3.4 Microscopic Investigation of Structured Surfaces
4.1.2 Theoretical Description and Modeling
The energy deposition by the source term S(r,z,t,Te, ω) into the electronic subsys-tem, its precise evolution over time and the coupling to the lattice is obtained by solving the differential equations of the TTM, described in detail in Section 2.3.4.
With a model forTedepending on the incident laser pulse, the next step is to relate the experimentally determined reflectivity change toTe. Under normal conditions at the surface between vacuum and a material the reflectivity is described by Eq. (2.30), depending on the real and imaginary part of the refractive index Eq. (2.29) which are described by the dielectric function Eq. (2.17). 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 by a model of free-and bound electrons by inter-bfree-and transitions (between the d- free-and s/p-bfree-and) free-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 ω of the incident photons [38] as introduced in Eq. (2.17). This approach is here extended by the out of equilibrium description of Te 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:
(Te, ω) ={inter}(Te, ω) +{intra}(Te, ω) . (4.1) The inter-band {inter}(Te, ω) part describing excitations from bound to unbound states can be obtained by calculating the imaginary part of the inter-band con-tribution to the three-dimensional dielectric tensor given by the above described Eq. (2.19) and is calculated here by WIEN2k (Version 13.1) [38], the calculation is done by E.S. Zijlstra. Its derivation and precise calculation method by DFT is described for equilibrium conditions in detail by Ambrosch-Draxl et al. [38]. In this code, the method of all-electron full-potential linearized augmented plane waves is used to calculate the Kohn-Sham eigenstates, taking into account the screening ef-fect of inner electrons, the influence of relativistic electrons close to the core, as well as the effect of spin-orbit coupling for optical transitions [134]. The calcula-tions of gold were performed in the local density approximation where 17 valence orbitals are used and a lattice parameter of 0.408 nm. The basis of 734 plane waves to describe the valence electrons was determined by a maximal valuekmax by R∗kmax = 8.0, whereRis the radius of the used muffin tins. In total, the electronic band structure was calculated at 816 k-points. To account for the width of the discrete energy levels a lifetime broadening of Γ = 10 fs is used. For laser excited reflectivity, the code of WIEN2k is modified and extended to allow the calculation of an electronic temperature-dependent reflectivity value under electron-phonon non-equilibrium conditions.
The intra-band part{intra}(Te, ω), described in the Drude-model [41, 42], describing the transitions within the free electron gas of the metal, derived in Section 2.1.3 and
is here made dependent on the electronic temperature, {intra}(Te, ω) = 1− ωp;ij2
ω2+iνe(Te)ω . (4.2)
In the Drude-Model the response of the free electrons in the model on the external laser field with ω is additionally damped by the collision rate νe(Te) describing collisions of the free electrons with bound electron states closer to the core, according to [40, 135]. Here the plasma frequency as defined in WIEN2k [38] is used,
ω2p;ij = ~2e2 πm∗2
X
n
Z
k
d3k pi;n,n,k pj;n,n,kδ(En,k−EF) , (4.3) with the Fermi-energyEF, and the Dirac-delta functionδ describing the excitation process.
Room Temperature conditions
The WIEN2k code determines the DOS by ab-initio calculation and from that the inter-band transitions as defined in Eq. (2.19). The Drude model of the intra-band term given in Eq. (2.19), however requires a value as damping parameter νe(Te;RT) =νeph at nearly room temperature (RT) here defined as 316 K (0.002 Ry).
It describes the electron phonon collision rate at RT conditions which is defined by the reflectivity value given by literature at a photon energy of 1.66 eV. In this model the Drude-damping parameter therefore is set to a value ofνeph= 0.088 fs−1 where the inter-band part is given by Eq. (2.19), and the plasma frequency is given by Eq. (4.3).
EDOS
DOS (states/eV/atom)
Energy - E F
(eV) Au
1 2 3 4 5 6
0.2 0.4 0.6 0.8 1.0
¬
Johnson et al.
DFT at RT
Reflectivity R
Photon energy ¬ (eV)
Figure 4.2 DOS of the valence electrons of crystalline gold with the atomic shell configuration [Xe]4f145d106s1 calculated at RT with WIEN2k, DOS calculated by E. Zijlstra. The broad d-band edge of 0.9 eV width is highlighted in gray. The Fermi-Dirac distribution for differentTe is included, and shifted by ∆µ(Te), and the electronic density of states (EDOS) is shown in light blue. The obtained RT reflectivity is shown in the inset compared to the literature value from Johnson et al. [42].
A problem of DFT calculation is that it can not precisely describe the correct abso-lute energy positions of the Fermi level band positions. Therefore, the literature data for the dielectric function and thusR(ω) from Johnson et al. [42] are used to define these parameters at RT conditions. To obtain the correct absolute energy positions the calculated DOS at RT, shown in Figure 4.2 needs to be shifted by ∆E = 0.41 eV.
The reflection edge is defined at the DOS spanning over 0.9 eV 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 1.7 eV to 2.6 eV shaded also in gray in Figure 4.2. At the center of this gray bar a photon energy of ~ω0 ≈ 2.15 eV is needed to excite in an unoccupied state aboveEF and marks the transition from high reflection (R≈0.97 ) for excitations from the s/p-band to low reflection (R≈0.34 ) for photon excitation from the d-band.
Modeling the Laser Excited State
The effect of an increase in the electronic temperature on the DOS due to the broad-ened Fermi distribution and the shift of the chemical potential µ(Te) is shown in Figure 4.3.
EDOS at 10 kK
DOS(states/eV/atom)
Energy - E F
(eV) Au
(a) occupied states atTe= 10kK
EDOS at 45 kK
DOS(states/eV/atom)
Energy - E F
(eV) Au
(T e
)
(b) occupied states atTe= 45kK
Figure 4.3 Occupied states in DOS of Au at different electronic temperatures ofTe= 10 kK in (a) andTe= 45 kK in (b) are shown and the shift ofµ(Te) versus the at RT definedEF.
First the influence of these smeared occupation of states in laser excited gold on the inter-band part of the dielectric function is shown separately in Figure 4.4 (a). In a second step the model is extended by the temperature-dependent collision frequency νe(Te), plotted in Figure 4.4 (b).
In Figure 4.4 a) (DFT) a constant collision frequency ofνe(Te;RT) =νeph= 0.088 fs−1 is considered over the whole shown reflectivity map. For the case of the electronic temperature near the equilibrium conditions the color changes in the plot in Figure 4.4 (a) from red to green in a range of ≈ 0.9 eV. For increasing temperatures, the midpoint of the reflectivity edge however shifts to larger photon energies, also the chemical potential (thick gray line) shifts from ~ω0 = 2.15 eV up to ~ω0 ≈5 eV an effect described previously by Holst et al. and others [70]. It can be understood in the DOS picture, Figure 4.3 where µ(Te) is located so that the number of holes below and electrons above it are equal. Since the DOS is higher below µ(Te) a re-distribution of electrons and holes due to a rising Te shifts the chemical potential towards higher energies. Another visible feature in Figure 4.4 (a) is a broadening or smearing of the reflectivity edge mentioned previously by Ping at al. [69, 74] from a range of≈0.9 eV above electron temperatures to a broad edge ranging roughly from 1.2 eV to 6 eV at Te ≈ 70 kK. A possible explanation for this effect is a smearing introduced by the Fermi-Dirac distribution appearing only above electronic temper-atures where the d-band is starting to be depopulated, while the s/p-band is further populated. The onset temperature of this effect of 10 kK matches roughly the width of the d-band edge itself shown in the DOS in Figure 4.3 (marked there in gray) and leads to an onset of the smearing only above (Te≈10 kK) Te≈1 eV.
In Figure 4.4 b) (DFT with eh-collisions), in addition the effect of
electron-hole-1 2 3 4 5 6 0
10 20 30 40 50 60 70
(T e
)-1/4 k B
T e
(T e
)
DFT
Electronic temperatureTe
(kK)
Photon energy ¬ (eV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Reflectivity
R
a) (T
e )+1/4 k
B T
e
1 2 3 4 5 6
0 10 20 30 40 50 60 70
4.98 eV (248 nm)
DFT w ith eh-collisions b)
Electronic temperatureT e
(kK)
Photon energy ¬ (eV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Reflectivity
R 1.66 eV (745 nm)
Figure 4.4 a) Calculated reflectivity mapR(Te,ω) as a function of the electron temperatureTe
and photon energy~ω obtained by DFT calculations using a temperature-dependent modifi-cation of the WIEN2k code, simulation by E. Zijlstra. The chemical potential curveµ(Te) is included (black solid line) and the corresponding Fermi-broadening is also included (dotted black lines). In b) the DFT calculations are extended by the effect of eh-collisions. Photon energies at 1.66 eV and 4.98 eV are marked with dotted lines at which the calculated results are compared to experiment.
collisions on the reflectivity mapR(Te, ω) is included by introducing aTe dependent damping parameter [40, 66, 67]. The results are compared with each other and later to experimental data. The dependent damping parameter enters Eq. (2.25) as the collision frequencyνe(Te) =νeph+νeh(Te)+νee. In this equation the electron-phonon collisionsνeph, representing the collisions also present at equilibrium conditions, are extended by the dynamic part of the electron-hole collisionsνeh(Te) and in addition the effect of electron-electron collisions νee is included. The influence of electron-electron collisionsνeeon the Drude damping term can be neglected when describing the dielectric function [40]. The electron-hole-collisionνeh(Te) can be further divided in collisions with free holes in the s/p-band which play a minor role even at elevated electronic temperatures [40] and the effect of collisions with bound holes in the d-band which do change the damping term in the Drude-model whenTe 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 byNhd(Te) =Nesp(Te)−1. 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
νeh(Te) =AehNed(Te)Nhd(Te) (4.4) is then given by the parameter Aeh (assumed constant over Te and ~ω) times the scattering electrons (which are the effective number per atom of all occupied states in the band 5d10) times the holes in the d-band [40]. The parameterAeh was found to fit best to our experimental data forAeh= 0.45 fs−1.
The two different reflectivity maps R(Te,ω) 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 it is possible to separate the effect of the the ab-initio DFT calculations presented in Figure 4.4 a). The effect of the additional inclusion of a Te dependent increase of eh-collisions is shown in Fig. 4.4 b). When comparing Figure 4.4 a) and b) the effect of the added dynamic damping term νeh(Te) is visible especially in the IR, where the increase of Te leads to an earlier and stronger drop in reflectivity with rising Te compared to the case without a temperature dependent damping term. The effect on the higher energetic photons is only a slight increase of the reflectivity. The results in Fig. 4.4 b) belowTe ≈10 kK around the absorption edge qualitatively agree with experimental thermo-reflectance results from the literature [13–15]. A decrease in reflectivity below an excitation energy of ~ω0 = 2.35 eV 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 νeh(Te) describing free electrons colliding with bound d-band holes explains this broadening, rather than 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 is assumed [13–15]. In our ab-initio simulation (Figure 4.4 a)) this effect is visible, but only appearing above Te ≈ 10 kK and attribute this to the width of the d-band edge itself as described
above. (or the shift of µ combined with the drop due to the collision rate). With this detailed knowledge of the effect of Te on the reflectivity map the next task is to relate this parameter with a model to the experiment. The model is described in Section 2.3.3, the description of the experiment at two wavelengths at 248 nm and 745 nm follows.