Rare-earth vs. heavy metal pigments and their colors from first principles

Abstract

Many inorganic pigments contain heavy metals hazardous to health and environment. Much attention has been devoted to the quest for nontoxic alternatives based on rare-earth elements. However, the computation of colors from first principles is a challenge to electronic structure methods, especially for materials with localized f-orbitals. Here, starting from atomic positions only, we compute the colors of the red pigment cerium fluorosulfide as well as mercury sulfide (classic vermilion). Our methodology uses many-body theories to compute the optical absorption combined with an intermediate length-scale modelization to assess how coloration depends on film thickness, pigment concentration, and granularity. We introduce a quantitative criterion for the performance of a pigment. While for mercury sulfide, this criterion is satisfied because of large transition matrix elements between wide bands, cerium fluorosulfide presents an alternative paradigm: the bright red color is shown to stem from the combined effect of the quasi-2D and the localized nature of states. Our work shows the power of modern computational methods, with implications for the theoretical design of materials with specific optical properties.

Light propagating inside a heterogeneous solid experiences (i) absorption and (ii) scattering. The light that is not absorbed is diffusely reflected and responsible for the perceived color. The visual appearance of a material is, hence, determined by selective absorption of light and sufficient (back)scattering. For a material to be, e.g., a luminous red pigment, two criteria must, thus, be satisfied. First, its absorption edge should be located at the appropriate energy (∼2.1 eV) so that the red component of the visible spectrum is not absorbed. Second, the absorption edge should be sharp so that most other photons within the visible range (green and blue) are absorbed.

The computation of these effects from first principles is faced with three fundamental difficulties. First, in view of the sensitivity of the human eye, the optical gap must be obtained with a precision of at least 100 meV. Conventional electronic structure methods yield a well-documented underestimation of the gap of conventional semiconductors. Second, the localized 4f states, which play a crucial role in optical properties of rare earth-based pigments (1–3), are poorly described by standard density-functional theory (4) or even GW approaches (5). Third, a realistic assessment of the coloration of a pigment must take into account scattering properties depending on concentration, granularity, and film thickness. Ab initio simulations so far have not ventured beyond calculating the optical conductivity of infinite bulk samples (refs. 6 and 7 discuss organic molecules). In this article, we address all these issues and develop a general methodology for the prediction of the color of narrow-band materials.

We investigate cerium fluorosulfide (CeSF), a typical example of the new class of rare-earth pigments (8, 9).

It crystallizes in the layered ThCr₂Si₂ structure sketched in Fig. 1 (which is, incidentally, also the structure of the recently discovered iron-based superconductors) (10). Fig. 1 displays the momentum-dependent many-body spectral function A_k(ω) that encodes the excitation energies associated with the addition or removal of an electron into the many-body ground state (SI Text). The localized Ce-4f states form quasiatomic multiplets that hybridize weakly with the rest of the solid. From Fig. 1 C–E displaying the specific orbital character of each electronic state, it is apparent that the top of the valence band has dominantly S-3p character. The occupied Ce-4f states are located at higher binding energies near the center of the S-3p bands. The lowest unoccupied states are found to be the almost localized empty Ce-4f states, with the dispersing Ce-5d conduction band lying higher in energy.

Fig. 1.

CeSF. (A) Crystal structure. The red, yellow, and gray spheres represent Ce, S, and F, respectively. (B) The total many-body momentum-dependent spectral function along high symmetry lines. (C–E) The corresponding partial spectral functions for (C) Ce-4f, (D) S-3p, and (E) Ce-5d. (F) The total momentum-integrated spectral function (black curve) as well as the partial momentum-integrated spectral functions for Ce-4f (red), S-3p (blue), and Ce-5p (green).

The calculated optical gap Δ = 2.14 eV (578 nm) is consistent with the red color of this material. Furthermore, our calculation reveals that the absorption edge is associated with the optical transition between S-3p and Ce-4f states. This identification is at variance with the standard lore associating the coloration of cerium-based pigments to the intraatomic Ce-4f to Ce-5d transition. Indeed, such a conventional assignment was proposed for CeSF in ref. 8, and X-ray photoemission spectra (XPS) were interpreted along the same lines in ref. 11. In Fig. 2, we show that our results are, nonetheless, fully consistent with XPS measurements (11). We propose a different orbital assignment of the observed spectral peaks: according to our calculations, the peak at −3 eV is caused by contributions from localized Ce-4f states, whereas the shoulder at −1.5 eV is mainly formed by S-3p states that are pushed upward in energy by the hybridization with Ce-4f states. The previously proposed intraatomic scenario can also be simulated theoretically (SI Text), and it is shown in Fig. 2 to be inconsistent with the measured spectra.

Fig. 2.

XPS. The many-body spectral momentum-integrated function (black curve) of CeSF compared with experimental XPS spectra (11) (thick green curve). To simulate experimental resolution, the theoretical spectral function has been convoluted with a Gaussian of full width at half-maximum of 0.3 eV. As a comparison, we show (red curve) the XPS spectrum of the alternative intraatomic scenario previously proposed. (Discussion is in the text and SI Text.) The shaded regions of the corresponding colors are the contribution from Ce-4f states in each case.

In contrast to CeSF, mercury sulfide α-HgS (also known as cinnabar red or vermilion) has been used as a red pigment since antiquity (12). It is a conventional semiconductor with a simple hexagonal structure. Although the additional complication of localized states is not present here, an accurate determination of the gap requires the use of GW calculations (SI Text). The onset of absorption is caused by transitions between broad and strongly hybridizing bands of mainly S-3p and Hg-6s character. Given the qualitatively different nature of the optical transitions involved, HgS and CeSF are good pigments for entirely different reasons, which is explained in detail below.

The absorption properties of these two compounds can be derived from the frequency-dependent optical conductivity, Re σ(ω). Using linear response theory and neglecting excitonic effects (discussion in SI Text), it can be expressed as (13, 14)

where V is the unit-cell volume, α labels the polarization, v_k,α are the optical transition matrix elements, and the Fermi functions f(ω) restrict transitions to take place between occupied and empty states. The absorption is described by the macroscopic absorption coefficient K(ω):

where k(ω) is the imaginary part of the refractive index and c_e is the speed of light.

The calculated optical conductivity and absorption coefficients of CeSF and α-HgS are displayed in Fig. 3. Although marked differences exist between the two compounds on a broad energy range, the magnitude of σ and K near the onset of absorption (Fig. 3 A Inset and B) is actually similar, with and for eV. The physical origin of the spectral weight just above the absorption edge (which in turn, determines K) is, however, fundamentally different for HgS and CeSF. Indeed, we will now show that the large absorption in HgS is caused by the strength of optical transition matrix elements, whereas for CeSF, it is caused by the large density of localized Ce-4f as well as weakly dispersing S-3p states.

Fig. 3.

Optical conductivity and absorption coefficient of CeSF and HgS. (A) Polarization averaged optical conductivity. Inset is a zoom into the energy region of the onset of absorption. (B) Macroscopic absorption coefficient K. The horizontal dashed lines are the quality criterion (3) as defined in the text and computed for /mm.

To substantiate this claim, we note that color is determined by a fairly small frequency range above the absorption edge (see below) and hence, we focus on , with . To disentangle density of states effects from transition probabilities, we compute the ratio between the optical conductivity, Eq. 1, and the joint density of states N(ω) [obtained from replacing the trace in Eq. 1 with ; i.e., by setting the Fermi velocities to unity and considering transitions between the total (traced) spectral weight] at the energy . We find that for HgS, whereas is almost three times smaller for CeSF (a₀ is the Bohr radius and m_e is the mass of the electron), which proves that transition matrix elements dominate for HgS, whereas density of states effects dominates for CeSF. The reason for this difference is the weak dispersion of the Ce-4f states (expected from their localized character) but also, the S-3p states at the top of the valence band in CeSF. Interestingly, the weak dispersion of the S-3p states along the c axis is caused by the quasi-2D environment of the sulfur atoms, which are located within planes parallel to those layers containing cerium atoms (Fig. 1A). As detailed in SI Text, the joint density of states N(ω) and the absorption K(ω) display a discontinuous increase at threshold for a strictly 2D dispersion. In contrast, a 3D parabolic dispersion (mimicking, e.g., the 3p or 6s bands in HgS) yields a less sharp frequency dependence above the absorption edge.

The macroscopic quantities σ and K describe the absorption properties of a perfectly crystalline bulk solid. Pigments are used, however (in, e.g., paints), in the form of small particles embedded into a transparent matrix. In this context, light propagation depends on the morphology of the sample, and a multiple scattering problem has to be solved to determine the diffuse reflectance R(ω) (in contrast to the simpler specular reflectivity) (15, 16). A commonly used (17) approach is the effective-medium description given by Kubelka and Munk (KM) (18, 19). The KM model treats the propagation of light through a homogeneous layer of matter with pigment concentration c, which absorbs light with rate cK(ω) and backscatters with rate β(ω) per unit length. The quantity β effectively contains all of the information on the microscopic structure of the sample. As shown in SI Text, the energy dependence of β can be neglected, and we use a typical value β = 50 mm⁻¹, consistent with the range of values reported in the literature for a wide range of industrial inorganic pigments (17). For a semiinfinite layer, the KM model yields a simple expression for the diffuse reflectance: , with . As expected, for (weak absorption) and for (strong absorption). The formula for a sample of finite thickness is given in SI Text.

We have used the KM model in conjunction with our first-principles absorption K(ω) to compute the diffuse reflectance of CeSF and HgS samples as well as their color (Fig. 4). The latter is obtained (e.g., as XYZ tristimulus values or xy chromaticities) by taking into account the spectral distribution of the light source as well as the sensitivity of the human eye to red, green, and blue light as encoded in the empirical color-matching functions displayed in Fig. 4, Upper (details in SI Text). We considered semiinfinite samples as well as realistic 10-μm layers on a white (R = 1) substrate, consisting of either pure () or diluted () pigments. The resulting diffuse reflectances and colors depicted in Fig. 4 reveal that the pure semiinfinite bulk samples of both materials have bright red colors, whereas the thin films with a pigment concentration of 20% (vol/vol) have a more orange tone, especially for HgS.

Fig. 4.

Diffuse reflectance and the colors. (Upper) Color-matching functions , , and of the human eye [given in the 1931 XYZ standard of the International Commission on Illumination (CIE)] as a function of wavelength λ (SI Text). (Lower) Calculated diffuse reflectances of CeSF and HgS. Shown are the diffuse reflectances of the bulk (R_∞) and a diluted (concentration 20%) thin film of 10 μm on a white substrate (R_10μm,20%) computed for β = 50 mm⁻¹. The corresponding color for each case is displayed in the legend).

We now introduce a simple performance criterion for the usability of such materials as pigments. For the color to be a bright red, the reflectance for should drop sufficiently quickly such that admixtures from the green and blue part of the visible spectrum are suppressed. This selection can be ensured by requiring that . Here, is the characteristic frequency interval over which the human eye distinguishes between the primary colors (corresponding to nm for red light) (Fig. 4, Upper). Because the reflection for arises from a finite absorption, the above requirement can be translated, using the KM model, into the criterion

This threshold is marked in Fig. 3B: for pure materials (), both CeSF and HgS largely satisfy the criterion. Indeed, with switches from to within only a few nanometers (Fig. 4). Criterion (3) can be turned, for a given , into a minimal thickness L_min, above which the pigment has the same apparent color as the bulk, i.e., its color is sufficiently stable. This correspondence is represented in Fig. 5. Using , the minimal thickness of a paint layer consisting of 20% CeSF is 3.2 μm, whereas it is 35 μm for 20% HgS. As a result, the reflectivity of 10-μm layers with 20% diluted HgS violates our quality criterion, which explains the notably different (orange) color of those samples, observed in Fig. 4.

Fig. 5.

Universal quality curves for thin films of CeSF and HgS. The thin films located above the corresponding curve in the (βL_min, c/β) coordinates pass the quality criterion for a good pigment (in the text). The top and right axes correspond to a fixed value of β of 50/mm, allowing for a direct translation of a given concentration into a minimal film thickness.

In conclusion, we presented a framework for the theoretical determination of the color of pigment materials. Our methodology combines first-principles calculations of the frequency-dependent absorption based on state-of-the art many-body theories, with an intermediate length-scale modelization of the scattering properties of realistic samples. We applied this methodology to classic vermillion HgS and the recently discovered rare-earth pigment CeSF. Our results reveal that the red coloration of these two materials has a very different origin at the atomic scale. While it is caused by large optical matrix elements in HgS, the key effect in CeSF is the large density of weakly dispersing states (resulting from both the low dimensionality of the crystal structure and the localized character of Ce-4f states). We also showed that the relevant optical transitions in CeSF are S-3p to Ce-4f interatomic transitions, in contrast to previous belief. Our methodology may lead to rational materials design in which first-principles calculations are used in synergy with solid-state chemistry to create new materials with specific optical properties.

Methods

Combining electronic structure and many-body techniques (20–26), we calculate the electronic structure of rare-earth pigments. Then, we take these approaches further by performing an ab initio calculation of the optical response functions and going all of the way to a prediction of the actual color [expressed, e.g., in red/green/blue (RGB) coordinates] (27, 28) of a sample with a given granularity using an intermediate length-scale modelization (18).

As discussed in the text, the strong Coulomb repulsion between the 4f electrons in CeSF is responsible for the splitting between the empty Ce-4f states at the bottom of the conduction band and the occupied Ce-4f states, which hybridize with the S-3p states to form the valence band. A proper treatment of this strong Coulomb interaction is essential to correctly reproduce the optical gap and electronic structure of this material. Hence, we have used the so-called local density approximation + dynamical mean-field theory (DMFT) approach (21, 22), which combines electronic structure calculations with a many-body treatment of local correlations on the Ce-4f shell in the framework of DMFT (20). The interaction vertex has been computed from first principles using the constraint random phase approximation (29, 30).

Within local density approximation + DMFT, the Ce-4f shell is treated in terms of an effective atom, self-consistently coupled to an environment describing the rest of the solid. From the Hamiltonian of this effective atom (which takes into account crystal-field effects, intraatomic Coulomb interactions, and the spin-orbit coupling), a many-body self-energy is computed and inserted into the Green’s function of the full solid. Self-consistency over the total charge density and the effective atom parameters is implemented (24, 25). Important technical points of our calculational approach are presented in SI Text.