Introduction

Manufacturability, the need to design products that are easy to manufacture, is one of the highest priorities in materials science. Yet, it remains a challenge in industrial production1. Additionally, high costs and sourcing raw materials further complicates the issue. One possible solution lies in multifunctional compounds, combining in a single system features usually attained in different classes of materials and fabrication processes. In particular, the discovery of optical-mechanical multifunctional materials would enable extending optical applications (e.g., radar and satellite, thermo-optical generators, thermophotovoltaics, magnetic recording) that require optical plasmon-based solutions (e.g., emitters2, modulators and detectors3,4, switches5, broadband absorbers6,7, etc.) in harsh environments (e.g., high temperature and pressure8,9, and chemical or mechanical abrasion10).

Plasmons—electronic collective excitations controlled by external electric fields—are exploited: i. to confine the electromagnetic radiation to nanometric regions shorter than the diffraction limits11,12, ii. to amplify local electric fields13, and iii. to generate “extraordinary” propagating waves in metamaterials and hyperbolic media14,15. Because of these capabilities, plasmonic materials16 are growing their appeal in a wide range of applications including light harvesting17, biosensing18, neuroscience19, telecommunications20, hyperbolic metamaterials21,22, and quantum optics. Other classes of thermal devices, e.g., photothermal emitters23,24 and heat-assisted magnetic recording (HAMR)25, exploit the de-excitation of plasmons and the consequent localized high-temperature gradient.

No single material satisfies the requirements of all the possible applications26. Standard plasmonic metals (Au, Ag, Cu) are ductile, have a reduced thermal stability, are not CMOS (Complementary Metal-Oxide Semiconductor) compatible, and are difficult to pattern into thin films or layers, making them unsuitable for high temperature or mechanical stress-working conditions27. Doped metal-oxides (AZO, ITO)28,29,30 and transition-metal nitrides (TiN and ZrN)31,32,33,34 have been proposed as plasmonic materials in the visible range. The former have tunable plasmonic energies but poor mechanical resistance; while the latter have good mechanical properties, but non-tunable plasmonic energy. Transition metal (TM) carbides, such as TiC, TaC, and WC, are simple systems, easy to grow, CMOS compatible, with high hardness, high thermochemical stability, and have been largely used for coating and anti-corrosion applications. Yet, they have poor optical properties (e.g., reflectance, a summary of the optical and electronic properties of TM-carbides is reported in Supplementary Figs. 1, 2).

Among TM-carbides, only TaC exhibits a plasmonic excitation in the visible range, with all others being non-plasmonic35. This carbide has a plasmonic resonance at E = 2.7 eV, high Vickers hardness (HV = 27.6 GPa), and represents an example of a multifunctional plasmonic-mechanical material. However, in view of the large application range, the need for tunability of optical properties becomes crucial36. In general, the possibility to combine tunable plasmonic properties with thermal stability and/or mechanical resistance would enable high temperature devices, novel optically active refractory, coating or wearing materials for aerospace or satellite applications. While a few examples to combine optical properties and extreme hardness have been achieved in superlattice metamaterials37,38, ours is an attempt to combine all the useful properties in a multifunctional material.

Here, we propose plasmonic transition-metal high-entropy carbides (PHECs) as a class of multifunctional materials able to couple thermal stability, mechanical resistance, and plasmonic properties in the near-IR and visible range. Indeed, they gather the good plasmonic properties of TaC, the superior mechanical properties of TM-carbides, and the tunability given by compositional freedom39. HECs were discovered in 2018 by some of the authors40, through an entropy-forming ability (EFA) descriptor, calculated on a set of randomized structures41. Along with high thermodynamic stability39,40,42,43, HECs showed increased hardness40,42,44,45,46, strength42,47,48, toughness42,48 and resistance to creep49—attributes that are maintained to high temperatures42,47,48,49—along with improved wear resistance39,50 and better resistance to oxidation and corrosion39,51,52,53,54. This work extends the discovery algorithm beyond synthesizability and hardness by employing time-dependent density-functional theory to calculate the dielectric function. The multi-element character of the high-entropy carbides offers a natural testbed to investigate the role of chemical composition and stoichiometry on the tunability of the plasmon excitation and of the plasma energy.

Results

Materials response

The response of materials to incoming electromagnetic waves is described by the complex dielectric function \(\hat{\epsilon }({{{{{{{\bf{q}}}}}}}},\, E)\,\equiv\, ({\epsilon }_{r}\,+\,i{\epsilon }_{i})\), where q is the transferred momentum and E is the energy of the incoming field55. The real part, ϵr, relates to polarization effects and distinguishes between dielectrics (ϵr > 0) and metals (ϵr < 0). The imaginary part, ϵi, is associated with dissipation of energy into the medium. For bulk systems, the electron-energy loss spectra (EELS) is defined as \(-{{{{{{{\rm{Im}}}}}}}}[{\epsilon }_{j}^{-1}({{{{{{{\bf{q}}}}}}}},\, E)]\) and it describes the energy loss due to inelastic electron scattering upon electromagnetic irradiation. In the low-loss energy region (E < 50 eV), the spectrum of EELS provides information about electronic structure, optical, and plasmonic properties. The plasmon excitation energies are the poles of \(\hat{\epsilon }({{{{{{{\bf{q}}}}}}}},\, E)\) and correspond to EELS’s peaks. Experimentally, plasmons are determined by measuring the characteristic electron-energy loss. The evaluation of \(\hat{\epsilon }({{{{{{{\bf{q}}}}}}}},\, E)\) and the corresponding EELS completely characterizes the plasmonic features, and provides a direct pathway to comparison with experimental results32.

Here, we adopt a high-throughput numerical approach that couples thermodynamic methods56, for the evaluation of the atomic structure and stability of PHECs, with first principles approaches based on time-dependent density-functional perturbation theory for the characterization of optical and plasmonic properties. In view of the optical properties of native TaC, we consider, as an example, the case of HfTa4C5, which is a well-established compound with exceptional thermal and mechanical properties.

Plasmonics and disorder modeling

The macroscopic disorder is factorized as the sum of many microscopic configurations, also called tiles, which are determined by the Partial Occupation module (POCC) method41, as implemented within the AFLOW computational materials design framework57. The method generates a factor-group weighted Boltzmann ensemble of Hermite normal form superlattices, having the minimum size compatible with the required stoichiometry and accuracy. Non-unique configurations are removed through symmetry58 and identification59 considerations. The capability of this approach in reproducing the electron and phonon density of states has already been proved60. Here we extend it to optical spectra of disordered systems.

For each optimized POCC j-structure, we calculate \(\hat{{\epsilon }_{j}}({{{{{{{\bf{q}}}}}}}},\, E)\,\equiv\, {({\epsilon }_{r}\,+\,i{\epsilon }_{i})}_{j}\), and the corresponding EELS, \(-{{{{{{{\rm{Im}}}}}}}}[{\epsilon }_{j}^{-1}({{{{{{{\bf{q}}}}}}}},\, E)]\). Unless otherwise specified, we consider only the optical limit at q = 0. Then, the spectra of the disordered material are obtained as an ensemble average on the POCC j-structures (see Methods Section). The poles (peaks) of the averaged dielectric function (EELS) defines the spectral position of the plasmon resonance. As an accuracy testbed, we compare our results to an archetype of plasmonic materials, the disordered solid-solution AuAg and the fcc constituents Au and Ag. The results are summarized in Supplementary Fig. 3. The excellent agreement between simulated and experimental findings confirms the accuracy of the POCC method in calculating optical properties of disordered systems. This is not surprising: POCC is expected to well reproduce all disorder properties which have characteristic lengths larger than the one of the radial distribution function, which is captured by the size cutoff of the POCC cells. On the basis of these results on well-known plasmonic materials, we use this method to predict the plasmonic properties of high-entropy carbides.

The complex dielectric function of HfTa4C5 is shown in Fig. 1a. At low energy (E → 0) the real part of the dielectric function ϵr is negative, while the imaginary part ϵi is positive and diverging, as in free-electron (i.e., Drude-like) metals. Overall, \(\hat{\epsilon }(E)\) maintains a metal-like behavior in the energy range from E ∼ 12.0 eV to Ep = 22.36 eV. Ep is the plasma energy of the system and corresponds to the excitation of a bulk plasmon, i.e., to a collective oscillation of the total free charge density. ϵr becomes definitively positive (i.e., dielectric behavior) for E > Ep. Despite interesting quasiparticle properties, bulk plasmons have excitation energies which are too high for any realistic application. On the contrary, optical properties in near-IR/visible part of the spectrum are more useful. In particular, at E0 = 2.41 eV and upon the application of an external electromagnetic field, the system undergoes an abrupt change of the optical response from metal- to dielectric-like. E0—so-called crossover energy—identifies the energy at which the real part of the dielectric function switches sign ( − → +, see Fig. 1). This behavior results from a balance between intraband transitions, responsible for the negative Drude tail and associated to free-electron density, and the excitation of interband transitions giving positive contributions to ϵr, which turns positive at E0. In the case of HfTa4C5 at the crossover energy, ϵi(E0) also has a minimum (see inset in Fig. 1a). This corresponds to the possibility of exciting a collective electronic oscillation of a reduced (screened) part of the free-electron density, known as a screened plasmon resonance. In this case, E0 corresponds also to the low-energy plasma energy. The condition \(\hat{\epsilon }(E)\,\simeq\, 0\) results in a peak in the electron-energy loss spectra.

Fig. 1: Optical properties of plasmonic HfTa4C5.
figure 1

a Real (ϵr, dark gray) and imaginary (ϵi, light purple) part of the complex dielectric function. E0 and Ep indicate the crossover energy and the plasmon energy, respectively. Inset zooms on low-energy range of the spectrum. b Simulated electron-energy loss spectrum (EELS) calculated by assuming two sets of structures with 10 (black) or 20 (cyan) atoms per POCC cell at T = 0 K. Simulated EELS of TaC crystal (light gray) is included in inset for comparison. c Comparison between simulated (dark gray) and experimental (green) EELS spectra. Inset zooms on low-energy range of the spectrum. Source data are provided as a Source Data file.

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The simulated optical EELS (Fig. 1b) is dominated by the high-energy peak at Ep and by the low-energy peak at E0, which are the fingerprints of the bulk and screened plasmons, respectively. We conclude that HfTa4C5 combines well a plasmonic response in the visible range with superior thermal and mechanical properties, making it the first instance of a multifunctional disordered carbide. The size and the number of the adopted POCC structures do not affect this result. The simulation of disordered HfTa4C5 with a double number of atoms per cell (cyan line in Fig. 1b) reproduces all the spectral features of the original system (black line) with, e.g., a difference in the crossover energy E0 smaller than 0.15 eV.

To corroborate our claims, a set of experimental EELS measurements on the archetype high-temperature carbide HfTa4C5 was performed from room temperature to 1500 K (Methods and Supplementary Information). Figure 1c compares the simulated (dark gray) and the experimental (green) loss functions. The two curves concur (except for minor details), reproducing the main optical features of the material, in particular the screened plasmon with its maximum at E0 = 2.7 eV—very close to the simulated value. In addition, the experimental spectrum has similar features to plasmonic TiN, characterized by the same EELS techniques61. The theoretical spectra also reproduces the low-intensity shoulder (inset, vertical arrow) close to the main E0 peak.

The comparison between HfTa4C5 and crystalline TaC (inset, panel b) indicates that the loss functions of the two systems are very similar, except for the values of E0 and Ep (Table 1). The inclusion of Hf imparts a red-shift of both the main spectroscopic features, suggesting a major role of composition and stoichiometry in the optical properties of the system. To exploit the composition-driven-shift, we try to understand its cause so we can then search for PHECs with (1) plasmonic resonances in the IR-vis range and (2) tunable crossover energy.

Table 1 List of simulated TM-carbides and PHEC structures at varying composition
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Understanding composition-driven red-shift

Let us consider two sets of virtual-crystal compositional studies, focused on the separate effects of stoichiometry and chemical composition. Stoichiometry: we consider HfxTa5−xC5 with x = [0−5], where x = 0 and x = 5 correspond to pure TaC and HfC, respectively. Chemical composition: we choose different transition metals in MTa4C5, where M ∈ {Ti, Zr, Hf, V, Nb, Cr, Mo, W}.

The tests have been calculated starting from a single reference POCC structure of HfTa4C5 and substituting Ta atoms with different transition metals. The results, summarized in Supplementary Figs. 3 and 4, indicate that the increase of Hf content is followed by degradation of the screened plasmon features up to the limiting case of HfC, whose optical properties are included in the Supplementary Information (Supplementary Fig. 5) for comparison. The latter does not exhibit any low-energy peak in the loss function, similar to the case of HfC rocksalt structure (Supplementary Fig. 1).

Substitution of Hf with other transition metals can have different effects on the optical spectrum: i. energy shifts for E0, i.e., red- and blue-shifts for group 4 and 6 metals, respectively; ii. plasmonic resonance deterioration with 3d metals (especially V and Cr), with strongly reduced intensity of the EELS peak; and iii. broadening of the EELS peak with group 6 metals, indicating that Ta, Zr, Hf, Nb, and W seem to be the most promising elements to obtain plasmonic carbide materials; while V, Cr, and Mo can be neglected.

Plasmonic high-entropy carbides

To corroborate the finding, we consider a set of 14 PHECs, whose stability and crystalline phase was recently demonstrated40,44. These are six element alloys: carbon and a combination of five different transition metals among the nine elements of groups 4 (Ti, Zr, Hf), 5 (V, Nb, Ta), and 6 (Cr, Mo, W) of the periodic table. The complete list of simulated compounds is reported in Table 1, where an integer label ({1, ⋯ , 14}) is also assigned to each system for simplicity. All EELS plots exhibit spectroscopic characteristics similar to the HfTa4C5 case of Fig. 1, e.g., the characteristic double peak in the loss function corresponding to the bulk plasmon resonance Ep in the far UV, and a crossover energy E0 in the IR-visible part of the spectrum (see Table 1 for numerical values and Fig. 2 for loss spectra).

Fig. 2: Simulated EELS spectra for selected HECs.
figure 2

E0 and h indicate the crossover energy and the effective intensity of the EELS spectra corresponding to energy E = E0. a 3-HfNbTaTiZrC5. b 6-HfNbTaTiVC5. c 8-NbTaTiVWC5. d 9-MoNbTaVWC5. e 10-CrMoTiVWC5. f 14-HfNbTaTiWC5. Source data are provided as a Source Data file.

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Let us now focus on the low-energy part of the spectrum (E < 5.0 eV) in which we refer to screened plasmons simply as plasmons. Even though a spectroscopic resonance around E0 is evident in all systems, the energy position, the width and the intensity of the peaks vary with composition. Peaks in systems 3-HfNbTaTiZrC5 and 14-HfNbTaTiWC5 are high, sharp and well defined. They reduce to low intensity and broad shoulders in systems 9-MoNbTaVWC5 and 10-CrMoTiVWC5. In systems 6-HfNbTaTiVC5 and 8-NbTaTiVWC5, peaks have intermediate shape.

Figure 3a shows the energy variation of E0 as a function of composition. The systems are grouped into four classes with respect to the TM groups (Table 1). The first class, 3g4 in Fig. 3, maximizes the number of group 4 elements (Ti, Zr, Hf); the second, 3g5, maximizes the group 5 metals (V, Nb, Ta); the third, 3g6, those of group 6 (Cr, Mo, W), with the remaining two elements per system from other groups; the last (2g42g5) has a mixed composition, with two elements from group 4 and 2 from group 5. Figure 3a shows some evident trends. Wide ranges. E0 spans a remarkable range of energies from near-IR to visible. This is quite interesting, as the possibility to control and tune the optical properties of materials by varying the composition is of critical relevance in the development of plasmonics and nanophotonics applications. Intragroup increase. Within each group, E0 increases monotonically with the increase of the atomic number Z of the constituents. For example, systems 1–3 have in common three elements of group 4 (Ti, Zr, Hf) and differ for the remaining two elements from the group 5, V-Nb (1), V-Ta (2), and Nb-Ta (3). Indeed, E0 increases from system-1 to 3 along with the atomic number of the constituents. Intergroup increase. Across different groups, the higher the Z of the elements, the higher becomes the crossover energy. E0 tends to increase moving from 3g4 to 3g6, i.e., increasing number of electrons. For mixed compositions, e.g., system 14-HfNbTaTiWC5, E0 spans the intermediate energy range.

Fig. 3: Trends in the simulated PHECs.
figure 3

Properties of simulated PHECs, ordered in classes according the number of group 4 (g4), group 5 (g5), and group 6 (g6) transition metals included in the compound. Labels are summarized in Table 1. a Crossover energy (E0), with horizontal dashed line indicating the IR/visible range. b effective EELS intensity h. Source data are provided as a Source Data file.

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The heights and the widths of the E0 peaks are also indicative of the properties of the plasmonic response: sharp and intense peaks (small ϵi) are representative of low-loss collective oscillations, i.e., long-living plasmons; broad and small peak shoulders (large ϵi) indicate single-particle dissipation, i.e., no plasmon excitations. For the ideal plasmonic resonance, the entire complex dielectric function should be zero at E0, like in the ideal free-electron metals, corresponding to a loss function being zero everywhere, except for E0, where a sharp peak rises. Here, the non-zero value of the EELS corresponds to imaginary part of the dielectric function ϵi different than zero, indicating dissipative interband optical absorption. If the latter becomes predominant, ϵi does not show any minimum and the corresponding peaks in the EELS spectra cannot be associated to collective plasmon-like oscillations, but rather to single-particle optical transitions and energy loss dissipative processes. Here, E0 would simply represent the threshold between metallic (E < E0) and dielectric (E > E0) behavior.

In order to estimate the intensity of the resonance peak with respect to background, we define an effective EELS intensity, h in Fig. 2, calculated as the convex distance of the peak to the linearized spectra around E0. This simple and phenomenological quantity allows us to separate plasmonic from the dissipative contributions (colored versus white areas in Fig. 2). The effective intensities are shown in Fig. 3b. Systems 3-HfNbTaTiZrC5 and 14-HfNbTaTiWC5 have the highest h and the lowest energy loss dissipation, i.e., good plasmonic properties. Systems 4–8 have mixed characters, with broad but still evident excitation peaks that can be associated to lossy plasmons. For the remaining systems, dissipation is too high and no collective oscillations can be associated to the E0 resonances. From the analysis, we discover five important facts: i. the presence of Ta is necessary but not sufficient for obtaining good plasmonic PHECs; ii. the best results are obtained combining high-Z elements from groups 4 and 5 (Zr, Hf, Nb, and Ta); iii. V, Cr and Mo are detrimental; iv. accurate choice of TMs allows for a modulation of the crossover energy over a large part of the spectrum; and v. the quasi-neutral effect of Ti and W on the final optical properties is also advantageous because it can be exploited to stabilize the PHEC and to change its mechanical or thermal properties.

The compositional variability of the plasmonic properties of PHEC can be interpreted in terms of their electronic structure. Figure 4a shows the electronic density of states, eDOS, of 3-HfNbTaTiZrC5, used as the testbed case. Near the Fermi energy, EF, the eDOS has two main peaks: b1 has a predominant TM character (cyan area), it crosses the Fermi level and is responsible for the Drude character of the dielectric function; b2 has mixed character with contributions from both sp states of carbon (light green line) and the d states of TMs; and it is mainly responsible for the optical excitations from valence to conduction band.

Fig. 4: Optical properties of plasmonic 3-HfNbTaTiZrC5 system.
figure 4

a Total (black area), TM- (cyan area), g4 - (thin blue line), g5 - (thin orange line) and C-projected (thick green line) eDOS. Zero energy reference is set to the Fermi level (EF). b1, b2 and δ indicate the band manifolds and the energy difference discussed in the text. b Simulated EELS for a representative POCC structure at increased transferred momentum q in units of 2π/a0. c Plasmon energy E0(q) dispersion relation. Yellow circles are the calculated crossover energies from b) while the black dashed line is the corresponding quadratic fit. Source data are provided as a Source Data file.

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The difference δ between the b2’s edge and the Fermi level can give an estimate of the crossover energy (Fig. 4a). Only b1 intraband transitions are possible for incoming radiation having energy E < δ, while for E > δ, both interband b2 → b1 and intraband b1 → b1 compete, leading to an increase of the real part of the dielectric function (dielectric screening) that crosses zero at E0.

While this behavior is common to all simulated PHECs, details of the E0 and h are composition-dependent and can be related to the electronic structure of the parent binary TM-carbides (e.g., TiC, VC, NbC, WC, etc.), as shown in Supplementary Fig. 238. The b1 and b2 peaks are observed in all the TM-carbides. Moving from group 4 to group 6 carbides, the higher number of electrons shifts the Fermi level deeper in the b1 band increasing distance δ from b2. This justifies the blue-shift of E0 observed in Fig. 3a. In addition, moving from 3d to 5d carbides, more bands belonging to b1 cross EF. This increases the relative amount of free-electron density and thus the intraband component of the excitation. This finding corroborates the enhanced plasmonic behavior in compounds with higher atomic number. In the specific case of Fig. 4a, the projection of the TM component of the eDOS on group 4 (blue line) and group 5 (orange line) elements highlights the shift of group 5 toward higher binding energies and the corresponding displacement of the b2 band from the Fermi level. This imparts an increase of δ and the blue-shift of the crossover energy. In addition, the high metal-derived component of the eDOS at the Fermi level is responsible for the good plasmonic character of the excitation at E = E0.

In the free-electron approximation, a plasmon-like excitation follows a parabolic dispersion of E0, function of the transferred momentum q. Figure 4b shows the evolution of the crossover peak for different transferred momenta, for the testbed system 3-HfNbTaTiZrC5. Increasing q causes a blue-shift of the E0 energy and a flattening of the EELS, which becomes broader and less intense. The dispersion of the plasma energy E0(q) is almost quadratic (parabolic fit—dashed line—superimposed in Fig. 4c). This behavior is a fingerprint of the collective plasmon-like character of E0. The upward energy dispersion reflects the broadening and the reduction of the intensity of the peak shown in panel b, and it is an indication of the deviation from the purely free-electron character of the system due to dissipative optical transitions.

Since HfTa4C5 and HECs have exceptional mechanical properties, high hardness, and super-high thermal stability, we investigate the thermal evolution of the plasmonic properties. Results for HfTa4C5 and system 3-HfNbTaTiZrC5 are summarized in Fig. 5. Panel a shows the experimental and theoretical EELS of spectra HfTa4C5 at different temperatures, in the range T ∈ [300–1500] K. Increasing temperature produces minor changes to the main plasmonic properties: the spectral feature corresponding to the low-energy plasmon remains clearly recognizable up to T = 1500 K. The plasmon resonance is surprisingly stable even at high temperature, much higher than the standard plasmonic metals' melting points (e.g., Ag and Au). Besides, the increased temperature causes an expected slight reduction of the intensity and a small broadening of the E0 peak, which may be attributed to an increase of interband effects. Simulations (inset, panel a) concur with experiments, representing plasmonic resonance even at high temperatures. It is worth noting that here the temperature is the conformational temperature used to average the POCC ensemble41. Beyond 1500K, it might be arduous to characterize experimental EELS due to equipment limitation. Experimental-computational agreement for HfTa4C5 makes us confident of the existence of plasmonic properties of 3-HfNbTaTiZrC5 even at ultra-high temperature (panel b)—temperature should have a minor effect on the spectral feature, causing a small reduction of the maximum intensity of E0, along with a broadening of the peak due to larger scattering effects. Since plasmons are purely electronic excitations, ionic temperature does not affect the energy position of the plasmonic band (i.e., the crossover energy), except for effects related to thermal expansion of the atomic structure55 (see the Supplementary Information for further details). The results of Fig. 5 indicate the possibility to excite a plasmon resonance even at ultra-high-temperature, where structural resistance of HECs can be advantageous for applications. Temperature could still affect lifetime and de-excitation of the plasmons through dissipative electron-phonon scattering. Our approach does not include time evolution—de-excitation and related energy release—which is beyond the purpose of this work.

Fig. 5: Thermal evolution of plasmonic properties.
figure 5

a Experimental EELS spectra of HfTa4C5 as a function of temperature. Inset reports the corresponding theoretical spectra, evaluated at the same temperatures. b Simulated EELS spectra of 3-HfNbTaTiZrC5 at different temperatures. Source data are provided as a Source Data file.

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Discussion

This article shows an investigation of optical properties of HfTa4C5 and its high-entropy carbide derivatives. Results are promising: for all systems, we identify the existence of a crossover energy in the infrared and visible range, which corresponds to a metal-to-dielectric transition, exploitable for optical and telecommunication applications. The optical response of plasmonic high-entropy carbides (PHECs) can be largely tuned from the near-IR to visible by changing the composition. For a few systems, this corresponds to the excitation of a low-energy screened plasmon. HfTa4C5, HfNbTaTiZrC5 (3), and HfNbTaTiWC5 (14) exhibit the best plasmonic properties (i.e., low-energy loss and high lifetime) among the investigated compounds. Other systems are also useful: their higher dissipative character can be used for exploiting de-excitation of plasmons and the localized release of energy, such as in photothermal applications. The analysis of the electronic structure allows us to identify the chemical elements (e.g., Ta, Hf, Nb, Zr) capable of optimizing the optical properties and to design high-entropy ceramics, whose optical properties are tailored to the specific application needs. The combination of plasmonic activity, high-hardness and extraordinary thermal stability, makes PHECs an example of multifunctional plasmonic-mechanical materials that can be used as optical systems in harsh environments.

Methods

Theory: structure generation

The macroscopic disorder is factorized as the sum of many microscopic ordered j-configurations which are determined by the POCC method41, as implemented within the AFLOW computational materials design framework57,62. The method generates a factor-group weighted Boltzmann ensemble of Hermite normal form superlattices having the minimum size compatible with the required stoichiometry and accuracy. Non-unique supercells are removed through symmetry58 and identification59 considerations. HfTa4C5 is simulated with two sets of structures, 5 unique 10-atom cells and 77 unique 20-atoms cells, used to check the effect of the structure size on the optical properties of the systems. For each investigated PHEC system, POCC generates 49 unique 10-atom structures. The unique structures are relaxed and the total are energies calculated within the AFLOW framework with standard parameters63, using the VASP density-functional theory package64 with the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional65, and a grid of at least 8000 k-points per reciprocal atom.

Theory: optical properties and ensemble integration

For each POCC configuration and for each HEC system, optical properties are evaluated in linear response by using the turboEELS code66 included in the Quantum ESPRESSO distribution67,68. turboEELS implements a Liouville-Lanczos approach to linearized time-dependent density-functional perturbation theory (TD-DFPT)69 for the evaluation of the complex susceptibility χ(q, E), where q is transferred momentum and E is the energy of the incoming radiation. The complex dielectric function \(\hat{\epsilon }(E)\) and the electron-energy loss function are calculated through the relations ϵ(q, E) = 1 + χ(q, E) and \(EELS({{{{{{{\bf{q}}}}}}}},\, E)\,=\,-{{{{{{{\rm{Im}}}}}}}}[{\epsilon }^{-1}({{{{{{{\bf{q}}}}}}}},\, E)]\). The optical response function \(\hat{\epsilon }(E)\,=\,\epsilon ({{{{{{{\bf{0}}}}}}}},\, E)\) is obtained in the limit q → 0. The capability of the present approach in simulating the optical properties of plasmonic materials has been previously established in refs. 32, 36. As a further accuracy test, in the Supplementary Information we include a comparison (Supplementary Fig. 7) between the dielectric function of the reference TaC rocksalt crystal and of 3-HfNbTaTiZrC5 high-entropy carbide, calculated with TD-DFPT and with a single-particle Drude-Lorentz approach70 often adopted for plasmonic studies29,34,38. PBE is also used here. Atomic potentials are described by ultrasoft pseudopotentials of Vanderbilt type71. 3s3p, 4s4p, 5s5p semicore electrons are explicitly included in the valence shell of period 4 (Ti, V, Cr), period 5 (Zr, Nb, Mo), and period 6 (Hf, Ta, W) atomic elements, respectively. Single-particle wavefunctions (charge) are expanded in plane waves up to a kinetic energy cutoff of 28 Ry (280 Ry). A uniform mesh of (12 × 12 × 12) k-points was used to sample the 3D Brillouin zone of each POCC configuration. The optical (L, ϵ) and electronic (eDOS) spectra for the PHECs are obtained along the lines of ref. 41. The ensemble average of the generic physical property \({{{{{{{\mathcal{O}}}}}}}}\) is given by the formula \({{{{{{{\mathcal{O}}}}}}}}\,=\,{\sum }_{j}\,{P}_{j}{{{{{{{{\mathcal{O}}}}}}}}}_{j}\), where \({{{{{{{{\mathcal{O}}}}}}}}}_{j}\) and \({P}_{j}\,=\,{g}_{j}{e}^{-\Delta {H}_{j}/{k}_{{{{{{{{\rm{B}}}}}}}}}T}/{\sum }_{j}\,{g}_{j}{e}^{-\Delta {H}_{j}/{k}_{{{{{{{{\rm{B}}}}}}}}}T}\) are the spectrum and the probability of the j-configuration, respectively. ΔHj and gj are the relative formation enthalpy with respect to the multidimensional convex hull72 and the symmetry degeneracy (factor-group cardinality)58 of the j-configuration, where kB is the Boltzmann constant, and T is the temperature.

Experiments: sample preparation

Tantalum carbide (99.5% purity, Stanford Advanced Materials) and hafnium carbide (99.0% purity, H.C. Starck) were blended at a 4:1 ratio, respectively, in a Nalgene plastic jar with 3/16” WC-Co satellites and ball-milled at a 1:1 ball-to-powder ratio for 24 h. Then, bulk HfTa4C5 was sintered using field assisted sintering technology (FAST), which allows for sintering at high heating rates and short processing times without sintering aids73. A 25 Ton FAST system (FCT Systeme GmbH) at the Penn State Applied Research Laboratory was used for sintering the powders. The powders were sintered in a 40 mm OD graphite die to a final pellet thickness of ∼4 mm. The sintering was carried out in two concurrent steps at temperatures (2100 °C/2400 °C), pressures (55 MPa/40 MPa), and hold times (40 min/30 min) and was completed at a uniform heating rate of 100 °C/min and under vacuum at ∼3 mTorr. Density was measured to be 94.4% of the theoretical value using the Archimedes principle on a precision digital analytical balance (AND HM-202, ±0.1 mg).

Experiments: EELS spectra

Cross-sectional scanning transmission electron microscopy (STEM) and EELS were performed using an aberration corrected ThermoFisher Titan3 G2 60–300 with a monochromator and an X-field emission gun source at a beam energy of 300 keV. The HfTa4C5 specimen used for experimental EELS data acquisition was prepared according to Supplementary Fig. 9. Spectral resolutions of ≤0.2 eV were achieved for all EELS measurements as calculated by the full width at half maximum (FWHM) of the zero loss peak (ZLP). Low-loss EELS spectra were collected from HfTa4C5 as a function of temperature, ranging from room temperature (∼25 °C) to 1200 °C, where the specimen was heated at a uniform rate of 10 °C/s. EELS spectra were collected after stabilizing at each target temperature. A power-law decay function was fitted to the tail of the ZLP in front of the first absorption feature in order to filter out ZLP background signal and resolve the features of interest in the spectra. This was carried out using the Gatan DigitalMicrograph software suite.