Exciton binding energies and luminescence of phosphorene under pressure
Abstract
The optical response of phosphorene can be gradually changed by application of moderate uniaxial compression, as the material undergoes the transition into an indirect gap semiconductor and eventually into a semimetal. Strain tunes not only the gap between the valence band and conduction band local extrema, but also the effective masses, and in consequence, the exciton anisotropy and binding strength. In this article, we consider from a theoretical point of view how the exciton stability and the resulting luminescence energy evolves under uniaxial strain. We find that the exciton binding energy can be as large as 0.87 eV in vacuum for 5 % transverse strain, placing it amongst the highest for 2D materials. Further, the large shift of the luminescence peak and its linear dependence on strain suggest that it can be used to probe directly the strain state of single-layers.
I Introduction
The discovery of graphene ten years agonovoselov2004electric ; neto2009electronic triggered findings of a plethora of novel two-dimensional materials with unprecedented physical phenomena and potential technological applicationsschwierz2010graphene . The wide variety of these novel 2D materials include insulators like hexagonal boron nitride (h-BN)alem2009atomically and graphaneelias2009control , superconductors like niobium diselenide (NbSe)novoselov2005two , and semiconductors like molybdenum disulfide (MoS) as well as other transition metal dichalcogenides (TMDC)novoselov2005two ; mak2010atomically .
Recently, the paradigm to obtain novel two-dimensional materials from the exfoliation of layered crystals gave rise to the single-layer black phosphorus, also known as phosphorene rodin2014strain ; liu2014phosphorene . Phosphorene is a two-dimensional semiconductor formed exclusively by phosphorus atoms in a puckered anisotropic rectangular lattice due to the hybridization with lone pairs [See Fig. 1(a)]. The single-layer and few-layer phosphorene can be obtained from mechanical exfoliationliu2014phosphorene or plasma-assisted exfoliation of 3D layered crystal of black phosphorus (BP)lu2014plasma weakly bound by van der Waals interaction. Since phosphorene is a homopolar semiconductor, it shows some advantages when compared with TMDC, such as electronic inactivity of grain boundaries and native point defectsliu2014two . Futhermore, the range of bandgaps from 0.3 eV (BP) to 1.6 eV (single-layer)PhysRevB.89.235319 and high hole mobility allow the production of field-effect transistors with on/off ratio up to at room temperatureli2014black ; liu2014phosphorene ; qiao2014high . The puckered anisotropic geometry of phosphorene also allows bandgap engineering with in-planePhysRevB.90.085402 ; elahi2014modulation ; wei2014superior and out-of-plane strainrodin2014strain ; jiang2014negative , resulting in direct-indirect bandgap and semiconductor-metal transitionsPhysRevB.90.085402 ; rodin2014strain .
One of the extraordinary phenomena observed in 2D semiconducting materials is the high stability of excitons, both due to the confinement of electron and hole and to the reduced screening of the Coulomb interaction. Concordingly, excitons dominate the optical spectra of 2D semiconductors, and can even be detected at room temperature.ye-nature-513-214 ; chernikov-PRL-113-076802 Measured exciton binding energies (EBEs) on monolayer transition metal dichalcogenides on SiO are about 0.3-0.4 eV for sulfides and 0.6 eV for selenides,ye-nature-513-214 ; chernikov-PRL-113-076802 ; he-PRL-113-026803 ; wang-1404.0056 ; stroucken-1404.4238 and in vacuum are expected to reach values as high as 0.7 eV in WSye-nature-513-214 and 1.1 eV in MoS.komsa-PRB-86-241201 These values are much higher than in bulk ( 0.1 eV for WSbeal-jphysc-5-3540 ) and three dimensional semiconductors, where excitonic effects can be negleted from optical spectra to a good approximation. In suspendend (freestanding) phosphorene, the EBE has also been predicted to be 0.8 eV,tran-PRB-89-235319 even though phosphorene has a much smaller quasi-particle bandgap than WS or MoS.tran-PRB-89-235319 ; ye-nature-513-214 ; komsa-PRB-86-241201 Also recently, the phosphorene EBE was calculated with in-plane strain in elastic regime peeters-2014 .
In this article, we consider the electronic and optical properties of phosphorene with uniaxial out-of-plane strain, and show that under such conditions the exciton binding energy can still be further increased. Moreover, we show that in an extended range of applied stress (both in the elastic and plastic regime), and specially near the point when the bandgap vanishes, the exciton binding energies are comparable to the quasi-particle bandgap. Thus, it is necessary to take into account the strength of the exciton binding when interpreting the evolution of the luminescence under stress.
The strain regime investigated here is comparable to what would be achieved in ideal conditions using a diamond anvil cell (DAC), as shown in Fig. 1(b). The compressive force is applied perpendicular to the phosphorene plane, and uniformized by a pressure-transmitting medium. We adopt the assumption that for a 2D material, the strain imposed by such a device setup can be considered uniaxial. This approximation is based on the small lateral cross section of phosphorene, which is orders of magnitude smaller than the area of mechanically exfoliated flakes. Due to phosphorene’s high flexibility, strain as high as 22% can be reached in the DAC, thus entering into the plastic regime of the material.
Ii Methods
The exciton binding energies for the 2D material with static dielectric constant were calculated with the dielectric screening induced by surrounding materials with dielectric constants (above) and (below). The screening effect of these dielectric materials on the phosphorene can be measured by mean dielectric constant . The effective interaction between electrons and holes is given by the Hamiltonian
(1) |
where and are the reduced effective masses (reciprocal mean of effective electron and hole masses) in zigzag and armchair directions, and is the Keldysh potential given by
(2) |
where and are the Struve function and Bessel function of the second-kind, , and is the 2D electric susceptibility. The EBE was calculated applying the Numerov method to solve the problem with the screening Keldysh potential (2), as described in Ref. rodin2014strain, .
First-principles calculations based upon density functional theory (DFT)PhysRev.136.B864 ; PhysRev.140.A1133 were performed as implemented in the PWscf code of the Quantum Espresso packagegiannozzi2009quantum . Norm-conserved pseudisation of external potential were performed with the Troullier–Martins parameterizationtroullier1991efficient . We used energy cutoff of 70 Ry for the Kohn–Sham orbitals and -points grid of in Monkhorst–Pack algorithmPhysRevB.13.5188 for phosphorene single-layer and for the weakly interacting black phosphorusPhysRevB.90.075429 . For the exchange-correlation functional we used the Perdew–Burke–Ernzerhof (PBE) approximationperdew1996generalized . The strained phosphorene were completely relaxed until reaching the convergence criteria of residual forces smaller than meV/Å and residual total energies smaller than Ry. For the dielectric tensor calculation, we applied a scissor operator of 0.72 eV for the Kohn–Sham eigenvalues obtained from the comparison of the DFT bandgap from the GW calculationsPhysRevB.90.075429 . Hybrid functional calculations based upon HSE06heyd2003hybrid ; heyd2006erratum functional were performed with PBE relaxed geometries. The small -vectors divergence to the Coulomb potential was treated in the Gygi–Baldereschi approachgygi1986self , and the three dimensional -vector mesh was used for the Fock operator.
Iii Results
In order to understand the uniaxial out-of-plane strain effect on phosphorene, we studied the phosphorene electronic properties by varying the layer height , i.e. the distance between phosphorus planes on both sides of the same layer. The atomic positions and cell vectors were relaxed in plane under this fixed height constraint. The resulting stress-strain curve is shown in Fig. 1(c).
The stress was calculated by Hellmann–Feynman theorem forces acting on the Born–Oppenheimer potential energy surfaces. The forces on the phosphorus atoms in a unit cell with area result in a stress that can evolve non-linearly with the strain , where is the phosphorene height in vacuum (no pressure). As the strain increases, the lattice parameters and of the relaxed variable-cell change so that the area of the strained unit cell is greater than area 15.34 Å (in vacuum).
The ab initio stress data was fitted by polynomial for the stress-strain curve, shown in Fig. 1(c) by the solid blue lineNote1 . The Young’s modulus is obtained from the first-order derivative of the stress-strain curve at , i.e.,
(3) |
The phosphorene elastic regime is inferred from the tangent line with slope equal to the Young’s modulus, shown by the dashed black line in the Fig. 1(c). For strains of 3–4 % (proportionality limit), the stress-strain curve is approximately linear. In this range, we say that phosphorene is in the elastic regime. Above , the phosphorene is in the plastic regime. The stress-strain curve increases monotonically until a local maximum at . The maximum stress, also called Yield strength, is about 25 GPa.
An alternative approach to the direct calculation of the stress can be conceived by spacial average of Nielsen–MartinPhysRevLett.50.697 . However, instead of normalizing using the supercell volume (with vacuum spacing), we normalized by the effective volume , where is the phosphorene height under pressure and is the phosphorene unit cell area.
The Young’s modulus for the -direction (out-of-plane) is slightly larger than the average of the Young’s moduli found in Ref. wei2014superior, for the zigzag and armchair directions (in-plane strains). Despite this hardening in the -direction, the low Young’s modulus results in high flexibility and strongly tunable electronic properties with the uniaxial strain engineering.
To calculate the Poisson’s ratio, we use the lattice constants and obtained from the variable cell relaxation dynamics with fixed strain . Using this method, we obtain the Poisson’s ratio and at . The auxetic property (negative Poisson’s ratio) in the -direction can be explained by the hinge-like geometry of the chemical bonds between the phosphorus atomsjiang2014negative .
The band structures of phosphorene in equilibrium and under strain from 0 to 24 % are shown in Fig. 2(a), (b) and (c). In equilibrium, phosphorene shows three valleys (local minima) in the conduction band, labeled , and . The evolution of these valleys with relation to the valence band maximum (VBM) is shown in Fig. 2(d). For strains smaller than %, the band structure presents a direct bandgap at -point. For strains close to %, there a direct-indirect bandgap transition, with conduction band valley between and point. For strains larger than % and smaller than %, tha bandgap is indirect with minimum located between and point. From strains larger than % there is a semiconductor-metal transitionrodin2014strain . Futhermore, for strains greater than % the conduction band valley is located at -point. This shift in the conduction band valley location is responsible for the change in the curve slope at 20 % in Fig. 2(d).
We focus our attention on the excitons, that are expected to give rise to a luminescence peak throughout the whole range of strain, independently of the existence of a smaller indirect gap. The direct bandgap at takes lower values in two regions: (i) an approximately elastic region between 0 and 5 %, and (ii) plastic region from 20 % to 24 %. The exciton binding energies were determined in this region, using as input the 2D electric susceptibility and the effective masses obtained using density functional theory.
The 2D electric susceptibility was calculated as described in Ref. PhysRevB.90.075429, ; keldysh1979, ; berkelbach2013theory, . The 2D screening is characterized by the and parameters given by the variation of the electric permittivities and as a function of spacing between phosphorene layers. These 2D electric susceptibility are obtained from the fitting
(4) |
where is the unit cell height, ranging from Å to Å, as shown in Fig. 3. Note that the 2D electric susceptibilities and are more anisotropic in plastic regime than elastic regime. However, the exciton binding energy calculation depends only on the geometrical mean .
The phosphorene anisotropy can also be seen from the large variation of the electrons’ and holes’ effective masses as a function of strain. Although the 2D electric susceptibility is averaged, the interaction between electrons and holes remains anisotropic through their effective masses in the zigzag and armchair directions, as shown in Fig. 4(a). While in the elastic regime the electrons’ effective masses are light in the armchair direction and heavy in the zigzag direction, in plastic regime this is reversed. This reversion is explained by the band crossing between two conduction bands ( and ), as shown in Fig. 4(b). This conduction band crossing occurs between % and %.
In the armchair direction, for low strains, the holes’ and electrons’ effective masses are roughly equal, resulting in reduced effective masses ( state are so large that the reduced effective masses is approximately equal to holes’ effective masses (). In the zigzag direction, the holes’ effective masses are orders of magnitude greater than electrons’ effective masses, so that we have . ). However, for strains larger than 8 %, the electrons’ effective masses of
Based on the parameters and effective masses, we calculate the exciton binding energies (EBE) and luminescence energies () as a function of the strain , shown in Fig. 5. While the EBE increases 10 % for % (from 0.79 eV to 0.87 eV), the luminescence increases almost 50 % for the same strain (from 0.81 eV to 1.21 eV). This variation of luminescence energies is explained by the conduction band valley behavior with the strain, that increase of 30 % for % (from 1.60 eV to 2.08 eV), as shown in Fig. 2(d). The EBE also depends strongly on the dielectric media into which phosphorene is immersed, parameterized by the mean permittivity . The EBE and luminescence energies are shown in Fig. 5(b) in elastic regime and Fig. 5(c) in plastic regime.
Iv Discussion
We have shown that the exciton binding energies are comparable in magnitude to the quasi-particle gaps and are sensitive to strain. Thus, the variation of the position of the luminescence peaks under strain in the ideal DAC experiment that inspired our work is affected by those two components. The resulting photon energy increases with strain in the elastic regime (up to 5 %), in the plastic regime the trend is inverted and a redshift is observed.
In the elastic regime, the photon energy blueshifts by up to 0.4 eV for only 5 % strain, and its variation is approximatelly linear, and nearly independent on the permittivity of the substrate. This suggests that the shift or broadening of the direct luminescence peak can be used as a direct way to probe the strain state of the material. This is an excellent alternative to Raman measurements, since the phosphorene Raman peaks are little changed by armchair strain,fei2014lattice and is also affected by the number of layers and the proximity to the edge.
Finally, we note that the exciton binding energy can be increased to 0.87 eV in vacuum for a modest strain of 5 %. This value is in the range of exciton binding energies predicted for transition metal dicalcogenides, which however, have larger quasi-particle bandgaps.
Acknowledgements
L.S. acknowledges financial support provided by “Conselho Nacional de Desenvolvimento Científico e Tecnológico” (CNPq/Brazil). The authors acknowledge the National Research Foundation, Prime Minister Office, Singapore, under its Medium Sized Centre Programme and CRP award “Novel 2D materials with tailored properties: beyond graphene” (R-144-000-295-281). The first-principles calculations were carried out on the CA2DM and GRC high-performance computing facilities.
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