Attosecond Core-Exciton Dynamics in Wide-Gap Insulators

Abstract

An option to overcome the limits of electronics are excitons, quasi-particles generated by the electron-hole Coulomb interaction. A fundamental prerequisite is the possibility to control their properties with optical fields. We adopted attosecond reflection spectroscopy to investigate the interplay between core excitons and ultrashort infrared pulses in MgF${ }_2$. Their ultrafast response is divided into two contributions, unfolding on different time scales, related to the exciton dual nature. Namely, we discern the atomic-like behaviour, dominating the few-femtosecond response, from the excitonic solid-like characteristics, with sub-femtosecond timing. Theoretical models highlight the role of the exciton binding energy in dictating their optical response.

1 Introduction

Limitations of current electronic devices have become evident during the last decade [1]. Several options, like valleytronics [\({ }_2\) (001) single-crystal sample at an angle of 73.5${ }^\circ$. The XUV radiation reflected by the sample is steered into an XUV spectrometer by a gold mirror. The relative XUV-IR time delay is actively stabilized with a residual standard deviation of 24 as, ensuring the required attosecond temporal resolution [13]. The propagation between the first and the second focus induces an additional (propagation) delay between XUV and IR pulses [14]. Its value, fundamental for an absolute calibration of the experimental delay axis, has been characterized by independent measurements and amounts to $240\pm 13$ as [11].

3 Results

The static (unpumped) experimental reflectivity $R_0(E)$ of MgF${ }_2$, the theoretical calculation and the reflectivity computed from data reported in literature [15] are presented in Fig. 1a. XUV-induced transitions probe the Mg L${ }_{2,3}$ absorption edge (${\sim }56$ eV), involving the Mg${ }^{2+}$ 2p states and the conduction band. A bright excitonic feature (A), replicated by spin-orbit splitting (${\sim }0.44$ eV, feature A’), is observed. The IR dressing of these excitons is studied by attosecond reflection spectroscopy. The measured quantity, reported in Fig. 1b, is the transient differential reflectivity, $\varDelta R / R(E,\tau ) = [R(E,\tau )-R_0(E)]/R_0(E)$, where R and $R_0$ are the reflectivity of the sample with and without the IR field, $\tau$ is the XUV-IR delay and E the XUV photon energy (for positive delays, the XUV pulse comes first). The squared modulus of the experimentally measured IR vector potential, $|A_{IR} (t)|{ }^2$, is reported in Fig. 1c.

Fig. 1

(a) Unpumped experimental (red) and calculated (black) reflectivity of MgF${ }_2$ at 73.5${ }^\circ$, compared to data reported in literature (blue dashed) [15]. The onset of the conduction band (dash-dotted) and the Mg${ }^{2+}$ semi-core states (A dashed, A’ dotted) are marked by the black horizontal lines. The dark shaded region extends over twice the standard deviation of the measurement. (b) Differential transient reflectivity of MgF${ }_2$. (c) Squared modulus of the simultaneously measured IR vector potential. (d) Sub-fs and (e) few-fs components of the differential transient reflectivity trace. (f) Conduction band structure (CB1, CB2, CB3), dark and bright excitonic states, semi-core (SC1, SC2) Mg${ }^{2+}$ states

The transient differential reflectivity trace can be decomposed in two main contributions: sub-fs oscillations in the reflectivity (Fig. 1d) at twice the frequency of the IR radiation, and a few-fs signal (Fig. 1e). Analyzing the band structure [16] (Fig. 1f), we can consider that an XUV pulse is inducing transitions involving the Mg 2p semi-core states (SC1 and SC2) and states above the Fermi level. The Coulomb electron-hole interaction forms a core-exciton (A and A’), but only s-symmetric states (bright) will be populated by dipole-allowed transitions, while the ones with p character are optically dark. For positive delays, the bright and dark states and the conduction band are coupled by the IR pulse, giving IR-induced variations of the reflectivity. For large negative delays, instead, the IR pulse precedes the XUV one, and the exciton is not formed yet. A nonlinear fit of the experimental trace gives a core exciton decay time constant of $2.35\pm 0.3$ fs, similar to those already reported in literature for other dielectrics [8, 9]. However, to obtain an accurate estimate of the parameters of the excitonic dipole, more refined ptychographic techniques should be adopted [17].

The experimental results (also reported in Fig. 2a) are compared with three different theoretical approaches: a single-particle single-hole Wannier-Mott model (Fig. 2b) describing both the exciton and crystal properties; a pure-exciton model (Fig. 2c), describing the exciton as a three-level atom; and a pure-crystal model (Fig. 2d) [18]. In the full and exciton-only calculations, we used an exciton binding energy of $1.4$ eV, matching the experimental value. Only the full and exciton-only calculations reproduce the few-fs dynamics, thus proving that they have an atomic-like origin, i.e. the IR-induced Stark-shift between the dark and bright exciton states. Considering instead the sub-fs dynamics, a precise characterization the phase delay between the oscillations and $|A_{IR}(t)|{ }^2$ (Fig. 2e) is possible. The exciton-only model fails in reproducing the data even qualitatively, although fast oscillations are still present due to the optical Stark effect [19]. In this picture, the Stark effect changes the exciton binding energy $E_b = -\hbar ^2/(2\mu a_0^2)$, where $\mu = 0.24m_e$ is the reduced mass of the electron-hole system. This modulation of the binding energy affects the excitonic Bohr radius ($1.65$ Å at equilibrium), as expected from the IR-induced acceleration of the electron and hole in opposite directions. However, the timing of these oscillations is inconsistent with the experimental one. Instead, a quantitative agreement is present only with the full model, while the crystal-only model reproduces a similar feature shifted to larger photon energies. This allows us to attribute those fast features to a pure bulk phenomena, namely intra-band motion [7]. Eventually, the full model shows both few-fs and sub-fs features within the same order of magnitude of the experimental measurement. Deviations from the experimental values might originate from underestimating the pump intensity, unavoidable experimental imperfections, and approximations in theoretical models. Nevertheless, these discrepancies should not affect the results we will discuss. Therefore, only the full Wannier-Mott model is thus able to show both the atomic- and solid-like nature of core excitons. Therefore, the effect of the IR dressing is to induce a motion of virtual charges (dynamical Franz-Keldysh effect) [6] in the conduction band, altering the exciton properties. This gives IR-induced oscillations in the reflectivity whose phase relation is close to the one of the crystal. In this picture the exciton motion in real space can be related to the experimental sub-fs features.

Fig. 2

(a) Experimental transient reflectivity trace compared to the simulated ones with (b) the full model, (c) the exciton-only model, and (d) the crystal-only model. The black lines mark the phase delay of the $2\omega _{IR}$ oscillations. We used the same color scale to represent the full, exciton-only and crystal-only calculations. (e) Phase delay of the oscillations for the experimental (brown), full model (orange), exciton only (light-blue) and crystal only (blue) traces. A $\pm \sigma$ interval is represented by shaded regions

To gain further insight on the exciton dynamics, we used the full Wannier-Mott model to study the dependence on the exciton binding energy. Increasing the binding energy, the radius of the exciton decreases and its properties are less influenced by those of the crystal, significantly changing the few-fs features (Fig. 3a–f). The energy-dependent shape of the timing of the oscillations, instead, is almost preserved, but shifted to lower energies, and the phase delay increases. Smaller binding energies, in turn, decrease the exciton localization: the exciton will thus react almost in phase with the external field, decreasing the phase delay (Fig. 3g). The exciton binding energy, that can be changed by dielectric screening or inducing a strain in the crystal, can thus be used as a tuning knob to control the interaction with the surrounding crystal [20,21,22] for Wannier-Mott excitons in general, including valence excitons.

Fig. 3

(a–f) Simulated transient reflectivity traces for different exciton binding energies (0.7, 1.4, 2.1, 2.8, 3.5 and 4.2 eV). Horizontal lines mark the onset of the conduction band (dash-dotted) and of the A exciton transition (dashed). (g) Phase delay of the oscillations in simulations for different exciton binding energies, compared to the crystal-only case (black). The onset of the A exciton transition (in colors) and the onset of the conduction band (black) are marked. Circles show the phase delay corresponding to the A exciton transition

4 Conclusions

We studied the interaction between conduction states and core excitons in MgF${ }_2$ on the attosecond time scale. The uniqueness of our experimental setup allowed for a precise comparison with three different theoretical models. The optical Stark effect is found to be mainly dictating the atomic-like few-fs optical response, while sub-fs dynamics are interpreted in terms of interaction of the IR-dressed conduction band with the exciton state. The dependence of the sub-fs optical response on the exciton binding energy is proposed to be exploited to control excitonic properties on ultrafast time scales.