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In this paper, we study the co-existence of and mutual conversion between excitons and trions in electrically gated monolayer and bilayer molybdenum ditelluride (MoTe2) (see the "Methods" section for details). Such electrical gate control was previously used to study trions in a MoS2 field-effect transistor17, to explore the conversion dynamics between excitons and trions18-20, or to study light-emitting diodes33, 34. We conducted systematic micro-photoluminescence (µ-PL) and reflectance spectroscopy (see the "Methods" section for details) on electrically gated MoTe2 devices using continuous wave (CW) lasers. Our intention was to probe the inner mechanisms of the co-existence of or transition kinetics among various excitonic complexes as the pumping increased and to investigate the possibility, origin, and physical mechanism of optical gain in such low-density regimes.
Figure 1 shows a schematic diagram (Fig. 1a) and an optical microscope image (Fig. 1b) of the electrically gated sample structure with a bottom Au/Ti electrode (Vg) separated by an ~50nm h-BN film from the 2D MoTe2 layer for electrical control of charges. The top contact (Vs) consisted of two stripes of graphite separated by ~6µm to enable laser excitation and collection of the reflectance signal from MoTe2. The absorption (α(e)) and optical gain (g(e)) spectra at a given pumping level, e, are related to the differential reflectance (see Supplementary Information, SI, Section 1 for further discussion) as follows:
$$ g\left( e \right) = - \alpha (e) \propto \frac{{\left[ {R\left( {e, p, s} \right) - R\left( {0, p, 0} \right)} \right] - R\left( {e, 0, s} \right)}}{{R\left( {0, p, 0} \right)}} $$ (1) Fig. 1 Sample structure and basic optical properties.
a Schematic of the electrically gated MoTe2 device. Monolayer or bilayer MoTe2 flakes of side lengths typically over 6μm were encapsulated with hexagonal boron nitride (h-BN). The entire structure was placed on top of a SiO2 (300nm)/Si substrate covered with Au/Ti electrodes. b Optical microscope image of a fabricated device with MoTe2 marked by the purple dashed lines. The grey solid line and blue dashed line indicate the regions of graphite contacts and h-BN, respectively. PL c, e and absorption d, f maps in the gate voltage and photon energy plane for bilayer c, d and monolayer e, f MoTe2. Different excitonic species are well resolved, including exciton (X1 and X2), electron-trion (T-) and hole-trion (T+) states. The green dashed lines indicate the maxima of the spectral features, while the horizontal white dashed lines indicate the maxima of excitons and minima of trionswhere R(e, p, s) represents the reflection spectrum in the presence of CW-laser excitation ("e"), a broadband probe ("p"), and the sample ("s"). A zero ("0") in the respective positions represents the absence of excitation, the probe, or the sample, as indicated in Fig. 1a. It is important that PL (R(e, 0, s)) is subtracted in the numerator of Eq. (1), since, by definition, optical gain or absorption is the response of a material to a weak probe. This subtraction is important in CW experiments, as opposed to ultrafast pump-probe measurements, where the PL process has not occurred during the short delay between the excitation and probe pulses. Thus, the intrinsic absorption (α(0)) of a material without laser excitation is given by
$$ - \alpha (0) \propto \frac{{\left[ {R\left( {0, p, s} \right) - R\left( {0, p, 0} \right)} \right] - R\left( {0, 0, s} \right)}}{{R\left( {0, p, 0} \right)}} = \frac{{R\left( {0, p, s} \right) - R\left( {0, p, 0} \right)}}{{R\left( {0, p, 0} \right)}} $$ (2) Equations (1) and (2) are referred to as the differential reflectance and are commonly used for the determination of the optical gain or absorption of a thin film. As shown in SI Section 2 in more detail, the determination of the optical gain can be complicated due to the effects of substrates with metal.
The PL and absorption (processed using Eq. (2)) spectra are shown in Fig. 1c-f for bilayer MoTe2 samples measured at 4K (Fig. 1c, d) and monolayer MoTe2 samples measured at 10K (Fig. 1e, f) as the gate voltage is varied over ±12V. Stacked line plots of the same data are reproduced in SI Section 3 (Fig. S6) to more clearly display the spectral features and mutual conversion of excitonic species. We see from both the PL and absorption spectra that the intensity and spectral positions of trions and excitons can be effectively controlled by the gate voltage. While the trion intensity increases with gate voltage, the exciton intensity decreases due to the conversion of excitons into trions with increasing number of electrons or holes. In both the monolayer and bilayer cases, the trion intensities show a minimum at negative voltages, denoted by Vm (-2.5V for the bilayer and -1V for the monolayer), indicating the initial existence of negative charges in the ungated samples. Both the PL and absorption spectra of the monolayer samples show certain symmetric features with respect to Vm, where the exciton feature is the maximum. However, the corresponding spectral features of the bilayer samples show more complicated behaviour. For almost all the bilayer samples measured, there are two neutral excitons (X1 and X2) in both the PL and absorption spectra that shift and convert with changing gate voltage. Similar two-exciton features in bilayer samples were also previously observed in MoTe234. The trion absorption peak is only observable at very high gate voltages. We also noted in Fig. 1c that the emission for hole trions (T+) for the bilayer sample is much stronger than that for electron trions (T-).
Figure 2a shows PL spectra for the same bilayer sample as in Fig. 1c, d as a function of excitation level at 10V. The PL spectra predominantly show two peaks centred at 1.137 and 1.124eV, labelled X2 and T-, which correspond to the exciton and trion for bilayer MoTe2, respectively, consistent with previous observations19, 34-37. Figure 2b shows a series of differential reflectance spectra with increasing excitation, processed using Eq. (1). We see that there is minimum bleaching of the absorption near the two excitons (X1 and X2) within the relatively weak pump range. However, there is a quite pronounced change in the absorption features at ~1.119eV; eventually, the absorption is above the background level (see further discussion in SI Section 2, especially Fig. S4 for data processing), or optical gain occurs near 1.119eV and increases with increasing pumping. A zoomed-in view of the gain spectra is shown in Fig. 2c to clearly display the pump dependence and peak positions of the gain spectra. The optical gain reaches a maximum at a pumping level of 30µW. It should be noted that the optical gain is different from the signal enhancement, which is a measure of absorption reduction by pumping, defined as the absorption change with and without laser excitation (see SI Section 4 for further discussion).
Fig. 2 Photoluminescence and optical gain spectra.
Pump power-dependent PL a and optical gain spectra b for the bilayer sample measured at a gate voltage of +10V. The gain peak is ~4.3meV below the trion PL peak, as indicated by the two dashed lines. The grey dashed curve is the absorption without pumping. c Zoomed-in view of figure b to more clearly display the pump dependence and peak positions. d-f Corresponding PL and gain spectra for the monolayer sample measured at a gate voltage of-9V. The gain peak is ~10.5meV below the trion PL peakFigure 2d-f shows the results of similar measurements for the same monolayer sample as in Fig. 1e, f at -9V. The PL spectra are dominated by hole-trion emission (T+), centred at 1.155eV. The exciton emission is extremely weak and not visible (see also Fig. 1e). Optical gain occurs near 1.145eV. The appearance of gain with increasing pumping and the trend of the change are very similar to those for the bilayer case. There is a general trend for both bilayer and monolayer samples that can be described as follows: with increasing pumping, the absorption features near the trion energy start to saturate (see Fig. 2b, e), and a peak emerges slightly below the trion emission energy above the background, indicating the appearance of optical gain. For a given gate voltage, the gain peak saturates at a sufficiently high pumping level (e.g., 30µW in Fig. 2c). However, the features of the gain spectra become visibly more irregular for the monolayer sample due to the existence of defect states below the trion peak, which are clearly visible in the PL spectra (see defect features in Fig. 2d below 1.14eV). We note throughout this research that the gain spectrum becomes significantly noisy whenever defects become visible in the PL spectrum. This is especially the case for monolayer samples with significant defect emission. The noisy features in the gain spectrum (e.g., in Fig. 2f) are mainly the results of error amplification in the data processing to obtain the differential reflectance using Eq. (1). Figure S8 in SI Section 5 shows the respective integrated PL intensities of excitons and trions as a function of pumping, showing a nearly linear dependence. Such linear dependence excludes exciton-exciton scattering8 or biexcitons10, 11 as the gain mechanism, since both feature a quadratic dependence of the PL on pumping. Within the range of pumping, no biexciton emission is observed. Thus, the appearance of optical gain, which is spectrally associated with the trion state, likely originates from trions. For a detailed discussion about trion dispersion and the varieties of trions in MoTe2, please see SI Section 6 for more details.
Typically, optical gain appears on the low-energy side of the PL peak and is ~ 4.3 and ~10.5meV below the trion peak for the bilayer and monolayer samples, respectively. This behaviour and the relative spectral features strongly suggest that the optical gain originates from trions. The physical mechanism of trionic gain was first studied in doped ZnSe quantum wells12. According to this understanding, the trion system can be described by a "two-band" model: a ground state of a doped (e.g., n-type) material in which the conduction band, Ee, is filled with a given number of electrons (which could be pre-doped due to defects, gate generated or intentionally doped) and an upper trion band, ET, which has a much heavier effective mass (mT=2me+mh for electron trions), as illustrated in Fig. 3a. The trion formation is more accurately described by the "four-band" model, as illustrated in Fig. 3b, which consists of three key steps: Excitons (bound electron-hole pairs) generated through optical pumping (step 1) quickly find their charged partners generated through gating (step 2) to form trions (step 3) by releasing a binding energy of EbT. For each pump-generated exciton, one electron in the lower band is consumed to form a trion in the upper band. The net effect of a pumping photon (at energy EP) is to decrease (increase) the population of the lower band, Ee (upper band, ET), by one. Such a three-step process can lead to population inversion between the trion and electron bands and achieve optical gain, as shown in step 3 of Fig. 3b. The upper limit of the optical gain is reached when all pre-existing electrons form trions with their exciton partners. The maximum gain is limited by the total number of pre-doped electrons (nD). More precisely, the occupation of the trion state (step 3) at a given pumping level (np) is determined by the relative distribution of trions, electrons, and excitons (at EX). Thus, the co-existence, mutual conversion, and resulting steady-state distribution of free electrons, holes, and all excitonic complexes determine the population inversion and the amount of achievable optical gain.
Fig. 3 Physical mechanisms and theoretical model for trionic gain.
a Parabolic bands (solid lines) and electron distributions (dashed lines) in the trion band (ET) and electron band (Ee). The pink inner band indicates the region around K where the absorption process is dominant, while in the outer green bands (separated at k=kc), local population inversion can occur. b Schematic of the three key steps of trion formation through exciton generation (EX) via optical pumping (EP) from the ground state (Eg) (step 1); pre-existence of electrons (Ee) due to gating or doping (step 2); and possible population distribution among three states (trion, electron, and exciton) and occurrence of population inversion (step 3). EbT and EbX denote the binding energies for trions and excitons, respectively. c Theoretical absorption and gain spectra at different nt/nD ratios (nt: trion density; nD: doping level) from the model (Eq. (3)). d Fitting result (solid lines) of the measured gain spectra (dotted lines) from Fig. 2c for the "four-band" model using Eq. (3)Interestingly, to achieve positive optical gain, there is no need for the total number of carriers in the trion band to exceed the number in the conduction band, so-called global population inversion. As seen in Fig. 3a, due to the much heavier effective mass in the trion state, the same number of carriers in the trion bands occupies a much larger range in k-space, leading to the existence of a crossover k point, kc, such that there is local population inversion for K-kc > k > K+kc. This leads to a situation where the optical gain provided in the green bands could exceed the absorption in the pink band before global inversion is reached, depending on the pumping and linewidth broadening for a fixed level of doping density. Such an occurrence of optical gain through local population inversion without global inversion was first observed in quantum cascade lasers38. However, here, the effect is more pronounced due to the much larger difference in effective masses. More quantitatively, the optical gain is modelled by
$$ G(\omega ) \propto \mathop {\sum}\limits_k {\left| {\left. {\mu _k} \right|} \right.^2L\left( {k, \omega } \right)} \left( {f_{\mathrm {t}} - f_{\mathrm {e}}} \right) $$ (3) where ft and fe represent the Fermi distributions of electrons in the trion band and conduction band, respectively. \(\left| {\left. {\mu _k} \right|} \right.^2\) is the optical dipole matrix element, and L(k, ω) is a lineshape function (see SI Section 7 for details). The calculated gain spectra for a sequence of increasing ratio, nt/nD, are shown in Fig. 3c, where we see that the optical gain appears on the slightly red side of the trion peak energy, consistent with the experimental results. For a fixed electron doping density nD, the critical condition for population inversion is nt=ne=0.5nD for a typical two-band system. Since nD can be of any low value, in principle, optical gain can occur at an extremely low carrier density. This is in strong contrast to the occurrence of optical gain based on an electron-hole plasma, which requires a very high transparency density. In addition, due to the possibility of local k-space population inversion before global inversion, optical gain occurs at a trion density slightly < 0.5nD, as seen in Fig. 3c.
To more quantitatively understand our experimental measurement of the gain spectrum in terms of the "four-band" model and the gain spectra calculated from Eq. (3), we need to determine the steady-state distributions of electrons, excitons, and trions using the well-known mass-action law3, 39. For electron trions formed by\(X + e \leftrightarrow T^ -\), we have \(n_{\mathrm {x}}n_{\mathrm {e}} = n_{\mathrm {t}}K\left(T \right)\), where K(T) is a temperature-dependent equilibrium constant (see SI Section 8 for details). From charge conservation, the trion density nt can be calculated as a function of np and nD. The pumping density, np, is determined by the measured absorption coefficient at the pumping wavelength. The doping density, nD, is determined by best-fitting experimentally measured spectra using Eq. (3) and the calculated trion density (see SI Section 8). The experimental spectra agree with the modelled spectra quite nicely with only a single nD (see Fig. 3d). It is important to emphasize that the optical gain occurs below a 5µW pump power, corresponding to a pumping density of ~3.6×107cm-2. The electron doping density obtained by best fitting is ~7.2×107cm-2, in good agreement with the "two-band" gain model discussed above in Fig. 3a, with all the populations determined by the "four-band" model in Fig. 3b (step 3). We note that electron-hole plasma gain in 2D materials typically appears at densities on the order of ~1013-1014cm-2 30-32, or several orders of magnitude higher than that required for the trion gain studied in this paper. The very low density level also excludes exciton-polarons40, 41 as a possible gain mechanism, which occurs in the presence of a highly degenerate electron gas with Fermi energy close to the trion-binding energy. As shown in Fig. S10 in SI section S7, the electron chemical potential is still negative in our cases, and we are still in the low density regime where the coupling of electrons with excitons mostly displays the character of three-particle trions, not the full many-body version of exciton-polarons.
Figure 4 presents similar gain measurement results for another bilayer MoTe2 sample, where both the pumping-dependent gain spectra at a fixed gate voltage (Fig. 4a) and the gain spectra at varying gate voltage for a fixed pumping level (Fig. 4b-d) are shown. Different from the gain features in Fig. 2, the gain spectra in Fig. 4a-c show a tilted negative background at the low photon energy side below 1.115eV, indicating strong absorption, likely due to the existence of defects. The results are discussed in more detail in SI Section 9. However, despite the negative background, the gain due to trion is clearly visible as a peak that starts to build at ~1.125eV. As shown in Fig. 4a, optical gain due to trions occurs at ~10µW pumping (np~7.2×107cm-2). A clear pumping-dependent increase in the optical gain is seen at a relatively large gate voltage of 8V, where a sufficiently high electron density (nD) is produced by gating. With increasing pumping, an increasing number of excitons are generated that form negative trions with their partner electrons. This translates into a higher population in the trion band, leading to a monotonous increase in the optical gain. The gain evolution for various gate voltages at a fixed pumping level of 40µW in Fig. 4b, c is especially interesting. The optical gain initially increases with the gate voltage from 3 to 7V. The 40µW excitation level produced enough excitons for gate-generated electrons to form trions up to 7V. Thus, with each increase in the gate voltage, there is an increase in the population in the trion band, leading to an increase in the optical gain in Fig. 4b. The gain starts to decrease when the gate voltage is further increased beyond 8V, as shown in Fig. 4c. This is because too many electrons are produced at higher gate voltages relative to the number of excitons produced at 40µW. The trion population in the upper band is capped by the given total exciton number generated by pumping, while the electron population in the lower band keeps increasing with increasing gate voltage beyond 8V. Such a decrease in the population difference (compared to Fig. 3b, step 3) due to the increased lower band population leads to a continuous decrease in the optical gain as the gate voltage is further increased. This feature is more clearly displayed in Fig. 4d, where the extracted peak gain values and the photon energies of the corresponding gain peaks are plotted versus the gate voltage. The overall negative gain values are due to the coupling to absorption for both defects and excitons. This is discussed in more detail in SI Section 9. The trionic gain model and physical picture presented in Fig. 3 can explain the data here quite satisfactorily. More results at different gate voltages or for other devices are shown in SI Section 10 to provide more evidence of optical gain.
Fig. 4 Optical gain of another bilayer sample.
a Optical gain spectra at several pumping levels for another bilayer sample measured at 4K and a gate voltage of 8V. b, c Optical gain spectra at several gate voltages at 40µW pumping. The gain spectra are presented in two groups in figures b, c, separately showing increasing b and decreasing c trends with the gate voltage. d Extracted peak gain values and photon energies of the gain peaks as a function of gate voltage