# Key role of electron-phonon interactions in the electronic conductivity of Ti$_3$SiC$_2$: Experiment and ab initio calculations

A. Nassour, V. Mauchamp, W. Yu, T. Cabioch, L. Piraux, V.

Gauthier-Brunet, S. Dubois

► To cite this version:

A. Nassour, V. Mauchamp, W. Yu, T. Cabioch, L. Piraux, et al. Key role of electron-phonon interactions in the electronic conductivity of Ti$_3$SiC$_2$: Experiment and ab initio calculations. Physical Review B, 2016, 93 (8), pp.085119. 10.1103/PhysRevB.93.085119. hal-04449682

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Key role of electron-phonon interactions in the electronic conductivity of
Ti₃SiC₂: experiment and *ab initio* calculations

A. Nassour<sup>a</sup>, V. Mauchamp<sup>a</sup>, W. Yu<sup>a</sup>, T. Cabioch<sup>a</sup>, L. Piraux<sup>b</sup>, V. Gauthier-Brunet<sup>a</sup> and S.
Dubois<sup>a</sup>

<sup>a</sup>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

<sup>b</sup>Institute of Condensed Matter and Nanosciences,
Université catholique de Louvain,
Place Croix du Sud,
B-1348 Louvain-la-Neuve, Belgium

Abstract

The electronic conductivity anisotropy of Ti₃SiC₂ is directly evidenced from data collected on (i) a thin film epitaxially grown on a (11-20)-oriented SiC single crystal and (ii) a single crystal. Density Functional Theory calculations, including linear response approach, and coupled to a Bloch-Grüneisen model show that the electron-phonon interactions are mainly responsible for the observed anisotropy. Detailed analysis of the electron-phonon coupling constants allows for the rationalization of these scattering processes in terms of the Ti₃SiC₂ nanostructure, giving insights into the possibility to modify the electron-phonon interaction in this system by substitution effects.
In the large family of M<sub>n+1</sub>AX<sub>n</sub> phases, a class of nanolaminated ternary carbides and nitrides (X) built on the stacking of n M<sub>6</sub>X octahedra layers separated by planes of A atoms (M being a transition metal and A an element from columns IIIA to VIA),<sup>1</sup> Ti<sub>3</sub>SiC<sub>2</sub> is probably one of the most attractive material. In addition to its outstanding electronic properties, which include negligible thermopower over a wide temperature range in its polycrystalline form,<sup>2,3</sup> Ti<sub>3</sub>SiC<sub>2</sub> could be a key element in the development of the SiC-based microelectronics industry because of its role in the formation of good ohmic contacts.<sup>4,5</sup> Ti<sub>3</sub>SiC<sub>2</sub> also evidences promising properties in terms of solar absorbers, electromagnetic interference shielding applications<sup>6</sup> and has been shown to improve the performances of lithium-ion battery electrodes.<sup>7,8</sup> All these applications rely on the fact that Ti<sub>3</sub>SiC<sub>2</sub> in particular, and MAX phases in general, 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 among others. In this context, the rationalization of the MAX phases' electronic properties (and particularly that of Ti<sub>3</sub>SiC<sub>2</sub>), which are intimately associated to their highly anisotropic structure, is of crucial interest.

To date, several techniques have been used to probe the electronic properties anisotropy of
these nanolaminated compounds at various scales. Electron energy-loss spectroscopy (EELS)<sup>9</sup>
was used to investigate their dielectric response at the nanometer scale,<sup>10,11</sup> and eventually
estimate their conductivity anisotropy<sup>12</sup> which was found to be around 2-2.5 for Ti<sub>2</sub>AlC and
Ti<sub>2</sub>AlN.<sup>12,13</sup> Other studies were based on polarized X-ray emission or absorption
spectroscopies<sup>14,15</sup> in order to investigate the anisotropy of the site projected density of states
of the different M, A and X elements.
Concerning transport properties, it is only very recently that a huge conductivity anisotropy could be evidenced on V<sub>2</sub>AlC and Cr<sub>2</sub>AlC single crystals<sup>16</sup> whereas previous studies, based on indirect estimations from data collected on polycrystalline samples and thin films, showed more modest anisotropy and even isotropic behaviors.<sup>17,18</sup> These results obviously call for theoretical support. Ground-state electronic structure calculations show that the electronic carriers at the Fermi level are mostly the transition metal *d* states leading some authors to predict better transport properties along the basal planes than along the *c*-axis of the hexagonal structure.<sup>19,20</sup> However, Hug *et al* mentioned that strongly localized *d* states are not expected to have high mobilities.<sup>21</sup> From these results: (i) it is by now well-accepted that MAX phases transport properties cannot be understood from ground-state electronic structure calculations, but are significantly affected by electron-phonon scattering processes, (ii) it is necessary to conduct a thorough study to understand the intrinsic transport properties of these materials.

In this paper we thus provide deep insights into the transport properties of Ti<sub>3</sub>SiC<sub>2</sub> by studying the electrical resistivity and electron-phonon coupling projected either within the basal plane or along the stacking axis of the nanolaminated structure. The basal plane and out of basal plane resistivity measurements were performed on the same (11-20)-oriented Ti<sub>3</sub>SiC<sub>2</sub> thin film deposited onto a SiC single crystal (SC). Experimental results are understood on the basis of Density Functional Theory (DFT) pseudopotential calculations performed within a generalized gradient approximation.<sup>22</sup> These calculations are coupled to a linear-response scheme<sup>23</sup> and a semi-classical Bloch-Grüneisen modeling of electron-phonon scattering<sup>24,25</sup> in order to extract and analyze the electron-phonon coupling constants and Eliashberg function.
The Ti₃SiC₂ thin film⁵ has been grown on n-type (11-20) 4H-SiC single crystal with a mirror-polished surface supplied by CREE Research Inc. Compared to reference 5, the use of a mirror-polished surface allows getting a (11-20)-oriented Ti₃SiC₂ thin film of good enough quality to get the intrinsic transport properties of Ti₃SiC₂ (see below and the supplemental material). The Ti₃SiC₂ film thickness is 130 nm. A Ti₃SiC₂ SC was synthesized from high temperature solution growth in the Materials and Physical Engineering Laboratory in Grenoble.²⁶ Resistivity measurements ρ(T) were carried out in the 15-295 K temperature range, in a four-point probe geometry. The inset of figure 1 shows that the thin film has been cut in order to measure, on two part of the same sample, the in-plane and out-of-plane resistivities. With sample 1 (on the left of figure 1), the current is flowing along the c-axis (along E<sub>z</sub>) and the voltage is measured in between traces of the basal planes (schematically presented as blue parallel lines in figure 1). With sample 2, the current is flowing in the basal plane (along E<sub>x</sub>) and the voltage is measured in between two points along the x-axis.

Fig. 1 shows ρ(T) measured with the current flowing in the [000ℓ] direction and in the basal plane of the same Ti₃SiC₂ (11-20)-oriented thin film. The resistivity of the Ti₃SiC₂ SC (current flowing in the basal plane) is also plotted for comparison. In the 100-300K temperature range, ρ(T) is a linear function of T as generally observed for MAX phases.²⁷ This typical metal-like behavior results from the large density of states at the Fermi level and from the predominant electron-phonon interactions.²⁸ The room temperature (RT) resistivities of the Ti₃SiC₂ SC and thin film (current flowing in the basal plane) are respectively 11 and 20 μΩ.cm and the dp/dT values in the temperature range 150-300K, are respectively 0.048 and 0.049 μΩ.cm.K⁻¹: the quality of the Ti₃SiC₂ (11-20)-oriented thin film is thus good enough to get the intrinsic transport properties of the material. The intrinsic RT anisotropy has been extracted from the experiments: although significant (ρᵢᵢz/ρᵢᵢxx~ 3), this anisotropy is much
lower than that reported for Ti₂AlC, V₂AlC or Cr₂AlC, evidencing the influence of the MAX phases nanostructure on their electronic properties.

In addition, the dp/dT value calculated in the same temperature range, in the case of the current flowing along the c-axis direction, is 0.118 µΩ.cm.K⁻¹. Such a value is more than twice the value measured for the current flowing in the basal plane. This result implies that the charge carriers mobility is more temperature dependent along the c-axis than along the basal plane, evidencing the key role of electron-phonon interactions. It is worth noticing that the average value of dp/dT extracted from our data (i.e.: (2dpₓₓ/dT+dρₓₓ/dT)/3 = 0.072 µΩ.cm.K⁻¹) is equal to the dp/dT measured on polycrystalline samples¹,²⁹.

In order to get a quantitative understanding of these scattering processes, the temperature dependent electrical resistivity for the two orientations is analyzed in terms of the Boltzmann transport theory using the Bloch-Grüneisen(BG) model:²⁴,²⁵

$$ \rho(T) = \rho(0) + \rho(\theta_{BG}) \left(\frac{2T}{\theta_{BG}}\right)^5 \int_{0}^{\frac{\theta_{BG}}{2T}} \frac{x^5}{\sinh^2(x)} dx; \quad (1) $$

where ρ(0) is the residual resistivity due to the electron-impurity and electron-defect scattering and the second term is the temperature-dependent intrinsic resistivity, calculated from electron-phonon scattering,³⁰ θ<sub>BG</sub> is the effective Debye temperature determined from experimental resistivity data, ρ(θ<sub>BG</sub>) = 4πλ<sub>tr</sub>ω<sub>D</sub>/ε₀ω<sub>p</sub><sup>2</sup> is the resistivity at the effective Debye temperature, λ<sub>tr</sub> is the electron-phonon coupling constant and ω<sub>D</sub>=θ<sub>BG</sub>k<sub>B</sub>/ħ and ω<sub>p</sub> are the Debye and Drude plasma frequencies.

The Drude plasma frequencies entering eq. (1), and corresponding to the two components of Ti₃SiC₂ dielectric tensor (i.e., ω<sub>p</sub><sup>xx</sup> and ω<sub>p</sub><sup>zz</sup>), have been obtained from linearized augmented plane wave electronic band structure calculations.³¹ It is worth noticing that the value deduced from the trace of the dielectric tensor, (ħω<sub>p</sub> = 4.0 eV) is in very good agreement with that
obtained from monochromated EELS experiments performed on polycrystalline Ti₃SiC₂ (ħωₚ = 4.13 eV),¹³ giving strong support to our approach.

The only two free-parameters ($ρ(θ_{BG})$ and $θ_{BG}$) involved in the BG model were then adjusted to minimize the relative error defined as $(ρ^{EXP} - ρ^{FIT})/ρ^{EXP}$. Fig. 2 shows the experimental and fitted ideal electrical resistivity variations as a function of temperature for the two orientations of the Ti₃SiC₂ thin film. The value of $θ_{BG}$ determined from the fits is 520 K (480K for the SC) and 425 K in the basal plane and out of the basal plane, respectively. One can notice that $θ_{BG}$ is different for the current flowing along the c-axis and along the basal plane: such an anisotropy has already been reported in hexagonal systems.²⁵ Despite this difference, the $θ_{BG}$ values extracted for the different samples show good agreement underlying the self-consistency of these results. Our values of $θ_{BG}$ are different from those obtained from the specific heat data³¹ ($θ_D$ = 620 K) or the second moment of phonon density of states obtained from the density functional perturbation theory³⁰ ($θ_D$ = 740 K). Such differences can be explained by the fact that the BG model involves several hypothesis and approximations: (i) the electron-electron scattering processes are ignored, (ii) Only longitudinal phonons are supposed to contribute to electron-phonon scattering, (iii) phonons are assumed to be in thermal equilibrium. In addition, the heat capacity was determined at constant pressure however Debye theory describes the heat capacity at constant volume; this may also explain the observed differences.

To identify the origin of the anisotropic electrical transport properties of Ti₃SiC₂, the electron-phonon coupling constants ($λ$) deduced from the $ρ(θ_{BG})$ and the components of the plasma frequency (i.e., $ω_p^{xx}$ and $ω_p^{zz}$) are extracted and analyzed. The experimental electron-phonon coupling constants of Ti₃SiC₂ are 0.34 (0.33 for the SC) and 0.70 in the basal plane and out of the basal plane, respectively. These results highlight an electron-phonon coupling twice larger
along the c-axis compared to the basal plane of the MAX phase. Since the components of the
plasma frequency in the basal plane ($\hbar\omega_p^{xx} = 4.12$ eV) and out of the basal plane ($\hbar\omega_p^{zz} =
3.75$ eV) obtained from *ab initio* calculations³¹ are almost identical, our results evidence the
crucial role of the electron-phonon coupling in the resistivity anisotropy of Ti₃SiC₂.

The average electron-phonon coupling calculated by first-principles,³⁰ $\lambda$= 0.35, is in
reasonable agreement with the value $(2\lambda_{xx} + \lambda_{zz})/3=0.46$ obtained from the experimental
resistivity. Moreover, these results are in very good agreement with electron-phonon coupling
constant (about 0.4) experimentally determined in a large number of polycrystalline MAX
phases samples¹. From the good agreement between the calculated and fitted values of $\lambda$, the
*ab initio* calculations can be used to identify the atoms contributing in the electron-phonon
coupling. This is done by focusing on the lattice dynamics obtained from DFT coupled to a
linear-response approach.²⁹ The calculated phonon dispersion relations are sketched in Fig. 3-
a: all optical phonon frequencies are positive throughout the Brillouin zone, which
demonstrates that Ti₃SiC₂ is dynamically stable. Fig. 3-b shows the total and partial phonon
density of states (PhDOS) together with the Eliashberg spectral function $\alpha^2 F(\omega)$ and integral
electron-phonon interaction $\lambda(\omega)$ as a function of frequency. The profile of the spectral
function is reasonably similar to that of the total PhDOS which indicates that all phonon
modes provide different contributions to the electron-phonon interaction. A study of the
partial PhDOS exhibits two well-separated bands: the low frequency region (100-430 cm⁻¹)
and a high optical region (550–700 cm⁻¹) which is mainly occupied by C atoms. As evidenced
in Fig. 3-b, nearly 80% of the electron-phonon coupling is due to the low frequency vibrations
in the region (100-430 cm⁻¹) which are typically ascribed to the Si and Ti atoms.
In order to understand the origin of the strong electron-phonon coupling, the phonon modes (Raman and Infrared spectra) at high-symmetry points of the Brillouin zone were calculated. The vibrational properties of Ti₃SiC₂ have also been analyzed before using group theory. A factor-group analysis at the zone-center gives 33 optical modes: 15 Raman active modes of 2A₁g+3B₁g+6E₂g+4E₁g and 18 infrared (IR) active modes of 3A₂u+3B₂u+6E₂u+6E₁u. In Table S2 of the supplemental material, the calculated zone-center modes of Ti₃SiC₂ are listed: the present *ab initio* results agree very well both with experiments³³ and previous calculations.³⁴

Fig. 4 illustrates the vibration pattern for the Raman and IR active optical modes of Ti₃SiC₂ at the Γ point. The length and direction of the arrows in Fig. 4 indicate vibration amplitude and direction parallel or perpendicular to the basal plane. The calculated vibration at 125 cm⁻¹ with E₂g symmetry involves all atoms in the unit cell with the maximum contribution from Si atoms. The second optical vibration E₁g at 180 cm⁻¹ is polarized following the basal plane with stationary Si atom. The A₁g mode at 272 cm⁻¹ is polarized along the c-axis with the vibration of Ti and C atoms. Only C atoms vibrate in the E₁g phonon modes of frequencies 580 and 601 cm⁻¹. We have observed an IR active phonon mode at 178 cm⁻¹ with E₁u symmetry which is created by the vibrations of Si atoms in the basal plane. The A₂u mode at 326 and 427 cm⁻¹ involves the motion of the Ti and Si atoms along the z direction. To the best of our knowledge, no experimental IR spectrum is available; the calculated frequencies could not be confirmed by experimental data. In spite of this, the predicted IR active frequencies are advantageous to identify Ti₃SiC₂.

The electron-phonon coupling parameters corresponding to these phonon modes are also listed in Table S2. All phonon modes have very small electron-phonon coupling parameters except the ones at 180, 202 and 272 cm⁻¹ with E₁g, E₂g and A₁g symmetry, respectively. More insight can be gained by inspection of the eigenvectors for the three symmetries E₁g, E₂g and A₁g: the Ti₂ atom contributes in the three vibration modes (see Fig. 4). The major role played
by the Ti₂ in the electron-phonon interaction in Ti₃SiC₂ is again a direct consequence of the fact that its *d* states dominate at E<sub>F</sub>. From these results, a profound modification of the electron-phonon coupling can be expected by selective substitutions of this Ti site with other elements. The fact that MAX phases offer the possibility for the modification of their physical properties by playing with their composition is now well-established, with new results evidencing the possibility to obtain ordered solid solutions.<sup>35, 36</sup> Controlling the substitution on this Ti₂ site could then lead to the tuning of the λ constant and open the way to new results in the search for superconducting MAX phases.<sup>37</sup>

In conclusion, the electronic conductivity anisotropy of Ti₃SiC₂ has been evidenced by focusing on model systems: (i) a thin film epitaxially grown on SiC substrate, and (ii) a single crystal. A detailed analysis of the experimental data in terms of the BG model based on DFT calculations coupled to a linear response approach allow to highlight electron-phonon interactions as the main driving parameter for the observed anisotropy. These scattering processes have been rationalized in terms of the MAX phase nanostructure giving insights into the possibility to tune the electron-phonon coupling by controlled substitution on one of the Ti sites.

The ANR is gratefully acknowledged for financial support: ANR project n°13-BS09-0024-02. Computations have been performed on the supercomputer facilities of the "Mésocentre de calcul de Poitou Charentes. Thierry Ouisse is acknowledged for providing the Ti₃SiC₂ single crystal sample.
**Figure captions**

**FIG. 1:** Temperature dependence of the resistivity of a highly (11-20)-oriented Ti₃SiC₂ thin film (current flowing in the basal plane and out of the basal plane) and a (0001)-oriented single crystal. Inset: schematic of the (11-20)-oriented thin film cuts in order to have the current flowing in the basal plane (along Eₓ) or out of the basal plane (along Eᵧ).

**FIG. 2:** Temperature dependence of the experimental and fitted ideal resistivities with current flowing in the basal plane and out of the basal plane of the (11-20)-oriented Ti₃SiC₂ thin film (see inset of figure 1). The solid line represents the fit using the Bloch-Grüneisen relation discussed in the manuscript.

**FIG. 3:** (a) Calculated phonon-dispersion curves of Ti₃SiC₂ along high-symmetry directions of the Brillouin zone. (b) Corresponding total phonon density of states (PhDOS) (shaded areas), Eliashberg spectral function $\alpha^2 F(\omega)$ (red dashed lines) and electron-phonon integral $\lambda(\omega)$ (blue dashed lines).

**FIG. 4:** Eigenvectors representations of zone-center optical phonon mode in Ti₃SiC₂. The arrows indicate the direction of motion (parallel or perpendicular to the basal plane) and the magnitude of the vibrations. The electron-phonon coupling parameter for these phonon modes is also given.
<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>
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