Discovery of carbon-vacancy ordering in Nb4AlC3–x under the guidance of first-principles calculations

The conventional wisdom to tailor the properties of binary transition metal carbides by order-disorder phase transformation has been inapplicable for the machinable ternary carbides (MTCs) due to the absence of ordered phase in bulk sample. Here, the presence of an ordered phase with structural carbon vacancies in Nb4AlC3–x (x ≈ 0.3) ternary carbide is predicted by first-principles calculations, and experimentally identified for the first time by transmission electron microscopy and micro-Raman spectroscopy. Consistent with the first-principles prediction, the ordered phase, o-Nb4AlC3, crystalizes in P63/mcm with a = 5.423 Å, c = 24.146 Å. Coexistence of ordered (o-Nb4AlC3) and disordered (Nb4AlC3–x) phase brings about abundant domains with irregular shape in the bulk sample. Both heating and electron irradiation can induce the transformation from o-Nb4AlC3 to Nb4AlC3–x. Our findings may offer substantial insights into the roles of carbon vacancies in the structure stability and order-disorder phase transformation in MTCs.

m (for n = 2) 10,11 space group, their crystal structures are closely related, which can be regarded as the periodically stacking of strongly bonded "M m C m-n " sheets and "A" atomic layers along [0001]. The building block of "M m C m-n " in the MTCs strongly resembles that in the binary carbides. The presence of structural carbon vacancies in the MTCs are widely postulated 12 since monolithic MTCs can be synthesized only with certain degree of carbon deficiency 13 . The carbon vacancies have been believed to be disordered before the pioneering work by Etzkorn and coworkers 14 on V 4 AlC 3-x single crystal (with a dimension of 0.2 × 0.2 × 0.01 mm) grown by the auxiliary metal bath technique. They pointed out that the V 4 AlC 3-x single crystal grown at 1500 °C holds 10% disordered carbon vacancies, while the carbon vacancies become ordered at 1300 °C, forming V 12 Al 3 C 8 . So far, the knowledge of the carbon vacancies in the interesting MTCs is quite limited 12  First-principles calculation is a powerful tool to investigate the point defects, crystal structure and properties of the MTCs 7, [15][16][17][18][19] . However, it is frustrating for the phase stability of stoichiometric Nb 4 AlC 3 . Theoretically, Wang et al. 20 argued that stoichiometric Nb 4 AlC 3 is unstable and decomposes to Nb 2 AlC and NbC above 57 K. Experimentally, Hu et al. 21 demonstrated that Nb 4 AlC 3 has a good stability at 2000 K. Since Nb 4 AlC 3 bears striking resemblance to V 4 AlC 3 , this puzzling and unsolved inconsistence necessitates the revisiting of the crystal structure of Nb 4 AlC 3 with considerations of carbon vacancies.
Here, under the guidance of the first-principles prediction, an ordered phase bearing structural carbon vacancies in Nb 4 AlC 3-x (x ≈ 0.3), o-Nb 4 AlC 3 (Nb 12 Al 3 C 8 ), is unambiguously identified in experiment, demonstrating the validity to investigate the carbon vacancies in the MTCs with the combination of first-principles calculations, electron diffractometry and Raman spectroscopy.

Results
Predication of ordered phase. In analogy with the carbon-vacancy ordered phase in V 4 AlC 3-x (Ref. 14), ten hypothetical carbon-vacancy configurations (VCs) with a vacancy concentration of 1/9 were constructed based on a × 3 3 supercell of Nb 4 AlC 3 ( Fig. 1a-j). There are two distinct types of Nb 6 C octahedrons in Nb 4 AlC 3 (P6 3 /mmc), involving the carbon atoms located at 4f sites with a Nb-C bond length of 2.21 Å (OCT-4f) and 2a sites with a Nb-C bond length of 2.28 Å (OCT-2a . E VC and E Nb AlC 4 3 are total energies of the VC and Nb 4 AlC 3 unit cell, respectively. The chemical potential of carbon, μ C , is assumed to be that in graphite. Table 1 lists the calculated values. With lower total energies, VC8 and VC10 are the energetically most possible VCs. The E f VC for VC8 and VC10 are negative (a brief discussion in the context of chemical potential is provided in Supplementary Note 1), indicating that stoichiometric Nb 4 AlC 3 is metastable and prone to spontaneously forming ordered phases. With a more negative E f VC , VC8 isostructural with As determined by electron-probe X-ray microanalysis, the molar ratio of Nb:Al in o-Nb 4 AlC 3 is 4:1.05 (see Supplementary Table 3). The energy dispersive X-ray spectroscopy (EDS) mapping of Al (Fig. 3c) and Nb (Fig. 3d) demonstrates that there is no compositional difference of Al and Nb between o-Nb 4 AlC 3 and Nb 4 AlC 3-x . In addition, carbon is 10% deficient in the starting materials to synthesize monolithic Nb 4 AlC 3 24 . Then, the chemical formula of o-Nb 4 AlC 3 is Nb 24 Al 6+δ C 18-n (δ = 0.3) since the unit cell of o-Nb 4 AlC 3 is three times that of Nb 4 AlC 3-x (Fig. 4a). As the lowest multiplicity for the Wyckoff sites of  Table 2. o-Nb 4 AlC 3 can be constructed readily by removing the carbon atoms at (0, 0, 0) and (0, 0, 1/2) of the × 3 3 supercell (Fig. 4b). The orientation relationship is: [12 -10]     Statistics on the TEM dark field morphologies imaged with superlattice diffraction spots indicate that o-Nb 4 AlC 3 accounts for ca. 81 vol.% of the as-prepared sample. The X-ray diffraction (XRD) pattern in Fig. 5a is indexed with o-Nb 4 AlC 3 . The superlattice peaks, (h0h l) with h = 3n ± 1, are unidentifiable in the XRD pattern due to their remarkably low intensities (see Supplementary Table 4).

Discussion
Formation of a carbon vacancy within the Nb 6 C octahedron breaks six Nb-C bonds and destabilizes the structure. Meanwhile, the redistribution of the electron charge within the vacancy neighbors through the dilatation of the carbon-vacant octahedron strengthens the remaining Nb-C bonds around the carbon vacancy and stabilizes the structure. The triumph of the stabilizing factor over the destabilizing one gives rise to carbon-vacancy ordered phases 26,27 . With weaker Nb-C bonds, forming a carbon vacancy in OCT-2a costs less energy than that in OCT-4f (Fig. 5b). Similar features have been confirmed in Ta 4 AlC 3 and Ti 4 AlN 3 (Ref. 16,17). Generally speaking, the more the diagonal distances of the Nb atoms in the carbon-vacant octahedron expand, the stronger the remaining Nb-C bonds around the carbon vacancy become, and then the more stable the carbon-vacant structure is (Fig. 5b). Therefore, VC8 and VC10 with carbon vacancies only in OCT-2a and most expansions of the carbon-vacant octahedrons have lower total energies than the other VCs.
The revisiting of the phase component in Nb 4 AlC 3-x confirms the presence of carbon-vacancy ordered phase predicted by our first-principles calculations: o-Nb 4 AlC 3 has the VC8 configuration. The difference of E f VC between VC10 and VC8 is only 0.07 eV, and it is therefore not unreasonable to anticipate the existence of VC10. Virtually, there are several weak Raman peaks belonging to neither o-Nb 4 AlC 3 (VC8) nor Nb 4 AlC 3-x in the Raman spectrum collected with a 1800 lines per mm diffraction grafting (Fig. 5c). These extra peaks are most likely generated by VC10 (see Supplementary Table 5). Since no EDPs belonging to VC10 (see Supplementary Fig. 2g-i) were identified in the present study, VC10 is believed to exist not in a highly ordered manner. The crystal structure information of VC10 is provided in Supplementary  Table 6.
Carbon-vacancy ordered phase is stable at low temperature 1 . When temperature rises and the contribution of entropy to the Gibbs free energy is appreciable, carbon vacancies tend to be in short-range order or disordered. Therefore, o-Nb 4 AlC 3 is a low-temperature phase; while Nb 4 AlC 3-x (with certain amounts of disordered carbon vacancies) is the corresponding high-temperature phase. As indicated by the first-principles calculations (Table 1, Fig. 5b) and Rietveld refinements of X-ray (neutron) diffraction patterns of V 4 AlC 3-x (Ti 4 AlN 3-x ) 14,28 , the disordered vacancies in Nb 4 AlC 3-x are most likely located at the 2a site of the P6 3 /mmc space group. The existence of carbon-vacancy disordered Nb 4 AlC 3-x at room temperature is due to the fact that the cooling rate during the sample synthesis is not slow enough, and brings about disordered domains (Figs 3a and 6a). When o-Nb 4 AlC 3 is heated above a critical temperature, transformation to Nb 4 AlC 3-x occurs with the nucleation and growth of new disordered nanodomains in the ordered phase (Fig. 6b). For the sample quasi-quenched from 1400 °C after keeping  30 min, the amount of disordered domains (with dark contrasts) increases from ca. 19 vol.% (Fig. 6a) to 36 vol.% (Fig. 6b). Dwelling for 10 s at 1500 °C, nearly all o-Nb 4 AlC 3 transforms to Nb 4 AlC 3-x , leaving some ordered nanodomains (with bright contrasts, Fig. 6c). Consequently, the EDP (Fig. 6d) exhibits the  features of short-range ordering. Thereby, similar to the carbon-vacancy disordering in V 4 AlC 3-x where the disordering occurs in the range from 1300 °C to 1500 °C 14 , that in Nb 4 AlC 3-x starts around 1400 °C and completes at 1500 °C.
Resembling the ordered phase in binary carbides 29,30 , transformation from o-Nb 4 AlC 3 to Nb 4 AlC 3-x can be induced by electron irradiation, as shown in Supplementary Movies 1,2. The intensity of the superlattice diffraction spots decreases dramatically as the irradiation proceeds (Fig. 6e). Irradiated for approximately 120 s, the superlattice diffraction spots disappear (Fig. 6f,g) with the transformation from o-Nb 4 AlC 3 to Nb 4 AlC 3-x . The extremely electron irradiation sensitive nature possibly hides o-Nb 4 AlC 3 and domains from being discovered before.
In summary, under the guidance of first-principles calculations, a new carbon-vacancy ordered phase, o-Nb 4 AlC 3 (Nb 12 Al 3 C 8 ) has been discovered. It crystalizes in the space group of P6 3 /mcm with a = 5.423 Å, c = 24.146 Å. Coexistence of ordered (o-Nb 4 AlC 3 ) and disordered (Nb 4 AlC 3-x ) phase brings about domains with irregular shape. Both heating and electron irradiation can induce the transformation from o-Nb 4 AlC 3 to Nb 4 AlC 3-x . The excellent consistency between the first-principles prediction and experimental results demonstrated in this work may inspire the theoretical investigation on the vacancies in over 70 machinable ternary carbides/nitrides. The unveiled domain structure likely ignites investigation enthusiasm on the order-disorder phase transformation as well.

Methods
First-principles calculations with CASTEP module. Electronic exchange-correlation energy was treated under the generalized gradient approximation (GGA-PBE) 31,32 . Interaction of electrons with ion cores was represented by norm-conserving pseudopotential 33 . The plane-wave cut off energy and Brillouin zone sampling were fixed at 770 eV and 5 × 5 × 2 Monkhorst-Pack-point meshes 34 , respectively. The Broyden-Fletcher-Goldfarb-Shanno minimization method was used for geometry optimization 35 , where the tolerances were selected as the difference in total energy within 1 × 10 −8 eV per atom, maximum ionic Hellmann-Feynman force within 0.001 eV Å −1 , maximum ionic displacement within 5 × 10 −4 Å, and maximum stress within 0.02 GPa. The elastic constants were determined by the method reported by Milman et al. 36 . Vibrational frequencies were determined with the finite displacement method 37 .

Sample preparation. Bulk Nb 4 AlC 3-x (x ≈ 0.3) sample was synthesized by the method reported in
Ref. 21,38. Briefly, Nb (− 300 mesh), Al (− 300 mesh) and graphite (D 90 = 6.5 μ m) powders with a molar ratio of 4 : 1.2 : 2.7 were homogenized with agate balls and absolute alcohol in an agate jar for 12 h, and then dried at 70 °C for 24 h. After that, the blended powders were cold compressed in a graphite mold. Finally, the green compact together with the mold were put into a hot pressing furnace and sintered at 1900 °C for 1 h under a uniaxial pressure of 30 MPa with flowing Ar as protective gas.
Composition and microstructure characterization. Composition of thirty points in different regions of the as-prepared sample was analyzed by an electron-probe microanalyser (Shimadzu EPMA-1610, Kyoto, Japan). The phase components were investigated by XRD in an X-ray diffractometer (Rigaku D/max-2400, Tokyo, Japan) with Cu Kα radiation. The microstructural characterizations were performed on a transmission electron microscope (FEI Tecnai G 2 F20, Oregon, USA) working at 200 kV with an energy dispersive spectroscopy detector and a high-angle annular dark-field detector in the scanning transmission electron microscopy system. Selected area electron diffraction and CBED patterns were taken in Tecnai T12 (FEI Tecnai T12, Oregon, USA).

Micro-Raman spectroscopic characterization.
Unpolarized and polarized Raman spectrums were collected at room temperature on a LabRAM HR800 (Horiba Jobin Yvon, France) equipped with an air-cooled CCD array detector in a backscattering geometry, and with diffraction gratings of 600 and 1800 lines per mm. A He-Ne laser (632.82 nm) with an incident power of ca. 20 mW was used as excitation source. Theoretical Raman shifts were obtained by lattice dynamics calculation. According to Zhang et al. 11 , the peaks with A 1g symmetry disappear when θ =  0 (θ is the angle between the z axis of hexagonal crystal and that of the system coordinates). Therefore, the grains with θ ≈  0 were chosen to collect the polarized and unpolarized Raman spectrums.
The cooling rate above 800 °C is near 200 °C s -1 , as shown in Supplementary Fig. 3.