Ultrafast Heating-Induced Suppression of d-Band Dominance in the Electronic Excitation Spectrum of Cuprum

The combination of isochoric heating of solids by free-electron lasers (FELs) and in situ diagnostics by X-ray Thomson scattering (XRTS) allows for measurements of material properties at warm dense matter (WDM) conditions relevant for astrophysics, inertial confinement fusion, and materials science. In the case of metals, the FEL beam pumps energy directly into electrons with the lattice structure of ions being nearly unaffected. This leads to a unique transient state that gives rise to a set of interesting physical effects, which can serve as a reliable testing platform for WDM theories. In this work, we present extensive linear-response time-dependent density functional theory (TDDFT) results for the electronic dynamic structure factor of isochorically heated copper with a face-centered cubic lattice. At ambient conditions, the plasmon is heavily damped due to the presence of d-band excitations, and its position is independent of the wavenumber. In contrast, the plasmon feature starts to dominate the excitation spectrum and has a Bohm–Gross-type plasmon dispersion for temperatures T ≥ 4 eV, where the quasi-free electrons in the interstitial region are in the WDM regime. In addition, we analyze the thermal changes in the d-band excitations and outline the possibility to use future XRTS measurements of isochorically heated copper as a controlled testbed for WDM theories.


Introduction
The study of matter under extreme densities and temperatures has emerged as a highly active research field due to the availability of modern laser facilities equipped with various X-ray diagnostic techniques.High-power laser facilities are routinely being used to explore the physics and chemistry at conditions relevant to planetary astrophysics, [1][2][3] and inertial confinement fusion, [4][5][6] and to explore new exotic materials. 7][10][11][12][13] In addition, ultrashort X-ray free electron laser (XFEL) capabilities e.g. at the European XFEL 14 and LCLS 15 have opened the way to study phenomena on femtosecond timescales. 16By heating the electrons without directly affecting the ions, an XFEL with a sub-100 fs duration provides a unique opportunity to generate and study a transient state with hot electrons within the unperturbed crystal structure of the ions. 17,18In these experiments, the X-ray Thomson scattering (XRTS) technique 19 can then be used to probe the electronic structure of a given system by measuring its electronic dynamic structure factor (DSF) S(q, ω), where q and ω are the change in momentum and frequency of the scattered photon.
In this way, it was shown that the laser-induced heating of electrons leads to the lattice instability and melting (disordering) of silicon due to the weakening of the inter-ionic bonds. 20,21In contrast to semiconductors, metals can remain stable under laser heating of the electrons, and, strikingly, can even manifest a more rigid lattice structure.For example, Descamps et al. 17 have recently reported the observation of a stable gold crystal lattice where the electrons have been heated by the FEL to a few electronvolts.In this experiment, a signature of phonon hardening has been observed, whereby the bonds stiffen between atoms.This effect was earlier predicted by ab initio Kohn-Sham density functional theory (KS-DFT) calculations 22 of the hot electrons within the cold ionic lattice using the local density approximation (LDA) for the exchange-correlation (XC) functional.A second example for the successful utilization of KS-DFT for predicting the properties of solids with laser-excited electrons is the calculation of the XRTS spectrum of an isochorically heated aluminium foil by Mo et al. 23 based on linear-response time-dependent DFT (LR-TDDFT) using an adiabatic LDA (ALDA) XC kernel.The same combination of LR-TDDFT with ALDA was shown to accurately describe the XRTS spectrum of aluminum at ambient conditions, where the plasmon was measured with ultrahigh resolution at the European XFEL. 24Therefore, LR-TDDFT can be expected to yield accurate results for the electronic dynamic structure factor of isochorically heated metals across temperature regimes.
Very recently, Moldabekov et al. 25 have used this approach to study the effect of electronic heating on the order of a few electronvolts on the expected XRTS spectrum; 25 this has revealed an interesting red shift of the plasmon energy by 0.1 eV for aluminium and by 1 eV for silicon as a consequence of thermal excitations.In the case of aluminium, the effect is small and only manifests at small wavenumbers q ≲ 0.1 −1 , making it very challenging to measure.For silicon, on the other hand, the plasmon shift of 1 eV at temperatures T ≃ 2eV is well within experimental measurement capabilities. 24,26However, the possible instability of the lattice due to the weakening of the inter-ionic bonds 20,21 can be a serious obstacle in practice.Therefore, it is important to ask if such a heating-induced red shift prominently manifests itself in other metals that are stable under FEL radiation.Going back to aluminium, an additional thermally induced feature is the formation of a double plasmon peak as the region of Landau damping is shifted to lower wavenumbers upon increasing the electronic temperature. 258][29] These results show that thermal excitations in X-ray-driven solids can generate a variety of new features in the XRTS spectrum at a finite momentum transfer.
In the present work, we carry out extensive new LR-TDDFT calculations to explore the XRTS spectrum of isochorically heated copper.In contrast to simple metals, the effect of d -states dominates over plasmon-type excitations in the DSF of electrons in transition metals. 30,31In gold and copper, excitations originating in the d -band lead to the formation of a prominent double peak structure at ω > ω p and a substantial broadening of the plasmon feature at ω = ω p .Interestingly, the presence of the d -state excitations leads to a plasmon dispersion that is nearly independent of the wavenumber for both materials. 30,31Here, we investigate in detail the interplay of these effects with thermal excitations on the DSF of copper at different temperatures, wavenumbers, and crystallographic directions.Indeed, thermal effects on the DSF are profound: we find an emerging collective plasmon excitation that becomes dominant over the d-band feature for T ≳ 4 eV and which starts to follow the familiar Bohm-Gross relation in this regime.In addition, we find a pronounced blue-shift of the plasmon with increasing T , which is in stark contrast to other isochorically heated metals such as Al. 25 Finally, we discuss the possibility to use XRTS experiments with isochorically heated copper as a rigorous testbed for the theoretical modeling of warm dense matter (WDM) [32][33][34] -an extreme state that occurs in astrophysical objects, 3,[35][36][37] and which plays an important role e.g. for inertial confinement fusion 4,6,38 applications.
The paper is organized as follows: In Sec. 2, we give an overview of the LR-TDDFT approach and provide computational details of our simulations.The results of the calculations are presented and discussed in Sec. 3. The paper is concluded by a summary of the main findings and an outlook over future works in Sec. 4.
2 LR-TDDFT approach to the dynamical structure factor

Theoretical Framework
The intensity that is measured in an XRTS experiment is given by a convolution of the combined source-and-instrument function R(ω s ) 12 and the electronic dynamic structure factor S(q, ω) = S(q, −∆ω) (with ∆ω being the energy loss of the scattered photon), where the latter accounts both for the finite width of the probing X-ray source, and for all effects of the detector. 39The momentum transfer q is determined from the scattering angle.
The state-of-the-art is given by the European XFEL in Germany, where XRTS measurements with the capability of resolving electronic features with a resolution of up to δω ∼ 0.1 eV have been recently demonstrated. 24 study the effect of thermal electronic excitations on the XRTS spectrum of X-raydriven copper, we use the LR-TDDFT method with an adiabatic XC kernel.Indeed, LR-TDDFT constitutes the most common method to study the DSF of solids, and there is a vast body of dedicated literature, see e.g.Ref. 40and references therein; here, we restrict ourselves to a concise overview of the main ideas.
As a first step, we consider the well-known fluctuation-dissipation theorem that connects the macroscopic dielectric function ϵ M (q, ω) with S(q, ω), 41,42 S(q, ω) where n denotes the electronic number density, and e the elementary charge.4][45] It is computed by taking the diagonal part of the inverse microscopic dielectric matrix ϵ −1 M (q, ω) = [ϵ −1 (k, ω)] GG , where q = G+k (with k being in the first Brillouin zone) and G is a reciprocal lattice vector. 46,47The latter is defined by the microscopic density response function , 40 The LR-TDDFT method allows one to compute χ G,G ′ (k, ω) in different approximations.
The lowest rank corresponds to the so-called independent particle approximation (IPA).In the IPA, the Kohn-Sham (KS) orbitals and eigenenergies are used to calculate the density response function χ 0 G,G ′ (k, ω) according to the ideal electron gas model. 48Since the KS eigenenergies from the self-consistent ground state (equilibrium state) calculations are employed, χ 0 G,G ′ (k, ω) already has information about excitations between different orbitals.However, being computed using a formula for the ideal Fermi gas model, χ 0 G,G ′ (k, ω) omits various correlation effects, such as screening due to the Hartree mean field, microscopic density inhomogeneities due to the field of the ions etc.The inclusion of correlation effects leads to a Dyson's type equation for the density response function χ G,G ′ (k, ω): 47,49 where is the XC kernel capturing electronic correlations; it is defined as the functional derivative of the XC potential in KS-DFT. 50e LR-TDDFT method provides the DSF of the electrons in the thermal equilibrium.
An alternative approach that can perform the simulation of electronic dynamics with a distribution different from the Fermi-Dirac distribution is real-time TDDFT (RT-TDDFT), where electronic wave functions are propagated according to time-dependent KS equations.This method was used by Silaeva et al. 51 to study ultrafast electron dynamics thermalization in metals driven by a 7-fs laser pulse.Silaeva et al. 51 showed that valence electrons reach a thermalized state within the time of the laser pulse.The RT-TDDFT method can also be used to compute the DSF.For example, Baczewski et al. 52 used RT-TDDFT to compute the DSF of warm dense beryllium in thermal equilibrium.We note that if the same XC functional were used in both, RT-TDDFT and LR-TDDFT are formally equivalent for linear response properties in thermal equilibrium . 40[55] The relevant thermal signatures explored in this work are characterized by a difference of δω ≳ 1 eV from the ground state features.If needed, a further fine-tuning can be achieved either by employing more advanced static XC kernels beyond ALDA 56,57 or by using an explicitly dynamic approximation (e.g., see Ref. 49 and references therein) in future works.
9][60] In principle, the XRTS spectrum and the EELS are directly related since From Eq. ( 2), one can see that S(q, ω) ∼ q 2 EELS(q, ω).This means that EELS is advantageous for measurements at small wavenumbers, whereas XRTS might be more suitable at large wavenumbers.However, EELS measurements are problematic for experiments with matter under extreme conditions due to its requirements for thin targets as well as long measurement times, 60 which are not realistic for the transient states that are of interest in the current work.
In consistency with measurements at FEL facilities, we consider the electronic response on subpicosecond time scales and treat the ions as being frozen in their crystal lattice positions surrounded by heated electrons.2][63][64][65][66] Furthermore, this approximation is corroborated by the predictions of increased melting temperature with electron heating in copper and other dband metals, 64,67 and by the recent observation of phonon hardening in gold. 17

Calculation parameters
We used the GPAW code, [68][69][70][71][72][73] which is a real-space implementation of the projector augmentedwave (PAW) approach. 74We used the ground-state LDA XC-functional by Perdew and   Wang. 75The simulations have been carried out for a face-centered cubic (fcc) lattice with the lattice parameter 3.61 Å set according to the experimental value. 76For the calculation of the KS states, we used the energy cutoff 1000 eV, the PAW dataset of copper provided by GPAW (with 1s − 3p orbitals treated as frozen core electrons), and the primitive cell combined with the k-point grid 40 × 40 × 40.We note that in the employed LR-TDDFT formalism, the momentum transfer must be the difference between two k-points times 2π/a.
To study the possible impact of inhomogeneity with respect to crystallographic directions on the DSF, calculations were performed along the [100], [111], and [011] directions.We considered electronic temperatures in the range 0.025 eV ≤ T ≤ 12 eV and the number of KS bands was set to N b = 100.The smearing of the occupation numbers was computed according to the Fermi-Dirac distribution.On the stage of the calculation of the density response matrix, the local field effect cutoff was set to 150 eV.In all calculations, we used η = 0.1 eV for the Lorentzian smearing parameter in χ G,G ′ (k, ω). 71For the density of state (DOS) calculations of copper, we used the same parameters as for the DSF.The DOS was plotted by setting the Gaussian width parameter to 0.2 eV.

Simulations Results and Discussion
3.1 DSF S(q, ω) in the X-ray driven copper We start our investigation by considering the electronic ground state (here represented by the results for T = 0.025 eV) in the limit of small wave numbers.In the left panel of Fig. 1, we show the EELS spectrum at q = 0.0754 Å -1 for different temperatures in the range from 0.025 eV up to 8 eV (with the temperature increasing in the subplots from bottom to top).
We focus on four main features of the EELS spectrum (which equivalently appear in the This is illustrated in more detail in Fig. 2, where we show the thermally induced density change in the 3D simulation box for both temperatures.The blue surface depicts δn(r) = 0; the red surface indicates δn(r) > 0 (for the illustration, we choose δn(r) = max[δn(r)]/70); the green surface indicates δn(r) < 0 (here we show δn(r) = min[δn(r)]/15).We can thus clearly see that the electronic density is reduced in the vicinity of the ions and accumulates in the interstitial region, which is more pronounced for the higher temperature, as it is expected.A more detailed, quantitative analysis of the increased free electronic density and the resulting plasmon blue-shift is presented in Sec.3.2 below.
Let us next investigate the DSF, which is the key property in XRTS experiments; it is shown in Fig. 3 along the [100] direction for 0.0754 Å -1 ≤ q ≤ 1.4794 Å -1 .Since feature A is strongly damped for T ≥ 2 eV, we will ignore it in the following discussion and instead focus on the thermally induced changes of features B, C, and D. At all considered temperatures, the position of features C and D is nearly independent of the wavenumber.In stark contrast, the plasmon feature B exhibits a considerably richer behavior.At T = 0.025 eV and at T = 2 eV, its position does not follow the Bohm-Gross type dispersion of the free electron gas. 42Overall, it is difficult to quantify its q-dependence for these two temperatures due to the comparably weak spectral weight and the possible overlap with other features due to local field effects created by the lattice structure. 30In contrast, we observe a pronounced increase with q for T ≥ 4eV.As a result, the plasmon eventually overtakes feature C, leading to the disappearance of the latter from the DSF at q ≳ 0.9 Å -1 in the cases of T = 4 eV and T = 6 eV, and at q ≳ 0.6 Å -1 for T = 8 eV.
An additional interesting research topic is due to the lattice structure, which is known to lead to an anisotropy of the DSF with respect to the crystallographic direction at certain wavenumbers. 30,77To quantify this anisotropy effect on X-ray driven copper, we show the DSF in the [011] direction at different temperatures and wavenumbers in Fig. 4. We find that the DSF in the [011] direction closely resembles the DSF in the [100] direction for q ≲ 0.9 Å -1 , and differences in the shape and peak positions emerge for q ≳ 1.2 Å -1 .For completeness, we note that the DSF in [111] direction is equivalent to the [100] direction (see Appendix).In summary, we conclude that the electronic DSF of copper is nearly isotropic at T ≥ 2 eV for q ≲ 0.9 Å -1 .
In addition to the discussed dominant features, we observe an increase in the DSF at low energies and a new peaked feature emerges in the DSF at ω < 10 eV in both [100] and [011] directions at T ≥ 4 eV and q ≳ 0.6 Å -1 .The increase in the temperature modifies the DSF due to the factor f (ω) = (1 − e − ̵ hω/k B T ) −1 in Eq. ( 2).This effect is particularly pronounced for the low-energy part of the DSF, and enhances subtle features in this regime.This is illustrated in Fig. 5 for T = 4 eV and T = 6 eV at wavenumbers q = 0.9044 Å -1 , q = 1.2058Å -1 , and q = 1.5073Å -1 .From Fig. 5 we observe that the factor f (ω) in Eq. ( 2) for the DSF results in the substantial increase of the DSF values at low energies leading to an amplification of the thermally induced features at ω < 10 eV.Physically, these features might originate from transitions between accumulated states located closely above the Fermi level, cf.Fig. 1.Indeed, one can observe that at T ≥ 4 eV states above the Fermi level become partially occupied allowing the emergence of new excitation features.

Plasmon dispersion and conditions in interstitial regions
Analyzing the DSF, we have found that the collective plasmon oscillations in X-ray driven copper overcome the dominance of the d-band excitations, eventually overtaking them with respect to the spectral weight.In addition, the plasmon position starts to exhibit a substantial dispersion with respect to the wavenumber for sufficiently high temperatures, which is in contrast to the ground state plasmon.To examine the character of the plasmon dispersion, we show the dependence of the plasmon energy (frequency) on the wavenumber, ω(q), in Fig. 6 for different temperatures.We consider q < 1 Å -1 , where the plasmon peak can be clearly identified at all considered temperatures, and independent of the crystallographic direction.The uncertainty in the plasmon position is evaluated by looking at the onset of a broadened peak.At T = 2 eV, we find that ω(q) is qualitatively similar to the results for T = 0.025 eV.For T ≥ 4 eV, the plasmon dispersion starts to follow the familiar quadratic dependence on q that is well-known from the free electron gas model. 78To further quantify this trend, we fit the LR-TDDFT results using the Bohm-Gross type dispersion relation, 79,80 where α and ω p are the free parameters.The results are shown as the solid ([011] direction) and dashed ([100] direction) lines in Fig. 6, which are nearly identical; the small differences are likely due to uncertainties introduced by the broadened peaks of the DSF.
In subsection 3.1, we have indicated that the increase in the plasmon energy with the temperature is a consequence of the excess electronic density in the interstitial regions between the ionic lattice, cf.Fig. 2. Here, we propose to utilize such forward scattering data for the DSF as a diagnostic for the free electronic density and for the effective charge state.Specifically, we define the effective density parameter rs by inverting the usual relation between the density and the plasmon frequency of a free electron gas, We note that the parameter rs also characterizes the coupling strength or, equivalently, the degree of non-ideality of the free electrons. 33,81 additional, related effective parameter of interest is given by the corresponding de- , where E F (ñ) = (3π 2 ñ) 1/3 denotes the Fermi energy of a free electron gas of density ñ. 42 Finally, we consider the effective ionic charge Z, which is a key ingredient to equation-of-state tables. 9,10The effective charge is computed using Z = (r s (T = 0)/r s (T )) 3 , which follows from the condition Zn i = ñ, with n i = const being the number density of ions, and setting Z = 1 for copper at T = 0 since we have one valence electron in the 4s state.
In Table 1, we provide an overview of these parameters for all selected temperatures.For T = 0.025 eV, we find an effective density parameter of rs = 2.64 ± 0.11, which agrees with the value r s = 2.668 that corresponds to the conduction electrons density in the fcc copper at room temperature.
Upon increasing the temperature, we find a monotonic increase in the density of effectively free electrons, leading to a corresponding decrease of rs and an increase in Z.The observed monotonic increase of Θ is less trivial.On the one hand, we have Θ ∼ r2 s , 81 which, by itself, would indicate a decrease of Θ with T .However, this effect is overridden by the relation Table 1 clearly shows that the effectively free electrons in the interstitial region of isochorically heated copper are in the WDM regime. 32In nature, WDM occurs in a variety of astrophysical objects such as giant planet interiors, 3 brown dwarfs, 36 and white dwarf atmospheres. 35In the laboratory, these conditions are encountered on the compression path of a fuel capsule and its ablator in inertial confinement fusion experiments, 4,6,38 and, in addition, they are relevant for material science, synthesis, and discovery. 82,833][34] Our new simulation results thus imply that future XRTS experiments with isochorically heated copper constitute a suitable and highly controlled testbed for the benchmarking of theoretical methods and simulations.

Conclusions
Motivated by experimental capabilities at modern FEL facilities, we have performed a detailed study of the changes in the electronic DSF in fcc copper due to isochoric heating.Our new LR-TDDFT simulations show that the heating induces a prominent plasmon feature that eventually becomes dominant over the d-band-signal.This is in marked contrast to the ground state, where the plasmon is strongly damped by the presence of d-band excitations, and where it does not exhibit a meaningful dispersion with respect to the wavenumber q.Indeed, we have shown that at T ≥ 4 eV, the plasmon dispersion follows the familiar Bohm-Gross type relation, and the plasma frequency ω p substantially increases with the temperature.This has been explained by the accumulation of effectively free electrons in the interstitial region between the ions.This, in turn, is a consequence of the availability of the electrons in the d-orbitals for filling states in the quasicontinuum when the temperature is increased.Interestingly, the reported behavior of the plasmon in copper is fundamentally different from isochorically heated aluminium (Al), where LR-TDDFT simulations have shown that heating to a few electronvolts (at T ≲ 7 eV) causes a red shift of ω p . 25 In this regard, we note that Al has 3 valence electrons from 3p3s orbitals and further ionization of the elec-trons from 2p requires temperatures about 70 eV.Therefore, in contrast to copper, thermal excitations in Al are not accompanied by an increase in the density of valence electrons with the increase in the temperature at T ≲ 10 eV.
From a physical perspective, we find that the quasi-free electrons in the interstitial space are in the WDM regime with an effective density parameter rs ∼ 2 and an effective degeneracy temperature Θ ∼ 0.5.In addition to being interesting in their own right, such extreme conditions occur in a wealth of astrophysical objects and play a key role in experiments with inertial confinement fusion and material science.In practice, there does not exist a single method that is capable of giving an accurate description of WDM states over the entire relevant parameter space, and the interpretation, modeling, and design of corresponding experiments is usually based on a number of de-facto uncontrolled approximations.
In this regard, we propose to use future XRTS experiments with isochorically heated copper as a controlled testbed for the rigorous assessment of different theoretical models and simulation tools for the description of WDM.Ideally, one might infer the temperature of the heated sample based on the model-free imaginary-time thermometry approach 11,12 as a first step.Indeed, even the inference of comparably moderate temperatures of T ∼ 1 − 10 eV is expected to be feasible using the new high-resolution set-up that has recently been demonstrated at the European XFEL in Germany. 24Second, we propose to carry out measurements at multiple scattering angles (i.e., multiple wavenumbers q) to observe the plasmon dispersion ω(q) and, in this way, to infer the effective charge state Z and the effective density parameter rs .Since the ambient density is a-priori known, this will give one full access to the most relevant parameters of the system.This is an important advantage over shock experiments, or XRTS measurements with backlighter sources, 39 where one or multiple of these parameters can only be inferred on the basis of the theoretical models which we aim to test in the first place.Isochoric heating can be achieved by employing a 400 nm optical short-pulse laser on thin targets and combining it with the delayed FEL probe. 84,85Alternatively, one can use X-ray pump and X-ray probe beams that are separated in both colour and time.Proper time separation allows heating to be completed by the pump before the target is probed.Such X-ray pump X-ray probe experiments can be performed at European XFEL 86 and at SACLA in Japan. 87,88A detailed discussion of the feasibility of measuring thermal excitations in the DSF of isochorically heated targets is provided in Ref. 25 Such a hypothetical XRTS dataset can then be used to benchmark the zoo of available theoretical methods such as the widely used effective chemical models. 19,89,902][93][94][95][96] Moreover, frequency-resolved inelastic X-ray scattering data for a set of finite wavenumbers q will be ideally suited to gauge the accuracy of different XC-kernels in LR-TDDFT calculations, including a rigorous assessment of the popular adiabatic approximation. 40,44,97As a WDM testbed, the observed isotropy of the DSF of electrons in copper is advantageous for achieving unambiguous conclusions in the assessment of various models.For completeness, we show the DSF in the [111] direction at the different considered temperatures and wavenumbers in Fig. 7.We find that the B, C, and D features behave similarly to our results for the [100] and [111] directions, both with respect to temperature and wavenumber.Indeed, the DSF in [111] direction is nearly identical to the DSF in [100]   direction over the entire depicted q-range.

Figure 3 :
Figure 3: LR-TDDFT results for the DSF of fcc Cu along the [100] direction at different wavenumbers for the ground state with T = 0.025 eV, and for isochorically heated electrons with T = 2 eV, T = 4 eV, T = 6 eV, and T = 8 eV.The wavenumber values are given in the units of Å -1 .

Figure 4 :
Figure 4: LR-TDDFT results for the DSF of fcc Cu along the [011] direction at different wavenumbers for the ground state with T = 0.025 eV, and for heated electrons with T = 2 eV, T = 4 eV, T = 6 eV, and T = 8 eV.The wavenumber values are given in the units of Å -1 .

Figure 5 :
Figure 5: The demonstration of the role of the factor f (ω) = (1 − e − ̵ hω/k B T ) −1 on the enhancement of the DSF features at small energies.Solid lines are the results for the DSF along the [100] direction and dashed lines are the same data divided by f (ω).The wavenumber values are given in the units of Å -1 .

EuropeanFigure 7 :
Figure 7: LR-TDDFT results for the DSF of fcc Cu along the [111] direction at different wavenumbers for the ground state with T = 0.025 eV, and for heated electrons with T = 2 eV, T = 4 eV, T = 6 eV, and T = 8 eV.The wavenumber values are given in the units of Å -1 .

Table 1 :
Plasmon energy at q = 0.0754 Å -1 extracted from the DSF calculations with corresponding effective density parameter rs , effective degeneracy parameter Θ, and effective charge of ions Z.