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Dynamical Excitonic Effects in Doped Two-Dimensional Semiconductors Shiyuan Gao, Yufeng Liang, Catalin D. Spataru, and Li Yang Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b02118 • Publication Date (Web): 01 Aug 2016 Downloaded from http://pubs.acs.org on August 2, 2016

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Dynamical Excitonic Effects in Doped Two-Dimensional Semiconductors Shiyuan Gao,† Yufeng Liang,‡ Catalin D. Spataru*,∗,¶ and Li Yang*∗,† Department of Physics, Washington University in St. Louis, St. Louis, Missouri 63130, USA, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA, and Sandia National Laboratories, Livermore, California 94551, USA E-mail: [email protected]; [email protected]

KEYWORDS: Exciton, 2D Material, Doping, Dynamical Effects, Bethe-Salpeter Equation Abstract It is well-known that excitonic effects can dominate the optical properties of two-dimensional materials. These effects, however, can be substantially modified by doping free carriers. We investigate these doping effects by solving the first-principles Bethe-Salpeter Equation. Dynamical screening effects, included via the sum-rule preserving generalized plasmon-pole model, are found to be important in the doped system. Using monolayer MoS2 as an example, we find that upon moderate doping, the exciton binding energy can be tuned by a few hundred meVs, while the exciton peak position stays nearly constant due to a cancellation with the quasiparticle band gap renormalization. At higher doping densities, the exciton peak position increases linearly in energy and gradually merges into a Fermi-edge singularity. Our results are crucial for the quantitative interpretation of optical properties of two-dimensional materials and the further development of ab initio theories of studying charged excitations such as trions. ∗ To

whom correspondence should be addressed of Physics, Washington University in St. Louis, St. Louis, Missouri 63130, USA ‡ Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA ¶ Sandia National Laboratories, Livermore, California 94551, USA † Department

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One of the most prominent features of two-dimensional (2D) materials is the enhanced manybody interactions because of quantum confinement and the reduced electronic screening. This is evident from the significant shift of optical absorption spectra of semi-metallic graphene 1,2 and the huge exciton binding energies in monolayer transitional metal dichalcogenides (TMDCs) and black phosphorus in an order of a few hundred meVs. 3–9 Meanwhile, doping is critical to the proper functioning of electronic and photonic devices. It is widely observed in 2D materials by either inevitable defects or intended electrostatic 10–16 and chemical 17–19 processes. The doped free carriers make the structure more “metallic" and impact a wide range of many-body interactions, including quasiparticles (QP), excitons, and higher-order excitations such as trions and biexcitons. 12–15 In particular, the enhanced van Hove singularity (vHS) in reduced-dimensional structures can boost doping effects, evidenced by the huge renormalization of excitonic effects in doped carbon nanotubes. 20,21 Recent experimental measurements have raised more questions about excitonic effects in doped 2D materials, e.g. settling the discrepancies in excited-state properties as measured under various doping conditions. 3,19 Unfortunately, limited progress has been made towards this goal. Effective Hamiltonian theories developed for quantum wells can only qualitatively explain the trend of the spectral evolution, 22–24 and models of effective static electron-hole (e-h) potential could reproduce the binding energy of small carrier complexes 25–27 but not their doping dependence. The ab initio GW+Bethe-Salpeter Equation (BSE) approach 28–30 has been very successful in predicting the QP band gaps and optical properties of intrinsic (undoped) 2D materials, 6–8 but it remains elusive how this framework can be extended to excitonic properties in doped 2D materials. It motivates us to further develop this approach to describe the full evolution of optical properties of 2D semiconductors from intrinsic to heavily-doped cases. In this work, we have developed a generalized plasmon-pole (GPP) model for capturing the essential dynamical screening, making it possible to efficiently calculate the excitonic properties (energies and oscillator strength) in doped 2D systems. With this methodological advancement, we focus on monolayer MoS2 , a prototypical 2D semiconductor of broad interest. We reveal the


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importance of including dynamical effects for describing the effective screened interaction within the e-h pairs in the presence of extra charge carriers. The calculation shows good agreement with experiments in the evolution of the bright exciton energy according to the doping density, while it also raises questions of interpretations of the observed exciton energy spectrum.

Figure 1: Schematic diagram showing the doping effects, including a reduction of QP band gap Eg , a reduction of exciton binding energy Eb , and a rise of e-h continuum energy Econt relative to the band gap due to Pauli blocking. Changes in the exciton energy Ω is a combination of these effects. Doping can impact excitonic effects and optical spectra through several mechanisms, as shown in the schematic diagram in Figure 1. First, within the single-particle picture, the Pauli blocking effect (Burstein-Moss shift 31,32 ) of doped carriers raises the e-h continuum energy (Econt ) relative to the band gap linearly, due to the constant 2D density of states. More importantly, reduceddimensional systems are susceptible to the changes of electronic screening, which can result in large renormalization in the excited state properties. 33–35 On one hand, screening from the doped free carriers can induce a large nonlinear QP band gap renormalization (BGR) due to the carrier plasmon, 36 and results in negative electronic compressibility. 37–39 On the other hand, screening 3 ACS Paragon Plus Environment

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results in the reduction of exciton binding energy, which is the center of our discussion. Interestingly, experimental measurements show that the energy of the neutral exciton stays nearly constant in various doping conditions, with a small and linear blue shift roughly following the Fermi energy, 12–16 suggesting an nearly exact cancellation between the change in QP band gap and exciton binding energy. As we will see, this cancellation cannot be trivially reproduced from the widely used static BSE; dynamical corrections, arising from the strong correlation effects of doped carriers, need to be treated carefully. The widely used GW+BSE approach 29,30 starts with the G0 W0 approximation to the QP self−1 energy, where the (inverse) dielectric function εGG ′ (q, ω ) and the screened Coulomb interaction −1 ′ WGG′ (q, ω ) = εGG ′ (q, ω )v(q + G ) is constructed from the random-phase approximation (RPA).

Beyond single-particle excitation, for studying excitonic effects, the BSE can be formulated as a generalized eigenvalue problem: (Eck − Evk )ASvck +

Kvck,v′ c′ k′ (ΩS )ASv′ c′ k′ = ΩS ASvck , ∑ ′ ′ ′

(1)

vck

where the correlated e-h excitation S of energy ΩS is expanded on the basis of e-h pairs |S⟩ = ∑ ASvck |vck⟩, and v and c stand for the valence and conduction band index, respectively. Here we have restricted our discussion within the e-h excitations under the Tamm-Dancoff approximation. 29 The e-h interaction kernel K is dominated by the attractive direct term: S ′ d S ∗ ′ ˜ −1 Kvck,v ′ c′ k′ (Ω ) = − ∑ Mc′ c (k, q, G)Mv′ v (k, q, G )ε GG′ ;cvc′ v′ k (q, Ω )v(q + G ),

(2)

GG′

where q = k′ − k and Mn′ n (k, q, G) is the plane-wave matrix element containing the band structure information. 40 The dynamical effects are incorporated into the effective dielectric function, which is given by (neglecting finite lifetime effects): 20,21,41,42 ∫

∞ 1 −1 S −1 −1 ε˜GG (q, Ω ) = ε (q, 0) − P d ω ImεGG ′ ;cvc′ v′ k ′ (q, ω )× GG′ π 0 2 1 1 + S ]. [ + S ω Ω − ω − (Eck+q − Ev′ k ) Ω − ω − (Ec′ k − Evk+q )


(3)

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The static approximation for BSE, which neglects the last term in this equation, is commonly used for intrinsic semiconductors and typically reproduces the excitonic properties accurately because the differences ΩS − (Ec − Ev ) can be neglected in comparison to the characteristic energy of the −1 29 loss function ImεGG ′ (q, ω ) (the plasmon energy).

However, the approach described above cannot fully capture the excitonic effects when doping is introduced, due to the following reasons: First, the acoustic carrier plasmon in the 2D electron gas (2DEG), which falls on a similar energy-scale as that of the exciton binding energy, dynamically couples with QP and e-h excitations. 43–45 This corresponds to a characteristic dynamical screening time similar as the e-h scattering time, which invalidates the static screening approximation. Second, beyond excitons, the success in describing the trion binding energy in doped monolayer TMDCs by an effective pairwise interaction of intrinsic systems 25,26 irrespective of the doping density, contradicts the simple picture of static free-carrier screening. It has been suggested that dynamical correlations of excitons could explain this puzzle. 46,47 Third, the correlation effects naturally grow stronger in the low doping limit as the interaction energy of electrons dominates over the kinetic energy, 48 making vertex corrections beyond RPA more important in the screening process. To date, the importance of dynamical effects has been noted in bulk noble metals 49 and doped semiconducting carbon nanotubes, 20,21 but has never been studied in 2D materials. The BSE is hardly solvable with the dynamical effects in the form of Eq.(3), but it can be greatly simplified if we make a plasmon-pole approximation (PPA) to the dielectric function. Assuming there is only a single plasmon-pole −1 εGG ′ (q, ω ) = δGG′ +

1 AGG′ (q) 1 [ − ], + π ω − ω˜ GG′ (q) + i0 ω + ω˜ GG′ (q) + i0+

(4)

Eq.(3) can be simplified into: S Eint 2 −1 S −1 ′ (q, E ) = (q, 0) − , ε˜GG ε A (q) ′ int GG′ S ) π GG ω˜ GG′ (q)(ω˜ GG′ (q) + Eint

(5)

where ω˜ GG′ (q) and AGG′ (q) are the frequency and (negative) amplitude of the pole, respectively. 5 ACS Paragon Plus Environment

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Figure 2: (a) The RPA static dielectric function at different doping densities. (b) The effective di−1 electric function ε˜00 (q, Eb ) from GPP-BSE for the primary excitonic state compared with intrinsic −1 and doped static dielectric functions ε00 (q, 0). Doping densities of 0.03 × 1013 cm−2 (green) and 0.60 × 1013 cm−2 (red) are shown. Here we have also approximated the terms on the denominator of Eq.(3) by (Eck+q − Ev′ k ) − ΩS ≈ S , where E S = (E − E )|AS |2 − ΩS is the average e-h interaction (Ec′ k − Evk+q ) − ΩS ≈ Eint ∑ ck vk int vck

energy which approximately equals to the binding energy EbS = Econt − ΩS for tightly bound states. It becomes clear from Eq.(5) that the dynamical effect serves as a positive correction to the static dielectric function, and weakens the screening to the e-h interaction compared with the static approximation. The BSE can now be solved self-consistently as a generalized eigenvalue problem with only a few self-consistent steps additional to the regular problem, if we are interested in a few optically active excitonic states which typically dominate the optical spectra. Due to the elusive nature of vertex corrections beyond RPA, we employ the single-pole HybertsenLouie GPP approximation 28 to the dielectric function in calculating e-h interactions, which preserves the generalized f-sum rule, an exact constraint to the dielectric function to all orders in the diagrammatic expansion, thus including correlation effects beyond RPA. 50 We will call this approximation the GPP-BSE approximation, as opposed to the commonly used static BSE (S-BSE) approximation. It is important to point out that a tentative alternative choice to the form of PPA


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may be the low-energy acoustic plasmon of the 2DEG that matches the full-frequency RPA. However, this approximation hugely underestimates the screening to e-h interactions down to almost the same level as the intrinsic system, thus leading to a large nonlinear variation of exciton energy due to BGR and almost no decay in exciton oscillator strength (see Supporting Information for detials). Note that the GPP approximations here is validated a posteriori through comparison with experiment, and its disagreement with the full-frequency RPA in both GW 51 and BSE should not be regarded as a drawback of GPP, but rather an indication of the importance of vertex corrections. With the GPP-BSE approximation, the head part of the effective dielectric function is simplified to be −1 S −1 ε˜00 (q, Eint ) = ε00 (q, 0) +

where ω p (q) ≈

S [1 − ε −1 (q, 0)]3/2 Eint 00 √ , −1 S ω p (q) + Eint 1 − ε00 (q, 0)

(6)

√ 2π n2 q is the 2D plasma frequency and n2 is the total 2D charge density. Con-

tributions from the parts with nonzero G-vector are small enough to be neglected. 36 The static −1 dielectric function ε00 (q, 0) of the intrinsic system, shown in Fig.2(a) for monolayer MoS2 , ap-

proaches 1 as q → 0, signifying the vanishing long-range screening effects in 2D systems, whereas it drops to 0 in the doped system, signifying the metallic screening. After including dynamical effects, however, the effective dielectric function of the doped system, shown in Fig.2(b), diverts from its static value and rises sharply to 1 as q → 0. It results from the 2D plasmon dispersion where plasmon energy vanishes as q → 0, which delineates the frequency-range within which the doped carriers can respond. The consequence is a reduced effective screening as the carriers are unable to catch up with the dynamics of the e-h pair. Figure 3 summarizes the resulting energy of the primary exciton state of n-doped monolayer MoS2 versus the doping density n, calculated from the GPP-BSE and S-BSE approximation. On the DFT level, rigid-band doping is used to mimic electrostatic doping. The QP band gap is determined using the G0 W0 approximation with the same GPP approximation to the frequencydependence of the dielectric function. The k-point sampling grid is 48 × 48 × 1 for calculating the QP energies, 24 × 24 × 1 for the coarse grid and 120 × 120 × 1 for the fine grid of e-h interaction


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a)

2.5

Exct(GPP-BSE) Exct(S-BSE) Econt

Energy (eV)

2.4 2.3 2.2 2.1 2.0 1.9 0.0

0.5

1.0 n(1013cm-2)

b) 0.6

Eb(eV)

0.2 0.0 -0.2

3.0

4.0

0.5 0.4 0.3

1s (GPP-BSE) 2s (GPP-BSE)

0.2 0.1

-0.4 -0.6 0.0

2.0

c) 0.6

0.4 ΔE(eV)

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1.0 2.0 3.0 13 -2 n(10 cm )

4.0

0.0 0.0

0.5 1.0 2.0 3.0 4.0 n(1013cm-2)

Figure 3: (a) The energy of the band continuum edge and exciton as a function of the doping density from S-BSE and GPP-BSE. (b) Cancellation between BGR (∆Econt , lower curve) and exciton binding energy reduction (−∆Eb , upper curve) which determines the change in exciton energy in GPP-BSE. Dashed line corresponds to the complete vanishing of exciton binding energy. (c) Binding energy of the 1s and 2s excitonic states as a function of the doping density from GPP-BSE.


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kernels, which converges the band gap and exciton binding energy within 20meV. A slab truncation scheme 52 is employed to avoid artificial interactions between suspended monolayers. A spin-orbit splitting of ∆SO = 160meV, which is not affected by the GW corrections, 53 has been included as a rigid shift to the exciton energy. Figure 3(a) shows the evolution of e-h continuum and exciton energy. The most striking feature is that the exciton energy within the GPP-BSE approximation is nearly flat at low doping densities and slightly increases at higher doping densities. This nearly fixed exciton energy at the low doping density comes from an almost exact cancellation between the nonlinear BGR and exciton binding energy reduction, as shown in Fig. 3(b). For higher doping densities, the reduction of the exciton binding energy is nearly saturated while the e-h continuum energy still increases linearly due to Pauli blocking. Thus the exciton energy also increases linearly. In comparison, the exciton energy calculated from S-BSE shows a wiggling blueshift at low doping densities, as shown in Fig. 3(a). This difference shows that the overestimation of screening from the static e-h interaction kernel, as seen by comparing the static versus effective dielectric function in Fig. 2(b), is indeed important in doped systems. It is worth noting that dynamical effects also reduce the exciton binding energy of intrinsic monolayer MoS2 by 40meV, which quantifies the margin of error expected from the common static approximation in intrinsic 2D semiconductors. It also needs to be noted that the electron-phonon interaction is neglected in our calculation, which may introduce an extra variation of optical properties. 54 However, phonon modes will not be abruptly changed by doping in MoS2 . 55 Thus we expect the electron-phonon coupling does not significantly change our results. Excitonic effects are also generally more robust when dynamical effects are included. For example, as marked in Fig. 3(a), in S-BSE the exciton merges into the e-h continuum and dissociates before the doping density reaches 0.6 × 1013 cm−2 , while in the GPP-BSE it survives until the doping density reaches 4 × 1013 cm−2 . Considering the screening from the dielectric environment that could further reduce exciton binding, 34,35 such a density (4 × 1013 cm−2 ) is expected to be the upper limit for exciton dissociation in experiments.


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Figure 4: (a) Exciton energy shift from different approximation compared with experi12,15,16 ments. (b) Exciton oscillator strength change from different approximations compared with WS2 reflectance contrast spectra experiment. 16 Interestingly, the exciton energy spectrum does not scale linearly with the doping density. The 2s exciton state is far more sensitive to the doping level, as shown in comparison to the 1s state in Fig.3(c). It quickly vanishes when the doping density reaches around 1012 cm−2 , leaving the 1s state as the only bound excitonic state. This is because a weaker bound state has slower dynamics and thus allows more time for the carrier screening to catch up, and consequently feels a stronger effective screening, as given by Eq.(6). As a result, we find the higher quantum-number excitonic states are more unstable against doping than the primary 1s state. This is different from a recently reported experimental finding in WS2 , 16 in which the higher excitonic states would survive large doping densities. We speculate from their extremely small amplitude that the experimental 2s and 3s states may be pinned to defects, although more works are needed to settle this discrepancy definitely. In Fig. 4, the change of exciton energy and oscillator strength relative to the intrinsic system in our calculation are directly compared to experiments on doped monolayer MoS2 and WS2 . The GPP-BSE approximation achieves remarkable agreement with experiment in exciton energy within our numerical accuracy, while the e-h binding energy have changed a few hundred meV . It also reproduces the oscillator strength of exciton better than the S-BSE approach, although there’s a small disagreement at large doping density, which may emerge from the many-body effects beyond 10 ACS Paragon Plus Environment

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GPP such as the coupling with trions. As experimentally observed in the optical spectra of quasi-2D semiconducting quantum wells 56–59 and monolayer TMDCs, 12–16 when the doping density increases, spectrum weights are first transferred from exciton to trion, and then the exciton and trion peak in the spectrum merge into an asymmetric line shape called the Fermi-edge singularity (FES). This transition from discrete bound states to FES has been studied in model systems for quantum wells. 22–24,60 As is shown in Fig. 5, the spectra obtained from the GPP-BSE approximation are in stark contrast with the ones obtained from the S-BSE. Apart from the aforementioned exciton energy levels, the spectra differs markedly in their absorption edges as they evolve from discrete symmetric excitonic peaks to continuous asymmetric FES. At high doping densities, the S-BSE predicts a complete absence of excitonic effects, as its absorption line merges with the single-particle prediction. However, in the GPP-BSE a broad FES is retained at higher doping densities, because the important dynamical many-electron response is rectified by our effective dielectric model. Fig. 5 also shows how the real-space exciton wavefunction evolves with doping. As the doping density increases, apart from a slightly wider spread due to weaker binding, an Airy-type pattern also emerges due to the Pauli blocking in k-space. Finally, we have to address that, in the GPP-BSE approximation, the dynamical effects are treated equally in GW and BSE, which also improves the cancellation between QP self-energy and excitonic correction. 49,61 It should be noted that the GPP-BSE is only a crude approximation to the complicated dynamical response of the many-body system, where more higher-order vertex corrections come into play at lower doping densities, as the dimensionless Wigner-Seitz radius rs increases with decreasing density. On the other hand, as the doping density increases, the effects of vertex corrections are lessened and RPA becomes a better approximation, and consistently the difference between GPP-BSE and S-BSE becomes smaller. Graphene, on the other hand, has a constant rs due to its linear dispersion, 62 which suggests that the vertex corrections to its dielectric response are small for all doping density. And indeed, static BSE has been able to capture the optical response of doped graphene. 63,64


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Figure 5: (left) The evolution of absorption line shape with doping, calculated from S-BSE (yellow) and GPP-BSE (blue). The dashed line represents the absorption calculated without e-h interaction. The arrows donate the e-h continuum energies. A 0.03eV gaussian broadening is applied to the spectrum. (right) Modulus squared real-space wavefunction of the primary exciton state from solving GPP-BSE. It is plotted as a function of electron position with the hole fixed at the center and is integrated out along the off-plane z-direction. 12 ACS Paragon Plus Environment

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In conclusion, we have shown that the inclusion of dynamical excitonic effects and beyondRPA screening in BSE is crucial for correctly studying the optical properties of doped 2D semiconductors. We have developed the sum-rule-preserving GPP-BSE approximation to these effects, which produces good agreement with experiments. Our result, in particular the evolution of exciton binding energy, is important for interpreting experimental measurements and quantitatively understanding and predicting the doping effects in 2D semiconductors. Moreover, our method paves the way of understanding electronic structures of doped 2D devices and further studies on charged excitations, such as trions, in doped materials.

Acknowledgement We acknowledge the fruitful discussions with Vy Tran, Ruixiang Fei, Giovanni Vignale and Willem H. Dickhoff. S.G. and L.Y. are supported by the National Science Foundation (NSF) CAREER Grant No. DMR-1455346 and NSF EFRI-2DARE-1542815. This work is also supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. S.G. thanks James Bartz for the internship opportunity at SNL. The computational resources have been provided by the Stampede of Teragrid at the Texas Advanced Computing Center (TACC) through XSEDE. The authors declare no competing financial interests.

Supporting Information Available We provide the ab initio calculation details and a discussion of the effective screening to eh interaction within RPA. This material is available free of charge via the Internet at http: //pubs.acs.org/.


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