# Rising photoluminescence from the dark-exciton phonon reproduction in monolayer WSe 2

### Emergence of the dark-exciton reproduction in monolayer WSe2

We fabricate the BN/WSe2/BN van der Waals (vdW) heterostructure by a dry pickup technique which avoids exposing any vdW interface to polymer27,28,29. Two items of few-layer graphene have been used because the contact electrode to monolayer WSe2 and the clear high gate electrode, respectively, with the highest BN layer working because the dielectric. A schematic illustration of the gadget is proven in Fig. 1a. With the continual wave (CW) laser excitation centered at 1.879 eV and a low excitation energy of 60 μW, low temperature (four.2 Okay) PL spectra of the gadget proven in Fig. 1b resolve distinct peaks from completely different excitonic complexes, together with the darkish exciton30,31,32,33,34. The presence of the darkish exciton arises from the distinctive band construction of monolayer WSe2, by which the conduction band minimal and valence band most are reverse in spin orientations and the lowest-energy electron-hole pairs kind spin-forbidden excitons (Fig. 1d). Consequently, gentle emission with an in-plane electrical dipole is strictly forbidden, and an in-plane magnetic area30 or the coupling to a plasmonic construction32 is required to brighten the darkish exciton. Nevertheless, utilizing an goal of enormous numerical aperture (N.A.), the PL of the darkish exciton can nonetheless be straight noticed in high-quality samples, for the reason that radiation from a small out-of-plane dipole33 of the darkish exciton could be collected by the target. On this case, a well-resolved darkish exciton PL peak seems with a slim linewidth. We carry out valley-resolved PL spectroscopy on our gadgets utilizing optical excitation with sure round polarization (σ+ or σ−), and detect the PL of the identical or reverse round polarization1,2,four,eight,35,36,37,38. Such a configuration of excitation and detection is labeled (σ±,σ±) in our work. With out making use of a high gate voltage and magnetic area, the circularly polarized PL spectra within the (σ−,σ−) configuration clearly resolve the charge-neutral exciton X0 and two well-separated unfavorable trions38,39,40,41, (mathrmX_1^ -) and (mathrmX_2^ -), indicative of an initially frivolously electron-doped pattern. The linewidths of the 2 trions are 2.1 and a couple of.2 meV, respectively, a lot smaller than their vitality splitting of ~7 meV30,38,39,40,42,43,44,45. It’s price noting that the linewidth of the darkish exciton is as slim as zero.9 meV, which demonstrates the standard of the spectra and is the important thing to our discovery of the dark-exciton reproduction. With the appliance of an out-of-plane magnetic area of 6 T, the exciton (X0), trion 1 ((mathrmX_1^ -)), and trion 2 ((mathrmX_2^ -)) peaks all are blue shift in vitality and stay a single peak within the PL spectra (magenta curve in Fig. 1b), indicating that their emission is intrinsically circularly polarized at every valley. Nevertheless, the darkish exciton PL at 1.689 eV splits into two peaks at 1.687 and 1.690 eV at 6 T. This splitting happens as a result of the small out-of-plane electrical dipole of darkish excitons is anticipated to lead to linearly polarized quite than circularly polarized gentle for every wavevector. Subsequently, the emissions from the 2 valleys might each be detected on this σ− assortment configuration, with their vitality distinction dictated by the valley Zeeman impact45,46,47,48. Remarkably, one other PL peak (indicated by arrows in Fig. 1b) emerges and displays habits much like the darkish excitons. This PL peak, situated at 1.667 eV, splits into two peaks with unequal heights at 1.665 and 1.669 eV, with the appliance of the magnetic area of 6 T. Because of this, we label this peak because the dark-exciton reproduction ((mathrmX_^)). We now have reproduced (mathrmX_^) in 4 completely different BN encapsulated WSe2 gadgets (see Supplementary Notes 7 and eight). The ability-dependent PL depth of the exciton, darkish exciton, and darkish exciton reproduction is proven in Fig. 1d (stable dots) on the absence of the magnetic area. The PL depth could be fitted with an influence regulation: I ~ Pa, the place I is the PL depth and P is the excitation laser energy. It’s evident that the darkish exciton reproduction and darkish exciton share comparable power-law exponent, completely different from that of the brilliant exciton (X0). With low excitation energy (P ≤ 40 μW), the α values are 1.19 and 1.24 for the darkish exciton and darkish exciton reproduction, respectively. With larger excitation energy, saturation habits begins to happen and the α values are zero.72 and zero.77 for the darkish exciton and darkish exciton reproduction, respectively. As compared, the PL-power scaling for the brilliant exciton (X0) could be described with an α worth of 1.13 all through the entire excitation energy vary studied. The slight super-linear habits of the excitation energy dependence for the brilliant exciton (X0) is in line with earlier experiences28,30.

Fig. 1

Darkish exciton PL splitting in magnetic area. a Schematic illustration of BN encapsulated monolayer WSe2 with graphene contact and high gate electrodes. b PL spectra of the gadget in (a) at four.2 Okay with out the appliance of the highest gate voltage, with no magnetic area (black) and with 6 T out-of-plane magnetic area (magenta) utilized. c Built-in PL depth of WSe2 as a operate of the excitation energy for the X0, XD and (mathrmX_^) PL peaks. d Schematic configurations of exciton and dark-exciton states with the stable and empty dots representing the electron and gap. Blue and orange colours stand for spin-up and spin-down bands, respectively

### Magneto-PL spectra of WSe2

The connection between (mathrmX_^) and XD can be revealed by circularly polarized magneto-PL spectra measurements taken within the (σ−,σ−) configuration (Fig. 2). We notice that the depth oscillation as a operate of the B area is a measurement artifact, which we attribute to the slight beam place shift as we enhance the magnetic area. As proven in Fig. 2a, it’s evident that each one the peaks, apart from XD and (mathrmX_^), bear a monotonic blue-shift as a operate of the out-of-plane magnetic area because of the valley Zeeman impact45,46,47,48. Quite the opposite, XD and (mathrmX_^) exhibit a splitting that will increase linearly with the magnetic area. The emission vitality within the presence of the magnetic area could be expressed as (E = E_0 pm fracgmu _), the place g is the Landé g-factor of the excitonic advanced of curiosity, μB the Bohr magneton. “+” and “−” correspond to the PL peak energies from the Okay and Okay′ valleys, respectively. For the brilliant exciton, solely Okay′ valley radiation (σ−) is allowed to be detected within the valley polarized PL spectra of the (σ−,σ−) configuration, and therefore, solely the blue-shifted emission is noticed. The Zeeman splitting between the 2 valleys, (Delta E = E^Okay – E^ = gmu _), is plotted in Fig. 2c (dots) the place g-factor could be obtained by a linear becoming (stable traces). The g-factor for the brilliant exciton, trion (mathrmX_1^ -), and trion (mathrmX_2^ -) are −three.7, −four.four, and −four.5, respectively, in line with earlier research28,43,44,49 and a theoretical expectation of −four based mostly on a noninteracting particle evaluation (see Supplementary Observe three). The g-factor for the darkish exciton, nonetheless, is −9.three, in line with earlier experiences28,31,49 and the theoretical expectation of −eight (see Supplementary Observe three). Curiously, the darkish exciton reproduction (mathrmX_^) has a g-factor of −9.four, much like that of the darkish exciton XD however distinctly completely different from these of the brilliant exciton and trions. This specific magnetic area dependence of (mathrmX_^) signifies that its spin-valley configuration is nearly similar to that of the darkish exciton XD (see Supplementary Observe 7). It’s price noting that the relative depth of the 2 branches of the darkish exciton reproduction within the magneto-PL spectra sensitively relies on the round polarization of shiny exciton within the (σ+,σ+) or (σ−,σ−) measurement (Fig. 2a). The high-intensity department switches because the helicity of circularly polarized excitation change (see Supplementary Fig. 5e, f) because it carefully follows the Zeeman-shifted circularly polarized shiny exciton. That is in stark distinction to the 2 branches of the darkish exciton, that are at all times the identical in depth whatever the helicity of the detection. The shut correlation of the excessive PL depth department of the darkish exciton reproduction and the brilliant exciton strongly helps our concept of the phonon-mediated mixing of the darkish exciton and shiny exciton. The slight distinction between the valley polarization of the darkish exciton reproduction and the brilliant exciton at finite magnetic area could be doubtlessly attributed to the upper order mixing course of, which is past the scope of this work.

Fig. 2

Valley-resolved magneto-PL spectra of the dark-exciton and its reproduction. a Valley-resolved PL spectra at four.2 Okay as a operate of the emission photon vitality and the utilized out-of-plane magnetic area, with the excitation of a CW laser centered at 1.879 eV and excitation energy of 60 µW. The colour represents the PL depth. The darkish exciton and its reproduction exhibit distinctively completely different magnetic area dependence in comparison with shiny excitonic complexes. b Line traces of the PL spectra as a operate of the utilized magnetic area. The dashed traces are the information for the attention. c g-factor for various excitonic complexes obtained from the Zeeman splitting between the EK and (E^), obtained from (a) and Supplementary Fig. 5c. The information units are offset 5 meV deliberately in y-axis for readability

### Gate-voltage-dependent PL of WSe2

To additional discover the origin of the reproduction, we examine the PL spectra as a operate of the highest gate voltage for a second gadget, and the outcomes are proven in Fig. three. (The gate dependence of gadget 1, proven in Figs. 1 and a couple of, is included in Supplementary Observe eight.) We make use of a CW laser centered at 1.959 eV with an excitation energy of 40 µW, underneath which the biexciton (XX) and negatively charged biexciton (XX−) can each be noticed28. The (mathrmX_^) peak of the second gadget is situated at 1.676 eV, barely larger in vitality than the (mathrmX_^) peak (1.667 eV) within the first gadget. Regardless of the small peak-energy shift, which probably arises from the residual pressure, the splitting between the (mathrmX_^) and XD within the second gadget stays virtually the identical as that within the first gadget, ~21.three meV. (The (mathrmX_^ – mathrmX_) vitality splitting worth is included in Supplementary Observe 5 for all of the 4 gadgets that we now have measured.) From Fig. 3a, b, it’s apparent that the spectrum weight of all of the resolved excitonic complexes relies upon sensitively on the highest gate voltage that successfully controls the density and kind of cost carriers within the monolayer WSe2. Whereas the (mathrmX^+) happens when the monolayer WSe2 is hole-doped, (mathrmX_1^ -), (mathrmX_2^ -), and (^ -) emerge when the WSe2 is electron-doped, and XX solely exists within the charge-neutral area28,43,44,49. It may be seen in Fig. 3a that the areas the place XD and (mathrmX_^) exist overlap considerably. For a quantitative understanding, we plot the built-in PL depth as a operate of the gate voltage for every excitonic advanced in Fig. 3c. We discover that the gate-voltage-dependent built-in PL depth of the darkish exciton reproduction precisely mimics that of the darkish exciton, each reaching the utmost close to the charge-neutral area and lowering quickly with both electron-doping or hole-doping (gate voltage > zero V or<−2 V, as indicated by the onset of serious PL depth from unfavorable trions or constructive trion). The charge-neutral area additionally strongly correlates with the PL depth of the darkish exciton. This gate-voltage-dependent measurement guidelines out the likelihood that the darkish exciton reproduction is a charged darkish exciton.

Fig. three

Gate-dependent PL depth of the dark-exciton and its reproduction. a PL spectra at four.2 Okay as a operate of the highest gate voltage for a second BN encapsulated monolayer WSe2 gadget. The colour represents the PL depth. The excitation is a CW laser centered at 1.959 eV with an excitation energy of 40 µW, underneath which the biexciton (XX) and the charged biexciton (XX−) are additionally seen. The gate dependence of the darkish exciton reproduction (mathrmX_^) is much like that of the darkish exciton. b The road traces from (a) for the gate voltages of 1.zero V (blue), −1.zero V (magenta) and −four.zero V (purple). c Built-in PL depth for various exciton complexes as a operate of the gate voltage. The non-zero PL depth areas for the darkish exciton and its reproduction are virtually similar, from ~−2.1 V to ~zero.9 V of the highest gate voltage

The gate-voltage-dependence and magnetic-field-and of the PL exhibit that XD and (mathrmX_^) are each charge-neutral excitations, and they need to additionally share comparable valley-spin configurations and wavefunctions. The sample-independent vitality distinction (~21.6 meV) between XD and (mathrmX_^) and their sharp PL peaks additional counsel that (mathrmX_^), a beforehand unrecognized excitation of monolayer WSe2, arises from coupling XD to a quasiparticle at ~21.6 meV which can’t be a cost service or a plasmon. Based mostly on these analyses, we attribute (mathrmX_^) to a phonon reproduction state fashioned by the coupling between XD and a phonon. To test this assumption and establish the phonon mode concerned, we first carry out a first-principles calculation of the phonon dispersions of monolayer WSe2. Our outcomes present that doubly degenerate E″ phonon modes seem with vibration vitality (hbar _^primeprime )=21.eight meV, in line with earlier experiences24,50,51. This vitality is in wonderful settlement with our commentary of the XD − (mathrmX_^) vitality distinction of 21.6 meV.

### Electron-phonon coupling in WSe2

Regardless of the vitality settlement, it’s totally sudden that the phonon reproduction PL exhibits an depth akin to the brilliant exciton PL (Figs. 1b and 2b), which usually would require a robust exciton-phonon coupling energy or massive phonon inhabitants. Moreover, on nearer examination of the optical spectra underneath magnetic fields (Figs. 1b and 2b), we discover that the higher-energy reproduction peak is way stronger than the lower-energy one, that means that the reproduction PL arising from every valley has finite round polarization. As an example, at 5 T, we estimate the diploma of round polarization of the phonon reproduction to be 72% (See Supplementary Observe 6). To grasp these options, we first look into the spatial symmetry of the WSe2 lattice within the presence of E″ phonons. In Fig. 4a (inset), we schematically plot one of many doubly degenerate E″ lattice vibrational modes that contain the other in-plane motion of the upper-plane and lower-plane Se atoms alongside x. This vibration breaks the mirror symmetry in regards to the 2D aircraft and the threefold rotation symmetry. Moreover, the vibration modes alongside the 2 in-plane instructions can assemble two chiral E″ phonon modes50 on the gamma level, with angular momentum of 1 and −1, respectively. Consequently, the excitons within the Okay or Okay′ valleys can purchase a finite angular momentum and be brightened up by coupling to one of many chiral mixtures (1 for Okay valley or −1 for Okay′ valley) (see Supplementary Observe 12). For example this impact on the electrons quantitatively, we plot the conduction bands (Fig. 4a) and their in-plane spin parts (Sx and Sy) on the Okay valley (Fig. 4b), in a construction with the upper-plane and lower-plane Se atoms displaced alongside x by zero.035 Å (i.e., zero-point movement amplitude) and −zero.035 Å, respectively. On the Okay level, we get hold of a conduction band splitting of 49 meV, 9 meV bigger than that of 40 meV within the equilibrium construction. Extra considerably, Sx of the 2 conduction bands at Okay, initially zero within the equilibrium construction, will increase to (zero.27,hbar) and (- zero.27,hbar) after the displacement. Such band splitting enhancement and spin realignment clearly exhibit an rising in-plane spin-orbit area arising from the lattice vibrations (particulars and an evaluation of the valence bands are included in Supplementary Observe 11). As such, the E″ phonons are analogous to a fluctuating in-plane efficient magnetic area that induces finite coupling between the 2 conduction bands and between the brilliant and darkish excitons in the identical valley30, thus vastly enhancing out-of-plane and circularly polarized PL of (mathrmX_^).

Fig. four

Phonon coupling and recombination pathway of the Okay-valley dark-exciton. a The conduction band construction within the Okay valley, with the 2 Se atoms in a single unit cell displaced by zero.035 and −zero.035 Å, respectively, calculated utilizing Kohn−Sham density purposeful concept. The conduction band backside is about to zero eV. The okay path is taken alongside the y course throughout the Okay level, i.e., kx = Kx. The decrease (c1) and higher (c2) conduction bands are coloured orange and blue, respectively. The black dashed traces are the 2 conduction bands within the equilibrium construction. Inset: a schematic of an E″ phonon eigenmode. Grey and inexperienced spheres are W and Se atoms, respectively. Arrows point out displacement. b Expectation values of conduction-band electron spin angular momentum within the x and y course, Sx (stable line) and Sy (dashed line), of c1 (orange) and c2 (blue) as a operate of okay. c The intense exciton (X0) band and darkish exciton (XD) band are denoted by the blue and yellow parabola, respectively. The stable circle and the empty circle signify the electron and the opening, respectively, whereas the arrows up and down point out the spin orientation. The yellow-shaded space above XD signifies a quasi-equilibrium inhabitants of darkish excitons at four.2 Okay. The dark-exciton phonon reproduction ((mathrmX_^)) state is labeled by a line with alternating blue and yellow shade, indicating coupling between X0 and XD by emitting a chiral E″ phonon of an vitality ħωE″ (purple wavy arrow). The photon emission by (mathrmX_^) is labeled by the black arrow, having an vitality (hbar _^). The emission course of from XD, within the second-order perturbation concept, is illustrated by the purple wavy and blue dashed traces, comparable to the emission of a chiral E″ phonon and a circularly polarized photon, respectively. The intermediate state is the brilliant exciton X0

### Perturbation concept of phonon-photon emission

We additional use a frozen phonon technique and a second-order perturbation concept (schematically proven in Fig. 4c) to calculate the photon emission likelihood of the phonon reproduction (mathrmX_^) relative to the brilliant exciton. The perturbing Hamiltonian consists of two phrases, (H_mathrm – mathrm) and (H_mathrm – l), arising from the exciton-phonon and exciton-photon interactions, respectively. The exciton-phonon coupling permits the darkish exciton to combine with the intravalley shiny exciton by emitting a chiral E″ phonon; exciton-photon coupling describes circularly polarized gentle emission of the brilliant exciton with an in-plane electrical dipole (particulars in Supplementary Notes 10 and 12). The general course of, i.e., concurrently emitting a chiral E″ phonon with an vitality ħωE″ and a circularly polarized photon with an vitality (hbar _^), have a transition likelihood given by the next equation,

$$Wleft( proper) = left| {frac{{ langle _left| {H_mathrm – mathrm} proper|_0 rangle langle _0left| {H_mathrm – l} proper|_mathrmG rangle }}{E_ – E_0 – hbar omega _}} proper|^2delta left( E_ – hbar omega _ – hbar _^ proper).$$

Right here, ΦD, Φ0, and ΦG are the wavefunctions of darkish exciton, shiny exciton, and floor state, respectively. On the experimental temperature of four.2 Okay, the thermal vitality of darkish excitons are very small (~zero.four meV). We due to this fact use the zero-momentum darkish exciton wavefunctions for ΦD. Within the equation, (left| langle {_0left| {H_mathrm – l} proper|_mathrmG} rangle proper|^2) is the photon emission likelihood from the brilliant exciton. Subsequently, the opposite time period(left| {frac{{ langle _left| {H_mathrm – mathrm} proper|_0} rangle }{E_ – E_0 – hbar omega _}} proper|^2), having a dimensionless worth ~zero.04, defines the ratio of photon emission likelihood between the reproduction state and the brilliant state (calculation particulars in Supplementary Notes 11 and 12). This huge ratio, along with the lengthy lifetime of darkish excitons30,31,52,53 of (~250 ± 20 ps in our gadget proven in Fig. 1, see Supplementary Fig. 7c), explains the numerous reproduction PL in our experiment. It’s price noting that the darkish exciton phonon reproduction possesses a lifetime of ~230 ± 20 ps, similar because the darkish exciton lifetime throughout the experimental uncertainty, confirming the phonon reproduction interpretation.