Effect of number and position of methoxy substituents on fine-tuning the electronic structures and photophysical properties of designed carbazole-based hole-transporting materials for perovskite solar cells: DFT calculations

3.1 Structural features of designed dyes (D1-D7, T8-T11, and H12)

From the parent molecule 5,7-bis(9-ethyl-9H-carbazol-2-yl)-2,3,5,7-tetrahydrothieno[3,4-b][1,4]dioxine (abbreviated as R01), twelve newly derivatives was designed. The derivatives (abbreviated as D1-to-D7) were designed by introducing methoxy groups at the carbazole rings. In D1-D7 (di-methoxy derivatives), two methoxy groups were introduced at different positions at the carbazole rings. In T8-T11 (tetra-methoxy derivatives), four methoxy groups were present. The maximum number of six methoxy groups was introduced in H12 (hexa-methoxy derivative). The common feature in the designed molecules is the presence of electron-rich methoxy substituents that will improve the contact with the perovskite layer, and by doing so result in an efficient hole-injection (Calió et al., 2016). The chemical structures of the R01 and its designed derivatives along with their IUPAC names are depicted in Scheme 1. On the other hand, the designed molecules could be differentiated according to the positions of the methoxy groups either they are substituted at the ortho or at the ortho and meta position all with respect to the dioxythiophene ring, see Scheme 1. The designed HTMs have a unique general configuration of AD₁-D₁-D₂-D₁-AD₁, which has the carbazole moieties as the significant electron donors (D₁), methoxy groups as an additional electron donor (AD₁), dioxythiophene core as a minor electron donor (D₂), and a repeated unit of D₁, and AD₁.

3.2 Optimized geometries

Parent (R01): The inter-ring distances and the torsional angles in the optimized geometries of the parent molecule (R01) in its neutral, cationic, and anionic states at the ground state (S_o) are collected in Table 1. Fig. 1 represents the HOMOs and LUMOs distribution along with the % contribution to the HOMO and LUMO from different fragments of the HTMs under investigation. Two (R01)/Three (derivatives) main fragments are defined for each molecule. The nature of each fragment as electron-donating/accepting can be demonstrated from its contribution to these orbitals. If the fragment is contributing more to the HOMO than to the LUMO, it is an electron-donating fragment and vice versa. As can be seen from Fig. 1, for R01, the two defined fragments are the dioxythiophene core and two carbazole arms. The two fragments are contributing to the HOMO and LUMO; however, dioxythiophene core is contributing with more significant extent to the HOMO than to the LUMO. Dioxythiophene core is contributing by 42% and 36% to HOMO and LUMO, respectively. In contrast, the carbazole arms are contributing with more significant extent to the LUMO (64%) than to the HOMO (58%). Thus, we could conclude that both groups could be considered as an electron-donating group, however, according to the extent of their contributions to the HOMO and LUMO the carbazole arms are the major ones, and dioxythiophene core is the minor one. In the meantime, no X-ray crystal data is available for this compound, we will focus on the effect of the substituted methoxy groups on these geometrical parameters for the same molecule in its different forms, and we will compare between these geometrical parameters in different derivatives, Table 1. The optimized R01 molecule exhibits a non-planar structure with the C1 symmetry, the non-planarity geometries of R01 regulates the packing of the molecule in the solid state and influences the optoelectrical properties of the carbazole-based materials (Iwan and Sek, 2011).

Table 1 Bond distances (in Å), bond angles, and torsional angles (in °) of the ground state of R01 and its di-methoxy derivatives (D1-D7) in their neutral, cationic and anionic forms, data in parenthesis belong to the first excited state.

R01		S−C5	C5−C31	S−C5–C31–C32
neutral		1.7582 (1.7752)	1.4637 (1.4215)	22.800 (2.462)
cationic		1.7622	1.4364	9.168
anionic		1.7817	1.4316	2.687
D1		S–C5	C5–C30	S–C5–C30–C31	C31–O66	C32–O66–C68
neutral		1.7592 (1.7834)	1.4702 (1.4272)	32.699 (17.373)	1.3866 (1.3800)	113.856 (114.133)
cationic		1.7596	1.4456	35.082	1.3491	119.237
anionic		1.7847	1.4389	20.240	1.3950	112.932
D2		S–C5	C5–C30	S–C5–C30–C31	C32–O66	C31–O66–C68
neutral		1.7558 (1.7790)	1.4694 (1.4283)	47.593 (29.372)	1.3669 (1.3569)	118.340 (119.185)
cationic		1.7596	1.4456	35.082	1.3491	119.237
anionic		1.7804	1.4420	30.897	1.3740	118.435
D3		S–C5	C5–C30	S–C5–C30–C31	C35–O67	C35–O67–C72
neutral		1.7574 (1.7757)	1.4645 (1.4214)	23.021 (2.357)	1.3659 (1.3633)	118.223 (118.490)
cationic		1.7601	1.4398	11.275	1.3526	118.788
anionic		1.7831	1.4302	2.641	1.3794	117.610
D4		S–C5	C5–C30	S–C5–C30–C31	C35–O71	C35–O71–C72
neutral		1.7579 (1.7781)	1.4652 (1.4213)	21.994 (1.970)	1.3663 (1.3614)	118.199 (118.530)
cationic		1.7656	1.4385	8.384	1.3518	118.807
anionic		1.7782	1.4338	2.805	1.3792	117.620
D5		S–C5	C5–C27	S–C5–C27–C28	C28–O65	C28–O69–C66
neutral		1.7581 (1.7966)	1.4638 (1.4120)	23.043 (1.444)	1.3734 (1.3733)	118.309 (118.315)
cationic		1.7615	1.4392	9.989	1.3590	119.263
anionic		1.7811	1.4319	2.900	1.3854	117.765
D6		S–C6	C6–C12	S–C6–C12–C13	C14–O60	C14–O60–C61
neutral		1.7566 (1.7978)	1.4692 (1.4259)	47.038 (25.717)	1.3815 (1.3780)	114.040 (114.539)
cationic		1.7587	1.4435	35.629	1.3461	119.441
anionic		1.7853	1.4390	27.625	1.3876	114.280
D7		S–C5	C5–C27	S–C5–C27–C28	C29–O60	C29–O60–C61
neutral		1.7568 (1.7970)	1.4691 (1.4268)	47.074 (26.583)	1.3811 (1.3777)	114.089 (114.573)
cationic		1.7577	1.4431	36.334	1.3495	119.184
anionic		1.7847	1.4395	28.557	1.3872	114.335

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Fig. 1

Electron density isocontours (0.020 au) of HOMO and LUMO orbitals distribution for the ground states of R01 and its methoxy derivatives.

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The S–C5 distances are 1.7582, 1.7622, and 1.7817 Å in the neutral, cationic and anionic forms, respectively. These bond distances are elongated in the cationic and anionic forms by 0.004 and 0.024 Å, respectively, compared to that in the neutral form. On the other hand, the C5–C31 distances are shortened in the cationic and anionic forms by 0.027 and 0.032 Å, respectively, compared to that in the neutral form. In the cationic form the electron is removed from the HOMO, and in the anionic form the electron will be added to the LUMO, thus, by visualizing the HOMO and LUMO distribution among the molecule, we can explain the change in the inter-ring distances (Lin et al., 2003), Fig. 1. A large part of the HOMO (−4.793 eV) is delocalized on the dioxythiophene ring, upon removing of an electron from the HOMO in the cationic form leads to elongation of the S–C5 bond distance. Part of the LUMO (−1.346 eV) is distributed over the C5–C31 (and C5–C11), thus adding an electron to the LUMO results in shortening of these bonds. The shortening of the inter-ring distance in the anionic state is due to the bonding interactions between the π-orbitals on the two carbazole rings. The torsional angle S–C5–C31–C32 that simply resemble the extent of the planarity of the molecule shows a significant decrease by 13.6 and 20.1° in the cationic and anionic forms, respectively, compared to that in the neutral form. Thus, the cationic and anionic forms showed more planner geometries compared to the neutral form, and the anionic form is even more planner than the cationic form. In fact, R01 and its derivatives (as will be discussed later) exhibit larger geometric deformation between the neutral and ionic forms, this implies that this type of molecules should be p-channel but not n-channel materials (Zhao et al., 2013).

Di-methoxy derivatives (D1-D7): A general notice is that the investigated geometrical parameters (bond lengths and angles) show somehow significant differences between the parent molecule (R01) and its derivatives. It is then evident that the presence of the methoxy groups makes a significant influence on the geometrical structure and subsequently on the electronic properties. On one hand, due to the steric hindrance effect of the methoxy group/s especially for that/those present in the ortho-position/s, and on the other hand, due to the electron-accepting (by resonance)/donating (inductively) nature of this group. Comparing the torsional angle S–C5–C31–C32 between the parent molecule (R01) and its derivatives those substituted by two methoxy groups (D1-D7), indicated that the planarity of the R01 is distorted with different extent in its derivatives. For instance, for the derivatives D1, D2, D6, and D7, this dihedral angle is in the range ∼33° (D1) to ∼47° (D2, D6, and D7) compared to 22.8° for that of R01. On the other hand, the distortion from planarity for the derivatives D3, D4 and D5 is insignificant, since the dihedral angles for these three molecules are very similar to that of R01, ranges from 22.0° (in D4) to 23.0° (in D3 and D5). Thus, the positions of the methyl groups significantly affect the planarity of geometries of the designed derivatives; the positioning of the methoxy group/s at the ortho-position/s with respect to the dioxythiophene ring will have a more important steric effect than positioning it/them at the meta-position/s (Jeon et al., 2014). For instance, in D3, the two methoxy groups are positioning at the meta-positions corresponding to each other at the two carbazole rings, this molecule shows one of the least deviations from planarity and thus a very similar geometry to that of R01. Another notice is that the significant deviation from planarity shown by D1, D2, D6 and D7 derivatives. The displacement of one/two methoxy groups at the ortho-position (D1 and D2) and the other one at the phenyl ring of the carbazole ring farthest from the dioxythiophene ring (terminal methoxys), either on the same side (D6) or at the opposite side (D7), seems to enhance the steric effect of the methoxy groups.

The change in inter-ring distances, i.e., S–C5 and C5–C32/C30/C28, shows similar trend showed by the parent molecule (R01). The S–C5 distances are elongated in the cationic and anionic forms compared to that of the neutral form, and the other bond distances are shortened. The concept used to explain these changes in the R01 can be applied to the designed derivatives. Since the distribution of the HOMO and LUMO on the thiophene ring and on the C5–C12/17/27/30/31 bond does not show significant differences between that of the R01 and those of its derivatives. Additionally, the geometrical parameters of one of the substituted methoxy groups in the derivatives have been tabulated in Table 1. For instance, for the D2 derivative, the C32–O66 bond distance shortened by 0.018 Å in the cationic form and elongated by 0.007 Å in the anionic forms, the C31–O66–C68 bond angle increases by 0.897 in the cationic form and decreases by 0.095° in the anionic forms. From Fig. 1, it is clear that the HOMOs are delocalized with large extent on the substituted methoxy group over the electronegative oxygen atoms, while the LUMOs are distributed with no/less extent. The total % contributions from the methoxy fragments to the HOMO and LUMO orbitals of the investigated HTMs are collected in Table 2. As appeared from Table 2 and Fig. 1, the methoxy groups are contributing more to the HOMOs (range from 0.84 to 7.77%) than to the LUMOs (0.1–1.62%), indicating that the electron-donating nature by resonance is more predominant. The fact that the C–O distances for these derivatives are shortened in cationic form and elongated in anionic form is consistent with the MO pictures (Lin et al., 2003). The distribution of the HOMOs orbitals on the substituted methoxy groups of the derivatives is not the same, while, LUMOs distribution shows similar observation. In other words, the LUMOs are generally not delocalized on these groups indicating their electron-donating nature (resonance effect). The HOMOs are delocalized on the oxygen atoms of the methoxy groups, but not on all the substituted methoxy groups. The delocalization of the HOMOs on some methoxy groups indicates that the electron-donating nature of the methoxy group is a predominant effect, and the non-delocalization of them indicates that the electron-withdrawing nature is the predominant effect. Due to the high electronegativity of the oxygen atom; this group could become electron-withdrawing in an inductive sense via the σ bonds. Thus, the electron-donating/withdrawing nature of the substituted methoxy group depends on its specific position and interaction with the conjugation system of the phenyl rings. Thus, from MO picture, the methoxy groups that contributed to the HOMO and not contributed to the LUMO, we expected that the C-O bond length would be decreased and increased in the cationic and anionic form, respectively; the expectation is matching the obtained result.

Tetra-methoxy derivatives (T8-T11): In this kind of designed derivatives, four methoxy groups are substituted either at the two (upper) ortho- and two meta- positions (like in T8), or at the four ortho-positions (like in T9), or at the two meta- and the two other (lower) ortho-positions (like in T10), or finally, at the two ortho-positions and the other two methoxy groups are at the phenyl ring farthest from the dioxythiophene ring (like in T11), see Scheme 1 and Table SD 1. Comparing the planarity of T8-T11 and that of the parent molecule (R01) shows that and similar to the effect of some derivatives substituted with two methoxy groups (D1, D2, D6, and D7), the planarity of all of these derivatives are less than that of R01. The S-C5-C28-C28/29 dihedral angle is ranging from 33° (T10)-to-58° (T9). Since the value of this dihedral angle in R01 is 22.008°. On the other hand, the inter-ring distances included the S–C5, and C5–C27/28/31 bond distances of T8-T11 are slightly shortened and elongated, respectively, compared to those of R01. For instance, the S–C5 bond distances are 1.7582 and 1.7565 Å in R01 and T8, respectively, while their C5–C28/31 bond lengths are 1.4637 and 1.4700 Å, respectively, shortening by 0.002 Å and elongation by 0.006 Å.

Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.arabjc.2018.06.014.

Tetra-methoxy derivatives (T8-T11): In this kind of designed derivatives, four methoxy groups are substituted either at the two (upper) ortho- and two meta- positions (like in T8), or at the four ortho-positions (like in T9), or at the two meta- and the two other (lower) ortho-positions (like in T10), or finally, at the two ortho-positions and the other two methoxy groups are at the phenyl ring farthest from the dioxythiophene ring (like in T11), see Scheme 1 and Table SD 1. Comparing the planarity of T8-T11 and that of the parent molecule (R01) shows that and similar to the effect of some derivatives substituted with two methoxy groups (D1, D2, D6, and D7), the planarity of all of these derivatives are less than that of R01. The S-C5-C28-C28/29 dihedral angle is ranging from 33° (T10)-to-58° (T9). Since the value of this dihedral angle in R01 is 22.008°. On the other hand, the inter-ring distances included the S–C5, and C5–C27/28/31 bond distances of T8-T11 are slightly shortened and elongated, respectively, compared to those of R01. For instance, the S–C5 bond distances are 1.7582 and 1.7565 Å in R01 and T8, respectively, while their C5–C28/31 bond lengths are 1.4637 and 1.4700 Å, respectively, shortening by 0.002 Å and elongation by 0.006 Å.

The anionic forms of these derivatives show significant changes from the geometries of their counterpart neutral forms, while the cationic forms show slightly different geometries of those of the neutral forms. For instance (for T11), the S-C5 bond lengths are 1.7575, 1.7572, and 1.7829 Å in the neutral, cationic and anionic forms, respectively. Thus, it has been slightly shortened by 0.0004 Å and significantly elongated by 0.025 Å in the cationic and anionic form, respectively. Unlike any other derivative, the HOMO (−4.907 eV) of T11 is not delocalized on the S-C5 bond, and like the other derivatives, part of the LUMO (−1.059 eV) is delocalized on the C5–C27 (Fig. 1) thus removing an electron from the HOMO to form the cationic state should not affect the S-C5 bond length that results in the slight difference that was observed in this bond length in the neutral and cationic form. In contrast, and since the LUMO is delocalized on the C5–C27, adding an electron to the LUMO to form the anionic form would be expected to weaken and thus lengthening this bond as observed.

Hexa-methoxy derivative (H12): The deviation from planarity in this molecule in comparison with R01 is more pronounced, Table SD 2. Since, the S−C5−C28−C29 dihedral angle is 56.2 and 22.8°, for H12 and R01, respectively, a difference of ∼33°. This could be easily attributed to the steric hindrance effect of the six substituted methoxy groups. The inter-ring distances (S–C5) are also elongated by 0.006 and 0.027 Å, while the C5–C28 bond distances are shortened by 0.024 and 0.026 Å in the cationic and anionic forms, respectively. The C–O bond distances of the methoxy group are shortened in the cationic forms and elongated in the anionic forms compared to those of the neutral forms, this combined by reverse trend observed in the C–O–C bond angle. In comparison with the neutral forms, the C–O–C bond angles of the methoxy group increase and decrease in the cationic and anionic forms, respectively. Due to the presence of the six methoxy groups (the maximum number of substitution among all derivatives); the HOMO is showing greater delocalization over more substantial parts of the molecule. Four of the methoxy groups are contributing to the HOMO, except the two methoxy groups at the (upper) ortho-positions.

Hexa-methoxy derivative (H12): The deviation from planarity in this molecule in comparison with R01 is more pronounced, Table SD 2. Since, the S−C5−C28−C29 dihedral angle is 56.2 and 22.8°, for H12 and R01, respectively, a difference of ∼33°. This could be easily attributed to the steric hindrance effect of the six substituted methoxy groups. The inter-ring distances (S–C5) are also elongated by 0.006 and 0.027 Å, while the C5–C28 bond distances are shortened by 0.024 and 0.026 Å in the cationic and anionic forms, respectively. The C–O bond distances of the methoxy group are shortened in the cationic forms and elongated in the anionic forms compared to those of the neutral forms, this combined by reverse trend observed in the C–O–C bond angle. In comparison with the neutral forms, the C–O–C bond angles of the methoxy group increase and decrease in the cationic and anionic forms, respectively. Due to the presence of the six methoxy groups (the maximum number of substitution among all derivatives); the HOMO is showing greater delocalization over more substantial parts of the molecule. Four of the methoxy groups are contributing to the HOMO, except the two methoxy groups at the (upper) ortho-positions.

Optimized S₁ geometries of R01 and its designed derivatives: The inter-ring distances and torsional angles in the optimized geometries of the parent molecule R01 and its designed derivatives at the first excited state (S₁) calculated with TDB3LYP/6-31G(d) are collected in Table 1, Table SD 1, and Table SD 2. It was found that the excited state structures are different from the ground state structures. The investigated bond angles, bond angles, and dihedral angles are significantly changed upon excitation. The most interesting observation is that the planarity of the molecule is improved due to the transfer from the ground state to the first excited state; this is evident from the significant decrease in the S−C−C−C dihedral angles, more obviously for R01, D3, and D4. Thus, the excited state structure is more planar than the ground state structure. For instance, the values of S−C−C−C dihedral angles in the ground state change from 22.800, 23.021, and 21.994° to 2.462, 2.357, and 1.970° in R01, D3, and D4, respectively. The modifications of the dihedral angle strongly affect the optical properties of the compounds such as the emission properties and radiative lifetimes (discussed later) (Chitpakdee et al., 2014; Wazzan et al., 2017).

Optimized S₁ geometries of R01 and its designed derivatives: The inter-ring distances and torsional angles in the optimized geometries of the parent molecule R01 and its designed derivatives at the first excited state (S₁) calculated with TDB3LYP/6-31G(d) are collected in Table 1, Table SD 1, and Table SD 2. It was found that the excited state structures are different from the ground state structures. The investigated bond angles, bond angles, and dihedral angles are significantly changed upon excitation. The most interesting observation is that the planarity of the molecule is improved due to the transfer from the ground state to the first excited state; this is evident from the significant decrease in the S−C−C−C dihedral angles, more obviously for R01, D3, and D4. Thus, the excited state structure is more planar than the ground state structure. For instance, the values of S−C−C−C dihedral angles in the ground state change from 22.800, 23.021, and 21.994° to 2.462, 2.357, and 1.970° in R01, D3, and D4, respectively. The modifications of the dihedral angle strongly affect the optical properties of the compounds such as the emission properties and radiative lifetimes (discussed later) (Chitpakdee et al., 2014; Wazzan et al., 2017).

The distribution patterns of HOMOs and LUMOs for R01 and of its derivatives in the first excited states (S₁) have been shown in Fig. 2. Unlike R01 and other derivatives, T11 derivative shows the significant difference in the HOMO distribution (and not the LUMO) between the S_o and S₁. In the HOMO of the excited state, the electronic charge is delocalized over a region that was not covered by the HOMO in the ground state geometry, i.e., the dioxythiophene core. Thus, at the S₁ the HOMO is distributed over more substantial parts of the molecule. We found general intramolecular charge transfer (ICT) in all investigated HTMs for S₀ as well as the S₁ state.

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Fig. 2

HOMO and LUMO orbitals distribution for the first excited state (S₁) of some investigated HTMs as representative examples.

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3.3 Frontier molecular orbital

In the present study, we have carried out a systematic quantum chemical calculations using density functional theory DFT to estimate the electronic and optical properties for 5,7-bis(9-ethyl-9H-carbazol-2-yl)-2,3,5,7-tetrahydrothieno[3,4-b][1,4]dioxin and some of its new designed methoxy derivatives (Scheme 1), which they may be used as an efficient HTMs in the perovskite solar cells. As is known examination of the frontier molecular orbitals (FMOs) is vital to understand the electronic structures of chemical compounds. In addition, FMOs are the backbone factors to illustrate the carrier transport properties of the molecules, which affect the electronic, optical, and conducting properties of materials. By controlling the HOMO and LUMO levels, it is feasible to design materials for PSCs with desirable charge carrier transport properties (Chen et al., 2014a, 2014b). Therefore, examination of HOMOs and LUMOs of the newly designed compounds is highly useful, by which a reasonable qualitative indication of the ability of charge injection and transport can be estimated by knowing the relative ordering of HOMOs and LUMOs. The calculated distribution patterns of the highest occupied molecular orbitals (HOMOs) and lowest unoccupied molecular orbitals (LUMOs) and the energy levels for the ground states of the parent compound and its new design derivatives are plotted in Fig. 3. For the parent compound (R01), as sketched in Fig. 3, the HOMO and LUMO are almost distributed over the whole molecule, indicating that the molecule possesses π-features. A good delocalization of HOMO is favorable to enhancing the hole transfer integral (Zhang et al., 2017a, 2017b). Similarly, other twelve designed molecules exhibit the same results with some exceptions. These exceptions mainly depend on the position and the number of the substituted methoxy groups. For example, new designed compound D4, the HOMO is located on the central and left-hand side of the molecule, and it has a very weak localization on the right-hand side of the molecule, while the LUMO is almost localized on whole the molecule. For D6 and D7 molecules, the indole group on the right-hand side have a week contribution to the HOMOs, while the LUMOs of these molecules are almost localized overall the backbone of the molecule. These results can be attributed to the position of the methoxy groups. In D6 molecule, two methoxy groups are substituted on the carbazole unit, which located on the left-hand side of the molecule. On the other hand, in D7 molecule, the two methoxy groups are substituted on both carbazole units. As is known the methoxy substituent actually impacts the ring electronics via two competing effects. The oxygen's lone pair is well-placed to delocalize and increase electron density within the ring's conjugated system. This allows delocalization to better stabilize positive charges. So the methoxy is electron-donating from a resonance perspective. Additionally, because oxygen is quite electronegative, the methoxy group is electron-withdrawing in an inductive sense via the σ bonds. As can be seen from the distribution of the HOMOs and LUMOs the resonance donation is much stronger than the inductive sense.

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Fig. 3

The calculated molecular orbitals for molecule R01 and its newly designed derivatives P1-D7 (blue ones are LUMOs and red ones are HOMOs), using DFT at the B3LYP/6-31G(d) level. HOMO and LUMO energies are in eV. Energy gap values are given in the middle part of the Fig (eV). Value ranges from (−0.2)-to-0.2 eV.

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Chi and Li (2015a, 2015b) suggested that the excellent HOMO delocalization is favorable for hole transport and hole transfer integral. As compared with the hole-transport, they have the relative inferior electron transport possibility that is determined by the delocalization of the LUMO. The result of FMO analysis suggests that the parent R01 and most of the designed molecules have good hole-transport properties, and consequently they may be very useful potential candidates as HTMs. Previous studies (Minaev et al., 2012; Baryshnikov et al., 2012) reported that the central principle of HTM in perovskite is similar with that of electron transfer to TiO₂ surface in DSSC except that the electron is injected in the latter.

In order to understand the influence of the electronic properties of the R01 compound and the newly designed molecules, the analyzing of the values of HOMO energies ( $E_{\mathit{HOMO}}$ ), LUMO energies ( $E_{\mathit{LUMO}}$ ) and the energy gap ( ${\mathrm{\Delta }}_{H-L}=E_{\mathit{LUMO}}-E_{\mathit{HOMO}}$ ) are considered. As is known the HOMO energy level of HTM should be higher than the valence band of perovskite (Zhang et al., 2017a, 2017b), and the lower HOMO level that is much closer to the valence band of the mixed perovskite (−5.65 eV) would results in efficient hole extraction and overall PCE of the devices (Rakstys et al., 2015).

Table 3 and Fig. 3 summarize the DFT B3LYP/6-31(d) values of $E_{\mathit{HOMO}}$ , $E_{\mathit{LUMO}}$ , ${\mathrm{\Delta }}_{H-L}$ energies, as well as the available experimental data (Wang et al., 2015). It should be mentioned that solid-state packing effects are not included in the DFT calculations, which tend to affect the HOMO and LUMO energy levels in a thin film compared to an isolated molecule as considered in the calculations (Mandal et al., 2017). However, according to Table 3 the calculate HOMO energy level of R01 (−4.79 eV) is in a good agreement with the experiment value (−5.30 eV) (Wang et al., 2015) revealing the constancy of theoretical results calculated at B3LYP/6-31(d) level. It is important to note that the HOMO of R01 is much more positive than that of the conduction band of mixed-perovskite (−5.44 eV) (Bi et al., 2016) and CH₃NH₃PbI₂ perovskite (−5.44 eV) (Krishnamoorthy et al., 2014; Vivo et al., 2017). Taken the HOMO of the R01 as a reference, as a general, the HOMO energy level is increased (destabilized) when two methoxy groups were substituted (ranges from −4.70 to −4.79 eV), the exception are D6 (−4.84 eV) and D7 (−4.83 eV). On the other hand, the introduction of four and/or six methoxy substituents results in decreasing the HOMO (stabilization) (ranging from −4.94 to (−5.07 eV)), except of T10 (−4.66 eV). These indicate that di-methoxy substitution leads to destabilize (higher) HOMO level and generally the di-methoxy derivatives are losing their electrons more easily, which implies the higher hole injection or transporting ability, while the tetra-/hexa-methoxy derivatives should be more chemically stable (Zhang et al., 2012a, 2012b).

Generally, the open-circuit voltage ${(V}_{\mathit{oc}})$ values of all the designed HTMs decreased (except T10), as the number of methoxy groups increased. Since, the $V_{\mathit{oc}}$ values of the di-methoxy derivatives is in the range 0.586–0.724 eV, for tetra-methoxy derivatives the values are in the range 0.335–0.523 eV, and it is only equals to 0.460 eV for H12 (derivative with hexa-methoxy). We could say that as the number of methoxy groups increased as the electron recombination would be enhanced. The elevation of HOMO energy level would result in a high $V_{\mathit{oc}}$ the value that is determined by the difference between the HOMO level of HTMs and quasi-Fermi levels of TiO₂ or the valence band of perovskite (−5.65 eV), a bigger difference implies a higher open-circuit voltage, indicating that the HTM has an efficient hole extraction capability (Jeon et al., 2013a, 2013b; Chi et al., 2016; Zhang et al., 2017a, 2017b). Indeed, the HOMO energy level of new designed molecule T10 (−4.66 eV), which is ∼0.13 eV higher than the parent compound (R01) has the largest $V_{\mathit{oc}}$ (0.768 eV), T10The lowest HOMO level (−5.075 eV) is reported for T9 (with four substituted methoxy groups), it is found to be the deepest one which mean that T9 based on PSCs would have the lowest $V_{\mathit{oc}}$ (0.355 eV), Table 3, than the other molecules (Liu et al., 2016), however it is still more positive than the mixed and CH₃NH₃PbI₂ perovskite. This is followed by the HOMO level of H12, the hexa-methoxy derivative. H12 has HOMO energy of −4.970 eV and $V_{\mathit{oc}}$ value of 0.460 eV.

The calculated $V_{\mathit{oc}}$ values collected in Table 3, shows that the decreasing order of $V_{\mathit{oc}}$ when comparing the R01 with the di-methoxy derivatives is D4 (0.727 eV) > D3 (0.724 eV) > D2 (0.706 eV) > D5 (0.689 eV) > P1 (0.659 eV) > R01 (0.637 eV) $\ge$ D7 (0.602 eV) > D6 (0.586 eV), and with the tetra-methoxy derivatives is T10 (0.768 eV) > R01 (0.689 eV) > P9 (0.523 eV) > T8 (0.0.491 eV) > T9 (0.0.355 eV), and the hexa-methoxy derivative (H12) has lower $V_{\mathit{oc}}$ value by 0.229 eV compared to that of the parent. As reported, the open-circuit voltage values are only quantitative, since the applied model does not consider the kinetics of back electron transfer. Most of the designed HTMs have larger $V_{\mathit{oc}}$ values (and higher HOMO energies) compared to the parent molecule, that demonstrated the effect of the number and position of substituted methoxy groups that may retarded the back electron transfer, hence improving the open-circuit voltage (Mandal et al., 2017).

Wang et al. showed that for a proper HTM material, the LUMO energy level should be higher than the conduction band of perovskite to inhibit the electron in perovskite back to metal electrodes (Liu et al., 2016). The lower LUMO effectively blocks the electron transport from the perovskite to the Au contact, and this causes an increase in the fill factor (FF) value and thus an enhanced PCE (Calió et al., 2016). Almost all of the calculated LUMO energy level values are located between the $E_{\mathit{HOMO}}$ of MAPbI₃ (5.45 eV) and Au anode (5.10 eV), indicating the efficient hole-transporting processes in the PSCs devices.

According to Table 3 and Fig. 3, the LUMO energy levels of the newly designed molecules are estimated to be destabilized (more positive) than the that of CH₃NH₃PbI₃ perovskite (−3.91 eV). In addition, our results show that the energy difference between the LUMO energy level and the conduction band of CH₃NH₃PbI₃ perovskite is ranged from 2.14 eV to 2.47 eV (Table 3), which is enough to block the charge recombination. Based on the above results, the new designed HTMs are assumed to not only perform as a hole transporting layer but also play as an electron blocking layer in the PSCs, leading to the reduction of an electron from perovskite layer and hole from hole-transporting layer (Liu, Zhu et al.). In order to explore the cause of the HOMO destabilization (energy raised) of most of the di-methoxy derivatives and the cause of its stabilization (energy lowered) for most of the tetra- and hexa-methoxy derivatives, also the cause of LUMO destabilization (energies lowered) in all derivatives, we make an analysis on the molecular orbital population, Fig. 1. Fig. 1 indicates the total % contributions of the substituted methoxy groups $(\%\sum )$ . The methoxy groups in the di-methoxy derivatives (except D6 and D7) contribute mainly to the HOMO orbitals, their contributions are in the range from 1.6 to 7.8%, in the tetra- and hexa-methoxy derivatives (except T10), the methoxy substituents contribute also to the HOMO, but the total contributions are more significant, and it is in the range 1.9–11.8%. The total % contributions to the LUMOs from these groups in all derivatives is less significant and in the range 0.1–2.7%. Clearly, the substituent contributes more to the HOMO than the LUMO in these compounds. Hence, the decrease of HOMO energies in tetra- and hexa-derivatives and the increase of LUMO energies in all derivatives is mainly attributed to the enhanced electron delocalization that arise from the electron lone pair of an oxygen atom (of methoxy) leads to resonance. Thus, the resonance effect will overcome the inductive effect due to high electronegativity of the oxygen atoms, as evident from HOMO and LUMO distribution pattern (Zhao et al., 2013).

To overcome the electron-injection barrier energy in R01, 3.754 eV ( $E_{\mathit{LUMO}}$ (R01) –W (gold) =  $-1.346-\left(-5.1\right)=3.754eV)$ , would be required; the work function (W) of gold is −5.1 eV, and the $E_{\mathit{LUMO}}$ level of R01 is −1.346 eV, Table 3. On the other hand, to overcome the hole-injection barrier energy in R01, 1.057 eV ( $E_{\mathit{HOMO}}\left(R01\right)-\left(-5.85\right)=-4.793-\left(-5.85\right)=1.057eV)$ would be required (Irfan et al., 2015). To overcome the electron-injection barrier energy for all new designed derivatives with higher LUMO levels (ranges from −0.923 to −1.316 eV) are smaller than that of R01, indicating that these compounds may not apply to electron-injection materials, but they may be good hole-injection materials as R01 because of their significantly high HOMO levels (Chen et al., 2014a, 2014b). Additionally, to overcome the electron-injection barrier energy for the designed derivatives (D1, D2, D3, D4, D5, D6, D7, and T10), with higher HOMO levels, ranges from −4.828 to (−5.075 eV), are larger than that of R01. It is anticipated that the designed derivatives (except T8, T9, T11, and H12 derivatives) would be a superior hole-transport material than the parent compound.

As is known the magnitude of the HOMO-LUMO gap has significant chemical implications, even if qualitatively evaluated. A large gap implies good thermodynamic stability of the compound, whereas a small gap suggests an easy electronic transition. Inspection of Table 3. indicates that the energy gaps, ${\mathrm{\Delta }}_{H-L}$ , of a new designed systems are found to be higher than the parent molecule, except of D3 and D5 molecules. The highest electronic energy gap is attributed to the T9 molecule, which contains four substituted methoxy groups, while the lowest one, which results from a more extended conjugation, is corresponded to D3 in which only two methoxy groups are introduced. In general, this means that introduction of methoxy groups increases the thermodynamic stability of the system in comparison to the parent molecule. Indeed, the thermodynamic stability increases as the number of methoxy groups increases. These results may be attributed to the presence of the lone pair of an electron on the oxygen, which release the electron towards the π-bond, through resonance, raising the energy of frontier molecular orbitals. The trend of the energy gap ${\mathrm{\Delta }}_{H-L}$ is as: T9 > T8 > H12 > T11 > D6 > D7 > D2 > D4 > T10 > D1 > R01 > D3 > D5. The ${\mathrm{\Delta }}_{H-L}$ value of each investigated molecule is larger than its corresponding ${\mathrm{E}}_g$ value, mainly due to the neglect of interelectronic interaction upon the single one-electron excitation in estimating ${\mathrm{\Delta }}_{H-L}$ , however, the variation trend is similar (Chen et al., 2014a, 2014b).

3.4 Absorption and emission energy

It is worth mentioning that there are no experimental data for newly designed compounds because they have never been synthesized except for the parent molecule R01, which has been described in reference Wang et al. (2015). In order to adventure the properties of electronically excited states of the studied compounds, electronic transitions, the calculated absorption and emission spectra, ${\lambda }_{\mathit{abs}}$ and ${\lambda }_{\mathit{em}}$ (in nm), the excitation energies, the oscillation strengths ( $f$ ) and their assignments are calculated for each molecule using TD-B3LYP/6-311++G(d,p)//B3LYP/6-311++G(d,p) level of theory. The calculated results are presented in Table 4, together with the available experimental results. An important feature of HTMs is their light absorbance in the near UV-region, between 300 and 400 nm, thus avoiding competition with perovskite absorption (Calió et al., 2016). It is worth mentioning that in some cases (R01, D1, D2, D3, D4, and H12) the purely electronic transition HOMO → LUMO is observed, the S₀ → S₁ electronic transitions in these compounds are combined by the larger oscillation strengths, indicating that these transitions are strongly allowed. On the other hand, in other cases (D5, D6, D7, and T10), the S₀ → S₅ electronic transitions are the transition with maximum $f$ values; these are due to HOMO/HOMO-1/HOMO-2 to LUMO, indicating the small energy differences between HOMO, HOMO-1, and HOMO-2. In T8 and T9 derivatives, the S₀ → S₇ electronic transitions are predominately by HOMO to LUMO with $f$ values of 1.00 and 0.83, respectively. In the case of T11, S₀ → S₈ is dominated by HOMO-2 → LUMO transition.

It is apparently evident from Table 4 that all electronic transitions are π-π^* type, and no localized electronic transitions are shown among the calculated singlet-singlet transition. In addition the oscillator strength for S₀-S₁ attributed to HOMOLUMO transition for all compounds leading to maximum absorption was also noted. According to Table 4, the high values of the oscillator strength ( $f$ ) (0.47–1.29) indicate that the electronic transitions of π-π^* type are strongly allowed. It is obvious from Table 4 that the calculated maximum absorption wavelength, ${\lambda }_{\mathit{abs}}$ , of R01 compound (413 nm) moderately agrees with the experimental results (360 nm) (Wang et al., 2015). As compared with R01, the ${\lambda }_{\mathit{abs}}$ values of all new designed molecules are blue shifted (and in the range 357–405 nm), except of D3 molecules, which is red shifted by only 1 nm ( ${\lambda }_{\mathit{abs}}$  = 414 nm). It is also observed that the ${\lambda }_{\mathit{abs}}$ of the di-methoxy molecules lies between 382 and 414 nm, meaning that it is limited to the visible region. The compound D3 shows the longest wavelength absorption at 414 nm, which was considerably red shifted in comparison to that of the parent compound (R01) (413 nm). For compounds with two substituted methoxy groups (D1, D2, D4, D5, D4, D2 and D7), our results note a maximum absorption wavelength in the visible region (382–405 nm), which were considerably blue shifted compared to R01. The results imply that for all derivatives the absorption takes place outside or close to the visible region and overlaps slightly with the absorption band of perovskites (Chi et al., 2016). Moreover, we found a blue-shift and a decrease in the absorption intensity in the near UV/UV region in the order T9 (344 nm) → T8 (356 nm) → H12 (357 nm) → T11 (364 nm), as presented in Table 4, which is consistent with the energy gap trend. Thus, the presence of the methoxy groups would decrease the electron donating ability of the carbazole ring/s, maybe to the predominant inductive effect of electronegative oxygen atoms, and shortened π-electron delocalization over the whole molecule, which blue-shifted the absorption maxima, compared with that of R01 (Kim et al., 2015). However, the number, and position of methoxy groups attached to the carbazole ring had a marginal effect on the maximum absorption of the dyes.

The conclusion can be derived from above that T10 with four-methoxy shows longer (λ_abs = 401 nm) but stronger maximum absorption wavelength $(f$  = 1.04 nm), compared to those of other four/six methoxy substituents, although the methoxy groups in T10 and T8 are in the ortho-ortho- and meta-meta (in T8 they are at o1o2m3m4 and T10 they are at o5o6m3m4), see Scheme 1, this indicates that the particular position of the methoxy group/s has a major effect on the electronic properties of these compounds. These results lead us to estimate that increasing the number of substituted methoxy groups leads to blue shift, whereas, introduction two methoxy groups keeps the ${\lambda }_{\mathit{abs}}$ in the visible region. As a result, the number and position of methoxy groups introduced on the carbazole ring/s affected the electron distributions, which may have contributed to the hypsochromic shift of the absorption spectrum, and the decreasing of the oscillation strengths.

3.5 Light harvesting efficiency

The ability of the materials and the molecules to capture photons of solar light is called as light harvesting efficiency (LHE). Higher the value of $f$ , higher is the light capturing ability. The LHEs of the parent is comparable to those of some of its derivatives, however, it has the largest value (0.949). Among the derivatives, D1, D3, D5, D6, T8, and T10 are the derivatives with comparable LHE values to that of R01, the difference is not more than 0.05. In contrast, the other derivatives especially H12 show significant lower LHE values in comparison to those of the former derivatives and to that of R01. For instance, the difference reaches 0.28 in case of H12.

3.6 Emission properties

In this study TD-DFT//B3LYP/6-31G(d) has been performed for optimized structures on an excited state to simulate the emission spectra of the compounds under study where the first singlet excited states are listed in Table 4. In all cases and similar to the absorption spectra, S₁-S₀ fluorescence peaks in emission spectra have the largest oscillator strengths in all molecules, and ∼70% arises predominantly from LUMO → HOMO transition. Based on Table 4, the calculated maximum emission wavelength, ${\lambda }_{\mathit{em}}$ , of R01 compound (465 nm) is strongly agrees with the experimental results (467 nm) (Wang et al., 2015), supporting our selection of the theoretical level of calculation. Comparing with the parent molecule (R01), the majority of the new designed molecules show a red shift in the emission spectrum, whereas some of them show a blue shift, with a significant increase in the oscillator strength as in the case of molecules D5, D6 and D7. The highest oscillator strength is found for D5. In addition, the emission spectra show a blue shift in the order D4 ( ${\lambda }_{\mathit{em}}$  = 453 nm) → T9 (454 nm) = D1 (λ_em = 454 nm) → H12 (460 nm). On the other hand, the emission spectra show a red shift in the order T8 (λ_em = 468 nm) → D2 (λ_em = 476 nm) = T11 (483 nm) → D7 (490 nm) = D6 (490 nm) → D3 (498 nm) → T10 (503 nm) → D5 (504 nm). Generally, for the parent and designed derivatives, the emission maxima are longer than the absorption maxima, since ${\lambda }_{\mathit{em}}$ is red-shifted by not less than 52 nm compared with λ_abs.

3.7 Stokes shift

Stokes shift can reflect the degree of structural deformation between the ground state and excited state geometries. A more substantial Stokes shift is beneficial for pore-filling of HTM (Chi et al., 2016). The computed results of the newly designed molecules yield a stoke shift values ( ${\lambda }_{\mathit{ss}}$ ), described as: ${\lambda }_{\mathit{ss}}={\lambda }_{\mathit{em}}-{\lambda }_{\mathit{abs}}$ . Based on the values of the calculated ${\lambda }_{\mathit{ss}}$ (Table 4), the new designed compounds can be classified into three categories as follows:

1. Relatively high λ_ss (from 106 nm to 119 nm) (D6, T8, T9, T8 and T11 compounds): The relatively high ${\lambda }_{\mathit{ss}}$ values indicate that electron transition breaks the strong conjugative effect and leading to a remarkable geometry distortion (bond lengths and bond angles (Table 1). and large reorganization energy as will be shown in the next subsection (Table 5).

2. Relatively moderate λ_ss (from 84 nm to105 nm) (D3, D2, D5, H12 and D7 compounds). The moderate ${\lambda }_{\mathit{ss}}$ value reflects that electron transition is partially breaking the strong conjugative effect, which leads to a moderate geometry distortion from a ground state to an excited state and a moderate reorganization energies (Table 1 and Table 5).

3. Relatively small ${\lambda }_{\mathit{ss}}$ (52–80 nm) (R01, D1 and D4 compounds), which indicates that electron transition doesn't break the strong conjugative effect, assigning a small geometry distortion and a small reorganization energies (Table 1 and Table 5).

Since larger Stokes shift is also typical feature for a good HTM; we could predict that D6, T8, T9, and P9 derivatives as HTMs are better for increasing the performance of PSCs compared with the parent (R01). D6 and T8 combine in the nature of the methoxy position, the two and four methoxy groups in D6 and T8, respectively, are positioning in the ortho- and meta- positions. In D6, the methoxy groups are in o1m7, and in T8 they are in o1o2m3m4, see Scheme 1. While the four methoxy groups in T9 are at the four ortho-positions (o1o2o5o6).

3.8 The radiative lifetimes

The Radiative lifetimes (τ) of the compounds under probe have been computed for a spontaneous emission using the Einstein transition probabilities (Lukeš et al., 2005). This time refers to the average time that molecules stay in their excited state before emitting a photon. If the compounds have a shorter lifetime in the excited state, it means they have more efficiency in the emitting of fluorescence or photons (Zhang et al., 2012a, 2012b), and longer lifetime indicate improved suppression of back reactions of the injected electrons, this results in improvement the $V_{\mathit{oc}}$ value, due to the reduced electron recombination rate (Kim et al., 2015). The calculate lifetime τ for the investigated compounds are listed in Table 4. It was pointed that the shorter radiative lifetimes, the high light-emitting efficiency, while the longer radiative lifetime facilitates the electron and energy transfer (Hlel et al., 2015). According to Table 4, it has been observed that as the radiative lifetime of compounds is shortened, the oscillator strength is increased. These results lead to an increase in the luminescent efficiency. The computed τ values follow the following order: H12 > D4 > D2 > T10 > D3 > T9 > T11 > T8 > D1 > D7 > R01 > D5 > D6. For compound H12, with six methoxy groups, we find the lowest oscillator strength and the longest radiative lifetime, whereas the shortest radiative lifetime and the highest oscillator strength is found for compounds D5 and D6, with only two substituted methoxy groups. Therefore, they should be good-emitting materials with a high efficiency (Zhang et al., 2012a, 2012b).

To better explain the contribution of the electron, the charge density differences between ground and excited states of R01 and some of its designed derivatives, D1, D2, T9, T11, and H12 (as representative examples) are plotted in Table 6. In this case, it is assumed to be the difference between HOMO and LUMO since the first transition is mainly from HOMO to LUMO. Electron density moves from the cyan region to the blue region when moving from the ground state to the first excited state. Cyan and blue colors indicate increase and decrease of charge densities, respectively. As evident from Table 6 the R01 and its derivatives demonstrated clearly intramolecular charge transfer (ICT).

Table 6 Density difference between ground state (S_o) and excited state (S₁) of some investigated HTMs (as representative examples), contour threshold is 0.02 au.

HTM	ED moves from So	ED goes to S1	Full picture
R01
D1
T11
H12

3.9 Reorganization energy, electron affinity, and ionization potential

One of the parameters to determine the carrier hopping rate is the reorganization energy (λ $\lambda$ ), which comes from the contributions of external reorganization energy and internal reorganization energy. The latter includes the molecular geometry modifications when an electron is added or removed from a molecule, and the former is the modifications in the surrounding medium due to polarization effects. Norton and Brédas (Norton and Brédas, 2008) reported that the external reorganization energy is significantly lower than that of the inner part based on a polarized force field calculation. It is also fundamental to calculate the ionization potential ( $\mathit{IP}$ ) and electron affinity ( $\mathit{EA}$ ), which characterize the reduction and oxidation properties (Chen et al., 2011). These descriptors can be used to evaluate the energy barriers of holes and electron injection including charge mobility and balanced charge. The calculated adiabatic and vertical $\mathit{IPs}$ ( ${\mathit{IP}}_a$ and ${\mathit{IP}}_v$ ), the adiabatic and vertical $\mathit{EAs}$ ( ${\mathit{EA}}_a$ and ${\mathit{EA}}_v$ ), as well as the reorganization energies for holes ( ${\lambda \lambda }_{\mathit{hole}}$ ) and electrons (λ ${\lambda }_{\mathit{elec}}$ ) for the compounds under probe at B3LYP/6-31G(d) are presented in Table 5 and Fig. 4 Also, Table 5 includes the values of $\mathit{HEP}$ (the hole extraction potential) as the energy difference from M⁺ (cationic) to M (neutral molecule), using M⁺ geometric structure in the calculation, and $\mathit{EEP}$ (the electron extraction potential) as the energy difference from M- (anionic) to M, using M- geometric structure in the calculation (Chen et al., 2014a, 2014b). The internal reorganization energy is the energy difference between two molecules due to geometric relaxation during electron transfer, and it is an important parameter for charge transfer and according Marcus theory (Nelsen et al., 2001), which provides to a qualitative indication of the charge-transport rate (the lower the $\lambda$ values, the bigger the charge-transport rate). For reorganization energy, as is seen in Table 5, the ${\lambda \lambda }_{\mathit{hole}}$ values of the investigated molecules are ranged from 0.152 to 0.664 eV, while the ${\lambda }_{\mathit{elec}}$ values are in the interval of 0.338–0.517 eV. As shown, the first important remark to be noted is that most of the investigated molecules exhibit ${\lambda }_{\mathit{elec}}$ values larger than the ${\lambda }_{\mathit{hole}}$ values, except of D1D1 and T9, implying that the hole transfer should be faster than the electron transport, which is reflected by the $k_{\mathit{hole}}$ and $k_{\mathit{elec}}$ values (1.03 × 10¹⁴–1.86 × 10¹⁵, and 4.91 × 10¹⁴–6.79 × 10¹⁵ s⁻¹, respectively). From the variations in the main geometrical parameters in the neutral to anionic/cationic states, see Section 3.1, we observed that the geometry relaxation between the neutral and anion states is greater than that for the cations, resulting in greater polarization in the former, which results in a larger value of ${\lambda }_{\mathit{elec}}$ than ${\lambda }_{\mathit{hole}}$ (Irfan et al., 2015).

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Fig. 4

Adiabatic and vertical ionization potentials ( ${\mathit{I}\mathit{P}}_{\mathit{v}}$ / ${\mathit{I}\mathit{P}}_{\mathit{a}}$ ), (b) Adiabatic and vertical electronic affinities ( ${\mathit{E}\mathit{A}}_{\mathit{v}}$ / ${\mathit{E}\mathit{A}}_{\mathit{a}}$ ), (c) hole reorganization energies ( ${\mathit{\lambda }}_{\mathit{h}\mathit{o}\mathit{l}\mathit{e}}$ ) and electron reorganization energies ( ${\mathit{\lambda }}_{\mathit{e}\mathit{l}\mathit{e}\mathit{c}}$ ), and (d) adiabatic hardness of investigated HTMs calculated at the B3LYP/6-31G(d).

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We find that the ${\lambda }_{\mathit{hole}}$ of the parent molecule, R01 is only 0.297 eV, while its ${\lambda }_{\mathit{elec}}$ reaches as high as 0.368 eV; the latter is higher than the former. Hence, R01 should be suit to transporting holes than electrons. These results indicate, from the reorganization energy point of view, that R01 is p-channel rather than n-channel. In addition, the λ_hole of R01 is approximately equal to that of a typical hole transport material N,N′-diphenyl-N,N′-bis(3-methylphenyl)-(1,1 0-biphenyl)-4,4′-diamine (TPD) with ${\lambda }_{\mathit{hole}}$  = 0.29 eV. This also means that its hole transfer rate, ( $k_{\mathit{hole}}$  = 1.9 × 10¹⁵ s⁻¹), should be approximately equal that of TPD, thus the potential of R01 compound as tested experimentally as profound hole transfer material, is also proved theoretically. Unlike the others, T9 and T11 displayed considerable hole reorganization energies (0.664 eV and 0.460 eV, respectively), which can be ascribed to their large structural distortions when the ionization process takes place (Table 1). In comparison with the parent molecule (R01), most R01 derivatives exhibit larger ${\lambda }_{\mathit{hole}}$ and ${\lambda }_{\mathit{elec}}$ values, except D3, D5 T8 and T8 D5exhibit smaller ${\lambda }_{\mathit{hole}}$ value and only D1 exhibits smaller ${\lambda }_{\mathit{elec}}$ value. These results indicate that most of the designed derivatives may possess lower charge mobilities compared to that of the parent, which can be easily noticed from the $k_{\mathit{hole}}$ values. On the other hand, the calculated ${\lambda }_{\mathit{hole}}$ values for D3, D5 T8 and T8 D5are 0.280, 0.152 and 0.264 eV, respectively, which are clearly smaller than that of the value of R01 (0.297 eV). D3T8D5, indicating that these designed HTMs have larger hole mobilities compared to the parent. While, D1 with ${\lambda }_{\mathit{elec}}$ (0.338 eV) which is smaller than that of R01 (0.368 eV), clearly demonstrating that this compound has larger electron mobility compared to that of the parent. The trend of increasing of the ${\lambda }_{\mathit{hole}}$ values of the di-methoxy derivatives is D5 < D2 = D4 < D3 < P1 < D6 < D7, thus among this series, D5 is expected to show the faster hole transfer. While, for the series composed of the tetra-methoxy derivatives the trend observed is: T8 < T10 < T11 < T9, and in this series T8 and T9 are the compounds that expect to show the faster and slower hole transfer, respectively. On the other hand, the trend of increasing of the ${\lambda }_{\mathit{elec}}$ values of the di-methoxy derivatives is D1 < D5 > D2 < D3 < D7 < D4 < D6, thus among this series, D1 and D6 are expected to show the faster and slower electron transfer, respectively. While, for the series composed of the tetra-methoxy derivatives the trend observed is: T10 < T9 < T11 < T8, and in this series T10 and T8 are the compounds which expect to show the faster and slower hole transfer, respectively. However, H12 the hexa-methoxy derivative has intermediate hole and electron mobilities. In order to attain high quantum efficiency, it is necessary to achieve efficient charge injection at right charge balance; thus, materials used as hole/electron transporter should have closed reorganization energy with bipolar character (Chen et al., 2014a, 2014b). Therefore, most of the designed derivatives (except five out of twelve compounds, i.e., D3, D4, D5, T8, and T9), have comparable/better potential as HTMs as R01, since the difference between the ${\lambda }_{\mathit{hole}}$ and ${\lambda }_{\mathit{elec}}$ is only in the range 0.012 to 0.097 eV, while the difference for the parent is 0.072 eV.

To compare the magnitudes of hole and electron mobilities, we calculate the relative hopping rates of holes versus electrons, $k_{\mathit{elec}}$ / $k_{\mathit{hole}}$ . The values of $k_{\mathit{hole}}$ and $k_{\mathit{elec}}$ have been calculated, assuming T to be 300 K and neglecting the difference in $\mathrm{\Delta }{H}_{\mathit{ab}}$ , (Table 7). The ratio is predicted to be in the range 2–30. Where most of the investigated molecules show small ratio (between 2 and 6), where D6, D7 and H12 show relatively large ratio (30, 22, 22, respectively), and P2, D5, and T9 show moderate values, 18, 13, and 15, respectively. Although some experimental data indicated that $\mathrm{\Delta }{H}_{\mathit{ab}}$ varies over a rather limited range for intramolecular charge-transfer processes, this does not guarantee complete cancellation in the comparison of relative charge-transfer rates. The hole mobility of R01 has been reported to be 2.05 × 10^–4 cm² V⁻¹ s⁻¹ (Wang et al., 2015). However, direct comparison of its reported mobility is not feasible because the value depends on the electric field and film preparation details.

As is known the ionization potential and electron affinity are good descriptors to interpret the charge transfer behavior. The energy barrier for the injection of holes and electrons can be predicted by the explicit properties such as adiabatic ionization potential ( ${\mathit{IP}}_a$ ) and adiabatic electron affinity ( ${\mathit{EA}}_a$ ). Evaluation of the nature of devices of carrier transport can be measured by the stability of the HTMs. The stability of p-type HTMs is directly related to the ${\mathit{IP}}_a$ . The lower ionization potential indicates that material would be better as hole transporter, while higher electron affinity leads to be a better electron transport (Chan and Liao, 1970). The adiabatic ionization potential ( ${\mathit{IP}}_a$ ), vertical ionization potential ( ${\mathit{IP}}_v$ ), adiabatic electron affinity ( ${\mathit{EA}}_a$ ), and vertical electron affinity ( ${\mathit{EA}}_v$ ) were calculated. The computed results are summarized in Table 5. The trend observed for the change in the adiabatic and vertical $\mathit{EA}$ values of the investigated HTMs is very similar. In contrast, there is a small difference between the trend of changes showed by the adiabatic and vertical $\mathit{IP}$ values, as illustrated in Fig. 4. Additionally, for all investigated HTMs, the adiabatic $\mathit{IP}$ values are smaller than the vertical $\mathit{IP}$ values, while the reverse is true with respect to the $\mathit{EA}$ values.

The data in Table 5 and Fig. 4 shows that the order of decreasing the adiabatic and vertical $\mathit{EA}$ values of the di-methoxy derivatives with respect to that of the parent is: R01 > D5 > D3 > D6 $\approx$ D7 > P1 > D4 > P2, while that for the tetra-methoxy HTMs is: R01 > T10 > P9 > T8 > T9, and the $\mathit{EA}$ values of R01 are larger than that of the hexa-methoxy (H12) derivative. Thus, the parent compound show higher $\mathit{EA}$ values compared to those of its derivatives, indicating that R01 seems to be a better electron transporting material than its methoxy-substituted derivatives. In addition, D2, T8, T9, and H12 derivatives have negative calculated ${\mathit{EA}}_v$ values, different from others, which was raised by the incomplete cancellation of the electronic self-interaction energy due to the use of an inexact density functional and a finite basis set (Chen et al., 2014a, 2014b). The four aforementioned HTMs are expected to show the weakest potential as electron transporting materials. However, from the adiabatic/vertical $\mathit{EA}$ values of D1 (0.360/0.185 eV), D3 (0.431/0.194 eV), and D5 (0.464/0.251 eV), these three di-methoxy derivatives are expected to show comparable potential as that of the parent (0.462/0.255 eV). On the other hand, the adiabatic and vertical $\mathit{IP}$ values of all designed methoxy derivatives are smaller than that of the parent (except those of D6, T8, T9, and H12), indicating that most of the designed HTMs are seemed to be better hole transporting materials than the parent compound. Both the values of the adiabatic/vertical $\mathit{IP}$ of T8 (6.066/6.138 eV) and H12 (5.824/5.978 eV) are larger than that of R01 (5.751, 5.907 eV), indicating that these tetra- and hexa-methoxy derivatives, respectively, and relying on these values only are the least potential designed molecules as hole-transporting materials. The first inspection of Table 5 shows that T11 has the lowest ${\mathit{IP}}_a$ (5.395 eV) and its ${\mathit{EA}}_a$ (0.267 eV) is considered as a medium value as compared with other eleven molecules. However, the parent compound R01 has the highest ${\mathit{EA}}_a$ (0.462 eV) and its ${\mathit{IP}}_a$ is relatively high (5.751 eV) compared to its derivatives. We can also find that molecule T8 (6.066 eV) has the biggest ${\mathit{IP}}_a$ value and a relatively small ${\mathit{EA}}_a$ value (0.267 eV). Importantly, our computed results show that ${\mathit{IP}}_a$ is higher when 4 and /or six methoxy groups have been introduced to the parent molecules. On the other hand, the biggest ${\mathit{EA}}_a$ value is reported for D5 (0.464 eV) while the smallest ${\mathit{EA}}_a$ value is found for H12 (0.115 eV) as compared with the other molecules. The vertical $\mathit{IP}$ and $\mathit{EA}$ values calculated at the geometry of the neutral molecule can be used to study the charge injection of the materials. The molecule with low $\mathit{IP}$ or high $\mathit{EA}$ is likely to improve electron or hole injection/transport (Chen et al., 2014a, 2014b). It can be found that the change trends of the vertical $\mathit{IP}$ and $\mathit{EA}$ values for the investigated HTMs are almost similar to their HOMO and LUMO energies. The correlation between ${\mathit{IP}}_v$ values and HOMO energies ( ${\mathit{EA}}_v$ values and LUMO energies) for the HTMs with di-methoxy substitution and tetra-methoxy substitution all with the parent can be obtained linearly in Fig. 5.

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Fig. 5

The correlation between the calculated HOMO and LUMO energies and the vertical $\mathit{I}\mathit{P}$ and $\mathit{E}\mathit{A}$ values in (a) Di-methoxy derivatives and (b) Tetra-methoxy derivatives.

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Di-methoxy derivatives (D1, D2, D3, D4, D5, D6, and D7); along with R01 $${\mathit{IP}}_v=1.0917\left(-E_{\mathit{HOMO}}\right)+0.6438;R^2=0.9085$$ $${\mathit{EA}}_v=0.9020\left(-E_{\mathit{LUMO}}\right)+0.7334;R^2=0.9682$$

Tetra-methoxy derivatives (T8, T9, T10, and T11); along with R01 $${\mathit{IP}}_v=1.101\left(-E_{\mathit{HOMO}}\right)+0.5976;R^2=0.8366$$ $${\mathit{EA}}_v=0.7621\left(-E_{\mathit{LUMO}}\right)+0.5663;R^2=0.9711$$

Above regression equations may provide a simple formula to predict the vertical $\mathit{IP}$ and $\mathit{EA}$ values of other similar derivatives from their HOMO and LUMO energies (Zhang et al., 2012a, 2012b; Chen et al., 2014a, 2014b). Previous studies have demonstrated that the ${\mathit{IP}}_a$ values of air-stable p-channel materials ranges from 5.680 eV to 6.786 eV, and the ${\mathit{EA}}_a$ values for air-stable n-channel ones are in the range 2.411 to 3.141 eV (Zhao et al., 2013). On the basis of this viewpoint, only R01, T8, and H12 should be air-stable p-channel materials, while non-compound is an air-stable n-channel one. Hence, the investigated HTMs are not suitable to being used as electron transport materials for their low air-stability.

Furthermore, the absolute hardness (η) also has been calculated. As is known, hardness (η) is the resistance of the chemical potential to the change in the number of electrons, and it’s controlled the stability of PSC, since the HTM could play a crucial role in controlling the stability of PSCs; an ideal HTM should be stable, i.e., it should be a hard molecule with large η value (Zhang et al., 2017a, 2017b). The η values of all molecules are also presented in Table 5 and Fig. 4. Inspection of Table 5 and Fig. 4 reveals that the η values of the designed molecules in the range 2.564–2.931 eV with largest deviation of 0.37 eV, indicating that the stabilities of the investigated molecules are almost similar. Indeed, we have found that T8 (η = 2.931 eV) molecule has the highest stability while T11 $($ η = 2.564 eV) molecule has the lowest stability among all the studied molecules. The previous result indicates that the stability of R01 derivatives is comparable to that of Spiro-OMeTAD and ATT derivatives (Chi et al., 2016).

3.10 First singlet excitation and exciton binding energies

As is known the exciton binding energy ( $E_b$ ) of samples is an important factor for the electroluminescence quantum efficiency of organic light-emitting diodes (OLEDs) (Franceschetti and Zunger, 1997; Cao et al., 1999). The computed First singlet excitation and excitation binding energies are presented in Table 3. As shown in Table 3, the investigated molecules have large exciton binding energy in the range 0.357–0.571 eV (Hlel et al., 2015). The molecule D5 (0.357 eV) has the lowest $E_b$ and molecule H12 (0.571 eV). Our results show that the calculated $E_b\ge k_{B}T0.331eV$ at room temperature, reflecting that the investigated molecules certainly allow the formation of exciton even at room temperature after excitation and leading to higher fluorescence quantum efficiency after recombination in OLEDs (Hlel et al., 2015).

3.11 NLO properties

It has found that compounds with nonlinear optical properties have good photoelectric conversion performance (Cai-Rong et al., 2009; Kumar et al., 2010; Garza et al., 2014). The polarizabilities and hyperpolarizabilities of the six investigated HTMs were calculated at the B3LYP/6-311++G(d,p), Fig. 6. In addition to the total hyperpolarizabilities, the isotropic polarizability, polarizability anisotropy invariant and hyperpolarizability.

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Fig. 6

Isotropic polarizability and total hyperpolarizability values (in au) of investigated HTMs calculated at the B3LYP/6-311++G(d,p).

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The calculated isotropic polarizability ( $\mathrm{\Delta }\alpha$ ) values of D1-D7, T9-T11, and H12 are in the range from 397-to-647 au However, the calculated isotropic polarizability of the parent molecule R01 is an intermediate value ( $\mathrm{\Delta }\alpha =568au$ ) compared to those of its derivatives. The isotropic polarizability values of the designed D1 and D3 are very comparable to that of the R01 (the parent molecule), therefore, the specific positions of the substituent methoxy groups in these two designed HTMs make them with comparable response to the external electric field compared to R01. The designed H12 shows the least value (397 au). As reported an inverse correlation was found between the isotropic polarizability values and photo-to-current conversion efficiencies, and this was attributed to the fact that larger delocalization of electrons enhanced the response to the external field, but it is less efficient in generating charge separated state (Cai-Rong et al., 2009). Thus, it induced the lower photo-to-current conversion efficiency. According to the values of the isotropic polarizability, the efficiencies of the investigated HTMs to convert photons into electricity can be estimated as: D5 (647 au) > R01 (568 au) > D3 (562 au) > D1 (557 au) > T10 (543 au) > D7 (528 au) ≥ D6 (525 au) > T11 (500 au) > D4 (484 au) > D2 (420 au) > T8 (414 au) > T9 (406 au) > H12 (397 au).

As shown in Fig. 6, the order of decreasing the total hyperpolarizabilities ( ${\beta }_{\mathit{tot}})$ of the investigated di-methoxy HTMs in comparison with R01 is: D5 (1420 au) > D4 (1399 au) > D2 (1324 au) > D6 (919 au) > D7 (866 au) > R01 (700 au) > D1 (644 au) ≫ D3 (174 au). The order of decreasing the total hyperpolarizabilities of the investigated tetra-methoxy HTMs in comparison with R01 is: T11 (959 au) > R01 (700 au) > T8 (587 au) ≫ T9 (256 au) ≫ T10 (32 au). In contrast, the hyperpolarizability of the parent (R01) is seven times greater than that of the hexa-methoxy derivative H12 (96 au). Generally, the di-methoxy derivatives have the largest hyperpolarizabilities (except D3), ranges from 1420-to-644 au, followed by the tetra-methoxy derivatives (except T10), ranges from 959-to-256 au, and then the H12 with six methoxy substituents has the least hyperpolarizability. Indeed, the major factor affecting the hyperpolarizability values is the number of methoxy groups (as the number of the methoxy groups increasing, the hyperpolarizability decreasing), while their positions seems to has a minor effect on the hyperpolarizability values. The total hyperpolarizabilities for R01 molecule is 700 au, which is smaller than that of the typical NLO material p-nitroaniline (pNA) ( ${\beta }_{\mathit{tot}}=1327au$ ) that has been calculated at B3LYP/6-31G(d,p) (Garza et al., 2014). D2 and D4 are found to have considerably larger hyperpolarizabilities compared to the parent molecule and very comparable values compared to pNA, and D4 shows even larger value by ∼72 au. The difference between the four designed HTMs D1-D4 is in the position of the substituted methoxy groups, since all of these molecules have two substituted methoxy groups at different positions in the carbazole rings. From the data above, it is possible to conclude that the specific positions of the methoxy groups is believed to contribute with different extent to electron donor ability to the CT, hence leading to different hyperpolarizability. It is interested to note that, H12 molecule shows the very small value of total hyperpolarizability equals 96 au. Thus, the substitution of the six hydrogen atoms with methoxy results in a remarkable decrease in the molecule’s total hyperpolarizability. As previously mentioned, the methoxy substituent has an electron-donating ability through resonance and electron-withdrawing ability inductively; it seems that when the number of methoxy groups becomes six (as in H12) the second effect becomes more pronounced, leading to lesser electron donor ability to the CT and smaller.