Bandgap control in two-dimensional semiconductors via coherent doping of plasmonic hot electrons

Bandgap control is of central importance for semiconductor technologies. The traditional means of control is to dope the lattice chemically, electrically or optically with charge carriers. Here, we demonstrate a widely tunable bandgap (renormalisation up to 550 meV at room-temperature) in two-dimensional (2D) semiconductors by coherently doping the lattice with plasmonic hot electrons. In particular, we integrate tungsten-disulfide (WS2) monolayers into a self-assembled plasmonic crystal, which enables coherent coupling between semiconductor excitons and plasmon resonances. Accompanying this process, the plasmon-induced hot electrons can repeatedly fill the WS2 conduction band, leading to population inversion and a significant reconstruction in band structures and exciton relaxations. Our findings provide an effective measure to engineer optical responses of 2D semiconductors, allowing flexibilities in design and optimisation of photonic and optoelectronic devices.

these resonances can be fitted using the Lorentz model, with cavity (plasmon) dissipation being 48 κ 1 ≈ 180 meV (PC-01) and κ 2 ≈ 200 meV (PC-02), exciton A (X A ) decay rate being γ A ≈ 60 meV 49 and X B decay rate being γ B ≈ 210 meV. 50 In order to analyse the coupling between these resonances, we then have fitted the transmission 51 spectra of the PC-WS 2 system [ Supplementary Fig.2 whereẼ κ (θ) = E pl (θ) − iκ(θ),Ẽ γ = E X − iγ andẼ p (θ) denote the complex frequency of plasmon 64 resonances, excitonic resonances and the eigen-frequencies of the coupled system respectively, the minimum at 2.02 eV in Fig. 4(f). These signify the absorption at this frequency. 155 We want to point out that the peak shift and negative magnitudes are naturally presented 156 in transient spectra. ∆T/T are the normalised subtractions between two spectra taken at 157 different delay times. As illustrated in Supplementary Fig.8, resonances may be broadenned 158 and shifted at late delay times due to many-body Auger recombination, i.e. a kind of non-159 radiative recommbination resulting from exciton-exciton or exciton-electron annihilation.

160
As a result, the feature positions and magnitudes in transient spectra vary from those in 161 steady-state spectra.

162
What's interesting is that the 22 and 30 degree spectra feature much stronger absorption 163 (negative magnitudes) than do the 0 degree spectra. As explained above, the negative mag-164 nitudes are the result of Auger recommbination, which is highly relevant to carrier density.

165
Different absorption magnitudes in spectra of different coupling states suggest that the car-166 rier densities become higher as the coupled system is approaching to the tuned state. [The

167
system at θ = 30 • is closer to the tuned state than the system at θ = 0 • , see Supplementary is a bit higher than the one (12 µJ·cm −2 ) that enables the broad maxima in PC-WS 2 samples.

182
This result, together with the ∆T/T spectra of bare WS 2 as shown in Supplementary Fig.6 (Fig. 6d, 6e and 6f) using a tri-exponetial function F(t) = Σ i A i e −t/τ i (i = 1, 2, 3); ‡ Percentages, being normalised A i , refering to population ratios of each fit decay component; † τ NR = Σ i A i τ i /Σ i A i with (i = 1, 2) represents amplitude-weighed lifetime for polariton formation and non-radiative decays.
According to other experiments with TMDC MLS, the delayed absorption maxima nearby 199 excitons arise when the sample is exposed to high-power pump. In our experiments with bare  In our study, coherent coupling builds upon fast energy exchange between plasmon waves 225 and 2D excitons, which leads to plasmon-exciton hybridization that is accompanied by charge 226 generation/transfer. As shown in Supplementary Fig.9, the PC-WS 2 hybrid systems acquire a 227 metal-insulator-semiconductor architecture, which is an extensively studied configuration [12][13][14] 228 that can efficiently harvest hot electrons. In general, two main mechanisms can cause the enhance-229 ment of carrier density in the semiconductor lattice.    To better understand the relaxation dynamics, we have used a tri-exponetial function , 3) to fit the decays, similar to the analysis in Table I, shown as solid curves 253 in Fig.2e and 2f in the main text. The returned fit parameters A i and τ i are listed in Table II.  (Table I), which is consistent with our previous PL 261 experiments [1], where PL from PC-WS 2 is highly enhanced compared to that from WS 2 MLs. 262 We also use amplitude-weighed parameter τ NR = (A 1 τ 1 +A 2 τ 2 )/(A 1 +A 2 ) to characterise average is almost identical to τ NR of bare exciton A in WS 2 MLs under low pump intensity (Table I). As shown in Supplementary Fig.11, ∆Temp for UP and LP branches exhibit opposite variation 290 traces within the first 1 ps after photoexcitation, but can all achieve above 100 K. This is because a local temperature model is not that applicable. 298 We note that there appears a turning point for both UP and LP at the time range of 2 -3 299 ps (indicated by the dash line in Supplementary Fig.11). After that, ∆Temp swiftly decreases, coupling with plasmon resonances, the PL is enhanced as we have reported in a recent study [1].
In this section we developed a model to understand the hot-electron generation on the surface Many body effects that introduce dissipations, such as electron-phonon interaction, are excluded.

312
Because within the 1-ps time scale of the discussed giant bandgap renormalization in our paper, 313 effects from phonons are much less significant.

314
The dynamics of an electron can be described by where ρ is the one-particle density matrix, H is the Hamiltonian of an electron, Γ(ρ) is the Lindblad 316 loss terms. In matrix notation we have where ρ nm = n|ρ|m are the matrix elements of ρ, H 0 is the unperturbed single-particle Hamilto-318 nian, |n is the eigenstate of H 0 with an eigenenergy ε n , V opt is the interaction of the electron and 319 the driving plasmonic field, and Γ nm is the dephasing rate (When m = n, Γ nn is then replaced by 320 the energy relaxation rate). In our model Γ nm ≡ γ Drude is taken from the loss term of Drude model 321 of the Ag dielectric functions.

322
ρ nm is related to the interaction of moving one electron from a state of energy ε m to a state of 323 energy ε n , and ρ nn stands for the probability of finding the electron in its state |n . When there is 324 no laser input V opt = 0, the electron gas stays in its equilibrium state where where ρ (0) nn is the equilibrium occupation probability, f (ε) is the Fermi-Dirac distribution, ε F 326 is the Fermi energy level, k B is the Boltzmann constant and T is the local temperature. Note that the set of |n constitutes a complete basis and n ρ nn = 1. The concept of density of state will be 328 introduced to the model later.

329
For the interaction between electrons and the plasmonic mode, where e is the electron charge, r is the displacement operator, E loc is local electric field, ω is 331 the frequency of the input light. Under the rotating wave approximation, the matrix elements 332 n|V opt |m can be written as The system is driven away from its equilibrium state by the plasmonic field. We consider the 334 driving as a weak perturbation and the time dependent ρ nm (t) can thus be written as Noting that ρ mn in Eq. (S8) has a time dependence of e iωt , we then have where Γ n is the energy relaxation rate of the state |n . By defining g n and d n as the generation rate 344 and the decay rate of the population in |n , we can rewrite the above equation as 345 ∂ ∂t δρ nn = g n − d n (S13) stands for the probability of exciting the electron to a state of energy ε n . As there are in total N 347 electrons in the system, the number-generation rate of electrons to the state |n is G n = N n=1 g n .
(S17) P(ε n , ε) works similarly as a delta function and will be used to introduce DOS into our system.

364
Calculation of V nm 365 We consider a metallic thin film, as shown in Supplementary Fig.12. The corresponding wave 366 functions of electrons are 367 Supplementary Fig. 12. A representation of a metal thin film.
The quantum number n x , n y and n z give a complete description of all quantum states. We then 370 replace n in Eq.(S15) with n = (n x , n y , n z ) and rewrite Eq.(S15) as where n = (n x , n y , n z ) lable the quantum numbers in x, y, and z directions. The spectral distribution of the rate of generated carries can then be written as where n = (n x , n y , n z ) lable the quantum numbers of x, y, and z directions.

374
Substituting Eq. (S18) into Eq. (S7), we can obtain where δ n x ,m x and δ n y ,m y are the Kronecker delta function and E z is the electric field along z-direction.
For our system, we have E F ω γ, k B T ( E F = 5.76 eV, ω ≈ 2.1 eV, γ = 0.02 eV, and at 378 room temperature k B T = 0.02 eV). We can thus make the following simplification In the above equation, the m has m x = n x and m y = n y , and the summation over n x , n y , n z are 381 limited within a range of |ε n − ε| < ∆ε. For a given set of ε n and n z , there are many possible Note that there are δ functions in Eq.(S28), δn(ε) can then be calculated by performing one-dimensional integrals. For hot electrons that are defined as electrons with energy E F + ∆φ TB < 389 ε < E F + ω, we obtain the rate of generation of hot electrons by integrating δn(ε) over the energy The rate per area is where E z is replaced by E norm. , which stands for the E field normal to the Al 2 O 3 -WS 2 interface.
Including losses

403
Eq.S31 can be used to calculate the hot electron density in PC-WS 2 systems when the pump 404 is on-resonance with plasmon frequency. However, the pump frequency (3.1 eV) in our ex-405 periments is higher than the plasmon frequency at the tuned state(2.05 eV), which means that 406 additonal factors in the down-conversion process have to be taken into account in the modelling 407 to obtain an accurate estimation of the hot electron density. We therefore have rewritten Eq.S31 as: where E 0 is the equivalent incident field at the tuned frequency and F = |E/E 0 | 2 is the spatial 410 distribution of intensity enhancement at this frequency as depicted in Fig.4b in the main text. In 411 the following we will discuss how E pump (the incident field at the pump frequency) can develop to 412 E 0 . In particular, three main factors take effect in this process.
Supplementary Fig. 13. Light absorption in WS 2 monolayers and PC-WS 2 systems (a) Field enhancement |E/E 0 | at the pump energy (3.1 eV or 400 nm). (b) measured absorption spectra of a bare WS 2 ML and a PC-WS 2 sample. The absorption spectra is acquired by A = 100% − T − R, where T refers to transmission and R denotes reflection.
[1] Absorption of pump energy

414
The pump energy can not be fully absorbed by the system. The optical absorption of a bare 415 WS 2 ML at the pump frequnecy is ∼ 10% [ Supplementary Fig.13(b)]. When deposited on 416 a plasmonic crystal, the total absorption of the PC-WS 2 system is enhanced to 80%. As 417 shown in Supplementary Fig.13(a), in this system, the intensity enhancement at the pump 418 frequency is no larger than ∼7 times at the position of the WS 2 monolayer, which, according 419 to our calculation, results in an average optical absorption of 55% in the monolayer. This is 420 a moderate absorption enhancement compared to that in the bare WS 2 monolayer, but not 421 high enough to induce a Mott-transition as in the work [24].

422
[2] energy losses in the Down-conversion process 423 Since the pump photon energy is higher than the exciton frequency, there will be energy 424 dissipation through intraband scattering during the polariton formation, though the total 425 quanta number is still conserved. In particular, the energy that has been transfered to form 426 polaritons follows ω X /ω pump ≈ 66%.

427
[3] excitation of the plasmonic component in polaritons 428 In the coupled PC-WS 2 systems, the pump energy is down-converted to form polartions at 429 a lower frequency, but only part of this converted energy will excite the plasmonic com-430 ponent of polaritons that generates hot electrons. As revealed by Supplementary Fig.3, the 431 formation of half-plasmon half-exciton hybrid state allows ∼ 50% of the converted energy 432 to excite the plasmon part at the tuned state (θ = 22 • ).

433
Given that the incident field at the pump frequency can be obtained by: where F pump = 12µJ·cm −2 is the fluence at the pump frequency, n = 1 is the refactive index in 435 vacuum, c is the speed of light and 0 is the vacuum permittivity. The equivalent incident field at 436 the tuned plasmon frequency (|E 0 | 2 ) can be written as: where η A = 55% is the absorption coefficient, η D = 66% is the down-conversion coefficient and 438 η pl = 50% is the plasmonic coefficient. As a result, Eq.S32 can be rewritten to a new form to 439 estimate the hot electron density that are injected into the WS 2 ML in our experiments: which then gives: In addition to the factors we have discussed, we take φ TB = 1 eV for the tunneling barrier at Based on Eq.S36, we will be able to calculate the hot electron distribution in WS 2 monolayers 457 in the PC-WS 2 systems, which is plotted in Fig.4c as a function of the projected distance along the 458 x-direction in the main text. It is noted that as the pump fluence increases ( Supplementary Fig.15  We also observe that under high pump fluence ( Supplementary Fig.16), UP and LP relaxations 514 exhibit maximum at 3 and 1 ps, respectively, which sharply contrast with the unchanged early 515 maximum (∼ 150 fs) of X B in the same sample and X A in bare WS 2 MLs ( Supplementary Fig.6), 516 indicating the coupling with plasmon modes gives rise to the delayed maxima, which is highly 517 relevant to hot electron population. ductor lattice, which competes with both UP and LP relaxation. As a result, the LP branch, which 537 has higher decay rates, shows maximum at a earlier time than does the UP branch.

538
As shown in Supplementary Fig.17, the peaking time of broad maximum is also dependent 32 on pump fluence. For example, for 100µJ·cm −2 pump (red curve), the maximum peaks at 0.96 540 ps; while for 12µJ·cm −2 pump (black curve), the maximum peaks at 0.33 ps; and they all rapidly 541 decay after the maxima. Because the accumulation of hot electrons naturally competes with the 542 relaxation dynamics of excitons, the peaks with hot electron injections typically appear later than 543 the relaxation maxima of bare excitons ( Supplementary Fig.6c and 6f) typically appear earlier 544 than the hot electron peaks. Specifically, in the case of low-power pump, the electron density 545 is relatively low (Supplementary Fig.17), only capable of slightly affecting the exciton/polariton 546 relaxation, i.e. extended lifetimes of polaritons (τ NR , Section 4). In contrast, for high-power 547 pump, the electron density are highly enhanced, which can significantly slow down the formation 548 of relaxation peaks, or namely delaying the maxima of polariton peaks ( Supplementary Fig.16).