# High-pressure dysprosium carbides containing carbon dimers, trimers, chains, and ribbons

Fariia Iasmin Akbar, Alena Aslandukova, Yuqing Yin, Andrey Aslandukov, Dominique Laniel, Elena Bykova, Maxim Bykov, Eleanor Lawrence Bright, Jonathan Wright, Davide Comboni, et al.

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Fariia Iasmin Akbar, Alena Aslandukova, Yuqing Yin, Andrey Aslandukov, Dominique Laniel, et al. High-pressure dysprosium carbides containing carbon dimers, trimers, chains, and ribbons. Carbon, 2024, 228, pp.119374. 10.1016/j.carbon.2024.119374. hal-05095787

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# High-pressure dysprosium carbides containing carbon dimers, trimers, chains, and ribbons

Fariia Iasmin Akbar<sup>a,b,*</sup>, Alena Aslandukova<sup>a</sup>, Yuqing Yin<sup>b,c</sup>, Andrey Aslandukov<sup>a,b</sup>, Dominique Laniel<sup>d</sup>, Elena Bykova<sup>e</sup>, Maxim Bykov<sup>f</sup>, Eleanor Lawrence Bright<sup>g</sup>, Jonathan Wright<sup>g</sup>, Davide Comboni<sup>g</sup>, Michael Hanfland<sup>g</sup>, Natalia Dubrovinskaia<sup>b,c</sup>, Leonid Dubrovinsky<sup>a,**</sup>

## ARTICLE INFO

### Keywords:

- High-pressure

- Diamond anvil cell

- Rare-earth carbides

- Carbides

- Rare-earth elements

- Dysprosium carbide

## ABSTRACT

Exploring the chemistry of materials at high pressure leads to discoveries of previously unknown compounds and phenomena. Here chemical reactions between elemental dysprosium and carbon were studied in laser-heated diamond anvil cells at pressures up to 95 GPa and temperatures of ~2800 K. *In situ* single-crystal synchrotron X-ray diffraction (SCXRD) analysis of the reaction products revealed the formation of novel dysprosium carbides, γ-DyC₂, Dy₅C₉, and γ-Dy₄C₅, along with previously reported Dy₃C₂ and Dy₄C₃. The crystal structures of γ-DyC₂ and Dy₅C₉ feature infinite flat carbon polyacene-like ribbons and *cis*-polyacetylene-type chains, respectively. In the structure of γ-Dy₄C₅, carbon atoms form dimers and non-linear trimers. Dy₃C₂ contains ethanide-type carbon dumbbells, and Dy₄C₃ is methanide featuring single carbon atoms. Density functional theory calculations reproduce well the crystal structures of high-pressure dysprosium carbides and reveal conjugated π-electron systems in novel infinite carbon polyanions. This work demonstrates that complex carbon homoatomic species previously unknown in organic chemistry can be synthesized at high pressures by direct reactions of carbon with metals.

## 1. Introduction

The chemistry of binary compounds drastically changes and expands at high pressures. Recent striking examples include the synthesis of novel polyhalides [1,2], polynitrides [3–10], polyborides [11,12], polyhydrides [13–15], whose structures possess various homonuclear chemical species, such as dimers, trimers, pentagonal and hexagonal rings, polymeric chains, atomic layers and 3D networks previously unobserved in binary compounds at ambient pressures. The observed tendency of forming homonuclear chemical bonds is well understood by theory [16]: in very general terms, with increasing pressure the distances between atoms decrease and the value of resonance integrals

increases, resulting in a larger gap between bonding and antibonding states and stronger bonds. Modern *a priori* structure prediction techniques, however, are not precise enough to provide an exhaustive list of possible chemical compositions and phases at given thermodynamic conditions. Even moderate-pressure experiments report phases and crystal structures never predicted or previously considered [1,4,5,8,13].

Homonuclear chemical bonding between carbon atoms is at the heart of organic chemistry and the number of arrangements for carbons connections (or catenation) known at ambient pressure is enormous. Strikingly, recent experimental observations indicate that carbon polyanions in metal carbides at high pressures may be unknown and even not foreseen at ambient pressure. For example, novel yttrium carbide γ-Dy₄C₅

<footnote><sup>a</sup> Bavarian Research Institute of Experimental Geochemistry and Geophysics (BGI), University of Bayreuth, Univesitaetsstrasse 30, 95440, Bayreuth, Germany</footnote>

<footnote><sup>b</sup> Material Physics and Technology at Extreme Conditions, Laboratory of Crystallography, University of Bayreuth, Univesitaetsstrasse 30, 95440, Bayreuth, Germany</footnote>

<footnote><sup>c</sup> Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83, Linköping, Sweden</footnote>

<footnote><sup>d</sup> Centre for Science at Extreme Conditions and School of Physics and Astronomy, University of Edinburgh, EH9 3FD, Edinburgh, United Kingdom</footnote>

<footnote><sup>e</sup> Institute of Geosciences, Goethe University Frankfurt, Altenhöferallee 1, 60438, Frankfurt Am Main, Germany</footnote>

<footnote><sup>f</sup> Institute of Inorganic and Analytical Chemistry, Goethe University Frankfurt, Max-von-Laue-Straße 7, 60438, Frankfurt Am Main, Germany</footnote>

<footnote><sup>g</sup> European Synchrotron Radiation Facility, CS 40220, 38043 Grenoble Cedex 9, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>** Corresponding author.</footnote>

<footnote>E-mail addresses: Fariia.Akbar@uni-bayreuth.de (F.I. Akbar), Leonid.Dubrovinsky@uni-bayreuth.de (L. Dubrovinsky).</footnote>
at 44–51 GPa possesses significantly bent carbon trimers [C₃] isoelec-
tronic with ozone O₃ [17]. Polycarbides of different compositions have
been claimed in the Li–C system: due to the polymerization of Li₂C₂
acetelenide upon compression [18] or formation of LiC₂ grapheneide and
Li₃C₄ polyacenide [19] after laser heating. Beyond these, theoretical
calculations for metal carbides at high pressure predict carbon poly-
anions with unusual geometry and chemical bonding [20–22], for
example, in numerous polycarbides in the Ca–C, Y–C and Th–C systems
[20,22–27]. Some of these novel carbon compounds are expected to
have quite exotic structures, in which carbon atoms are polymerized to
form infinite quasi-1D ribbons built of fused six-membered rings, and
the synthesis of high-pressure CaC₂ phase (HP–CaC₂) with
polyacene-like ribbons was recently reported [28]. All these predictions
and discoveries call for systematic experimental studies of high-pressure
carbides.

Recently, we reported the synthesis of Dy₃C₂ and Dy₄C₃ at high temperatures and pressures below ~58 GPa [29]. Here we have extended the pressure range up to 95 GPa and report the results of our systematic high-pressure high-temperature (HPHT) investigations of the Dy–C system using laser-heated diamond anvil cells (DACs).

## 2. Materials and methods

Nomenclature

For the sake of clarity for readers, in this paper we use simplified nomenclature of anionic units in the crystal structures of Dy–C compounds. In particular, if not specified otherwise we will mean completely deprotonated species. So, by using the term ethanide, we will mean ethanehexaide C26-. Polyacetylide stands for fully deprotonated polyacetylene poly-[ethene-1,2-diyl]. Polyacenide is derived by full deprotonation of polyacene or poly(buta-1,3,-diene-1,4:3,2-tetrayl).

2.1. Sample preparation

In our experiments, we used BX90-type diamond anvil cells with a
large X-ray aperture [30]. As anvils, we employed Boehler–Almax-type
diamonds with culets diameter of 80/120 µm. Rhenium gaskets with an
initial thickness of 200 µm were indented to ~16/22 µm and a hole of
~35/55 µm laser-drilled in the center of the indentation. The dyspros-
ium (99.9 % purity, Merc Inc.) flakes were loaded between one of the
diamonds and a layer of dry sodium chloride (99.999 % purity, Chem-
PUR) that played a role of a thermal insulator and pressure transmitting
medium; diamond anvils were used as a carbon source. Samples were
compressed to the desired pressure and laser-heated up to ~2800 K.
Laser heating of the samples was carried out using an *in house*
double-sided YAG laser (1064 nm wavelength) heating setup [31].
Thermal emission spectra from the heated area were collected via Iso-
Plane SCT 320 spectrometer with a 1024 × 2560 PI-MAX 4 camera [31].
The pressure was determined using the NaCl equation of states (EoS)
[32,33].

## 2.2. X-ray diffraction

The reaction products were analyzed by single-crystal X-ray
diffraction measurements at the two synchrotron beamlines: ID11 of
ESRF, Grenoble, France (λ = 0.2846 Å, beam size ~ 0.75 × 0.75 µm²);
ID15B of ESRF, Grenoble, France (λ = 0.4100 Å, beam size ~1.5 × 2
µm²). Powder XRD images were collected upon continuous rotation of
the sample in a range ±1° around the vertical ω axis at ESRF. During
single-crystal collection, the cell was rotated from -38° to +38° with
narrow 0.5° steps. Creating maps with XDI software [34] helps to
visualize the phase distribution within the pressure chamber and to
locate areas where the step-scans should be performed. The CrysAlis<sup>Pro</sup>
software package [35] was used for the analysis of the single-crystal
XRD data (peak hunting, indexing, data integration, frame scaling,

and absorption correction). To calibrate an instrument model in the
CrysAlis<sup>Pro</sup> software, i.e. the sample-to-detector distance, detector’s
origin, offsets of the goniometer angles, and rotation of both the X-ray
beam and detector around the instrument axis, we used a single crystal
of orthoenstatite [(Mg<sub>1.93</sub>Fe<sub>0.06</sub>)(Si<sub>1.93</sub>,Al<sub>0.06</sub>)O<sub>6</sub>, Pbca space group, a =
8.8117(2) Å, b = 5.18320(10) Å, and c = 18.2391(3) Å]. The DAFi
program was used for the search of reflection’s groups belonging to the
individual single crystal domains [36]. Using the OLEX2 software
package [37], the structures were solved with the ShelXT structure so-
lution program [38] using intrinsic phasing and refined with the ShelXL
[39] refinement package using least-squares minimization. Crystal
structure visualization was made with the VESTA software [40]. The
equations of state were obtained by fitting the pressure-volume depen-
dence data using the EoSFit7-GUI [41].

Accurate determination of the positions of light carbon atoms in the
presence of heavy dysprosium atoms is challenging. In order to improve
both precision and accuracy in determining C–C distances, 12 to 27
domains with R<sub>int</sub> < 5 % and R<sub>1</sub> < 8 % for each of the studied carbides
were identified, their structures were refined, and C–C distances for each
corresponding domain were averaged. That allows reduce the standard
error of C–C distances to smaller than 0.009 Å, and the corresponding
data presented in the main text of the paper.

2.3. Theoretical calculations

The properties of the synthesized compounds were determined
through the first-principles calculations using the framework of density
functional theory (DFT) as implemented in the VASP (Vienna ab initio
simulation package) code [42]. To expand the electronic wave function
in plane waves we used the Projector-Augmented-Wave (PAW) method
[43]. The Generalized Gradient Approximation (GGA) functional was
used for calculating the exchange-correlation energies, as proposed by
Perdew–Burke–Ernzerhof (PBE) [44]. The PAW potentials with the
following valence configurations of 5s5p6s5d for Dy (“Dy_3”) and 2s2p
for C (“C”) were used to describe the interaction between the core and
the valence electrons in frozen f-electrons approximation for Dy [42].
Convergence tests with a threshold of 1 meV per atom in energy led to an
energy cutoff for the plane wave expansion of 750 eV for all phases and a
Monkhorst-Pack [45] k-point grid of 15 × 7 × 6 for γ-DyC<sub>2</sub>, 4 × 4 × 6 for
Dy<sub>5</sub>C<sub>9</sub>, 3 × 5 × 4 for γ-Dy<sub>4</sub>C<sub>5</sub>, 6 × 6 × 11 for Dy<sub>3</sub>C<sub>2</sub>, and 4 × 4 × 4 for
Dy<sub>4</sub>C<sub>3</sub>. Computations were performed for eight volumes that cover the
pressure range of 0–100 GPa. Harmonic lattice dynamics calculations
were performed with the PHONOPY software [46] using the finite
displacement method for 3 × 3 × 3 (γ-DyC<sub>2</sub>), 2 × 2 × 3 (Dy<sub>5</sub>C<sub>9</sub>), 2 × 2 ×
2 (γ-Dy<sub>4</sub>C<sub>5</sub>), 2 × 2 × 3 (Dy<sub>3</sub>C<sub>2</sub>) and 2 × 2 × 2 (Dy<sub>4</sub>C<sub>3</sub>) supercells with
respectively adjusted k-points. The tetrahedron method was used for
Brillouin zone integrations, employing a mesh of 16 × 16 × 16 k-points
for γ-DyC<sub>2</sub>, 8 × 8 × 12 k-points for Dy<sub>5</sub>C<sub>9</sub>, 10 × 10 × 8 k-points for
γ-Dy<sub>4</sub>C<sub>5</sub>, 12 × 12 × 22 k-points for Dy<sub>3</sub>C<sub>2</sub> and 8 × 8 × 8 k-points for
Dy<sub>4</sub>C<sub>3</sub> [47,48]. The integrated values of the crystal orbital bond index
(ICOBI) [49] and Mulliken charges were calculated using LOBSTER
v4.1.0 software [50]. The charge distribution in the ionic approximation
based on a generalization of Pauling’s concept of bond strength [51] was
made using CHARDI2015 [52]. In our calculations, temperature,
configurational entropy, and the entropy contribution due to lattice
vibrations were neglected.

3. Results and discussion

A summary of our experiments is presented in Table S1. High-
pressure reactions between dysprosium and carbon in DACs at pres-
sures of ~70 and ~95 GPa and temperatures of ~2800 K resulted in the
formation of five dysprosium carbides, namely γ-DyC<sub>2</sub>, Dy<sub>5</sub>C<sub>9</sub>, γ-Dy<sub>4</sub>C<sub>5</sub>,
Dy<sub>3</sub>C<sub>2</sub>, and Dy<sub>4</sub>C<sub>3</sub> (Table S2). The first three are novel polycarbides. The
full crystallographic datasets for all compounds were deposited to the
Cambridge Structural Database (CSD), the Cambridge Crystallographic
Data Centre (CCDC) [53], and their deposition numbers are provided in the corresponding Tables in Supporting Information. The reconstructed reciprocal lattice planes obtained upon processing the synchrotron single-crystal XRD for all synthesized phases are shown in Fig. S1.

The DyC₂ phase synthesized at ~70 GPa has a crystal structure different from that of the two previously known DyC₂ polymorphs, α-DyC₂ (CaC₂-type structure, tetragonal space group I4/mmm) [54] and β-DyC₂ (KCN-type structure, cubic space group Fm-3m) [55,56]. Naturally, we named it γ-DyC₂. The γ-DyC₂ phase (space group Immm, #71, Z = 4, Tables S2 and S3) is isostructural to the previously observed HP-CaC₂ (Immm) [28] and the predicted YC₂ (Immm) [23]. Its orthorhombic unit cell contains four dysprosium atoms occupying the Wyckoff site 4i and eight carbon atoms occupying two distinct crystallographic positions: C1 – 4h and C2 – 4g (Fig. 1, Table S3).

In the structure of γ-DyC₂ (Fig. 1, Tables S2 and S3), dysprosium atoms form distorted, slightly buckled closed packed layers in the (0 0 1) plane (Fig. 1b). Dysprosium atoms have 12 carbon neighbors arranged in a hexagonal prism with unequal bases (Fig. 1c). Carbon atoms lie in the (0 0 2) plane in the middle between planes of Dy atoms (Fig. 1). They are arranged in polymerized hexagonal rings – flat one-dimensional ribbons propagating along the [1 0 0] direction (Fig. 1). Thus, the carbon atoms form exotic [C₄] one-dimensionally infinite polyanions.

The structure of the Dy₅C₅ carbide (space group P4/mnc, #128, Z = 2) (Tables S2 and S4) is unprecedented, for we found neither any structural analogue, nor a relevant prediction from theoretical methods. Its unit cell contains two crystallographically distinct dysprosium atoms (Dy1 and Dy2 occupying 8h and 2b Wyckoff positions, respectively) and two types of carbon atoms (C1 and C2 occupying 16i and 2a Wyckoff positions, respectively). Dysprosium atoms form slightly distorted square layers in the (0 0 1) plane at a distance of 1/2c; each layer is rotated relatively to an adjacent one by ~40° (Fig. 2b and S2). The distinct dysprosium atoms have different environments – Dy1 atoms are in irregular polyhedra with 11 vertices, while Dy2 atoms are located in bicapped square prisms (Fig. 2c and d). The carbon atoms C2 have isolated positions in the structure like the carbon in methanides (Fig. 2). The C1 carbon atoms form zig-zag chains with four atoms per repeating element (Fig. 2). Chains are propagating along the [0 0 1] direction.

The dysprosium carbide isostructural with the recently discovered γ-Y₄C₅ phase [17] is referred to as γ-Dy₄C₅ (orthorhombic space group Cmce, #64, Z = 8) (Tables S2 and S5, Fig. S3).

Its structure described previously [17] contains carbon dimers [C₂] (dumbbells) and non-linear trimers [C₃] (the angle ∠(C2–C4–C2) is of ~132°, Fig. S3). Two other carbides observed in this work (Table S2), dysprosium (III) methanide Dy₄C₃, and dysprosium (II) ethanide Dy₃C₂, have been synthesized at ~55 GPa in our previous study [29]. For completeness, their structures at ~70 GPa (Dy₃C₂) and ~95 GPa (Dy₄C₃) are shown in Figs. S4 and S5 and structural data are given in Tables S6 and S7.

The analysis of the chemical nature of the novel dysprosium carbides containing polyanions requires a complex approach, especially because *ab initio* methods (see *Methods* section for details) do not always accurately reproduce experimental observations for the *f*-element Dy [29,57, 58]. Therefore, we first conduct the crystal-chemical analysis and then complement the analysis with theoretical calculations for isostructural (experimentally observed or hypothetical) compounds of analogous elements devoid of *f*-electrons (like Ca or Y).

In γ-DyC₂, the carbon ribbons are flat. The C–C distances in the hexagons, refined at 70 GPa, are equal to 1.451(9) Å and 1.508(5) Å for the C2–C2 and C1–C2 bonds, respectively. Two angles in the hexagon are found to be of 122.5(7)°, while the other four angles are of 118.8(3)° (Fig. 1d). The angles are close to 120°, and C–C distances are quite similar indicating the *sp*² hybridisation of carbon atoms. In the ideal polyacene chain with ordered single- and double-order carbon-carbon bonds [59], the distribution of shorter and longer C–C contacts is incompatible with the experimental observations for γ-DyC₂ (and for HP-CaC₂ as well [28]). Similar C–C bond lengths within the hexagons in γ-DyC₂ (and HP-CaC₂) suggest a conjugated π-electron system in polyacene-like ribbons. The integrated crystal orbital bond indexes (ICOBi) [49] calculated for γ-DyC₂, HP-CaC₂, and isostructural YC₂ (Table S8) vary from ~0.7 to ~1.0. These values, being sufficiently close to 1, confirm strong covalent bonding between carbon atoms. The ELF maps for γ-DyC₂, HP-CaC₂, YC₂ (Fig. S6) also demonstrate localization of electrons on C–C bonds in hexagons, and ionic bonding between Dy and C units. Calculated electron density of states (eDOSes) (Fig. S6) for these phases show the presence of carbon *p*-electrons at the Fermi level that agrees with the formation of the conducting conjugated π-electron system.

According to the experimental data, the length of common edges of hexagons in γ-DyC₂ and HP-CaC₂ is slightly (by ~0.05 Å) shorter than

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Crystal structure of DyC₂ at 66(3) GPa. Dy1 atoms are blue, C1 atoms are red, C2 atoms are brown; grey thin lines outline the unit cell. (a) A general view of the crystal structure. (b) The crystal structure viewed along the *c*-axis. (c) The coordination environment of the dysprosium atom. (d) A view of the carbon ribbon. (e) The electron localization function calculated in the (1 0 0) (top) and (0 0 1) plane containing carbon ribbons (bottom). (A colour version of this figure can be viewed online.)</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Crystal structure of Dy₅C₉ at 68(3) GPa. All Dy atoms are blue, C atoms forming chains are brown, discrete C atoms are red; grey thin lines outline the unit cell. (a) A general view of the crystal structure. (b) Arrangement of parallel planes comprised of Dy atoms rotated relative to each other by ~40°. (c) and (d) The coordination environment of the Dy1 and Dy2 atoms. (e) A carbon chain geometry (top) and the electron localization function calculated in the plane containing the carbon chain (bottom). (f) The electron localization function calculated in the (0 0 2) plane. (A colour version of this figure can be viewed online.)</figcaption></figure>

the length of non-shared edges. This agrees with the calculations for all three isostructural high-pressure carbides (DyC₂, CaC₂, and YC₂), which result in the shorter common edges by ~0.02–0.03 Å. This observation is in contrast with what is known for crystals of organic polyaromatic hydrocarbons (naphthalene, anthracene, tetracene, pentacene, pyrene [59–61]) where common edges of hexagon carbon rings are slightly longer (up to ~0.1 Å) than those with carbon connected to hydrogen. A similar effect is known, for example, for acetylene and acetylides [62]. The reason could be the strong electrostatic interaction between carbon and the metal ion: indeed, at ambient pressure, the difference in electronegativity of C and H is ~0.3 eV/e [63–65]. At 70 GPa, this difference significantly increases, reaching ~5.9 eV/e for C and Dy, ~17 eV/e – for C and Ca, and ~15.7 eV/e – for C and Y [64,65]. According to calculations and available experimental data, the average C–C distances in hexagons of γ-DyC₂ and YC₂ (~1.49 Å) are slightly longer than in HP-CaC₂ (~1.45 Å). This qualitatively aligns with some additional electron transfer from cation to carbon ribbons in rare earth carbides in comparison to HP-CaC₂: there is a small decrease of the C–C bond order and higher formal charges on Dy and Y in rare earth carbides than in HP-CaC₂.

According to SCXRD for Dy₅C₉ at ~70 GPa, the C–C distance in the carbon chains is of ~1.46 Å and the C–C–C angle is of ~132°. All atoms of the chain lie in the same plane, as the torsion angle is zero (Fig. 2e). The crystal structure is well reproduced by theoretical calculations (Table S4). According to the theory, both in Dy₅C₉ and in a hypothetical Y₅C₉ with the same structure, the lengths of C–C bonds are very similar (alternating lengths along the chain are of ~1.42 Å and ~1.44 Å at 70 GPa), and the angle between carbon atoms is of ~135°. That implies cis-polyacetylene type deprotonated carbon chains with predominantly sp²-hybridized carbon and conjugated π-electron system. The ICOBI calculated for carbon atoms in the chains of Y₅C₉ and Dy₅C₉, are equal to ~0.7 and ~1.1 respectively. The presence of carbon’s p-electrons at the Fermi level, as in the calculated eDOS (Fig. S7, Table S8) supports this assignment. Strong covalent bonding between carbon atoms in the chains, and the ionic nature of Dy and isolated C are evident from ELF

maps (Fig. 2 and S7). At the same time, C–C distances are significantly longer than in cis-polyacetylene – 1.37 Å [66], and the bond angle is significantly larger than 120° expected for sp²-hybridized carbon. Remarkably, in γ-Dy₄C₅ and γ-Y₄C₅ the geometry of [C₃] groups (in γ-Dy₄C₅, C–C contact is of ~1.43 Å, bond angle ~132°) is very similar to the geometry of elements forming cis-polyacetylene-type chains. Experimental and theoretical data on γ-Y₄C₅ [17] suggest that the bond order of carbon atoms in [C₃] is ~1.31. Thus, carbon chains discovered in Dy₅C₉ can be considered as polymerized in cis conformation [C₃] groups found in γ-Dy₄C₅ and γ-Y₄C₅.

As discussed above, the analysis of ELF maps (Figs. 1 and 2, and S3-S7) suggests the ionic bonding between dysprosium and carbon species in γ-DyC₂, Dy₅C₉, γ-Dy₄C₅, and Dy₃C₂. that allows considering them, along with Dy₄C₃, as salt carbides [62]. In fact, the synthesis of all these compounds from elements also makes them similar to salt carbides, so that one can assign a formal charge to a carbon atom or to carbon groups.

There is a correlation between the formal charges of nitrogen and carbon dimers and their lengths. Recently we demonstrated this using as an example [N₂]⁻ dimers in the high-pressure Na₃(N₂)₄, Ca₃(N₂)₄, Sr₃(N₂)₄, and Ba(N₂)₃ compounds [3] and, for [C₂]⁻ dimers, in yttrium carbide γ-Y₄C₅ containing both [C₂] and nonlinear [C₃] units [17]. In the structures of the carbon compounds synthesized in this study, there are more complex carbon arrangements, including infinite ribbons of fused carbon rings, that imposed a necessity to introduce the effective bond order into consideration to comprehend the relationship between the C–C bond lengths and formal charges of individual carbon atoms or their groups. We have collected the literature data for a number of binary metal-carbon compounds featuring [C₂] or [C₃] units [17,29,54,67–84] with reported C–C interatomic distances. For each of these compounds, we calculated the effective bond order (EBO) between the carbon atoms using two equations introduced below. For a compound containing both isolated “methanide”-type carbon atoms and carbon dimers, with the chemical formula Meₙ(C₂)mCₖ (m is number of isolated “methanide”-type carbon atoms, and *m* is the number of carbon dimers), the effective bond
order (EBO) between carbon atoms is calculated as follows:

$$EBO = \frac{(2 \times 4m - nZ_{Me} + 4k)}{2m}, \qquad (1)$$

where “4” is the number of valence electrons in a free carbon atom, $Z_{Me}$ is the formal charge of the cation. Carbons’ formal charge in dimers is $x = 4-EBO$. For example, for barium acetylide, $BaC_2$, according to this formula, the effective bond order of carbon atoms is ‘3’, and formal charge is ‘-1’. Similarly, for $Me_n(C_3)_mC_k$ carbides containing only $[C_3]$ units, we used the following equation:

$$EBO = \frac{(3 \times 4m - nZ_{Me} + 4k)}{4m}. \qquad (2)$$

The results of our calculations for the binary metal-carbon compounds found in the literature [17,29,54,67–84] are presented in the graphical form in Fig. S8, showing a correlation between the C–C distances in $[C_2]$ and $[C_3]$ units, $d(C–C)$, and the bond order in the pairs of C atoms (green and blue dots). The data points shown by green and blue dots in Fig. S8 were fitted by the linear equation: $EBO = 10.81 – 6.57 \times d$ (C–C) (dashed line in Fig. S8).

With the linear equation obtained above, we determined the bond order for C–C bonds for all dysprosium carbides studied here, using the values of $d(C–C)$ obtained experimentally from SCXRD. For $Dy_3C_2$ we found the bond order ~1 in dimers, implying it to be an ethanide featuring $[C_2]^{6-}$ units, which is in perfect agreement with the reasoning presented in Ref. [29]. In dimers of $\gamma$-Dy$_4C_5$, the EBO is of ~1.71 (implying the formal charge of the dimers as $[C_2]^{4.6}$), whereas in trimers the EBO is of ~1.40, suggesting $[C_3]^{6.4-}$ units. For $Dy_5C_9$, all C–C bonds in the chains are similar, and the EBO is of ~1.22 (thus one-dimensionally infinite polyanions $1/\infty \times [(C_4)^{6.2-}]$). For $\gamma$-DyC$_2$, possessing ribbons of fused hexagon rings, the estimated C–C EBO is of ~1.27 for shared hexagon edges and of ~0.90 for unshared ones. This gives a formal charge of ~ -3.0 for two carbons per formula unit. Remarkably, the same reasoning implies a formal charge of ~ -2.4 per two carbon atoms in HP-CaC$_2$ [28], that is about 1e less than expected for $Ca^{2+}$ vs $Dy^{3+}$. Knowing the EBO (thus formal charges) of the carbon polyanions, we can estimate the oxidation state of dysprosium atoms in the studied compounds: dysprosium (III) in $\gamma$-DyC$_2$, $Dy_5C_9$, $\gamma$-Dy$_4C_5$, and $Dy_4C_3$, and dysprosium (II) in $Dy_3C_2$.

The Mulliken charges of dysprosium in the carbides at ~70 GPa were found to be 1.66 for Dy1 in $\gamma$-DyC$_2$; 1.88 for Dy1 and Dy2 in $Dy_5C_9$; 1.69 for Dy1, 1.75 for Dy2 and 1.55 for Dy3 in $\gamma$-Dy$_4C_5$; 1.01 for Dy1 and 1.13 for Dy2 in $Dy_3C_2$; and 1.64 for Dy in $Dy_4C_3$ (Table S9). The values for $\gamma$-DyC$_2$, $Dy_5C_9$, $\gamma$-Dy$_4C_5$, and $Dy_4C_3$ are consistent with the Mulliken charges reported for other dysprosium(III)-containing compounds [29, 85, 86]. The lower Mulliken charges of dysprosium in $Dy_3C_2$ are in agreement with the assessment of the cation in this compound as $Dy^{2+}$ [29]. The model Y–C system shows similar results (Table S10).

Experimental studies of the equations of state of dysprosium carbides were beyond the scope of this work, however the pressure-volume relations were simulated and described with the second order Birch-Murnaghan equation of state (Table S11). The calculated bulk moduli decrease with an increase in dysprosium content and a simultaneous decrease in the number of homoatomic bonds in carbon anions, progressing from carbon ribbons, chains, trimers, and dumbbells to discrete carbon atoms, as follows: $K_0(\gamma$-DyC$_2$) = 189(2) GPa > $K_0(Dy_5C_9$) = 164 (2) GPa > $K_0(\gamma$-Dy$_4C_5$) = 123.8(8) GPa > $K_0(Dy_3C_2$) = 96.7(9) GPa > $K_0(Dy_4C_3$) = 91.6(1.2) GPa (Table S11, Fig. S9). Although the dysprosium content in $Dy_4C_3$ is slightly less than in $Dy_3C_2$, its bulk modulus is relatively lower. This is likely due to the different nature of carbon anions: $Dy_3C_2$ contains carbon dumbbells, whereas the structure of $Dy_4C_3$ features discrete carbon atoms. There is also an inverse correlation between the compressibility and the ratio $R = \langle d \rangle^3/(Z_{Dy} * Z_{C})$ (<d> is the average Dy–C distance in the first coordination sphere; $Z_{Dy}$ and $Z_{C}$ are the formal charges of dysprosium and carbon) (Fig. S10a). For the

studied compounds, with the increase of the R ratio the compressibility decreases (Fig. S10a), whereas for ionic and ionic-covalent compounds the correlation is known to be direct [3,87]. This indicates an unusual compression mechanism for the studied dysprosium carbides that is in line with the observation of the almost linear correlation between compressibility and the average bond order in carbon pairs (Fig. S10b).

Computations of phonon dispersion relations in harmonic approximation at 0 K show that at 70 GPa $\gamma$-DyC$_2$, $Dy_5C_9$, $\gamma$-Dy$_4C_5$ and $Dy_4C_3$ exhibit dynamical stability (Fig. S11), while $Dy_3C_2$ shows tiny instability at 70 GPa (Fig. S11g) similar to that found previously for this compound at the synthesis pressure of 55 GPa [29]. $\gamma$-DyC$_2$, $\gamma$-Dy$_4C_5$ and $Dy_4C_3$ are predicted to be dynamically stable at 0 GPa (Figs. S11b, f, j) and thus potentially quenchable to ambient conditions, but one should consider this prediction with caution. Notably, $\gamma$-Dy$_4C_5$ discovered earlier was found dynamically unstable at ambient conditions [17] in contrast to the isostructural $\gamma$-Dy$_4C_5$ from the current study (Fig. S11f).

The calculated convex hull diagram at 0 K and 70 GPa (Fig. 3) shows that according to the theory, $\gamma$-DyC$_2$, $\gamma$-Dy$_4C_5$ and $Dy_3C_2$ are thermodynamically stable at the synthesis pressure. Two dysprosium carbides, $Dy_5C_9$ and $Dy_4C_3$, lie above the convex hull by 70 and 16 meV per atom, respectively (Fig. 3). These differences in energy turn out to be smaller than $k_B T$ (241 meV) at the synthesis temperature (~2800 K), thus suggesting that these phases might be metastable at ambient temperature and high pressures.

## 4. Conclusion

To summarize, novel dysprosium carbides $\gamma$-DyC$_2$, $Dy_5C_9$, $\gamma$-Dy$_4C_5$, $Dy_3C_2$, and $Dy_4C_3$ were synthesized by the direct reaction of metallic dysprosium and carbon originated from diamond anvils upon laser heating to ~2800 K at ~70 GPa. Their crystal structures were studied in situ. The most notable features of the new carbides are polyacene-like ribbons in $\gamma$-DyC$_2$ and cis-polyacetylene-type chains in $Dy_5C_9$. This study demonstrates the drastic effect of high pressures on the chemistry of the Dy–C system with the general tendency to formation of carbides with complex polyanions.

<figure><img src="image_6.png" /><figcaption>Fig. 3. Calculated convex hull diagram constructed for the Dy–C binary system at 70 GPa. Only known Dy–C phases are included. Dashed lines indicate the convex hull; carbides synthesized in this work are given in red; $Dy_2C_3$ (shown in blue) has been previously observed both at extreme [29] and ambient conditions [84]; carbides shown in green have been reported at ambient pressure ($Dy_2C$ [88], $Dy_4C_5$ [73,89,90], $Dy_3C_2$ [91,92], and $DyC_2$ [54]). (A colour version of this figure can be viewed online.)</figcaption></figure>
## Data availability

The details of the crystal structure investigations may be obtained from The Cambridge Crystallographic Data Centre (CCDC, https://www.ccdc.cam.ac.uk/structures/) by quoting the deposition numbers 2311063, 2311064, 2311065, 2311066, and 2311067. All other information is available in the main text or the Supplementary materials.

## CRediT authorship contribution statement

**Fariia Jasmin Akbar:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Conceptualization. **Alena Aslandukova:** Methodology, Investigation. **Yuqing Yin:** Investigation. **Andrey Aslandukov:** Writing – review & editing, Investigation. **Dominique Laniel:** Writing – review & editing, Investigation. **Elena Bykova:** Investigation. **Maxim Bykov:** Writing – review & editing, Investigation. **Eleanor Lawrence Bright:** Resources, Methodology. **Jonathan Wright:** Resources, Methodology. **Davide Comboni:** Resources, Methodology. **Michael Hanland:** Resources, Methodology. **Natalia Dubrovinskaia:** Writing – review & editing, Supervision, Conceptualization. **Leonid Dubrovinsky:** Writing – review & editing, Validation, Supervision, Methodology, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The authors acknowledge the European Synchrotron Radiation Facility (ESRF) for the provision of beamtime at the ID15b and ID11 beamlines. Computations were performed at the Leibniz Supercomputing Center of the Bavarian Academy of Sciences and the Humanities, and the research center for scientific computing at the University of Bayreuth. F.I.A., N.D., and L.D. thank Gerd Steinle-Neumann (Bavarian Research Institute of Experimental Geochemistry and Geophysics (BGI), University of Bayreuth, 95440 Bayreuth, Germany) for the valuable discussions and advice regarding the theoretical calculations. N.D. and L.D. thank the Deutsche Forschungsgemeinschaft (DFG projects DU 393–9/2, DU 393–13/1; DU 945/15–1; LA 4916/1-1) for financial support. N.D. also thanks the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFO-Mat-LiU No. 2009 00971). M. B. acknowledges the support of Deutsche Forschungsgemeinschaft (DFG Emmy-Noether Program project BY112/2–1). Co-funded by the European Union (ERC, HIPMAT, 101077963). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. D.L. thanks the UKRI Future Leaders Fellowship (MR/ V025724/1) for financial support. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) license to any Author Accepted Manuscript version arising from this submission.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.carbon.2024.119374.

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