# Solid solution effects in the Ti2Al(C N ) MAX phases: Synthesis, microstructure, electronic structure and transport properties

W. Yu, V. Mauchamp, T. Cabioc'h, D. Magne, L. Gence, L. Piraux, V.

Gauthier-Brunet, S. Dubois

► To cite this version:

W. Yu, V. Mauchamp, T. Cabioc'h, D. Magne, L. Gence, et al., Solid solution effects in the Ti2Al(C N ) MAX phases: Synthesis, microstructure, electronic structure and transport properties. Acta Materialia, 2014, 80, pp.421-434. 10.1016/j.actamat.2014.07.064. hal-05490548

HAL Id: hal-05490548

https://hal.science/hal-05490548v1

Submitted on 4 Feb 2026

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Solid solution effects in the Ti₂Al(CₓNᵧ) MAX phases: synthesis, microstructure, electronic structure and transport properties.

W. Yuª, V. Mauchampª, T. Cabioc'hª, D. Magneª, L. Genceᵇ, L. Pirauxᵇ, V. Gauthier-Brunetª and S. Duboisª,\*

ª Institut PPRIME • Département de Physique et Mécanique des Matériaux

CNRS • Université de Poitiers • ENSMA • UPR 3346

SP2MI, Téléport 2, Boulevard Marie et Pierre Curie • BP30179

86962 FUTUROSCOPE CHASSENEUIL Cedex, France

ᵇInstitute of Condensed Matter and Nanosciences,
Université catholique de Louvain,

Place Croix du Sud,

B-1348 Louvain-la-Neuve, Belgium

Abstract

Ti₂AlCₓNᵧ solid solutions are synthesized by using Hot Isostatic Pressing. The X site solid solution effects are investigated by focusing on the microstructure, electronic structure and transport properties of the Ti₂AlCₓNᵧ MAX phases. Combining X-Ray Diffraction with Scanning Electron Microscopy and Wavelength Dispersive x-ray Spectroscopy measurements, it is shown that solid solutions can be synthesized with well controlled chemical compositions in the entire composition range (x = 0 to 1). The measured *a* and *c* lattice-parameter values are shown to be in good agreement with the ones calculated by using the cluster-expansion formalism. Combining electron energy-loss spectroscopy (EELS) and band structure calculations, it is demonstrated that solid solution effects induce weak perturbations on the electronic structure. Solid solution effect thus mainly results in a rigid shift of the Fermi energy in a flat part of the electron density of states of Ti₂AlC (or Ti₂AlN). From these observations, the variations of two key parameters of the conductivity are rationalized. The relative variations of the residual resistivity in the whole composition range are shown to be rather small which well agrees with the weak disorder evidenced from EELS. However, the values of the slope of the resistivity versus temperature variation are shown to vary quite significantly, evidencing a deviation from Matthiessen's rule in these systems. By comparing the experimental data with
calculations based on the semi-classical Boltzmann equation, it is demonstrated that the
observed variations are due to a combination of band structure effects and changes in the
scattering mechanisms. Finally, vacancies were also shown to have a prominent effect on the
transport properties of Ti₂AlCₓNᵧ solid solutions. In particular, both RR and dp(T)/dT values are
shown to be significantly higher than those of the corresponding stoichiometric compounds.
Introduction

M$_{n+1}$AX$_{n}$ phases (with n=1, 2 or 3) form a wide class of nanolaminated ternary carbides or nitrides, with a hexagonal structure, where M stands for an early transition metal, A for an A-group element and X for either carbon or nitrogen [1-9]. In addition to the considerable possibilities of changing the chemistry of these systems by playing on the M, A or X elements, solid solutions can also be synthesized on every one of these three atomic sites thereby allowing the tuning of their electronic, mechanical, optical or thermal properties [10-17]. In these so-called MAX phases, the M$_{n+1}$X$_{n}$ layers, characterized by mostly strong covalent M-X bonds, are interleaved with A layers through weaker M-A bonds. This inherent nanolayered structure provides a unique combination of metal-like and ceramic-like properties: machinability, good electrical and thermal conductivity, high thermal shock resistance, good oxidation resistance, stiffness at high temperature [9] which opens a way to diverse potential applications.

Among the most investigated applications to date are: (i) the realization of ohmic contacts on wide gap semiconductors, to achieve high quality devices using Ti-based MAX phase compounds[18-21], (ii) the use in future nuclear power plants[22]. Moreover, because of the combination of their nanolaminated structure and versatile chemistry, these materials were also shown to exhibit promising properties as magnetic inherent nanolaminates[23], and are precursors for a new class of two-dimensional materials, the MXenes[24-27]. In this context, solid solution effects play a key role in MAX phases properties: they induce magnetism and the possibility to tune some properties[17], allow the stabilization of MXene carbonitrides or of new MAX phases compounds (as (Cr$_{0.5}$V$_{0.5}$)$_{n+1}$AlC$_{n}$ with n=1, 2 and 3)[28]. A corollary to these effects is a modification of the electronic properties of the materials compared to the reference system. In this context, our purpose is to systematically investigate these solid solution effects in the Ti$_{2}$Al(C$_{x}$N$_{y}$) (x,y = 0 to 1) system by focusing on the microstructure, electronic structure and related transport properties.

MAX phases carbonitrides, in particular Ti$_{2}$AlC$_{x}$N$_{y}$ phases, are one of the most important example of MAX phase solid solutions on the X site. Since Ti-N and Ti-C have similar chemical bonding characteristics, resembling those of TiC and TiN binary compounds, it is possible to form a wide range of Ti$_{2}$AlC$_{x}$N$_{y}$ solid solutions[29, 30] with different physical properties. For example, Ti$_{2}$AlC$_{0.5}$N$_{0.5}$ has been reported [29] [31, 32] to be significantly harder than either of its end-members Ti$_{2}$AlC and Ti$_{2}$AlN, and there may be other ratios of C and N with even better
properties. As a prerequisite to their application for ohmic contacts, MAX phases electronic
properties and especially their modifications resulting from solid solution and/or vacancy
effects have to be properly studied and understood. Scarabozi et al. have shown that the
Ti₂AlC₀.₅N₀.₅ solid solution has a higher residual resistivity than the Ti₂AlC and Ti₂AlN end-
members[33]. Such a higher residual resistivity has been attributed to scattering by impurities,
vacancies or other defects. Results also suggest that vacancies are potent electron scatterers
as it is well established in binary transition carbides[36, 37]. Indeed, it has recently been
proposed that Ti₂AlCₓNᵧ solid solutions can behave a quite important amount of vacancies on
the X site [30]. Even if the stoichiometry of the Mₙ+₁AXn compounds is generally assumed to
be 2:1:1; 3:1:2 and 4:1:3 for n=1, 2 or 3 respectively, small difference from this “ideal”
stoichiometry can exist and can strongly influence the physical properties. In other words, it is
of great interest to know, not only the microstructure, but also the exact stoichiometry of the
MAX phase compounds when comparing properties of MAX phases samples (see [38] for a
recent review).

In order to understand the microstructure/stoichiometry/transport properties
relationships, we report on the synthesis, microstructural and electronic structure
characterizations with special emphasis on transport properties of Ti₂AlN, Ti₂AlC and a series
of Ti₂AlCₓNᵧ solid solutions. The samples' microstructure has been finely characterized
combining X-ray diffraction (XRD), scanning electron microscopy (SEM) including wave length
dispersive X-ray spectroscopy (WDS) to accurately determine the samples' composition. The
electronic structure and local order induced modifications have been investigated using
electron energy-loss spectroscopy (EELS) performed in the transmission electron microscope
(TEM) [39]. This technique allows the analysis of core electron excitations (very similarly to X-
ray absorption spectroscopy) with the spatial resolution inherent to the TEM, thereby giving
insights into the site-projected unoccupied density of states (DOS) of the excited element at
the single grain level. That EELS, and especially electron energy-loss near edge structures
(ELNES) is sensitive to disorder has been demonstrated on various compounds: (i) in yttria
stabilized zirconia, (Y₂O₃)ₓ(ZrO₂)₁-x, the blurring-out of the O-K edge fine structures was shown
to be characteristic of the disorder on the oxygen sublattice[40], (ii) in CaTi₁-xZrₓO₃ solid
solutions, the O-K edge fine structures were shown to be sensitive to the x content as well as
the pre-edge structures in the Ti-K edge which was shown to be a semi-quantitative probe for
the short-range cation order[41]. In the present study, we will mainly focus on the X element
K-edge (i.e. excitation of a carbon or nitrogen 1s electron) of the solid solutions. The X element core loss spectra were recently shown to be a valuable probe to investigate the local disorder induced in Ti₃AlC₂ by light ion irradiation[42] and the carbon and nitrogen K edges have been used to investigate alloying effects in the TiCₓNᵧ system[43]. The transport properties were investigated using a four points probe method for a temperature range going from 20 to 300K. Both EELS and transport properties were analyzed using *ab initio* calculations performed in the density functional theory framework.

The paper is organized as follows. In a first part, the experimental methods (synthesis and characterization) are described. In a second part, we describe the theoretical methods that are used in order to interpret both the EELS spectra and the transport data. Finally, we present our results and discussions which evidence: (i) the possibility to finely control the solid solution stoichiometry, (ii) the rather low disorder induced in the structure when making solid solutions on the X site, (iii) the important role of electron-phonon interactions in the transport properties variations observed for the solid solutions leading to deviations from the Matthiessen's rule in these compounds, and (iv) the major role of vacancies as scattering centers for the electrons.

## II. Experimental methods

Powders of Ti (<45µm, 99.5% purity), Al (<45µm, 99.5% purity), AlN (<45µm) and TiC (<45µm, 99.5% purity) were used to synthesize stoichiometric (x+y=1, first series) and substoichiometric (x+y<1, second series) Ti₂AlCₓNᵧ compounds, and related end-members. To fabricate stoichiometric Ti₂AlCₓNₓ(1-x) compounds (x=0.75, 0.5, 0.25) and related end-members (x=0, 1), 2Ti:1.1Al:xC:(1-x)N nominal compositions were used. Note that the powder mixtures used in this study contain an excess of 10 at.% of Al to compensate the loss of aluminium by preferential out-diffusion during the sintering process. To prepare homogeneous reactant mixtures, the powders were milled for 4h in a Turbula® mixer (WAB). The reactant mixtures were then cold-compacted into cylindrical steel dies (12mm in diameter and 10mm in length) using an uniaxial pressure of 100MPa.

Afterwards, the green samples were encapsulated into pyrex containers under high vacuum for reactive sintering in the Hot Isostatic Press (HIP) machine. The green samples were subjected to the following temperature and pressure cycles:

- The samples were heated up to 800°C before applying the isostatic Ar pressure
- Once the processing temperature (1400°C) and pressure (80MPa) were reached, the samples were held for 4 h.

- Cooling the furnace to room temperature and decreasing the pressure to atmospheric pressure proceeds in 90 min.

After HIPing, samples were machined to remove the encapsulating glass container and sliced using a diamond wheel. Samples were thus grinded using silicon carbide paper and then polished with a diamond suspension. Finally, a chemo-mechanical polishing has been performed using a neutral suspension of alumina particles. Such a chemo-mechanical polishing allows to produce a very flat surface.

The detailed procedure used to produce high-purity sub-stoichiometric Ti₂AlCₓNᵧ (x+y<1) solid solutions has been described elsewhere [44].

Phase identification was performed by X-Ray Diffraction (XRD) using a Bruker D8 diffractometer with Cu-Kα radiation. XRD data were refined using the Materials Analysis Using Diffraction (MAUD) software[45] in order to extract the lattice parameters and the content of the different phases of the Ti₂AlCₓNᵧ solid solutions and related end-members. Microstructures were examined by Scanning Electron Microscopy (SEM, JEOL 5600LV) coupled with Energy Dispersive X-ray Spectrometry (EDXS, Oxford Isis 300) for the chemical analysis (Ti over Al atomic ratio). Wavelength Dispersive x-ray Spectroscopy (WDS, CAMECA SX100) was used to determine the composition of the Ti₂AlCₓNᵧ solid solutions resulting from HIPing treatment. The uncertainty of the carbon and/or nitrogen content is considered to be close to 0.02-0.03 by using our analysis method.

Parallelepiped-shaped specimens with approximate dimensions 0.3×1.2×10mm³ were machined for the resistivity measurements. The resistance of the samples was measured as a function of temperature in the 20-300K range. The measurements were performed using a Quantum Design Physical Properties Measurements System (QDPPMS) with a current of 30mA. Four probe method was adopted to eliminate contact resistance for these low-resistivity materials.

For the TEM experiments, the HIPed samples were scratched using a diamond tip to get a small amount of powders. These last were hand milled with an agate mortar in an agate crucible before being collected on a holey carbon grid. EELS spectra were recorded on thin areas (thickness < 50 nm) on the border of microscopic grains, the probe size being on the
order of 100 nm. Given the nanolaminated structure of the MAX phases, most of the grains
were oriented along the [0001] zone axis. TEM-EELS experiments were performed in a JEOL
2200 FS microscope operating at 200 kV and equipped with an in-column omega filter. The
energy resolution determined from the zero-loss peak full width at half maximum (FWHM)
was 0.8 eV. Spectra were dark count corrected and deconvoluted from multiple scattering
using a Fourier-ratio technique[39]. Spectra were recorded in almost parallel illumination and
using a 7.5 mRad semi-collection angle β.

### III. Theoretical methods

Ab initio calculations were used to simulate the C-K and N-K ELNES, to investigate the solid solution effects on the density of states (DOS) and finally to analyse the transport properties. All calculations were performed in the density functional theory framework. For all calculations, the experimental unit cell parameters determined from X-ray diffraction (see table II) and canonical positions for the M, A and X atoms were used.

#### III-1 ELNES calculations

The simulations of the C-K and N-K ELNES of the end-members (i.e.: Ti₂AlC and Ti₂AlN) were performed with the Feff9 code. Feff9 is an *ab initio* self-consistent multiple scattering approach based on a Green's function formalism and implemented in the muffin-tin approximation[46, 47]. We used the EELS module which allows an accurate description of the EELS scattering geometry (zone axis, collection angles, acceleration voltage, experimental broadening) and includes relativistic corrections[48]. In the present study we used a real space approach where the system is described as a cluster centered on the excited atom (C or N in our case). The final state effects inherent to the excitation process induced during the EELS experiment, *e.g.* the creation of a core-hole, were treated using the random phase approximation (RPA) screened core-hole. Inelastic losses were disregarded: given the good agreement between the experiments and the calculated spectra, these effects are of minor importance in the present context. Note that Feff calculations were already proved to give a very good description of the X element core excitations in MAX phases[42]. The parameters used for the Feff calculations are summarized in table I.

#### III-2 Electronic structure and transport properties calculations
Ab initio band structure calculations were performed for the different stoichiometric (i.e. vacancy-free) solid solutions with the WIEN2k code[49]. WIEN2k is an all electron full potential approach based on a Linearized augmented plane waves (LAPW) or Augmented Plane Waves+local orbitals (APW+lo) method. APW+lo were used with a converged basis set obtained for a R<sub>min</sub>K<sub>max</sub> value of 7.5 (R<sub>min</sub> being the smallest sphere radii in the system, and K<sub>max</sub> the plane-wave momentum cutoff). 800 k-points were used to sample the full Brillouin zone (FBZ) for the self-consistent field calculations of potentials. Calculations were performed in the generalized gradient approximation (GGA) considering the Perdew-Burke-Ernzerhof parametrization[50]. The virtual crystal approximation (VCA) has been used in order to describe solid solution effects. VCA has been successfully used to investigate alloying effects in the chemical bonding of TiC<sub>x</sub>N<sub>1-x</sub> systems[43]. In this approach, the X site is described using an effective atom with Z<sub>eff</sub>=7-x (e.g. x = 0.5 for the Ti<sub>2</sub>AlC<sub>0.5</sub>N<sub>0.5</sub> system). The VCA is valid in our case since C and N are direct neighbors in the Periodic Table. As will be shown in this paper, the VCA gives results in very good agreement with more elaborated calculations based on the cluster expansion method[51] : in particular, the density of states around the Fermi level are very similar with both methods.

The band structure calculations were then used as inputs for the transport properties
calculations. These calculations are performed in the framework of the semiclassical
Boltzmann theory using the BoltzTrap code[52].Given that the relaxation time is hardly
accessible from ab initio calculations, we restrict ourselves to the Fermi surface dependent
quantity ξ=σ/τ, which is the ratio of the trace of the conductivity tensor (σ) to the relaxation
time (τ). The calculation of ξ was converged when considering 55000 k-points in the FBZ.

IV. Results and discussion

IV.1. Microstructural characterization

Figures 1a-e show SEM micrographs recorded in backscattered mode of the polished surface of samples HIPed at 1400°C and 80MPa for 4 h. WDS analyses show that, except for pure carbide for which the C content has been reduced from 1 to 0.88 after reactive sintering (figure 1e), the Ti, Al, C and N contents in the stoichiometric Ti₂AlCₓNᵧ solid solutions are very closed to the ones introduced in the starting powders. Moreover, for all samples, WDS analyses show that the overall Ti/Al ratio is very close to 2, which confirms the expected loss of approximately 10 at.% of the aluminum during reactive sintering.
WDS analyses coupled with SEM observations demonstrate that, starting with 2Ti:1.1Al:xC:(1-x)N nominal composition where x=0, 0.25, and 0.5, the resulting Ti₂AlN (Figure 1a), Ti₂AlC₀.₂₃N₀.₇₆ (Figure 1b), and Ti₂AlC₀.₄₉N₀.₄₉ (Figure 1c) HIPed samples are predominantly single-phased. Starting with reactant mixtures containing high C content, i.e. x=0.75 and 1, some Ti₃AlC₁.₄₈N₀.₄₉ grains are detected in addition to Ti₂AlC₀.₇₄N₀.₂₅ (Figure 1d) whereas large areas of Ti₃AlC₁.₈₁ are observed in the Ti₂AlC₀.₈₈ matrix (Figure 1e). It is worth noticing that for the carbide, the two MAX phase compounds (usually called (211) and (312) compounds) present some carbon substoichiometry, both having about 10% of vacancies on the X site. WDS analyses coupled with SEM observations also reveal the presence of Al₂O₃ (round shape black areas) in the Ti₂AlN (Figure 1a), Ti₂AlC₀.₂₃N₀.₇₆ (Figure 1b), Ti₂AlC₀.₄₉N₀.₄₉ (Figure 1c), Ti₂AlC₀.₇₄N₀.₂₅ (Figure 1d), and Ti₂AlC₀.₈₈ (Figure 1e) HIPed samples. Image analysis, performed using the imageJ software, gives a content of Al₂O₃ in the range 1.5-2.1 vol.%. The Al₂O₃ content increases with the increase of the AlN content in the initial reactant powder mixture. Thus, the presence of alumina likely results from the surface oxidation of AlN starting powders. About 1-2 vol.% of TiₓAlᵧ is detected in the Ti₂AlC₀.₇₄N₀.₂₅ solid solution (Figure 1d) and in the Ti₂AlC₀.₈₈ matrix (Figure 1e) using EDXS and the imageJ software. Phases' compositions and respective volume contents determined in the various HIPed samples are summarized in Table II.

Figure 2 shows XRD patterns recorded on the different stoichiometric solid solutions and related end-members. From figure 2, Ti₂AlCₓNₐₓ is the only phase detected after reactive sintering of the different investigated compositions. One can notice that, starting with 2Ti:1.1Al:C nominal composition, some traces of 312 are also detected after HIPing treatment. This secondary phase has also been evidenced by SEM (see figure 1e). Comparing the XRD patterns, a progressive shift of the diffraction peaks from Ti₂AlCₓNₐₓ towards lower 2θ values is observed when the C content increases. Given that the C covalent radius (Rc = 0.76 Å) is about the same as the one of N (RN = 0.71 Å)[53], and assuming that Ti-C or Ti-N bonds show covalent bonding[54], the cell parameters of Ti₂AlCₓNᵧ should decrease compared to the ones of Ti₂AlC: this is in reasonable agreement with our results.

Figure 3 shows the variation of the *a* and *c* values of the Ti₂AlCₓNₐₓ solid solutions and related end-members as a function of the C content measured by WDS. The measured *a* and *c* values are in quantitative good agreement with the ones calculated by using the cluster-expansion formalism[55]. From Rietveld refinements of the XRD data, quite accurate values
of *a* and *c* were determined using the MAUD software. Uncertainties of 5.10⁻⁴ Å and 2.10⁻³ Å were typically obtained for *a* and *c* values, respectively. The obtained values are given in Table II and compared to others obtained from the literature. From figure 3, it can be observed that the *a* value of the carbonitrides increases linearly with the C content; the *a* value of these solid solutions being larger than the one of pure nitride (2.9886 Å) but smaller than the one of pure carbide (3.0607 Å). Such a linear variation was also observed for TiCₓN₁₋ₓ carbonitride[56]: it gives evidence of the close relationships between the Mₙ₊₁Xₙ layers of the MAX phase and that of the corresponding binary MX compounds. The linear variation of the *a* value with composition is consistent with the formation of a continuous substitutional solid solution in which atoms that substitute for each other are randomly distributed on the X site. The *a* values obtained in this study are in good agreement with the experimental ones from the literature, especially for x=0, 0.25, 0.5 and 0.75 (see Table II). For x=1, some discrepancies exist for values reported in literature. One has to notice that the mean value of *a* and standard deviation in *a* are respectively *a*=2.992±0.006 Å and *a*=3.0525±0.0125 Å for Ti₂AlN (mean value and standard deviation obtained from our work and ref.[29] [30], [57] [58], [59] and Ti₂AlC (mean value and standard deviation obtained from our work and ref. [10, 29, 30, 57, 60]). Such discrepancies are difficult to understand if one considers that Ti₂AlCₓ has a narrow range of carbon composition[44] and that vacancy content slightly modifies lattice parameter values[44]. From figure 3, one can observe that the *c* value does not vary monotonically as a function of the C content in the solid solution. Furthermore, it is lower for carbonitrides (carbon content < 0.5) than for pure carbide and pure nitride. Similar results were also reported in [29-31]. Arroyave and Radovic have obtained the same kind of variation by investigating the energetics of the C-N alloying in the X sublattice using the cluster-expansion formalism[51]. The deviation from Vegard's law is attributed to the stronger M-X interactions that result from carbon and nitrogen mixing in the X sublattice. Figure 3 also shows that the lattice contraction of the Ti₂AlCₓN₁₋ₓ compounds is strongly anisotropic: it is much more reduced, going from Ti₂AlC to Ti₂AlN, along the *a*-axis than along the *c*-axis.

## IV.2. Electronic structure and local order

As mentioned in the introduction, EELS, and particularly ELNES, is used as a complementary tool in order to characterize, the perturbation of the electronic structure and
related disorder induced in the MAX phase because of solid solution effects. In a first approximation (i.e. neglecting final state effects) and owing to the dipole selection rule[42], these near edge structures are characteristic of the C (or N) unoccupied *p*-DOS which, as will be shown in the following, are directly related to the octahedra layers electron density in the MAX. By comparing the experimental spectra with *ab initio* calculations, it is thus possible to directly connect changes observed in the EELS spectra with the site-projected density of states and corresponding electronic structure. From this point of view, EELS is very complementary to XRD data in the sense that, owing to the unparalleled spatial resolution of the TEM, one can easily investigate single grains and access to local information. From the references cited in the introduction[40-43], the solid solution effects will be considered in terms of three major consequences on the ELNES: changes in the relative intensities of the peaks, energy shifts or broadening of the ELNES features.

The C-K and N-K edges recorded for the different solid solutions and related end-members are plotted in figure 4. The Feff simulations corresponding to Ti₂AlC and Ti₂AlN are also given (grey dashed and dotted lines). For the sake of comparison, the spectra were aligned and normalized on the most intense peak (B). It is worth noting that, as mentioned in the experimental details section, the C and N-K edges were recorded on grains oriented close to the [0001] zone axis. In these conditions, the momentum transfer from the electron beam to the sample is mainly oriented within the basal plane of the structure: this could raise questions concerning the effect of the anisotropy of the unit cell on the recorded signal. However, contrary to the low loss spectra of MAX phases which exhibit a significant anisotropy related to the anisotropy of the dielectric response of these materials [61-64], the anisotropy observed at the X element K edge by EELS in the MAX phases is weak. This is evidenced in Fig. 4 where the C-K edge recorded on a Ti₂AlC thin film prepared in cross section is also given (dashed black lines). This last spectrum is clearly similar to that obtained on the bulk sample: the main reason is that the X element in the MAX phase is an octahedral environment. This is in agreement with recent results obtained at the C-K edge in Cr₂AlC [65]. For this reason, we do not expect any major restriction related to anisotropy effects in the results that will be presented in the following.

Focusing first on the carbon K-edges (Fig. 4 (a)), one can see that the ELNES consist in 6 different peaks labeled A to F. These six structures are evidenced in all experimental spectra
and confirmed by the simulations which reproduce nicely their relative intensities (except for
peak A) and energy positions. Note that these spectra are in very good agreement with those
previously published by Hug *et al.* [55], with a better resolution, thus allowing the identification
of the A shoulder. Our calculation is also very consistent with the one presented in reference
[55]. The comparison between the experimental and theoretical spectra calls for two
comments: (i) the shoulder A is more intense in the simulations than in the experiments and
(ii) the structures C and D are broadened in the experimental spectrum corresponding to
Ti₂AlC₀.₈₅. Concerning point (i), a slight inaccuracy in the calculation of the charge transfers
between the X and M elements which would then modify the electronic levels filling could well
explain this difference. That charge transfers are quite complex in MAX phases is now well
documented: these materials exhibiting covalent, ionic and metallic-like bonds. Another
possibility is that a slight change in the carbon content (as it is obviously the case in Ti₂AlC₀.₈₅)
could also change the intensity of peak A since, as will be detailed below, this structure is
representative of the carbon sites surrounding the excited carbon atom. Concerning point (ii),
this broadening could be a consequence of some disorder induced by the 15% vacancies
measured on the X site. Such an observation is in line with the previously mentioned
broadening of ELNES observed in many materials when increasing disorder. This argument is
also consistent with the fact that reducing the cluster size from 7.4 to 6.1 Å results in a
broadening of these two structures (see the grey dotted line in Fig. 4 (a)) and that for
vacancies-free systems (Ti₂AlC₀.₂₅N₀.₇₅, Ti₂AlC₀.₅N₀.₅ and Ti₂AlC₀.₇₅N₀.₂₅), structures C and D
recorded experimentally are very close to the calculations.

From the comparisons between experiments and calculations, one can safely assume
that the C-K edge is sensitive to the local order up to approximately c/2 around the excited
carbon atom since theoretical spectra are converged for cluster radii of approximately 7 Å.
Considering the three main criteria showing the influence of solid solutions on the ELNES, the
here presented C-K edges evidence rather weak impact on the carbon atoms DOS. There is no
significant energy shift, weak broadening and weak changes in the relative intensities of the
different peaks. Still, having a closer look to the data, one can see a slight broadening of peaks
A and B for the C₀.₅N₀.₅ sample (this is evidenced in the inset of Fig. 4 (a)). This broadening goes
together with the fact that slight ELNES intensity variations were observed in the same sample
when focusing on different grains. As an example, the blue dotted curve is a spectrum recorded
on another grain of the sample. Interestingly this particular edge becomes very close to that
recorded for the nitrogen atoms in the same sample (see the blue curve in Fig. 4 (b)): these
changes could thus arise from the fact that for C<sub>0.5</sub>N<sub>0.5</sub> the C and N atoms adopt almost
identical surroundings which in turn gives identical ELNES at these two edges.

Focusing now on the N-K edges and the corresponding calculations, one can draw
conclusions rather similar to those related to the C-K edges. One can hardly observe any
change in the experimental fine structures of the different compounds (as evidenced in the
inset and contrary to the C-K edge, peak B is unaffected by solid solution effects). From the
ELNES point of view the weak perturbation observed on the carbon atoms is thus confirmed
for nitrogen atoms. Concerning the comparison with the calculations, it is worth mentioning
that the agreement is less satisfactory than for the C-K edges. In particular, peaks A and B,
which are clearly resolved in the calculations, are not separated experimentally. This is in
complete agreement with the results published by Hug *et al.* [55] Apart from this point, the
different structures C', C, D, E and F are observed. Here again, given the cluster size considered
in the calculations to reproduce all these structures, the N-K edges are shown to probe the
local order around the N atoms up to approximately c/2. It is important to compare our results
to those of Calmels *et al.* obtained for TiC<sub>x</sub>N<sub>y</sub> system. In these last systems, solid solution effects
induce peak broadening, energy shifts and variations of the relative intensities of the ELNES
features [43]. From this point of view, the electronic structure of the MAX phase is clearly less
perturbed by solid solution effects than the corresponding binary compounds.

In order to get more insights into the sensitivity of the ELNES on the local order in the
Ti₂AlCₓN₁₋ₓ systems, one can have a look at their decomposition among the different
coordination shells around the excited atom. It is given in Fig. 5. For the sake of clarity, we only
focus on the C-K edge decomposition since it is equivalent for the N-K edge. The sensitivity of
the ELNES to the different coordination shells is quite obvious from Fig. 5: the energy position
of peak B changes as a function of the number of coordination shells considered in the
calculation: it is only stabilized when considering more than five coordination shells (the fifth
coordination shell being constituted of 12 titanium atoms). Peak A is due to the presence of
the tenth coordination shell made of 6 carbon atoms what confirms the possible dependence
of its intensity on the carbon content in the material. The splitting of the C-D structures occurs
at the ninth shell built on 6 titanium atoms whereas the E-F double structure already appears
at the third coordination shell. These different features thus give information on the local order
at different distances from the carbon site. More interestingly, excepted for the fourth
coordination shell which slightly shifts the C-D peaks, the aluminum atoms do not participate to the ELNES: the C-K edge (and N-K edge as well) is thus characteristic of the electronic density within the MX octahedra layers. This is important in terms of transport properties since these states are precisely those mainly involved in the transport properties of the MAX phase as evidenced by the densities of states discussed in the next part (see figure 8). Although a direct comparison between EELS results and transport properties is not straightforward since the energy ranges investigated are not the same, one can expect from the EELS results that the solid solution effects are of second order when considering the band structure part of the transport properties (e.g. ignoring the relaxation mechanisms). This is discussed in details in the following.

IV.3. Transport properties

The resistivity variations as a function of temperature, ρ(T), for the different Ti₂AlCₓNᵧ solid solutions is given in Fig. 6. In the 100-300K temperature range, ρ(T) depends linearly on the temperature as generally observed for MAX phases [14, 33, 66, 67]. This typical metal-like behaviour simply results from the large density of states at the Fermi level and from the main electron scattering mechanism (i.e.: electron-phonon interaction)[68-70]. Fig. 6 evidences two major consequences of the solid solution effect on ρ(T): (i) the residual resistivity (RR), *i.e.* the low temperature resistivity, is modified and (ii) the slope of the ρ(T) curve is also changed. The residual resistivity of the different samples, extracted from the curves of Fig. 6, is plotted as a function of the carbon content in Fig. 7 (a). The substitution of X element leads to an increase of the RR in comparison to the end members compounds. Such an effect likely results from the increase of disorder when substituting carbon by nitrogen or nitrogen by carbon as previously observed for M substitution in (Ti₀.₅Nb₀.₅)₂AlC [55]. This disorder increases in turn the electron scattering. However, the RR in the Ti₂AlCₓNᵧ system only varies between 6 and 18 µΩ.cm in the entire composition range. Such a relative variation is much smaller than that observed in other systems such as Ti₁-xAlx disordered alloys in which the resistivity is vertically shifted by about 1 order of magnitude for x ranging from 0 to 0.06[71] or such as sub-stoichiometric TiCₓ single crystal where the added resistivity per atomic percent of carbon vacancy at RT is 16 µΩ.cm[36]. Even higher relative variations are found in highly pure Al/Ag/Cd or Mg metals [72]. Although, the RR in our samples is much higher than that reported in ref. [71]and [72], our data suggest that making solid solutions in the Ti₂AlCₓN₁-x systems
induces rather small perturbation of the RR: if we assume that the RR is linked to the induced disorder in the sample, the weak RR variations are in agreement with the conclusions obtained from the C-K and N-K edges in these samples.

As $\rho$ varies linearly with temperature in the range 100–300 K, one can calculate the slope of the $\rho(T)$ curve (Fig. 7(b)). Such a slope is representative of the intrinsic transport properties of the solid solutions and should not depend, in a first approximation, on defects and impurity contents. For the Ti$_2$AlN and Ti$_2$AlC samples, our values are compared to data from the literature. One can notice that the RT ideal resistivity (22 $\mu\Omega$.cm) and slope of the linear variation (0.1 $\mu\Omega$.cm.K$^{-1}$), given the experimental errors, are the same whatever the Ti$_2$AlN sample. Such a result is not surprising as ideal resistivity and electron scattering mechanisms are the same for a same compound. It also means that the different Ti$_2$AlN synthesized in the literature[32] likely have the same nitrogen stoichiometry. Moreover, experimental characterization of Ti$_2$AlN compounds indicates that the $\alpha$-parameter standard deviation is small which likely results from the small variation of the N stoichiometry in the different Ti$_2$AlN$_y$ synthesized samples. The dispersion of the results is higher for the Ti$_2$AlC samples: as will be discussed latter, this could be explained by different sub-stoichiometry's on the carbon site in these systems.

Fig. 7 (b) also shows that the slope of $\rho(T)$ are larger for solid solutions than for the Ti$_2$AlN and Ti$_2$AlC end-members. In a single band model, such a slope is given by $dp/dT = m_e^* / ne^2[d(1/\tau)/dT]$, where $m_e^*$ is the electron effective mass, $n$ is the charge carrier density and $\tau$ is the relaxation time. In a two band model and a compensated conductor for which $n=p$ [53, 54, 56], such a slope is given by $dp/dT = 1/ne^2[d(m_e^*\cdot m_h^*)/(m_e^*\tau_e + m_h^*\tau_h)]/dT$. Although these models are probably oversimplified when considering materials with such a complex structure as MAX phases are, these equations show that the observed variations depend on many parameters: the charge carrier density, the carrier effective mass and the derivative of the inverse of the relaxation time. In order to clarify the relative importance of these different factors, *ab initio* calculations of the band structure dependant quantity $\xi=\sigma/\tau$ were performed.

As detailed in section II, the theoretical investigation of the solid solution effects were performed using the VCA. This approximation is reasonable since EELS experiments evidence a weak perturbation of the electronic structure when substituting carbon with nitrogen atoms. The electronic density of states obtained for the different solid solutions are given in Fig. 8 (a). Compared the cluster expansion calculations of Arróyave et al., [51], our VCA calculations give
a good description on the DOS around the Fermi level: the Fermi energy is found to lie within
the plateau region between structures A and B, and these structures are found to shift by 0.5
eV when going from Ti₂AlC to Ti₂AlN as observed in the calculations of ref. [51]. That the VCA
gives a good description of the electronic structure in this particular case could be traced back
to the fact that Ti *d* states dominate near E<sub>f</sub> in both Ti₂AlC and Ti₂AlN (see Fig. 8 (b) and (c)): a
modification in the potential of the X atom only acts as a small perturbation of the DOS in this
energy range. Fig. 8 (a) evidences that the major effect induced by a modification of the X
element is a 0.5 eV rigid shift of the Fermi energy within the Ti *d* bands. It is worth noting that
the shift observed on the states lying below the Ti *d* bands is more important (these low lying
states disappear from the considered energy range for x = 0.5). This can be understood from
the fact that these states, which are shaded in figure 8 (a) and (b), correspond to Ti-X
hybridizations which are probably more sensitive to an accurate description of the X atom
potential.

The band structure calculations were used as inputs for the calculation of σ/τ. We obtained
the values given in table III. These values vary from 2.54 10²⁰ to 2.94 10²⁰ (Ω.m.s)⁻¹, the
maximum relative variation in the whole composition range is thus 15%; such a relatively
weak variation is consistent with the fact that the rigid shift observed in the electronic
structure is within a flat part of the DOS. From these calculations, it is fair to conclude that a
significant part of the variations observed experimentally for dp/dT, which is one the order of
40% in the whole composition range, can be attributed to relaxation time effects. In MAX
phases these effects are dominated by electron-phonon interactions. We thus evidence here
deviations from Matthiessen's rule in these compounds. This important interplay between
electronic structure and scattering mechanisms has already been evidenced as a crucial point
explaining the conductivity anisotropy of Ti₂AlC[73].

Before concluding, it is worth noting that vacancies also play a crucial role in the
transport properties of MAX phases. As an illustration, the ρ(T) curves recorded on Ti₂AlCₓNᵧ
systems presenting vacancies on the X sites are also shown in figure 6. The synthesis and
microstructural characterization of these samples are discussed elsewhere[44]. The RR and
ρ(T) slopes extracted from these curves are shown in figure 9 (a) and (b) respectively. In order
to evidence the influence of vacancies, the dp(T)/dT values are plotted on top of those
reported in figure 7 (b). Figure 9 (a) shows that the residual resistivity monotonically increases
with vacancy content. Such a result is in line with the fact that vacancies are strong electron
scattering centres as reported in the 60’s for titanium carbide[36]. Nevertheless, vacancy
effect seems to be much less pronounced in Ti₂AlCₓNᵧ MAX phases than in TiCₓ for which the
added resistivity per atomic percent of carbon vacancy are respectively about 2 µΩ.cm and 16
µΩ.cm. The smaller value measured for Ti₂AlCₓNᵧ comes partly from the large number of
conduction electrons in Ti₂AlCₓNᵧ compared with TiC. Vacancies also play a major role in the
temperature dependant scattering mechanisms (e.g. dp(T)/dT). Focusing on Fig. 9 b, it appears
clearly that putting 10 to 15 % vacancies on the X site in Ti₂AlN, Ti₂AlC or Ti₂AlC₀.₅N₀.₅ results
in the doubling of the ρ(T) curve slope. Such a result is in contrast with the ones obtained on
sub-stoichiometric TiCₓ for which the temperature-dependent mobility has been assumed to
be independent of the carbon vacancy content. [36, 74]. The physics behind these
observations is still unclear at the moment and deserves more theoretical investigations.
Finally, such a variation of the resistivity with vacancy content could explain the Ti₂AlC
resistivity’s values discrepancies noticed in the literature if ones assume that vacancy content,
which have not been measured, of the synthesized Ti₂AlC samples are different.

V. Summary

In summary, the solid solution effects on the X site of Ti₂AlCₓNᵧ MAX phases have been
investigated by focusing on their microstructure, electronic structure and transport properties.
Combining XRD with SEM and WDS measurements, it has been shown that solid solutions can
be synthesized with well controlled chemical compositions in the entire composition range (x
= 0 to 1). The *a* value of the carbonitrides increases linearly with the C content whereas the *c*
value does not vary monotonically. The measured values are moreover in quantitative good
agreement with the ones calculated using the cluster-expansion formalism[55]. Combining
electron energy-loss spectroscopy with *ab initio* calculations, it has been demonstrated that
such solid solution effects induce very weak perturbations on the octahedra layers electron
density; this is particularly true when comparison is realized with equivalent experiments
performed on the related binary compounds TiCₓNᵧ[43]. Such a weak perturbation has been
further confirmed by *ab initio* calculations performed in the virtual crystal approximation.
Indeed, band structure calculations show that solid solution effect mainly results in a rigid shift
of the Fermi energy in a flat part of the electron density of states of Ti₂AlC (or Ti₂AlN). Near
the Fermi level, solid solution effects do not significantly affect the DOS. This is traced back to
the fact that the electronic structure is dominated by the Ti *d* bands around the Fermi energy.
From this argument, it is likely that performing solid solution on the M site would much more
alter their electronic band structure and especially their transport properties as experimentally
demonstrated on (Ti,Nb)₂AlC and (Ti,Hf)₂InC solid solutions [75] [76].

From these observations, the variations of two key parameters of the conductivity, the
residual resistivity and the slope of ρ(T), have been rationalized. The relative variations of the
RR in the whole composition range were shown to be rather small. It gives evidence of a weak
disorder which is in good agreement with the EELS results. However, the dp(T)/dT values are
shown to vary quite significantly. By comparing the experimental data with calculations based
on the semi-classical Boltzmann equation, it is demonstrated that the observed variations are
due to a combination of band structure effects and changes in the scattering mechanisms,
which in MAX phases are dominated by electron-phonon interactions. Such a dependence of
the relaxation time on impurity evidences a deviation from Matthiessen's rule in these
compounds. Vacancies were also shown to have a prominent effect on the transport properties
of Ti₂AlCₓNᵧ solid solutions. In particular, both RR and dp(T)/dT values are shown to be
significantly higher than those of the corresponding stoichiometric compounds.

We expect these results to be relevant for many applications of MAX phases given the
importance of solid solutions effects in their synthesis (as bulk materials or thin films), for their
applications in terms of ohmic contacts, magnetic systems or as precursors for the MXenes.

## Acknowledgements

The authors from PPRIME Institute thank the Agence Nationale pour la Recherche (ANR
project n°13-BS09-0024-02) for financial support.

## Figure captions

**Figure 1:** Typical SEM micrographs (back-scattered electrons) of polished sections of HIPed Ti₂AlCₓN*(1-x)* samples obtained from (a) 2Ti:1.1Al:N, (b) 2Ti:1.1Al:0.25C:0.75N, (c) 2Ti:1.1Al:0.50C:0.50N, (d) 2Ti:1.1Al:0.75C:0.25N, (e) 2Ti:1.1Al:C

**Figure 2:** XRD patterns recorded on different Ti₂AlCₓN*(1-x)* stoichiometric solid solutions and related end-members (color online).
**Figure 3:** Evolution of the α (red triangles) and c (blue squares)-lattice parameters as a function of the carbon content in the Ti₂AlCₓNystox (color online). T

**Figure 4:** (a) C-K edges and (b) N-K edges recorded for the different solid solutions: C₀.₇₅N₀.₂₅ (green lines), C₀.₅N₀.₅ (blue full and dotted lines), C₀.₂₅N₀.₇₅ (red lines). These spectra are compared to those obtained on the end-members of the series - Ti₂AlC₀.₈₅ or Ti₂AlN and the corresponding Feff calculations obtained considering different cluster sizes (grey dashed and dotted lines). For the Ti₂AlC₀.₈₅ sample, a spectrum recorded on a thin film prepared in cross section (i.e. [1-210] zone axis) is also shown for comparison (thin dashed black lines). The energy range corresponding to the peaks A and B is enlarged in the insets.

**Figure 5:** Decomposition of the C-K fine structures in Ti₂AlC among the different coordination shells around the excited atom. Full black lines correspond to a Ti shell, red dotted-dashed line to carbon shell and blue dotted line to an aluminum shell. The numbers on the right give the multiplicity of the atom in the considered coordination shell. Vertical dashed lines are guide for the eyes.

**Figure 6:** Resistivity of the Ti₂AlCₓNᵧ solid solutions as a function of the temperature (color online). The curve corresponding to a Ti₂AlN₀.₉ sample is also given for comparison.

**Figure 7:** (a) Residual resistivity as a function of the carbon content extracted from the curves given in Fig. 6. (b) Slope of the ideal resistivity vs temperature curve (α=dp/dT) as a function of the carbon content in Ti₂AlCₓNᵧ solid solutions (color online).

**Figure 8:** (a) Total electronic density of states (DOS) calculated for the various solid solutions. From bottom to top: Ti₂AlC (black line), Ti₂AlC₀.₈N₀.₂ and Ti₂AlC₀.₇N₀.₃ (red full and dashed lines respectively), Ti₂AlC₀.₅N₀.₅ (green line), Ti₂AlC₀.₃N₀.₇ and Ti₂AlC₀.₂N₀.₈ (orange full and dashed lines respectively) and Ti₂AlN (blue line). Site projected DOS for (b) Ti₂AlC and (c) Ti₂AlN.

**Figure 9:** (a) Residual resistivity as a function of the vacancy content in Ti₂AlCₓNᵧ systems (x+y<1). (b) Comparison between the slopes of the ρ(T) curves recorded on the stoichiometric (x+y=1) (black squares) and substoichiometric (x+y<1) (red dots) Ti₂AlCₓNᵧ systems.
<table><thead><tr><th></th><th>Orientation</th><th>β (mRad)</th><th>Core-hole</th><th>Imax</th><th>Cluster size (Å)</th></tr></thead><tbody><tr><td>Ti<sub>2</sub>AlC<sub>0.85</sub></td><td>[0001]</td><td>7.5</td><td>RPA</td><td>4</td><td>7.4 and 6.1</td></tr><tr><td>Ti<sub>2</sub>AlN</td><td>[0001]</td><td>7.5</td><td>RPA</td><td>4</td><td>7.45 and 6.85</td></tr></tbody></table>

Note that for both compounds, the cluster size has been varied in order to see its influence on the calculated fine structures.

<caption>**Table I:** Feff parameters used for the present calculations: β is the collection semi-angle (mRad) and Imax the angular momentum cutoff.</caption>
<table><thead><tr><th>Nominal composition</th><th>Processing</th><th>Phase composition</th><th>a (Å)</th><th>c (Å)</th><th>Reference</th></tr></thead><tbody><tr><td>2Ti:1.1Al:C</td><td>HIP<br>1400°C/80MPa/4h</td><td>Ti<sub>2</sub>Al<sub>0.99</sub>C<sub>0.88</sub> (85 vol.%)<br>Ti<sub>3</sub>Al<sub>0.99</sub>C<sub>1.81</sub> (13 vol.%)<br>Ti<sub>2</sub>Al<sub>y</sub> (2 vol.%)</td><td>3.0630</td><td>13.680</td><td>This work</td></tr><tr><td>2Ti:Al:C</td><td>Arc melted, annealed at<br>1000°C</td><td>Ti<sub>2</sub>AlC*,&amp;</td><td>3.0560</td><td>13.620</td><td>[10]</td></tr><tr><td>Ti<sub>2</sub>AlC powder</td><td>Pressureless sintering of<br>commercial Ti<sub>2</sub>AlC<br>powder</td><td>Ti<sub>2</sub>AlC*,&amp;</td><td>3.065±0.004</td><td>13.71±0.03</td><td>[57]</td></tr><tr><td>2Ti:1Al:1C</td><td>HIP<br>1300°C/40MPa/30h</td><td>Ti<sub>2</sub>AlC* (96 vol.%)<br>Al<sub>2</sub>O<sub>3</sub>, Ti<sub>3</sub>P (4 vol.%)</td><td>3.051</td><td>13.637</td><td>[29]</td></tr><tr><td>2Ti:1.1Al:C</td><td>Pressureless sintering<br>1400°C/4h</td><td>Ti<sub>2</sub>AlC<sub>0.98</sub></td><td>3.0625</td><td>13.668</td><td>[30]</td></tr><tr><td>2Ti:Al:C</td><td>Hot pressed at<br>1400°C/1h/30MPa</td><td>Ti<sub>2</sub>AlC*,&amp;</td><td>3.0400</td><td>13.600</td><td>[60]</td></tr><tr><td>2Ti:1.1Al:0.75C:0.25N</td><td>HIP<br>1400°C/80MPa/4h</td><td>Ti<sub>2</sub>Al<sub>0.99</sub>C<sub>0.74</sub>N<sub>0.25</sub> (96 vol.%)<br>Ti<sub>3</sub>Al<sub>0.98</sub>C<sub>1.48</sub>N<sub>0.49</sub> (3 vol.%)<br>Ti<sub>2</sub>Al<sub>y</sub> (1 vol.%)</td><td>3.0483</td><td>13.629</td><td>This work</td></tr><tr><td>2Ti:1.1Al:0.75C:0.25N</td><td>Pressureless sintering<br>1400°C/4h</td><td>Ti<sub>2</sub>AlC<sub>0.72</sub>N<sub>0.25</sub></td><td>3.0464</td><td>13.615</td><td>[30]</td></tr><tr><td>2Ti:1.1Al:0.50C:0.50N</td><td>HIP<br>1400°C/80MPa/4h</td><td>Ti<sub>2</sub>Al<sub>0.99</sub>C<sub>0.49</sub>N<sub>0.49</sub> (98.5 vol.%)<br>Al<sub>2</sub>O<sub>3</sub> (1.5 vol.%)</td><td>3.0234</td><td>13.603</td><td>This work</td></tr><tr><td>2Ti:1Al:0.50C:0.50N</td><td>HIP<br>1300°C/40MPa/15h</td><td>Ti<sub>2</sub>AlC<sub>0.5</sub>N<sub>0.5</sub> (96 vol.%)<br>Al<sub>2</sub>O<sub>3</sub>, Ti<sub>3</sub>P (4 vol.%)</td><td>3.021</td><td>13.610</td><td>[29]</td></tr><tr><td>2Ti:1.1Al:0.50C:0.50N</td><td>Pressureless sintering<br>1400°C/4h</td><td>Ti<sub>2</sub>AlC<sub>0.49</sub>N<sub>0.51</sub></td><td>3.025</td><td>13.609</td><td>[30]</td></tr><tr><td>2Ti:1.1Al:0.25C:0.75N</td><td>HIP<br>1400°C/80MPa/4h</td><td>Ti<sub>2</sub>Al<sub>0.99</sub>C<sub>0.23</sub>N<sub>0.76</sub> (98.3 vol.%)<br>Al<sub>2</sub>O<sub>3</sub> (1.7 vol.%)</td><td>3.0084</td><td>13.59</td><td>This work</td></tr><tr><td>2Ti:1.1Al:0.25C:0.75N</td><td>Pressureless sintering<br>1400°C/4h</td><td>Ti<sub>2</sub>AlC<sub>0.24</sub>N<sub>0.76</sub></td><td>3.007</td><td>13.6063</td><td>[30]</td></tr></tbody></table>
<table><tr><td>2Ti:1.1Al:1N</td><td>HIP<br/>1400°C/80MPa/4h</td><td>Ti<sub>2</sub>Al<sub>0.99</sub>N<sub>0.99</sub> (97.9 vol.%)<br/>Al<sub>2</sub>O<sub>3</sub> (2.1 vol.%)</td><td>2.9897</td><td>13.610</td><td>This work</td></tr><tr><td>Ti<sub>2</sub>AlN powder</td><td>Pressureless sintering of<br/>commercial Ti<sub>2</sub>AlN<br/>powder</td><td>Ti<sub>2</sub>AlN<sup>*,&amp;</sup></td><td>2.986±0.003</td><td>13.60±0.02</td><td>[57]</td></tr><tr><td>2Ti:1Al:1N</td><td>Hot pressing<br/>1400°C/25MPa/1h</td><td>Ti<sub>2</sub>AlN<sup>*,&amp;</sup></td><td>2.9980</td><td>13.6130</td><td>[58]</td></tr><tr><td>2Ti:1Al:1N</td><td>HIP<br/>1400°C/40MPa/48h</td><td>Ti<sub>2</sub>AlN<sup>*</sup> (80-89 vol.%)<br/>Ti<sub>4</sub>AlN<sub>3</sub>, Al<sub>2</sub>O<sub>3</sub>, Ti<sub>3</sub>P (11-20<br/>vol.%)</td><td>2.9890</td><td>13.6140</td><td>[29]</td></tr><tr><td>2Ti:1.1Al:1N</td><td>Pressureless sintering<br/>1400°C/4h</td><td>Ti<sub>2</sub>AlN</td><td>2.9890</td><td>13.6157</td><td>[30]</td></tr><tr><td>2Ti:1.1Al:1N</td><td>Pressureless sintering<br/>16h, under vacuum</td><td>Ti<sub>2</sub>AlN<sup>*,&amp;</sup></td><td>2.989(2)</td><td>13.614(5)</td><td>[59]</td></tr></table>

* C and N content non determined ; & secondary phases non determined

<caption>Table II</caption>
<table><caption>**Table III:** Ratio of the conductivity by the relaxation time obtained from the *ab initio* calculations for the different solid solutions.</caption><thead><tr><th>Composition</th><th>Ti<sub>2</sub>AlN</th><th>N<sub>0.75</sub>C<sub>0.25</sub></th><th>N<sub>0.5</sub>C<sub>0.5</sub></th><th>N<sub>0.25</sub>C<sub>0.75</sub></th><th>Ti<sub>2</sub>AlC</th></tr></thead><tbody><tr><td>σ/τ (1/(Ω.m.s)).</td><td>2.80 10<sup>20</sup></td><td>2.63 10<sup>20</sup></td><td>2.54 10<sup>20</sup></td><td>2.70 10<sup>20</sup></td><td>2.94 10<sup>20</sup></td></tr></tbody></table>
<figure><img src="image_2.png" /><figcaption>Figure 1</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6</figcaption></figure>
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 8</figcaption></figure>
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Figure 9</figcaption></figure>
References

[1] Jeitschko W, Nowotny H. Monatsh Chem 1967;98:329.

[2] Jeitschko W, Nowotny H, Benesovsky F. Monatsh Chem 1963;94:672.

[3] Jeitschko W, Nowotny H, Benesovsky F. Monatsh Chem 1964;95:431.

[4] Wolfsgruber H, Nowotny H, Benesovsky F. Monatsh Chem 1967;98:2403.

[5] Nowotny H. Progress in Solid State Chem 1971;5:27.

[6] Barsoum MW, El-Raghy T. J Am Ceram Soc 1996;79:1953.

[7] Barsoum MW, Farber L, Levin I, Procopio A, El-Raghy T, Berner A. J Am Ceram Soc 1999;82:2545.

[8] Rawn CJ, Barsoum MW, El-Raghy T, Procopio A, Hoffmann CM, Hubbard CR. Mater Res Bull 2000;35:1785.

[9] Barsoum MW. Progress in Solid State Chem 2000;28:201.

[10] Schuster J, Nowotny H, Vaccaro C. J Solid State Chem 1980;32:213.

[11] Gupta S, Barsoum M. J Electrochem Soc 2004;151:D24.

[12] Ganguly A, Zhen T, Barsoum M. J Alloys Compd 2004;376:287.

[13] Zhou Y, Chen J, Wang J. Acta mater 2006;54:1317.

[14] Finkel P, Seaman B, Harrell K, Palma J, Hettinger J, Lofland S, Ganguly A, Barsoum M, Sun Z, Li S. Phys Rev B 2004;70:085104.

[15] Dubois S, Bei GP, Tomas C, Gauthier-Brunet V, Gadaud P. Int J Appl Ceram Tech 2010;7:719.

[16] Rosén J, Dahlqvist M, Simak S, McKenzie D, Bilek M. Appl Phys Lett 2010;97:073103.

[17] Cabioch T, Eklund P, Mauchamp V, Jaouen M, Barsoum MW. J Eu Ceram Soc 2013;33:897.

[18] Buchholt K, Ghandi R, Domeij M, Zetterling C-M, Lu J, Eklund P, Hultman L, Spetz AL. Appl Phys Lett 2011;98:042108.

[19] Wang Z, Tsukimoto S, Saito M, Ito K, Murakami M, Ikuhara Y. Phys Rev B 2009;80:245303.

[20] Borysiewicz MA, Kaminska E, Piotrowska A, Pasternak I, Jakiela R, Dynowska E. Acta Phys Pol, A 2008;114:1061.

[21] Drevin-Bazin A, Barbot JF, Cabioch T, Beaufort MF. Investigation of Al-Ti Ohmic Contact to N-Type 4H-SiC. Trans Tech Publ, 2012; 711:184.

[22] Hoffman E, Vinson D, Sindelar R, Tallman D, Kohse G, Barsoum M. Nucl Eng Des 2012;244:17.

[23] Ingason AS, Mockuté A, Dahlqvist M, Magnus F, Olafsson S, Arnalds UB, Alling B, Abrikosov I, Hjörvarsson B, Persson PÅ. Phys Rev Lett 2013;110:195502.

[24] Naguib M, Mashtalir O, Carle J, Presser V, Lu J, Hultman L, Gogotsi Y, Barsoum MW. ACS nano 2012;6:1322.

[25] Naguib M, Mochalin VN, Barsoum MW, Gogotsi Y. Adv Mater 2014;26:992.

[26] Mauchamp V, Bugnet M, Bellido EP, Botton GA, Moreau P, Magne D, Naguib M, Cabioc'h T, Barsoum MW. Phys Rev B 2014;89:235428.

[27] Halim J, Lukatskaya MR, Cook KM, Lu J, Smith CR, Näslund L-Å, May SJ, Hultman L, Gogotsi Y, Eklund P. Chem Mater 2014;26:2374.

[28] Zhou Y, Meng F, Zhang J. J Am Ceram Soc 2008;91:1357.

[29] Barsoum M, El-Raghy T, Ali M. Metall Mater Trans A 2000;31:1857.

[30] Cabioch T, Eklund P, Mauchamp V, Jaouen M. J EuroCeram Soc 2012;32:1803.

[31] Radovic M, Ganguly A, Barsoum M. J Mater Res 2008;23:1517.

[32] Du Y, Sun Z, Hashimoto H, Barsoum M. Phys Lett A 2009;374:78.

[33] Scabarozi T, Ganguly A, Hettinger J, Lofland S, Amini S, Finkel P, El-Raghy T, Barsoum M. J Appl Phys 2008;104:073713.
[34] Wang X, Zhou Y. Z Metallkd 2002;93:66.

[35] Wang P, Mei B, Hong X, Zhou W. Trans Non ferr Met Soc China 2007;17:1001.

[36] Williams WS. Phys Rev 1964;135:A505.

[37] Shacklette LW, Williams WS. J Appl Phys 1971;42:4698.

[38] Eklund P, Beckers M, Jansson U, Högberg H, Hultman L. Thin Solid Films 2010;518:1851.

[39] Egerton RF. Electron energy-loss spectroscopy in the electron microscope, second Ed. New York: Plenum Press; 1996.

[40] Pennycook TJ, Oxley MP, Garcia-Barriocanal J, Bruno FY, Leon C, Santamaria J, Pantelides S, Varela M, Pennycook S. Eur Phys J Appl Phys 2011;54:345.

[41] Krayzman V, Levin I, Woicik J, Yoder D, Fischer D. Phys Rev B 2006;74:224104.

[42] Bugnet M, Mauchamp V, Eklund P, Jaouen M, Cabioc'h T. Acta mater 2013;61:7348.

[43] Calmels L, Mirguet C, Kihn Y. Phys Rev B 2006;73:024207.

[44] Yu W, Gauthier-Brunet V, Cabioc'h T, Dubois S. J Am Ceram Soc 2014;97:2308.

[45] Lutterotti L, Matthies S, Wenk H. NRC Press Ottawa, Canada, 1999.1:1599.

[46] Rehr JJ, Kas JJ, Vila FD, Prange MP, Jorissen K. Physl Chem Chem Phys 2010;12:5503.

[47] Rehr JJ, Kas JJ, Prange MP, Sorini AP, Takimoto Y, Vila F. Comptes Rendus Phys 2009;10:548.

[48] Jorissen K, Rehr J, Verbeeck J. Phys Rev B 2010;81:155108.

[49] Blaha P, Schwarz K, Madsen GKH, Kvaniscka D, and Luitz J, WIEN2k, an augmented plane wave+local orbitals program for calculating crystal properties, Technische Universität WIEN, Austria, 2001.

[50] Perdew JP, Burke K, Ernzerhof M. Phys RevLett 1996;77:3865.

[51] Arróyave R, Radovic M. Phys Rev B 2011;84:134112.

[52] Madsen GK, Singh DJ. Comput Phys Commun 2006;175:67.

[53] Cordero B, Gómez V, Platero-Prats AE, Revés M, Echeverría J, Cremades E, Barragán F, Alvarez S. Dalton transac 2008:2832.

[54] Zhou Y, Sun Z. Phys Rev B 2000;61:12570.

[55] Hug G, Jaouen M, Barsoum M. Phys Rev B 2005;71:024105.

[56] Wokulska K. J Alloys Compd 1998;264:223.

[57] Manoun B, Zhang F, Saxena S, El-Raghy T, Barsoum M. J Phys Chem Solids 2006;67:2091.

[58] Yan M, Mei B, Zhu J, Tian C, Wang P. Ceram Int 2008;34:1439.

[59] Gamarnik M, Barsoum M, El-Raghy T. Powder Diffr 2000;15:241.

[60] Lin Z, Zhuo M, Zhou Y, Li M, Wang J. Acta Mater 2006;54:1009.

[61] Mauchamp V, Hug G, Bugnet M, Cabioc'h T, Jaouen M. Phys Rev B 2010;81:035109.

[62] Magnuson M, Mattesini M, Van Nong N, Eklund P, Hultman L. Phys Rev B 2012;85:195134.

[63] Hug G, Eklund P, Orchowski A. Ultramicroscopy 2010;110:1054.

[64] Haddad N, Garcia-Caurel E, Hultman L, Barsoum MW, Hug G. J Appl Phys 2008;104:023531.

[65] Bugnet M, Jaouen M, Mauchamp V, Cabioch T, and Hug G, submitted to Phys. Rev. B.

[66] Hettinger J, Lofland S, Finkel P, Meehan T, Palma J, Harrell K, Gupta S, Ganguly A, El-Raghy T, Barsoum M. Phys Rev B 2005;72:115120.

[67] Finkel P, Hettinger J, Lofland S, Barsoum M, El-Raghy T. Phys Rev B 2001;65:035113.

[68] Sun Z, Zhou Y. Phys Rev B 1999;60:1441.

[69] Lofland S, Hettinger J, Harrell K, Finkel P, Gupta S, Barsoum M, Hug G. Appl Phys Lett 2004;84:508.

[70] Hug G. Phys Rev B 2006;74:184113.

[71] Mooij J. Phys Status solidi A1973;17:521.

[72] Seth R, Woods S. Physl Rev B 1970;2:2961.

[73] Mauchamp V, Yu W, Gence L, Piraux L, Cabioc'h T, Gauthier V, Eklund P, Dubois S. Phys Rev B
2013;87:235105.

[74] Otani S, Tanaka T, Ishizawa Y. J Mater Sci 1986;21:1011.

[75] Barsoum M, Salama I, El-Raghy T, Golczewski J, Seifert H, Aldinger F, Porter W, Wang H. Metall Mater Trans A 2002;33:2775.

[76] Barsoum M, Golczewski J, Seifert H, Aldinger F. J All Comp 2002;340:173.