Nonequilibrium band occupation and optical response of gold after ultrafast XUV excitation

Free electron lasers offer unique properties to study matter in states far from equilibrium as they combine short pulses with a large range of photon energies. In particular, the possibility to excite core states drives new relaxation pathways that, in turn, also change the properties of the optically and chemically active electrons. Here, we present a theoretical model for the dynamics of the nonequilibrium occupation of the different energy bands in solid gold driven by exciting deep core states. The resulting optical response is in excellent agreement with recent measurements and, combined with our model, provides a quantitative benchmark for the description of electron–phonon coupling in strongly driven gold. Focusing on sub-picosecond time scales, we find essential differences between the dynamics induced by XUV and visible light.

www.nature.com/scientificreports/ the energy transfer between the electron and the phonon systems. Comparing these data with our results clearly rules out the most commonly used model for the coupling constant 34 but gives good agreement for a model predicting much slower energy transfer 35 . Comparing excitations with XUV and visible light, we observe a qualitatively different evolution of the optically active sp-density which can be traced back to different relaxation pathways on sub-picosecond time scales.

Results
Three-band rate equations. We study the relaxation behaviour in solid gold excited by short XUV and X-ray pulses as provided by FELs. Such radiation can promote electrons from deeper levels into the conduction band which requires to consider the dynamics between, at least, three optically active bands or levels. As an example, we use here the 4f-state in addition to the 5d-and 6sp-bands that always need to be considered as there is no band gap between the d-electrons and the delocalised sp-electrons in gold.
The fully kinetic stage of the relaxation process lasts only a few femtoseconds as has been shown by solving kinetic equations 29,36 and time-dependent DFT simulations 37 . Afterwards, a Fermi distribution has been established in each band. Here, we assume that the distributions in the d-and sp-bands have a common temperature but still nonequilibrium occupations, that is, different chemical potentials. Accordingly, the occupation of each band and the total energy content of the electrons are required to describe the system dynamics. For times beyond a picosecond, the energy transfer from the electrons to the lattice must be included as well.
The dynamics in the electron system is initiated by the photo-excitation of a core level, the 4f-level in our case. These electrons increase the density of the sp-band and the excess energy heats the electron system. The holes in the 4f-states are filled by d-electrons within 2.27 fs 38 . This process has a large impact on the occupation and energy density of the d-electrons as well as the sp-electrons: one d-electron fills the hole while in an Auger-like process another d-electron is promoted into the conduction band, the sp-band. Both, the initial photo-electrons as well as the Auger electrons can be excited far above the Fermi energy into quasi-free states of the sp-band. Both processes also release a large amount of kinetic energy into the electron system. After photo-and Augerexcitation, the occupations of the d-and sp-bands are not in equilibrium with the new energy content, i.e., the temperature. The time of the subsequent equilibration process between the two bands is quantified here by a relaxation time that describes the imbalance of the rate of impact ionisation of d-electrons by hot sp-electrons and the rate of deexcitation of sp-electrons into the d-band.
The processes described above are cast into a system of rate equations that can also be easily extended to include more states. The energy in the electron system is traced by enforcing energy conservation in each process. Moreover, we couple the electrons to the phonons via a well-established two-temperature model 39 with the electron-phonon energy transfer rate g ei as the critical parameter. Thus, our prediction for the evolution of the band occupation spans the large range of relaxation stages after the fully kinetic phase of a few femtoseconds to the equilibration of electron and phonon temperatures. Including all processes and boundary conditions mentioned above, we obtain Here n a labels the electron density in the a-band and T e and T i are the electron and lattice (ion) temperatures, respectively. S = S(t) describes the original energy source, that is, the energy input from photons with frequency ω L . Thus, S/ ω L is the creation rate of direct photo-electrons. τ Auger is the Auger time for the deep level and τ relax is the time constant for the equilibration between the densities in the d-and sp-bands. The details of the band structure, which has been calculated from DFT simulations 40 , enter the model via the equilibrium densities at elevated temperatures n eq a and the electron heat capacity C e . n ini f labels the density of the 4f-state before excitation. E sp f is the energy difference from the 4f-state to the Fermi energy in the sp-band and E d f is the energy gap from the 4f-state to the d-edge.
Except for two parameters, the relaxation time between the d-and sp-bands and the electron-phonon coupling, all input parameters are measured quantities or can be generated by ab initio simulations. In turn, models for the important parameters τ relax and g ei can be benchmarked by a quantitative comparison of model predictions www.nature.com/scientificreports/ and experimental data for the electron dynamics as we will demonstrate on the example of electron-phonon energy equilibration.
Density and temperature relaxation. We demonstrate the capabilities of the three-band model (1) by a direct comparison with a recent experiment 12 . To that end, we consider excitation of gold with an XUV pump pulse of 150 fs duration (full width at half maximum), a wavelength of 13.6 nm and a total excitation energy of 0.89 MJkg −1 . For this photon energy, the photo-electrons are excited 5.45 eV above the Fermi energy while the Auger electrons will be >70 eV more energetic. For the model parameters, we choose a density relaxation time of τ relax = 200 fs (see "Methods" section) and two models for the electron-phonon coupling parameter g ei according to Refs. 34,35 , respectively. Figure 1a-c show the temporal evolution of the electron density in the three active bands, i.e., the 6sp-and 5d-bands as well as the 4f-core state. The external excitation partially depopulates the 4f-state, but these core holes are quickly filled again by Auger processes. With an Auger time of 2.27 fs , the population of the 4f-state roughly follows the laser intensity.
The direct changes in the sp-and d-band densities due to laser excitation and Auger recombination are pretty small. Much more important is the large amount of energy each process brings into the sp-band that induces a strong nonequilibrium between its energy content, thus temperature, and its occupation. This imbalance triggers very effective impact ionisation of d-electrons that dominate the changes visible in Fig. 1a,b until a quasi-equilibrium is reached after roughly 700 fs . For longer times, electron-phonon coupling cools the electrons and the sp-density decreases accordingly. To quantify the effect of the relaxation between the sp-and d-bands, we included the result assuming instant equilibration between these bands. Although the data point from Ref. 12 is matched well, it cannot be used to distinguish between the models due to the large error bars in this inferred quantity.
The electron temperature, describing the energy content of the upper bands, is shown in Fig. 1d. Both direct laser excitation and Auger processes strongly drive it during the laser pulse. Then, it decays on a picosecond time scale due to electron-phonon coupling. The increasing lattice temperature exceeds melting temperature long before it equilibrates with the electron temperature, again in rough agreement with the experimental data of Ref. 12 . Note that the material can be in the solid phase after the melting temperature has been reached as thermal melting requires considerable time 4,41 .
The nature of the nonequilibrium created can be captured best when focusing on the chemical potentials of the sp-and d-bands as shown in Fig. 1e. The nonequilibrium band occupation is evident by the strong differences between the two chemical potentials during and shortly after the excitation. Similar to the densities, the chemical potentials equilibrate on a time scale of 700 fs . However, the energy loss to the phonons slightly drives data from Ref. [12] (a) n eq sp with g ei from Ref. [35] with g ei from Ref. [34] 0.0 www.nature.com/scientificreports/ the band occupation out of equilibrium, persistent even on the long time scale of electron-phonon equilibration. Such behaviour has also been observed for the very different case of ultrafast magnetisation of itinerant ferromagnets 42 . It proves that signatures of electronic nonequilibrium can be present on much longer time scales than suggested by the relaxation time 29,43 . Of course, this effect is more prominent for the case with stronger electron-phonon coupling.

Time-resolved DC electrical conductivity and scattering frequencies.
The optical properties, that can be directly observed in experiments, provide a much better test for the quality of our model than the spdensity. Ref. 12 provides data for the DC electrical conductivity. Within the Drude model, it can be expressed by the conduction band density n sp and a density-and temperature-dependent collision frequency ν tot as where e is the electron charge and m sp the effective electron mass in the sp-band.
For the complex band structure in gold, the scattering rate ν tot has two components: scattering of conducting sp-electrons with phonons and with d-band electrons. We apply the low-temperature form of electron-phonon scattering and take ν ei to increase linearly with the phonon (ion) temperature T i . For the electron-electron scattering rate ν ee , we follow Fourment et al. 44 and take it to be proportional to the densities of both electrons and holes in the d-band. Thus, we have where n full d describes a full d-band and the coefficients A and B have been determined by experimental data (see "Methods" section). The time-dependent temperature and density are provided by our relaxation model (1) which, through Eqs. (2) and (3), also determine the conductivity of the evolving system.
In Fig. 2, we compare the predictions for the evolution of the DC conductivity σ 0 and the scattering rates of excited gold with experimental data from Ref. 12 . The shaded area accounts for the uncertainty of the absorbed energy in the experiment around a mean value of 0.89 MJkg −1 represented by the line. Figure 2a shows an abrupt and fast decrease of the DC conductivity directly during the XUV laser pulse, followed by a more shallow decay on picosecond time scales. The strong conductivity drop is mainly caused by a strong increase of the scattering rate ν tot shown in Fig. 2b together with the contributing frequencies ν ee and ν ei . In cold gold, the rates are very small since Pauli blocking prevents scattering. The XUV pulse creates holes in the d-band by Auger recombination of the 4f-state and, in particular, further impact ionisation by energetic sp-electrons and is thus opening an efficient scattering channel. This newly opened way of scattering leads to a strong increase of ν ee already during the excitation process.
On a picosecond time scale, energy is transferred to the phonons (ions) leading to an almost linear rise of ν ei with time. The increase of ν tot is more shallow due to a slight decrease of ν ee in this phase. Together with a with g ei from Ref. [35] with g ei from Ref. [34] 10 www.nature.com/scientificreports/ decreasing sp-density, this behaviour leads to a weak decay of the DC conductivity, cf. Fig. 2a. We have employed two different models for the electron-phonon coupling: the often-used model of Lin et al. 34 and the model of Medvedev & Milov 35 which predicts much slower energy transfer. We find that the experimental data 12 agree over the complete time scale very well with the theoretical curve using the coupling strength of Ref. 35 for both the conductivity and the scattering rates. In contrast, the Lin model 34 neither reproduces the qualitative nor the quantitative behaviour of the conductivity (note the logarithmic scale in the figure). It decreases too strongly and saturates too early, resulting from a too fast and too strong increase of the electron-phonon scattering rate ν ei . In contrast to the sp-density, the error bars for the more directly obtained conductivity are small enough to clearly distinguish between these energy transfer models. This comparison demonstrates the strength of our model to provide an important benchmark for the controversially discussed electron-phonon coupling parameter 34,35,45,46 . The model presented here was developed for a material with a stable lattice and melting occurs within the time scale shown. In the experiment, a loss of lattice stability was found after 14.5 ps for the excitation conditions used here 12 . Although our model is not strictly applicable, the DC conductivity calculated compares well with the experimental data far beyond this time. This interesting fact suggests that the collision rates and conduction band density do not change considerably during the phase transition. The inset of Fig. 2a highlights the requirements if the second parameter of our model, τ relax , should be determined by experiments as well. For that end, we also have included calculations assuming instant relaxation, that is, both bands share a temperature and chemical potential. Unfortunately, present experimental data do not have the time resolution and data quality to distinguish these curves. However, dedicated experiments should be, in principle, able to resolve the sub-picosecond relaxation with the pulse structure of existing FELs 47,48 .
XUV versus optical excitation. We find strong qualitative differences in the relaxation behaviour after ultrafast excitation with an optical laser pulse as compared to the XUV excitation discussed above. In the XUV case, the entire laser energy is absorbed within the 4f-core shell. The Auger process creates highly energetic electrons and, accordingly, a high temperature in the conduction band. As a result, further ionisation of the d-band occurs when the chemical potentials equilibrate (cf. Fig. 1). In contrast, visible light is absorbed by d-and sp-electrons and a reduced system of rate equations can be set up for this case 49 . At 400 nm laser wavelength, most of the energy is absorbed by 5d-electrons, exciting them to free states of the conduction band at energies slightly above Fermi level. In this case, much more electrons are excited than in the XUV case if the same total energy is absorbed. Although we have the same relaxation term as in Eqs. (1a) and (1b), after optical excitation the relaxation drives electrons back into the d-band 49 , even before the coupling to the phonons further cools the conduction electrons. Figure 3a,b quantify the differences of optical and XUV excitation as described above for the electron density in the 6sp-band and the DC conductivity. For both excitations, the absorbed energy is the same and, thus, both systems relax to the same equilibrium. However, the intermediate phase is very different: we observe an underpopulation of the sp-band (compared to the quasi-equilibrium state determined by the electron temperature at this time) after XUV excitation, whereas the optical laser strongly overpopulates the conduction band initially, i.e., it creates a density above the quasi-equilibrium value. In the latter case, the very strong increase of density is followed by a slightly longer relaxation to its equilibrium value. These differences are also reflected in the DC conductivity which is mainly determined by the conduction band density and the free states in the d-band.
To further elucidate the origin of the different relaxation behaviour, we investigate the different excitation channels and relaxation terms in our rate equations in more detail. In Fig. 3c,d, we plot the integrated densitychange of the 6sp-band for different processes separately. For XUV photons, the initial photo-ionisation, as well as the Auger processes, give very small contributions. They are equal in magnitude and their integrated contribution remains constant after the pulse or Auger decay time, respectively. However, these processes created few highly excited electrons that efficiently trigger collisional excitation promoting many more d-electrons to the sp-band. Thus, this process is the main contributor in the relaxation phase and even throughout the XUV pulse. In contrast, photo-excitation of d-electrons is the only, but very efficient, process driving the sp-density increase for optical excitation (see Fig. 3d). Since the resulting density exceeds its equilibrium value by far, the subsequent relaxation is mainly driven by the recombination into d-band holes. Of course, the observable deviation from the equilibrium density is strongly influenced by the relaxation time τ relax which is here a free parameter. Recent data 50 showing strong recombination into the d-band already during the laser pulse thus indicate a shorter relaxation time than used here.

Conclusion
In summary, we have studied the dynamics of band occupation and DC conductivity in gold excited by ultrashort pulses of high-energy photons. For that goal, we have developed a system of rate equations that describes the evolution of electron density in the sp-and d-bands as well as a core state. Our model also tracks the energy content in the upper bands and couples this energy to the phonon systems. The results were employed to calculate the dynamics of the DC conductivity which allows for a more direct comparison to experiments. This comparison yields excellent agreement for both the sub-picosecond dynamics governed by the equilibration of electron densities within the sp-and d-bands and the evolution on several tens of picoseconds driven by the electron-phonon energy transfer. The latter allows us to clearly rule out an often-applied model for the electron-phonon coupling parameter. This fact demonstrates the strength of our model to bridge between theoretical estimates of important transport and relaxation parameters, which are usually done for static conditions, and highly dynamic experiments. This tool is particularly important for states with high energy density that prohibit static conditions in the laboratory. We also elucidate the different relaxation pathways for excitations with XUV and visible light: whilst secondary impact ionisation is the dominant process for XUV irradiation, direct www.nature.com/scientificreports/ photo-excitation prevails for optical lasers. These insights highlight the complex physical processes driving the transient electronic occupation and material response at a microscopic level that need to be accounted for when predicting and interpreting experiments with short-pulse excitations. Our model is also directly applicable to other noble metals, e.g., copper and silver. However, an application of the rate equations for transition metals will require adjustments, in particular, to the description of the excitation process. Combined with experiments having improved time resolution, the modelling of the occupational nonequilibrium can thus establish the time scales and pathways of the relaxation processes after ultrafast excitations.

Methods
The three-band model. The density response is described by the rate equations (1) with the assumption of a Fermi distribution in each band. In the XUV case, we consider heating by a pulse of photons with a wavelength of 13.6 nm ( 91.2 eV ) that inject 0.89 MJkg −1 of energy in solid gold. We consider a Gaussian pulse form with a full width at half maximum of 150 fs . Due to their high energy, each photon excites one electron from the 4f-state directly into the sp-band. The energy landscape for the electrons is given by the band-resolved density of state which have been calculated by density functional theory 40 . Prior to the excitation, the electron configuration of gold is [Xe]4f 14 5d 10 6s 1 corresponding to a fully occupied d-band and one electron in the conduction band. Although, we consider the density of state to be unchanged by the relatively small excitation, the equilibrium chemical potential will shift due to the increased temperature. Both the partial density of states and the temperature effects on the Fermi functions and chemical potential are shown in Fig. 4. The density of states was calculated for the ground state by the allelectron full-potential linearised augmented-plane wave code Elk using an 80 × 80 × 80 k-point grid, spin-orbit coupling and the local density approximation 40 . We consider the DOS to remain unchanged from the ground state which has been shown to be a good approximation for the range of electron temperatures used here 51 .
The set of coupled rate equations (1) requires a number of material constants. The rate of direct excitation of core electrons into the conduction band is given by the absorbed energy and the photon energy. The core holes in the 4f-states considered here are filled rapidly by d-electrons via Auger processes with decay times of 2.35 fs and 2.19 fs for the 4f 5/2 and 4f 7/2 states, respectively 38 . For practical reasons, we use the average of these times.
After the initial excitation and Auger recombination, we have an imbalance of electron occupation in the sp-and d-bands. The balance of recombination into free d-states and further impact ionisation into the sp-band will establish equilibrium densities that can be determined by (4) n eq a = dE f E, µ eq , T e D a (E), www.nature.com/scientificreports/ where a labels the band, D a is the partial density of states, f (E, µ eq , T e ) is the common Fermi distribution for both bands with the chemical potential µ eq being obtained by conservation of total number of electrons 52 . The relaxation of the band occupations towards this equilibrium density is described by the time scale τ relax and, without any other processes, we would observe an exponential decay of the imbalance in the d-and sp-band densities on this time scale. Here, we have chosen τ relax = 200 fs . This time scale is within existing bounds: it should be considerably longer than the fully kinetic phase that establishes Fermi-like distributions in each band and takes a few femtoseconds in the given energy range 29,36,37 and it must be shorter than the time resolution of previous measurements ( 540 fs ) that could be described by equilibrium densities 9,53 . The value we have taken here is thus near the upper limit to yield estimates for the minimum time resolution an experiment must have to investigate these effects. The energy balance between the electrons and phonons is described within the two-temperature model 39 . The ion heat capacity is taken to be constant C i = 2.327 MJm −3 K −1 (see Ref. 54 ). In contrast, the electron heat capacity C e is strongly temperature-dependent and it is calculated as the temperature derivative of the internal energy at each time step where the total density of states is expressed by the partial density of states of d-and sp-electrons, D sp and D d , respectively.
For the binding energies of the 4f-states used in the three-band-model (1d), we apply the average value of the 4f 5/2 and 4f 7/2 doublet with energies relative to the Fermi energy of 87.6 eV and 83.9 eV , respectively 55,56 .
The integration of the rate equations (1), that represent coupled nonlinear ordinary differential equations, has been performed by the Runge-Kutta scheme of order five. We also tested an implicit scheme with the Adams-Bashforth-Moulton multistep method of order five which yielded the same results for the integrated quantities.
Optical excitation. To describe gold excited by an optical laser pulse with 400 nm ( 3.1 eV ), we must only consider two optically active bands, namely the 5d-and 6sp-electrons as shown in Fig. 4. Moreover, the energy can be absorbed by both the 5d-and the 6sp-electrons. For such system, we have developed a dynamic two-band model 49 similar to Eqs. (1) that account for photo-absorption, excitation of d-electrons and deexcitation of spelectrons as well as the energy transfer to the phonons. The ratio of intra-and interband photo-absorption is set to be proportional to the number of electrons available, as well as to the number of free states for these electrons after excitation in order to account for Pauli blocking. As in the three-band model, the time scale for the equilibration of the band occupation is included as a free parameter. www.nature.com/scientificreports/ DC conductivity and scattering rates. The Drude model for the electrical conductivity (2) and the scattering rates (3) require a number of parameters that have been determined from experimental data. For the effective electron mass of the sp-band, we use the free-electron mass 57 . The constant B for the electron-phonon scattering is determined via its cold value B = ν cold ei /300 K , where ν cold ei = 0.084 fs −1 is taken from data of Refs. 57,58 . The parameter A for the electron-electron scattering (3) is found to be almost constant at elevated electron temperature and set to be A = V 2 · 0.36 fs −144 with the volume of the unit cell V = 1.69 × 10 −29 m 3 .

Data availability
The data supporting the findings of this study are available from the corresponding author upon reasonable request.