Electron recoil effect in electrically tunable $\mathrm{Mo}{\mathrm{Se}}_2$ monolayers

Radiative recombination of excitons dressed by the interactions with free charge carriers often occurs under simultaneous excitation of either electrons or holes to unbound states. This phenomenon, known as the electron recoil effect, manifests itself in pronounced, asymmetric spectral line shapes of the resulting emission. We study the electron recoil effect experimentally in electrically tunable monolayer semiconductors and derive it theoretically using both trion and Fermi polaron pictures. Time-resolved analysis of the recoil line shapes is employed to access transient, nonequilibrium states of the exciton-carrier complexes. We demonstrate cooling of the initially overheated populations on picosecond timescales and reveal the impact of lattice temperature and free carrier density. Both thermally activated phonons and the presence of free charges are shown to accelerate equilibration. Finally, we find similar values of relaxation times from recoil analysis and luminescence rise times, providing additional insight into trion dynamics.

Figure 1

(a) Representative, time-integrated PL spectra of hBN-encapsulated $\mathrm{Mo}{\mathrm{Se}}_2$ monolayer under electrical gating in the $p$-doped, neutral, and $n$-doped regimes at 5 K. (b) False-color plot of the PL intensity as function of emission energy and gate voltage. Dashed lines represent spectra shown in panel (a). The gate voltage of $\pm 1$ V corresponds to the electron and hole density of about $1.6\times {10}^{11}{\mathrm{cm}}^2$ and $2.9\times {10}^{11}{\mathrm{cm}}^2$, respectively, estimated from the trion-exciton energy separation (at higher voltages the scaling is $2\times {10}^{11}{\mathrm{cm}}^2$/V for holes and $4.8\times {10}^{11}{\mathrm{cm}}^2$/V for electrons). (c) Time-integrated PL spectra at different lattice temperatures in the $p$-doped regime. The hole density is set to $1\times {10}^{11}{\mathrm{cm}}^{-2}$ for all temperatures. The spectra are presented as function of the emission energy relative to the zero-momentum trion peak energy $E_0^{tr}$ at each temperature. Fit functions are convolutions of a Lorentzian peak with an exponential low-energy flank as well as a Gaussian with a fixed linewidth of 1 meV. (d) Schematic illustration of the electron recoil effect during trion recombination. The illustration presents the case of negatively charged trions and is equally applicable to positively charged ones. (e) Exponential low-energy constant ${\epsilon }^{*}$ extracted from the fits in panel (c) (filled triangles), presented in the form of $k_B/{\epsilon }^{*}$ as function of the inverse lattice temperature for $T\ge 15$ K. Filled circles correspond to a second measurement for temperatures up to 70 K at a different sample position. Fit of both data sets using Eq. (2) is shown in red. Corresponding values obtained from time-resolved measurements at 50 ps are indicated by open circles.

Figure 2

Top panels schematically illustrate overlap between trion/Fermi polaron wave function with that of the final state of the free electron. Presented are two cases of high and low kinetic energies resulting in small and large overlap integrals, respectively. The main panels show calculated dipole matrix elements as function of the kinetic energy of the quasiparticle, using either trion or Fermi polaron wave functions. The dashed black line is calculated after Eq. (8) and the dashed magenta line is calculated after Eq. (11). A single-exponential approximation is presented by the solid line for direct comparison. For the trion we set $a_{tr}=3a_x$ and obtain the binding energy of 17 meV, close to the experimental values (we note that such deviations are common for variational models). For the Fermi polaron approach we set $E_{b,tr}$ to the experimental value of 25 meV. The exponential approximation corresponds to ${\epsilon }_1=7$ meV.

Figure 3

Fermi polaron interaction with photons. (a) The full diagram describing the Fermi polaron PL in the Keldysh technique. (b) Self-energy responsible for the radiative decay. The wavy line shows the photon Green's function. The exciton Green's function ${\mathcal{G}}_x$ renormalized by the Fermi sea is shown by the thick green line. Blue boxes show the electron-exciton scattering amplitude $T$ and $\mathrm{\Pi }$ denotes the Fermi sea polarization loop composed of the bare electron Green's functions (black lines with arrows). Signs $+$ and $-$ denote the components of the Green's functions in Keldysh technique.

Figure 4

(a) Series of time-resolved PL spectra of the $X^{+}$ resonance at 5 K for different times after near-resonant excitation of the neutral exciton at 1.663 eV. Estimated pump density of the injected electron-hole pairs is $4.5\times {10}^{11}{\mathrm{cm}}^{-2}$, considering absorption of 2% at the pump energy. The density of the free holes is set to about $7\times {10}^{10}{\mathrm{cm}}^{-2}$. (b) A corresponding streak camera image of the PL intensity in a false-color plot, as function of time and emission energy. Dotted and dashed lines represent the time-dependent exponent ${\epsilon }^{*}$ of the low-energy flank ($\times 4$) and the trion peak energy $E_0^{tr}$, respectively. They are obtained by an automated fit procedure of spectra in intervals of 1 ps using Eq. (1). (c) Extracted, transient temperatures of the trion measured for different lattice temperatures between 5 and 50 K. (d) Characteristic $1/e$ cooling times of the trions as function of lattice temperature. Rise time of the PL transients is presented for direct comparison. (e) Corresponding PL transients of the trion emission. A laser scattering signal representing instrument response function with 4 ps width is shown by the gray area. An exponential rise-decay model was used for fitting, convoluted with a Gaussian to account for the instrument response.

Figure 5

(a) Representative, time-integrated PL spectra of $X^{+}$ at different doping densities up to $5\times {10}^{11}{\mathrm{cm}}^{-2}$ (using the hole mass of 0.6 $m_0$, a 1 meV energy shift corresponds to a change of doping by $2.5\times {10}^{11}{\mathrm{cm}}^{-2}$). All experimental parameters are kept the same as for the measurements presented in Fig. 4: photon energy of 1.663 eV, pump density of $6\mu \mathrm{J}$/${\mathrm{cm}}^{-2}$, estimated injected electron-hole density of $4.5\times {10}^{11}{\mathrm{cm}}^{-2}$, and $T=5$ K. (b) Transient excess temperatures of the $X^{+}$ trion for different densities of the free holes with respect to the equilibrium temperature at 50 ps. (c) The extracted cooling time from the exponential fit as function of hole density. The rise time of the trion PL is shown for comparison. (d) Corresponding PL transients of the trion emission together with exponential rise-decay fits $\propto {({\tau }_{\mathrm{decay}}-{\tau }_{\mathrm{rise}})}^{-1}[exp(-t/{\tau }_{\mathrm{decay}})-exp(-t/{\tau }_{\mathrm{rise}})]$.

Figure 6

(a) Transient trion temperature for different injected electron-hole pair densities $n_{eh}$ at a fixed hole density of $n_h\approx 1\times {10}^{11}{\mathrm{cm}}^{-2}$, near-resonant excitation at 1.663 eV, and lattice temperature $T=5$ K. Solid lines correspond to single-exponential fits of the data. The extrapolated zero-density trion temperature after 50 ps is 7.6 K. The inset shows the corresponding $X^{+}$ peak energy shift $\mathrm{\Delta }{E}_0^{tr}$ as a function of pump density. (b) Extracted $1/e$ cooling times from (a) as function of pump density $n_{eh}$. The rise time of the trion PL is shown for comparison.

Figure 7

Transient trion/Fermi polaron temperature for near-resonant and nonresonant excitation of the neutral exciton transition. The pump density is set to a few ${10}^{11}{\mathrm{cm}}^{-2}$ and the hole density to $n_h\approx 1\times {10}^{11}{\mathrm{cm}}^{-2}$. Single-exponential fit (solid line) corresponds to a cooling time of 9.5 ps.

Figure 8

(a) Transient excess trion temperature for estimated free carrier densities of $n_e\approx n_h\approx 7\times {10}^{11}{\mathrm{cm}}^{-2}$. The excess temperature is plotted with respect to the equilibrium temperature at 40 ps. (b) Corresponding normalized trion PL transients in the $p$- and $n$-doped regime. Extracted exponential cooling times as well as PL rise and decay times are illustrated.

Figure 9

(a) Relative exciton-trion energy separation as function of temperature corresponding to the measurements presented in Fig. 1 (c). (b) The relative exciton-trion energy separation as function of pump density $n_{eh}$ corresponding to the measurements presented in Fig. 6. The remaining experimental fluctuation of the doping level is estimated by setting the exciton-trion energy variations equal to the Fermi energy shift using a hole mass of $0.6m_0$. In both (a) and (b) the average hole density is $1\times {10}^{11}{\mathrm{cm}}^{.2}$.

Figure 10

(a) Energy-resolved reflectance contrast derivatives $\partial R_c/\left(\partial E\right)$ as a function of gate voltage at $T=5$ K. Trion and exciton emissions are labeled by $X^{+}$ and $X$, respectively. Note that due to local doping inhomogeneities the data shows weak trion features also at 0 V. (b) The reflectance contrast derivative at fixed gate voltage of $-0.7$ V corresponding to an estimated hole density of about $6\times {10}^{11}$ ${\mathrm{cm}}^{-2}$. The data is fitted by a transfer matrix model. (c) Extracted oscillator strengths as function of doping density normalized to the exciton oscillator strength at zero doping. (d) The radiative recombination time of the trion/attractive Fermi polaron estimated from the $X^{+}$/$X$ oscillator strength ratio in (c) and setting the radiative decay time of the exciton at zero doping to a typical value of 1 ps. Error bars originate from the fit of the trion oscillator strength. The corresponding PL rise time from Fig. 5 is plotted for comparison.

Figure 11

Diagrams describing the Fermi polaron interaction with phonons. We consider an $n$-type system with resident electrons. The dash-dotted line is the phonon Green's function. (a)–(c) Basic diagrams showing individual contributions resulting in ${({\mathrm{\Xi }}_c-{\mathrm{\Xi }}_v)}^2$, ${\mathrm{\Xi }}_c^2$, and $2{\mathrm{\Xi }}_c({\mathrm{\Xi }}_c-{\mathrm{\Xi }}_v)$ terms in ${\mathcal{B}}_{tr}\left(q\right)$, Eq. (18). (d)–(f) Fermi polaron self-energies. Note that the diagrams with at least one of the phonon vertices describing the Fermi sea hole interaction with phonons (not shown) are negligible at small electron densities. The diagrams with the phonon line encompassing ${\mathcal{G}}_x$ (not shown) have an additional smallness $\propto N_e/\left(\mathcal{D}{E}_{b,tr}\right)$.