit does decrease slightly for large interlayer separations and strong electric fields.In constrast to the results in Fig. 5, our calculations show that the difference in the optical transition energy betweenFSE and encapsulated Si is quite small, on the order of 5%, even at small interlayer separations and large electricfields. Indeed, the optical transition energies calculated using the Coulomb potential at large electric fields and at N = 1 are remarkably similar for each of the three FSE materials as well as encapsulated Si, with all quantities fallingbetween 90-110 meV. On the other hand, when calculated using the RK potential, the transition energy at N = 1 andat the maximal electric field shows a larger variation between the FS Xenes and encapsulated Si, falling between 600meV for FSE Sn, 85 meV for FSE Si, and 90 meV for encapsulated Si. This result suggests that the optical transitionenergy is not as sensitive to the choice of material parameters as the indirect exciton binding energy. B. Optical properties of indirect excitons
The results of calculations of the optical properties of indirect excitons in X-BN-X heterostructures are presentedin Fig. 8. The oscillator strength f of the 1 s → p optical transition of spatially indirect A excitons in encapsulatedSi increases monotonically with both E ⊥ and D , as shown in Fig. 8a. It was previously reported that f is expectedto increase monotonically with N for spatially indirect excitons in TMDC-BN-TMDC heterostructures, and thisphenomenon is observed for encapsulated Si, as well as for the three FSE materials. We find that the oscillatorstrength approaches 0.5 at large electric fields and interlayer separations, suggesting that the 1 s → p transition isvery strongly suppressed in this regime. f increases quickly for small values of E ⊥ and grows much more slowlybeyond around E ⊥ = 1 V / ˚A. At large N , f quickly approaches 0 .
5, which implies that the 1 s → p optical transitionis very strongly suppressed.The three FSE materials, not shown in Fig. 8, are quantitatively very similar to encapsulated Si. This is anotherexample of a quantity which is mostly insensitive to the choice of material parameters used in the calculations.Unlike the dramatic difference between the Coulomb and RK potentials seen in the eigenenergies of Fig. 6, thedifference in f between the Coulomb and RK potentials is quite small. In general, while there is some variation usingthese potentials between the materials studied here, the quantitative difference is very slight overall, except in FSESi, where there is still a noticable difference even as the electric field approaches its maximum.The oscillator strengths of the 1 s → p transitions in encapsulated Si were also calculated. We find that f s → p isapproximately one-tenth the magnitude of the corresponding f s → p for a given electric field and interlayer separation,very similar to the case of direct excitons in Xene monolayers.Also noteworthy is the unusual, and unique, behavior of f at zero electric field for the four materials. In FSE Siand Ge, the oscillator strength of the 1 s → p transition can exceed 0.5, an unphysical result which would appear toviolate the oscillator strength sum rule. In FSE Si, with its very small intrinsic gap and very large Fermi velocity,we sometimes observe unusual results at very small electric fields, such as the unreasonably large oscillator strengthobserved here. This may be due to the extremely small exciton mass at these small fields, which in turn leads to ahuge excitonic radius, which then may run into problems with our computational framework, specifically the size ofour computational “box”. FIG. 8: Optical properties of indirect excitons in X-BN-X heterostructures. The dependence of (a) the oscillator strength, f , (b) the absorption coefficient, α , and (c) the absorption factor, A , for indirect A excitons in encapsulated silicene, on theinterlayer separation, N , and the external electric field, E ⊥ . Calculations are performed using the RK potential. Figs. 8b and 8c demonstrate how optical absorption is suppressed by increasing the external electric field, just asis the case with direct excitons, as shown in Figs. 4b and 4c. As the interlayer separation is increased, the absorptionincreases by a small amount. These calculations for encapsulated silicene are quantitatively very similar to the threeFSE materials. Our calculations show that at large electric fields, encapsulated Si, as well as the three FSE materials,should absorb less than 2% of incoming resonant light. Encapsulated Si shows much weaker absorption than FSE Si,with encapsulated Si more closely resembling FSE Ge in terms of its absorption properties. As is the case with theother optical quantities, we find that the absorption for encapsulated Si decreases sharply for E ⊥ . E ⊥ > E ⊥ = 0 . N = 1 to about 3.5% at N = 5.Our calculations also show that there is a small difference in the absorption factor when comparing the Coulomband RK potentials. This difference is on the order of a few tenths of a percent at E ⊥ . A and B excitons can be quite large at small electricfields, and furthermore than B excitons always absorb more strongly than A excitons, due to their slightly smallermass. Finally, we note that the reduction in f by a factor of 10 carries over to the calculated values of α s → p and A s → p , as well. V. COMPARISON BETWEEN DIRECT AND INDIRECT EXCITONS
The binding energies of direct excitons are, of course, stronger than the binding energies of the spatially indirectexcitons in the same materials, but this drop in binding energy when moving from direct excitons to indirect excitonsis huge. For example, the direct exciton binding energy in FS Si is on the order of 900 meV at E ⊥ = 2 . N = 1 in FSE Si is only 140 meV (155 meV) for the RK (Coulomb) potential,a staggering reduction in the binding energy of over 80%. This change is not as drastic in FS Ge and FS Sn, wherethe binding energy drops by slightly less than 80% in FS Ge (from ≈
700 meV to ≈
160 meV) and by about 75% inFS Sn (from ≈
550 meV to ≈
170 meV). The dramatic difference between direct and indirect binding energies in thefreestanding materials is due to both the change in dielectric environment as well as the increase in the electron-holedistance. The difference between direct and indirect exciton binding energies is not as severe in encapsulated Si, wherethe change in binding energy is only about 50% at the maximum electric field. The significantly smaller change inencapsulated Si can be partially explained by κ remaining constant at 4.89 between the direct and indirect excitoncases.The difference in the optical transition energy for direct and indirect excitons is not as large as the aforementioneddifference between the binding energies. In FS Si, we see a drop in the transition energy of about 75% when movingfrom direct excitons to indirect excitons at N = 1. In FS Ge, the same change is approximately 66%, and in FSSn, the difference is only about 50%. Unlike the binding energies, where encapsulated silicene exhibited the smallestdirect-to-indirect change, when comparing the transition energies we find that encapsulated silicene shows a changeof about 66%, comparable to FS Ge.The oscillator strengths follow a consistent pattern, with the direct excitons having the smallest f at any given E ⊥ ,and f increasing as the interlayer distance is increased. As mentioned in Sec. II C, we observe that while the oscillatorstrength increases monotonically with E ⊥ , increasing quickly at first before slowly leveling off above E ⊥ & α decreases monotonically with E ⊥ but still increases monotonically with the interlayerseparation. The same is true of the absorption factor, A .On the other hand, the optical properties behave in the opposite way compared to the eigenenergies with respectto the electric field. At small electric fields, the difference in, for example, the oscillator strength can be significant,on the order of 10% or more. As the electric field is increased, this difference decreases, and the magnitudes covergetowards each other. The absorption coefficient and absorption factor exhibit this same behavior, but these differencescan be traced directly back to how the oscillator strength changes, since there are no other terms in the analyticalforms of α and A which would change depending on the choice of interaction potential. Due to the oscillator sumrule, we know that the maximum value of the oscillator strength for a given symmetric, photon-absorbing transitionmust be 0.5. Therefore, as the electric field increases, the oscillator strength must approach 0.5, regardless of theinteraction potential used.Regarding the choice of the RK or Coulomb potentials, we find huge differences in the binding and optical transitionenergies for interlayer separations N <
2, but this difference decreases sharply beyond N = 3. This significantdifference at small interlayer separations is clearly due to the way in which the two potentials treat the surroundingdielectric environment. When using the Coulomb potential, the dielectric constant is effectively ǫ h − BN = 4 .
89, whilethe RK potential still takes into account the screening length of the Xene monolayers. Since the Xenes have muchlarger dielectric constants than the h -BN, using the RK potential for indirect excitons results in much smaller bindingenergies when compared to the Coulomb potential.Analyzing the relationship between f and E ⊥ in the context of Coulomb and RK potentials is not as straightforwardas our analysis of the eigenenergies. This is because f is directly proportional to both the transition energy and thedipole matrix element, both of which depend directly on the choice of interaction potential. Ultimately, we observethat f calculated with the RK potential is always larger than f calculated using the Coulomb potential at smallelectric fields – therefore, despite the fact that the optical transition energy is always larger for the Coulomb potentialthan for the RK potential, it must be the case that the dipole matrix element integral is always much larger for theRK potential than for the Coulomb potential.This difference in behavior – where the difference between RK and Coulomb increases in the eigenenergies as theelectric field increases, while the difference in the optical properties decreases as the electric field increases – suggests2a complicated relationship between the choice of interaction potential and the external electric field. With regards tothe differences in the eigenenergies, we can understand why that difference increases as the electric field increases ifwe recall that the exciton radius is proportional to the excitonic reduced mass. At small electric fields, the excitonhas a small mass and therefore a large excitonic radius. At large separations, the RK potential converges towards theCoulomb potential, and therefore the difference in eigenenergies calculated using the two potentials is small at smallelectric fields. As the electric field increases, the excitonic reduced mass increases, which reduces the exciton radius,which in turn causes the eigenenergies calculated using the RK and Coulomb potentials to diverge from each other.We note that there is significant disagreement in the literature as to the exact value of the material parametersfor FS Xenes given in Table I. For example, the intrinsic band gap of silicene has been reported to be in the range1 . − . , the germanene band gap has been cited as between 24 −
93 meV , and the band gapin stanene has been reported in Ref. 66 to be between 30 −
123 meV. At large electric fields, these huge discrepanciesin the band gap would have a minor effect on the eigenenergies and optical properties of both direct and spatiallyindirect excitons in silicene, while the differences in germanene and stanene are noticable but minimal. At smallelectric fields, however, these differences in the intrinsic band gap can completely change the type of behavior onewould expect to observe.The Fermi velocity of charge carriers in Xene monolayers also shows significant variation between results. Forexample, ab initio calculations performed in Ref. 67 found that in FS Si, v F = 5 . × m/s, while in FS Ge, v F = 5 . × m/s. These values are considerably smaller than the parameters given in Table I and used in ourcalculations, though these v F are comparable in magnitude to v F in encapsulated Si . The significant difference inthese values of v F has a major effect on the charge carrier mass – while the two values of v F in FS Si only differ byabout 20%, the carrier masses in FS Si calculated with v F = 5 . × m/s are 49% larger than the carrier massescalculated with v F = 6 . × m/s. This difference of nearly 50% in the carrier masses would therefore noticeablyincrease the exciton binding energy while decreasing the absorption.Finally, data on the Xene monolayer thickness is scarce, and the data that does exist can vary wildly in magni-tude. For example, experimental measurements of Si monolayer thickness on various substrates using atomic forcemicroscopy (AFM) yield thicknesses of 0.3 nm , 0.37 nm , and 0.4 nm . It seems reasonable to expect that afreestanding germanene monolayer would be thicker than a freestanding Si monolayer, since Ge has a larger atomicradius, R Ge = 1 .
25 ˚A , than silicene, R Si = 1 .
11 ˚A , and germanene has a larger buckling constant by about0.2 ˚A. Likewise, freestanding stanene should similarly be thicker than freestanding germanene by roughly the sameamount, again because it has a larger atomic radius, R Sn = 1 .
45 ˚A , and larger buckling constant, again by about0.2 ˚A. Using l Si = 0 . as a baseline, we then arrive at rough estimates of the monolayer thicknesses of FS Geand FS Sn of 0 .
45 nm and 0 . VI. CONCLUSIONS
In this paper we demonstrate that an external electric field can be used to tune the eigenenergies and opticalproperties of direct and indirect excitons in Xene monolayers or X-BN-X heterostructures. Reflecting upon ourresults, we see that this is generally true, with the condition that most quantities in the FS Xenes reach a saturationpoint at some value of the electric field, beyond which the value of the quantity does not change by much as theelectric field continues to increase. Specifically, we find that in the freestanding Xenes, the optical transition energiesand oscillator strengths saturate at low electric fields, while in encapsulated Si, it is the absorption coefficient andabsorption factor that become saturated at low electric fields. For indirect excitons in X-BN-X heterostructures, weobserve saturation of the oscillator strengths, absorption coefficients, and absorption factors.In addition, our study of indirect excitons using both the Coulomb and RK potentials to describe the electrostaticinteraction of the electron and hole has indicated that the choice of interaction potential can cause huge changes in themagnitude of the binding energies and optical transition energies, making it imperative that theorists determine whichinteraction potential yields physically accurate results. The eigenenergies calculated using the Coulomb potential arealways larger than the corresponding quantities calculated using the RK potential, and this difference increases asthe electric field increases. Conversely, the optical properties calculated using the RK potential are always of highermagnitude than the corresponding values calculated using the Coulomb potential, though this difference is negligibleat large electric fields.Finally, our comparison of the properties of indirect excitons calculated using the material parameters of freestandingSi and using the material properties of Si with h -BN as a substrate show that the choice of material parameters doesindeed have a significant effect on the eigensystem, and that it would therefore be physically inaccurate to treat the3Xene parameters as unchanged between the freestanding monolayer and an X-BN-X heterostructure.These calculations provide a reference for future theoretical and experimental studies of intraexcitonic opticaltransitions. In addition, our calculations demonstrate that further studies are necessary to expand and refine ourunderstanding of the tunability of excitons in 2D Xenes. The comparison of the exciton properties in FSE Si andencapsulated Si demonstrate that it is necessary to correctly identify the material parameters of the Xenes, in particularthe band gap, Fermi velocity, and effective monolayer thickness. It is especially important to examine how theseproperties change when the Xene monolayer is placed on different substrates, and how, if at all, these parameterschange as a function of the external electric field. The difference in the eigenenergies and optical properties of indirectexcitons calculated used the Coulomb and RK potentials provides an opportunity for further study of the role ofscreening effects. These interesting topics will need to be explored further, as they may play an important role in theuse of 2D Xenes in novel nanodevices. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science , 666 (2004). C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard,et al., Nat. Nanotechnol. , 722 (2010). K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. , 136805 (2010). A. Molle, J. Goldberger, M. Houssa, Y. Xu, S. C. Zhang, and D. Akinwande, Nat. Mater. , 163 (2017). S. B. Fagan, R. J. Baierle, R. Mota, A. J. R. da Silva, and A. Fazzio, Phys. Rev. B , 9994 (2000). S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, Phys. Rev. Lett. , 236804 (2009). S. Leb`egue and O. Eriksson, Phys. Rev. B , 115409 (2009). B. Aufray, A. Kara, S. Vizzini, H. Oughaddou, C. Leandri, B. Ealet, and G. Le Lay, Appl. Phys. Lett. , 183102 (2010). P. De Padova, C. Quaresima, C. Ottaviani, P. M. Sheverdyaeva, P. Moras, C. Carbone, D. Topwal, B. Olivieri, A. Kara,H. Oughaddou, et al., Appl. Phys. Lett. , 261905 (2010). B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet, and B. Aufray, Appl. Phys. Lett. , 223109 (2010). P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. Le Lay, Phys.Rev. Lett. , 155501 (2012). C. L. Lin, R. Arafune, K. Kawahara, N. Tsukahara, E. Minamitani, Y. Kim, N. Takagi, and M. Kawai, Appl. Phys. Express , 045802 (2012). L. Li, S. Z. Lu, J. Pan, Z. Qin, Y. Q. Wang, Y. Wang, G. Y. Cao, S. Du, and H. J. Gao, Adv. Mater. , 4820 (2014). Z. Ni, Q. Liu, K. Tang, J. Zheng, J. Zhou, R. Qin, Z. Gao, D. Yu, and J. Lu, Nano Lett. , 113 (2012). L. Stille, C. J. Tabert, and E. J. Nicol, Phys. Rev. B , 195405 (2012). N. D. Drummond, V. Z´olyomi, and V. I. Fal’Ko, Phys. Rev. B , 075423 (2012). M. Fadaie, N. Shahtahmassebi, and M. R. Roknabad, Opt. Quant. Electron. , 440 (2016). N. Gao, J. C. Li, and Q. Jiang, Phys. Chem. Chem. Phys. , 11673 (2014). T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, Phys. Rev. B , 045318 (2013). A. Korm´anyos, G. Burkard, M. Gmitra, J. Fabian, V. Z´olyomi, N. D. Drummond, V. I. Fal’ko, and V. Fal’ko, 2D Mater. ,022001 (2015). M. M. Fogler, L. V. Butov, and K. S. Novoselov, Nat. Commun. , 4555 (2014). F. C. Wu, F. Xue, and A. H. MacDonald, Phys. Rev. B , 165121 (2015). O. L. Berman and R. Ya. Kezerashvili, Phys. Rev. B , 245410 (2016). O. L. Berman and R. Ya. Kezerashvili, Phys. Rev. B , 094502 (2017). M. N. Brunetti, O. L. Berman, and R. Ya. Kezerashvili, J. Phys.: Condens. Matter , 225001 (2018). C. Robert, M. A. Semina, F. Cadiz, M. Manca, E. Courtade, T. Taniguchi, K. Watanabe, H. Cai, S. Tongay, B. Lassagne,et al., Phys. Rev. Mater. , 011001 (2017). J. Horng, L. Zhang, E. Y. Paik, H. Deng, T. Stroucken, and S. W. Koch, Phys. Rev. B , 241404(R) (2018). G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Rev. Mod. Phys. , 21001(2018). M. Kuwata-Gonokami, M. Kubouchi, R. Shimano, and A. Mysyrowicz, J. Phys. Soc. Jpn. , 1065 (2004). M. Kubouchi, K. Yoshioka, R. Shimano, A. Mysyrowicz, and M. Kuwata-Gonokami, Phys. Rev. Lett. , 016403 (2005). M. J¨orger, T. Fleck, C. Klingshirn, and R. Von Baltz, Phys. Rev. B , 235210 (2005). R. Huber, B. A. Schmid, Y. R. Shen, D. S. Chemla, and R. A. Kaindl, Phys. Rev. Lett. , 017402 (2006). R. Huber, R. A. Kaindl, B. A. Schmid, and D. S. Chemla, Phys. Rev. B , 161314(R) (2005). R. Huber, B. A. Schmid, R. A. Kaindl, and D. S. Chemla, Phys. Stat. Sol. (b) , 1041 (2008). S. Cha, J. H. Sung, S. Sim, J. Park, H. Heo, M. H. Jo, and H. Choi, Nat. Commun. , 10768 (2016). C. Poellmann, P. Steinleitner, U. Leierseder, P. Nagler, G. Plechinger, M. Porer, R. Bratschitsch, C. Sch¨uller, T. Korn, andR. Huber, Nat. Mater. , 889 (2015). P. Steinleitner, P. Merkl, P. Nagler, J. Mornhinweg, C. Sch¨uller, T. Korn, A. Chernikov, and R. Huber, Nano Lett. , 1455(2017). C. J. Tabert and E. J. Nicol, Phys. Rev. B , 195410 (2014). A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109 (2009). D. S. L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. Chakraborty, Adv. in Phys. , 261 (2010). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801 (2005). L. Matthes, P. Gori, O. Pulci, and F. Bechstedt, Phys. Rev. B , 035438 (2013). L. Tao, E. Cinquanta, D. Chiappe, C. Grazianetti, M. Fanciulli, M. Dubey, A. Molle, and D. Akinwande, Nat. Nanotechnol. , 227 (2015), URL . L. Li, X. Wang, X. Zhao, and M. Zhao, Physics Letters A , 2628 (2013). S. Balendhran, S. Walia, H. Nili, S. Sriram, and M. Bhaskaran, Small , 640 (2015). N. S. Rytova, Proc. MSU Phys., Astron. , 30 (1967), URL . L. V. Keldysh, Sov. Phys. JETP , 658 (1979). G. Bergh¨auser, A. Knorr, and E. Malic, 2D Mater. , 015029 (2017). D. Y. Qiu, F. H. Da Jornada, and S. G. Louie, Phys. Rev. Lett. , 216805 (2013). G. Wang, X. Marie, I. Gerber, T. Amand, D. Lagarde, L. Bouet, M. Vidal, A. Balocchi, and B. Urbaszek, Phys. Rev. Lett. , 097403 (2015). A. T. Hanbicki, M. Currie, G. Kioseoglou, A. L. Friedman, and B. T. Jonker, Solid State Commun. , 16 (2015). T. Stroucken and S. W. Koch, J. Phys.: Condens. Matter , 345003 (2015). B. Zhu, X. Chen, and X. Cui, Sci. Rep. , 9218 (2015). F. Wu, F. Qu, and A. H. MacDonald, Phys. Rev. B , 075310 (2015). D. W. Snoke,
Solid State Physics: Essential Concepts (Addison-Wesley, 2009). Yu. E. Lozovik and A. M. Ruvinskii, JETP , 979 (1997). C. R. Dean, A. F. Young, P. Cadden-Zimansky, L. Wang, H. Ren, K. Watanabe, T. Taniguchi, P. Kim, J. Hone, and K. L.Shepard, Nat. Phys. , 693 (2010). K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, K. Kim, L. Colombo, P. R. Gellert, M. G. Schwab,and K. Kim, Nature , 192 (2013). A. K. Geim and I. V. Grigorieva, Nature , 419 (2014). K. S. Thygesen, 2D Mater. , 022004 (2017). L. Kou, Y. Ma, Z. Sun, T. Heine, and C. Chen, J. Phys. Chem. Lett. , 1905 (2017). E. V. Calman, C. J. Dorow, M. M. Fogler, L. V. Butov, S. Hu, A. Mishchenko, and A. K. Geim, Appl. Phys. Lett. ,101901 (2015). T. P. Kaloni, M. Tahir, and U. Schwingenschl¨ogl, Sci. Rep. , 3192 (2013). C. C. Liu, H. Jiang, and Y. Yao, Phys. Rev. B , 195430 (2011). C. C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. , 076802 (2011). X. Chen, R. Meng, J. Jiang, Q. Liang, Q. Yang, C. Tan, X. Sun, S. Zhang, and T.-L. Ren, Phys. Chem. Chem. Phys. ,16302 (2016). F. Bechstedt, L. Matthes, P. Gori, and O. Pulci, Appl. Phys. Lett. , 2010 (2012). C. Grazianetti, E. Cinquanta, and A. Molle, 2D Mater. , 012001 (2016). H. Oughaddou, H. Enriquez, M. R. Tchalala, H. Yildirim, A. J. Mayne, A. Bendounan, G. Dujardin, M. Ait Ali, andA. Kara, Prog. Surf. Sci. , 46 (2015). M. Derivaz, D. Dentel, R. Stephan, M. C. Hanf, A. Mehdaoui, P. Sonnet, and C. Pirri, Nano Lett. , 2510 (2015). J. C. Slater, J. Chem. Phys. , 3199 (1964). J. C. Slater,