Wavelength-dependent reflectivity changes on gold at elevated electronic temperatures

Wavelength-dependent reflectivity changes on gold at elevated electronic temperatures

A. Blumenstein Laser-Laboratorium Göttingen e.V., Hans-Adolf-Krebs-Weg 1, D-37077 Göttingen, Germany University of Kassel, Heinrich-Plett-Straße 40, D-34109 Kassel, Germany    E.S. Zijlstra University of Kassel, Heinrich-Plett-Straße 40, D-34109 Kassel, Germany    D.S. Ivanov University of Kassel, Heinrich-Plett-Straße 40, D-34109 Kassel, Germany Department of Physics and OPTIMAS Research Center, Technische Universität Kaiserslautern, Erwin-Schrödinger-Straße 46, D-67663 Kaiserslautern, Germany    S.T. Weber Department of Physics and OPTIMAS Research Center, Technische Universität Kaiserslautern, Erwin-Schrödinger-Straße 46, D-67663 Kaiserslautern, Germany    T. Zier University of Kassel, Heinrich-Plett-Straße 40, D-34109 Kassel, Germany    F. Kleinwort Laser-Laboratorium Göttingen e.V., Hans-Adolf-Krebs-Weg 1, D-37077 Göttingen, Germany    B. Rethfeld Department of Physics and OPTIMAS Research Center, Technische Universität Kaiserslautern, Erwin-Schrödinger-Straße 46, D-67663 Kaiserslautern, Germany    J. Ihlemann Laser-Laboratorium Göttingen e.V., Hans-Adolf-Krebs-Weg 1, D-37077 Göttingen, Germany    P. Simon Laser-Laboratorium Göttingen e.V., Hans-Adolf-Krebs-Weg 1, D-37077 Göttingen, Germany    M.E. Garcia University of Kassel, Heinrich-Plett-Straße 40, D-34109 Kassel, Germany
July 13, 2019

Upon the excitation by an ultrashort laser pulse the conditions in a material can drastically change, altering its optical properties and therefore the relative amount of absorbed energy, a quantity relevant for determining the damage threshold and for developing a detailed simulation of a structuring process. The subject of interest in this work is the d-band metal gold which has an absorption edge marking the transition of free valence electrons and an absorbing deep d-band with bound electrons. Reflectivity changes are observed in experiment over a broad spectral range at ablation conditions. To understand the involved processes the laser excitation is modeled by a combination of first principle calculations with a two-temperature model. The description is kept most general and applied to realistically simulate the transfer of the absorbed energy of a Gaussian laser pulse into the electronic system at every point in space at every instance of time. An electronic temperature-dependent reflectivity map is calculated, describing the out of equilibrium reflectivity during laser excitation for photon energies from , including inter- and intra-band transitions and a temperature-dependent damping factor. The main mechanisms are identified explaining the electronic temperature-dependent change in reflectivity: broadening of the edge of the occupied/unoccupied states around the chemical potential , also leading to a shift of and an increase of the collision rate of free s/p-band electrons with bound d-band holes.

I Introduction

The unique visible appearance of gold has fascinated not only scientists since its discovery. The origin of its typical colorful metallic look lies in the electronic properties resulting from the position of the absorbing d-band approximately 2.35  below the Fermi level , which leads to a high reflection of photons in the red to yellow spectral range and to a low reflection of photons in the blue to purple colored frequency interval. At photon energies too low to excite electrons from the d-band, the Drude-model can be applied Drude (1900). It describes the s/p-band electrons as freely oscillating in an electron gas resulting in a high reflectivity, typical for a metal and allows the use of gold as a reflective coating in the infrared (IR) energy range. Higher photon energies can excite bound electrons from the d-band leading to a strong increase in the absorption.
The extreme conditions during strong laser excitation however can influence the electronic system in a material and thus change its reflection/absorption behavior. Especially at the non-equilibrium conditions necessary for a precise surface structuring of a metal where the laser energy is introduced below the electron-phonon relaxation time in a picosecond regime Corkum et al. (1988); Mueller and Rethfeld (2013). These extreme conditions lead to electrons at elevated temperatures in a relatively cold lattice and can be described by a two-temperature model (TTM) Anisimov et al. (1974). Pump-probe experiments around ablation conditions confirm that the electronic temperature is the main parameter determining transient reflectivity changes Hohlfeld et al. (2000); Sun et al. (1994); Schoenlein et al. (1987); Fourment et al. (2014); Ping et al. (2006). They show that the reflectivity, even after the fast heating of the electronic system by a pump pulse is staying in a transient state, due to the delayed transfer of energy to the lattice and its related decrease in electronic temperature.
A complete picture of the wavelength dependent reflectivity changes around the absorption edge of gold and near the ablation threshold is to our knowledge not presented yet. What has been done are thermo-reflectance studies describing the effect very close to the absorption edge and only up to Hohlfeld et al. (2000); Sun et al. (1994); Schoenlein et al. (1987). Other studies describe the changes in , by describing in detail the effects valid for transitions within the s/p-band Petrov et al. (2013) up to Lin et al. (2008); Holst et al. (2014) and in agreement with experiments Fourment et al. (2014); Zhang et al. (2015) but only around the excitation energy . Also the conditions far above the ablation threshold of gold , Preuss et al. (1995); Furusawa et al. (1999) with conditions are well studied allowing the use of a plasma state model, applicable for describing the reflectivity for a wide class of materials and wavelength Milchberg and Freeman (1989); Fedosejevs et al. (1990); Price et al. (1995); Eidmann et al. (2000). However at the parameters usually used for metal surface structuring a plasma state is not reached and the model can not be applied.
The reflectivity map of gold calculate in this work covers typical surface structuring conditions. It allows a precise description of the transient change of reflected and absorbed energy during the excitation by a laser pulse for a variety of parameters. With the limitations of the calculation of a maximum electronic temperatures of and photon energies from (). Therefore the electronic system of gold is modeled by density functional theory (DFT) using the WIEN2k code Ambrosch-Draxl and Sofo (2006). A modification of the code allows an ab-initio calculation of the density of states (DOS) under elevated electronic temperature conditions. At an increased also the effect of a change in the occupation of states is taken into account as well as a change of the electron-hole (eh)-collision rate entering the Drude-term Fourment et al. (2014); Zhang et al. (2015); Petrov et al. (2013).
A problem one faces now is to precisely relate the calculated reflectivity map to experimental results since the transient state of in a bulk material is a parameter difficult to control experimentally and also to simulate in a model. A common way to overcome this problem is the use of thin gold foils Ping et al. (2010); Chen et al. (2013) or films Hohlfeld et al. (2000); Sun et al. (1994); Schoenlein et al. (1987) (10  to 100 ) in which the electronic temperature is homogeneously distributed in the probed area of the layer due to fast ballistic electrons. Another common approach is the use of a 1-D model simulating the electronic temperature only in depth assuming negligible difference in the incident laser fluence along the lateral direction Fourment et al. (2014); Fedosejevs et al. (1990).
For experiments with self-reflecting laser pulses on a bulk target and for determining the introduced energy for structuring processes Ivanov et al. (2015); Wu and Zhigilei (2014), a different approach is needed. Here, we provide a state of the art model being able to describe an arbitrary pulse with a symmetry around the -Axis, reflected on a gold bulk target surface (thick film ) with a pulse length shorter than the characteristic electron-phonon equilibration time Corkum et al. (1988). To describe the heat transport after absorption in detail, a TTM is used including the electron-phonon relaxation Anisimov et al. (1974). The symmetry around the -axis is used to reduce the simulation to a 2-D model of the cross section of the pulse shape. In addition ballistic electrons are considered leading to an effective increase of the laser energy deposition depth. In the conducted experiments, pulse energies from to for () and () are applied.
The combination of this macroscopic description of energy transfer during laser pulse self-reflection by a TTM combined with the microscopic calculation of the excitation in the crystal based on DFT, gives us the possibility to directly relate the parameter of electronic temperature to the reflectivity, applicable to a wide variety of geometries and parameter sets. Allowing also a description of the physical phenomena leading to the reflectivity changes under electron phonon non-equilibrium conditions.

Ii Physical picture of integrated laser pulse reflection

In the following a general theoretical description of the integrated reflectivity is given, resulting from the transient process of a strong ultra-short laser pulse reflected by a bulk material surface. For a realistic simulation of the phenomenon the precise evolution of the electronic temperature during the self-reflection and the microscopic understanding of the electron dynamics under elevated , at a certain photon energy in the bulk and its effect on the reflectivity are crucial. Experimentally the reflectivity is simply determined by the reflected laser pulse energy divided by the incident laser pulse energy . This gives the integral reflectivity of the pulse since every photon is reflected at a slightly different location and time and thus is related to a different electronic temperature . For a theoretical model can also be expressed as a function of the absorbed energy


can be obtained by integrating the source term of the laser pulse: over the radius , the depth , the time and around the angle , assuming that the incident laser pulse has a symmetry around the -Axis,


with the simulation height and diameter . The laser source describes the absorbed intensity distribution by the material at any point in space and time, as a function of the electronic temperature for a given at the surface and by the incident intensity distributions given by the functions: along the radial distribution, in depth and along the temporal shape,


The energy deposition by into the electronic subsystem, its precise evolution over time and the coupling to the lattice can now be obtained by solving the differential equations of a TTM. Described in detail in section V. With this precise description of the parameter depending on the incident laser pulse, the next step is to relate the reflectivity change to .
Under normal conditions however at the surface between vacuum and a material the reflectivity is


given by the refractive index of the material


the extinction coefficient, respectively. The complex dielectric function is given by . It describes the permittivity of a material for electro-magnetic waves, and is determined by the configuration of the valence electrons in the bulk. In gold excitations are described in a model of free- and bound electrons by inter-band transitions (between the d- and s/p-band) and intra-band transitions (within the s/p-band), respectively. The different excitations can be expressed separated by the dielectric function depending on the frequency of the incident photons Ambrosch-Draxl and Sofo (2006),


and is extended by the out of equilibrium description by relevant under the extreme conditions of laser excitation when the DOS and also its occupation of states can change having significant influence on the dielectric function. The inter-band part describing excitations from bound to unbound states can be obtained by calculating the imaginary part of the inter-band contribution to the dielectric tensor given by:


it is determined by the momentum transition-matrix elements describing the excitation probabilities from a bound state in the band to a free state in the band with a crystal momentum k. Also including the electronic temperature dependent Fermi-Dirac distribution in which the single particle energies in the bands are described by . Its derivation and precise calculation method by DFT is described for equilibrium conditions in detail by Ambrosch-Draxl et al. Ambrosch-Draxl and Sofo (2006).
The intra-band part , describing the transitions within the free electron gas of the metal, can be described in the Drude-model Drude (1900); Johnson and Christy (1972)


Using the WIEN2k code, a plasma frequency


can be extracted as described by Ambrosch-Draxl et al. Ambrosch-Draxl and Sofo (2006). In the Drude-Model the response of the free electrons in the model on the external laser field with is additionally damped by the collision rate describing collisions of the free electrons with bound electron states closer to the core.

Iii DFT Calculation of Broadband Reflectivity Change at Elevated Electronic Temperature

The dielectric function of crystalline gold is obtained by first principles DFT calculations using WIEN2k (Version 13.1) Ambrosch-Draxl and Sofo (2006). The code is modified and extended to allow the calculation of an electronic temperature-dependent reflectivity value under electron-phonon non-equilibrium conditions. In WIEN2k, the method of all-electron full-potential linearized augmented plane waves is used to calculate the Kohn-Sham eigenstates, taking into account the screening effect of inner electrons, the influence of relativistic electrons close to the core, as well as the effect of spin-orbit coupling for optical transitions Glantschnig and Ambrosch-Draxl (2010). The calculations of gold were performed in the local density approximation where we used valence orbitals and a lattice parameter of nm. The basis of plane waves to describe the valence electrons was determined by a maximal value by , where is the radius of the used muffin tins. In total the electronic band structure was calculated at -points. To account for the width of the discrete energy levels a lifetime broadening of is used.

Figure 1: DOS of the valence electrons of crystalline gold with the atomic shell configuration [Xe]4f5d6s calculated at RT with WIEN2k. The broad d-band edge of width is highlighted in gray. The Fermi-Dirac distribution for different is included, and shifted by . The obtained RT reflectivity is shown in the inset compared to the literature value from Johnson et al. Johnson and Christy (1972).

iii.1 Room Temperature conditions

The WIEN2k code determines the DOS by ab-initio calculation and from that the inter-band transitions (Eq. (7)), the influence of these transitions are normally described in a Drude-model as the core polarization term Fourment et al. (2014). The intra-band transitions however given in the model from Ambrosch-Draxl et al. also use a Drude-term which requires a damping parameter at nearly room temperature (RT) here defined as 316 (0.002 ). In addition DFT calculation can not precisely describe the correct absolute energy positions of the Fermi level. Therefore the literature data for the dielectric function and thus from Johnson et al. Johnson and Christy (1972) are used to define these two parameters at RT conditions. To obtain the correct absolute energy positions the calculated DOS at RT, shown in Fig. 1 needs to be shifted by . The reflection edge is defined at the DOS spanning over marked in gray from the onset of the d-band to the first peak in its DOS. In the inset showing the shifted reflectivity it spans from to shaded also in gray in Fig. 1. At the center of this gray bar an photon energy of is needed to excite in an unoccupied state above and marks the transition from high reflection () for excitations from the s/p-band to low reflection () for photon excitation from the d-band. The second parameter, the Drude-damping parameter . which describes the electron phonon collision rate at RT conditions is defined by the reflectivity value given by literature at an photon energy of . Defining in our model the Drude-damping parameter to a value of where the inter-band part is given by Eq. (7), and the plasma frequency is given by Eq. (9).

iii.2 Elevated Electronic Temperatures

The effect of an increase in the electronic temperature on the dielectric function introduced in Eqs. (6) and (8) is implemented in our DFT simulation. First only the influence of the inter-band part and the plasma frequency shown in Fig. 2 a) are discussed. In a second step our model is extended by a temperature-dependent collision frequency plotted in Fig. 2 b).

In Fig. 2 a) (DFT) when considering a constant collision frequency of over the whole reflectivity map shown in Fig. 2 (a) for the case of the electronic temperature near the equilibrium conditions the color plot changes from red to green in a range of . For increasing temperatures, the midpoint of the reflectivity edge however shifts to larger photon energies, also the chemical potential (thick gray line) shifts from up to an effect described previously by Holst et al. and others Holst et al. (2014). It can be understood in the DOS picture where is located so that the number of holes below and electrons above it are equal. Since the DOS is higher below a redistribution of electrons and holes due to a rising shifts the chemical potential towards higher energies. Another visible feature in Fig. 2 (a) is a broadening or smearing of the reflectivity edge mentioned previously by Ping at al. Ping et al. (2010, 2006) from a range of above electron temperatures to a broad edge ranging roughly from 1.2  to 6  at . A possible explanation for this effect is a smearing introduced by the Fermi-Dirac distribution appearing only above electronic temperatures where the d-band is starting to depopulated, while the s/p-band is further populated. The onset temperature of this effect of 10  matches roughly the width of the d-band edge itself shown in the DOS in Fig. 1 (marked there in gray) and leads to an onset of the smearing only above () .

Figure 2: a) Calculated reflectivity map in dependence of the electron temperature and photon energy obtained by DFT calculations using a temperature-dependent modification of the WIEN2k code. In b) the DFT calculations are extended by the effect of eh-collisions. Photon energies at 1.66  and 4.98  are marked with dotted lines at which the calculated results are compared to experiment.

In Fig. 2 b) (DFT with eh-collisions) in addition the effect of electron-hole-collisions on the reflectivity map is included by introducing a dependent damping parameter Petrov et al. (2013); Fourment et al. (2014); Zhang et al. (2015). The results are compared with each other and later to experimental data. The dependent damping parameter enters Eq. (8) as the collision frequency which includes the electron-phonon collisions , representing the collisions also present at equilibrium conditions, extended by the dynamic part of the electron-hole collisions plus the effect of electron-electron collisions . The influence by electron-electron collisions on the Drude damping term can be neglected when describing the dielectric function Fourment et al. (2014). The electron-hole-collision can be further divided in collisions with free holes in the s/p-band which play a minor role even at elevated electronic temperatures Fourment et al. (2014) and the effect of collisions with bound holes in the d-band which do change the damping term in the Drude-model when is rising. The reason is the formation of unoccupied states in the d-band and an equal increase of occupied states in the s/p-band due to the Fermi-Dirac distribution and can be described by . These effective numbers of holes and electrons per atom in the d-band and sp-band respectively are calculated by the DOS from the DFT calculation. The collision frequency


is then given by the parameter (assumed constant over and ) times the scattering electrons (which are the effective number per atom of all occupied states in the band 5d) times the holes in the d-bandFourment et al. (2014).

The two different reflectivity maps can now be used to describe the effects on the transition edge, marking the change from high to low reflectivity values, at elevated electronic temperatures where we can separate the effect of the the ab-initio DFT calculations presented in Fig. 2 a), from the effect of the additional inclusion of an dependent increase of eh-collisions shown in Fig. 2 b). When comparing Fig. 2 a) and b) the effect of the added dynamic damping term is visible especially in the IR, where the increase of leads to an earlier and stronger drop compared to the case without a temperature dependent damping term while the effect on the higher energetic photons is only a slight increase of the reflectivity. The results in Fig. 2 b) below around the absorption edge qualitatively agree with experimental thermo-reflectance results from the literature Hohlfeld et al. (2000); Schoenlein et al. (1987); Sun et al. (1994). Where a decrease in reflectivity below an excitation energy of and an increase above that excitation energy is measured, and related to a broadening of the edge. As an explanation for this broadening our calculation results suggest, that the dynamic damping term describing eh-collisions with the d-band explains this broadening, rather then the broadening of the excitation edge itself by a smearing of the edge of the occupied states in the s/p-band when excitation in a sharp d-band are assumed Hohlfeld et al. (2000); Schoenlein et al. (1987); Sun et al. (1994). In our ab-initio simulation (Fig. 2 a)) we can see this effect, but only appearing above and attribute this to the width of the d-band edge itself as described above. With this detailed knowledge of the effect of on the reflectivity map the next task is to relate this parameter with a model to the experiment.

The maps from with eh-collisions includes and also , , respectively are included in the supplementary material presented in matrix form.

Iv Experimentally Determined Reflectivity

The integral-reflectivity is obtained under common experimental conditions of a spatially and temporally shaped Gaussian laser pulse. The incident and reflected energy is measured to obtain the integral reflectivity by the use of Eq. (1). Two different photon energies are probed 1.66  (IR) and 4.98  (UV) with a pulse length of and respectively. The pulse energy is varied between and , to obtain typical structuring conditions covering the ablation threshold of gold in the UV , Preuss et al. (1995) and the IR , Furusawa et al. (1999) which both show also a pulse length dependency, described in the present work by the parameter . The IR experiments are conducted with pulses from a Ti:Sapphire regenerative amplifier system. For the UV experiments the Ti:Sapphire pulses were frequency tripled and amplified in a KrF excimer module to reach sufficient pulse energy Békési et al. (2002); Szatmari et al. (1991). The UV pulses pass a 3  aperture and are focused with a fused silica lens to a nearly Gaussian spot of FWHM. For the IR experiments a lens is used to create the same spot size for a larger Gaussian beam profile. An ultrashort single laser pulse is reflected under nearly normal incidence by a thick gold target and the incident and reflected energy of the pulse are measured. A x-y-z-stage is used to move the sample in the beam waist and to allow every pulse to hit a new undamaged area of the gold sample. The polycrystalline gold films are deposited by evaporation technique on a polished glass surface with a 50  chromium layer for enhanced lattice matching between gold and the substrate. The gold film has a thickness of 400  and an estimated grain size of  75 , determined by analyzing the cross section of TEM measurements. To ensure a high precision of the data, four different energy detectors in overlapping scales are used: A Polytec RjP-735, Ophir PE10BF-C, Ophir PE9-ES-C and Ophir PD10-C. In the setup one pulse energy sensor measures the Fresnel-reflection of the focusing lens while a second sensor measures the reflected pulse and is calibrated prior to the measurement. The pulse length is determined using a home made single-shot UV-FROG described in Nagy and Simon (2009), and in the IR by a ”Positive Light” single shot autocorrelator.

V TTM Simulation of Integral Reflectivity

The connection between the reflectivity map obtained in the DFT calculations and the experimentally determined integral value of self-reflectivity is realized by solving the time-dependent differential equations of a TTM simulation Anisimov et al. (1974). Therefore the temperature-dependent reflectivity is implemented in the source term as a tabulated value obtained from line profiles at the experimental probe photon energy from the map in Fig 2 a) and b). The volume of interaction is modeled by a disk with a thickness of and a diameter of shown in Fig. 3. The lateral size of the model was chosen so that the electronic temperature would not raise for more than by the end of the pulse.

Figure 3: Schematic picture of the nearly Gaussian laser focus spot with its experimental parameters. The inset shows the 2D-model geometry in cylindrical coordinates with the source term . The model boundary conditions are: At the -axis a periodic boundary (PB), in depth free boundary (FB) and at the surrounding circle a constant boundary (CB).

A Gaussian distribution of the pulse in the temporal and spatial domain is used, with the pulse length and the focus diameter at FWHM with using the values of the pulse from experiment described in sec. IV.


The effective energy attenuation function in depth for the pulses is given by:


With the characteristic penetration depth depending on the electronic temperature and the photon energy


Where the refractive index from Eq. 5 and the imaginary part of the dielectric function from Eq. 7 is inserted. The values are calculated by the described model including the effect of eh-collisions and inserted as tabulated data in the TTM. The values for the penetration depth at room temperature are for the probed wavelength and while for the highest calculated electronic temperature of the values are and varying in between not more than .
The Eq. 12 includes the sample thickness which is needed to account for possible losses of a thin sample relevant for low pulse energies with significant range of ballistic electrons . It describes the non-thermalized electrons quickly proceeding deeper into the material and their thermalization process by means of electron-electron-collisions which occurs at a greater distance than the optical penetration depth . The ballistic transport therefore can effectively increase the laser energy deposition by one order of magnitude in depth described by . Neglecting the kinetics of this process, we include the ballistic range into the source term, Eq. (3), as it was suggested by Hohlfeld et al. Hohlfeld et al. (2000).
The temporal shape of the pulse is given by:


with . The laser pulse is defined by its duration . At a single pulse shifted to is used to inscribe a Gaussian temporal profile with a length of 1.6  and 0.6  respectively.

The inclusion of the ballistic transport can alter the electron temperature distribution in the target upon the laser pulse absorption. We estimate its additional temperature dependence according to the transport relaxation time via collision rates, Landau (1957); Inogamov et al. (2009). The resulting relaxation time will thus be a function of the electron, , and phonon, , temperatures respectively with the corresponding contributions as and Anisimov and Rethfeld (1997); Wang et al. (1994). The influence of can be neglected since the electron hole collisions take place only at temperatures when holes are created in the d-band at which the penetration depth is already below a few nm. With the value of () for the Fermi-velocity of free electrons, in our simulations therefore the ballistic range in Eq. (12) is dynamically changing during the pulse and ranges from 300  at nearly zero intensity down to 5  at the peak intensity, where the temperature of electrons reaches its maximum level. During our simulations, however, we noticed that the inclusion of the ballistic transport does not alter the integral reflectivity value for more than in the range of incident energies greater than , which is in agreement with our previous publications Hohlfeld et al. (2000); Ivanov et al. (2015) and theoretical predictions by Petrov et al. Petrov et al. (2015). At lower incident energies, the ballistic range is comparable with the thickness of the modeled target and the first multiplier in Eq. (12) is accounting for its finite size. This parametric description of the source is now implemented in the TTM which in cylindrical coordinates can be written as:


where indexes and are standing for the electrons and lattice correspondingly. Since working with a short laser pulse, the thermal transport due to phonon conductivity is negligible when compared to electron conductivity and thus the corresponding parts in Eq. (15) can be omitted. and are the strength of the electron-phonon coupling with the value for the lattice taken as given by experimental data Lide (1992). By utilizing DFT calculated density of states with the effect of the d-band included, Lin et al. Lin et al. (2008) determined the electron temperature dependence for the electron-phonon coupling and the electron heat capacity functions and . These quantities therefore were considered in the present work in the form of tabulated data Lin et al. (2008). The complex behavior of the electron heat conductivity was approximated as it is suggested by Anisimov and Rethfeld Anisimov and Rethfeld (1997); Rethfeld et al. (2017), as a function of the electron and lattice temperatures and respectively:


with the parameters , , and for gold. This dependence shows linear behavior with at low excitation level, a significant decay at the excitation level comparable with , and a steep increase like plasma conductivity at a higher excitation level.
Finally, during the simulations the laser pulse, centered at , is irradiating the target At the front and rear sides of the target in the case of free boundary conditions when the energy conservation law was applied for controlling the accuracy of the calculations. A spatial resolution of in and 1  in direction is used. To test the accuracy of the model the energy conservation law was utilized for the case of fixed boundary conditions at the front, rear and lateral sides.

Vi Comparison of Simulation with Experiment and Discussion

The simulated integral values of reflectivity described by Eq. (1) are directly compared to the experimental results of the self-reflectivity by using the TTM and implementing both of the reflectivity maps shown in Fig. 2 a) and b). The results are plotted In Fig. 4 versus the incident pulse energies resulting from the spatial and temporal integration showing atop the corresponding incident peak fluence. A change in reflectivity is observed in both experiment and simulation above in the IR for the 0.6  laser pulses and above for the 1.6  pulses in the UV when looking at the simulation where DFT with eh-collisions are implemented, shown in (red diamonds) and (magenta circles), respectively. The difference of the onset of the reflectivity change in the IR and UV can be explained by the initial reflectivity difference at equilibrium conditions and therefore the difference of the absorbed amount of energy and thus the reached electronic temperatures. Even though the rise of R in UV appears at much higher than the drop in R in the probed IR range. When looking at the simulation results where only the ab-inito results of the modified WIEN2k code are used shown in (orange diamonds) and (blue circles), for IR and UV respectively, it appears that the shift of the chemical potential and the broadening of the Fermi-Dirac distribution alone is not able to describe precisely the change in reflectivity observed in experiment. To describe the strong decrease of reflectivity in Fig. 4 in the IR and the only small increase of R in the UV a temperature-dependent eh-collision obtained from the calculated DOS for the intra-band excitations needs to be included, resulting in an remarkable agreement of experiment and simulation at both probed wavelength even though one has to mention that therfore a parameter to the ab-initio calculations needs to be included.

The electronic temperature thus seems to be the key parameter describing the optical response around a fluence relevant for metal surface structuring. In the IR the rise of explains an increases of the rate of free electrons colliding with bound states of the d-band, where the depopulation of states by a broadened Fermi-Dirac distribution increases the probability of free electrons to collide with. Also the effect of the broadening of the excitation edge of the d-band itself plays a role as shown in the DFT results in In Fig. 2 a). In the UV an increase of reflectivity is observed in experiment and the DFT simulations described in section III.2 and suggest that in this case mainly the extent of the broadening of the Fermi-Dirac distribution explains the change in reflectivity. The explanation is that a de-population of the d-band which creates more occupied states in the s/p-band and leads to an effective shift of the now broad chemical potential to higher energies and thus also increases the relative depth of the d-band. The UV photons thus have an increased probability to excite from a bound state since excitations from bound states to unoccupied states even at an excitation energy of is possible.

Figure 4: Single-shot self-reflectivity of gold measured at different incident pulse energies for laser pulses with (IR) and (diamonds in red and orange) in a), (UV) and (circles in blue and pink) in b) respectively, focused on a spot with compared to simulations obtained by using temperature-dependent DFT calculations, combined with the effect of eh-collisions. The literature values of Johnson and Christy for low pulse energies are (black bar) and (gray bar) Johnson and Christy (1972).

An increase of the electronic temperature up to will already change the reflectivity around the absorption edge and is referred to as thermo-reflectance Hohlfeld et al. (2000); Sun et al. (1994); Schoenlein et al. (1987). In literature the effect is normally described by a simplified picture of a smearing of the excitation from the Fermi level (chemical potential ) to a sharp d-band Hohlfeld et al. (2000). At these elevated no material changes after equilibrating with the lattice will appear. However, when the laser induced energy is sufficient to ablate material after its transfer to the lattice electronic temperatures up to can be reached during transient self-reflectance. This state is often referred to as warm dense matter Ping et al. (2010); Chen et al. (2013). Under these conditions the density of states (DOS) itself changes, the occupation around the chemical potential smears out spanning a few on the photon energy axis and a shift of is observed Lin et al. (2008); Holst et al. (2014). At these conditions a dynamic decrease in reflectivity during the pulses interaction with the surface can even produce a self strengthening effect altering the onset of the ablation threshold of a pulse and the introduced amount of energy. Parameters crucial for a precise simulation of the nano-structuring of metal surfaces Ivanov et al. (2015); Wu and Zhigilei (2014).

Vii Conclusion

The reflectivity change under electron-phonon non-equilibrium conditions is measured, simulated and described for a wide range of photon energies at pulse energies relevant for structuring purposes. A map describing is introduced and visualises the effect of a boradeining of the Fermi-Dirac distribution on the typical d-band absorption edge of gold as well as the effect of introducing an additional dependet damping factor on the IR reflectivity of gold. At two probe wavelength in the IR and UV a good agreement between experiment and simulation was shown. Our model is also in agreement with experimental thermo-reflectance data around the absorption edge described by Hohlfeld et al. and others Hohlfeld et al. (2000); Sun et al. (1994); Schoenlein et al. (1987), and gives a more detailed picture of the involved processes than described before. Agreeing also with experiments at warm dense matter conditions, a decrease in reflectivity in the IR described by Fourment et al. and others Fourment et al. (2014); Zhang et al. (2015). In the UV the increase in reflectivity described by Fedosejevs et al. Fedosejevs et al. (1990) also agrees qualitatively with our model. The effect is distinguished from effects related to the formation of a plasma mirror. The relevant phenomena assumed here appearing at elevated electronic temperatures are a smearing of the excitation into the d-band combined with a eh-collision rate increase of free electrons with bound d-band holes being responsible for the dynamic change in reflectivity around the ablation regime, especially in the IR. The shown approach represents a powerful tool, allowing the description of the most general case of laser self-reflectivity and its precise absorbed energy at a certain time and location for elevated electronic temperatures.

The present work was supported by the Deutsche Forschungsgemeinschaft (DFG) grants IH 17/18-1, IV 122/1-1, IV 122/1-2, RE 1141/14, RE 1141/15 and 600 GA 465/15-1, as well as, the Carl-Zeiss Foundation. We thank V. Roddatis from University of Göttingen for TEM measurements, determining the sample thickness. The calculations for this work were performed on Lichtenberg Super Computer Facility within the project 242.


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