The Formation of Warm Dense Matter: Experimental Evidence for Electronic Bond Hardening in Gold

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Science  20 Feb 2009:
Vol. 323, Issue 5917, pp. 1033-1037
DOI: 10.1126/science.1162697


Under strong optical excitation conditions, it is possible to create highly nonequilibrium states of matter. The nuclear response is determined by the rate of energy transfer from the excited electrons to the nuclei and the instantaneous effect of change in electron distribution on the interatomic potential energy landscape. We used femtosecond electron diffraction to follow the structural evolution of strongly excited gold under these transient electronic conditions. Generally, materials become softer with excitation. In contrast, the rate of disordering of the gold lattice is found to be retarded at excitation levels up to 2.85 megajoules per kilogram with respect to the degree of lattice heating, which is indicative of increased lattice stability at high effective electronic temperatures, a predicted effect that illustrates the strong correlation between electronic structure and lattice bonding.

Photo-induced structural dynamics of crystals have been extensively studied with the use of femtosecond spectroscopic methods based on all-optical pump-probe techniques. Although this approach can probe the electronic response of metals and semiconductors to optical excitation with high time and energy resolution, it only provides indirect information on the nuclear response of a system. In contrast, the more recently developed techniques of femtosecond electron (1) and x-ray (2, 3) diffraction combine atomic-scale spatial and femtosecond temporal resolution.

Modern laser technology allows the creation of highly excited, rarified states of matter. As visible and near-infrared laser light is absorbed by the electrons, intense short-pulsed optical excitation initially forms highly nonequilibrium conditions (that is, very hot electrons within a cool lattice). There has been considerable theoretical interest in this transient state of matter (4, 5), which is the precursor in the formation of high-density plasmas, also referred to as warm dense matter. Time-resolved diffraction experiments enable us to follow the evolution of the atomic structure under these conditions, which is governed by the instantaneous electronically induced change of the lattice potential, as well as the strength of the electron-phonon coupling. In particular, the dynamics of the subsequent order-to-disorder structural transition directly reflect the lattice stability under metastable conditions.

In the case of canonical free-electron metals like aluminum, the lattice stability appears to be mostly unaffected by electronic excitation (4). Subsequent to the optical excitation, electron-phonon scattering heats the lattice and results in thermal melting on the picosecond time scale (1, 6).

In contrast, the excitation of semiconductors weakens the covalent bonding, softens the lattice, and, at the excitation level of ∼10% of the valence electrons, leads to the collapse of the transverse acoustic phonon branch (4, 7, 8), resulting in electronically driven disordering of the lattice. With the use of time-resolved diffraction techniques, this order-to-disorder transition has been experimentally observed for various systems like Si (9) and InSb (2, 3).

In this context, Peierls-distorted crystals like bismuth (i.e., lattices with reduced symmetry stabilized by the ground state electronic structure) can be considered a third class of systems. Impulsive electronic excitation of those crystals shifts the minimum of the potential energy surface and launches coherent, large-amplitude optical phonons (10), equivalent to excited-state wave packet motion in a molecular system. For excitation below the melting threshold, a softening of the interatomic potential with increasing carrier density has been observed (11).

For all of the systems mentioned, electronic excitation either softens or does not substantially affect the interatomic potential. In the case of gold, however, the opposite effect has been predicted in recent theoretical studies based on electronic structure calculations (4, 5): Strong electronic excitation reduces the screening of the attractive internuclear potential, resulting in a steepening of the phonon dispersion and a hardening of the lattice. Observing this effect requires probing the lattice stability on extremely short time scales, faster than equilibration between the excited electrons and the underlying lattice that occurs on picosecond-to-subpicosecond time scales.

Laser-induced melting of gold has also been studied by molecular dynamics simulations, but only for lower excitation levels where electronic effects on lattice stability are not expected (12). Experimental studies on strongly excited gold films mostly employed all-optical pump-probe spectroscopy. Gou and Taylor concluded from time-resolved reflectivity and surface second-harmonic generation experiments that there is an electronic (nonthermal) contribution to the driving force of the solid/liquid phase transition (13). Ng et al. observed a quasi–steady state in the optical response of warm dense gold films that has been attributed to the existence of a nonequilibrium liquid state formed by nonthermal melting (14, 15). In contrast, on the basis of ab-initio simulations, Mazevet et al. attributed the experimentally observed optical response to a superheated state of ordered, electronically stabilized warm dense gold (16); however, this finding has been disputed (17). Hitherto, the structural response of gold has been studied with lower excitation (18) and for nanoparticles by electron diffraction under reversible excitation conditions (19) and by synchrotron-based x-ray diffraction with a time resolution of 100 ps (20).

In the present work, we studied the structural evolution of 20-nm-thick free-standing 111-oriented polycrystalline gold films upon 387-nm laser excitation up to 2.85-MJ/kg excitation levels, using synchronously timed femtosecond electron pulses in diffraction mode to provide atomic resolution of the nuclear response. Solid-density matter with such energy density is often referred to as dense plasma or warm dense matter. The highest excitation level employed corresponds to ∼14 times the energy required to melt the sample starting from room temperature. Under such irreversible conditions, the sample must be moved to a new position for each laser shot. This puts severe constraints on the electron gun design. The experiment requires high time resolution to access the ultrashort time scale under which a nonequilibrium electron distribution persists and sufficient electron number density to provide near single shot structure determinations to record a full time sequence of events within a limited sample area. Our latest femtosecond electron diffraction (FED) setup (Fig. 1) was designed to achieve these conditions by minimizing the propagation time to the sample and thereby reduce the time for coulombic repulsion between electrons to broaden the electron pulses. The electron pulse duration under the experimental conditions (areal density = 4 × 107 cm–2) was directly determined at the sample position to be 400-fs full width at half maximum using grating-enhanced laser pondermotive scattering (21). Another critical element to FED experiments under high excitation conditions is the suppression of strong background signals originating from photoelectrons and ions emitted from the sample, as well as stray pump light. We achieved this by using a thin mesh-supported film of amorphous carbon placed directly in front of the detector as a filter.

Fig. 1.

Schematic depiction of the FED setup. Photoelectrons are emitted via two-photon photoemission from a Au photocathode through back-illumination with a 40-fs visible pulse. The electrons are accelerated over a distance of 6 mm by a static electric field over a potential difference of 55 kV and subsequently collimated by a magnetic lens before arriving at the sample that is placed less than 30 mm from the photocathode. The sample (depicted by a photograph taken after the experiment) is excited by a counter-propagating (angle = 10°) optical pulse (λc = 387 nm, 200 fs) with either flat top–like or Gaussian intensity profile. The flat-top pump profile (diameter = 265 μm) is shown in the image and is obtained by 1:1 imaging of an aperture outside the vacuum chamber by a single lens in 4f configuration. The root mean square intensity variation within the probed area (indicated by a white circle) is below 4%, ensuring homogeneous excitation conditions while minimizing the sample area exposed to a single pump pulse. The two-dimensional diffraction pattern is detected by a pair of multichannel plates and a phosphor screen, and the signal of each electron pulse is captured individually by a charge-coupled device camera. In addition to the actual time-resolved diffraction image taken with pump and probe pulses at given time delay, the static diffraction patterns at every sample position before and after irradiation with the pump pulse are recorded. Between 5 and 60 images were recorded and averaged per time point.

Although the skin depth of 387-nm light in gold is below 10 nm, the range of ballistic electron transport is on the order of 100 nm (22), resulting in a homogeneous, isochoric excitation of the gold films within the duration of the optical pump pulse. Figure 2A shows raw data [radially averaged scattering intensity, I(s); s, scattering vector] of the oriented Au(111) film excited with 470 J/m2 (absorbed fluence) for selected time points. Because of the normal incidence of the electron pulses and the 111-morphology of the sample, several Bragg peaks—particularly the (111) peak—are strongly suppressed compared with the pattern formed by an isotropic polycrystalline sample. This feature was deliberately exploited so that the (111) order, in particular, would not obscure the onset of lattice disordering that is most pronounced near this scattering direction (vide infra). The static diffraction patterns show a substantial background, arising from inelastic (e.g., phonons, single-electron excitations, collective electronic excitations) and multiple elastic scattering. In the case of a high–atomic number material like gold, thermal diffuse scattering (DS) is expected to dominate the inelastically scattered intensity (23). Upon photo-excitation, this background and the intensity of the unscattered electrons (center of the diffraction image) show a dynamic increase and decrease, respectively. The decrease of the zero-order peak reflects the reduction of the inelastic mean free path of the probe electrons. For polycrystalline gold excited modestly above the melting temperature (Tm), we indeed found that the diffraction intensity between the Bragg peaks as well as the zero-order diffraction intensity show the same dynamics compared to the evolution of the lattice temperature as extracted from the Debye-Waller effect. The diffraction pattern of liquid gold (Fig. 2A) exhibits increased DS background over the whole detected scattering range and one single broad diffraction peak near 0.43 Å–1, corresponding to the liquid structure factor. By using 111-oriented gold, this peak (the signature of the product state of the phase transition) arises within a range of the scattering vector s that is essentially free of Bragg peaks in the crystalline state. This allows us to distinguish heating and disordering of the lattice directly from the data obtained in reciprocal space, rather than from real-space data obtained by Fourier transformation (18). For unoriented polycrystalline gold under strong optical excitation, the distinction between a strongly superheated lattice and a disordered state based on the real-space reduced density function is less definitive, especially in the presence of a time-dependent scattering background.

Fig. 2.

Temporal evolution of the diffraction signal. (A) Raw radially averaged diffraction pattern of 111-oriented gold excited with an absorbed fluence of 470 J/m2 at selected time points (solid black lines), the equivalent diffraction signals without excitation (dashed red lines), and the vertically shifted baselines (dotted black lines). a.u., arbitrary units. (B) Temporal evolution of the diffraction signal shows three distinct features in the scattering range from 0.3 to 1.1 Å–1: (i) the rise of the diffraction peak characteristic for a liquid, (ii) the decay of the (220) Bragg peak, and (iii) an overall rise in diffuse background. (C and D) Time traces of the rise of the liquid peak (red squares), the decay of (220) (black circles), and the rise of DS (blue triangles) for absorbed excitation fluences of 470 and 1100 J/m2 fit with convolutions of single exponentials and the system response function (solid lines). In case of the weaker excitation, the onset of the rise of the liquid peak is visibly delayed (Δt) compared with the heating effects. The error margins give the standard error obtained from the fits with equal weighting of the data points.

Figure 2B shows the temporal evolution of the diffraction pattern after excitation with 470 J/m2 plotted as relative diffraction intensity I(s,t)/I(s,t <0), where I(s,t< 0) denotes the diffraction signal averaged over all negative delay points. In the range from 0.3 to 1.1 Å–1, there are three distinct effects: (i) the increase in DS, which spans the whole displayed range but that is most obvious in the Bragg peak-free region between 0.9 and 1.05 Å–1, (ii) the decay of the (220) Bragg peak, and (iii) the rise of the liquid structure factor at ≈0.43 Å–1, which dominates over the DS rise in this range. The temporal evolutions of these three features differ significantly, as shown in Fig. 2, C and D, for two excitation levels. Lacking a reliable model to describe the various effects contributing to the transient signals, we performed monoexponential fits to quantify the dynamics. The increase in DS shows the fastest dynamics. In comparison, the decay of the Bragg peak, being a probe for lattice heating as well as disordering, is slightly but consistently slower. For an absorbed excitation fluence of 470 J/m2 (Fig. 2B), the rise of the liquid structure factor is delayed by 1.4 ± 0.3 ps compared with the DS and (220) dynamics. This retardation of the onset reflects the time required to heat the lattice above Tm, which is an indication of a thermal melting process. Employing a two-temperature model (TTM) that takes the electronic structure of gold specifically into account (24)—in particular, the dependence of the electron heat capacity Ce(Te) and the electron-phonon coupling G(Te) on the electron temperature Te—the lattice temperature Tl is expected to reach the lattice melting temperature Tm within 1 ps at this excitation level. Subsequently, the lattice is expected to rapidly superheat and melt homogeneously on the time scale of ∼1 ps (25); i.e., for Tl ≥ 1.4 Tm, the lattice disorders within a few vibrational periods. In the case of aluminum, this mechanism has been experimentally observed (1, 6, 26). In contrast, the rise time of the liquid peak in gold excited with 470 J/m2 (corresponding to an absorbed energy density of ∼1.2 MJ/kg) is 7 ps, clearly separated from the other two time constants that mostly reflect lattice heating. This behavior is also in contrast to that of InSb excited above the ablation threshold, where the decay of the (111) Bragg peak and the rise of DS were found to be synchronous (27). We would like to emphasize that the appearance of the liquid structure factor in our data reflects the transition from an anisotropic polycrystalline state to a fully isotropic disordered state.

With increasing excitation up to 2.85 MJ/kg, the overall dynamics accelerate, whereas the different time constants remain in their relative order (Fig. 3A), and the retardation Δt of the liquid structure factor drops to 0.3 ps (close to the experimental time resolution). Figure 3A also shows the time t(Tl = 1.4Tm) (gray circles) when Tl reaches 1.4Tm, according to the decay of the (220) peak. The significant difference between these points in time when ultrafast melting is expected (tT = 1.4 Tm) and the time constant τliquid indicates enormous superheating that cannot be explained by classical nucleation theory (12, 25). Recent ab initio calculations predict a hardening of the phonon modes in gold due to increased Te, resulting in an increase of Tm (4, 5, 16). This effect has been qualitatively explained by increased electronic delocalization and decreased screening of the attractive nuclear potential (4, 5). The highest excitation conditions employed in this work correspond to an initial Te of ∼4 eV, which instantaneously increases Tm from 1340 to 2400 K (or, equivalently, the Debye temperature θD from 180 to 250 K), according to ab initio calculations (4). Alternatively, the effect of electronic excitation has been discussed in terms of electronically induced pressure, resulting in a reduction of the concentration of monovacancies (seeds for homogeneous nucleation) (5).

Fig. 3.

Fluence-dependence of heating and disordering. (A) Summary of the time constants shown in Fig. 2 for five different data sets. The gray circles indicate the times when the lattice temperature reaches 1.4Tm. Error bars indicate the standard error. (B) Experimental data of the (220) decay (solid blue circles) compared with two different models for the DWF—DWF[θD(Te)] (solid red lines) and DWF(θD = constant) (dashed black lines)—for different excitation levels. Te and Tl are obtained from a TTM. See text and SOM for details.

In general, the temporal evolution of Tl before melting can be estimated from the decay of the Bragg peaks. By describing electronic lattice hardening by a Te-dependent Debye temperature, however, the Debye-Waller factor (DWF) becomes a function of Te as well [see supporting online material (SOM)]. Figure 3B shows comparisons of the decay of the (220) peak for all employed excitation levels with the simulated decay according to two different models: DWF{Tl(t), θD[Te(t)]} takes the effect of bond hardening through θD[Te(t)] explicitly into account, whereas DWF[Tl(t), θD = constant] ignores the effect of bond hardening. We apply an advanced TTM based on a nonlinear electron heat capacity Ce(Te) and a Te-dependent electron-phonon coupling G(Te) (24), which we solve for Tl(t) and Te(t). The simulations account for the experimental temporal resolution. Although both models are free of any adjustable parameter, the model taking electronic bond hardening into account shows significantly better agreement with the experimental data.

The good agreement between the simulation and the experimental results justifies estimating Tl(t) according to the relation between DWF(Tl, Te) and Tl(t) obtained from the model. At the points in time, when the DWFs drop to 1/e of the room temperature value, the lattice temperatures reach ∼3000, 3600, and 4100 K for the three employed excitation levels, respectively, indicative of strong superheating of the face-centered cubic lattice (see also SOM). The smooth decay of (220) within the first picoseconds, which is temporally well resolved in our experiments, supports the conclusion of a substantial superheating of the crystal. In this stage, isochoric heating of the lattice can be expected to further increase the stabilization of the crystal structure (Clapeyron effect) (28).

The above experiments and SOM provide an atomic-level perspective on the formation of warm dense matter. The observations demonstrate that warm dense gold is formed in a purely thermal process. Previously observed quasi–steady state signatures in the optical response cannot be assigned to instantaneous, nonthermal formation of a liquid-like state. The difference in the rate of substantial superheating and lattice disordering, at the predicted electronic temperatures for increased lattice stability, supports a photo-induced bond hardening mechanism and the concept of a Te-dependent melting temperature. The increased lattice stability will also increase the barrier to nucleation as the electron distribution and nuclear configurations are strongly coupled. Full-scale ab initio molecular dynamics calculations are now needed to understand this phenomenon to which the experiment provides rigorous benchmarks for comparison.

Supporting Online Material

SOM Text

Figs. S1 and S2


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