# Thermoelectric efficiency enhancement in ScN-based multilayers by Nb diffusion-driven doping

Joris More-Chevalier, Urszula Wdowik, Jiří Martan, Xavier Portier, Stanislav Cichoň, Esther de Prado, Petr Levinský, Ladislav Fekete, Jan Pokorný, Dejan Prokop, et al.

► To cite this version:

Joris More-Chevalier, Urszula Wdowik, Jiří Martan, Xavier Portier, Stanislav Cichoň, et al. "Thermoelectric efficiency enhancement in ScN-based multilayers by Nb diffusion-driven doping." Applied Surface Sciences Advances, 2025, 29, pp.100821. 10.1016/j.apsadv.2025.100821. hal-05244129

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https://normandie-univ.hal.science/hal-05244129v1

Submitted on 23 Jan 2026

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Full Length Article

# Thermoelectric efficiency enhancement in ScN-based multilayers by Nb diffusion-driven doping

<figure><img src="image_2.png" /></figure>

Joris More-Chevalier<sup>a, *, D</sup>, Urszula D. Wdowik<sup>b, *, D</sup>, Jiří Martan<sup>c</sup>, Xavier Portier<sup>d</sup>, Stanislav Cichoň<sup>a, D</sup>, Esther de Prado<sup>a, D</sup>, Petr Levinský<sup>a</sup>, Ladislav Fekete<sup>a</sup>, Jan Pokorný<sup>a</sup>, Dejan Prokop<sup>a, e, D</sup>, Petr Hruška<sup>a, e, D</sup>, Markéta Jarošová<sup>a</sup>, Jan Kejzlar<sup>a, D</sup>, Dominik Legut<sup>b, e, D</sup>, Michal Novotný<sup>a</sup>, Ján Lančok<sup>a</sup>

## ARTICLE INFO

### Keywords:

Thermoelectricity

Scandium nitride

Multilayers

Seebeck coefficient

## ABSTRACT

The thermoelectric properties of ScN/Sc₁ₓNbₓN multilayers deposited on MgO (001) substrates were studied using a combined experimental and theoretical approach based on density functional theory. Four ScN/Sc₁ₓNbₓN multilayers with varying doped layer thicknesses were produced, resulting in an overall Nb atomic percentage in the samples of 0.4 %, 1.2 %, 1.8 %, and 4.8 %, respectively. Structural characterization confirmed the epitaxial growth of multilayers with sharp interfaces. Raman spectroscopy revealed a redshift of optical phonon modes with increasing Nb concentration. Temperature-dependent Raman measurements showed an irregular shift in transverse optical (TO) modes, while longitudinal optical (LO) modes experienced a blueshift, indicating complex phonon dynamics influenced by temperature and multilayer structure. Thermoelectric measurements indicated an increase in the Seebeck coefficient and a decrease in thermal conductivity with Nb-doped ScN interlayers. The figure of merit (ZT) potentially increased to over 0.3. This improvement highlights the potential of this approach for enhancing the thermoelectric performance of scandium nitride.

## 1. Introduction

Thermoelectric power generation devices convert heat to electricity through a temperature difference, allowing charge carriers to circulate in semiconductors. These devices have long been used in niche applications like, for example, powering spacecraft [1]. In recent years, thermoelectrics have gained renewed interest due to their potential for waste-heat recovery and increasing global energy efficiency. In addition to the traditional thermoelectric materials based on tellurides, other materials such as Zintl phases, Heusler alloys, silicides, oxides, and selenides have also been investigated [2–4]. Recently, metal nitrides based on ScN and CrN have emerged as promising thermoelectric materials. Their high-symmetry rocksalt crystal structure offers an opportunity for band structure engineering, which could lead to low electrical resistivity (ρ) and low thermal conductivity (κ) by phonon

anharmonicity [5].

Thermoelectric material efficiency is determined by the figure of merit (ZT), which corresponds to a dimensionless value expressed as $ZT = (S^2\sigma/(\kappa_e + \kappa_L))T$, where S, σ, S²σ, k_e and k_L stand for the Seebeck coefficient, electrical conductivity, power factor, electronic and lattice thermal conductivities, respectively. For ScN, the reported S values vary between ~ -20 and ~ -40 μVK⁻¹ at room temperature (RT), between ~ -40 and ~ -160 μVK⁻¹ at 800 K. The σ ranges between ~ 0.1 and ~ 1.8 mΩ cm at RT, and k between 8 - 12.5 W m⁻¹ K⁻¹ at RT [6–10]. Multiple factors influence ScN properties variations, such as nitrogen-oxygen substitution, nitrogen and scandium vacancies, and structural defects introduced during film growth [10–14]. The most prevalent effect observed is the substitution of nitrogen atoms with oxygen atoms, which predominantly occurs during the film growth process and, to a smaller degree, during subsequent oxidation in air [15–17]. Under optimal

<footnote><sup>a</sup> Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, 18200 Praha 8, Czech Republic</footnote>

<footnote><sup>b</sup> IT4Innovations, VSB - Technical University of Ostrava, 17. listopadu 2172/15, CZ 708 00 Ostrava-Poruba, Czech Republic</footnote>

<footnote><sup>c</sup> New Technologies Research Centre (NTC), University of West Bohemia, Univerzitni 8, 301 00 Plzeň, Czech Republic</footnote>

<footnote><sup>d</sup> CIMAP Normandie Université, ENSICAEN, UNICAEN, CEA, UMR CNRS 6252, 6 Boulevard Maréchal Juin, 14050 Caen Cedex 4, France</footnote>

<footnote><sup>e</sup> Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague 2, Czech Republic</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: morechevalier@fzu.cz (J. More-Chevalier), urszula.danuta.wdowik@vsb.cz (U.D. Wdowik).</footnote>
growth conditions, ScN typically contains 1-2 % oxygen, which has a minimal impact on thermoelectric properties compared to pure ScN [10]. Other phenomena usually occur through other conditions of growth, such as temperature deposition or N₂/Ar gas ratio [16,18–20]. For example, ScN films grown on MgO substrates at 700°C in Sc-rich conditions exhibited two types of defects: small Sc and N vacancy clusters of 3-4 V<sub>Sc</sub> and 4-13 V<sub>N</sub>, respectively, indicating nitrogen deficiency [18].

Various approaches have been investigated to improve the thermoelectric performance of ScN, including doping/alloying, defects induced by implantation, and multilayer growth. Doping with elements such as Cr or Nb has been employed to reduce thermal conductivity from approximately 10 W m⁻¹ K⁻¹ to 2 - 3 W m⁻¹ K⁻¹, by increasing the phonon scattering [21,22]. Alloying ScN with Mg below 2 % improves the power factor, while ion implantation reduces its thermal conductivity by a factor of 3 [23,24]. Conversely, the Mg content higher than 3 % in Sc₁ₓMgₓN transforms the material into a p-type semiconductor with a Seebeck coefficient and a resistivity of ~200 µV K⁻¹ and ~50 Ω cm at 825K, and a thermal conductivity of 15 to 9 W m⁻¹ K⁻² between RT and 600 K [24]. Recently, the electronic structure and thermoelectric properties of Sc₁ₓVₓNₓ films grown on MgO (001) substrates have been studied [6]. These studies have shown that increasing the vanadium content beyond 15 % results in a transition from n-type to p-type semiconducting behavior. The introduction of defects into ScN via ion implantation with Li⁺, Ar⁺, He⁺, and O⁻² increases resistivity, but also enhances Seebeck coefficient and reduces thermal conductivity, leading to an overall improvement of ZT [9,25–27].

In addition to doping and alloying strategies, epitaxial growth of multilayer structures combining ScN (or doped ScN) with metal nitrides (ZrN, HfN) has been achieved on MgO substrates [28–32]. Recently, metal/semiconductor nitride multilayers based on ZrN/carrier-compensated Sc₀.₉₉Mg₀.₀₁N were successfully deposited on MgO for thermionic energy conversion [32]. Multilayer structures are a promising approach to enhance thermoelectric performance by reducing thermal conductivity [29]. However, none of the existing studies have consistently improved all thermoelectric properties, ultimately resulting in a significant increase in their values.

This paper reports the thermoelectric properties of ScN/Sc₁ₓNbₓN multilayers deposited on MgO (001) substrates. We employ a combined experimental and theoretical approach to elucidate the impact of Nb concentration on these multilayer systems' structural and thermoelectric characteristics.

## 2. Method

### 2.2. Experimental details

The ScN/Sc₁ₓNbₓN multilayers were deposited on double-side polished (10×10×0.5 mm³) MgO (001) substrates in an ultra-high vacuum (UHV) system (~10⁻⁸ Pa) by DC reactive magnetron sputtering using a mixture of Ar (99.9999 % pure) and N₂ (99.9999 % pure) discharge with a ratio of 60/40, respectively. The MgO substrates (99.95 % purity, from Alineason Materials and Technology) were cleaned using several steps, first with acetone and then in isopropanol in an ultrasonic bath, and blown dry with an N₂-gun. The sample holder was electrically heated first to be degassed at 573 K for 30 min and then to reach the deposition temperature of 1123 K. Before starting the deposition process, the substrate was kept for 1 h at the deposition temperature for the surface reconstruction process. The temperatures were measured by two thermocouples attached to the sample holder. The working pressure of 2 Pa was used to sputter ScN/Sc₁ₓNbₓN from two Sc targets (NEYCO: Sc 99.9 %) and one Nb target (NEYCO: 99.99 %) of one-inch diameter each. A sputtering power density of 2.8 W·cm⁻² was used for Sc targets, and a density of 1.2 W·cm⁻² for the Nb target, to deposit films with thicknesses of 520±50 nm. The thickness of the ScN layers was fixed at ~30 nm, and the thickness of ScNbN interlayers was varying depending on

the deposition time from around 2 to 6 nm (More details in the discussion part 3.3 Transmission electron microscopy). A total deposition time of 8 h was performed, and every 30 min, doping with Nb was done for a time varying from 0 s to 180 s. In total, 13 interlayers were deposited for each sample, and the final layer deposited was systematically a pure ScN layer. The distance between the targets and the sample holder was fixed at 5 cm. The thicknesses were measured by a profilometer. The oxygen content was found to be around 1 % for each sample.

Wavelength dispersive X-ray spectroscopy (WDS) was carried out by the microprobe JEOL JXA-8230. The Stratagem software processed data. The nitrogen content was calculated by stoichiometry to get the Nb/ (Sc+Nb) ratio. In each sample, three points at different positions were measured. Results are presented in Table 1. The column "formula" refers to the names of each multilayer that will be used for the next parts of the paper and discussion.

The X-ray diffraction measurements were performed on two different diffractometers. We measured high-resolution 2θ/ω scans, rocking curves, and reciprocal space maps (RSMs) on a SmartLab SE Multipurpose diffractometer (Rigaku Corp., Tokyo, Japan) using Cuα₁ radiation and a 2-bounce Ge (220) monochromator, while the pole figures were measured on a PANalytical X'Pert PRO (PANalytical B.V., Almelo, Netherlands) powder diffractometer without any monochromator.

The Raman spectra were recorded in a backscattering and parallel-polarization (VV) configuration using a Renishaw Raman spectrometer (System 1000) equipped with Bragg filters. The spectrometer was calibrated using the silicon F1g peak at 520.2 cm⁻¹. An Ar-laser with an excitation wavelength of 488 nm, power ~10 mW, and spot size of 5 µm (× 50, NA = 0.55, Olympus microscope objective) was used. The spectra were scanned at 10 s per frame, × 5 accumulations, in a temperature chamber Linkam HFS600E-PB4 stage at temperatures ranging from 300 to 800K. The measured spectra covered the frequency range of 15 - 1000 cm⁻¹.

An atomic force microscope (AFM Dimension ICON, Bruker) was used to investigate the surface morphology and roughness. The measurements were performed under ambient conditions, and images were obtained in the Peak Force Tapping mode using ScanAsystAir tips with scan areas of 5×5 and 1×1 µm². The root mean square (RMS) value of the surface roughness of each film was extracted from the measurements.

Transmission electron microscopy (TEM) observations were carried out on a double corrected cold FEG JEOL ARM200 operated at 200 kV and using DigitalMicrograph software by GATAN for image processing. Chemical maps were performed using STEM EDX data. The EDX spectrometer was a CENTURIO equipment from JEOL. TEM foils were prepared by a dual-beam SEM-FIB FEI Helios nanolab 660.

Electrical resistivity and Seebeck coefficient were measured simultaneously using a custom-made instrument that employs the four-probe method [33]. Platinum electrodes were used as current probes, and Pt-PtRh (type S) thermocouples served as voltage probes and measured the sample temperature and its gradient. Measurements were performed

<caption>**Table 1**</caption>

WDS analysis of the multilayers, presenting the total (Sc/(Nb+Sc))×100 atomic percentage and the corresponding sample names referenced in this paper. The third column reports the Nb % focused only on the ScNbN interlayers from cross-sections.

<table><thead><tr><th>Nb doping time (s)</th><th>Sc % total</th><th>Nb % total</th><th>Nb % focused on the interlayers</th><th>Sample name</th></tr></thead><tbody><tr><td>0</td><td>100</td><td>0</td><td>0</td><td>ScN</td></tr><tr><td>15</td><td>99.58</td><td>0.42 ± 0.04</td><td>4.2 ± 2.5</td><td>ScN-Nb 0.4 %</td></tr><tr><td>30</td><td>98.79</td><td>1.21 ± 0.03</td><td>12.3 ± 4.9</td><td>ScN-Nb 1.2 %</td></tr><tr><td>60</td><td>98.76</td><td>1.84 ± 0.03</td><td>25.6 ± 8.6</td><td>ScN-Nb 1.8 %</td></tr><tr><td>180</td><td>95.20</td><td>4.80 ± 0.03</td><td>40.5 ± 4.0</td><td>ScN-Nb 4.8 %</td></tr></tbody></table>
in a nitrogen atmosphere on samples typically 10 mm long and 3 mm wide.

Thermal effusivity in various depths under the sample surface is determined by a measurement system based on pulsed photothermal radiometry, described in detail elsewhere [34,35]. This technique uses a pulsed laser to heat the material surface and a fast infrared detector to observe the temperature decrease after the laser pulse impact. From the temperature decrease with time in the nanosecond time range, the thermal effusivity in depth of hundreds of nm is evaluated. As described in Ref [36], the three-layer 1D model with surface absorption of laser light without thermal interface resistances between layers is employed. Semi-transparency correction is applied according to Ref. [35].

## 2.3. Computational details

To represent the random distribution of Nb dopants in the calculated Sc₁ₓNbₓN compositions with x = 1.85 % and x=4.63 %, a supercell method was used along with a special quasi-random structure (SQS) approximation [37]. Each composition was modeled by a 216-atom supercell, which contained 108 N, 106 Sc, and 2 Nb atoms for the system with x=1.85 %, and 108 N, 103 Sc, and 5 Nb atoms for the system with x = 4.63 %. These supercells matched quite well the experimentally explored samples with x = 1.84 % and x= 4.80 % (see Table 1). Our simulations were carried out within the framework of density functional theory (DFT) implemented in the VASP code [38]. The projector augmented wave (PAW) method [39,40] for description of electron-ion interaction and the generalized gradient approximation with parame-trization of Perdew, Burke, and Ernzerhof (GGA-PBE) [41] for the ex-change and correlation potential were applied. The wave functions were expanded in a plane-wave basis set with the energy cut-off of 520 eV. The PAW pseudopotentials with three valence electrons for Sc, five valence electrons for N, and eleven valence electrons for Nb were taken from the VASP database. The Brillouin zones of structures were sampled with eight irreducible k-points generated according to the Monkhorst-Pack scheme, and the structures were fully optimized with convergence criteria for the total energy and residual Hellman–Feynman forces of 10⁻⁷eV and 10⁻⁵ eV/Å, respectively. The *ab initio* molecular dynamics (AIMD) simulations were performed to study the temperature evolution of the Sc₁ₓNbₓN Raman spectra. Each system was heated from

300 to 800 K with a temperature step of 100 K in the NpT (isothermal-isobaric) ensemble during 20 ps and subsequently equilibrated at each temperature for 10 ps. The Langevin thermostat was used to control the temperature during the NpT simulations. The shape constraints for the volume fluctuations experienced under Parinello-Rahman dynamics were imposed. Production runs were carried out within the NVT scheme (canonical ensemble) for 20 ps. The temperature was controlled by the Nosé-Hoover thermostat. To compare the results of our AIMD simulations with the measured Raman spectra, the power spectra of the autocorrelation function were projected onto the Brillouin zone centers of the cubic structures of Sc₁ₓNbₓN compositions. Details of the formalism and applied procedure are provided and discussed elsewhere [42,43].

# 3. Results and discussion

## 3.1. X-ray diffraction

Symmetric 2θ/ω scans of ScN-Nb samples deposited at 1123 K on MgO (001) substrates are shown in Fig. 1 (a). The peaks are indexed according to their phase: S for the substrate and L for the layers, followed by their (hkl) indices. The inset in each figure presents the rocking curve analysis of the 002 reflection. The ScN-Nb samples with Nb percentages of 0.4 %, 1.2 % and 4.8 % show a distinct reflection connected to the 002 orientation, which corresponds to a cube-on-cube epitaxial growth with respect to the MgO substrates. These results are similar to those reported for the ScN films deposited on MgO by different PVD techniques [8–10, 16, 19, 44–46]. It can be noticed that the presence of small humps on the right side of the 002 and 004 peaks for the sample containing 4.8 % of Nb. This effect is coherent with the multilayer presenting the ScN peaks at the expected position and small extra ones corresponding to the ScNbN layers shifted due to the Nb substituting the Sc in the structure. The 2θ/ω scan for the layer with x = 1.8 % presents a (001) and (111) poly-orientations with a high texturation along the [001] direction. The poly-orientation is commonly observed in ScN thin films, usually connected to a change in the adatom mobility during the film growth [16, 17, 19]. In the present work, the change in the adatom mobility comes from an unexpected interruption of one of the Sc cathodes during the depositions. However, as this interruption occurred near the end of the

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** In (a), the 20/ω scans of each ScN-Nb sample in the 20 range from 32° to 97°. The inserts include the 002 rocking curves of each sample, including the FWHM, which are equal to 0.576°, 0.510°, 0.593°, and 0.698° for the multilayers containing 0.4 %, 1.2 %, 1.8 %, and 4.8 % of Nb, respectively. In (b), 111 pole Fig. of the ScN-Nb 1.8 %.</figcaption></figure>
deposition process, explaining the very low intensity of the 111 phase, it
only resulted in a minor change without significantly affecting the
overall multilayer structure or the material properties. The full width at
half maximum (FWHM) values extracted from rocking curves of 002
peaks are equal to 0.576°, 0.510°, 0.593°, and 0.698° for the multilayer
samples containing 0.4%, 1.2%, 1.8%, and 4.80% of Nb, respectively.
These narrow values correspond to very low crystallite out-of-plane
misorientation, confirming the high quality of the multilayer crystal
structures in all samples.

Fig. 1 (b) shows the pole figure of the 111 orientations in the ScN-Nb 1.8 %. All pole figures of the 002, 022, and 111 orientations in each ScN-Nb sample are presented in Figs. S1 and S2 (supplementary information). Similarities between the pole figures of ScN-Nb with compositions of 0.4 %, 1.2 %, and 4.8 % are clearly observed, confirming their epitaxial growth. The center of the 002 pole figures, displayed in Figures S1 and S2, (a) and (d), corresponds to the main 002 orientation peak, while the rest of the weaker spots are due to the tails of other MgO reflections. Then, four spots at $\chi \approx 45°$ for the 022 pole figure and at $\chi \approx 55°$ for the 111 pole figure are observed, as expected for the cubic structure of ScN [10,13,14]. Pole figures of the layer containing 1.84 % of Nb are slightly distinct from those observed for the layers with the remaining concentrations of Nb regarding the intensity of the central spot in the 111 pole figure. However, the other pole figures have no additional spots or rings. These findings confirm the reported result in Fig. 1 (b) and the slight presence of the phase composed of 111-oriented crystallites with their diffraction vectors aligned perpendicular to the sample surface.

3.2. *Atomic force microscopy*

The focused 5×5 µm² images of the ScN-Nb sample surfaces are shown in Fig. 3, while focused 1×1 µm² images of each sample are presented in Fig. S3, along with the surface images of the ScN layer at both 5×5 µm² and 1×1 µm² (supplementary information). The RMS surface roughness values are 5.7 nm, 5.8 nm, 11.8 nm, and 22.5 nm for the multilayers with 0.4 %, 1.2 %, 1.8 %, and 4.8 % Nb, respectively. Flat, square-shaped grains are visible in the samples with x = 0.4 % and x = 1.2 %, as shown in Fig. 2 (a) and (b). These are consistent with the expected structures of face-centered cubic materials like ScN [10,14], which agrees with the XRD results previously shown in Fig. 1. In addition to the square grains in the ScN-Nb samples with 1.8 % and 4.8 % Nb, a few grains with isosceles-pyramidal and equilateral shapes are observed. The isosceles-pyramidal grains are caused by misoriented crystals undergoing twinning in the 111 planes, while the equilateral-shaped grains grow with a 111 orientation. This behavior is expected for Sc<sub>1-x</sub>Nb<sub>x</sub>N with x = 1.8 %, which shows poly-oriented crystallization; however, it is surprising for layers with x = 4.8 %, which exhibit single-orientation growth. This implies that poly-orientation may be limited to the film's surface or that the amount of material aligned along [111] is too small to detect by XRD. The change in orientation, leading to mound formation, is a common feature in ScN and other nitride films [16, 19, 20, 22, 44–46]. It is related to the mobility of adatoms on the film surface, influenced by the presence of defects that hinder downward atom movement due to the Ehrlich-Schwoebel barrier, while promoting upward migration on

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Focused 5×5 µm² AFM images from the surface of ScN-Nb x % samples with (a) x = 0.42 %, (b) x = 1.21 %, (c) x = 1.84 %, and (d) x = 4.80 %. The RMS values of multilayer surface roughness: 5.7 nm, 5.8 nm, 11.8 nm, and 22.5 nm for ScN-Nb x % with 0.4, 1.2, 1.8, and 4.8 %, respectively.</figcaption></figure>
terraces.

### 3.3. Transmission electron microscopy

Fig. 3 (a) and (b) show bright-field TEM images of the ScN-Nb 0.4 % and ScN-Nb 4.8 %. Both images are along the [001] direction of the MgO substrate. Both images are along the [001] direction of the MgO substrate. The multilayer thicknesses are equal to 521 nm and 565 nm, respectively. Dark horizontal lines pointed out by arrows are clearly visible in Fig. 3 (b), indicating the presence of Nb in these layers (Z contrast). In Fig. 3 (a), due to the lower dopant concentration, the Z contrast is less obvious, but looking at the image carefully, we also do see some darker lines indicated by parallel arrows. One can notice Bragg contrasts in the samples, particularly in the ScN-Nb 4.8 %. These contrasts result from stresses induced by the presence of Nb in the ScN matrix. The fast Fourier transforms (FFTs) of images reveal in each case a perfect epitaxial growth of the multilayer with respect to the MgO substrate. The arrow in the FFT inset of Fig. 3 (a) shows two separated spots of the (0-40) planes of the MgO and ScN cubic structures, whose lattice parameters amount to 0.42 nm and 0.45 nm, respectively.

Fig. 4 (a) shows a typical STEM HAADF image of the ScN-Nb 0.4 % sample at a relatively low scale. The Z contrast observed in this mode confirms the presence of heavier elements in thin layers within the ScN film. No Nb atoms are present between these brighter lines, as shown in Fig. S4 and Fig. S5 (supplementary information). Fig. 4 (b) is a high-resolution (atomic scale) STEM HAADF image containing an Nb-enriched sublayer. The thickness of these sublayers is about 2 nm. One can notice that the atomic structure is similar to that of ScN.

Fig. 5 (a) and (b) are the counterpart images for the ScN-Nb 4.8 % sample, containing more than ten times more dopants in the sublayers. These layers are about 6 nm thick, and as in the previous case, the atomic structure remains unchanged compared to that of ScN. However, more extended oblique defects are visible to release the stress induced by the presence of the dopants in ScN. As mentioned for the lower doping level, no Nb atom is detected between two successive Nb-enriched layers. According to the N and Sc maps (shown in the supplementary section), Nb replaces Sc in the ScN structure since the N map is uniform, whereas that of Sc clearly shows the Nb-enriched layers, meaning that the Sc concentration in these sublayers is lower than in ScN. Globally, the atomic percentages measured by TEM of Sc and Nb in the interlayer for each sample are consistent with those measured by scanning electron microscopy, with much greater accuracy.

<figure><img src="image_5.png" /><figcaption>Fig. 4. (a) STEM HAADF image of the ScN-Nb 0.4 % sample along the [002] direction of the MgO substrate. (b) High-resolution STEM HAADF image of a region containing a Nb-rich layer. Sc and Nb concentration percentages of 96.4 % and 3.6 %, respectively, were measured in the interlayers.</figcaption></figure>

### 3.4. Raman spectroscopy

The Raman spectroscopy was employed to support structural and phonon dynamics characterizations of the ScN-Nb x % samples, which are relevant for the thermoelectric properties of these multilayers. We note that strictly stoichiometric ScN shows no Raman-active modes as each atom in its $Fm\bar{3}m$ structure occupies an inversion-symmetry site, and hence atomic vibrations at the Brillouin-zone center are only associated with the triply degenerate infrared mode of $T_{1u}$ symmetry. On the other hand, the first-order Raman scattering spectra from almost stoichiometric ScN films (ScN with 1 at % of N-vacancies) [47], ScN films with various concentrations of oxygen impurities [13], and epitaxial $Ti_{1-x}Sc_xN$, $Al_{1-x}Sc_xN$, and $Mg_{1-x}Sc_xN$ layers, clearly show peaks due to the Raman-active phonons [48,49]. Indeed, ScN in the form of epitaxial layers is known from its affinity to be an off-stoichiometric compound, which can easily accommodate different types of impurities. Furthermore, slight atomic disorder resulting from the preparation procedure can also be present in ScN layers. Both defects and disorder destroy local inversion symmetry of a system, usually at sites of nearby atoms, which leads to the appearance of the Raman-active modes. Disordered systems usually show broad Raman bands as defects lead to a breakdown of

<figure><img src="image_6.png" /><figcaption>Fig. 3. Bright field TEM images of the 0.4 at. % (a) and 4.8 at. % (b) Nb-doped ScN samples.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** (a) STEM HAADF image of the ScN-Nb 4.8 % sample along the [002] direction of the MgO substrate. (b) High-resolution STEM HAADF image of a region containing a Nb-rich layer. Sc and Nb concentration percentages of 60.7 % and 39.7 %, respectively, were measured in the interlayers.</figcaption></figure>

selection rules, and such behavior has already been observed for transition-metal nitrides and carbides [50].

The Raman spectra of ScN and ScN-Nb samples at RT, as well as the temperature evolution of the measured spectra of ScN-Nb samples with Nb atom concentrations of 0.4, 1.2, 1.8, and 4.8 % are presented in Fig. 6. Each graph presented is baseline corrected. In general, all spectra are very similar to those reported in our previous studies on ScN films [10,13]. Therefore, the main Raman spectrum for each temperature covers the energy range from 300 to 800 cm⁻¹, and an additional shallow peak observed above 800 cm⁻¹ represents the second-order Raman modes. All spectra are composed of two weakly separated broad bands. The first one, extending between 300 and 550 cm⁻¹, corresponds to the transverse optical (TO) modes (noted as (1) and (2) in Fig. 6), and the second one, extending between 550 and 800 cm⁻¹, which is due to the longitudinal optical (LO) modes (noted as (3) and (4) in Fig. 6). The peak position evolutions are presented in Fig. S6 and Fig. S7. The fits of each Raman spectrum for each temperature were performed using Gaussian and Lorentz profiles, shown in Figures S8–S11 (supplementary information).

The positions of particular peaks resulting from fits are listed in Tables S1-S4 as well as shown in the respective figures given in the supplementary information. The peaks shift their positions downward with increased concentration of Nb dopants (Fig. 6 (a)). Each peak position seems to follow the same tendency, which consists of a linear red shift until a plateau over 1 % of Nb concentration. The observed softening in the frequency of modes is consistent with the substitution of a Sc atom with a heavier Nb atom. This tends to reduce the frequency of the optical mode vibrations. Furthermore, the Raman spectra of cubic NbN show peaks of much lower frequencies than those in ScN (~200 and ~500 cm⁻¹) [51,52]. The plateau between 1 % and 5 % of Nb may be explained by a very similar percentage substitution between the multilayers with Nb content of 1.2 %, 1.8 %, and 4.8 %. The same Nb power cathode was used for each sample during the doping steps, and only the deposition time varied. Therefore, the thickness of the Sc₁-xNbxN layers in the ScN/Sc₁-xNbxN multilayer structure increases, which increases the total concentration of Nb, and the actual substitution percentage of Nb within these layers may not necessarily increase proportionally.

The temperature variation of peaks' positions in ScN-Nb x %, which is presented in Fig. 6 (and in Fig. S7 in the supplementary information), can be confronted with the behavior of particular peaks in ScN [10] (see Fig. S12, supplementary information). Generally, the temperature dependence of phonon mode frequency can be written as [53–55]: $\omega(T)$

= $\omega_0 + \Delta_E + \Delta_A$, where $\omega_0$ is the harmonic frequency, $\Delta_E$ is the frequency shift through the thermal expansion of the lattice, and $\Delta_A$ is the phonon coupling factor, which is defined as $\Delta_A = \Delta^{(3)} + \Delta^{(4)}$ with $\Delta^{(3)}$ corresponding to the cubic anharmonicity phonon coupling factor and $\Delta^{(4)}$ standing for the quartic anharmonicity phonon coupling factor. In ScN, the temperature-mediated redshift of the theoretically determined optical phonons has been assigned to the thermal expansion term $\Delta_E$, which dominates over the remaining contributions [10]. On the other hand, the experimentally observed redshift was found for the transverse optical (TO) phonons, without a clear shift in the longitudinal optical (LO) phonons. Such an effect has been attributed to a strong positive quartic term for the highest-frequency phonons, which overcomes the cubic and thermal expansion terms at elevated temperatures.

The shifts in the Raman peaks (phonon modes) upon temperature and the Nb concentration in ScN-Nb x % samples show two tendencies. There is an irregular change in the shifts of the TO modes, which is accompanied by a blue shift of the LO modes, the latter ranging between 10 to 15 cm⁻¹, and these observation applies to each measured sample. Although this effect is distinct from that noticed for ScN, both are likely to have a similar origin, i.e., a strong positive quadratic term, which is significantly higher in the Nb-doped than in undoped layers, and hence it leads to the increase in frequencies of the LO phonons. Apart from the anharmonic term in lattice vibrations, which plays a more significant role at high temperatures, a coupling between phonons due to multilayer structuration of the ScN-Nb samples may appear and result in the blueshift of the LO modes. Indeed, the multilayer structure is evidenced by the STEM images demonstrated in Fig. 4 and Fig. 5.

The present experimental observations of the temperature evolution of the Raman spectra in the Nb-doped ScN layers are to some extent supported by the results of AIMD simulations carried out for bulk Nb-containing ScN, which are illustrated in Fig. 7. Theoretical spectra reflect the separation of phonon modes into the TO and LO bands, covering ranges of 350-500 cm⁻¹ and 580-680 cm⁻¹, respectively. They correspond to those determined from experiments. The calculated spectral patterns of the TO bands in both systems (1.85 % and 4.63 % of Nb) at 300 and 400 K show a larger number of well-resolved peaks than the LO band. They are also richer in peaks than those obtained from experiments. With increasing temperature, the widths of peaks undergo broadening, which results in smearing out the spectral patterns, being especially pronounced above 400 K. A small downward shift of the bands, i.e., decrease in phonon frequencies, is observed below 600 K. This is connected with thermal expansion of the Nb-doped ScN lattice. However, at 800 K and probably at higher temperatures, the bands shift upward, which is equivalent to an increase in phonon frequencies. Such an increase is perhaps mediated by a strong anharmonicity of lattice vibrations at high temperature that overcomes the effect associated with lattice dilation.

We also note that both experimental and theoretical spectra reveal a reduction of the intensities of Raman peaks within the TO and LO phonon bands with increasing temperature. This is a typical behavior driven by the Boltzmann distribution function [10]. On the other hand, the peaks constituting the LO band and arising from high-frequency lattice vibration undergo enhancement in their intensities above 600 K. This effect is revealed by both experiment and simulations and follows from the increase of phonon occupation with the temperature, which is governed by the Bose-Einstein function.

### 3.5. Thermoelectric properties

Thermoelectric properties of ScN-Nb multilayers are found to be sensitive to the Nb doping level and the thickness of the Sc₁-xNₓN interlayers, as reflected by temperature changes of the measured and computed Seebeck coefficient (S) and electrical resistivity $\rho$ of the ScN-Nb samples, shown in Fig. 8. For all investigated Nb concentrations in the ScN layers, a linear increase in the magnitude of the Seebeck
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Raman spectra of ScN and ScN-Nb x % in (a) at RT. In (b) to (e), ScN-Nb x % samples with x = 0.4, 1.2, 1.8, and 4.8 % measured at the VV polarization and temperatures ranging from 296 K to 800 K. Baseline corrections were applied. The vertical lines denote the positions of the main fitted peaks for the room temperature spectra. For Nb concentration of x = 0.42 %: (1) 378.8 cm⁻¹, (2) 469.5 cm⁻¹, (3) 631.3 cm⁻¹, and (4) 673.5 cm⁻¹. For Nb concentration of x = 1.2 %: (1) 379.1 cm⁻¹, (2) 447.2 cm⁻¹, (3) 624.6 cm⁻¹, and (4) 670.2 cm⁻¹. For Nb concentration of x = 1.84 %: (1) 383.6 cm⁻¹, (2) 460.2 cm⁻¹, (3) 626.63 cm⁻¹, and (4) 674.4 cm⁻¹. For Nb concentration of x = 4.80 %: (1) 382.9 cm⁻¹, (2) 451.7 cm⁻¹, (3) 624.4 cm⁻¹, and (4) 673.8 cm⁻¹.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Temperature evolution of the Brillouin zone projected power spectra of the autocorrelation function computed for Nb-doped ScN with a concentration of dopants of 1.85 % and 4.63 %.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** (a) Temperature dependence of the Seebeck coefficient *S* in ScN-Nb samples. (b) Temperature dependence of electrical resistivity *ρ* in ScN-Nb samples. Solid lines represent the calculated *S* and *ρ* of Nb-doped ScN with concentrations of Nb-dopants of 1.85 % and 4.63 %. For clarity, the experimental errors are given for selected data above 700 K. At lower temperatures, the uncertainty errors are smaller, varying from 3 to 5 %.</figcaption></figure>

coefficient and a simultaneous increase in resistivity (decrease in electrical conductivity σ) is observed between RT and 800 K. The values of *S* and *ρ* for the measured concentrations of Nb in ScN at RT and 800 K are gathered in Table 2.

In general, the ScN-Nb x % samples show improved Seebeck coeffi-

cient and higher (lower) resistivity (electrical conductivity) in comparison to the Nb-free (x = 0 %) layers. According to the observed changes of *S* and *ρ* as a function of Nb concentration, the ScN-Nb x % samples with x = 0.42 % and x = 4.80 % reflect behavior typical for lightly- and heavily-doped n-type semiconducting systems. In a non-degenerate
<caption>**Table 2**</caption>

Values of the Seebeck coefficient *S* and resistivity *ρ* of ScN-Nb *x* % samples for different concentrations *x* of Nb measured at 300 K and 800 K.

<table><thead><tr><th>x (%)</th><th>S (μV K<sup>-1</sup>)</th><th colspan="2">ρ (mΩ cm)</th></tr></thead><tbody><tr><td>0</td><td>300 K<br/>-30 ± 1.0</td><td>800 K<br/>-64 ± 3.2</td><td>300 K<br/>0.21 ± 0.01</td><td>800 K<br/>0.39 ± 0.02</td></tr><tr><td>0.42</td><td>-59 ± 1.8</td><td>-118 ± 11.0</td><td>1.24 ± 0.06</td><td>2.48 ± 0.20</td></tr><tr><td>1.21</td><td>-50 ± 1.5</td><td>-110 ± 10.0</td><td>0.47 ± 0.02</td><td>1.04 ± 0.08</td></tr><tr><td>1.84</td><td>-54 ± 1.6</td><td>-116 ± 10.8</td><td>0.85 ± 0.04</td><td>1.62 ± 0.20</td></tr><tr><td>4.80</td><td>-44 ± 1.3</td><td>-82 ± 7.5</td><td>0.56 ± 0.03</td><td>0.89 ± 0.05</td></tr></tbody></table>

semiconductor, the Seebeck coefficient can be expressed as [1]: $S = -1/(eT) (E_c - E_F)$, where $e$, $T$, $E_c$, and $E_F$ denote the charge of carriers, temperature, mean energy of charge carriers, and Fermi energy, respectively. The doping of ScN with a small concentration of Nb atoms ($x = 0.42$ %) causes the Fermi level to shift downward, while leaving the mean energy of electrons in the conduction band practically unchanged. Hence, the difference between the mean energy of electrons and the Fermi energy $|E_c - E_F|$ gets larger, which leads to an increase in the absolute value of *S* and *ρ*, i.e., decrease in *σ*. On the other hand, the Sc<sub>1-x</sub>Nb<sub>x</sub>N with *x* = 4.80 % represents a degenerate heavy-doped semiconducting system, in which the Fermi level lies deeply inside the conduction band. In this case, the difference between $E_c$ and $E_F$ becomes smaller ($E_c \approx E_F$), and thus results in a reduced value of *S* and an enhanced value of *σ* (lower value of *ρ*). Finally, the ScN-Nb samples with concentrations 1 % < *x* < 2 % show intermediate behavior between these two extremes, having similar Seebeck coefficient values but distinct electrical conductivities in the whole measured temperature range.

Our experimental observations of variations of Seebeck coefficient and resistivity in ScN-Nb *x* % are supported by theoretical simulations performed for Sc<sub>1-x</sub>Nb<sub>x</sub>N systems with *x* = 1.85 and 4.63 %, which closely correspond to experimental Nb concentrations of 1.8 % and 4.8 %, respectively. These computations were performed by using the BoltzTrap2 code [56], with the modified constant relaxation time approximation $\tau_T = \tau (300/T)$, where $\tau$ stands for the lifetime estimated from the present experimentally measured resistivity $\rho$. This approach has been demonstrated to give reasonable results in the efficient calculation of carrier scattering rates from first principles [57,58]. The equations used to determine theoretically electronic transport properties, namely, electrical conductivity $\sigma = \rho^{-1}$, Seebeck coefficient *S*, and the electronic component of thermal conductivity $k_e$ are given in the Appendix. Fig. 11 shows the calculated temperature dependences of *S* and $\rho = \sigma^{-1}$ in Sc<sub>1-x</sub>Nb<sub>x</sub>N with *x* = 1.85 and 4.63 %. They correlate well with those determined experimentally for the ScN-Nb *x* % samples with *x* = 1.84 and 4.80 %.

Theoretical studies of thermal conductivity of Sc<sub>1-x</sub>Nb<sub>x</sub>N systems with *x* = 1.85 and 4.63 % have also been carried out to elucidate the effect of Nb doping on the thermoelectric efficiency to convert heat to electricity, which is governed by the dimensionless Fig. of merit *ZT* = ($S^2\sigma/k$)T, where $k = k_e + k_L$ is the total thermal conductivity, being a sum of the electronic $k_e$ and lattice (phonon) $k_L$ contributions. The electronic term is determined as mentioned above [56] (see also Appendix), whereas the lattice term has been obtained by applying a modified version of the relaxation-time model [59] based on the model of three-phonon anharmonic scattering developed by Slack et al. [60–62]. Compared to the original Slack model, its modified version takes into account the full *ab initio* calculated phonon spectrum to evaluate required quantities, such as mean velocity *v*, Grüneisen parameter *γ*, and Debye temperature *Θ*, and thereby avoids a certain arbitrariness in making a Debye model for a given system [59]. Thus, it provides a fitting-parameter-free expression for calculating the lattice thermal conductivity of a material. Details of this model can be found in the Appendix.

Fig. 9 shows the resulting temperature variation of the total thermal conductivity along with its particular contributions ($k_e$ and $k_L$)

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Calculated temperature dependence of thermal conductivity in Sc<sub>1-x</sub>Nb<sub>x</sub>N with *x* = 1.85 % and *x* = 4.63 %. Inset: electronic ($k_e$) and lattice ($k_L$) contributions to the total thermal conductivity. Experimental values of thermal conductivity at 300 K are given in Table 3.</figcaption></figure>

calculated for Nb-free [10] and Nb-doped ScN ($x = 1.85$, 4.63 %). As evidenced from our theoretical simulations, the Nb-doped material exhibits 2.3 (for *x* = 1.85 %) and 2.6 (for *x* = 4.64 %) times lower total thermal conductivity at RT as compared to the undoped one. The thermal conductivity for all computationally investigated Sc<sub>1-x</sub>Nb<sub>x</sub>N systems decreases with increasing temperature. In the temperature range of 300 – 800 K, the highest reduction in *k* is revealed for undoped ScN (40 %) [10], while it remains smaller for the ScN-Nb *x* %, i.e., 20 % and 30 % for *x* = 1.85 % and *x* = 4.63 %, respectively. As indicated by the results of our calculations depicted in the inset of Fig. 9 the lattice thermal conductivity is the leading term in the overall thermal conductivity below 540 K for *x* = 0 %, 460 K for *x* = 1.85 %, and 620 K for *x* = 4.63 %, whereas at higher temperatures the electronic thermal conductivity dominates in the heat transfer. We also note that the computed values of thermal conductivity in Sc<sub>1-x</sub>Nb<sub>x</sub>N at RT agree well with those determined experimentally for ScN-Nb *x* % samples with corresponding stoichiometries (cf. Table 3).

Upon having determined the temperature dependence of the Seebeck coefficient, electronic conductivity, and thermal conductivity, one may easily obtain the values of the figure of merit *ZT* at the given temperature. The behavior of computed *ZT* along with its theoretical and experimental lower limits as a function of temperature is given in Fig. 10. The lower limit of *ZT* corresponds to a constant value of thermal conductivity measured at room temperature, and thus it is generally lower than the actual *ZT*, as confirmed by the present and our previous works [10] (cf. solid and dashed curves in Fig. 10). Such an approximation is usually made when measurements are limited to room temperature [10], similar to the present experimental study. However, it becomes useful while making direct comparisons between the results of experiments carried out for a very limited temperature range and

<caption>**Table 3**</caption>

Comparison of the experimental ($k_{exp}$) and calculated ($k_{cal}$) total thermal conductivities in ScN-Nb *x* % samples at 300 K. Numbers in brackets denote calculated concentrations of Nb. The data for *x* = 0 % are adopted from our previous research [10].

<table><thead><tr><th>x (%)</th><th>k<sub>exp</sub> (Wm<sup>-1</sup>K<sup>-1</sup>)</th><th>k<sub>cal</sub> (Wm<sup>-1</sup>K<sup>-1</sup>)</th></tr></thead><tbody><tr><td>0</td><td>10.06 ± 0.5</td><td>10.48</td></tr><tr><td>0.42</td><td>7.0 ± 0.5</td><td></td></tr><tr><td>1.21</td><td>4.5 ± 1.0</td><td></td></tr><tr><td>1.84 (1.85)</td><td>4.0 ± 1.0</td><td>3.84</td></tr><tr><td>4.80 (4.63)</td><td>4.7 ± 1.0</td><td>4.32</td></tr></tbody></table>
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Comparison of the calculated temperature dependence of the figure of merit ZT (solid lines) and its lower limit (dashed lines) for Nb-doped ScN with Nb concentrations of 0, 1.85, and 4.63 % with temperature dependence of the experimental lower limit of ZT (solid symbols) in ScN/MgO(001) layers containing 0, 0.42, 1.21, 1.84, and 4.80 % of Nb. The data for almost stoichiometric ScN are adopted from our previous work [10]. For clarity, the experimental errors are given for selected data above 700 K.</figcaption></figure>

theoretical calculations. The experimental and simulated lower limits of ZT in Sc<sub>1-x</sub>Nb<sub>x</sub>N compositions with comparable concentrations of Nb dopants (x~2 % and x~5 %), show very good agreement between each other at temperatures not exceeding about 500 and 600 K for the lower and higher Nb concentrations, respectively. At higher temperatures, a departure from the calculated lower limit of ZT from that yielded by experiments is observed. Nevertheless, the ZT lower limits of ScN-Nb x % samples with concentrations of doping element ranging between 1.2 % and 4.8 % exhibit higher values of ZT than those for undoped ScN films (0.05-0.2) [8,10,26]. Indeed, the experimental results of ScN-Nb 1.2 % and 1.8 % present ZT values of approximately 0.2 and 0.25 at 800 K. Assuming the thermal conductivity decreases as illustrated in Fig. 9, the true ZT values could potentially exceed 0.3. This value is comparable to ZT typically reported in thin films at similar temperatures, varying from 0.05 to 0.7, and concerning oxides and telluride materials [63–66]. Furthermore, ZT values close to or exceeding 1 have been reported for superlattices (such as PbTe-, Bi<sub>2</sub>Te<sub>3</sub>, or SiGe-based structures), at different temperature regimes, below 400 K for tellurides, or over 1000 K, for SiGe [67]. Interestingly, the ZT values of the ScN sample with the lowest concentration of Nb dopants (x = 0.42 %) do not differ too much from the values encountered for pure ScN below 540 K, but at higher temperatures, they even fall below the values of the almost pristine ScN layers. The observed improvement in the Fig. of merit upon doping the ScN with Nb and concentration of the doping element between 1.2 % and 4.8 % is mostly due to the enhancement of the Seebeck coefficient and lower electrical and thermal conductivities of the doped multilayers.

## 4. Conclusion

In summary, epitaxial ScN/Sc<sub>1-x</sub>Nb<sub>x</sub>N multilayers with varying Nb concentrations were successfully grown on MgO (001) substrates, and their structural and thermoelectric properties were systematically examined. X-ray diffraction and TEM experiments confirmed the high crystalline quality and layered structure of the samples, with Nb

preferentially incorporated into specific sublayers within the ScN matrix. The thickness and Nb concentration in the interlayers of ScN-Nb 0.4 % and ScN-Nb 4.8 % were measured as 2 nm for 3.6 % Nb and 6 nm for 39.3 % Nb, respectively. Raman spectroscopy showed a concentration-dependent redshift of optical phonon modes, due to the substitution of Sc by the heavier Nb atoms. Notably, temperature-dependent Raman studies revealed distinct behaviors of TO and LO modes, with the former showing irregular shifts and the latter a blueshift, indicating a complex interplay of thermal expansion and anharmonicity, further influenced by the multilayer structure. Thermoelectric measurements indicated that Nb multilayer doping enhances the Seebeck coefficient and decreases the thermal conductivity of ScN. For example, the Seebeck coefficients reached -110 and -116 µV/K at 800K for the ScN-Nb 1.2 % and 1.8 % samples, respectively, compared to -64 µV/K for undoped ScN, while the thermal conductivity decreased from 10.06 Wm<sup>-1</sup>K<sup>-1</sup> to about 4 Wm<sup>-1</sup>K<sup>-1</sup>for 1.2 % and 1.8 % Nb doping at room temperature. Importantly, theoretical calculations of the Seebeck coefficients along with electrical and thermal conductivities match well with the experimental results. The reduction in thermal conductivity and the increase in Seebeck coefficient improve the Fig. of merit, ZT, which can potentially surpass 0.3 at 800 K for ScN-Nb 1.2 % and 1.8 % multilayer samples. These findings highlight the potential of ScN/Sc<sub>1-x</sub>Nb<sub>x</sub>N multilayer structures for thermoelectric applications, offering a pathway to tailor thermoelectric performance through controlled doping and multilayer engineering.

## CRediT authorship contribution statement

**Joris More-Chevalier:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Urszula D. Wdowik:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Jiří Martan:** Writing – original draft, Visualization, Validation, Investigation, Formal analysis, Data curation.
**Xavier Portier:** Writing – original draft, Visualization, Validation, Investigation, Funding acquisition, Formal analysis, Data curation.
**Stanislav Cichón:** Writing – review & editing, Writing – original draft, Visualization, Validation, Investigation, Formal analysis, Data curation.
**Esther de Prado:** Writing – review & editing, Writing – original draft, Visualization, Validation, Investigation, Formal analysis, Data curation.
**Petr Levinský:** Visualization, Validation, Investigation, Formal analysis, Data curation.
**Ladislav Fekete:** Formal analysis, Data curation.
**Jan Pokorný:** Visualization, Formal analysis, Data curation.
**Dejan Prokop:** Formal analysis, Data curation.
**Petr Hruska:** Formal analysis, Data curation.
**Markéta Jarošová:** Formal analysis, Data curation.
**Jan Kejzlar:** Formal analysis, Data curation.
**Dominik Legut:** Writing – review & editing, Validation, Resources, Project administration, Funding acquisition, Conceptualization.
**Michal Novotný:** Writing – original draft, Validation, Resources, Project administration, Funding acquisition, Formal analysis, Data curation.
**Ján Lančok:** Visualization, Resources, Project administration, Funding acquisition, Formal analysis, Data curation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The authors acknowledge the Czech Science Foundation (GAČR) project No. 23-07228S. This work was also supported by the Ministry of Education, Youth and Sports of the Czech Republic through the SEN-DISO - CZ.02.01.01/00/22_008/0004596, the e-INFRA CZ (ID:90254),
and the MeBioSys - no. CZ.02.01.01/00/22_008/0004634. This work has also been supported by the Charles University Grant Agency (GA UK – 222125). The Interdisciplinary Centre of Mathematical and Computational Modelling (ICM), Warsaw University, Poland, is acknowledged for providing computer facilities to perform part of the present calculations. This work was partially supported by the CNRS Federation IRMA- FR 3095. The FIB equipment belonged to the GENESIS platform

funded by the French National Research Agency under the "Investissements d'avenir" program with the reference ANR-11-EQPX-0020.

Data availability

The data as well as the figures are available through the following link https://doi.org/10.57680/asep.0637995.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.apsadv.2025.100821.

Appendix

Electronic transport properties, such as electrical conductivity $\sigma$, resistivity $\rho = \sigma^{-1}$, Seebeck coefficient $S$, and electronic contribution to thermal conductivity $k_L$ are calculated through Onsager coefficient $\mathcal{L}_{a\beta}^n$ as follows [56]:

$$ \sigma_{a\beta} = \mathcal{L}_{a\beta}^0 $$

$$ S_{a\beta} = \frac{1}{eT} \frac{\mathcal{L}_{a\beta}^1}{\mathcal{L}_{a\beta}^0} $$

$$ k_{a\beta} = \frac{1}{e^2 T} \left[ \frac{(\mathcal{L}_{a\beta}^1)^2}{\mathcal{L}_{a\beta}^0} - \mathcal{L}_{a\beta}^2 \right] $$

where $e$ is the electron charge, $\alpha$ and $\beta$ denote Cartesian coordinates, $\mathcal{L}_{a\beta}^n$ are the $n$-th moments of the generalized transport coefficients expressed in the following way:

$$ \mathcal{L}_{a\beta}^n = e^2 \int \Sigma_{a\beta}(E)(E - E_F)^n \left[ -\frac{\partial f^0}{\partial E} \right] dE $$

where $f^0$ and $\Sigma_{a\beta}$ stand for the Fermi-Dirac distribution function and the spectral conductivity, respectively. The spectral conductivity is computed as:

$$ \Sigma_{a\beta}(E) = \sum_i \int \frac{dq}{8\pi^3} v_{iq,a} v_{iq,\beta} \tau_{iq} \delta(E - E_{iq}) $$

where $E_{iq}$ and $v_{iq,a}$ are the energy and group velocity of electron in band $i$ and wave vector $q$, respectively. The lattice contribution to thermal conductivity $k_L$ follows from the following expressions [59–62,68,69].

$$ k_L = \frac{k_B}{2\pi^2\nu} \left(\frac{k_B T}{\hbar}\right)^3 \int_0^{\Theta_T} \tau \frac{x^4 e^x dx}{(e^x - 1)^2} $$

$$ \tau^{-1} = p\omega^2 \frac{T}{\Theta} e^{-\Theta/3T} $$

$$ p = \frac{1 - 0.514\gamma^{-1} + 0.228\gamma^{-2}}{0.0948} \frac{\hbar^2\gamma^2}{k_B\theta MV^{1/3}\nu} $$

with $V$ and $M$ denoting unit cell volume and average atomic mass, respectively. The parameter $\theta = n^{-1/3}\theta_D$, where $n$ represents the number of atoms per unit cell [68]. The $\theta_D$ is determined from the second moment $\langle\mu^2\rangle$ of the whole phonon spectrum as [69]:

$$ \Theta_D = \frac{\hbar}{k_B} \sqrt{\frac{5}{3} \langle \mu^2 \rangle}, $$

$$ \langle \mu^2 \rangle = \frac{\int \omega^2 g(\omega) d\omega}{\int g(\omega) d\omega}, $$

where $g(\omega)$ denotes the computed density of phonon states.
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