Attosecond screening dynamics mediated by electron localization in transition metals

Abstract

Transition metals, with their densely confined and strongly coupled valence electrons, are key constituents of many materials with unconventional properties1, such as high-temperature superconductors, Mott insulators and transition metal dichalcogenides2. Strong interaction offers a fast and efficient lever to manipulate electron properties with light, creating promising potential for next-generation electronics3,4,5,6. However, the underlying dynamics is a hard-to-understand, fast and intricate interplay of polarization and screening effects, which are hidden below the femtosecond timescale of electronic thermalization that follows photoexcitation7. Here, we investigate the many-body electron dynamics in transition metals before thermalization sets in. We combine the sensitivity of intra-shell transitions to screening effects8 with attosecond time resolution to uncover the interplay of photo-absorption and screening. First-principles time-dependent calculations allow us to assign our experimental observations to ultrafast electronic localization on d orbitals. The latter modifies the electronic structure as well as the collective dynamic response of the system on a timescale much faster than the light-field cycle. Our results demonstrate a possibility for steering the electronic properties of solids before electron thermalization. We anticipate that our study may facilitate further investigations of electronic phase transitions, laser–metal interactions and photo-absorption in correlated-electron systems on their natural timescales.

Main

The characteristic thermalization time of laser-excited hot electrons in solids is in the femtosecond regime and becomes faster with stronger electron interactions7. Attosecond time resolution is thus required to resolve coupled electron dynamics during the laser–matter interaction before electron thermalization has occurred. Attosecond transient absorption spectroscopy has revealed electric-field-guided electron dynamics in simple dielectrics and semiconductors9,10,11,12,13,14. However, transient absorption studies of localized and strongly interacting electrons, such as on the d and f orbitals of transition metals, have been limited to the few-femtosecond regime15. Transition metal elements are the key constituents of many materials exhibiting remarkable properties, such as Mott insulators, high-Tc superconductors and transition metal dichalcogenides2. Understanding the coupled-electron dynamics in these systems is central for their applications in optoelectronics, energy-efficient electronics, magnetic-memory devices, spintronics and new solar cells16. Transition metal elements have been studied with attosecond photoemission spectroscopy17,18,19. However, state-of-the-art theoretical treatment20 of photoemission from metals on the attosecond timescale neglects any electronic perturbation induced by the optical field beyond the skin effect. This approximation is based on efficient screening of the optical field by surface electrons21 and space-charge limitation of the maximum optical field intensity22. In contrast, femto- and picosecond core-level photo-absorption of metals systematically shows signatures of excitation-induced electronic structure modification at high intensities23,24,25,26.

To bridge this gap in intensities and timescales, we have used attosecond transient absorption spectroscopy to study the collective electron dynamics in the transition metals Ti and Zr. In the following, we discuss the phenomena observed in Ti, while the physically analogous behaviour of Zr is presented in the Supplementary Information. A comprehensive description of our experimental apparatus is provided in refs. 12,27. In the pump–probe experiments (Fig. 1a), few-cycle near-infrared (NIR) laser pulses excite electrons in a 50-nm-thick, free-standing Ti film. The induced change in the sample’s transmission around the M2,3 absorption edge is probed by single attosecond pulses in the extreme-ultraviolet range (XUV, ~35 eV, Fig. 1b) with a pulse duration of 260 as. The attosecond pulses are produced via high-harmonic generation in Kr. In addition, a Ne gas jet is placed in front of the metal film sample, and photoelectrons from the jet are collected with a time-of-flight spectrometer in a streaking experiment, providing precise information on the electric field of the NIR pump pulse.

Fig. 1: Attosecond transient absorption spectroscopy.
figure1

a, Schematic of the experimental apparatus for acquisition of photon and photoelectron spectra to measure both the incoming transient NIR pump laser field and the spectrally resolved probe absorption in the XUV. Photo-ionized electrons from the Ne gas nozzle are accelerated by the optical field and collected with a time-of-flight spectrometer. Transmitted probe radiation is recorded with an XUV spectrometer. b, The blue area shows the calculated electronic density of states of Ti. The violet area indicates the probe spectrum with respect to the 3p level. The vertical dashed line is the Fermi energy position. c, Experimental absorption spectrum of Ti (blue line; the shaded area shows the s.d., see Methods) compared to TDDFT calculations. The red line shows the absorption coefficient from a full simulation and the black line shows the results of the independent-particle (IP) model. d, Calculated band structure of Ti. Red and violet arrows indicate the pump and probe transitions, respectively.

Figure 1c shows the absorption spectrum of Ti, obtained with attosecond pulse trains covering the photon energy range from 25 to 75 eV (see Methods). We compared this with ab initio calculations based on time-dependent density functional theory (TDDFT) in the local density approximation (LDA) for a hexagonal close-packed (hcp) Ti cell (see Methods for details). Note that this theoretical framework has been shown to accurately describe electronic screening effects28 as well as attosecond electron dynamics in solids10,12,29. The full calculation (shown in red, Fig. 1c) closely reproduces the experimental spectrum, whereas the independent-particle approximation (Fig. 1c, black), which ignores electron–electron interactions (see Methods for definition), fails to describe it. The underlying many-body process is known as the local field effect30, which is ascribed to the additional potential induced locally by charge displacement. The Ti M2,3 absorption edge, located at 32.6 eV (ref. 31), is dominated by the intra-shell 3p–3d electronic transition (Fig. 1d), which features a large overlap of initial- and final-state wavefunctions8. The M1 absorption edge at 56.1 eV caused by 3s–4p transitions is much less prominent due to the delocalized character of the 4p final states. A strong interaction of electrons participating in the transition and screening processes leads to collective excitation, which manifests as a giant resonance32 above the M2,3 absorption edge. Our probing mechanism is thus sensitive to the dynamics of screening and collective electron motion8,33 via many-body effects. In contrast, a conventional probing mechanism is usually based on the independent-particle picture and only senses the final state occupation. We note that the experimental data in Fig. 1c exhibit no prominent peaks in the region of 32–52 eV, in contrast to the TDDFT calculations (red curve, Fig. 1c). We suggest that this discrepancy originates from the smoothening role of Auger relaxation, which is not included in the calculation.

We recorded the transmitted probe spectra with the pump on and off in a fast sequence (using a mechanical shutter) and calculated the induced optical density (ΔOD, equation (1), Methods). Figure 2a shows the pump-induced change in absorption as a function of pump–probe delay in a 20 fs window with 250 as delay steps. The pump peak intensity is Ipump = 7.5 ± 0.7 × 1011 W cm−2 and the centre photon energy is 1.55 eV. In the presented data, we have subtracted the delay-independent thermal background (see Methods and Supplementary Information).

Fig. 2: Experimental pump–probe results.
figure2

a, Transient absorption spectrogram showing pump-induced optical density change ΔOD, probed with a single attosecond pulse. Positive delays indicate the XUV probe arriving after the NIR pump pulse. b, Profile of the induced optical density at positive delays (0–10 fs). The blue area indicates s.e.m. (see Methods). c, Delay-dependent signal extracted from the data in a in the photon energy band of 31.7–32.9 eV (dashed horizontal lines in a and b) with incident pump fluence reconstructed from the streaking experiment (red curve). The blue area shows s.e.m. (see Methods). d, Electric field of the NIR pump pulse retrieved from the streaking experiment and associated to the fluence in c.

The pump–probe spectrogram in Fig. 2a reveals an optical density profile that rapidly increases in amplitude and exhibits several peaks along the energy axis (Fig. 2b). Previous studies of attosecond electron dynamics in semiconductors and dielectrics revealed field-driven, oscillatory dynamics9,10,11,12,13,14. In contrast, in the metallic system of Ti, we observe a quasi-instantaneous linear response. This behaviour was further confirmed by ab initio calculations of the induced absorption (Supplementary Fig. 8). Whereas intra-band motion in bandgap materials induces a periodic modulation of the absorption34, the optical response of Ti is determined by a different mechanism. Figure 2c shows that the optical density transient follows the laser fluence extracted from the streaking experiment (Fig. 2d). By analysing several independent experimental datasets with a linear convolution model, we estimated an upper limit of the microscopic response time to the pump fluence as 1.1 fs with an experimental accuracy of 290 as (see Methods and Supplementary Information). Although the time evolution of the signal is guided by pump absorption, the spectrogram does not display any induced transparency features (Fig. 2a, blue false colour), indicating that the expected state-filling effect is completely cancelled by a many-body phenomenon.

To provide a more detailed understanding of the underlying electronic dynamical rearrangement and its effect on Ti absorption, we simulated the pump–probe experiments with ab initio TDDFT calculations12,14,35. Figure 3a (red line) presents the absorption change in hcp Ti induced by a 10 fs NIR pump pulse with centre photon energy of 1.55 eV and incident intensity of Ipump = 1 × 1012 W cm−2. Our simulation successfully reproduces the main absorption feature around 32 eV in a highly non-equilibrated state of Ti. We compared these results with the electron-thermalized state of Ti and found that a particular excited electron distribution has little influence on the induced spectral feature (Fig. 3a,c). Accordingly, the experiments show that XUV absorption at the end of the pump pulse and 12 fs later differ only slightly, so thermalization has little influence on our observables (Fig. 3b). The independent-particle model (Fig. 3a, black), which ignores electron–electron interactions, merely reflects the electron–hole population (Fig. 3c) and exhibits transparency above the M2,3 edge. In contrast, induced transparency is suppressed in the full model, in agreement with the experiment.

Fig. 3: Ab initio calculation results.
figure3

a, Calculated change of the absorption coefficient in non-equilibrium, immediately after excitation with an incident pump intensity of 1 × 1012 W cm−2 (red line). Also shown are the thermo-induced absorption changes with an electron temperature of 0.315 eV from a full model (blue line) and independent-particle (IP) approximation (black line, scaled down by a factor of 0.25). b, Experimental optical density profiles 10 fs and 22 fs after the pump–probe overlap (blue and red lines; shaded areas show s.e.m., see Methods). c, Non-equilibrium electron–hole population after NIR pump excitation (red line) and corresponding thermalized distribution (blue line). Light blue and dark blue areas show the electronic density of states and electron population in the electron-thermalized phase, respectively. d, Decomposition of the induced absorption coefficient for a thermalized system into three physical components: state filling (blue line), electronic structure renormalization (red line, scaled up by a factor of 5) and local-screening modification (black line).

To gain further insight, we separated the TDDFT calculation into three physical factors: electronic structure renormalization, state-filling and local-screening modification (see Supplementary Information). Figure 3d shows that the effects of state-filling (blue line) and local-screening modification (black line) dominate the absorbance change. Importantly, the induced transparency above the M2,3 edge due to the state-filling effect is primarily cancelled by the local-screening modification. As a result, a single positive peak around the absorption edge is formed, indicating a breakdown of the independent-particle picture by the strong screening of localized d orbitals. The renormalization effect is well approximated by a redshift of the absorption edge (see Supplementary Information). This fact indicates enhancement of the static screening around the Ti atom due to increased electron localization, in agreement with previous calculations of thermo-induced renormalization36. Finally, a modified electron interaction also influences the core-hole lifetime, which is determined by super-Coster–Kronig Auger decay, involving two interacting 3d electrons. An increase in d-electron population thus shortens the semicore-hole lifetime. We find that additional empirical broadening of 0.15 eV qualitatively reproduces the second, weaker absorption maximum (35.5 eV, Fig. 3b), although it is slightly shifted to higher energies (Supplementary Fig. 11), thus capturing the essential features of the experimental absorption spectrum.

Based on the agreement between theory and experiments, we proposed that the pump absorption results in an ultrafast modification of screening in the electronic system of Ti. To provide additional support for our hypothesis, we investigated in detail the underlying microscopic dynamics in real time and real space. Figure 4 shows the TDDFT simulation results of a pump field interacting with an hcp Ti cell. The laser electric field is parallel to the c axis of the Ti crystal (Fig. 4a), and we visualize the electronic density transients in the ca plane.

Fig. 4: Real-time and real-space theoretical investigation of the laser-driven microscopic electron dynamics in Ti.
figure4

a, The simulated laser field is parallel to the c axis of the hcp Ti cell. The blue plane shows the ca axis cut. b, Electron density dynamics in the ca plane during the laser cycle, displayed in false colour (red for increased electron density). c, Localized charge around the Ti atom (blue line), which closely follows the laser fluence (red line). The localization region is marked with a black circle in b.

Figure 4b shows the electron density difference induced by the NIR pump pulse. On top of the field-induced polarization dynamics we observe a build-up of charge density, which acquires a distinct d-orbital shape by the end of the pulse. Therefore, the pump creates an inflow of electron density towards the Ti atom and localizes it on the d orbitals. To illustrate this, we integrated the localized electron density around a Ti atom and plotted it as a function of time along with the pump fluence (Fig. 4c). As seen from the figure, the electron localization closely follows the laser fluence in time, similar to the peak amplitude of the experimental transient absorption (Fig. 2c). This supports our hypothesis that the transient absorption peak originates from ultrafast electron localization due to the pump pulse. The NIR pump modifies the atomic screening, resulting in the absorption peak structure around the M2,3 absorption edge, as discussed.

We note that the observed transient absorption features are robust against variation in experimental parameters such as sample thickness, surface oxidation state and pump photon energy (see Supplementary Information). Additionally, TDDFT calculations show no significant dependence on the crystal structure. We also observe identical induced spectral features at the N2,3 absorption edge of zirconium, suggesting a universal character of this phenomenon for transition metals (see Supplementary Information). The possibility of charge localization in the metallic system of Ti is based on the relatively inefficient screening by 4s/4p electrons, because their wavefunctions weakly penetrate the 3d spatial region37. This effect has essentially an atomic origin, suggesting that a photo-induced change in electron localization must also govern the first step of the interaction of light with transition metal compounds. A change in the localization of valence electrons is expected to vary the macroscopic properties of transition metals, such as conductivity, reflectivity, plasmonic behaviour, magnetization and catalytic activity at surfaces. A particularly high electron density in transition metals, associated with the fastest screening rates in the valence band, thus allows us to manipulate these macroscopic properties at unprecedented rates, in agreement with recent studies38,39. Our results unambiguously demonstrate photomanipulation of screening in transition metals via ultrafast electron localization below the electron thermalization timescale.

Methods

Pump–probe set-up

The pump–probe experiments were carried out on an attosecond set-up as extensively described in refs. 12,27. In this set-up, the pump–probe interferometer is actively stabilized by introducing a weak co-propagating continuous-wave 473 nm laser beam. A fast feedback loop acts on the piezo delay actuator, stabilizing the pump–probe delay to a desired value with 15 as r.m.s. uncertainty. A mechanical shutter in the pump arm of the interferometer is used for data acquisition, with or without the pump, in a fast sequence. We define the pump-induced optical density as follows:

$$\begin{array}{*{20}{c}} {\Delta {\it{\mathrm{OD}}} = {\mathrm{ln}}\frac{{I_{\mathrm{pr}}^{\mathrm{no}\,{pump}}}}{{I_{{\it{\mathrm{pr}}}}^{{\it{\mathrm{pump}}}}\left( \tau \right)}}} \end{array}$$
(1)

where \({\it{I}}_{{\it{\mathrm{pr}}}}^{{\it{\mathrm{pump}}}},{\it{I}}_{{\it{\mathrm{pr}}}}^{{\it{\mathrm{no}}}\,{\it{\mathrm{pump}}}}\) are the transmitted probe spectral intensities with and without pump radiation, and τ is the pump–probe delay with positive values for the probe arriving after the pump.

Samples

The samples consist of passivated 48.6-nm-thick Ti films (Luxel, https://luxel.com) covered with 4.9 nm of amorphous carbon on both sides (without breaking the vacuum). The films are mounted on copper transmission electron microscopy grids with nine windows. The unsupported window areas are 300 µm in diameter. The pump/probe spot sizes on the sample have dimensions of ~50 µm.

Static absorption

To characterize the absorption of our samples in a broad energy region, we generate attosecond pulse trains in Xe, Kr and Ar. By changing the double-filament compression parameters we are able to tune the attosecond pulse train spectrum to close the gaps between harmonics. We acquire 10 pairs of signal (sample in) and reference (sample out) spectra in sequence by moving the motorized sample manipulator. Each signal/reference spectrum is an accumulation of 104 XUV pulse trains, acquired rapidly in ~10 s. We then calculate the total optical density, 〈α〉, as a weighted average over different probing spectra:

$$\begin{array}{*{20}{c}} {\left\langle {\it{\alpha }} \right\rangle = \frac{{{\sum} {\alpha _iw_i} }}{{\frac{{n - 1}}{n}{\sum} {w_i} }},w_i = \frac{1}{{\sigma _{{\it{\mathrm{tot}}}\,i}^2}}} \end{array}$$
(2)

For each shape of the probe i and the corresponding absorption spectrum αi the variance comprises fast and slow fluctuations:

$$\begin{array}{*{20}{c}} {\sigma _{\mathrm{tot}\,i}^2 = \left\langle {\sigma _{{\it{\mathrm{fast}}}\,j}^2} \right\rangle _i + \sigma _{{\it{\mathrm{slow}}}\,i}^2} \end{array}$$
(3)

The fast fluctuation term reflects the carrier-envelope phase and laser intensity fluctuation on a timescale of tens of milliseconds:

$$\begin{array}{*{20}{c}} {\sigma _{{\it{\mathrm{fast}}}\,j}^2 = \left( {\sigma _j^{I\,{\mathrm{ref}}}/I_j^{\mathrm{ref}}} \right)^2 + \left( {\sigma _j^{I\,{\mathrm{sig}}}/I_j^{\mathrm{sig}}} \right)^2} \end{array}$$
(4)

where j [1,10] is the index for each signal/reference pair and \(\sigma _j^I\) is the s.d. over 104 laser shots. The slow fluctuation term accounts for drifts between subsequent sample insertions (timescale of tens of seconds) and takes into account the position uncertainty of wrinkled metal foils:

$$\begin{array}{*{20}{c}} {\sigma _{{\it{\mathrm{slow}}}\,{\it{i}}}^2 = \frac{{\mathop {\sum }\nolimits_j \left( {\alpha _{j,i} - \left\langle \alpha \right\rangle _i} \right)^2}}{{{\it{N}} - 1}}} \end{array}$$
(5)

where αj,i is the optical density for each of the 10 signal/reference pairs, taken with the ith probe spectrum. The final uncertainty of the optical density is estimated as

$$\begin{array}{*{20}{c}} {\sigma _\alpha ^2 = \frac{{{\sum} {\left( {\alpha _i - \left\langle \alpha \right\rangle } \right)^2} w_i}}{{\frac{{{\it{n}} - 1}}{{\it{n}}}{\sum} {w_i} }}} \end{array}$$
(6)

Evaluation of the dynamics

In the course of the pump–probe delay scan the samples are exposed to an average power of ~0.5 mW, resulting in elevated temperature. This effect manifests as a fully reversible, delay-independent background with a lifetime of ~5 ms, which is separated from the transient signal at negative pump–probe delays (see Supplementary Information for more details). By keeping the pump intensity sufficiently low, we have ensured that no permanent sample modification takes place during the experiments. Therefore, in the presented data the background signal has been subtracted.

To characterize the material response time with sub-cycle resolution, we reconstructed the temporal profile of the pump electric field using a photoelectron streaking experiment40,41. In this investigation, XUV-photoionized electrons are accelerated in the NIR pump field and collected in a time-of-flight spectrometer. Their kinetic energy as a function of XUV–NIR delay directly reflects the NIR field vector potential. In contrast to refs. 12,14, we performed the streaking experiment after the transient absorption measurement, with the metallic sample removed so as to avoid the electron background in the time-of-flight measurement. The actively stabilized pump–probe delay ensures a timing accuracy between the experiments to a precision of 15 as r.m.s.

We assume that the optical density response is linear and follows the absorbed laser intensity:

$$\begin{array}{*{20}{c}} {\Delta {\it{\mathrm{OD}}}_{{\it{\mathrm{model}}}}\left( t \right) = a\mathop {\smallint }\limits_{ - \infty }^t I_0\left( {t^\prime - t_{{\it{\mathrm{jitter}}}}} \right)h_\tau \left( {{\it{t - t}}^\prime } \right){\it{dt}}^\prime } \end{array}$$
(7)

with the response function having the form

$$\begin{array}{*{20}{c}} {h_\tau \left( t \right) = 1 - e^{ - \frac{t}{\tau }}} \end{array}$$
(8)

where tjitter accounts for possible jitter between the streaking and transient absorption data due to acquisition noise, delay drift and spatial separation of the gas and solid targets.

We find the response time by simultaneously fitting seven different pairs of streaking and transient absorption experiments, with parameters a and tjitter being varied, and the response time τ being common for all scans. This approach reflects the fact that τ is a constant of the system being studied, while a and tjitter are attributable to experimental noise contributions. We minimize the sum of χ2 of seven scans:

$$\begin{array}{*{20}{c}} {\left\{ {\mathrm{A}} \right\},\left\{ {{\it{t}}_{{\it{\mathrm{jitter}}}}} \right\},\tau :{\mathrm{min}}\left( {\mathop {\sum }\limits_{{\it{\mathrm{scan}}}\,{\it{\mathrm{pairs}}}} \chi ^2} \right)} \end{array}$$
(9)
$$\begin{array}{*{20}{c}} {\chi ^2 = \mathop {\sum}\nolimits_i {\frac{{\left( {\Delta {\it{\mathrm{OD}}}_{{\it{\mathrm{exp}}}} - \Delta {\it{\mathrm{OD}}}_{{\it{\mathrm{model}}}}} \right)^2}}{{\sigma _i^2}}} } \end{array}$$
(10)

where \(\Delta {\it{\mathrm{OD}}}_{{\it{\mathrm{exp}}}}\) is the experimental pump-induced optical density change in the energy band from 31.7 to 32.9 eV and \(\sigma _\tau ^2\) is the standard deviation of the optical density. We use the simplex-downhill algorithm to find the minimizing solution, starting with uniformly distributed initial guess parameters to check for local minima effects. To find the uncertainty of the response, a Monte Carlo simulation is used. The experimental uncertainty on the optical density change is calculated as the s.e.m. for 300 pairs of signal (pump on)/reference (pump off) spectra:

$$\begin{array}{*{20}{c}} {\sigma _{\Delta {\it{\mathrm{OD}}}} = \sqrt {\frac{{\mathop {\sum }\nolimits_1^{\it{N}} \left( {\Delta {\it{\mathrm{OD}}}_{\it{i}} - \left\langle {\Delta {\it{\mathrm{OD}}}} \right\rangle } \right)^2}}{{{\it{N}}\left( {{\it{N}} - 1} \right)}}} ,\Delta {\mathrm{OD}}_i = \ln \left( {I_i^{\mathrm{ref}}/I_i^{\mathrm{sig}}} \right)} \end{array}$$
(11)

where N is the number of signal/reference pairs per delay step. From this distribution, we generate 1,000 transient absorption signals for each attosecond transient absoprtion spectroscopy/streaking pair and run the fitting algorithm. The results of the Monte Carlo simulations give a mean response time of 1.12 fs with a standard deviation of 290 as (see Supplementary Information for more details).

Theoretical modelling details

In the framework of the TDDFT, the electron dynamics is described by a single-particle Schrödinger-like equation—the so-called time-dependent Kohn–Sham (TDKS) equation. For laser-induced electron dynamics in solids, the TDKS equation has the following form:

$$\begin{array}{*{20}{c}} {i\hbar \frac{\partial }{{\partial {\mathrm{t}}}}u_{b{\mathbf{k}}}\left( {{\mathbf{r}},t} \right) = h_{{\mathrm{KS}},{\mathbf{k}}}\left( t \right)u_{b{\mathbf{k}}}\left( {{\mathbf{r}},t} \right)} \end{array}$$
(12)

where ubk(r,t) describes electron orbitals with band index b and Bloch wavevector k. The Hamiltonian hKS,k(t) is the so-called Kohn–Sham Hamiltonian, and is given by

$$\begin{array}{*{20}{c}} {h_{{\mathrm{KS}},{\mathbf{k}}}\left( t \right) = \frac{1}{{2m}}\left( {{\mathbf{p}} + \hbar {\mathbf{k}} + \frac{e}{c}{\mathbf{A}}\left( t \right)} \right)^2 + \hat v_{\mathrm{ion}} + v_{\mathrm{H}}\left( {{\mathbf{r}},t} \right) + v_{\mathrm{xc}}\left( {{\mathbf{r}},t} \right)} \end{array}$$
(13)

where \(\hat v_{\mathrm{ion}}\) is the ionic potential, vH(r,t) is the Hartree potential and vxc(r,t) is the exchange-correlation potential. Note that the Hartee and exchange-correlation potentials are functionals of the electron density

$$\begin{array}{*{20}{c}} {\rho \left( {{\mathbf{r},t}} \right) = \mathop {\sum }\limits_{b{\mathbf{k}}} n_{b{\mathbf{k}}}\left| {u_{b{\mathbf{k}}}\left( {\mathbf{r},t} \right)} \right|^2} \end{array}$$
(14)

where nbk is the occupation factor. In this work, we employ the adiabatic local density approximation (ALDA) for the exchange-correlation potential42. We describe the laser fields with a spatially uniform vector potential, \({\mathbf{E}}\left( t \right) = - \frac{1}{{{c}}}\frac{{{{{\mathrm{d}}A}}\left( t \right)}}{{{\mathrm{d}}t}}\), assuming that the wavelength of the laser fields is much longer than the spatial scale of electron dynamics (this is nothing but the dipole approximation).

To describe the laser-induced electron dynamics in Ti metal, we employ the hcp structure of Ti. Ti atoms are described with a norm-conserving pseudopotential method, treating 3s, 3p, 3d and 4s electrons as valence43,44. In the practical calculations, we solve the TDKS equation (6) in real space and real time: the spatial coordinate r is discretized into uniform grid points with spacing h = 0.3 a.u., and the electron orbitals ubk(r,t) are propagated under the influence of the laser fields with a time step Δt = 0.03 a.u. The first Brillouin zone is also discretized into 203 k-points. All the TDDFT simulations in this work are carried out with the Octopus code45.

One of the most important outputs of the TDDFT calculations is the macroscopic current density, J(t), which is directly related to the optical properties of solids. We compute the current with

$$\begin{array}{*{20}{c}} {{\mathbf{J}}\left( t \right) = - \frac{e}{{m\Omega }}\mathop {\smallint }\limits_\Omega {\mathrm{d}}{\mathbf{r}}\mathop {\sum }\limits_{b{\mathbf{k}}} n_{b{\mathbf{k}}}\left[ {u_{b{\mathbf{k}}}^ \ast \left( {{\mathbf{r}},t} \right)\left\{ {{\mathbf{p}} + \hbar {\mathbf{k}} + e\mathbf{A}\left( t \right)/c} \right\}u_{b{\mathbf{k}}}\left( {{\mathbf{r}},t} \right)} \right] + {\mathbf{J}}_{\mathrm{PS}}} \end{array}$$
(15)

where Ω is the volume of the unit cell and JPS is the contribution from the pseudopotential. If the electric current under an impulsive distortion, E(t) = E0δ(t), is computed, the optical conductivity of the system can be evaluated as

$$\begin{array}{*{20}{c}} {\sigma \left( \omega \right) = \frac{{{\int} {{\mathrm{d}}t\,e^{i\omega t - \gamma t}{\mathbf{J}}\left( t \right)} }}{{E_0}}} \end{array}$$
(16)

where γ is a numerical damping factor. We set this to 0.5 eV −1 in this work. From the optical conductivity, the absorption coefficient can be computed.

The absorption coefficient in Fig. 1c was computed with TDDFT using this scheme. If the time dependence of the Hartree and exchange-correlation potentials is ignored in the time propagation, this scheme provides the absorption coefficient with the independent-particle approximation in Fig. 1c because the TDKS equation is simply reduced to the Schrödinger equation of an independent-particle system.

Optical response of pumped systems in the non-equilibrium phase

To investigate the modification of the absorbance immediately after the pump laser pulse, we performed a numerical pump–probe simulation based on the TDDFT35. For the pump pulse, we employ the form

$$\begin{array}{*{20}{c}} {{\mathbf{A}}_{{\it{\mathrm{pump}}}}\left( {\it{t}} \right) = - \frac{{{\it{cE}}_{{\it{\mathrm{pump}}}}}}{{{\it{\omega }}_{{\it{\mathrm{pump}}}}}}\cos ^2\left[ {\frac{\pi }{{{\it{T}}_{{\it{\mathrm{pump}}}}}}t} \right]{\mathrm{sin}}\left[ {{\it{\omega }}_{{\it{\mathrm{pump}}}}{\it{t}}} \right]} \end{array}$$
(17)

in the domain −Tpump/2 < t < Tpump/2, and zero outside. Here, ωpump is the mean frequency and Tpump is the full duration of the pump pulse. We set ωpump to 1.55 eV −1 and Tpump to 20 fs. The corresponding full-width at half-maximum of the laser intensity is ~7 fs. We also set the maximum field strength Epump to 9.7 × 108 V m−1, which corresponds to the maximum field strength at the front surface of the sample under an irradiation of 1012 W cm2, taking the surface reflection into account with the stationary solution of the Maxwell equation. For the probe pulse, we employ the impulsive distortion at the end of the pump pulse (t = Tpump/2).

To extract the optical property of the pumped system, we performed two TDDFT simulations. One employs both the pump and the probe pulses, while the other employs only the pump pulse. We then define two kinds of electric current. One is computed with the pump–probe calculation and the other with the pump-only calculation. We denote the first the pump–probe current Jpump-probe(t) and the latter the pump-only current Jpump-only(t). By subtracting Jpump-only(t) from Jpump-probe(t), one may extract the current induced by the probe pulse in the presence of the pump pulse. We shall call it the probe current, Jprobe(t) = Jpump-probe(t) − Jpump-only(t). Applying a Fourier analysis of equation (16) to the probe current, the optical property of the laser-excited system can be computed. As a result of this pump–probe scheme, the absorption coefficient in the non-equilibrium phase is provided.

Optical response of pumped systems in the electron-thermalized phase

To theoretically investigate the effect of electron thermalization, we computed the optical properties of hcp Ti with a finite electron temperature46. In this process, we first compute the ground state, assuming the Fermi–Dirac distribution for the electron occupation nbk for the density in equation (14). As a result, the initial electron orbitals ubk(r,t = 0) and the occupation nbk with finite electron temperature can be prepared consistently. Applying the impulsive distortion and computing the electron dynamics based on the above scheme, we then compute the optical properties of hcp Ti with the finite electron temperature. Here, we set the electron temperature Te to 0.315 eV so that the excess energy of the finite electron-temperature system becomes identical to that of the system pumped by the laser used in the above analysis.

To clarify the microscopic mechanism of the absorption change induced by the pump pulse, we performed a theoretical decomposition of the computed transient absorption coefficient. Given that the thermal and non-equilibrium electronic distributions provide qualitatively equivalent results, we focused on the thermal distribution for simplicity.

To construct the theoretical decomposition, we first revisit the linear response in the TDDFT with a weak electric field. In the weak perturbation limit, the Kohn–Sham Hamiltonian (equation (13)) under the ALDA can be rewritten as

$$\begin{array}{l}h_{{\mathrm{KS}},{\mathbf{k}}}\left( t \right) = \frac{1}{{2m}}\left( {{\mathbf{p}} + \hbar {\mathbf{k}} + \frac{e}{c}{\mathbf{A}}\left( t \right)} \right)^2 + \hat v_{\mathrm{ion}}\\ + v_{H_{{\mathrm{xc}}}}\left[ {\rho \left( {{\mathbf{r}},t = 0} \right)} \right] + \mathop {\smallint }\nolimits {\mathrm{d}}{\mathbf{r}}^\prime f_{{H_{\mathrm{xc}}}}\left( {{\mathbf{r}},{\mathbf{r}}^\prime } \right){\mathrm{\delta}} \rho \left( {{\mathbf{r}}^\prime ,t} \right)\end{array}$$
(18)

where \(v_{H_{{\mathrm{xc}}}}\)[ρ(r,t = 0)] is the sum of the Hartree and the exchange-correlation potentials evaluated with the initial electron density ρ(r,t = 0). The dynamical part of the Kohn–Sham potential can be evaluated with the Hartree–exchange-correlation kernel \(f_{H_{{\mathrm{xc}}}}\)(r,r′) and the induced density δρ(r,t) = ρ(r,t) − ρ(r,t = 0). An essential difference between the optical response in the electron-thermalized phase and in the ground state is the choice of the occupation factor nbk. In the present work, we use the Fermi–Dirac distribution with finite temperature Te = 0.315 eV for the electron-thermalized phase and almost zero temperature Te = 0.01 eV for the ground state. Therefore, the change in the absorption due to the electron temperature can be analysed based on the occupation factor contribution. In the linear response TDDFT calculation, the occupation factor affects the following three features: (1) electronic structure renormalization via the initial Hartree–exchange-correlation potential \(v_{H_{{\mathrm{xc}}}}\)[ρ(r,t = 0)] and the initial density ρ(r,t = 0); (2) modification of the local-screening effect via modification of the induced potential and the induced density δρ(r,t); (3) the state-filling effect via the current evaluation with equation (15).

Quantifying the electron localization

To quantify the localization dynamics under a pump pulse, we compute the local induced density as

$$\begin{array}{*{20}{c}} {S\left( t \right) = {\int} {{\mathrm{d}}r\left[ {\rho \left( {{\mathbf{r}},t} \right) - \rho \left( {{\mathbf{r}},t = 0} \right)} \right]e^{ - \left( {{\mathbf{r}} - \mathbf{R}_{\mathrm{Ti}}} \right)^2/2\sigma ^2}} } \end{array}$$
(19)

where ρ(r,t) is the electron density under the pump pulse and RTi is the position of a Ti atom. The width of the Gaussian window σ was set to 1 a.u.

Data availability

The data that support the findings of this study are available via https://doi.org/10.3929/ethz-b-000345468 or from the corresponding author upon reasonable request.

Code availability

The Octopus code for TDDFT is available at https://octopus-code.org.

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Acknowledgements

The authors acknowledge discussions with E. Krasovskii. S.A.S. and A.R. thank M. J. T. Oliveira for helping with the generation of a transferable pseudopotential for Ti, dealing with semicore electrons. This work was supported by the National Center of Competence in Research – Molecular Ultrafast Science and Technology (NCCR MUST) funded by the Swiss National Science Foundation. The authors acknowledge financial support from the European Research Council (ERC-2015-AdG-694097) and the European Union’s Horizon 2020 Research and Innovation programme under grant agreement no. 676580 (NOMAD). S.A.S. acknowledges support from the Alexander von Humboldt Foundation.

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M.V., S.A.S., L.G., A.R. and U.K. supervised the study. M.V., F.S., N.H., L.K. and M.L. conducted the experiments. M.V. and F.S. analysed the experimental data. S.A.S. and A.R. developed the theoretical modelling. All authors were involved in the interpretation of data and contributed to the final manuscript.

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Correspondence to M. Volkov.

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Volkov, M., Sato, S.A., Schlaepfer, F. et al. Attosecond screening dynamics mediated by electron localization in transition metals. Nat. Phys. 15, 1145–1149 (2019). https://doi.org/10.1038/s41567-019-0602-9

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