Unveiling structural effects on the DC conductivity of warm dense matter via terahertz spectroscopy and ultrafast electron diffraction

Domain: physics
Published: 2025-11-26
Editorial outcome: minor_revision

Abstract
Understanding how materials under far-from-equilibrium conditions conduct electricity is vital for modeling planetary interiors, fusion energy, and other high-energy-density environments. Yet direct measurements of electrical conductivity in these states are challenging, as experiments must capture changes in both electronic conditions and atomic arrangement. Here we show, using laser-heated aluminum films, how the electrical conductivity of materials driven to the warm dense matter regime is influenced by temperature and structure. By directly measuring the electrical conductivity using terahertz time-domain spectroscopy and observing the atomic arrangement using mega-electron-volt ultrafast electron diffraction studies, we separate the impact of these different contributions on the observed sharp drop in the conductivity after laser heating. This approach is broadly applicable for measuring the electrical conductivity of matter laser heated to high-energy-density conditions. Our results are used to benchmark leading theoretical models and highlight the importance of accurately treating both electron and ion dynamics.

Introduction
Many unresolved challenges in physics, from understanding the generation of planetary magnetic fields1,2 to successfully driving a burning plasma for Inertial Fusion Energy3,4, require a rigorous understanding of transport properties of matter in high-energy-density conditions. A critical parameter for simulating these processes is the zero-frequency (DC, or electrical) conductivity of materials in the warm dense matter (WDM) regime5–7. Ideal plasma models, which assume the electron thermal energy, E k, greatly exceeds the Fermi energy, E F, exist for determining the DC conductivity8,9. In the WDM regime, E k is on the order of both E F and the mean potential energy, 〈V〉. This leads to a near unity degeneracy parameter, Θ=E k /E F~1, and Coulomb coupling, Γ=〈V〉/E k~1. Consequently, ideal plasma theories break down as quantum mechanical effects need to be considered, and the strong Coulomb coupling between ions lead to correlations reminiscent of crystal-like and liquid-like structures. Computational methods such as time-dependent density functional theory (TD-DFT), density functional theory coupled with molecular dynamics (DFT-MD) using the Kubo-Greenwood formalism10–17, or average atom density functional theory (DFT-AA)18–20 have been developed to calculate the DC conductivity in WDM. These approaches require validation; thus, experiments that directly determine transport properties are essential.

Ultrafast laser pulses can drive thin metal films to WDM conditions21; however, as there is rapid heating of the electrons followed by much slower transfer of energy to ions, the resulting states are very far from equilibrium. The heated state is also highly transient, persisting for only picosecond durations22–25. This places strict requirements on the temporal resolution of any probe used to investigate laboratory-generated WDM. Furthermore, there is the challenge of developing accurate and consistent models that account for the electrons and ions being out of equilibrium. This always occurs on short timescales and is essential for comparison with experiments. Benchmark quality data must thus independently determine the atomic configuration, ion temperature, and electron temperature. Attempts have been made to estimate the DC conductivity from measurements of the high-frequency (AC, or optical) conductivity using femtosecond optical laser pulses22,24,26–31, or by X-ray Thomson scattering measurements12,32–34. However, both approaches require models to infer the DC conductivity, and as such are susceptible to uncertainties in the models used. This highlights the need for direct measurement of the DC conductivity on ultrafast timescales.

Terahertz time-domain spectroscopy (THz-TDS) has been recently developed for studies of highly transient states, including WDM35–39. THz fields have such low frequencies that they effectively probe DC-like material properties. THz pulses can have durations on the order of picoseconds, allowing interrogation of highly transient phases. These properties make THz pulses an ideal probe of the conductivity of WDM. Similarly, mega-electron-volt ultrafast electron diffraction (MeV-UED) is well-suited for measuring the ionic structure of WDM23,40–43 due to the high scattering cross section and electron mean free path comparable to the sample thickness required for laser excitation of thin films.

In this work, we combine THz-TDS with MeV-UED to interrogate the electrical conductivity of warm dense aluminum (WD-Al) driven by a short optical laser pulse. Our measurements use a custom table-top THz setup44 and obtain data with excellent signal-to-noise with single-shot detection and direct measurement of the THz spectrum. We find that upon irradiation, there is a rapid monotonic decrease in the conductivity, with larger changes for increasing energy deposition. By combining THz and MeV-UED measurements, we disentangle changes in the conductivity due to increases in the electron scattering rate, which dominates on sub-ps timescales, from the concurrent atomic rearrangement as the sample transitions from solid to liquid, which takes place on the few-ps timescales. Finally, we use our data to assess state-of-the-art computational methods, finding that DFT-MD and DFT-AA calculations with the appropriate choice of exchange correlation functional and ionic structure match closely with our experimental data. This represents an important step towards providing benchmark-quality data to further improve theoretical models. Our results offer a direct measurement of the quantity of interest, the DC conductivity, near the static limit, while tracking the ionic structure. Consequently, the data offer constraints on calculations in the warm dense matter regime.

Results
Ultrafast electron diffraction
To investigate the evolution of ionic structure in the laser-generated WD-Al, UED measurements were performed at the SLAC MeV-UED facility40,45. The experimental setup is illustrated schematically in Fig. 1a. Free-standing 35nm thick Al foils were irradiated by λ=800nm, 70fs laser pulses with fluences, F, of up to 550mJcm−2. The absorbed energy density, ρ E, is defined as the amount of absorbed laser energy by the ambient material normalized to the mass of the material and is given by ρ E=FA/d 0 ρ M, where A=13.5% is the absorption ratio, d 0 is the film thickness, and ρ M is the mass density of Al. For the fluences used here, A has been shown to remain unchanged46. The maximum ρ E reached was ~6.5MJkg−1. Figure 1b–e show false-color diffraction patterns that are the average of 20 single-shot measurements at selected time delays, Δt, following irradiation with 199±20mJcm−2 fluence laser pulses. The corresponding absorbed energy density was 2.42MJkg−1. Comparing Δt=0ps and Δt=2ps, there is a loss of higher-order rings in the diffraction patterns consistent with the onset of melting. By Δt=20ps, only one broad feature is visible, suggestive of the formation of a liquid-like state.

Figure 2a shows radial lineouts plotted against momentum transfer Q=4πsin(θ)/λ , with λ ~0.34pm the de Broglie wavelength of the 3.7MeV electrons, resulting from azimuthal integration of the measured diffraction patterns with background signals from inelastic and multiple scattering and dark current subtracted (see methods and supplementary text note 5). At Δt=2ps, the 111 Debye-Scherrer ring, located at Q=2.69±0.01Å−1, remains visible but appears broadened when compared with the diffraction pattern from an unperturbed sample. Additionally, there is a clear loss of intensity in the 220 ring (Q=4.40±0.01Å−1). After 5ps, a single broad feature appears in a similar location as the 111 ring, and a weaker second broad feature can also be seen around the position of the initial 220 ring. These changes indicate loss of long-range order as the sample transitions from a crystalline solid into a more disordered liquid phase.

To quantify these changes, the radial lineouts were fit to either three Gaussians corresponding to the 111, 200, and 220 rings of the crystalline material or to two Gaussians for liquid diffraction. Parameters extracted from these fits are shown in Fig. 2b, c for different pump fluences and time delays. To follow the melting dynamics, we examined the full width at half maximum (FWHM) of the lowest order peak extracted from our Gaussian fits, shown in Fig. 2b. Of particular relevance to this study, in the liquid state, increased temperature leads to peak broadening47–50. We observe that the FWHM remains unchanged for Δt<2ps before increasing. The initial delay is attributed to the time required to sufficiently increase the lattice temperature to the melting temperature. After melting is complete, the width continues to change as the ions are heated. We determined when the sample has melted by comparing when the FHWM matches previously measured values. Based on analysis of neutron scattering experiments on liquid Al50, we set a threshold FWHM, the dashed gray line in Fig. 2b, of 0.67±0.01Å−1 (see supplementary text note 6).

Details of changes in the lowest-order diffraction peak position are shown in Fig. 2c. For Δt<3ps, the lowest-order peak position remains essentially unchanged before decreasing and reaching an apparent minimum 10ps after irradiation. The initial delay in the density change can be ascribed to an acoustic wave originating within a crystallite and traveling at the sound velocity. Given a longitudinal sound speed of v s=6.4kms−151, and assuming a crystallite size on the order of the film thickness, d 0=35nm, the time required to travel half the film thickness is d 0/2v s=2.8ps, consistent with our observation. Equilibration then continues over longer time delays, indicated by the shift of the peak back to higher Q at 20ps delay at all but the highest irradiation.

Considering both the FWHM and lowest-order peak position, we infer that for Δt<2ps, our Al films retain a solid structure with near-solid density for all fluences of interest. At 3ps delay, the Al film is largely composed of liquid-like material, and by 5ps the sample has completely melted for energy densities exceeding 2.42MJkg−1. For Δt≥3ps, hydrodynamic expansion should be considered. These results are consistent with other recent MeV-UED measurements on Al52, with our data confirming these observations over the energy densities relevant to our measurements. We use these details for interpreting changes in the THz data. Changes observed for Δt=1ps will thus primarily report on temperature effects, while after Δt~2–3ps will also be influenced by the modification of the ionic structure.

Terahertz spectroscopy
To determine the conductivity, we performed single-shot THz spectroscopy measurements of 33nm Al films heated to WDM conditions using intense femtosecond laser pulses36,37,39,53. Figure 3a shows a schematic of the experimental setup. The films were heated using λ=800nm pulses focused over a ~2mm diameter area. To ensure probing of heated material, a thick sample card with holes <1mm in diameter was used to mask the Al films. The highest fluence achieved was ~330mJcm−2, and the corresponding ρ E was ~5.1MJkg−1. The near single-cycle THz pulses transmitted through the sample were used to probe the conductivity. In our experiments, we varied the timing such that the THz pulse arrived 1ps before the pump pulse (Δt=−1ps) and up to 5ps after the pump pulse (Δt=5ps).

Figure 3b shows example THz time-domain waveforms collected on unheated and laser-heated films at Δt=1ps. The THz time-domain waveforms were collected using an echelon-based electro-optic sampling method, which recorded the waveforms in a single shot36,38,39,54,55. Different colors correspond to different ρ E, and measurements on an unheated film and a reference collected with no sample (the latter scaled by 0.1) are shown for comparison. Upon heating, the magnitude of the THz signal increases, |E(t)|, indicating an increase in the sample transmission. For the highest incident fluence, we observe an almost 4× increase in the transmission due to the changes in the electronic properties of Al (|E(t)|~0.16kVcm−1 for the unheated film compared to almost 0.60kVcm−1).

To quantify the change in transmission, the frequency-domain spectra were calculated using a numerical Fourier transform of the time-domain data. When taking the Fourier transform, a hyper Gaussian window function, exp[-((t-t 0)/τ)20], was used56 to remove contributions from signals outside the main cycles of the THz pulse; doing so limits the frequency resolution of the measurement to ~0.6THz, with additional points in the frequency domain arising from the length of the original time window. We normalized the spectra of the heated film, E~ s(ω), to the spectrum taken with no sample, E~ r(ω), and this was used to determine the frequency-dependent transmission, t(ω)=E~ s(ω)/E~ r(ω). The transmission magnitudes |t(ω)| are plotted in Fig. 3c. For the ambient film, the initial transmission is ~2%, and this increases to almost 7% when the film is excited to the highest energy density, consistent with the estimates derived by using only the peak of the THz electric field.

Figure 3d shows plots of the real part of the conductivity extracted via the Tinkham formula (see methods). We observe a largely frequency-independent conductivity over our THz bandwidth. For the ambient film, at 1THz we determine a conductivity of 14.6±1.1×106Sm−1. Independent four-point resistance measurements of the ambient film yield a conductivity of 17.4±3.5×106Sm−1, closely matching our data and thus supporting the use of THz spectroscopy for probing WD-Al. We note that the lower conductivity is expected in thin films, as there is increased electron scattering by grain boundaries and surfaces57–60; as such, care should be taken to ensure when extracting the conductivity of low-dimensional films. The increase in transmission as the sample is excited leads to a decrease in the conductivity, with values as low as 3.4×106Sm−1 for the film excited to 5.1MJkg−1.

Given the weak structure in the frequency dependence of the conductivity, we determined the average conductivity over the THz spectrum between 0.5–1.5THz. The results of this averaging are shown in Fig. 4 for various energy densities and time delays. Beyond Δt=1ps, hydrodynamic expansion leads to a change in the film thickness, d, which is an input to the Tinkham formula. Frequency domain interferometry (FDI) measurements were performed to determine the change in the thickness which were then used to determine the conductivity (see supplementary text note 4). We observe a monotonic decrease in the conductivity with increasing energy density, consistent with faster electron scattering at higher electron temperatures. For a fixed energy density, there is an up to 2× decrease in the conductivity between Δt=1ps and Δt=3ps and a much smaller decrease from Δt=3ps to Δt=5ps. From the MeV-UED measurements, we expect that the ion temperature of our Al films exceeds the melting temperature (~933K, ~0.08eV) within 5ps. The conductivity trends observed here are expected to be due to changes in both the electronic and ion temperatures. The former impacts the energy distribution of scattering electrons, whereas the latter impacts the electron-ion scattering cross sections.

Methods
Sample preparation
Thin films of Al were made by two methods—evaporation onto a sodium chloride (NaCl) substrate or sputtering onto 20nm of silicon nitride (Si 3 N 4) on a silicon (Si) wafer. Evaporation onto NaCl substrates was performed at the Stanford Nano Shared Facility (SNSF). Sputtering onto Si 3 N 4/Si wafers was performed by Silson Ltd., UK. Films evaporated onto NaCl were transferred onto a thick aluminum plate with 1mm diameter holes, resulting in free-standing films. For films on Si 3 N 4, the Si wafer was back etched to produce 600μm x 600μm square holes. Measurements on both films yielded consistent results. 

Ultrafast electron diffraction
The diffraction measurements were performed at the SLAC MeV-UED facility40,45,82. An amplified Ti:sapphire laser system capable of operating at 120Hz supplied 12mJ pulses used to generate the ultrafast electron bunches and the optical pump pulses. To produce the electron bunches, approximately 1mJ of the Ti:sapphire output was frequency tripled and directed onto a copper photocathode; the resulting photoelectrons were accelerated to a kinetic energy of 3.2MeV using a Linac Coherent Light Source–type photocathode radio frequency (rf) gun. The RF gun was powered by a pulse-forming network–based modulator and a 50-megawatt S-band klystron. The relativistic electrons were focused onto the target by two separate solenoids installed after the RF gun. 60fC electron bunches were delivered in a 200μm diameter spot at the sample, with a bunch length at the sample estimated to be 1ps FWHM based on Debye-Waller measurements. Diffracted electrons were deposited onto a P43 phosphor screen located 3.2m after the sample. A 1.6-mm-diameter hole was cut in the middle of the phosphor screen, and undiffracted electrons passed through this hole. An f=40mm focal length lens was used to image the phosphor screen onto an electron-multiplying charge-coupled device camera (Andor iXon Ultra 888).

The optical pump pulses were time-delayed using a mechanical delay stage, and a pinhole was utilized to select a uniform portion of the pump pulse profile with energy fluctuations <10% over the region probed by the electrons. A polarizer and a computer-controlled half-waveplate were used to modulate the pump pulse energy. The pump pulse was directed at ~10° to the sample normal.

For data collection, the laser was chopped to 30Hz and computer-controlled shutters were used to select individual pulses. Each data point was the average of single-shot measurements on 20 distinct targets. For each target, undriven and driven diffraction patterns were collected and averaged offline.

THz spectroscopy
THz measurements were performed using a home-built single-shot THz-TDS setup based on a previous experimental design36,39. The setup was driven by a Coherent Astrella Ti:sapphire laser system capable of producing 8.5mJ, 50fs pulses at a maximum repetition rate of 1kHz. The THz and heating pulses were derived from the same laser source, and as such Δt could be set. The laser output was split, delivering up to 4.5mJ pulses to heat the samples. Their energy was adjusted using a computer-controlled half-waveplate and a pair of thin-film polarizers. The pump pulses were then directed to a delay stage to set their arrival time relative to the THz pulses and focused using a 75cm lens. Like the UED experiments, it was important to ensure uniform illumination. Consequently, the focal point of the lens was placed after the target plane. Using an upstream aperture to select a uniform portion of the pump beam would have significantly reduced incident fluence on the target, as the long THz wavelength prevents tight focusing.

4mJ of laser energy was used for THz generation and detection. Of this energy, a beam splitter selected 3.5mJ, which was used to produce THz pulses via optical rectification in a N-benzyl-2-methyl-4-nitroaniline (BNA) optically glued to a sapphire substrate83,84. The resulting THz pulses were directed using a set of 90° off-axis parabolic reflectors to the target plane, where they probed the sample. A pair of downstream off-axis parabolic reflectors was used to direct the transmitted pulses to a 2mm thick (110)-cut zinc telluride (ZnTe) crystal. Holes in the off-axis parabolic reflectors and a downstream imaging system were used to check sample placement and overlap the THz and pump pulses at the target position.

THz detection was accomplished using the remaining 0.5mJ pulses for echelon-based single-shot electro-optic sampling35,36,38,39,54. The sampling pulses were attenuated using a combination of a half-waveplate and polarizer before being directed to a second delay stage. The beam was expanded to overfill an 18mm × 18mm square echelon. The echelon, produced by Sodick F.T., consisted of 120 steps with a width and height of 150μm and 7.5μm, respectively. Upon reflection off the echelon, a single sampling pulse was split into a series of time-delayed beamlets, which were focused into the 2mm thick ZnTe crystal using an f=100mm focal length lens. An NA=0.1 objective and f=75mm focal length lens were then used to image the echelon onto an Allied Vision Manta camera. A quarter waveplate and Wollaston prism were used in the single shot electro-optic sampling. An illustration of the experimental setup is shown in supplementary Fig. S1a.

For data collection, the Astrella was set to a 1Hz repetition rate and computer-controlled shutters were used to block specific pulses. For each target, 5 frames were collected with the pump and THz pulses blocked, 5 with pump pulses blocked and THz pulses unblocked, 1 with the pump and THz pulses unblocked, and 5 with pump pulses blocked and THz pulses unblocked.

As the sampling pulse propagated through the Wollaston prism, each frame contained two images of the echelon, denoted with the superscripts L and R. The relative brightness change ΔIL,R /IL 0 ,R was determined using images collected with the THz pulse present, IL and IR, and images collected with the THz pulse absent, IL 0 and IR 0 , according to ΔIL,R /IL 0 ,R = IL,R/IL 0 ,R - 1. In these images, the information about the THz time-domain waveform is encoded along the horizontal axis, and so the time-domain waveform was extracted by averaging the ΔIL,R /IL 0 ,R images along the vertical axis of the image. Finally, the resulting signal S(t) was determined by subtracting the waveform obtained using the right image from the corresponding waveform resulting from the left image, i.e. S(t)=ΔIL/IL 0 - ΔIR/IR 0 36,39.

This procedure yielded THz time-domain waveforms transmitted through the unpumped film, the heated film, and a reference. Each data point was the average of three targets. The experiments were performed in a nitrogen atmosphere with a relative humidity of <5%. Δt=0 was determined from optical-pump THz probe measurements on a Si wafer (see supplementary text note 1).

From the transmission, we used the Tinkham formula85, which accounts for multiple reflections within a film whose thickness is much smaller than the wavelength and when the conductivity is large, to determine the conductivity, σ(ω): σe(ω) = (ns+1)/Z 0 d (1/t(ω) - 1) (4)
ns is the refractive index of the substrate, 2.7 for Si 3 N 4 86, Z 0=377Ω is the impedance of free space, and d=33nm is the thickness of the film. For Δt=1ps, we do not expect significant expansion of the film, as the MeV-UED measurements show no change in density. For Δt≥3ps, we accounted for hydrodynamic expansion using frequency domain interferometry measurements (see supplementary text note 4) to determine the film expansion22,87,88.

Two-temperature modeling
In the form of the TTM described in the main text, we assume uniform heating through the thickness of the film, as electrons initially heated by the laser within the skin depth travel ballistically through the film, rapidly thermalizing with cold electrons through collisions. Furthermore, given the large spot size of the beam, we do not expect lateral thermal diffusion on the ps timescales of this work.

At each timestep of the simulation, we calculated the electron specific heat capacity according to C e(T e)=γT e, with γ=91.2Jm−3K−2 61, while the ion specific heat capacity was held constant at C i=2.5×106Jm−3K−1, the constant volume heat capacity in the Dulong-Petit limit, consistent with the expected isochoric heating. Calculations using the constant pressure heat capacity, which differed by ~5%, can be found in the supplementary text note 9 and Fig. S10. The electron-temperature dependent electron-ion coupling parameter, G ei(T e), is taken from DFT calculations by Lin et al.61 and Simoni et al.62 with different values used for the solid and liquid. Details of the error in the temperature are given in supplementary note 7.

Average atom DFT calculations
The DFT-AA/Slater conductivities given in Fig. 6 of the main paper were calculated using Bemuze, a semi-relativistic version of the average-atom code Purgatorio89 modified to include self-consistent ion structure following the quantum Ornstein-Zernicke approach from Starrett and Saumon90. Given an element, mass density, electron temperature, and exchange-correlation potential, average-atom models find self-consistent orbital wavefunctions from the Schrodinger or Dirac equations, orbital occupations and chemical potentials based on Fermi statistics and charge conservation, and electron-ion potentials from the Poisson equation. These orbitals were used to generate T-matrix differential cross sections that can be integrated over the energy-dependent electronic density of states to produce Q-dependent scattering terms suitable for the Ziman formulation of conductivity18.

Data availability
The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information file. All other relevant data supporting the findings of this study are available on request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Review of manuscript “Unveiling Structural Effects on the DC Conductivity of Warm Dense Matter via Terahertz Spectroscopy and Ultrafast Electron Diffraction”\nSubmitted to Nature Comm.\nIn this manuscript the authors employed terahertz spectroscopy with mega electron volt ultrafast electron diffraction to obtain direct measurement of the DC conductivity on ultrafast timescales. The measurements remove the requirement for additional parameters for this. This is the main and important contribution in this manuscript.\nThe authors focus on the electrical conductivity of warm dense aluminum for this.\nComments:\n1) WDM could be explicitly defined on pg. 2.\n2) The authors correctly point out that other derivations of conductivity approaches require models to infer the DC conductivity and claim that the method employed in the submitted manuscript is a direct measurement without this requirement. DFT-MD and DFT-AA calculations with the appropriate choice of exchange correlation functional and ionic structure match closely with the authors experimental data. Does this not imply that the match of experiment to model requires a theoretical method to evaluate the exchange correlation functionals and ionic structures?\n3) The authors critical description of the experimental results with regard to phase transitions is clear and excellent. Figure captions are all clear and describe well the details.\n4) The real component of the conductivity is correctly employed the Tinkham formula and the frequency dependence is obtained via a critical application of the Drude model.\n5) Minor comments: Pg. 43, There is only a reference number 8 at the sentence ending on line 886 but no simple details as to what approximation was employed. An incomplete sentence beginning only with the word “There” on line 922 needs completion.\nIn summary, this manuscript clearly describes experimental results that improve on earlier work due to the reduction of parameters needed to obtain conductivity information. Although there remains a need for improvement of a few parameters that are actually pretty well accepted already, this work provides excellent new data that will definitely aid in the improvement of theoretical understanding of conductivity models.", "reviewer_2": "The manuscript contains experimental data for the ionic structure of warm dense liquid aluminium and the dc conductivity of the same state of matter. The experimental results are compared to a number of theoretical results and also to different experimental results. They cover temperatures up to about 10eV.\nI have several issues with this manuscript:\n1.) The electron diffraction measurements and the THz conductivity measurements seem to be two different setups/experiments that were done at different times and with different pump lasers. Is that correct? Are both systems so stable and comparable that shot-to-shot and day-to-day variations do not play a role? I assume the different thicknesses of the Al foil of 33 nm and 35 nm for the two setups do not matter even though hydrodynamic motion (expansion) needs to be taken into account?\n2.) There is no temperature measurement. There is an estimate of possible electron and ion temperatures using a two-temperature model. Several of the input parameters seem not trustworthy. For instance, a quick check in the NIST database shows quite some variation of the heat capacity of solid aluminum with temperature but the authors state that they used a single value for all cases. The absorption of light in matter is a notoriously difficult problem, in particular in warm dense matter, so the manuscript should acknowledge this difficulty and at the very least give a citation for the simple formula (S10) and note in which limit it is valid.\n3.) The comparison with DFT results opens up a number of questions as well. The citation given is Witte et al. (PoP 2018). In it, all conductivity calculations seem to be for equilibrium aluminum. It is however not clear to me, which results (from THz spectroscopy) in this manuscript stem from equilibrated runs (featuring a large delta t) or stem from non-equilibrium runs (delta t < 2ps or so). In particular, since the main figure (Fig. 6) has the abscissa labeled as \"electron temperature\". Similar problems exist in Fig. 5, where the ion temperature dependence is shown. What is the respective electron temperature?", "reviewer_3": "The authors combine ultrafast electron diffraction and terahertz spectroscopy to monitor the evolution of electrical conductivity of an aluminum foil melted by an ultrashort laser pulse. The strong point here is the use of the ultrafast electron diffraction which unambiguously provides the evolution of the structure of the photoexcited material. It is thus possible to accurately correlate the retrieved terahertz conductivity with instantaneous structure of the aluminum.\nThe manuscript is comprehensible and well written; I appreciate the faithful and careful documentation of all results and methods as well as the scientifically sound analysis. Altogether, since the manuscript well advances the understanding of behavior of photoexcited materials at the shortest time scales, I recommend it for publication in Nature Communications.\nThe following technical issues are minor (they do not have the potential to compromise the conclusions):\n* My major objection concerns the frequency resolution displayed in Figs. 3c,d. The employed time window indicated in Fig. 3b is ~1.6 ps; this means that the frequency resolution is 1/1.6 ps = 600 GHz, i.e., there are exactly two experimental points in Figs. 3c,d. All other points are artificial since they appear as a result of zero-padding (or more precisely, almost zero): in turn, they represent just an interpolation of the “experimental” points and do not carry out any information supported by experiments. I do not think that using a longer time-window would change the conclusions on THz conductivity (and I’m aware that it would lead to artifacts due to the ultrafast evolution of the system, [J. Opt. Soc. Am. B 17, 327 (2000)]). However, absolutely smooth data from any THz system (the more from a single-shot system which is generally noisier; in fact, the noise is clearly seen in Fig. 3b) is a highly visible hint that some suspicious operations were done with the experimental data. Please reconsider the analysis or just include a sentence on what is the real realistic frequency resolution.\n* Although THz conductivity is confirmed to be close to the DC conductivity in this particular system, one should be aware that charge localization may occur in certain systems [e.g. in thin gold layers, Phys. Rev. B 76, 125408 (2007) or in most semiconductor nanoparticles], i.e., THz frequencies may not be generally low enough to permit a direct connection to DC conductivity. A certain reservation should thus be expressed when discussing equality between THz and DC conductivity in the manuscript.\n* When presenting THz conductivity, consider whether it would not be more relevant to show (and easier to discuss) the sheet conductivity instead – the sheet conductivity should be insensitive to plasma expansion.\nMinor remarks related to the presentation (these are formal only and it makes no sense to publish them along with the rest of the report)\n* Consider transfer of the equations governing the two-temperature model into the main text (or at least refer to the equations explicitly) – now it is really difficult to decode what the two-temperature model is about. It seems to me that the G_ei value is temperature-dependent: if this is the case, it would be helpful to indicate this dependence explicitly, like in the case of the thermal capacitances.\n* Although excitation conditions are described in the methods section, it would be helpful if the authors presented their opinion to which extent the conditions are compatible for ultrafast electron diffraction and THz measurements and point out if there are some possible aspects in which they may differ.\n* Fig. 3b-d: Please label the curves for individual powers unambiguously – the current continuous color scale (with obscure number like 4.58) permits a readout with ±~1 MJ/kg which translates to >100% error for the lowest fluences. I also expect that the curves with lowest transmission/highest conductivity corresponds to unexcited sample, but the corresponding color implies 2±1 MJ/kg – it would be good to clarify this.\n* I have nothing against showing absolute THz field intensities in kV/cm, nevertheless, what matters is the relative change (text at lines 158 - 159 could be slightly rephrased in this respect) – this avoids the need to specify whether this is the field just after the sample, inside the sample or at some particular position in front of/inside the sensor."}, "author_response": "Response: We thank the reviewer their review of our work and concise summary of the results. Below we respond to the comments.\n1) WDM could be explicitly defined on pg. 2.\nResponse: We thank the reviewer for suggestion. We have modified the main text to include the following description of WDM on page 2:\nIdeal plasma models, which assume the electron thermal energy, E , k greatly exceeds the Fermi energy, E , exist for determining the DC F conductivity8,9. In the WDM regime, E is on the order of both E and the k F mean potential energy ⟨V⟩. This leads to a near unity degeneracy parameter, Θ = E /E ~ 1, and Coulomb coupling Γ = ⟨V⟩/E ~ 1. k F k Consequently, ideal plasma theories break down as quantum mechanical effects need to be considered and the strong Coulomb coupling between ions leads to correlations reminiscent of crystal-like and liquid-like structure.\n2) The authors correctly point out that other derivations of conductivity approaches require models to infer the DC conductivity and claim that the method employed in the submitted manuscript is a direct measurement without this requirement. DFT-MD and DFT-AA calculations with the appropriate choice of exchange correlation functional and ionic structure match closely with the authors experimental data. Does this not imply that the match of experiment to model requires a theoretical method to evaluate the exchange correlation functionals and ionic structures?\nResponse: We thank the referee for this observation. It is certainly true that all DFT-based models require some choice of exchange correlation potentials – there are a wide range of theoretically motivated potentials to choose from – and models that calculate the DC conductivity directly may require additional choices: for example, DFT-MD models require care with respect to finite-size eKects and extrapolation of the dynamic Kubo-Greenwood response to the DC limit, and DFT-AA models require care with respect to the ion structure factor.\nWe believe that the present work addresses the reviewer’s questions and is uniquely constraining for models, in two important ways:\n1. First, the THz probe enables a measurement of the dynamic response very near the DC (static) limit. As such, we do not have to invoke a model to translate the measured observable (e.g. via implicit or explicit extrapolation of a dynamic conductivity) into the quantity of interest – instead we measure the quantity of interest directly with the THz probe.\n2. Second, the UED measurements provide a clear constraint on the ionic structure, eliminating one degree of freedom necessary for calculating the DC conductivity.\nIn a very important sense, the interpretation of other experiments are constrained by models; our experimental can data provide constrains to theoretical predictions. To highlight this distinction, we are adding the following sentences to the manuscript:\nIn the introduction, last sentence of the last paragraph:\nThis represents an important step towards providing benchmark quality data to further improve theoretical models. Our results offer a direct measurement of the quantity of interest, the DC conductivity, near the static limit, while tracking the ionic structure. Consequently, the data offer constraints on calculations in the warm dense matter regime.\nIn the conclusion, middle of the last paragraph:\nThese measurements thus provide an important advance towards obtaining benchmark quality data for verifying theoretical models, especially in the selection of structure factors and exchange-correlation models. While most previous approaches measure observables that require interpretation to obtain the transport property of interest, our approach measures the conductivity directly.\n3) The authors critical description of the experimental results with regard to phase transitions is clear and excellent. Figure captions are all clear and describe well the details.\nResponse: We deeply appreciate this positive assessment!\n4) The real component of the conductivity is correctly employed the Tinkham formula and the frequency dependence is obtained via a critical application of the Drude model.\nResponse: We thank the reviewer for this positive endorsement of our work!\n5) Minor comments: Pg. 43, There is only a reference number 8 at the sentence ending on line 886 but no simple details as to what approximation was employed. An incomplete sentence beginning only with the word “There” on line 922 needs completion.\nResponse: We thank the reviewer for catching these errors. We have removed the “There” at the end of the caption and adjusted the text, page 41:\nFor solids, we follow Potekhin et al.8 and approximate the inelastic scattering in solids using Coulomb logarithms defined in terms of the inelastic component of the dynamic structure factor and the scattering potential.\nIn summary, this manuscript clearly describes experimental results that improve on earlier work due to the reduction of parameters needed to obtain conductivity information. Although there remains a need for improvement of a few parameters that are actually pretty well accepted already, this work provides excellent new data that will definitely aid in the improvement of theoretical understanding of conductivity models.\nResponse: We thank and appreciate the reviewer for offering this supportive appraisal of our work!\nReviewer #2 (Remarks to the Author):\nThe manuscript contains experimental data for the ionic structure of warm dense liquid aluminium and the dc conductivity of the same state of matter. The experimental results are compared to a number of theoretical results and also to di[erent experimental results. They cover temperatures up to about 10eV.\nResponse: We thank the reviewer for taking their time to read and review the manuscript. Below we respond to the comments.\nI have several issues with this manuscript:\n1) The electron diffraction measurements and the THz conductivity measurements seem to be two different setups/experiments that were done at different times and with different pump lasers. Is that correct? Are both systems so stable and comparable that shot-to-shot and day-to-day variations do not play a role? I assume the different thicknesses of the Al foil of 33 nm and 35 nm for the two setups do not matter even though hydrodynamic motion (expansion) needs to be taken into account?\nResponse: We thank the reviewer for raising this point. This is correct – the experiments were performed on different setups. The laser systems used in both experiments are commercial systems with shot-to-shot energy fluctuations <1%. In addition, for each set of measurements calibrations of the laser energy on target were performed to ensure that the absorbed energy density could be accurately determined and any drift in the laser energy was compensated. The largest source of error here was due to the spatial inhomogeneity of the excitation laser, the details of which are described in the supporting materials.\nRegarding the sample thickness, measurements provided in the supplementary information show an estimated variation of ±4.7 nm, which is larger than the nominal difference of 33 nm vs 35 nm. As such any variation in this regard we believe is already considered. The thickness variation did contribute to the error in the absorbed energy density, which was calculated assuming different film thicknesses. However, this was error was propagated as described in the supplementary information.\n2) There is no temperature measurement. There is an estimate of possible electron and ion temperatures using a two-temperature model. Several of the input parameters seem not trustworthy. For instance, a quick check in the NIST database shows quite some variation of the heat capacity of solid aluminum with temperature but the authors state that they used a single value for all cases. The absorption of light in matter is a notoriously difficult problem, in particular in warm dense matter, so the manuscript should acknowledge this difficulty and at the very least give a citation for the simple formula (S10) and note in which limit it is valid.\nResponse: We thank the reviewer for raising this point. Indeed, the lack of a direct temperature measurement is something we are keenly aware of as an issue. We do mention briefly in the concluding remarks that the combination of the THz probing with XFEL diagnostics, such as inelastic X-ray scattering would provide a means to a direct temperature measurement, either X-ray Thomson scattering to study the electron temperature, or milli-electron-volt inelastic X-ray scattering to measure the ion temperature. We have added some references of possible ideas.\nRegarding the choice of heat capacity, the value chosen is the constant volume specific heat capacity, C , in the Dulong-Petit limit. We chose this because the v sample is isochorically heated, and so initially the temperature changes occur under constant volume conditions. Additionally, the ion temperature very quickly exceeds the Debye temperature. We also checked the results of the TTM calculations with the heat capacity on the NIST database, which is the constant pressure heat capacity, C , and using the fully temperature p dependent C under the Debye model, C (T), and find a 5% difference in the v v temperature. We have included plots of both cases in the supplementary materials as well as in this response for clarity.\nFigure S10. Comparison of constant pressure and constant volume heat capacities on TTM calculations. Results of calculations of the electron temperature are shown in a; corresponding plots of the ion temperature are shown in b. Solid lines show the results of TTM calculations using the constant pressure heat capacity, and dashed lines are for the constant volume heat capacity.\nWe have added the following text to the supplementary information, page 42:\nComparison of effect of constant pressure vs constant volume heat capacities on TTM\nFigure S10 shows the results of TTM calculations using the constant volume and constant pressure heat capacities. The constant pressure heat capacity is taken from the NIST database and is parametrized by the ion temperature using the Shomate equation:\nC = A+BT +CT2 +DT3 +ET 2. (S12)\np i i i i→\nThe constants A, B, C, D, and E are specified for the solid and liquid phase. The constant volume heat capacity is investigated in two cases: 1) the value from the Dulong-Petit limit, C 0, and 2) the temperature dependent C using v v the Debye model. In this case,\nC (T ) = 3C0 dx. (S13)\nv i v T (ex 1)2\n! D\" #0 →\nHere T = 428 K is the Debye temperature of Al. As expected, the higher D constant pressure heat capacity results in a lower final temperature of the system. The absolute difference is largest for the highest energy density simulated and corresponds to an ~5% difference.\nThe other values used in the TTM are drawn from computations and are widely used: the electron heat capacity is computed using density functional theory by Lin et al. [Physical Review B 77, 075133 (2008)]. Notably, this value is also in good agreement with analytical models [Ashcroft, N. W. & Mermin, N. D. Solid State Physics. (Brooks/Cole, Belmont, CA, 1976)]. We also added the following text to the description of the Two-Temperature modeling in the methods section, page 18:\nAt each timestep of the simulation, we calculated the electron specific heat capacity according to C (T ) = γT , with γ = 91.2 J/(m3 K2)61, while the ion e e e specific heat capacity was held constant at C = 2.5×106 J/(m3 K), the i constant volume heat capacity in the Dulong-Petit limit, consistent with the expected isochoric heating. Calculations using the constant pressure heat capacity, which differed by ~5%, can be found in the supplementary materials. The electron-temperature dependent electron-ion coupling parameter, G (T ), is taken from DFT calculations by Lin et al.61 and ei e Simoni et al.62, with different values used for the solid and liquid\nThe formula in equation S10, which is also given in the main text of the manuscript, used to compute the absorbed energy density is defined as the amount of absorbed laser energy by the ambient material over the mass of the material. It can be converted to the form in the manuscript by considering:\nAbsorbed energy Incident Energy Absorption Coe!cient\nω = = →\nE\nmass mass density sample volume\n→\nIncident Fluence sample area Absorption Coe!cient\n= → →\nmass density sample area sample thickness\n→ →\nIncident Fluence Absorption Coe!cient\n= →\nmass density sample thickness\n→\nThis is valid for our measurements and is true provided in initial density and the absorption are known. We agree that the absorption of light by WDM remains challenging but this complication is not relevant here since the absorption takes place before the sample has a chance to be driven to WDM conditions. As our fluences/intensities are relatively modest (I < 1013 W/cm2, F < 600 mJ/cm2), we assume a flat absorption constant. Work by Genieys et al. [Appl Phys A 126 263 (2020)] has shown the reflectivity of Al remains unchanged until about 1 J/cm2 fluence, supporting this assumption. We have added to the sentence in the main text where the energy density is defined, and included the reference justifying the use of a constant absorption coefficient, page 4:\nThe absorbed energy density, ρ , is defined as the amount of absorbed laser E energy by the ambient material normalized to the mass of the material and is given by ρ = FA/d ρ , where A = 13.5% is the absorption ratio, d is the E 0 M 0 film thickness, and ρ is the mass density of Al. For the fluences used here, M A has been shown to remain unchanged46.\n3) The comparison with DFT results opens up a number of questions as well. The citation given is Witte et al. (PoP 2018). In it, all conductivity calculations seem to be for equilibrium aluminum. It is however not clear to me, which results (from THz spectroscopy) in this manuscript stem from equilibrated runs (featuring a large delta t) or stem from non- equilibrium runs (delta t < 2ps or so). In particular, since the main figure (Fig. 6) has the abscissa labeled as \"electron temperature\". Similar problems exist in Fig. 5, where the ion temperature dependence is shown. What is the respective electron temperature?\nResponse: We thank the reviewer for raising this concern and apologize for the unintended ambiguity. The data in Figs. 5 and 6 are drawn from the same measurements and are plotted against different temperatures. For the data in Fig. 5, the corresponding electron temperatures are plotted in Fig. 6. In Fig. 6, for the data labeled as solid the electrons and ions are not in equilibrium; the liquid data are essentially in thermal equilibrium. The intended point of comparison is between the THz data at 5 ps and the theory curves, since these most closely match the theoretical conditions. We have modified the main text to highlight this, page 12:\n“In evaluating the DFT simulations with THz experiments on liquid WD-Al, the intended focus is on the ∆t = 5 ps data, plotted as red squares, as the electrons and ions are expected to be in near thermal equilibrium; hence, these experimental conditions most closely match those of the simulations.”\nBelow we use the results of TTM calculations to plot the data with the x- and y- axes are the ion and electron temperature, respectively:\nThe blue colored band is for the case where ∆t = 1 ps, and is equivalent to using the values between the blue dotted lines shown in main figure 5 for the x-axis, and the blue solid lines shown in main figure 5 for the y-axis; the red band is the equivalent for ∆t = 5ps. The bands are bounded by different G from Lin et al. ei [Physical Review B 77, 075133 (2008)], and Simoini et al. [Physical Review Letters 122, 205001 (2019)]. The dots on the plot are the T and T values where the e i measurements were made; the labels are the conductivities. We have added this figure to the supplementary materials as well.\nReviewer #3 (Remarks to the Author):\nThe authors combine ultrafast electron diffraction and terahertz spectroscopy to monitor the evolution of electrical conductivity of an aluminum foil melted by an ultrashort laser pulse. The strong point here is the use of the ultrafast electron diffraction which unambiguously provides the evolution of the structure of the photoexcited material. It is thus possible to accurately correlate the retrieved terahertz conductivity with instantaneous structure of the aluminum.\nThe manuscript is comprehensible and well written; I appreciate the faithful and careful documentation of all results and methods as well as the scientifically sound analysis. Altogether, since the manuscript well advances the understanding of behavior of photoexcited materials at the shortest time scales, I recommend it for publication in Nature Communications.\nResponse: We thank the reviewer for this positive endorsement and recommendation of our publication! Below we address the minor technical issues and remarks raised.\nThe following technical issues are minor (they do not have the potential to compromise the conclusions):\n* My major objection concerns the frequency resolution displayed in Figs. 3c,d. The employed time window indicated in Fig. 3b is ~1.6 ps; this means that the frequency resolution is 1/1.6 ps = 600 GHz, i.e., there are exactly two experimental points in Figs. 3c,d. All other points are artificial since they appear as a result of zero-padding (or more precisely, almost zero): in turn, they represent just an interpolation of the “experimental” points and do not carry out any information supported by experiments. I do not think that using a longer time-window would change the conclusions on THz conductivity (and I’m aware that it would lead to artifacts due to the ultrafast evolution of the system, [J. Opt. Soc. Am. B 17, 327 (2000)]). However, absolutely smooth data from any THz system (the more from a single-shot system which is generally noisier; in fact, the noise is clearly seen in Fig. 3b) is a highly visible hint that some suspicious operations were done with the experimental data. Please reconsider the analysis or just include a sentence on what is the real realistic frequency resolution.\nResponse: We thank the reviewer for raising this concern. We agree that applying the filter does reduce the frequency resolution of the method, and comply with the suggestion to add a sentence which states the frequency resolution of the measurement in the main text of the manuscript, page 7:\n“When taking the Fourier transform, a hyper Gaussian window function, exp[-((t-t )/𝜏)20], was used55 to remove contributions from signals outside the 0 main cycles of the THz pulse; doing so limits the frequency resolution of the measurement to ~0.6 THz, with additional points in the frequency domain arising from the length of the original time window.”\n* Although THz conductivity is confirmed to be close to the DC conductivity in this particular system, one should be aware that charge localization may occur in certain systems [e.g. in thin gold layers, Phys. Rev. B 76, 125408 (2007) or in most semiconductor nanoparticles], i.e., THz frequencies may not be generally low enough to permit a direct connection to DC conductivity. A certain reservation should thus be expressed when discussing equality between THz and DC conductivity in the manuscript.\nResponse: We thank the reviewer for raising this point. We are aware of this concern and furthermore we believe that this discrepancy remains relevant on short timescales if the sample is made up of grains or crystallites with transverse dimensions shorter than the electron mean free path. A detailed description of this goes beyond the scope of the current work, but we plan to investigate this in the future. We have strengthened our original reference to this issue in the manuscript when we discuss the conductivity values that we observed, page 8:\nWe note that the lower conductivity is expected in thin films, as there is increased electron scattering by grain boundaries and surfaces57–60; as such, care should be taken to ensure when extracting the conductivity of low- dimensional films.\n* When presenting THz conductivity, consider whether it would not be more relevant to show (and easier to discuss) the sheet conductivity instead – the sheet conductivity should be insensitive to plasma expansion.\nResponse: We thank the reviewer for this suggestion. While it is true that the sheet conductivity is insensitive to plasma expansion, the quantity is extrinsic which makes it difficult to compare with simulations which calculate the intrinsic conductivities. Additionally, most HED applications use the conductivity as an input; because of this and to avoid the complication of still having to normalize by the thickness to compare with the simulated values, we instead show the conductivity.\nMinor remarks related to the presentation (these are formal only and it makes no sense to publish them along with the rest of the report)\n* Consider transfer of the equations governing the two-temperature model into the main text (or at least refer to the equations explicitly) – now it is really difficult to decode what the two-temperature model is about. It seems to me that the G_ei value is temperature-dependent: if this is the case, it would be helpful to indicate this dependence explicitly, like in the case of the thermal capacitances.\nResponse: We thank the reviewer for this suggestion. We have moved the two- temperature model equations into the main text, along with the definitions for the various terms in the equations. Additionally, we have explicitly labeled the electron temperature dependence of Gei. The equations now read:\nωT\ne\nC (T ) = G (T )(T T )+S(t)\ne e ei e e i\nωt → →\nωT\ni\nC = G (T )(T T )\ni ei e e i\nωt →\n* Although excitation conditions are described in the methods section, it would be helpful if the authors presented their opinion to which extent the conditions are compatible for ultrafast electron diffraction and THz measurements and point out if there are some possible aspects in which they may differ.\nResponse: We thank the reviewer for this suggestion. We have added some discussion of this in the methods section:\nLike the UED experiments, it was important to ensure uniform illumination. Consequently, the focal point of the lens was placed after the target plane. Using an upstream aperture to select a uniform portion of the pump beam would have significantly reduced incident fluence on target, as the long THz wavelength prevents tight focusing.\nBoth experiments used relatively modest fluences (< 600 mJ/cm2). As it is possible to focus the electron beam to ~200 um, it would be possible to achieve higher fluences for the UED measurements. Additionally, we expect better spatial for the UED measurements. Using an upstream aperture to select a uniform portion of the pump beam would significantly reduce achievable energy densities for THz measurements because of the large THz focal spot size.\n* Fig. 3b-d: Please label the curves for individual powers unambiguously – the current continuous color scale (with obscure number like 4.58) permits a readout with ±~1 MJ/kg which translates to >100% error for the lowest fluences. I also expect that the curves with lowest transmission/highest conductivity corresponds to unexcited sample, but the corresponding color implies 2±1 MJ/kg – it would be good to clarify this.\nResponse: We thank the reviewer for this suggestion. We have made this change, and added an extra label for the unheated film.\n* I have nothing against showing absolute THz field intensities in kV/cm, nevertheless, what matters is the relative change (text at lines 158 - 159 could be slightly rephrased in this respect) – this avoids the need to specify whether this is the field just after the sample, inside the sample or at some particular position in front of/inside the sensor.\nResponse: We thank the reviewer for this suggestion. We have adjusted the sentence to shift the focus more to the sample transmission, with the THz field values as parentheticals to offer supporting information, page 7:\nUpon heating, the magnitude of the THz signal increases, |E(t)|, indicating an increase the sample transmission. For the highest incident fluence, we observe an almost 4× increase in the transmission due to the changes in the electronic properties of Al (|E(t)| ~ 0.16 kV/cm for the unheated film compared to almost 0.60 kV/cm)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have succeeded in convincing this reviewer that they have provided direct measurement of the DC conductivity of warm dense aluminum on ultrafast timescales.\nThe data presented therefore show that it is now possible to obtain definitive benchmark data for theoretical methods.\nIn particular, the authors have provided clarification regarding what is required in order for a full theoretical understanding of their experimental data. This is now greatly helped by the authors’ responses to all reviewers’ questions regarding experimental details such as sample temperatures, thicknesses, heat capacity models employed, DFT calculation references, The comments by the experimental reviewers were mostly directed at technical issues that were clarified by the authors, thus making theoretical work much easier since less assumptions are now required.\nIn summary, this manuscript will encourage the theory community to take new and further looks at the properties of warm dense elements since it is important for a large number of subjects ranging from astrophysical to fusion physics.", "reviewer_2": "The authors have answered my questions satisfactorily. I do not have any more comments or questions. I can recommend publication."}, "author_response": "Response: We deeply appreciate this positive assessment and perspective of the impact of our work and thank the reviewer for recommending publication!\nResponse: We appreciate this positive review of our work and thank the reviewer for their recommendation!\nResponse: We appreciate the strong support for our manuscript, and thank the reviewer for their recommendation to publish!"}]

Decision letter