Optical and magneto-optical properties of metal phthalocyanine and metal porphyrin thin films

In order to support the experimental results and to further investigate the electronic structure of the materials we performed all-electron DFT calculations of all individual free molecules and for the bulk structures of the MePcs.

The calculations for the individual free molecules were carried out using the NRLMOL program package [28]. NRLMOL combines large Gaussian orbital basis sets, numerically precise variational integration and an analytic solution of Poisson's equation in order to accurately determine the self-consistent potentials, secular matrix, total energies and Hellmann–Feynman–Pulay forces. To include exchange and correlation effects, the generalized gradient functional developed by Perdew, Burke, and Ernzerhof (PBE) was applied [29]. First we relaxed the structures of the individual molecules by performing a full geometry optimization. The symmetries of the initial and the final state are of decisive character for the calculation of the possible optical transitions. The corresponding dipole matrix element is obtained for any possible transition and the intensity of every transition is weighted with the value of the dipole matrix element. This offers a direct mapping of the spectral signals to the electronic structure of the molecule. In order to reach qualitative understanding of our experimental data we added a Gaussian broadening of about 0.15 eV to the theoretical spectra.

For the bulk structure calculations we used the ELK program which is an implementation of the all-electron full potential linearized augmented plane-wave (FP-LAPW) method [30]. The starting structures correspond to the α-phase of MePc and were obtained from the Cambridge Structural Database. The calculations were performed using a basis set with 440 valence states and an additional 562 local orbitals together with the same PBE functional as for the NRLMOL calculations. After geometry relaxation of the structures (F_max < 0.05 eV  Å⁻¹) the full dielectric tensor was calculated according to [31]. The Voigt constant can be retrieved directly from the elements of the dielectric tensor. The effective k-point mesh size used for Brillouin-zone integration in the case of calculation of optical properties was 400. The effect of spin–orbit coupling and a magnetic field normal to the molecular plane were included in the calculations.

The basic assumptions within the perimeter model (PM) are as follows.

• (i)  
  The π-system under investigation can be treated independently of the remaining electrons of the molecule.
• (ii)  
  Spin–orbit coupling is negligible small.
• (iii)  
  Full aromaticity of the investigated π-system (i.e. containing (4N + 2) electrons, with $N \in \mathbb {N}$, with N being a natural number) is assumed, describing a closed path lying approximately in a plane.
• (iv)  
  σπ^∗ and nπ^∗ state-mixing into π→π^∗ transitions will be ignored.

Assumptions (i) and (ii) assure the separability of the molecular wavefunctions (WFs) into three components, one perpendicular to the plane of the molecule z, one polar angle ϕ in the molecular plane and an in-plane radius r. Assumption (iii) validates that outer shell π→π^∗ transitions are sufficient for the main part of the optical spectra. Note that the model is no longer valid if the number of electrons in the investigated π-system deviates from the number claimed above. The discussion may then be carried out for a stable aromatic anion or cation instead. Now that we know about the limitations of the PM and can ensure its validity, we will focus on the definition of the orbital angular ring quantum numbers (OARNs).

The wavefunctions predicted by the PM are complex e^ikϕ with the imaginary unit i, the angular variable ϕ and the OARN k. Commonly the PM is combined with classic molecular orbital theory, which restricts the values of k to $k \in \left \{0, \pm 1, \pm 2,\ldots , \pm \left (\frac {n}{2}-1\right ), \frac {n}{2}\right \}$, where n is the number of electrons in the considered π-system. However, one has to note that the selection rules are defined with respect to orbital angular quantum numbers (OANs). If spin–orbit coupling is negligible, the selection rules split up in statements for spin s and angular momentum l resulting in the well known dipole selection rules Δl = ±1, if at the same time spin flips are not allowed. But why is it not possible to state that the OARN is equal to the OAN? Since an arbitrary closed path along a π-system never possesses a C_∞-symmetry, $\hat {L}_z$ is not a constant of motion. In other words if there is no potential with circular symmetry, the OARN is not equal to the OAN. However, identifying the OARN as the OAN of $\hat {L}_z$ is often misleadingly done in literature. The interpretation of the OARN as an eigenvalue of the linear momentum of the 1D problem along the circle is invalid, due to the periodic potential resulting from the atoms, along the polar coordinate ϕ. Hence the linear momentum is not a constant of motion in the 1D case.

The molecular WFs are either linear combinations of eigenstates or not eigenstates at all of the angular momentum operator $\hat {\vec L}$. That leaves basically three options: (1) direct computation of $\hat {\vec {L}}$ applied on the molecular WF at distinct points, (2) using linear combinations of atomic orbitals (LCAOs) to model the molecular WF at distinct points and obtain the linear combinations of the OAN or (3) using the perimeter model, to gain OARNs k, see [32]. In contrast to the first two, the third option is unique for a distinct WF, even if the WF is not an eigenstate or linear combination of eigenstates of $\hat {\vec {L}}$. The OARN can be used instead of the OAN, since a one-to-one correlation exists. Thus knowing the OARNs for the initial and final WFs and knowing how OARNs relate to the OANs, we are able to characterize transitions in optical absorption and MOKE spectra. The OARN becomes a measure of the OAN by defining the modulus of the OARN as half the number of nodes with sign change of the WF Ψ_k(ϕ), along the outermost possible path of the investigated π-system. The sign of the OARN can be defined in two ways. The first possibility would be to identify the sign with the sign of the following equation:

$$\begin{eqnarray} &&\frac {1}{2\pi } \int _0^{2\pi } \Psi _k^{*} \left (\phi \right ) \hat {\Xi }\, \Psi _k \left (\phi \right ) \, {\rm d} \phi \nonumber \\ &&= \frac {1}{2\pi } \int _0^{2\pi } \Psi _k^{*} \left (\phi \right ) \frac {\hbar }{{\rm i} } \frac {\partial }{\partial \phi } \Psi _k \left (\phi \right ) \, {\rm d} \phi = \frac {\hbar k}{2\pi }. \label {lbl_sign_definition} \end{eqnarray} \tag{ 6 }$$

This gives the mean expectation value of the operator $\hat {\Xi } := \frac {\hbar }{{\rm i} } \frac {\partial }{\partial \phi }$. $\hat {\Xi }$ can be interpreted either as linear momentum operator of the free electron model, i.e. along the circle with sufficient boundary conditions and the variable ϕ, or as $\hat {L}_z$ of the restrained 3D problem. Note that the 3D problem can be substituted by a 1D problem with coinciding z-axes. Since the radius is fixed, the molecule is approximately planar and the cylinder coordinates are separable.

The second way of determining the sign of the OARN is by using classic electrodynamics and Lorentz law. It follows that k < 0 if the magnetic moment, induced by the moving electrons, and the z-axis, are anti-parallel, and k > 0 otherwise. This suggests the illustration of the sign as the circulating sense of the electrons. Thereby the case of maximal k is implicitly defined as positive.

Note that the OARNs k are adding up like scalars to a total ring quantum number K, which follows from the interpretation of $\hat {\Xi }$ as a linear momentum operator in 1D or from power laws and the form of the molecular WF (a simple product of exponential functions) of n electrons. Therefore it is possible to apply the selection rules on k instead of l, corresponding to a replacement of $\hat {\vec {L}}$ by $\hat {L}_z$, because $\hat {L}_x^2 + \hat {L}_y^2 = \mbox {const}$ on the circle of the perimeter model. The change of the OARN Δk, resulting from a transition, can be recognized as a measure of the circular charge distribution. For example a greater OARN of the excited state can be interpreted as formation of a negatively charged particle. It is now clear that magnetic moments, induced by the circular currents, are proportional as well to Δk.

In order to assign OARN one can start from the geometry of the molecule with the corresponding real-valued representations of the WF of interest and condense it using the following three steps.

• (i)  
  Determine whether the molecule in question is aromatic or not. If it is not, use any stable aromatic anion or cation derivable from the molecule.
• (ii)  
  Determine the number of electrons in the π-system under investigation.
• (iii)  
  Plot the real-valued WFs and count the number of nodes with sign change along the path of the π-system of interest, this would equal figure 1. Now half the number of counted nodes equals the number of nodal planes, i.e. the absolute value of the OARN k.

Finally we are going to remark on possible influences of the center atoms, which will support our discussion and can be understood with the help of the WF symmetry. All metal atoms have lower electronegativities than hydrogen and therefore the binding σ-electrons will be shifted towards the ring. This will increase the electron densities on the nitrogen sites and increase the energies of states with electron density there by Coulomb interaction. This effect is known as the inductive effect.

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The next effect is known as the conjugative effect and influences primarily the π-system of the TMPPs. As the p orbitals, especially p_z, of the d-block elements are empty, transition group elements increase the size of the π-system by conjugation. The energy of a WF with according symmetry, i.e. without nodal points on the center atom, is lowered.

The B-band was defined as the energetically higher feature rising at ∼2.8 eV for TMPPs in the optical spectrum based on an a_1u→e_g transition. For porphyrins, experiments indicate, in agreement with theory, the B-band being related to π→π^∗ transitions of the aromatic system, nearly independent of the center atom(s) (see [33, 22]). Also the methoxy sidegroups do not influence the lineshape of the B-band significantly.

The Q-band is lower in energy at ∼2.25 eV for TMPPs and based on an a_2u→e_g transition. It was also found that the Q-band is dependent on the central atom(s), since a WF of a_2u symmetry has no nodal point at this location. Figures 2 and 5 show this dependence by comparing H₂- and the MeTMPPs. The former shows a four-fold splitting of the Q-band, since the total spin is 0 and the unoccupied states are approximately degenerated within 10 meV. This can be related to the C₂ symmetry of the molecule. Please note that the definition of Q- and B-band by using the energy can be misleading, as in the case of Pcs the character of the two features exchanges with the a_2u→e_g transition occurring at higher energy in that case.

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The influence of the methoxy sidegroups on the shape of the Q-band is shown in figure 2(b). One can see a slight redshift of the functionalized molecule and a change in the ratio between in-plane and out-of-plane components of the extinction coefficient. The redshift of the H₂TMPP absorption is most probably an effect of stronger intermolecular interactions, an effect known as crystallochromy, which might be an indication of a better degree of crystallinity of these films.

As can be seen from DFT calculations, each of the a_1u→e_g and a_2u→e_g transitions ends in the same final pair of states with e_g symmetry. This is true for all investigated molecules, see figure 3. The reversal of Q- and B-band ordering between TMPP and Pc is related to a change in level ordering of the corresponding occupied states. Accordingly the a_1u orbital is once above (TMPP) and once below (Pc) the a_2u orbital, as was already observed in the case of manganese derivates [34]. These four states account for the Q- and B-bands of porphyrins [35] and the description is known as Gouterman's four orbital model [36], which was used successfully for various porphyrinoids [19, 22, 33, 36–38].

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In figure 4(b) it can be seen that the Q-band of all Pcs consists of at least two main features. H₂-, Co-, Ni-, Cu-, and ZnPc all show significant splitting of the Q-band into main features at ∼1.75 and ∼2.1 eV. The Q-bands of Mn- and FePc show an even more complex character and in contrast to the other Pcs significant absorption at low energies (<1.5 eV ) is observed. The structure of the MePc Q-band (1.5−2.5 eV ) has been studied using different experimental and theoretical techniques, but there is so far no agreement as to what are the main effects that determine the observed line shapes. The proposed models discuss intramolecular excitation, vibronic coupling, lifting of degeneracy due to lowered symmetries in a crystal, or Davydov splitting (see e.g. [39] and references there). However, none of the approaches is able to explain the experimental spectra quantitatively, and even the qualitative agreement between experiment and theory is often unsatisfactory. Knupfer [39] and Wojdyla [40] therefore proposed the inclusion of charge transfer excitations which can explain the splitting of the Q-band in CuPc very well. The inclusion of different charged species to explain the experimental optical data of MnPc was also discussed before [41]. Here, we will extend this idea to all Pcs.

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In figure 4(c), we show our DFT results of ε₂ for the different Pcs. The bandgap of standard DFT calculations is systematically too small [42, 43], and therefore the calculated optical spectra are rigidly shifted by ∼0.5 eV. The overall agreement of the theoretical data with the measurements is rather good. The most apparent difference, however, is the lack of the Q-band splitting for Co-, Ni-, Cu- and ZnPc in the calculations.

To understand this major difference we estimated the size of all previously proposed effects on free molecules. We found for example that breaking the symmetry of the molecules (Davydov splitting) cannot explain the magnitude and the direction (i.e. shifts to higher instead of lower energies) of the Q-band splitting of the Pcs. Following the idea of charged species [41] we performed calculations on doped molecules. Our results are shown in an exemplary manner for ZnPc in figure 6 and are similar for all investigated Pcs. With reduction of the electron count, we see a broadening and splitting of the Q-band. At the same time a shift of ∼0.3 eV to lower energies can be observed, which is a perfect match to experiment. Combining spectra from doped and undoped ZnPc with different weights can easily reproduce the experimental lineshape. Thus we conclude that the main effect of the observed Q-band splitting has its origin in the existence of charged species. The question about the exact origin of the charged species needs to be addressed in further investigations. It is possible that the charged species arise during the excitation experiment, that thin-film defects lead to intrinsic doping of the material or that the observed charged species arise from the coupling of charge transfer states with Frenkel excitons, analogous to the situation found in perylene pigments [44].

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As discussed earlier (see equation (5)) the optical spectra allow for an estimation of average tilt angles by using the in-plane A_in and out-of-plane A_out components of the extinction coefficient k. For H₂Pc, CoPc, NiPc and CuPc, A_out is significantly larger than A_in as shown in figure 4. This indicates a standing configuration of the molecules on the substrate (molecular plane close to perpendicular to the substrate plane). For MnPc, FePc, and ZnPc, A_out is only slightly larger than A_in, which points to a slight preference for crystallites with standing molecules, i.e. average molecular tilt angles between 60° and 75°.

In the case of the investigated TMPPs A_in is higher than A_out. The estimated average tilt angles of porphyrins range between 35° and 47°. Figure 2(b) shows a higher ratio between the A_in and A_out of H₂TMPP compared with H₂TPP. This indicates, together with the crystallochromy, that the presence of methoxy sidegroups seems to introduce a better ordering of the molecules. The surface roughness values are obtained as stated in the experimental section. In the case of Pcs they range from (2.5 ± 1.0)% of the total film thickness for H₂Pc and MnPc to (9.5 ± 1.5)% for CuPc. For PP films, the roughness is in the range of 1–5% of the film thickness.

For the Pcs the spectral dominance of the Q-band is even more pronounced in the Voigt constant Q than in the extinction coefficient k. This is a bit surprising because both Q-band (∼2 eV) and B-band (∼3.5 eV) are supposed to absorb into the e_g LUMO of the metal Pcs (or into two states arising from the LUMO splitting, b_2g and b_3g, in the symmetry reduced H₂Pc [45]). Transitions from or into degenerated orbitals are responsible for features in magneto-optical spectra [11, 46]. The degeneracy of the excited state is lifted in the magnetic field due to the Zeeman effect [11]. The magnitude of the Q-band feature therefore depends on the orientation of the xy-degenerate band with respect to the incident light. If the molecular plane is parallel to the light polarization, the light can most effectively couple with the dipoles. Therefore in the polar MOKE geometry (analogous to the Faraday geometry) a lying molecule exhibits the largest magneto-optical response and the magnitude of the Voigt constant will scale with the cosine of the molecular tilt angle. This explains the larger magnitude of the signal of the porphyrins which were already found by ellipsometry to align more parallel to the substrate.

Figures 9(a) and (b) summarize the results of the DFT calculations on single molecules of Pcs and TMPPs, yielding to the same values of total spins for identical central atoms. The chemical environment of the central atom is the same in both cases and all differences between TMPPs and Pcs can be accounted for by perturbations resulting from the substituents. This makes these two classes perfect for analyzing the influence of different substituents. However, the molecular spin appears to have either no, or only a small, influence on the value of Q, as the comparison between figure 4 and the single molecule spin-states in figure 9(a) shows. A definite statement regarding this is difficult as the influence of the tilt angles seems to be too large.

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For the presented molecules the calculated Voigt constant dispersion was very similar between the samples of different film thickness. Therefore only one representative spectrum of the real and the imaginary part for each material is shown in figures 7 and 8.

The trend of increasing magnitude with decreasing tilt angles found for the ellipsometry data does not match the magnitude variations of the Voigt constant of the Pcs (figure 7). If solely the effect of the molecular orientation on the magnitude of the Voigt constant is considered, the magnitude should be smallest for CuPc and largest for FePc, which is not the case. Another effect influencing the magnitude of the Voigt constant is inter-molecular interaction in the film. This can induce slight deviations from the single molecule symmetry, which can be the reason for the smaller Q-band feature of MnPc in the Voigt constant. In the literature, hints for the symmetry reduction in the case of MnPc are given, although they were previously dismissed as computational artifacts [11, 47]. Another reason for the small Voigt constants of MnPc and FePc is developed in [48], namely that multiple electronic configurations are competing in a very sensitive way with regard to the chemical environment.

The effect of the center atom(s) on the Q- and B-band of TMPP is essential for the understanding of Porphyrinoid spectra. For MePPs with D_4h symmetry it was found [33], that there is a linear correlation of electronegativity and transition energy of the Q-band. As the electronegativity drops the transition energy increases. This correlation results from inductive and conjugative effects. The inductive effect increases the orbital energies of e_g and a_2u if the electronegativity of the center atom(s) drops. The conjugative effect, on the other hand, lowers the energy of only a_2u if electronegativity is lowered, as a_1u and e_g have nodal points on the central atom site.

Now we will discuss the application of the perimeter model to describe the π→π^∗-transitions of TMPPs and Pcs. Following the algorithm of the theoretical section, we have to check for aromaticity and determine the electron numbers. There are 16 atoms in the circumference of the TMPPs (hydrogen is always omitted) and therefore an adequate perimeter model would be based on the (16)-annulene di-anion, [C₁₆ H₁₆]²⁻. For TMPPs we obtain the same result neglecting the substituents, which can be done as they do not interfere with the π-system. The chosen anion has then in total 18 electrons in the π-system, which agrees with values obtained for TPP in literature [16, 19, 38]. As only the main π-system is of interest the values for TPP and TMPP coincide.

The same procedure gives 36 atoms for Pc and therefore the PM should be based either on the (36)-annulene di-anion, [C₃₆ H₃₆]²⁻, or on the (36)-annulene di-cation, [C₃₆ H₃₆]²⁺. Michl [16] has shown that the total number of electrons in the π-system should be 34. Hence the latter choice is the best approximation, although a di-cation of Pc may not be existent in nature. Using Gouterman's 4-orbital model [36], the perimeter model has to be applied to only four different WFs of correct symmetries with energies near the Fermi level.

This gives, according to figure 10(b), the following absolute values of the OARN for the investigated porphyrinoids [21]: Ψ₁, Ψ₂ have k = 4 and Φ₁, Φ₂ have k = 5 for the TMPP. For the Pc it is k = 8 and k = 9, respectively. The sign has to be chosen by following experimental findings, since the WFs, like in figure 10(b), are linear combinations of positive as well as negative OARNs.

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As we can see in figure 5 the B-band is a very strong transition for TMPPs at higher energies than the Q-band. Thus the B-band has to be a transition between molecular WFs of the same sign, i.e. ± 4→ ± 5. This is the only choice possible, which is dipole allowed according to the selection rule Δk = ±1. The Q-band will be weakly forbidden as it will correspond to transitions ± 4→ ∓ 5, yielding Δk = ±9. The distorted character of the porphyrinoids with respect to the perimeter model is obvious as the Q-band does not vanish completely. The assumption of dipole transitions is not crucial here, because even quadrupole moments are not sufficient for Δk = ±9. A symmetry classification and interpretation of DFT data confirmed that the stronger transition at higher energies is of a_1u→e_g character for the TMPPs. Thus the a_2u→e_g electronic transition will be weakly symmetry forbidden for TMPPs (Q-band) and Pcs (B-band).

In figure 5 one can see that this holds for all TMPPs investigated. For Pcs (figure 4) however, the forbidden transitions seem to be allowed nearly completely. If the perimeter model would hold strictly, then all molecular orbitals belonging to the same OARNs would be degenerated, see figure 10(a). As discussed the final states of the main electronic transitions of Q- and B-bands are the same and of equal OARN. Since a_1u and a_2u also share the same OARN, they would have to be degenerated for Pcs as well as for TMPPs. In fact the degeneracy is always lifted because the porphyrinoids can be represented only by perturbed perimeters. Therefore we can take the energy difference of both bands as an indicator of the validity of the perimeter model. Since figures 4(b) and 5(b) show clearly that the energy difference is much larger for the Pcs (e.g. NiPc with ∼2.25 eV; NiTMPP with ∼0.5 eV), the perimeter model is assumed to hold less strictly. In other words, the selection rules based on the OARNs are weakened and intensities of the Q-band should be more alike for the different Pcs.