# Anisotropy of the resistivity and charge-carrier sign in nanolaminated Ti 2 AlC: Experiment and ab initio calculations

Vincent Mauchamp, Wenbo Yu, Loïk Gence, Luc Piraux, Thierry Cabioc'H, Véronique Gauthier, Per Eklund, Sylvain Dubois

► To cite this version:

Vincent Mauchamp, Wenbo Yu, Loïk Gence, Luc Piraux, Thierry Cabioc'H, et al.. Anisotropy of the resistivity and charge-carrier sign in nanolaminated Ti 2 AlC: Experiment and ab initio calculations. Physical Review B, 2013, 87 (23), pp.235105. 10.1103/PhysRevB.87.235105. hal-05487661

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# Anisotropy of the resistivity and charge carriers sign in nanolaminated Ti₂AlC: experiment and ab initio calculations.

Vincent Mauchamp†,¹,* Wenbo Yu,¹ Loïk Gence,² Luc Piraux,² Thierry Cabioc'h,¹ Véronique Gauthier,¹ Per Eklund,³ and Sylvain Dubois¹

¹Institut Pprime, UPR 3346 CNRS - Université de Poitiers - ISAE-ENSMA, SP2MI Téléport 2, BP 30179, F86962 Futuroscope-Chasseneuil Cedex, France

²Institute of Condensed Matter and Nanosciences, Université Catholique de Louvain, BE-1348 Louvain la Neuve, Belgium

³Thin Film Physics Division, Linköping University, IFM, 581 83 Linköping, Sweden

The anisotropy of Ti₂AlC transport properties is investigated focusing on the Hall effect and resistivity vs temperature measurements performed on a highly (000l)-oriented thin film and a bulk polycrystalline sample. Experimental data are interpreted on the basis of Density Functional Theory calculations including transport coefficients obtained with the Boltzmann semi-classical transport equation in the isotropic relaxation time approximation. It is shown that the Hall constant is independent of the temperature and that the charge carrier sign depends on the investigated crystallographic orientation. Charge carriers exhibit a hole-like character along the basal plane of the MAX phase whereas the bulk sample Hall constant is negative. The resistivity anisotropy is also evidenced: using an effective medium approach, the room temperature basal plane resistivity is shown to be more than one order of magnitude lower than that along the *c*-axis. This very important anisotropy is shown to result from the anisotropy of the Fermi surface increased by electron-phonon interactions. These interactions are much more important along the *c*-axis than within the basal plane, a situation opposite to that observed in Ti₂GeC which resistivity was reported to be isotropic.

Keywords: 72.15.Eb, 73.50.h, 71.15.Mb

## I. INTRODUCTION

In the search for on-purpose designed materials, the ternary carbides and nitrides with general formula $M_{n+1}AX_n$ (n = 1, 2 or 3), M being an early transition metal, A an A-group element (from IIIA to VIA) and X being either C or N, are promising.¹,² These compounds, named MAX phases from their chemical formula, crystallize in the hexagonal P6₃/mmc space group and can be described as the stacking of $M_6X$ octahedra layers interleaved with pure A element layers. These structures exhibit large c/a ratios, typically 3, 6 and 8 for n = 1, 2, 3 what leads to a very anisotropic unit cell (see the unit cell of Ti₂AlC, which corresponds to n=1, sketched in figure 1(a)). Due to this nanolaminated structure, they combine the best properties of metals and ceramics demonstrating, for instance, high sustainability in extreme environment (high temperature, corrosive, oxidizing) and very good electronic properties.³⁻⁶ This unique combination of properties opens a way to diverse potential applications. Of special interest are the titanium-based compounds, intensively investigated for high temperature ohmic contacts on wide gap semiconductors.⁷⁻¹⁰

MAX phases electronic properties and especially the anisotropy resulting from the nanolaminated structure is not yet fully understood. Although intensively studied, it remains a complex issue since the majority of transport experiments were performed on polycrystalline samples thereby averaging the basal plane and *c*-axis transport properties.⁶,¹¹,¹³,³² To circumvent this point, several alternative approaches have been investigated such as the probe of single grains plasmon excitations anisotropy

<figure><img src="image_2.png" /><figcaption>FIG. 1. (Color online) (a) Unit cell and (b) Fermi surface of Ti₂AlC. For details about the calculation, see section III.</figcaption></figure>

using Electron Energy-Loss Spectroscopy (EELS),¹⁴⁻¹⁷ the comparison between data obtained on (000l) oriented thin films and bulk polycrystalline samples,¹⁸ or the comparison between thin films with different grain populations.¹⁹ In the two latter cases, the conductivity anisotropy is evidenced since the (000l) thin film room temperature (RT) resistivities and electron-phonon interactions are different from either the data recorded on the polycrystalline sample or on the thin film exhibiting a different grain population. Moreover, comparison between these results also evidence the importance of the chemical composition of the MAX phase since Scabarozi *et al.* report a basal plane RT resistivity of 0.24 µΩ m in Ti₂GeC whereas the Cr₂GeC basal plane RT resistivity reported by Eklund *et al.* is twice larger, 0.51 µΩ m.

In all cases, the transport properties were qualitatively understood using phenomenological two band models or,
at best, by comparing with band structure calculations. However, the calculations of transport properties is sometimes mandatory for a quantitative understanding of these compounds electronic properties. This was demonstrated by Chaput *et al.* who showed that the negligible Seebeck coefficient of a polycrystalline $Ti_3SiC_2$ sample could be understood from the compensation of electron-like states along the c-axis by hole-like states within the basal plane orientation at the Fermi level.⁵,²⁰,²¹ An accurate description of both the electronic structure and the transport properties is all the more desirable that MAX phases exhibit complex Fermi surfaces with generally more than two bands (see figure 1(b) for $Ti_2AlC$),²² and that the chemical bonding in these materials is based on complex charge transfers involving metallic, ionic and covalent bonds.²³

The aim of this paper is to understand the anisotropy of $Ti_2AlC$ transport properties by characterizing both charge carriers and the anisotropy of the resistivity. Temperature dependent Hall coefficients and resistivity measurements were performed on two samples : an epitaxial (000l) $Ti_2AlC$ thin film and a bulk polycrystalline sample. Experimental results are understood on the basis of band structure Density Functional Theory (DFT) calculations which are used to compute the transport properties in a semi-classical framework. The paper is organized as follows. First, the experimental details concerning sample elaboration and transport properties measurements are described. In the second part, the calculation details are given. Then, experimental results are given and compared to calculations and the anisotropy of the transport properties is discussed.

## II. EXPERIMENT AND MATERIAL PROCESSING

The epitaxial (000l) $Ti_2AlC$ MAX-phase thin films was synthesized under ultra-high vacuum (UHV) conditions by unbalanced (type II) DC magnetron sputtering using high purity elemental targets.²⁴,²⁵ The deposition process was carried out at a constant pressure of 0.5 Pa in high purity Ar (99.9999%) discharges. Thin film was grown at 900°C on $Al_2O_3$ (0001) substrates pre-seeded with a thin epitaxial TiC layer.

Powders of Ti (<45µm, 99.5% purity), Al (<45µm, 99.5% purity) and TiC (2µm, 99.5% purity), with the nominal composition 1.15Ti:1.05Al:0.85TiC, were used to synthesize $Ti_2AlC$ bulk sample. Powders were milled for 4h in a Turbula mixer (WAB). The reactant mixtures were then cold-compacted into cylindrical steel dies 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 1400°C (heating rate = 35°C/min) while the Ar pressure was in-

creased up to 80MPa (pressure rate = 1.2MPa min⁻¹).

* Once the processing temperature and pressure were reached, the samples were held for 4h.

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

After HIPing, samples were machined to remove the encapsulating glass container and sliced using a diamond wheel. Samples were thus grinded, down to about 30 µm, 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 producing a very flat surface.

Measurements of the resistivity were carried out on both samples, in the temperature range 15-295 K, in a Van der Pauw geometry. The (000l)-oriented film thickness is 140 nm whereas the polycrystalline bulk sample is 26.5 µm thick. Wavelength Dispersive x-ray Spectroscopy (WDS) was used to determine the carbon stoichiometry in the two samples. Experiments were performed at ICMCB (Bordeaux, France) with pyrolytic carbon as a standard: the carbon stoichiometries were found to be almost identical, 0.82 and 0.85 for the thin film and bulk sample respectively, with a standard deviation of 0.05. The carbon stoichiometry of the bulk sample was confirmed by WDS measurements performed at Pprime Institute (Poitiers, France) with SiC as a standard (pure SiC (> 99.9 %) provided by Micro-Analysis Consultants Ltd. (MAC) (number : S59-04-14173, part number : S4-5030-M)): the carbon content was found to be 0.84.

## III. CALCULATION DETAILS

Band structure calculations were performed with the WIEN2k code, an all electron full potential approach based on a linearized augmented plane wave (LAPW) method.²⁶ Augmented plane waves (APW) combined with local orbitals (APW+lo) were used with a converged basis set obtained for a $RK_{max}$ value of 7.5.²⁷ Calculations were performed considering the Perdew-Burke-Ernzerhof Generalized Gradient Approximation,²⁸ 800 k-points were used in the full Brillouin zone for the self-consistent field determination of the potentials. Optimized unit cell parameters were considered : a = 3.04 Å, c = 13.60 Å and $z_{Ti}$ = 0.0845, $z_{Ti}$ being the only free parameter in the space group. These values are in very good agreement, to within less than 1 %, with the experimental ones deduced from x-ray diffraction and extended absorption fine structures refinements.²³

The band structure calculations were used to compute the Fermi surface using the Xcrysden software,²⁹ and as
an input for the Hall and conductivity tensors calculations. These transport coefficients were computed in the framework of the semi-classical Boltzmann theory, considering the relaxation time approximation, in a rigid band approach using the BOLTZTRAP code.³⁰ Transport coefficient are computed from the transport distribution:

$$ \sigma_{\alpha,\beta}(\varepsilon) = \frac{1}{N} \sum_{i,\mathbf{k}} \sigma_{\alpha,\beta}(i, \mathbf{k}) \frac{\delta(\varepsilon - \varepsilon_{i,\mathbf{k}})}{d\varepsilon} \quad (1) $$

which is expressed in terms of the $\mathbf{k}$-dependent transport tensor, given by:

$$ \sigma_{\alpha,\beta}(i, \mathbf{k}) = e^2 \tau_{i,\mathbf{k}} v_{\alpha}(i, \mathbf{k}) v_{\beta}(i, \mathbf{k}) \quad (2) $$

where $\tau_{i,\mathbf{k}}$ is the relaxation time and $v_{\alpha,\beta}(i, \mathbf{k})$ the group velocity. The transport coefficients are then calculated as a function of the temperature and the chemical potential by integrating the transport distribution (see ref 30 for details). Since the details of the scattering mechanisms are unknown, so is $\tau_{i,\mathbf{k}}$. In the present approach, it is assumed to be constant and isotropic so that the Hall coefficient becomes independent of $\tau$ and can be compared directly to the experiment. Concerning the conductivity, we only have access to the Fermi surface dependent quantity: $\zeta_{ii} = \sigma_{ii}/\tau_{ii}$. Note that, although $\zeta_{ii}$ cannot be compared directly to the experimental conductivity, for isotropic relaxation time the ratio $\zeta_{xx}/\zeta_{zz}$ is independent of $\tau$ and can be compared to the ratio of the experimental conductivities. The isotropic relaxation time approximation was successfully used for the calculation of the anisotropy of 312 MAX phases thermopower. Calculations were performed using a fine k-mesh of 200000 k-points in the full Brillouin zone ensuring a convergence of the Hall coefficient to within 10 %.

# IV. RESULTS AND DISCUSSION

## A. Hall constants

The Hall resistivity recorded as a function of the magnetic field at 25K, 150K and RT on the Ti₂AlC (000l) thin film and at RT on the bulk sample is sketched in figure 2. Only RT measurements could be performed on the bulk sample since the Hall voltage was very weak (typically a hundred times lower than that for the (000l) film), so that temperature variations led to important fluctuations. In all cases, the variation of the Hall resistivity is linear whereas its sign depends on the relative orientation of the current and the Ti₂AlC crystal. For the current flowing in the Ti₂AlC basal plane, the Hall resistivity is definitely positive whereas it is negative for a current flowing in the Ti₂AlC bulk sample. The Hall constants extracted from these curves are -3.4 $10^{-10}$ and +2.2 $10^{-10} m^3 C^{-1}$ for the bulk sample and

<figure><img src="image_3.png" /><figcaption>FIG. 2. (Color online) Hall resistivity recorded as a function of the magnetic field at 25K (black squares), 150K (blue triangles) and RT (red diamonds) on the (000l) Ti₂AlC thin film and at RT on the bulk sample.</figcaption></figure>

thin film respectively. The experimental bulk value is in good agreement with the one measured by Scabarozi et al. on a similar sample, $-2.8 \ 10^{-10} m^3 C^{-1}$ with very weak variations with the temperature.¹³

Phenomenologically, the weak temperature dependence of the basal plane Hall constant, as measured on the film, justifies the interpretation of basal plane MAX phase transport properties in terms of a single band model (hole-like states in Ti₂AlC). On the other hand, the opposite sign of the trace of the Hall tensor compared to the one of the basal plane Hall constant justifies the interpretation of MAX phase transport properties in terms of a two band model, with electron-like and hole-like states respectively along c-axis and in the basal plane. This model is consistent with the hole-like and electron-like behaviors observed respectively in the basal plane and along the c-axis of M₃AX₂ phases.²² It is thus possible to extract, from the basal plane Hall constant measurement, the charge carriers density. In this single band model, hole-like carriers density (p) can be estimated from the relation $R_H = 1/pe$ which leads to $2.8 \ 10^{+28} m^{-3}$. This values is in quite good agreement with results deduced from heat capacity measurements ($\approx 3.8 \ 10^{28} m^{-3}$).³¹,³² The deduced charge carrier mobility in the basal plane is thus, at RT, $1.1 \ 10^{-3} m^2 V^{-1} s^{-1}$ (estimated from the ideal resistivity at RT, see Fig. 4b).

By comparing these results with the ones obtained by Scabarozi et al. who assume a two band model,¹³ one can notice that our values are one order of magnitude higher for the hole-like concentration and five times lower for the mobility. Such a discrepancy can be ascribed to their hypothesis of equal electron-like and hole-like mobilities. This equality is questionable given the important band dispersion anisotropy evidenced in the band structure sketched in Fig. 3: the bands show an important dispersion along reciprocal lattice directions corresponding to the basal plane ($\Gamma \to M$ or $\Gamma \to K$) whereas their dispersion is very weak along directions corresponding to
<figure><img src="image_4.png" /><figcaption>FIG. 3. (Color online) Ti$_{2}$AlC site projected density of states and the corresponding band structure calculated along the high symmetry points of the first Brillouin Zone.</figcaption></figure>

the c-axis ($\Gamma \to A, H \to K$ or $L \to M$). This results in an important anisotropy in the band velocities. Moreover, as will be seen in the next paragraph, the electron-phonon interaction is also very anisotropic.

The experimental Hall constants are compared to the *ab initio* calculations in table I. Assuming that the bulk sample displays randomly oriented grains, the $R_H$ value is compared to the trace of the Hall tensor whereas the thin film data are compared to the $R_{H,xyz}$ component determined from the geometry of the experiment. The calculated values are in qualitative agreement with the experiments. The $R_H$ signs are confirmed for both the (000l) thin film and the bulk sample, confirming that the transport properties along the MAX phase basal plane are governed by hole-like electron states. However, the relative error between the experimental data and the calculations in the thin film is much higher than that for the bulk sample: 50 and 10 % respectively. The error for the bulk sample is very similar to that resulting from the k point sampling which gives strong support to the calculation. In contrast, the error between the theoretical value of $R_{H,xyz}$ and the experimental one is almost 50 %. One possible explanation is the failure of the isotropic relaxation time approximation used for the transport properties calculations. The good results obtained for the bulk sample could then be due to cancellation of errors between the different components of the Hall tensor when computing the trace. This is in agreement with recently published results on the Seebeck coefficient of Ti$_{3}$SiC$_{2}$,²¹ where it is shown that the in-plane component obtained with similar calculations is 25 % lower than the experimental value at room temperature whereas the trace of the Seebeck coefficient gave very good results compared to experimental data obtained on a polycrystalline sample.²⁰

The validity of the isotropic relaxation time approximation is however not directly comparable between Hall constants and Seebeck coefficient. Indeed, the application of a magnetic field in the former case modifies the

<figure><img src="image_5.png" /><figcaption>FIG. 4. Resistivity vs temperature: (a) raw data obtained on the (000l) thin film and the bulk polycrystalline sample, (b) corresponding ideal resistivities.</figcaption></figure>

symmetry constraints that the relaxation time must fulfill. Chaput et. al. investigated this problem in elemental zinc and showed that if the relaxation time is assumed to be constant for the calculation of the Hall tensor, then it must also be isotropic.³³ Our result suggests that the relaxation time is most probably anisotropic so that a constant relaxation time approximation is not satisfactory for the description of the Hall tensor. In particular, although dominated by the titanium d bands, the complex electronic structure at the Fermi level of Ti$_{2}$AlC also involves aluminum and carbon character (see the site projected density of states in Fig. 3), so that one would expect a k dependent relaxation time (i.e. dependent of the Bloch state character). Such an approach involving different relaxation times for the different angular momenta of the wave functions at the Fermi level was demonstrated to greatly improve the calculation of the aluminum Hall constant when compared to the experiment.³⁴

## B. Resistivity

Figure 4 (a) shows the Ti$_{2}$AlC resistivity variations as a function of the temperature. The RT and residual resistivities are 36 µΩ cm and 16 µΩ cm respectively for the (000l)-oriented thin film and 65 µΩ cm and 27 µΩ cm for the bulk sample. The RT and residual resistivities of the (000l)-oriented film are in the same range as the ones determined on bulk samples by Hettinger et al. and very close to that reported for a thin film by Magnuson et al. (40µΩ cm).¹¹,³⁵ The values obtained on the bulk sample are larger. This implies that defects limit the mean free path at low temperature. By using Mathiessen's rule, one can extract the ideal resistivity $\rho_i$ which is given by:

$$ \rho_i(T) = \rho(T) - \rho_0 \quad (3) $$

$\rho_0$ being the residual resistivity. The ideal resistivity, depicted in figure 4 (b), only depends on the charge carrier scattering mechanism (i.e. it does not depend on
<table><caption>TABLE I. Ti₂AlC temperature dependent Hall constants ($R_H$), room temperature ideal resistivities ($\rho$) and slopes of the $\rho(T)$ curves measured on the bulk sample and the (000l) oriented thin film. The Hall constants obtained from the ab initio calculations are also given for comparison. The resistivity along the c-axis deduced from the effective medium (E.M.) approach considering elongated spheroids with an aspect ratio of 2.5 is also given (see section IV B for details).</caption><thead><tr><th>Sample</th><th>R<sub>H,exp</sub> (m<sup>3</sup> C<sup>-1</sup>)</th><th>R<sub>H,Th</sub> (m<sup>3</sup> C<sup>-1</sup>)</th><th>ρ (μΩ cm)</th><th>∂ρ/∂T (μΩ cm K<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Bulk sample</td><td>-3.4 10<sup>-10</sup> at RT</td><td>-3.7 10<sup>-10</sup></td><td>37</td><td>0.18</td></tr><tr><td>(000l)-oriented sample</td><td>+2.2 10<sup>-10</sup> at 25K, 150K and RT</td><td>1.0 10<sup>-10</sup></td><td>20</td><td>0.10</td></tr><tr><td>c-axis (deduced from E.M.)</td><td></td><td></td><td>347</td><td>1.75</td></tr></tbody></table>

defect and impurity densities). At RT, the ideal resistivities are 20 and 37 µΩ cm for the (000l) film and the bulk sample respectively (the bulk sample resistivity is in excellent agreement with the one reported by Scabarozi et. al.¹³). These values are summarized in table I. This evidences the resistivity anisotropy of this nanolaminated compound. As usually observed in MAX phases, $\rho_i$ varies linearly with the temperature in the range 150-300 K and one can calculate the slope of the linear variation. The slopes are 0.10 and 0.18 µΩ cm K⁻¹ for the (000l)-oriented thin film and the bulk polycrystalline sample respectively.

The thin film resistivity is in good agreement with measurements carried out on other Ti₂AlC samples for which $\partial\rho/\partial T$ is in the range 0.067-0.1 µΩ cm K⁻¹. Nevertheless, one can notice that the slope obtained for the bulk sample is higher than previously reported values. Such a behavior can be ascribed to the presence of vacancies and the disorder that they induce in the structure. That vacancies are strong electron scatterers is evidenced in the high residual resistivity that we find, 27 µΩ cm, compared to that reported by Scabarozi et al., 7 µΩ cm. Although, the details of the scattering mechanism is unknown, equivalent increase of the $\rho(T)$ slope with disorder or vacancies concentration is also observed in metals

or semi metals exhibiting the same linear $\rho(T)$ tendency as that observed in MAX phases.³⁶,³⁷

Given that the thin film and the bulk sample exhibit very similar compositions, one can extract the c-axis resistivity ($\rho_{zz}$) from these two curves. As Scabarozi et al. did for Ti₂GeC,¹⁸ we use an effective medium (E.M.) approach assuming spherical grains:³⁹

$$ \frac{2(\sigma_b - \sigma_{xx})}{2/3\sigma_b + 1/3\sigma_{xx}} + \frac{(\sigma_b - \sigma_{zz})}{2/3\sigma_b + 1/3\sigma_{zz}} = 0 \quad (4) $$

where $\sigma_b$ is the bulk conductivity. The corresponding curve is plotted in Fig. 5. Contrary to what is observed for Ti₂GeC, the E.M. approach evidences a very important anisotropy of the resistivity, $\rho_{zz}$ being more than 20 times higher than $\rho_{xx}$. This value most probably overestimates the anisotropy because of the assumption of spherical grains which is not realistic considering the scanning electron microscope (SEM) image of the sample (see Fig. 5). In order to investigate the influence of the grains shape, $\rho_{zz}$ has been extracted from an E.M. approach considering elongated spheroid grains with an aspect ratio $a/b = 2.5$ or $a/b = 5$, systems much closer to the true configuration. In such cases, the equation fulfilled by $\sigma_{xx}$, $\sigma_{zz}$ and $\sigma_b$ is:³⁹

$$ p \left[ \frac{\sigma_{xx} - \sigma_b}{\sigma_b + g_{//}(\sigma_{xx} - \sigma_b)} + \frac{2(\sigma_{xx} - \sigma_b)}{\sigma_b + g_{\perp}(\sigma_{xx} - \sigma_b)} \right] + (1-p) \left[ \frac{\sigma_{zz} - \sigma_b}{\sigma_b + g_{//}(\sigma_{zz} - \sigma_b)} + \frac{2(\sigma_{zz} - \sigma_b)}{\sigma_b + g_{\perp}(\sigma_{zz} - \sigma_b)} \right] = 0 \quad (5) $$

In this expression, p is the volume fraction of grains oriented along the basal plane. Supposing that the grain orientation is averaged in the bulk sample, which according to the results obtained on the trace of the Hall constant seems to be a reasonable approximation, we took $p = 2/3$. $g_{//}$ and $g_{\perp}$ are the depolarization factors corresponding to the major axis and the transverse direction of the spheroid. The expressions of these factors can be found in Ref. 40 sec. 4. For the particular case of spherical grains, one has $g_{//} = g_{\perp} = 1/3$ and Eq. 5 reduces to Eq. 4. The $\rho_{zz}$ deduced from this Eq. 5 for aspect ratios $a/b = 2.5$ and 5 are compared to the E.M. with spherical grains in Fig. 5. Although $\rho_{zz}$ is reduced when considering elongated grains, it is still far more important than

the basal plane resistivity with RT resistivities ranging from 347 µΩ cm to 290 µΩ cm depending on the aspect ratio considered in the effective model. Compared to $\rho_{xx}$ the deduced resistivity is more than one order of magnitude larger. The slope of $\rho_{zz}(T)$ is also very important compared to that obtained along the basal plane. The values vary from 1.97 µΩ cm K⁻¹ for the model with spherical grains to 1.59 µΩ cm K⁻¹ for the spheroids with $a/b = 5$. These values are here again much higher than that obtained on the thin film (0.10 µΩ cm K⁻¹).

Although our E.M. approach can only give a crude estimate of the anisotropy, the results obtained are in reasonable agreement compared to the RT optical conductivities obtained from ellipsometry.³⁸ In these experiments
<figure><img src="image_6.png" /><figcaption>FIG. 5. (Color online) $\rho_{zz}$ obtained from the effective medium approach considering spherical grains (blue dotted-dashed line), spheroids with an aspect ratio $a/b=2.5$ (red dots) and $a/b=5$ (green dashed line). For comparison $\rho_{xx}$ and $\rho_{bulk}$ are also given: grey and black curve respectively. Inset: Resistivity anisotropy, $\rho_{zz}/\rho_{xx}$, deduced from these curves and compared to the Fermi surface anisotropy obtained from $\zeta_{xx}/\zeta_{zz}$. A scanning electron microscopy micrograph is also given to show the grain shape of the bulk sample.</figcaption></figure>

performed in reflexion, the signal retrieved from the poly-
crystalline sample is the average between the basal plane
and c-axis response: $\rho_{bulk} = \frac{1}{3}(2*\rho_{xx}+\rho_{zz})$. This is a big
difference with transport measurements where the cur-
rent flow is influenced by the respective resistivity of the
different grains and explains why the resistivity deduced
from ellipsometry is generally superior to that obtained
from four points probe measurements in MAX phases.
Taking the $\rho_{xx}$ and $\rho_{zz}$ values deduced from our models,
we obtain an average resistivity of 130 $\mu\Omega$ cm taking an
aspect ratio $a/b = 2.5$ and 110 $\mu\Omega$ cm taking an aspect
ratio $a/b = 5$. These values are in qualitative agreement
with the ellipsometry data: 84 $\mu\Omega$ cm. Here again, the
higher value that we obtain can be explained by the dis-
order induced by the vacancies. This phenomenon has
already been observed in niobium carbides for instance
where the optical conductivity is reduced by almost a
factor of three for 13 % of vacancies on the carbon site
when compared to the stoichiometric compound.⁴¹

In order to get more insights into Ti₂AlC resistivity
anisotropy, the ρ<sub>zz</sub>/ρ<sub>xx</sub> ratio is plotted as a function
of the temperature in the inset of figure 5 and com-
pared to the anisotropy of the Fermi surface velocities
distribution as deduced from the ratio ζ<sub>xx</sub>/ζ<sub>zz</sub>. It ap-
pears clearly that ρ<sub>zz</sub>/ρ<sub>xx</sub> is large, between 14 and 18,
and weakly dependent on the temperature in the 150-
300 K temperature range. This anisotropy is a combi-
nation of the Fermi surface anisotropy (see Fig. 1) and
the scattering mechanisms. From the electronic struc-
ture point of view, Ti₂AlC already evidences an impor-
tant resistivity anisotropy since the ζ<sub>xx</sub>/ζ<sub>zz</sub> ratio gives

a resistivity almost five times higher along the c-axis
than within the basal plane. This anisotropy is dras-
tically increased to more than one order of magnitude
by the scattering mechanisms which, from the linear de-
pendence of the ρ(T) curves, are most probably domi-
nated by electron-phonon interactions: these results re-
veal a stronger electron-phonon interaction along the c-
axis than within the basal plane of the MAX phase as pre-
viously observed in Cr₂GeC.¹⁹ They also evidence the im-
portant interplay between electronic structure and scat-
tering mechanisms in the MAX phases transport proper-
ties. Note that in other materials such as elemental zinc,
the electron-phonon interaction tends to compensate the
Fermi surface anisotropy so that this material exhibits a
very weak resistivity anisotropy.³³ Such a mechanism is
a possible explanation for the very different behavior ob-
served between Ti₂GeC and Ti₂AlC: in Ti₂GeC the slope
of ρ(T) for the bulk sample is lower than that recorded
on (000l) thin films (0.084 μΩ cm K⁻¹ and 0.1 μΩ cm
K⁻¹ respectively) evidencing weaker electron-phonon in-
teractions along the c-axis than within the basal plane in
this compound.¹⁸ This is totally different from what we
observe in Ti₂AlC.

V. SUMMARY AND CONCLUSION

The anisotropy of Ti₂AlC transport properties has
been investigated focusing on a (000l) oriented thin film,
giving access to the basal plane response of the MAX
phase, and a bulk highly pure polycrystalline sample of
similar composition. Both temperature dependent Hall
effect and resistivity measurements were performed and
interpreted thanks to *ab initio* calculations.

Concerning Hall effects measurements, the anisotropy
of the Fermi surface resulting from the nanolaminated
structure is evidenced through a change of the charge
carriers sign with the crystallographic orientation. Given
that the Hall constants of both samples are weakly de-
pendent on the temperature, our results justify the in-
terpretation of MAX phases polycrystalline bulk samples
transport properties in terms of a two band model, one
describing the hole-like behavior within the basal plane
and one describing the electron-like behavior along the
c-axis of the structure.11,13 This behavior is moreover
consistent with that observed in 312 MAX phases.20,22
Considering a single band model and a hole-like state
to account for the basal plane Hall constant, the charge
carriers density is found to be 2.8 10<sup>+28</sup> m<sup>-3</sup>, a value in
good agreement with others deduced from heat capacity
measurements.31,32 The charge carriers mobility is found
to be 1.1 10<sup>-3</sup> m<sup>2</sup> V<sup>-1</sup> s<sup>-1</sup>, a value five times higher
than that reported by other authors.13 We attribute this
difference to the fact that they considered equivalent mo-
bilities for electron and holes in their two band models
whereas a careful analysis of either the Fermi surface or
the bandstructure, reveals an important band velocity
anisotropy when focusing on reciprocal directions corre-
sponding to the basal plane (hole like states) or c-axis (most probably electron-like states).

The calculations qualitatively reproduce the experimental results: the charge carrier signs are confirmed for both the thin film and the bulk sample. The agreement on the absolute value of the Hall constant is however dependent on the crystallographic orientation. It is in good agreement for the bulk sample but the discrepancy is much more pronounced for the $R_{hxyz}$ component corresponding to the thin film data (the relative error is worse than 50 %). This discrepancy is understood in terms of a failure of the isotropic relaxation time approximation used in our calculations, which would be canceled in the calculation of the Hall tensor trace. This explanation is coherent with $Ti_3SiC_2$ Seebeck coefficient data which were recently published.²¹

The resistivity measurements also clearly evidence a strong anisotropy of $Ti_2AlC$ electronic properties. Comparing the data recorded on the thin film to those obtained on the bulk sample, which are interpreted in terms of an effective medium approach, it is shown that the room temperature resistivity $\rho_{zz}$ is more than one order of magnitude larger than $\rho_{xx}$. The slopes of the $\rho(T)$ curves confirm such an anisotropy, $\partial\rho_{zz}/\partial T \approx 1.75 \ \mu\Omega$ cm $K^{-1}$ whereas $\partial\rho_{xx}/\partial T \approx 0.1 \ \mu\Omega$ cm $K^{-1}$. Compar-

ison of the experimental data with the anisotropy of the Fermi surface deduced from the *ab initio* calculations reveal that the observed anisotropy is a cumulated effect of the band structure anisotropy and the electron-phonon interaction. This behavior is significantly different from that evidenced in $Ti_2GeC$ with a similar approach where the resistivity was found to be almost isotropic. These results evidence the crucial role of the chemical composition on the transport properties in the MAX phase family. Since in $Ti_2GeC$, the electron-phonon interaction is found to be weaker along the c-axis than within the basal plane, it can be an explanation for the reduced anisotropy observed in this sample. Similar effect was observed in elemental zinc for instance.³³

## VI. ACKNOWLEDGEMENTS

Jenny Frodelius is acknowledged for providing the $Ti_2AlC$ thin film sample. P. E. acknowledges financial support from the Swedish Research Council (VR) through the Linnaeus LiLi-NFM Strong Research Environment and Project Grant no. 621-2009-5258. Michel Lahaye from ICMCB (Bordeaux) and Dominique Eyidi from Pprime Institute (Poitiers) are acknowledged for performing WDS experiments on the $Ti_2AlC$ samples.

* vincent.mauchamp@univ-poitiers.fr

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