Effect of Mn doping in mechanical properties of Cd_1−xMn_xTe₂

Transition metal dichalcogenides (TMDCs), with chemical formula MX₂ (where M is a transition metal and X is a chalcogen, such as S, Se and Te) have gained much attention due to their novel characteristics, that can be widely used in industrial and engineering fields [1]. TMDCs' crystal structure was first determined by Linus Pauling in 1923 [2], where hexagonally packed atoms form weakly coupled sandwiched X-M-X layers, with one M layer enclosed within two layers of X [3].

The bulk TMDCs possess diverse properties ranging from insulators (such as HfS₂), semiconductors (MoS₂ and WS₂), semimetals (WTe₂ and TiSe₂) and metals (NbS₂ and VSe₂) [1] _. The relatively high earth abundance of TMDCs and their direct bandgaps in the visible range make them an attractive light-absorbing material that can be used as an alternative thin-film solar cell [4], including flexible photovoltaics that could be used for coating buildings and curved structures. Since, they possess high electron mobilities they can also be used as a catalyst and a good lubricant (for e.g. MoS₂ and MoSe₂) [5]. The recent reports on TMDCs also suggest their thermoelectric applications with temperature dependent figure of merit defined by (ZT) in range of 0.1 to 0.7 [6, 7]. The ZT value was later on enhanced to 0.8 to 0.95 with doping at moderate temperature [8].

Among the significant member of TMDCs, Cd based candidates (CdX₂) are also appreciable and interesting. These materials are mechanically stable in pyrite structure [9] where CdS₂ has the strongest resistance to deformation while CdTe₂ has the least of them [10]. CdTe₂ in its pyrite structure that coexist with CdTe under certain conditions acts as an ohmic contact with large work function. Furthermore, its high absorption coefficient (4 × 10⁴ cm⁻¹–8 × 10⁵ cm⁻¹) with cohesive energy −0.485 eV/c.u. and low band gap semiconducting nature is also regarded as a good photon absorber in the visible range [11]. Similarly the low band gap (0.7 eV) semiconductor MnTe₂ [12] also exhibit good thermoelectric properties with high seebeck coefficient of 400 μV K⁻¹ and low hole concentration of 10⁻¹⁹ cm⁻³ [12]. It also exist in pyrite structure and undergoes a second-order antiferromagnet-paramagnet phase transition at T_N = 86.5 K [13].

There exist limited studies on CdTe₂ and similarly the previous reports on MnTe₂ are also unable to predict the mechanical properties. From the literature it has been observed that p type doping are efficient to enhance the thermoelectric performance and knowing the effect of doping an effort has been made to understand the stability and structural properties of CdTe₂ and consequence of subsequent Mn doping. For this purpose an efficient density functional theory (DFT) based method is implemented to explore the structural and elastic properties of Cd_1−xMn_xTe₂ (x = 0.03125, 0.125, 0.25, 0.5, 0.75 and 1) and the results so obtained may provide the reference data for the experimentalists for further investigations.

The result reported in the manuscript, are carried out by using full potential linearized augmented plane wave [FP-LAPW] method [14] within the framework of DFT as implemented in the wien2k code [15]. The exchange-correlation effects are treated by using the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) [16]. The self consistent potentials are calculated on a 15 × 15 × 15 k-mesh in the Brillouin Zone and the self consistency convergence criteria is set to 10⁻⁴ Ry. The energy eigen values were converged with a plane wave cut off parameter of R_MT x K_max = 8 (where R_MT is the smallest atomic radii of all atomic spheres and K_max is the maximum value of the wave vector K = k+G) in the interstitial region and the charge density was Fourier expanded upto G_max = 13 a.u.⁻¹. The different MT sphere radii (R_MT ) used were 2.2 a.u, 2.3 a.u and 2.5 a.u for Cd, Mn and Te, respectively. A 2 × 2 × 2 supercell was constructed for lesser concentration generating a total 32 Cd and 64 Te atoms where a subsequent replacement of Cd atoms by Mn leads to Cd_1−xMn_xTe₂ (x = 0.03125, 0.125) structure and for higher concentration Mn was added in a unit cell with 4 Cd atoms. For x = 0.125 we need to replace 4 Cd atoms which can be done in 35,960 ways which has a high computational cost. After investigating the structure for many possible configuration considering the corner atoms specifically as for these atoms the environment is symmetrically same there was not much difference in the energy, however the energetically most favourable configurations for the Mn atoms was chosen. The same procedure was followed for other lesser concentration.

The (Cd/Mn)Te₂ crystallize in pyrite ( space group Pa $\bar{3}$) structure (figure 1) where one Cd/Mn atom takes the atomic positions of 4a (0, 0, 0) and one Te atom is located at 8c (0.389, 0.389, 0.389). The experimental lattice constant of 7.25 Å and 6.95 Å for CdTe₂ and MnTe₂ were optimized by calculating the total energy of the system by varying the crystal cell volume. The optimized lattice constants are calculated then by fitting the data into Murnaghan's equation of state and a comparitive analysis of the experimental and theoretical results are presented in table 1. The structures were optimized for the magnetic and non-magnetic phases (figure 1), and the optimization curve depicts that CdTe₂ crystallizes in non-magnetic phase, whereas the addition of Mn results the magnetic ground state (figure not included). The effect of GGA, that overestimates the results is distinct in CdTe₂, however in MnTe₂ it is slightly underestimated as compared to experimental report, which can be attributed to the temperature factor existing in experimental and our calculations.

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The structure Cd_1−xMn_xTe₂ so obtained by adding Mn in CdTe₂ supercell was also optimized by minimising their energy corresponding to their volume and the forces between two atoms and as we increase the concentration of Mn atom with slightly small atomic radii (0.67 Å) as compared to Cd (0.95 Å) the lattice constant decreases in regular trend. In lack of experimental results for the complex doped structure , the lattice constant was also verified from the empirical vegard's rule [20] as given by equation (1),

$$\begin{eqnarray}&&{{\rm{a}}}_{{\rm{vegards}}}=\left(1-x\right){a}_{CdT{e}_{2}}+x{a}_{MnT{e}_{2}}\end{eqnarray} \tag{ 1 }$$

where a_vegards is the lattice constant calculated by vegards rule, ${a}_{CdT{e}_{2}},{a}_{MnT{e}_{2}}$ is the lattice constant of CdTe₂ and MnTe₂ compound where (1−x) and x denotes the number of Cd and Mn atoms respectively. The qualitative agreement between a and a_vegards also validates the present data, however a slight deviation (0.2%–0.5%) in the results may be due to mismatch of lattice constants of CdTe₂ and MnTe₂. In order to find the stable crystal structure that minimises the total energy the internal parameter (u) was optimized, which for CdTe₂ agrees well with experimental report. The value of u shows a jump from 0.389 to 0.305 at small x (table 1), and jump back at large x as for lower concentration doping, a supercell was chosen which reduces the value of the internal parameters that defines the position of the Te atom in the cell with minimum forces betwen them. Whereas, for higher concentration a unit cell was considered that does not affect the position of Te atom and hence the internal parameter, however MnTe₂ has slightly lower value that follows the trend observed in lattice parameter .

The possibility of doping a crystal structure with a particular dopant and their thermodynamical stability is verified by estimating their formation energy (ΔE_F ) [21, 22] as given by equation (2),

$$\begin{eqnarray}&&{\rm{\Delta }}{{\rm{E}}}_{{\rm{F}}}=\displaystyle \frac{{E}_{F}-\left(x{E}_{Cd}+y{E}_{Mn}+z{E}_{Te}\right)}{x+y+z}\end{eqnarray} \tag{ 2 }$$

where E_F is the total energy of the compound, E_Cd, E_Mn and E_Te are the equilibrium energy of the individual Cd, Mn and Te atoms, which is found to be –11192.07 Ry, −2316.65 Ry and −13593.64 Ry, and x, y, z denotes the number of Cd, Mn and Te respectively. The formation energy so obtained has been listed down in table 2 and their negative value indicates that they are energetically stable. In addition, the negative values of ΔE_F also indicates stronger bonding between the atoms and more alloying stability of the crystal [23]. From the table one can note that as the concentration of Mn increases the value of formation energy also increases suggesting a more stable structure.

Elastic constant tells us about the bonding forces and mechanical strength of a system, which will be of great importance in technological applications. Thus, the independent elastic constants C_ij which is reduced to C₁₁,C₁₂,C₄₄ due to cubic symmetry of the crystal are estimated at ambient condition after their volume deformation that follow the stability criteria [9], laid by C₁₁ + 2C₁₂ > 0, C₄₄ > 0, C₁₁−C₁2 > 0. For all the compounds the value of C₁₁ is greater than C₁₂ and C₄₄ which indicates that the compound is hard to compress along the x axis and as C₄₄ relates to shear stress and strain in the same direction, the compound has a weaker resistance against a pure shear deformation.

The C_ij(C₁₁, C₁₂, C₄₄) derived parameters such as B, Y, G (table 2) of CdTe₂ are in qualitative agreement with the previous reports [10] and similar technique has been applied to compute their values for these materials listed in table 1 that may validate the reliability of the present calculation in the absence of previous theoretical and experimental report .

To shed more light in the mechanical properties of the structure, we calculated the Bulk modulus (B), Young's modulus (Y) and shear modulus (G) using the equations (3)–(5) [24–26],

$$\begin{eqnarray}&&{\rm{B}}=\displaystyle \frac{1}{3}\left({{\rm{C}}}_{11}+2{{\rm{C}}}_{12}\right)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\rm{Y}}=\displaystyle \frac{9BG}{3B+G}\end{eqnarray} \tag{ 4 }$$

Poisson's ratios depicted in table 1 were calculated from the Voigt-Reuss-Hill (VRH) approximation shown in Eqs.

$$\begin{eqnarray}&&{{\rm{G}}}_{{\rm{V}}}=\displaystyle \frac{{C}_{11}-{C}_{12}+3{C}_{44}}{5}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{{\rm{G}}}_{{\rm{R}}}=\displaystyle \frac{5\left({C}_{11}-{C}_{12}\right){C}_{44}}{4{C}_{44}+3\left({C}_{11}-{C}_{12}\right)}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\rm{G}}=\displaystyle \frac{{G}_{V}+{G}_{R}}{2}\end{eqnarray} \tag{ 7 }$$

here G_V and G_R are Voigt's and Reuss's shear moduli respectively.

In comparison with analogous CdX₂ (X = S,Se,Te) compound, CdTe₂ has the least Bulk modulus specifying that it has the least power of withstanding compressibility. The results also implies that B increases with x (concentration of Mn) and thus the material becomes stiffer.

Higher the value of Y than bulk modulus (B), the material is hard to be broken [27]. The Y value of compounds like CdS₂ and CdSe₂ has value of 70.075 GPa and 56.445 GPa [10] which is greater than CdTe₂. Shear modulus or modulus of rigidity G is also an important parameter that relates to the hardness of a crystal. The calculated values of G for CdTe₂ are in qualitative agreement with the theoretical values of analogous compounds and varies as CdS₂ > CdSe₂ > CdTe₂ [10]. Unfortunately for MnTe₂ no elastic results are available for comparison.

Poisson's ratio (ν) identifies the compressibility and stretchability of a material before it collapses. The ratio is found to vary as 0.1 ≤ ν ≤ 0.5, which agrees well with the other cadmium based dichalcogenides. It also characterizes the bonding nature of solid material, where for covalent materials the ratio has a small value (typically at 0.1) and for ionic compounds, its value is greater than or equal to 0.25. As the ν value for Cd_1−xMn_xTe₂ (for $x\leqslant 12.5 \%$) is greater than 0.25, the ionic nature of bonding can be expected whereas for x >12.5% covalent bonding exists.

The value of pugh ratio (B/G) measures the ductility or brittleness of a material with cut off value of 1.75. The pure CdTe₂ and Mn doped CdTe₂ (for x < 0.25) suggest a ductile nature whereas as the concentration of Mn dopant increases the material starts to show a brittle nature. The variation of B,Y and G with respect to concentration are presented in figure 2.

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Another important parameter, which closely relates to the specific heat and melting point of the solid is the debye temperature (θ_D) which can be calculated using equation (8),

$$\begin{eqnarray}&&{\theta }_{{\rm{D}}}\displaystyle \frac{h}{k}{\left[\displaystyle \frac{3n}{4\pi }\left(\displaystyle \frac{\rho {N}_{A}}{M}\right)\right]}^{\displaystyle \frac{1}{3}}{V}_{m}\end{eqnarray} \tag{ 8 }$$

here h is the planck's constant, k is the Boltzmann's constant, n is the number of atoms per molecule, ρ is the density of the unit cell , N_A is the Avagadro's number, M is the molar mass and v_m is the average sound velocity. High value of Debye temperature indicates that the crystal is stiffer and have a high melting point. The Debye temperature is in reasonable agreement with other theoretical work and other analogous compound like CdS₂ and CdSe₂ with values 287.25 K and 212.4 K respectively [10]. As the Mn concentration increases the θ_D also increases with highest value for MnTe₂ indicating that MnTe₂ is stiffest of all.

We have calculated the structural and elastic properties of Cd_1−xMn_xTe₂ using first principles method. The optimized lattice constant and internal parameter are in close agreement with the available results and that have also been validated by Vegards rule. The complex structure Cd_1−xMn_xTe₂ so formed after the addition of Mn in CdTe₂ are mechanically and thermodynamically stable and their elastic properties are calculated at stable structure at volume deformation. The Bulk modulus, Young's modulus and Shear modulus of the CdTe₂ is in good agreement with available experimental and theoretical work. The elastic constant reveals the ductile nature and ionic bonding of CdTe₂ compound with stiffness lower than that of MnTe₂ and analogous compound. The debye temperature was also found to be highest for MnTe₂ indicating that MnTe₂ is stiffest of all. The formation energy calculated reveals that the structure is thermodynamically stable and the stability increases with the increase in the concentration of Mn dopant. Since there have not been any experimental or theoretical works in complex doped structure to compare with our results to the best of our knowledge, we hope these results will be useful for further investigations of their electronic, optical and thermoelectric properties.

A Shankhar acknowledges SERB New Delhi (EEQ/2017/000319).