Bright excitons with negative-mass electrons

Bound electron-hole excitonic states are generally not expected to form with charges of negative effective mass. We identify such excitons in a single layer of the semiconductor WSe2, where they give rise to narrow-band upconverted photoluminescence in the UV, at an energy of 1.66 eV above the first band-edge excitonic transition. Negative band curvature and strong electron-phonon coupling result in a cascaded phonon progression with equidistant peaks in the photoluminescence spectrum, resolvable to ninth order. Ab initio GW-BSE calculations with full electron-hole correlations unmask and explain the admixture of upper conduction-band states to this complex many-body excitation: an optically bright, bound exciton in resonance with the semiconductor continuum. This exciton is responsible for atomic-like quantum-interference phenomena such as electromagnetically induced transparency. Since band curvature can be tuned by pressure or strain, synthesis of exotic quasiparticles such as flat-band excitons with infinite reduced mass becomes feasible.

One Sentence Summary: Unconventional high-lying bound excitonic states comprising negative-mass electrons and positive-mass holes are found in monolayer WSe2, which shows radiative recombination in the UV at an energy of almost twice the bandgap.

Main Text:
Mass is perhaps the most tangible and the most elusive concept of physics. For stable matter, mass is strictly positive. In quantum mechanics, however, interacting electrons in a medium can behave like free particles with an effective mass related to the curvature of their energymomentum dispersion relation. In periodic potentials, such as that of the crystal lattice of a semiconductor, the effective mass can become negative in certain parts of the bands, implying counterintuitive effects such as a tendency for opposite charges to accelerate apart rather than to attract via the Coulomb interaction. At sufficiently low temperatures, an electron excited to the conduction band and a hole in the valence band of a semiconductor can interact to form a stable correlated state -an exciton. Typically, excitons with stable bound states are made up of a positive-mass electron in the lowest-energy conduction bands and a positive-mass hole in the highest valence bands. In monolayer WSe2, such excitons can exhibit large optical transition dipoles and thus interact strongly with incident radiation(1,2). When driven by an intense laser pulse, discrete band-edge excitons in monolayer WSe2 have been found to exhibit signatures of quantum interference in their transition pathways (3). In addition, efficient Auger-like upconversion has been attributed to transitions to higher conduction bands (4). These experiments suggest that the lowest-energy, optically bright exciton couples strongly to a discrete long-lived state of approximately twice the energy of the lowest bandedge exciton. The resulting multilevel excitonic state structure of WSe2 is therefore reminiscent of that of optically driven atomic systems exhibiting electromagnetically induced transparency (EIT) (5). This purported high-energy state appears to behave like a bound -or very strong resonant -exciton, but it is not clear how such an exciton would remain stable for timescales exceeding 100fs as implied by the EIT phenomenon (3,6), at energies reaching far into the free-particle continuum around the K-points. Such an exciton appears particularly improbable when the independent-particle interband transition picture suggests that the only available electronic states with consistent energy and momenta are of negative effective mass. Figure 1A sketches the calculated abinitio GW quasiparticle band structure of monolayer WSe2 around the K-points in momentum space. The magnitude of the interband transition oscillator strength coupling the spin-orbit split top valence band to the different conduction bands is coded in color. The full band structure is shown in Fig.S1A. The lowest-energy band-edge "A-exciton" is characterized by a dipole-allowed optical transition between the valence-band maximum and the upper spin-split conduction-band minimum at the K-points in the Brillouin zone. The A-exciton can recombine radiatively, emitting a photon with an energy reduced with respect to the quasiparticle bandgap by the exciton binding energy, which can be as large as a fraction ofan eV for transition-metal dichalcogenide (TMDC) monolayers(1,2). The energy ofthis transition is indicated bythe lower red double-headed arrow. Signatures of excitonic quantum interference are observed in optical second-harmonic generation (SHG) under pumping by ultrashort pulses with photon energies just below the energy ofthis transition (3,6), suggesting the involvement of a second excitonic state approximately 1.7eV above the A-exciton (higher-energy red double-headed arrow). Within the independent-particle picture, the only electronic state close to this energy range is the lower spin-split CB+2 band in Fig.1A. This band has negative curvature. Our calculations in are allowed(2,7), with approximately 4% of the oscillator strength of the band-edge transition (CB + to VB + ) atthe K-points. While the idea ofa stable exciton involving a negative-mass electron seems counterintuitive, such a complex is possible within a simple effective-mass hydrogenic model provided the hole mass * is positive and smaller in magnitude than the negative electron mass * . The reduced mass of the exciton (1/ 1/ * 1/ * ) is then positive (see Fig.S2). Drawing an analogy to classical orbital motion, the semiclassical motion of the electron and hole may then be thought of as orbiting around a common center which does not lie geometrically between the two particles, so that, as indicated inthe inset of Fig.1B, the electron accelerates in the same direction as the hole.
A crucial difference between excitons comprising electrons of positive and negative effective mass lies in the coupling to phonons: since the band-edge exciton forms at the energetic minimum, the exciton cannot dissipate energy further to a final state by emitting phononsluminescence is therefore generally dominated by a "zero-phonon" line. The opposite is true for an exciton involving an electron from a band of negative curvature since it is in resonance with the finite-momentum electron-hole continuum: electrons tend to scatter into lowerenergy band states by emitting phonons. Assuming sufficient radiative decay rates(8), a phonon progression should appear inthe luminescence and the zero-phonon transition needs not be the maximum.
To selectively excite electron-hole pairs in the vicinity of the fundamental bandgap at the Kpoints, we use a linearly polarized narrow-band continuous-wave (CW) laser at 716nm (1.732eV), resonantly driving the A-exciton of monolayer WSe2 encapsulated in hexagonal boron nitride (hBN) (see Fig.S3 for sample details). As sketched in the bottom-left inset of Fig.1C, Auger-like exciton-exciton annihilation (4,9,10) can raise the electron to a higher band as momentum conservation localizes it around the K-points. The resulting upconversion photoluminescence (UPL) spectrum at 5K (Fig. 1C) shows a narrow peak at twice the excitation energy, i.e. at 3.46eV, limited in width bythe spectral resolution of the monochromator. This feature arises from CW SHG(11) originating from the broken inversion symmetry of monolayer WSe2. At an energy 60meV below this peak, UPL is observed. Ten narrow peaks are resolved with a mean linewidth of 11.4meV. SHG and UPL are discriminated by measuring the change in emission intensity copolarized with the laser as the laser polarization is rotated with respect to the WSe2 crystal (12). The right inset of Fig. 1C shows the characteristic six-fold symmetry of the SHG polarization dependence, arising from the three-fold rotational crystal symmetry. In contrast, UPL is isotropic and appears as a circle. We label this emissive species the "high-lying exciton" HX. The linewidth of the dominant HX peak can beas narrow as 5.8meV at low pump fluences as shown in Fig. S3C, and places a lower limit on the exciton coherence time of ~100fs. Resolving such narrow discrete luminescence peaks at almost twice the bandgap is unexpected since excitonic linewidth generally increases with transition energy as plotted in Fig.S3; higher-lying transitions such as theB, A' andB' excitons (13) are typically subject to a broader range of effective non-radiative relaxation channels, reducing lifetime. The connection between HX UPL and the A-exciton transition canbe established by sweeping the excitation energy over the A-exciton resonance. The top-left inset of Fig.1C shows the PL spectrum of the fundamental band-edge exciton (red line) in comparison to the PL excitation (PLE) spectrum of the HX emission (blue dots); PL and PLE spectra are virtually identical. As discussed in  (14). We propose that strong inelastic resonant electron-phonon scattering occurs between +K and K valleys (15,16), as sketched in Fig. 2B, C, and note that such intervalley scattering can occur in <100fs through a deformation potential (17)(18)(19). As in previous monolayer TMDC studies, ourGW band-structure calculations of monolayer WSe2 reveal that all the conduction bands and the valence bands around the K-points are spin split because of spin-orbit coupling. The combination of mirror-plane and time-reversal symmetry dictates that spin in the out-of-plane direction is a good quantum number with opposite orientations in the +K and K valleys (20). Two different phonon-scattering processes between valleys are thus conceivable to move a negative-mass electron of the high-lying exciton down in energy: one-phonon scattering ( Fig.2B), which requires a spin-flip(21,22); and a double-resonance two-phonon mechanism ( Fig.2C) (15), which conserves spin. The two-phonon process should be favored in luminescence over the one-phonon process, although we note that the helicity of the excitation may be destroyed by the fast exchange interaction(23,24) arising during the Auger-like population of the HX state. The spectrum in Fig.1C demonstrates that evennumbered peaks are indeed more intense than odd-numbered ones, implying a higher transition probability for two-phonon processes. Repeated electron-phonon scattering not only moves the electron downwards in energy, but also away from the K-symmetric points. The spin-valley locking is then relaxed, which, for higher-lying bands, persists only over a limited region of momentum space. The alternation in peak intensity is pronounced up to peak5.
Inspection of the Brillouin zone of single-layer WSe2, inset in Fig.2A, together with the phonon progression, supports the assertion that HX PL originates from the K-symmetric points. If instead the radiative state were formed around the or Mpoint, which lack spinvalley locking, there would beno obvious reason why the phonon intensity should alternate.
We note that the phonon progression can also be interpreted from the perspective of polaronic excitons. The question of whether theHX phonon progression should be rationalized in terms of excitonically bound charges, which emit phonons, or as polaronic excitons in split subbands (16), is mostly a matter of electron-phonon interaction strength and is left for further study.
A signature of the excitonic nature of optical transitions in TMDC monolayers is the existence of a series of excited states(1). Besides Auger-like excitation, it is also conceivable that twophoton absorption (TPA) can populate dark states that then undergo conversion to the radiative HXstate. As sketched in Fig.3A, for systems undergoing dipole-allowed interband transitions, TPA can address odd-parity p-like excitonic states, while one-photon absorption and emission probe even-parity s-like ones(25,26). In order for the two-photon excitation to relax to the radiative HX, the TPA energy must be no lower than the HX zero-phonon transition energy, i.e. around 3.4eV. There is no need for TPA to occur at exactly twice the energy of the A-exciton, i.e. to overlap with the one-photon resonant Auger-like mechanism in PLE. Figure 3B Fig.S6 and TableS1, the separation between p-like (two-photon transition) and s-like (one-photon emission) HX states varies between 27meV and 37meV across samples. No such change is seen in the spacing of the phonon progression, which is almost identical for all samples.
To establish the contribution of higher-lying conduction bands to the HX, we performed an ab initio GW-BSE calculation (27-29). Figure  -. The lower conduction bands have a much smaller contribution to the HX peak due to the relatively large energy separation between higher (CB+2) and lower conduction bands (CB and CB+1) in the K-valleys. As seen in the band structure in Fig.1,

CB+2
corresponds to an electron of negative mass. The inset in Fig.4D shows the envelope function of the HX wavefunction in momentum space. Its comparison with the A-exciton is shown in Fig.S8. FigureS8D projects the HX exciton envelope function onto the quasiparticle band structure and shows that the HX is localized around the K-points, where the effective-mass approximation is valid. The HX has an exciton radius(1) of 1.2nm, which is smaller than the 1.5nm radius of the1s A-exciton, consistent witha heavier reduced exciton mass arising from the negative-mass electron.
Since the electronic structure of many semiconducting monolayer TMDCs is qualitatively

Materials
We prepared the samples following the same procedure described previously (3,30). In brief, we exfoliated the monolayer WSe2 and thin layers of hBN from bulk crystals (WSe2, HQ Graphene; hBN, NIMS) on PDMS films (Gel-Pak, Gel-film® X4) using Nitto tape (Nitto Denko, SPV 224P). We stamp-transferred the flakes onto either Si/SiO2, sapphire or diamond substrates while heating the substrate to 65°C. The hBN-encapsulated WSe2 samples were annealed under high vacuum at 150°C for 5 hours.

Experimental methods
The setup is illustrated in Fig. S9. We used an objective of 0.6 numerical aperture (Olympus, LUCPLFLN) to focus the laser onto the sample and collect the signal. The sample was placed under vacuum on the cold finger of a helium-flow cryostat (Janis, ST-500). We used a grating of 600 grooves mm -1 or 150 grooves mm -1 to disperse the signals, and a CCD camera (Princeton Instruments, PIXIS 100) to record them. A 50:50 beam splitter was used to separate excitation and detection pathways. The photon energy-dependent instrument response in Fig.  S9B was measured with a calibration light source (LS-1-CAL, Ocean Optics) placed in front of the objective. The original spectra are shown in the main text and the supporting information, without correction for the instrument response.
We measured the photoluminescence (PL) of monolayer WSe2 by exciting samples with an argon-ion laser (Spectra Physics, 2045E) at 488 nm and filtering out the laser line from the signal with a 488 nm long-pass edge filter. The reflectance contrast of monolayer WSe2 was measured using a broad-band Xenon lamp (EQ-99X, Energetiq). We measured the upconverted PL and SHG of monolayer WSe2 by exciting samples with a tunable continuous-wave laser (Sirah, Matisse CR) and filtering out the laser line using a 680 nm short-pass filter.

Ab initio GW and GW-BSE Calculations
Density-functional theory (DFT) Kohn-Sham wavefunctions for monolayer WSe2 are calculated using the Quantum ESPRESSO (31) package as the starting orbital energies and wavefunctions for our GW calculation. For the DFT calculations, we use a plane-wave basis set and norm-conserving pseudopotentials. Scalar-relativistic (SR) pseudopotentials are used for spin-unpolarized scalar-relativistic calculations, and fully relativistic (FR) pseudopotentials are used for fully relativistic noncollinear spinor calculations. The generalized gradient approximation (GGA-PBE) (32)(33)(34) is used for the electron exchange and correlation energy. To accurately capture the exchange contribution to the GW quasiparticle (QP) self-energy of the monolayer WSe2, semi-core 5s, 5p, and 5d states are included in the pseudopotentials of W in addition to the 6s and 6p valence electrons. The plane-wave cutoff for the DFT calculation is set to 80 Ry for the plane-wave expansion of the wavefunctions. The length of the periodic supercell is set to be Lz = 120 Å. The crystal structure of monolayer WSe2 (35) has in-plane lattice constants of 3.2820 Å and an atomic-plane to atomic-plane Se-Se distance of 3.3411 Å. The quasiparticle (QP) self-energies and exciton excitation energies are, respectively, computed using the ab initio GW and GW plus Bethe-Salpeter equation (GW-BSE) approaches as implemented in the BerkeleyGW (27-28, 36) package. In the GW and GW-BSE calculations, the Coulomb interaction beyond 60 Å in the z-direction (i.e., out-of-plane direction) is truncated to prevent spurious interactions between periodic images (37). The calculated QP band structure is plotted in Fig. S1 and Fig. 1A.

S-3
To calculate the excitation energies of the excitons, the Bethe-Salpeter equation (BSE) (28, 36) is solved by using the above-calculated QP energies and spinor DFT wavefunctions. The electron-hole interaction kernel of the BSE Hamiltonian is first calculated on a uniform kgrid of 72×72, using two valence and eight conduction bands and a dielectric matrix that is calculated on a uniform q-grid of 72×72, summed over 1400 bands and using a 5-Ry-planewave cutoff. The BSE Hamiltonian is diagonalized using interaction kernel matrix elements that are interpolated (28, 38) from the uniform 72×72 k-grid to a finer uniform k-grid of 120×120, using directly calculated matrix elements for q-points of a density equivalent to 150×150. Using the calculated exciton wavefunctions and excitation energies, the imaginary part of the dielectric function is calculated using Im ∑ | ⋅ ⟨0| | ⟩| Ω , where ⟨0| | ⟩ is the optical velocity matrix element between exciton | ⟩ and ground state |0⟩, Ω is the energy of state | ⟩, is the direction of the polarization of light, and is the crystal volume. The velocity matrix element of the exciton, ⟨0| | ⟩, is a coherent superposition of transitions between non-interacting electron-hole pairs ⟨ | | ⟩ , given by ⟨0| | ⟩ Ω ∑ , where and | ⟩ are the Kohn-Sham eigenvalues and eigenfunctions, respectively. To obtain the absorption spectrum from a selectively excited part of momentum space, we calculate the mode-decomposed (MD) imaginary part of the dielectric i.e., the band-and k-resolved contributions to the dielectric function. To calculate the absorbance, the real part of the dielectric function is firstly obtained from the imaginary part using the Kramers-Kronig relation. Using the dielectric function , we calculate the extinction coefficient . Finally, the absorbance spectra A ω , as defined by A ω log / 100%, are plotted in Fig. 4 of the main text and Fig. S7.
In Fig. 4D, the plots are made for contributions to the absorbance due to the subset of that corresponds to parts of momentum space that are circular patches of 0.2 Å -1 radius centered at the K-valleys (highlighted in Fig. S1A), involving only transitions from the highest valence band. In Fig. 4E, we plot the mode-decomposed oscillator strengths for each contributing conduction band, where the area of each disk is proportional to the integrated oscillator strengths from contributing excitons, where the oscillator strength of each excitonic state, | ⟩, is given by ⋅ 0 . To analyze the exciton radius, we calculate the root mean square radius of the exciton envelope function in real space. In Fig. S8C, we show the momentum-space envelope functions of three individual excitons arising primarily from CB, CB+1, and CB+2 in the vicinity of the K-valleys. Notably, the CB band curvature is smaller than the CB+1 band curvature, and CB+2 has negative curvature. Therefore, the reduced mass of the CB exciton is smaller than that of the CB+1 exciton, and the reduced mass of the CB+1 exciton is smaller than that of the CB+2 exciton (HX). This increase in mass is consistent with the fact that the CB exciton has a larger exciton radius than the CB+1 exciton and the CB+1 exciton has a larger exciton radius than the CB+2 exciton. The CB+2 exciton also has a larger binding energy of 0.6 eV than the CB+1 exciton (0.55 eV) and the CB exciton (0.45 eV). These values will decrease with hBN encapsulation because of increased dielectric screening.

Effective-mass model
In order to investigate the stability of the HX within the effective-mass limit, we restrict ourselves to the two bands with the largest contribution, i.e. VB + and CB+2around the Kpoints, based on the GW-BSE calculations given in Fig. S1, Fig. 1A and Fig. 4. The effective S-4 masses are calculated from the GW band structure, leading to h * 0.3636 for VB + and e * 0.4604 for CB+2 -. These values fulfill the condition | * | | * | and thus the reduced mass of the exciton, * h * / * * , remains positive. For illustration purposes, we investigate the effect of | * | | * | by simply arbitrarily interchanging the two values. The excitonic states are obtained by solving the effective BSE (28, [39][40], given by where e ⃗ 0 ℏ 2 2 /2 * , h ⃗ ℏ 2 2 /2 h * , and ⃗ is the envelope function of the N-th exciton state. The electron-hole interaction ⃗ ′ ⃗ is described by the Rytova-Keldysh potential (41-43) in which is the unit area, e is the electron charge, 0 is the vacuum permittivity, 0 is the screening length of the 2D material, and ε is the effective dielectric constant. For the calculations, we assume a bare WSe2 monolayer, i.e. ε 1 and r 0 45.1 Å (44). We solve the effective BSE numerically in a 2D k-grid from -0.5 to 0.5 Å -1 in kx and ky directions with 121ⅹ 121 points (leading to a spacing between k-points of ∆ = 0.5/60 ≈ 0.0083 Å −1 ). To improve convergence, we average the Coulomb potential around each k-point in a square region of −∆ /2 to ∆ /2 sampled with 121ⅹ121 points.
In Fig. S2A, we present the electronic band structure within the effective-mass approximation for the two conditions | * | | * | (solid blue curves) and | * | | * | (dashed grey curves). In Fig. S2B we show the energy difference between conduction and valence bands, e ⃗ h ⃗ , i.e. the diagonal contribution of the BSE equation in Eq. (1). The two different mass conditions thus lead to distinct effective curvatures. From this analysis we can already envision localized excitons around 0 for | * | | * |, whereas for | * | | * | the excitons would be delocalized at the edges of the k-region we considered in the calculations. The fundamental exciton (N = 1) envelope function is shown in Fig. S2C, supporting our expectations of bound excitons around 0 for the condition of | * | | * | and no bound exciton for the condition of | * | | * |. We note that the s-like shape of the exciton envelope function obtained within this effective description is consistent with the results obtained from the ab initio GW-BSE calculations.

Phonon calculation and electron-phonon coupling
Based on state-of-the-art DFT, we perform first-principles calculations using the Vienna ab-initio simulation package (VASP) (45). We use the projector-augmented-wave potential method with tungsten 5p 6 5d 4 6s 2 and selenium 4s 2 4p 4 valence states, together with the PBE-GGA parameterization for the exchange correlation functional (32). A plane-wave basis set with a kinetic energy cutoff of 500 eV and a 15×15 k mesh over the electronic Brillouin zone (BZ) leads to converged results. For structural relaxation of the unit cell, we consider energy differences converged to within 10 -6 eV and Hellmann-Feynman forces converged to within 10 -4 eV/Å. We obtain the harmonic interatomic force constants with density functional perturbation theory using a 5×5 supercell with a k-point sampling of 3×3 of the electronic BZ. The phonon dispersion and eigenvectors are calculated using the PHONOPY package (46).

S-5
We estimate the strength of electron-phonon coupling by building a smallest-possible nondiagonal supercell that includes the K-point in the vibrational BZ (47): where as and bs are the lattice parameters of the supercell, and ap and bp are the lattice parameters of the unit cell. Using this supercell, we calculate the electronic energy levels at the electronic K-points as a function of the normal mode amplitude of the LA phonon mode at the vibrational K-point. The normal-mode amplitude is defined as (48) 1 where and are reciprocal-space phonon wavevector and branch, respectively, Np is the number of primitive cells in the real-space supercell, is the position vector of unit cell p, is the nuclear mass of atom α, i runs over Cartesian coordinates, ℎ is the displacement coordinate, and ; is the corresponding eigenvector.   Fig. 4. A detailed analysis of exciton PL linewidth is provided in Table S1.
S-9 ). The dependence is much stronger for the dark exciton X D than for the neutral A-exciton AX and the trion Xintravalley, since X D is characterized by an out-of-plane dipole. The dependence on iris size is much weaker for HX than for X D , implying that HX emits via an in-plane dipole.

Fig. S7.
Contributions of the spin-split valence band to exciton formation. Calculated absorbance spectrum (in arbitrary units) of bare monolayer WSe2 from transitions around the K-points from either VB + (blue) or VB ± (yellow) to all eight conduction bands CB ± , CB+1 ± , CB+2 ± and CB+3 ± . Top panel: plot showing full absorbance spectrum. Bottom panel: close up to show more clearly the absorbance near the HX transition. In addition to the HX from VB + , a shoulder from VBwith much weaker oscillator strength overlaps with the HX (black arrow).  . The opacity is proportional to the amplitude of the envelope function and the color denotes the phase according to the scale bar. The black dot marks the K-point. The HX evidently has a larger radius in reciprocal space and thus a smaller Bohr radius compared with the A-exciton. The scale bar is 0.2 Å -1 . (D) Even though the HX transition has a dominant contribution from CB+2, the HX also has certain wavefunction amplitudes from CB+1. The figure shows the momentum-space envelop function of the CB+1 contribution to the HX. The wavefunction shows nodes revealing its p-like (i.e. odd-parity) nonemissive nature. The contribution of CB+1 to the HX in the absorbance spectrum is therefore minor as seen from the yellow curve in panel B. (E) Conduction bands of monolayer WSe2 with the projection of the modulus-squared exciton envelope function of HX onto the CB+2band and of AX onto the CBband, normalized at the K-point. The momentum space is set to the range used to identify the HX in the calculated absorbance spectrum in (A), i.e. within a range of 0.2 Å -1 around the K-points corresponding overall to 6 % of the total area of the Brillouin zone. The HX is clearly localized around the K-points, as seen by the red amplitudes in valence and conduction bands. The effective-mass approximation is therefore still valid.
S-14  S-15 Table S1. Energies and linewidths (FWHM) of the excitonic states A, B, A', B' and HX extracted from the PL and UPL spectra in Fig. 4