Ring and Linear Structures of CdTe Clusters

: We report the results of an ab initio study of the linear and ring structures of cadmium telluride clusters [CdTe] n (Cd n Te n ) n ≤ 10 within the generalized gradient approximation (GGA) and Purdue–Burke–Ernzerhof (PBE) parameterization with Hubbard corrections (GGA+ U ). We optimized the linear and ring isomers for each size to obtain the lowest-energy structures and to understand their growth behavior. The cases of n < 8 for ring-type structures and n = 6 and 9 for linear-type structures were found to be the most favorable. All observed clusters with a linear structure were found to have a small highest-occupied–lowest-unoccupied molecular orbital (HOMO–LUMO) gap. The CdTe clusters with ring structure showed larger values of the HOMO–LUMO gaps than the band gap value for the bulk crystal. Structural and electronic properties like bond length, the HOMO–LUMO gap, binding energy, and electronegativity were analyzed.


Introduction
Semiconductor materials are of great importance in the development of technology.In particular, the materials of the A II B VI crystal group have applications such as in solar cells [1][2][3], gas sensors [4][5][6], photocatalysts [7,8], or quantum devices [9,10].This has led to extensive investigation of these materials.Many theoretical studies have been reported on the bulk electronic structure of these compound semiconductors [11][12][13][14].The values of band gaps and some other optical properties of these compounds (CdTe [15], CdS [16], ZnO [17], and their modifications via doping or substitutions) make them especially interesting for solar cells.A good material for use as an absorber in a solar cell must have a band gap close to the range of the Sun's irradiation (~1.75-3 eV [10]).Bulk cadmium telluride has a narrow direct optical band gap of approximately 1.44 eV [18] and a high absorbance (above ~10 5 cm −1 ).That is why p-CdTe is used as a typical absorber for solar cells.
The dependence of energy and structural properties on the size and shape of semiconductor nanoparticles (NPs) have become the focus of intensive research in recent years.Composite systems of NPs, quantum dots (QDs), and clusters of II-VI group materials (such as CdS, CdSe, and CdTe) have the potential to be used in the production of photoactive molecular devices [19].In addition, the application arrears of CdTe QDs or clusters vary from biological labeling to third-generation solar cells and hybrid organic-inorganic light-emitting diodes [20][21][22][23].CdTe QDs are potentially attractive as emitters in LEDs due to their tunable luminescence properties [24].CdSe is commonly used in core-shell structures [21,22], for which the valence band level is approximately −6.8 eV [25].On the other hand, for CdTe QDs, the valence band level is approximately −5.6 eV [26], which is significantly higher and results in a significant reduction in the hole injection barrier compared to CdSe.
In the present work, the results of the studies of the low-dimensional (ring and linear) structures of CdTe were analyzed.These results can help in understanding the growth behavior of the stable CdTe clusters in linear or ring structures.Some results of studies of the physical properties of CdTe clusters were found in the literature [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42].In these works, there were also reports on the density functional theory (DFT) calculations of the electronic properties of Cd m Te n (m ̸ = n [35,40], m = n [32,35,[38][39][40][41][42]) clusters.Particularly, the functional LDA [40], GGA [32,35,40], B3LYP [32,38,41], and B3LYP/LanL2DZ [42] were used for DFT calculations.Nonetheless, the properties of these materials still need further investigation.This work is focused on establishing the energy properties and stability factors of Cd n Te n clusters (n ≤ 10) as a function of their size.Establishing the main energy parameters and stability of cluster formations is crucial for determining optimal cluster sizes, which can be utilized in further studies for the formation of nanotubes.Also, we note that for such types of calculations, the PBE approximation with Hubbard corrections (GGA+U) was used.Note that all the above characteristics are important for optimizing the materials for photovoltaic engineering.The difference between the studies presented in this work and the works of Refs.[35,40,42] is based on the selection of the calculation method and the use of Hubbard corrections, which aligned our findings more closely with the experimental conditions.Additionally, the studies in [35,40,42] did not explore clusters with sizes n > 7.

Materials and Methods
All calculations including geometric optimization and electronic structures were performed using DFT, which was implemented in the Quantum Espresso package [43].GGA and PBE parameterization were used to describe the exchange-correlation energy of the electronic subsystem with Hubbard corrections (GGA+U).Unfortunately, for strongly correlated materials including CdTe, standard DFT with functional GGA (PBE) tends to underestimate the band gap [4,6,12,15,43].This is largely because standard GGA overestimates the delocalization of Cd-4d electrons, thereby pushing up the valence electrons (the Te-5p valence band) [44].The easiest way to obtain results closer to experimental ones is to use a so-called 'scissor' operator, which leads to the band gap changing by shifting the conduction band into a region of higher energies [15] or to using Hubbard U correction [4].The 'scissor' operator is based on the proximity of the E(k) dispersion dependences of the energies of conduction bands, which are determined from the Kohn-Sham equations [45].The conduction bands of the calculated electronic structure are usually shifted until the experimental value of the minimum energy gap bandwidth E g of the crystal is reached.Using Hubbard U correction on Cd d-orbitals is an efficient and computationally cost-effective method for addressing the significant underestimation of a band gap and correcting the energy position of the Cd atoms' d-states.
Firstly, the structural optimization and calculation of electron band energy structure were performed for bulk CdTe.This calculation was performed to estimate the value of the Hubbard corrections.The value E cut-off = 660 eV for the energy of cutting off the plane waves was used in our calculations.The electronic configurations of 5s 2 4d 10 for Cd and 5s 2 5p 4 for Te atoms formed the valence electronic states.The self-consistent convergence of the total energy was taken as 5.0 × 10 −6 eV/atom.Geometric optimization of the lattice parameters and atomic coordinates was performed using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization technique with the maximum ionic Hellmann-Feynman forces within 0.01 eV/Å, the maximum ionic displacement within 5.0 × 10 −4 Å, and the maximum stress within 0.02 GPa.
The convergence criteria for energy and force were set to ~3 × 10 −4 eV and ∼5 × 10 −2 eV/Å, respectively, for all the calculations.To accurately describe the electronic spectrum, two Hubbard corrections were selected for the studied objects: for d-orbitals Cd (U 4d = 5.80 eV) and p-orbitals Te (U 5p = 2.55 eV).Similar Hubbard correction values for CdTe were used in [27].
Analysis of the results of theoretical calculations of the energy band spectrum showed that the smallest band gap was localized in the center of the Brillouin zone (the Γ point) for the GGA and GGA+U calculations.This means that the crystal is characterized by a direct energy band gap.The estimated band gap for the GGA calculation was 0.494 eV.This is less than the appropriate value obtained experimentally (~1.44 eV [18]).Using the Hubbard correction U 4d = 5.80 eV for Cd and U 5p = 2.55 eV for Te atoms, we obtained the band gap of 1.438 eV for the bulk CdTe, perfectly consistent with the experimental data [18].

Linear Structures of CdTe Clusters
The optimized structures of some Cd n Te n clusters with linear structures are shown in Figure 1.The details of the structures and the energy properties of different isomers are given in Table 1.Before optimization, all the structures of CdTe clusters had similar CdTe bond lengths (l = 2.75 Å).Analysis of the average bond length (see Table 1) between atoms of Cd and Te after the optimization of the structures of CdTe clusters led to the value being near 2.74 Å.This value is close to the bond lengths between Cd and Te for the non-optimized clusters.
Quantum Rep. 2024, 6, 3 Analysis of the results of theoretical calculations of the energy band spectrum showed that the smallest band gap was localized in the center of the Brillouin zone (the Г point) for the GGA and GGA+U calculations.This means that the crystal is characterized by a direct energy band gap.The estimated band gap for the GGA calculation was 0.494 eV.This is less than the appropriate value obtained experimentally (~1.44 eV [18]).Using the Hubbard correction U4d = 5.80 eV for Cd and U5p = 2.55 eV for Te atoms, we obtained the band gap of 1.438 eV for the bulk CdTe, perfectly consistent with the experimental data [18].

Linear Structures of CdTe Clusters
The optimized structures of some CdnTen clusters with linear structures are shown in Figure 1.The details of the structures and the energy properties of different isomers are given in Table 1.Before optimization, all the structures of CdTe clusters had similar CdTe bond lengths (l = 2.75 Å).Analysis of the average bond length (see Table 1) between atoms of Cd and Te after the optimization of the structures of CdTe clusters led to the value being near 2.74 Å.This value is close to the bond lengths between Cd and Te for the non-optimized clusters.The energy positions of the HOMO, LUMO, and HOMO-LUMO gap dependences on the sizes of CdTe clusters with a linear structure are shown in Figure 2. Observed was the decrease in the HOMO-LUMO gap with an increase in the cluster size (increasing in the number of atoms in the cluster).This can be connected with the overlap of the s-and d-orbitals of the Cd atom with the p-orbital of the Te atom (known as the surface passivation effect) [42].The energy positions of the HOMO, LUMO, and HOMO-LUMO gap dependences on the sizes of CdTe clusters with a linear structure are shown in Figure 2. Observed was the decrease in the HOMO-LUMO gap with an increase in the cluster size (increasing in the number of atoms in the cluster).This can be connected with the overlap of the sand d-orbitals of the Cd atom with the p-orbital of the Te atom (known as the surface passivation effect) [42].To estimate the stability of the studied clusters, the binding energy was calculated.The binding energies of the CdnTen clusters for n = 1-10 obtained using Equation (1) [42] are listed in Table 1.In Table 1 are also listed the electronegativities, which were calculated based on Equation (2) [46]: where Etotal(Cd) is the energy of the cadmium atom, Etotal(Te) is the energy of the tellurium atom, and n is the number of atoms in the cluster.The dependence of the binding energy on the cluster size for linear structures is shown in Figure 3.The binding energy of CdTe clusters with a linear structure increased with cluster size increasing.This is connected with the increasing stability of the linearly structured CdTe clusters.Another property used as an indicator of the stability of geometrically optimized CdTe clusters is their relative stability (stability factor, SF, or second-order change in energy), which can be calculated by the use of the Equation (3): To estimate the stability of the studied clusters, the binding energy was calculated.The binding energies of the Cd n Te n clusters for n = 1-10 obtained using Equation (1) [42] are listed in Table 1.In Table 1 are also listed the electronegativities, which were calculated based on Equation (2) [46]: where E total (Cd) is the energy of the cadmium atom, E total (Te) is the energy of the tellurium atom, and n is the number of atoms in the cluster.The dependence of the binding energy on the cluster size for linear structures is shown in Figure 3.The binding energy of CdTe clusters with a linear structure increased with cluster size increasing.This is connected with the increasing stability of the linearly structured CdTe clusters.To estimate the stability of the studied clusters, the binding energy was calculated.The binding energies of the CdnTen clusters for n = 1-10 obtained using Equation (1) [42] are listed in Table 1.In Table 1 are also listed the electronegativities, which were calculated based on Equation (2) [46]: where Etotal(Cd) is the energy of the cadmium atom, Etotal(Te) is the energy of the tellurium atom, and n is the number of atoms in the cluster.The dependence of the binding energy on the cluster size for linear structures is shown in Figure 3.The binding energy of CdTe clusters with a linear structure increased with cluster size increasing.This is connected with the increasing stability of the linearly structured CdTe clusters.Another property used as an indicator of the stability of geometrically optimized CdTe clusters is their relative stability (stability factor, SF, or second-order change in energy), which can be calculated by the use of the Equation (3): Another property used as an indicator of the stability of geometrically optimized CdTe clusters is their relative stability (stability factor, SF, or second-order change in energy), which can be calculated by the use of the Equation ( 3): where E is the total energy of the CdTe clusters and n is the number of atoms in the clusters.
According to this definition, large positive values of the SF indicate enhanced stability, as they signify a gain in energy during formation from the preceding size and a lower gain in energy to the next size [47].
Figure 4 shows the dependence of the calculated relative stability for the linear structures of the CdTe on cluster size.One can notice that most favorable CdTe cluster sizes are n = 6 and 9.The calculated energy properties showed good agreement with some known parameters obtained by using different DFT approximations for the CdTe clusters.The correlation analysis is presented in Table 2.
lower gain in energy to the next size [47].
Figure 4 shows the dependence of the calculated relative stability for the linear structures of the CdTe on cluster size.One can notice that most favorable CdTe cluster sizes are n = 6 and 9.The calculated energy properties showed good agreement with some known parameters obtained by using different DFT approximations for the CdTe clusters.The correlation analysis is presented in Table 2.

Ring Structures of CdTe Clusters
As reported in Ref. [42], the CdTe structure that was truncated at both ends (linear structure) revealed less stability than the ring or 3D structures.The next step of the CdTe cluster study was the estimation of the most stable ring structure for forming nanorods from this material.Some optimized CdnTen clusters with a ring structure are shown in Figure 5.The details of the structural properties of CdTe clusters with a ring structure are given in Table 3.

Ring Structures of CdTe Clusters
As reported in Ref. [42], the CdTe structure that was truncated at both ends (linear structure) revealed less stability than the ring or 3D structures.The next step of the CdTe cluster study was the estimation of the most stable ring structure for forming nanorods from this material.Some optimized Cd n Te n clusters with a ring structure are shown in Figure 5.The details of the structural properties of CdTe clusters with a ring structure are given in Table 3.
Analysis of the structural properties of the CdTe clusters with a ring structure showed an increasing tendency for angles between Te-Cd-Te and Cd-Te-Cd atoms (see Figure 6).However, average bond lengths between Cd and Te atoms showed decreasing behavior for clusters with a size n ≤ 8 (see Table 3).Such behavior can be an influence on the structural stability of large-ringed CdTe clusters (n ≥ 8).Also, ring structures are not planar, suggesting that the bonding nature in CdTe clusters has some covalent character.
The dependences of the energy positions of the HOMO, LUMO, and HOMO-LUMO gaps on the sizes of the CdTe clusters with a ring structure are shown in Figure 7. Based on these results, it was observed that all the CdTe clusters with ring structures were semiconductors.However, the cluster size dependence of the value of the HOMO-LUMO gaps showed more difficult behavior for the CdTe clusters with a ring structure than for linearly structured clusters.The obtained value of the HOMO-LUMO gap for CdTe clusters with a ring structure (the minimum value for a ring structure was 1.798 eV (n = 2), see Table 4) was much higher than the one for a linear structure (the maximum value for a linear structure was 0.969 eV (n = 1), see Table 1).Such behavior can be related to the chemical inertness of CdTe clusters with a ring structure [48,49].The HOMO-LUMO gap for the CdTe clusters with a ring structure increased with increasing cluster size below n = 5 and decreased above n = 5.  3.
Table 3.Bond lengths (l in Å), angles between Cd-Te-Cd (α in degree), Te-Cd-Te (β in degree), and their average value (ε in deg.) for CdnTen clusters with a ring structure, n = 2-10.Analysis of the structural properties of the CdTe clusters with a ring structure showed an increasing tendency for angles between Te-Cd-Te and Cd-Te-Cd atoms (see Figure 6).However, average bond lengths between Cd and Te atoms showed decreasing behavior for clusters with a size n ≤ 8 (see Table 3).Such behavior can be an influence on the structural stability of large-ringed CdTe clusters (n ≥ 8).Also, ring structures are not planar, suggesting that the bonding nature in CdTe clusters has some covalent character.3. Table 3.Bond lengths (l in Å), angles between Cd-Te-Cd (α in degree), Te-Cd-Te (β in degree), and their average value (ε in deg.) for Cd n Te n clusters with a ring structure, n = 2-10.Using the same methods (see Equations ( 1)-( 3)) as for the CdTe clusters with a linear structure, their binding energy, electronegativity, and relative stability were estimated.Their binding energy showed the same behavior as the HOMO-LUMO gap (see Figure 8).The maximum value of the binding energy was obtained for the CdTe cluster with n = 5. Figure 9 shows the dependence of the calculated relative stability for the ring structure of the CdTe on cluster size.The ring-type structures of the CdTe clusters with n = 3 and 5 were found to be the most favorable.The correlation analysis of the energy properties of the CdTe clusters with a ring structure are presented in Table 5.The dependences of the energy positions of the HOMO, LUMO, and HOMO-LUMO gaps on the sizes of the CdTe clusters with a ring structure are shown in Figure 7. Based on these results, it was observed that all the CdTe clusters with ring structures were semiconductors.However, the cluster size dependence of the value of the HOMO-LUMO gaps showed more difficult behavior for the CdTe clusters with a ring structure than for linearly structured clusters.The obtained value of the HOMO-LUMO gap for CdTe clusters with a ring structure (the minimum value for a ring structure was 1.798 eV (n = 2), see Table 4) was much higher than the one for a linear structure (the maximum value for a linear structure was 0.969 eV (n = 1), see Table 1).Such behavior can be related to the chemical inertness of CdTe clusters with a ring structure [48,49].The HOMO-LUMO gap for the CdTe clusters with a ring structure increased with increasing cluster size below n = 5 and decreased above n = 5.  2. Please change the hyphen (-) into a min (−, "U+2212"), e.g., "-1" should be "−1".
Changes have been made.The dependences of the energy positions of the HOMO, LUMO, and HOMO-LUMO gaps on the sizes of the CdTe clusters with a ring structure are shown in Figure 7. Based on these results, it was observed that all the CdTe clusters with ring structures were semiconductors.However, the cluster size dependence of the value of the HOMO-LUMO gaps showed more difficult behavior for the CdTe clusters with a ring structure than for linearly structured clusters.The obtained value of the HOMO-LUMO gap for CdTe clusters with a ring structure (the minimum value for a ring structure was 1.798 eV (n = 2), see Table 4) was much higher than the one for a linear structure (the maximum value for a linear structure was 0.969 eV (n = 1), see Table 1).Such behavior can be related to the chemical inertness of CdTe clusters with a ring structure [48,49].The HOMO-LUMO gap for the CdTe clusters with a ring structure increased with increasing cluster size below n = 5 and decreased above n = 5.Using the same methods (see Equations ( 1)-( 3)) as for the CdTe clusters with a linear structure, their binding energy, electronegativity, and relative stability were estimated.Their binding energy showed the same behavior as the HOMO-LUMO gap (see Figure 8).The maximum value of the binding energy was obtained for the CdTe cluster with n = 5. Figure 9 shows the dependence of the calculated relative stability for the ring structure of the CdTe on cluster size.The ring-type structures of the CdTe clusters with n = 3 and 5 were found to be the most favorable.The correlation analysis of the energy properties of the CdTe clusters with a ring structure are presented in Table 5.

Conclusions
First principles theoretical studies of the structures and electronic properties of Cd n Te n (n ≤ 10) clusters with linear and ring structures were carried out using the reliable techniques of density functional theory and known approximations.Based on these calculations, the energy positions of the HOMO, LUMO, and HOMO-LUMO gaps, binding energies, and electronegativities were obtained for the studied clusters.The decreasing tendency of the HOMO-LUMO gap with increasing cluster size for a linear structure is connected with the overlapping of the sand d-orbitals of the Cd atom with the p-orbital of the Te atom.The stability of the CdTe clusters with a linear structure increased with the size of the clusters increasing.The most favorable linearly structured CdTe cluster sizes were n = 6 and 9.The obtained values of the HOMO-LUMO gaps for CdTe clusters with a ring structure were much higher than those corresponding with linearly structured ones.After comparing linear and ring structures, the more favorable structures were determined to be ring-type ones.The highest relative stability of CdTe clusters with a ring structure corresponded to the cases n = 3 and 5.

Figure 2 .
Figure 2. Energy positions of the HOMO, LUMO (a), and HOMO-LUMO gaps (b) for the CdTe clusters with a linear structure.

Figure 3 .
Figure 3. Binding energy per CdTe molecule for the CdTe clusters with a linear structure.

Figure 2 .
Figure 2. Energy positions of the HOMO, LUMO (a), and HOMO-LUMO gaps (b) for the CdTe clusters with a linear structure.

Figure 2 .
Figure 2. Energy positions of the HOMO, LUMO (a), and HOMO-LUMO gaps (b) for the CdTe clusters with a linear structure.

Figure 3 .
Figure 3. Binding energy per CdTe molecule for the CdTe clusters with a linear structure.

Figure 3 .
Figure 3. Binding energy per CdTe molecule for the CdTe clusters with a linear structure.

Figure 4 .
Figure 4. Variation in the stability with the cluster size for the CdTe clusters with a linear structure.

Figure 4 .
Figure 4. Variation in the stability with the cluster size for the CdTe clusters with a linear structure.

Figure 6 .
Figure 6.Variation in angles between Te-Cd-Te and Cd-Te-Cd atoms with a cluster size for the CdTe clusters with a ring structure.

Figure 7 .
Figure 7. Energy positions of the HOMO, LUMO (a), and HOMO-LUMO gaps (b) for the CdTe clusters with a ring structure.

Figure 6 .
Figure 6.Variation in angles between Te-Cd-Te and Cd-Te-Cd atoms with a cluster size for the CdTe clusters with a ring structure.

Figure 6 .
Figure 6.Variation in angles between Te-Cd-Te and Cd-Te-Cd atoms with a cluster size for the CdTe clusters with a ring structure.

Figure 7 .
Figure 7. Energy positions of the HOMO, LUMO (a), and HOMO-LUMO gaps (b) for the CdTe clusters with a ring structure.

Figure 7 .
Figure 7. Energy positions of the HOMO, LUMO (a), and HOMO-LUMO gaps (b) for the CdTe clusters with a ring structure.

Figure 8 . 8 Figure 8 .
Figure 8. Binding energies per CdTe molecule for the CdTe clusters with a ring structure.

Figure 8 .
Figure 8. Binding energies per CdTe molecule for the CdTe clusters with a ring structure.

Figure 9 .
Figure 9. Variation in the stability with the cluster size for the CdTe clusters with a ring structure.

Funding:
This work was supported by the Project for Young Scientists, No. 0124U000760, granted by the Ministry of Education and Science of Ukraine.