The optimized geometries of ZnPc and TPA-ZnPc by DFT method (B3LYP/6-311G(d,p)) [20, 21] are shown in Fig. 2. Our calculations indicate that both molecules exhibit a planarity characteristic of this type of structure. Furthermore, we observed that ZnPc is more planar than TPA-ZnPc, with this change attributed to the substitution by the triphenylamine group. A more homogeneous distribution of charge across the molecule can enhance the overlap between the HOMO of the HTM and the valence band maximum of the perovskite, thereby facilitating more efficient charge extraction and limiting recombination [22].

Ionization potential (IP), electron affinity (EA), and reorganization energy are crucial parameters that help us understand the behavior of HTM materials [23] for efficient charge transfer and optimized photovoltaic conversion in solar cells [24, 25].

During the operation of a perovskite solar cell, an electron from the HOMO of the HTM must fill the hole left in the valence band of the perovskite after the excitation of another electron [26, 27]. To facilitate this hole transfer, the IP of the HTM must be sufficiently low [26]. The lower the IP value, the easier it is to form holes in the analyzed molecules [28]. ZnPc and TPA-ZnPc have ionization energies of 6.204 eV and 6.073 eV, respectively, allowing their use as HTMs with MAPbI₃ perovskite.

Similarly, the EA indicates the ability of a material to undergo reduction. The electron affinity of the materials is 1.959 eV and 2.4 eV, respectively. These IP and EA values were obtained directly from the frontier molecular orbital energies (HOMO and LUMO) using Koopmans’ theorem, as indicated in Eqs. (1) and (2).

IP = E_c – E_n, (1)

EA = E_n – E_a. (2)

The reorganization energy is one of the fundamental criteria to consider when designing an HTM [29]. A low reorganization energy promotes efficient charge transfer [30]. The total reorganization energy (λ_total) corresponds to the sum of the reorganization energies of the electrons (λ_e) and holes (λ_h) [31]. It is defined by the following expression:

λ_e = (E_n^– – E_a) + (E_n⁰ – E_n); (3)

λ_h = (E⁺_n – E_c) + (E_c – E_n); (4)

λ_total = λ_e + λ_h, (5)

where E_c, E_a, and E_n correspond to the total energies of the cation, anion, and neutral system in their optimized geometries, respectively. E_n⁺ and E_n⁻ represent the energies of the cationic and anionic systems, calculated at the optimized geometry of the neutral state. Finally, E_c⁰ and E_a⁰ denote the energies of the neutral system evaluated at the optimized geometries of the cation and anion, respectively [32].

HTM materials are responsible for transporting holes (positive charge), so the study of Mulliken charges of the cation is particularly essential [33]. The charges represent in Fig. 3. The positive charges for ZnPc and TPA-ZnPc are well distributed, indicating that the molecule will have good hole mobility, which is crucial for applications in solar cells. The x-axis represents the atoms numbered according to their order in the Gaussian output file. The value “0” corresponds to the reference level of atomic charge used for comparison, and numbering starts from the central Zn atom and proceeds clockwise along the macrocycle.

Marcus theory is used to confirm the nature of charge transfer, as studying the reorganization energy alone is not sufficient [13]. According to the calculation of the hole and electron transfer rates of ZnPc and TPA-ZnPc using Eqs. (6) and (7), it is observed that ZnPc has a faster hole transfer rate compared to TPA-ZnPc, but a lower electron transfer rate.

kh=V2h(πλhkBT)1/2exp(λh4kBT) (6)

ke=V2h(πλekBT)1/2exp(λe4kBT) (7)

where V represents the charge transfer integral, λ corresponds to the reorganization energy, k_B is the Boltzmann constant, and T is the temperature, typically set to 300 K. Based on the equation above, the hole hopping rate (k_h) primarily depends on two key parameters: V and λ. These parameters are actually divided into two simple components: the electron transfer integral (V_e) and the hole transfer integral (V_h) [34]. These are calculated respectively using the following equations [35]:

V_h = (E_HOMO − E_HOMO−1)/2, (8)

V_e = (E_LUMO – E_LUMO+1)/2. (9)

Calculations of the chemical potential and hardness confirm the chemical stability of the studied compounds [36]. Equations (10) and (11) were used to calculate the chemical potential (μ) and hardness (η). The results, presented in Table 1, show that ZnPc has a lower electrochemical potential than TPA-ZnPc, indicating greater electrochemical stability.

μ = (IP + EA)/2; (10)

η = (IP − EA)/2. (11)

The distribution of the HOMO and LUMO of the studied HTM molecules is shown in Fig. 4 below. This representation provides a general understanding of the electronic structure [25]. From these results, we can say that both HTM molecules exhibit good charge transport properties. It is observed that the electronic density of the HOMO and LUMO orbitals extends over the entire molecule.

Table 2 presents the results obtained for the HOMO, LUMO, and E_g energies of the two molecules, ZnPc and TPA-ZnPc. For a molecule to be designed as an HTM, the HOMO must be lower than the maximum valence band level of the perovskite to promote hole extraction and minimize energy losses at the interfaces [37]. Additionally, the LUMO must be sufficiently high to block electrons and thus reduce charge recombination [38, 39]. ZnPc and TPA-ZnPc have respective bandgap energies of 2.178 eV and 2.062 eV. We observed a decrease in the bandgap energy following the addition of the triphenylamine group. This reduction is due to the increase in the energy of the HOMO.

To analyze the electronic density and possible hydrogen bonds with the HTM material, it is necessary to estimate the MEP (Fig. 5) through a calculation based on the DFT/B3LYP/6-311(d,p) method [41].

Based on this representation, and using the electrostatic potential values illustrated by a colored diagram, we observed that there are no negative values or red regions, indicating the absence of potentially electrophilic sites [41, 42]. On the other hand, a region with a positive charge, indicated in blue, is observed around the nitrogen atom.

We obtained the absorption properties using the TD-DFT technique with a 6-311G basis set on the optimized gas-phase structures (Fig. 6). Additionally, we calculated the maximum absorption wavelengths (λ_max) and identified the electronic transitions. All of these results are presented in Table 3. We observe that both ZnPc and TPA-ZnPc exhibit absorption wavelengths in the visible region. The absorption peak intensities of TPA-ZnPc in the visible range are higher than those of ZnPc. This increase may be due to the new electronic transitions induced by the TPA group. Furthermore, we observe a shift in the TPA-ZnPc spectrum. This difference is attributed to the presence of the triphenylamine group as a substitution.