1 Introduction

Many research groups are now focusing on the quest for novel materials that can be used for energy conversion applications. Thermoelectric (TE) materials have the ability to convert heat directly into useful electrical energy according to the Seebeck effect and to pump heat according to the Peltier effect. Thus, TE materials have received increasing global attention since the discovery of bismuth telluride Bi2Te3 in 1960 [1], which is the reference thermoelectric material. Many families of materials, such as clathrates [2], skutterudites [3, 4], and Zintl phases [5], have proven to be effective TE materials since this first discovery.

The efficiency of TE materials is governed by a dimensionless physical parameter known as “figure of merit,” which is defined as follows: \(\mathrm{ZT}=\frac{{S}^{2}\sigma }{{\kappa }_{\mathrm{e}}+{\kappa }_{\mathrm{l}}}\), where σ represents the electrical conductivity, S represents the Seebeck coefficient, κe represents the electronic thermal conductivities, κL represents the lattice thermal conductivities, and T represents the absolute temperature. The thermoelectric performance of a TE material can be enhanced in different ways such as through convergence of electronic band valleys [6], adding impurities or by creating point defects through alloying [7,8,9], and making superlattices from thin films [10]. On the other hand, searching for materials having low thermal conductivity, a large Seebeck coefficient and high electrical conductivity present a serious dilemma. This is because these parameters are inversely dependent on the charge carriers.

Here, we report the results of a detailed investigation of the electronic structure and thermoelectric properties of the CuTlSe2 compound, which is a Weyl semimetal material and possess a correlated-electron system [11]. Furthermore, the CuTlSe2 compound belongs to the Cu-based chalcopyrite family, including the CuGaTe2, CuInTe2, CuAlSe2, and CuAlTe2 compounds, which exhibit interesting thermoelectric properties [12,13,14]. Some researcher groups tried to enhance the thermoelectric performance of chalcopyrite by doping them with different transition metal atoms, such as doping CuFeS2 with Zn, Mn, Co, Ni, and Pd atoms [15, 16], CuInTe2 [17] and CuGaTe2 [18] with Ag element, and CuGaTe2 with Mn atom [19]. Besides, it is reported that the sodium substitution affects the electronic properties and improves the TE character of semiconductors [20] and metallic materials [21]. To the best of our knowledge, to date neither detailed experimental measurements nor theoretical investigations regarding the Na-doped CuTlSe2 compound have been reported. In this context, we propose a strategy to boost the thermoelectric efficiency of CuTlSe2 by alloying it with the Na atom, which may enhance the Seebeck coefficient (S) and probably reduce the thermal conductivity (k), i.e., increase the ZT value, which quantifies the thermoelectric performance.

2 Crystal structure and computational details

The CuTlSe2 compound adopts the chalcopyrite crystal structure, which is a tetragonal system with space group I-42d (no. 122) [22]. In this structure, the Cu, Tl, and Se atoms are located the 4a (0, 0, 0), 4b (0, 0, 1/2), and 8d (0.23, 1/4, 1/8) Wyckoff positions, respectively. To model the crystalline structure of the Cu0.75Na0.25TlSe2/Cu0.50Na0.50TlSe2 compounds, we substituted one/two Cu atoms in the CuTlSe2 (1 × 1 × 2) supercell with one/two Na atoms as shown in Fig. 1, panels b and c. The full potential linearized augmented plane wave (FP-LAPW) approach [23, 24] as incorporated in the WIEN2k package [25] was used to carry out all ab initio calculations. The exchange–correlation interactions were modeled using the Wu and Cohen [26] generalized gradient approximation GGA-WC functional to determine the optimized structural parameters, including the lattice parameters and atomic position coordinates, and the Tran and Blaha [27] modified Becke–Johnson TB-mBJ potential to calculate the electronic structure. Note that the TB-mBJ potential was specially developed to produce a bandgap value closer to the experimental counterpart than that provided by the standard GGA [28, 29]. There are some convergence parameters that should be set carefully for precise calculation, namely \(R_{{{\text{MT}}}}^{\min } K_{\max }\), where Kmax is the largest wave vector and \(R_{{{\text{MT}}}}^{\min }\) is the smallest muffin-tin sphere radius, angular momentum cut-off (ℓmax), muffin-tin radii (RMT), and number of the k-points in the irreducible Brillouin zone (IBZ). For good convergence of the performed calculations, we choose \(R_{{{\text{MT}}}}^{\min } K_{\max }\) = 8, RMT (Cu) = 2.1 a.u., RMT (Tl) = 2.36 a.u., RMT (Se) = 1.9 a.u., RMT (Na) = 2.1 a.u., ℓmax = 10, and 99 k-points in IBZ. The crystal structure was optimized by minimizing the forces acting on each constituent atom with convergence criteria set to 10–4 Ryd/Bohr. The total energy convergence criterion was set to 10−4 Ryd. The BoltzTrap code [30], which is an implementation of the Boltzmann transport theory with the rigid-band model and constant-scattering time approximation (CSTA), was used to calculate the thermoelectric parameters. For precise calculation of the thermoelectric parameters, a dense mesh of 6083 k-points in the IBZ was used.

Fig. 1
figure 1

Crystal structures of CuTlSe2 (a), Cu0.75Na0.25TlSe2 (when Cu1 is replaced) (b), and Cu0.50Na0.50TlSe2 (when Cu1 and Cu2 are replaced) (c) Cu (black filled circle), Tl (red filled circle), Se (green filled circle), Na (yellow filled circle)

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3 Results and discussion

3.1 Structural properties

The lattice parameters (a and c) along with the bulk modulus B, and its pressure derivative B′ for the Cu1–xNaxTlSe2 (x = 0.00, 0.25, 0.50) compounds, calculated by fitting total energy versus unit cell volume to Murnaghan equation of state [31], are collected in Table 1 together with the available experimental counterparts. The obtained lattice parameters of CuTlSe2 agree well with the available experimental results [32]. Note that there are no available data for the structural parameters for the Cu1–xNaxTlSe2 (x = 0.25, 0.50) compounds in the literature. Table 1 shows that both a and c increase with increasing composition of the Na atom (x) in the Cu1–xNaxTlSe2 alloy, while the bulk modulus decreases with increasing x.

Table 1 The calculated lattice parameters (a and c, in Å), bulk modulus (B, in GPa), bulk modulus pressure derivative (B′), formation energy (Ef, (in kJ/mol), bandgap (Eg, in eV), and electron/hole effective mass (me/mh, in m0 “the rest mass of an electron” unit)
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Besides, the formation energy Ef per atom is calculated in order to examine the thermodynamic stability of the Cu1–xNaxTlSe2 alloys using the following relations:

$$ E_{{\text{f}}} \left( {{\text{Cu}}_{1 - x} {\text{Na}}_{x} {\text{TlSe}}_{2} } \right) = 1/N\left( {E_{{\text{T}}} \left( {{\text{Cu}}_{1 - x} {\text{Na}}_{x} {\text{TlSe}}_{2} } \right) - \left[ {\left( {1 - x} \right)E_{{{\text{Cu}}}} + xE_{{{\text{Na}}}} + E_{{{\text{Tl}}}} + 2E_{{{\text{Se}}}} } \right]} \right) $$
(1)

Here, ET is the total energy of the Cu1–xNaxTlSe2 alloys, N is the number of atoms, ECu, ETl, ENa and ESe are the energies of Cu (cubic, Pm3m), Tl (cubic, Fm3m), Na (cubic, I43m), and Se (cubic, Pm3m) atoms, respectively, in their solid state. Calculated formation energies for all considered Cu1–xNaxTlSe2 alloys are negative as can be seen from Table 1, indicating their thermodynamic stability. Note that the calculated Ef value for CuTlSe2 compound is in good agreement with that estimated by Kumar and Sastry [33].

3.2 Electronic properties

Both GGA-WC and TB-mBJ were used to calculate the energy band dispersions (Fig. 2) of the Cu1–xNaxTlSe2 compounds. The GGA-WC band structure reveals that CuTlSe2 is of metallic character, while the TB-mBJ one with including spin–orbit coupling (SOC) (Fig. 2) shows semi-metallic character, which is consistent with the work of Ruan et al. [11]. However, both Cu0.75Na0.25TlSe2 and Cu0.50Na0.50TlSe2 alloys are found to be semiconductors with direct narrow band gap at (Γ–Γ) of about 0.17 eV (0.06 eV) and 0.35 eV (0.15 eV), respectively, using TB-mBJ (GGA-WC). Note that the inclusion of SOC does not affect the bandgap values of the considered alloys. The transition from semi-metallic character to semiconductive nature after Na-doping results from the shift of the minimum conduction band (MCB) toward higher energy. From Table 1, it can be seen that the bandgap value increases with increasing Na composition in the Cu1–xNaxTlSe2 alloys; the bandgap value increased of about 0.177 eV when x increased from 0.25 to 0.50. This increase can mainly be attributed to the change in the valence states and the difference in electronegativity [34] between Cu (1.9) and Na (0.93) atoms.

Fig. 2
figure 2

Calculated band structure of CuTlSe2 (a), Cu0.75Na0.25TlSe2 (b), and Cu0.50Na0.50TlSe2 (c)

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We calculated the electron effective mass (me*) at the MCB and the hole effective mass (mh*) at the maximum valence band (MVB) for Cu1–xNaxTlSe2 alloys via the following equation:

$$ m^{*} = \hbar^{2} \left( {\frac{{{\text{d}}^{2} E}}{{{\text{d}}k^{2} }}} \right)^{ - 1} $$
(2)

where the continuous function E(k) was obtained by fitting the energies E at the band edge positions k to the parabolic function and \(\hslash \) is the reduced Planck constant [35, 36].

It appears from Table 1 that the effective mass of the electron at MCB is smaller than that of the hole at MVB for the CuTlSe2 alloys. This may be attributed to the fact that the minimum conduction band is more dispersive than the maximum valence band. The hole effective mass increases with increasing Na composition in the Cu1–xNaxTlSe2 alloys, while the electron effective mass is slightly changed, which is due to the fact that the slopes of the MCB remain somewhat unchanged (see Fig. 2).

To understand the electronic structure of Cu1–xNaxTlSe2 (x = 0.00, 0.25 and 0.50) materials, it is essential to calculate the density of states (DOS) diagram. Figure 3 depicts the calculated DOS spectra using the TB-mBJ potential. It is clear from Fig. 3 that the valence band close to the Fermi level of the CuTlSe2 material is mainly originated from the Cu-3d states hybridized with the Se-4p and Tl-6p states, and the conduction band close to the Fermi level is dominated by the Tl-6s states hybridized with the Se-4p and Cu-3d states. When the copper is substituted by the sodium, we remark the apparition of new states in the MVB due to the ‘2p’ states of Na, and the conduction states in the energy range 0–1.3 eV, formed mainly by Tl-6s and Se-4p states, are pushed toward higher energy. Consequently, Cu0.75Na0.25TlSe2 and Cu0.50Na0.50TlSe2 have semiconductor character with a bandgap energy of about 0.17 eV and 0.35 eV, respectively.

Fig. 3
figure 3

Density of states of Cu1–xNaxTlSe2 (x = 0.00, 0.25, and 0.50) alloys

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For a better understanding of charge distribution in the studied materials, the electron charge distribution map was calculated in the (112) plane, which involves all constituent atoms. Figure 4a shows that the charge in CuTlSe2 is shared between the cations (Cu, Tl) and the anion atom Se, confirming the formation of a covalent bond between them. When the Na substitutes the Cu atom, a high charge accumulation in the vicinity of the anion occurs, which is caused by the electron transfer from the Na atom to the Se atom (Fig. 4b, c), and since there is a high electronegativity difference between Na and Se, which is about 1.62, a Na–Se ionic bond is formed. This new Na–Se ionic bond is responsible for the metal–semiconductor transition in the Cu1–xNaxTlSe2 alloy. By increasing the concentration of Na, stronger ionic bonds are formed (Fig. 4c) and hence, the bandgap value is increased.

Fig. 4
figure 4

Contour plots of the charge density distribution in color scale, \(\Delta n\left(r\right)\): nuances of rendering resolution of the charge density, for Cu1–xNaxTlSe2 (x = 0.00, 0.25, and 0.50) alloys. Red color (+ 0.0) represents no charge, while pink color (+ 1.0) represents the maximum charge accumulation site

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3.3 Thermoelectric coefficients

Knowledge of material’s thermoelectric properties is essential for device manufacture. We computed temperature-dependent thermoelectric transport coefficients of Cu1–xNaxTlSe2 alloys using the semi-classical Boltzmann transport theory. Figure 5 illustrates the chemical potential dependence of the Seebeck coefficient, electrical conductivity, thermal conductivity, and power factor at absolute temperatures of 300 K, 600 K, and 800 K for Cu1–xNaxTlSe2 compounds.

Fig. 5
figure 5

Seebeck coefficient, electronic conductivity, electronic thermal conductivity, and power factor of Cu1–xNaxTlSe2 (x = 0.00, 0.25, and 0.50) alloys at absolute temperatures of 300 K, 600 K, and 800 K

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Figure 5a shows that the Seebeck coefficient value at Fermi level is  − 19.27 µV/K for CuTlS2, 148.84 µV/K for Cu0.75Na0.25TlSe2, and 180.77 µV/K for Cu0.50Na0.50TlSe2. Knowing that the Seebeck coefficient value for n-doped materials is negative and positive for p-doped materials, we conclude that CuTlSe2 is an n-type conductivity material with electrons as the majority charge carriers, whereas Cu0.75Na0.25TlSe2 and Cu0.5Na0.5TlSe2 alloys are of p-type conductivity materials proving that the majority charge carriers are holes. Consequently, alloying CuTlSe2 with Na leads to a transition from n-type to p-type conductivity, which can be elucidated by the shift of the Fermi level into the edge of the valence band for Cu1–xNaxTlSe2 alloys (Fig. 2). Knowing that the Seebeck coefficient is proportional with the effective mass, \(S = \frac{{8\pi k_{B} T}}{3eh}m*(\frac{\pi }{3n})^{2/3}\) [37]; the increase in S with increasing Na composition can be attributed to the flatness of the valence band maximum (VBM) in the case of Cu0.50Na0.50TlSe2 alloy, suggesting a higher hole effective masse (mh*) (Table 1). Moreover, this increase in the S value can also be attributed to the increase in DOS near the Fermi level with increasing Na atom composition (see Fig. 3). Besides, it is also observed that the Seebeck coefficient value decreases with increasing temperature.

Panels of Figure 5d–f show that the electrical conductivity decreases with increasing Na composition in Cu1–xNaxTlSe2 and is less affected by temperature. This behavior can be explained by the increase in the ionic character with the increase in the Na concentration due to the increase in the ionic bonds Na–Se. The electronic thermal conductivity (κe) is related to the electrical conductivity (σ) through Wiedemann–Franz relation κe = LσT, where L is the Lorenz number; thus, κe will vary with temperature (T) and Na composition as σ. Low thermal conductivity is required for efficient TE materials. Figure 5j–l represents the variation of the power factor (PF) with the chemical potential at the absolute temperatures of 300 K, 600 K and 800 K for the Cu1–xNaxTlSe2 alloys. It appears from Fig. 5j–l that CuTlSe2 has a maximum value of PF for n-type, whereas both alloys Cu1–xNaxTlSe2 (x = 0.25, 0.50) have higher peak for p-type at room temperature. Similarly, PF has a maximum for n-type in the case of CuTlSe2 and Cu0.75Na0.25TlSe2 at high temperature.

3.3.1 Figure of merit

As mentioned in the introduction, to determine the figure of merit ZT, one needs to estimate the total thermal conductivity (κ = κe + κl), and since the BoltzTrap code yields only the electronic thermal conductivity (κe), we calculated the thermal lattice conductivity (κl) using the Slack model formula [38]:

$$ \kappa_{{\text{l}}} = A \cdot \frac{{\overline{M}\theta_{{\text{D}}}^{3} \delta }}{{\gamma^{2} Tn^{2/3} }} $$
(3)

where \(\overline{M}\) is the average atomic mass of all constituent atoms, A is a constant determined as [39]: \(A = \frac{{2.43.10^{ - 8} }}{{1 - \frac{0.514}{\gamma } + \frac{0.228}{{\gamma^{2} }}}}\), δ is the average atomic volume to the power of (1/3), T is the absolute temperature, n is the number of atoms in primitive cell, γ is the Grüneisen parameter, and θD is the Debye temperature. γ and θD can be estimated through the quasi-harmonic Debye model integrated in the GIBBS2 program [40]. The estimated lattice conductivity for Cu1–xNaxTlSe2 alloys is illustrated in Fig. 6. On the other hand, the relaxation time τ is estimated based on deformation potential theory [41] in which the acoustic phonon modes coupling with the free carrier is considered. According to DP theory, τ is given by:

$$ \tau = \frac{{2\sqrt {2\pi } \hbar^{4} C_{ii} }}{{3\left( {m^{*} k_{{\text{B}}} T} \right)^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} E^{2} }} $$
(4)

where ћ, m*, Cii, T, kB, and E are, respectively, the reduced Planck constant, effective mass [hole mass (mh*) for p-type or electron mass (me*) for n-type], elastic constants, absolute temperature, Boltzmann constant, and deformation potential constant of the MCB for electron or MVB for hole. The obtained electron/hole relaxation time (τ) for Cu1–xNaxTlSe2 alloys is presented in Fig. 7. It is evident that the temperature dependence of τ is similar in the three alloys (see Fig. 7); it decreases when the temperature increases. This behavior can be explained by the fact that the thermal speed of electrons increases when the temperature increases; thus, the collisions between electrons become more important, and then, the amplitude of vibration of the atoms around their equilibrium positions increases. The calculated figure of merit ZT considering both electronic and lattice thermal conductivities for Cu1–xNaxTlSe2 alloys at the absolute temperatures 300 K, 600 K and, 800 K is presented in Fig. 8. At room temperature, the maximum ZT is about 0.09 and 0.10 for CuTlSe2 of p-type and n-type. When part of the Cu atoms is replaced by Na atoms, the ZT value increases to reach a maximum of about 0.46 and 0.87 for p-type Cu0.75Na0.25TlSe2 and n-type Cu0.50Na0.50TlSe2, respectively. The aforementioned ZT maximums, i.e., 0.46 for p-type Cu0.75Na0.25TlSe2 and 0.87 for n-type Cu0.50Na0.50TlSe2, correspond to carrier concentrations of about 1.7 × 1020 cm−3 and 2 × 1020 cm−3, respectively. By increasing the temperature, the ZTmax values gradually decrease and the corresponding peaks move toward the low chemical potential region.

Fig. 6
figure 6

The estimated lattice conductivity κl for Cu1–xNaxTlSe2 (x = 0.00, 0.25, and 0.50) alloys

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Fig. 7
figure 7

The calculated hole/electron relaxation time for Cu1–xNaxTlSe2 (x = 0.00, 0.25, and 0.50) alloys

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Fig. 8
figure 8

Electronic figure of merit at absolute temperatures of 300 K, 600 K, and 800 K for Cu1–xNaxTlSe2 (x = 0.00, 0.25, and 0.50) alloys

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4 Conclusion

The equilibrium structural parameters, electronic structure, and thermoelectric parameter properties of the Cu1–xNaxTlSe2 (x = 0.00, 0.25, and 0.50) were explored using first-principles calculations based on density functional theory. Calculated TB-mBJ band structure shows that CuTlSe2 is a semimetal material and undergoes a metal–semiconductor transition when Cu is replaced by Na atoms. As a result, a direct band gap is observed for Cu1–xNaxTlSe2 (x = 0.25 and 0.50) alloys. Such band gap energy was found to increase with increasing Na concentration, from 0.17 eV for Cu0.75Na0.25TlSe2 to 0.35 eV for Cu0.50Na0.50TlSe2. This behavior is caused by the formation of new Na–Se bond Cu1–xNaxTlSe2 alloys (x = 0.25 and 0.50). Moreover, the effect of alloying CuTlSe2 with Na on thermoelectric coefficients was studied in detail. The results show that Cu0.50Na0.50TlSe2 marked the highest Seebeck coefficient and power factor, which leads to the highest ZT of about 0.87 at room temperature. Consequently, the Na substitution enhances the thermoelectric properties of CuTlSe2, making it a promising compound for power generation.