Stability

The PSO structure predictions were performed by running CALYPSO ^[21] based on density‐functional theory for unit cells containing up to four formula units TCN₃ and T′ ₂CN₄, employing VASP together with projector augmented waves (PAW), ^[22] the generalized‐gradient approximation (GGA), ^[23] and the Monkhorst‐Pack scheme. ^[24] Twelve low‐energy compounds were studied in detail whose well optimized structural parameters are shown in Table S1. Their stability was confirmed by three criteria, namely, phonon band structure by finite displacements (Phonopy), ^[25] elastic constants, and synthetic route. That is to say that, first, all twelve phases are dynamically stable with no imaginary modes in the phonon bands (Figure S1), even at zero pressure. Second, as regards mechanical stability, the calculated elastic constants all satisfy the corresponding Born elastic stability criteria ^[26] as shown in Table S2. With respect to chemistry, the selected metathetic pathway for the composition TCN₃ was chosen as [Eq. 1]

assuming convenient (i.e., high‐energy Na₃N) starting materials. For the T′ ₂CN₄ composition, the metathetic route was targeted as [Eq. 2]

Fortunately, the negative formation energies shown in Figure S2 indicate that all TCN₃ and T′ ₂CN₄ are exothermic phases, so successful synthesis should be tried. Other routes, better still, might be possible as well. Structurally, all TCN₃ are predicted to crystallize in the hexagonal system with space group P $\bar{6}$ c2 (see Figure 2) whereas, for T′ ₂CN₄, three different types of compounds should be observable, trigonal P $\bar{3}$ m1, tetragonal P $\bar{4}$ 2₁ c, and orthorhombic Cmc2₁ (see Figure 3), all three differing in and indicative of their chemical characters.

Figure 2

a) Crystal structure of TCN₃ guanidinates (T=V, Nb, Ta) with space group P $\bar{6}$ c2 and b) bond‐length histogram of various interactions.

Figure 3

The crystal structures of the a) nitride carbodiimides, b) nitride guanidinates, and c) ortho‐nitrido carbonates of the T′ ₂CN₄ (T′=Ti, Zr, Hf) compositions using the Hf₂CN₄ example. The corresponding bond‐length histograms are given in d), e), and f).

Structural Details

Starting with the TCN₃ types crystallizing in P $\bar{6}$ c2, the compounds VCN₃, NbCN₃, and TaCN₃ adopt the same structure as shown in Figure 2 a), and we select the heavy TaCN₃ for a detailed internal description, see Figure 2 b). First, and in perfect harmony with the sum of Shannon's ionic radii for Ta⁵⁺ and N³⁻, ^[27] there are six equidistant Ta−N bonds of 2.10 Å inside a trans face‐sharing TaN₆ octahedron stacked along c, a bit similar to the wider SrN₆ octahedron (Sr−N=2.67 Å) in SrC(NH)₃ but more twisted. ^[18] As for the important CN₃ core with D _3h symmetry, there are no H atoms as in SrC(NH)₃ while C−N=1.35 Å and N−C−N=120° are practically identical. ^[18] The shortest N−N distance of 2.35 Å is nonbonding, and Ta−Ta=2.8 Å is also far beyond the sum of the effective ionic radii for Ta⁵⁺ (0.64 Å), indicating essentially no N−N and Ta−Ta interactions. That being said, this first predicted transition‐metal guanidinate without H atoms appears as showing a “layered” motif with guanidinate anions and metal cations alternately stacked on top of each other, Figure 2 a).

As for the T′ ₂CN₄ formula, there are three different chemical motifs, corresponding to three different compound classes, and they are shown in Figures 3 a) to 3 c). For reasons of convenience, we select T′=Hf to study their chemical and structural peculiarities, as depicted in Figures 3 d) to 3 f). Similar to the previous discussion of TCN₃, the shortest N−N distance of 2.32 Å and Hf−Hf distance of 2.97 Å are far beyond any significant interaction and will not be discussed any further. In contrast, the Hf−N and C−N bond lengths and connectivities help to separate the three different types of T′ ₂CN₄:

Figure 3 a) depicts the first P $\bar{3}$ m1‐type T′ ₂CN₄ representative which is predicted to crystallize with two spatially separated Hf−N and N−C−N layers, a nitride carbodiimide T′ ₂N₂(NCN). There are three shortest Hf−N=2.09 Å bonds in a plane around each Hf⁴⁺ by nearest N³⁻ neighbors to generate a heterographene‐like Hf−N layer, and two such layers form a double layer. For comparison, the Hf−N distances in Hf(NCN)₂ lie between 2.03 and 2.24 Å. ^[28] The second shortest Hf−N=2.17 Å distance is the one connecting the upper and lower layers, see Figures 3 a) and 3 d). The third Hf−N=2.52 Å distance is essentially nonbonding. For the isolated carbodiimide unit, see Figure 3 d), there are two C−N=1.24 Å double bonds and a linear N=C=N shape, just as expected. In a sense, this crystal structure is topologically reminiscent of the recently reported bismuth oxide carbodiimide, Bi₂O₂(NCN), consisting of layers of [Bi₂O₂]²⁺ and [N=C=N]²⁻. ^[29] In addition, Bi₂O₂NCN has been confirmed suitable as a photoanode for photochemical water oxidation, just like the predicted T′ ₂N₂(NCN) as will be discussed later.

The second Cmc2₁‐type T′ ₂CN₄ candidate, depicted in Figure 3 b), is easily identified as a nitride guanidinate T′ ₂N(CN₃) but crystallizes with a lower symmetry than the previously predicted TCN₃ guanidinates, mirrored by the irregular T′N₇ decahedron and the slightly distorted planar CN₃ ⁵⁻ anion, see Figure 3 e). The Hf−N distances in the HfN₇ decahedron range from 2.09 to 2.34 Å, wider than before. As for the CN₃ ⁵⁻ unit, the C−N bonds arrive at 2×1.35 Å and 1.40 Å, the angles being 115° and 128°, similar to Yb(CN₃H₄)₃. ^[18]

Third, there is the primarily sought T′ ₂CN₄ class of phases crystallizing in P $\bar{4}$ 2₁ c, given in Figure 3 c), the one that has never been observed before. In that crystal structure, Hf is coordinated by six N with 2×2.11 Å and 4×2.23 Å to form a distorted edge‐ and corner‐sharing HfN₆ octahedron. The crucial CN₄ ⁸⁻ unit, corresponding to an ortho‐nitrido carbonate (onc) anion, contains four identical C−N bonds of 1.49 Å, slightly larger than those of the known carbodiimides and guanidinates. Judged by the N−C−N angles of 106° and 117°, the onc unit is almost tetrahedral and conforms to D _2d symmetry, see Figure 3 f). We will further analyze the different chemical behavior to be expected from those differing structures.

Chemical Bonding and Electronic Structure

Because we are mostly interested in the behavior of the complex C/N‐based anions of the T′ ₂CN₄ formula, let us focus on them first. For the nearest intraionic C−N bonds (dubbed C−N1) listed in Figure 4 a), the bond lengths slightly increase as we go from nitride carbodiimides T′ ₂N₂(NCN) to nitride guanidinates T′ ₂N(CN₃) to onc T′ ₂(CN₄), a trivial function of the increasing coordination number of the central C atom; a similar course is not found for the T′−N distances. That is to say that the C−N1 bond slightly weakens but the larger number of C−N bonds upon going from carbodiimide (2) to guanidinate (3) to onc (4) must increase covalency as a whole. On the other side, there is a changing and likewise trivial trend of the T′−N distances, consistent with the changing ionic radii. If we take the onc structure, for example, in which T′ is sixfold coordinated, Shannon's ionic radii for such coordination are 0.61 Å for Ti⁴⁺, 0.72 Å for Zr⁴⁺, and 0.71 Å for Hf⁴⁺, ^[27] and the course runs parallel to what is found theoretically. In fact, the situation is a bit more complex because one finds two types of a T′−N bond, the slightly longer T′−N1 and slightly shorter T′−N2, Figure 3 f). For T′ ₂N₂(NCN) and T′ ₂N(CN₃), the T′−N1 distance is significantly longer than the T′−N2 distance, so the existence of the isolated N³⁻ anion in T′ ₂N₂(NCN) and T′ ₂N(CN₃) is quite obvious even from geometry.

Figure 4

(a) Bond lengths and (b) integrated crystal orbital Hamilton population (ICOHP) of (the sum of) various bonds in T′ ₂N₂(NCN), T′ ₂N(CN₃), and T′ ₂CN₄. For simplicity, we designate C−N1 as the shortest C−N bond whereas T′−N1 is the shortest T′−N bond. The inset in (b) is the electron localization function (ELF) of anionic groups, with an isosurface level of 0.8.

The strengths of the chemical bonds are directly quantified by the Integrated Crystal Orbital Hamilton Populations (ICOHP) as projected by LOBSTER, ^[30] plotted in Figure 4 b). For T′ ₂N₂(NCN) and T′ ₂N(CN₃), the covalent part of the T′−N bonding is not too large, as expected for a metal‐nitrogen bond. As for the much more covalent and stronger intraionic C−N1 bond, the corresponding strength in the entire complex anions increases from carbodiimide to guanidinate to onc, see Figure 4 b), as a function of increasing condensation. This can also be illustrated in color by the socalled electron localization function (ELF) ^[31] of the N=C=N²⁻, CN₃ ⁵⁻ and CN₄ ⁸⁻ units (for T′=Hf), as displayed in Figure 4 b) using an isosurface level of 0.8. In the language of ELF, the clouds around the N atoms indicate “lone‐pair” electrons while the “localized” ones are visible, at least in principle, between C and N. ICOHP directly and numerically quantifies the higher covalency of the CN₄ ⁸⁻ anion.

Questions of relative stability as a function of condensation (or volume) are most easily answered from energy‐volume plots. In order to do so, Vinet equations‐of‐state were calculated for Hf₂N₂(NCN), Hf₂N(CN₃), and Hf₂CN₄ to directly provide that information, see Figure 5 a). ^[32] It immediately turns out that, under standard conditions, the nitride carbodiimide Hf₂N₂(NCN) may be considered the most stable compound (see also convex‐hull discussion below), the chemical ground state, while the nitride guanidinate Hf₂N(CN₃) and the ortho‐nitrido carbonate Hf₂CN₄ are the metastable ones. As the pressure increases, see Figure 5 b), Hf₂N₂(NCN) will transform into Hf₂N(CN₃) at about 26 GPa and, at about 168 GPa, into Hf₂CN₄. On the other hand, it is puzzling that the spatial requirement of the different anions, as given at zero pressure, does not run parallel to the condensed nature of the complex anions. As seen from Figure 5 a), a nitride guanidinate is more densely packed than the onc while onc is still better packed than the nitride carbodiimide, so the mutual fit of the Hf−N bonds or the packing itself also must play a role. Given sufficient pressure, however, the nitride carbodiimide will condense into a nitride guanidinate, and a nitride guanidinate will condense into an ortho‐nitrido carbonate, so the highly covalent bonds eventually determine the effective volume.

Figure 5

(a) Equation‐of‐state (EOS) fits for the total energies per atom as a function of volume for Hf₂N₂(NCN), Hf₂N(CN₃), and Hf₂CN₄; b) enthalpy‐pressure course of the three predicted compounds.

Before discussing other physical properties, two additional chemical questions must be answered, at least tentatively, as regards absolute thermochemical stability and chemical inertness under laboratory conditions. With respect to the first question, we have calculated possible decomposition pathways and theoretical phase diagrams (Figure S3). Confirming the prior arguments, Hf₂N₂(NCN) is the true ground state by −0.24 eV per Hf atom, thermodynamically stable against the convex hull, and should be straightforward to make. Surprisingly enough, Hf(NCN)₂, ^[28] previously made by the Meyer group, turns out as being unstable by +0.08 eV. Its existence and strong inertness, even against water and air, points towards large activation barriers, a common phenomenon involving complex C/N‐containing anions (see below).

As regards Hf₂N(CN₃) and Hf₂(CN₄), they are prone to decay by +0.55 and +0.62 eV per Hf atom, whereas TaCN₃ is unstable by +1.41 eV per Ta atom. That being said, the Ta phase is indeed less likely (but not impossible) whereas the two Hf phases would just need substantial kinetic barriers, larger than in the case of the known Hf(NCN)₂. While we have been unable to carry out the necessary activation‐barrier calculations, partly due to the sheer complexity (far more complex than, say, the graphite‐diamond problem), partly due to our restricted computational facilities, other semiquantitative arguments are very much in favor of such large kinetic barriers. First, any decomposition of a carbodiimide, guanidinate, or ortho‐nitrido carbonate will involve C−N or C=N covalent bond breaking, on the order of 305–615 kJ mol⁻¹ (3.2–6.4 eV), ^[33] and this is unlikely to begin with; this is also what makes diamond being inert for eternities. Second, electrostatic reasoning points into the same direction, as the LOBSTER‐calculated Madelung energies (per Hf atom) arrive at −31.9 eV for Hf₂N₂(NCN), −28.1 eV for Hf₂N(CN₃), and −32.0 eV for Hf₂(CN₄). Not only must these impressive energies be overcome for decomposition, they go back to the highly charged CN₃ ⁵⁻ and CN₄ ⁸⁻ anions and favor such densely packed high‐pressure phases. As a side note, we reiterate that the (even smaller) Madelung energy of Cr₂(NCN)₃ makes this unstable carbodiimide inert even at high temperatures as well as in acidic to alkaline media between pH 1–14. ^[14b]

This brings us to the second question targeting chemical stability which can only be answered experimentally. For example, some carbodimiides such as Hf(NCN)₂, Cr₂(NCN)₃, PbNCN, etc. are perfectly inert in water,[ ^14b , ²⁸ , ³⁴ ] sometimes simply due to surface passivation, others such as Li₂NCN, Na₂NCN, or CaNCN are not.[ ^13a , ³⁵ ] Hence, we truly need the experiment to corroborate the aforementioned signposts as regards activation barriers and to test the surface stability against a typical laboratory atmosphere.

Coming back to physical properties, the metal‐nonmetal interactions (i.e., covalent part of the Hf−N bonds) mirror what goes on between conduction and valence band, so they determine the band gap, as shown in the calculated densities of states (DOS) for Hf₂N₂(NCN), Hf₂N(CN₃) and Hf₂CN₄, given in Figure 6. Clearly, the nitrogen DOS spreads over the entire energy window through interaction with Hf and C, and the difference between an isolated N³⁻ nitride and a C‐bonded nitrogen is easy to spot. As regards the band centers (shown as arrows on the right), the energetic proximity of Hf and N also indicates some covalent interactions in general, as already seen from COHP analysis, and the DOS shapes clearly broaden upon going from a) to b) to c), so the C−N covalency also strengthens and becomes maximized for the most condensed ortho‐nitrodo carbonate, as expected.

Figure 6

Calculated densities of states (DOS) for a) nitride carbodiimide Hf₂N₂(NCN), b) nitride guanidinate Hf₂N(CN₃), and c) ortho‐nitrido carbonate Hf₂CN₄. Orange, black and green lines represent Hf, C and N, respectively. Arrows on the right axis represent the atomic band centers below the Fermi level.

The hybrid functional HSE06 was chosen to arrive at the most reliable band‐gap values that are depicted in Figure 7. ^[36] For each composition and crystal structure, the band gap (grey and orange patterns) obviously increases with an increasing atomic number of the metal atom. It is straightforward to correlate this behavior with the course of the lowering Pauling electronegativities (Ti > Zr > Hf, and V> Nb > Ta), so the metal−N interactions become more ionic (larger band gap) upon going down each group of transition metals. What is puzzling, however, is the fact that the band‐gap character generally differs between the different classes of compounds. Clearly, the pure guanidinates TCN₃ show the smallest band gaps, followed by the nitride guanidinates T′ ₂N(CN₃), then followed by the nitride carbodiimides T′ ₂N₂(NCN) and ortho‐nitrido carbonates T′ ₂(CN₄) which are comparable to each other.

Figure 7

Calculated HSE06 band gaps for stable compounds of the composition TCN₃ (T=V, Nb, Ta) and T′ ₂CN₄ (T′=Ti, Zr, Hf), the latter grouped into nitride carbodiimides, nitride guanidinates, and ortho‐nitrido carbonates. The band‐edge potentials are referenced to the reversible hydrogen electrode. Grey and orange colors indicate the band gap itself whereas green and blue colors represent the conduction band minimum (CBM) and valence band maximum (VBM), respectively.

Photochemical Water Splitting

Likewise, the band‐edge potentials were calculated with respect to the reversible hydrogen electrode (RHE), including the conduction band minimum (CBM) and the valence band maximum (VBM) estimated by the semiconductor electronegativity concept (Supporting Information), ^[37] see again Figure 7. The VBM and CBM of the nitride carbodiimides T′ ₂N₂(NCN) and the ortho‐nitrido carbonates T′ ₂CN₄ bracket the water redox energy range for T′=Zr and Hf, and for these elemental combinations they would achieve water splitting without any external bias voltage (EBV). The CBM of the other materials, however, locate under the H⁺/H₂ energy level, indicating the necessary EBV for water splitting. On the other hand, the band‐gap value limits the type of light being harvested. As for T′=Zr and Hf, T′ ₂N₂(NCN) and T′ ₂CN₄ are the near ultraviolet (from 305 to 367 nm) light‐harvesting materials. As for the materials with a slightly too narrow band gap, thereby requiring a small EBV, that is, nitride carbodiimide Ti₂N₂(NCN) and onc Ti₂CN₄, they are capable to harvest the red (689 nm) and green light (502 nm), respectively. As such, the nitride carbodiimides T′ ₂N₂(NCN) and ortho‐nitrido carbonates T′ ₂CN₄ have been identified as potential candidates for water splitting. Especially for the first ones, the layered structures should turn out as particularly useful for making almost 2D‐like crystals, which would provide additional potential as multifunctional compounds. ^[38]

Nonlinear Optics

Within the ortho‐nitrido carbonates of the T′ ₂(CN₄) compounds, the complex CN₄ ⁸⁻ anion itself comprises a large number of valence electrons (32) that may be shifted around and lead to electronic polarization. In combination with the tetrahedral character of this unit and a significant band gap, those onc compounds are therefore expected as being good candidates for nonlinear optical (NLO) applications, the NLO data calculated within modern polarization and density‐functional perturbation theory (DFPT) as implemented in ABINIT. ^[39] As for the visual spectrum from 400 to 700 nm, Ti₂CN₄ is predicted to show a second harmonic generation (SHG) coefficient of |d ₃₆|=10.35 pm V⁻¹ for the cutoff wavelength of λ=502 nm, which is comparable to the known AgGaS₂ with d ₃₆=13.0 pm V⁻¹ for λ=454 nm. ^[40] As for the ultraviolet‐visible (UV/Vis) regime extending from 100 to 400 nm, T′ ₂CN₄ with T′=Zr and Hf have SHG coefficients of 3.96 and 2.62 pm V⁻¹, with λ=390 and 367 nm, respectively. This is lower than for the known NLO phase BaGaS₂ with 12.6 pm V⁻¹ for λ=346 nm. ^[41] Nonetheless, onc‐type T′ ₂CN₄ with the distinct CN₄ ⁸⁻ anion opens yet another NLO option in this field. In addition, the calculated birefringence shown in Table 1 is larger than 0.1, as derived from their refractive index (Figure S4).

Table 1

Calculated NLO properties of ortho‐nitrido carbonates T′ ₂CN₄

	λ (nm)	SHG coefficients (pm V−1)	Δn (nm)
Ti2CN4	502	d 36=−10.35	0.123
Zr2CN4	390	d 36=−3.96	0.160
Hf2CN4	367	d 36=2.62	0.127

Mechanical Properties

For practical fabrication and device applications, the mechanical properties of the T′ ₂CN₄ composition which are computationally accessible by VASPKIT ^[42] must be studied (Table 2). For being brief, we here focus on the Hf compounds. Despite the sp³‐like carbon within ortho‐nitrido carbonate Hf₂CN₄, the calculated bulk modulus K=212 GPa and Young's modulus Y=250 GPa, indicating its resistant ability to compression and stiffness, are much smaller than diamond's sp³‐carbon (K=435, Y=1120 GPa), simply due to the much softer Hf−N bonds. The compound's universal elastic anisotropy A _u=0.30, however, is close to that of diamond (0.27). In addition, the “softness” of the Hf−N interaction differs, as a function of the different chemical functionality, between the three types of compounds. For example, the negative Cauchy pressure or K/G<1.75 indicate the brittleness of the nitride guanidinate Hf₂N(CN₃). ^[43] The nitride carbodiimide Hf₂N₂(NCN) and the ortho‐nitrido carbonate Hf₂CN₄, however, are expected to show a more ductile behavior. Finally, the values of the minimum lattice thermal conductivity as derived from the elastic constants indicate the potential of all three compounds in terms of good thermal conductivity.

Table 2

Mechanical properties of the Hf₂CN₄ compositions adopting different chemical structures. Calculated bulk modulus K (GPa), shear modulus G (GPa), Young's modulus Y (GPa), universal elastic anisotropy index A _u, Cauchy pressure C (GPa), brittleness <1.75 < toughness K/G, minimum lattice thermal conductivity κ _min (W mK⁻¹).

	K	G	Y	A u	C	K/G	κ min
Hf2N2(NCN)	168	40	111	1.7	130	4.2	1.0
Hf2N(CN3)	182	113	280	0.6	−44	1.6	1.4
Hf2(CN4)	212	96	250	0.3	37	2.2	1.4