Charge Carrier Mobility in Phthalocyanines: Experiment and Quantum Chemical Calculations

The main goal of this chapter is to show how and why quantum chemistry modeling can /should be applied on class of organic materials with relatively high carrier mobility – phthalocyanines (H2Pc, NiPc and NiPc(SO3Na)x) . It will be shown how Density Functional Theory (DFT) can be used to calculate/model main parameters that influence the group of material properties that are crucial from practical point of view.

motivation for creating such elements has been two-fold: first, to create nanoscale laboratories to explore physics in a new way, and second, to develop novel devices with significant applications. The architecture of molecular-scale electronic devices can be designed starting from molecular segments whose properties have been known from experiment and/or suitable theoretical models. The field of ''Molecular Electronics'' has been opened by the seminal proposition of A. Aviram and M. Ratner in 1974 to build a diode from a single molecule [1][2]. Since then, it took almost 20 years before the first molecular diode was experimentally realized. By now, molecular electronics is a broad field of research world-wide [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
Some examples of molecular nanostructures that might be used as switching units, memories, logic elements and devices embodying a negative resistance have recently been demonstrated (e.g., [1][2][3][4][5]). A pioneering construction of a molecular switch was based on the electron tunneling principle [1]. An electron travels along a 'molecular wire' (e.g., a conjugated polymer chain) containing a finite series of periodic potential walls. The tunnel switch is 'on' if the transmission coefficient of the electron is close to unity, i.e., if the electron energy matches pseudostationary energy levels of the walls, and can be turned off by either changing a barrier height or the depth of a potential well, which can be controlled by the dipole moment of polymer side groups.
As we have already mentioned the charge transport in molecular electronic materials is a very complex process which can be affected by many physical and chemical parameters; the knowledge of its nature is crucial for the development and optimization of molecular-based devices. Each step forward in understanding and controlling the charge transport is extremely important for the practical applications of molecular systems in electronics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].
One of the reasons for the difficulties in charge-transport phenomena/conditions description is the lack of a well-understood mechanism of charge transport in organic materials. The charge-carrier transport in molecular systems is a very complicated and comprehensive event affected by many parameters. In the early days, band theory was applied to predict the charge-carrier mobility in organic materials. However, it was repeatedly pointed out by several authors [18][19][20][21] that this approach is not suitable. Comparing organic and inorganic semiconductors, the materials from the latter class are usually less disordered and have molecular sites closer to one another. For disordered materials, the hopping mechanism seems to be more appropriate for the description of the charge-carrier behaviour. In this approach, the charge carrier is localised on the molecular site and jumps to the other site by overcoming some energy barrier. However, for materials possessing a preferred direction of charge-carrier transport like phtalocyanines we need a combination of different approaches for the description of the charge-carrier motion along the preferred paths and for hopping among them.
Molecular materials are generally not very good conductors. Main limitation follows from low charge carrier "on-chain" mobility which mostly using the microwave photoconductivity was found to be 10 -5 m 2 V -1 s -1 and 10 -4 m 2 V -1 s -1 , for and -conjugated molecular materials, respectively. The mobility is limited by polaron formation and by the dispersion of transfer integrals among the monomer units the wire. Electrical current through a single molecule is influenced by charge tunnelling -the Fowler-Nordheim model seems to be a good approximation for the description of charge transport. The presence of dipolar species results in the mobility decrease due to the increase of the transfer integral dispersion. Polar group chemically attached to the molecular wire, can cause the orbital localization. The charge transport in 3D samples can be described by the theory of disordered polarons which postulates that the activation energy of the charge carrier mobility is composed of contribution both from the dynamic disorder, i.e. the polaronic barrier, and from the static disorder, i.e. the variation of the energy of transport states as a result of the environment. The main contribution to the polaron binding energy results from molecular deformation; electron-phonon term makes for 20 % only. Dipolar additives make the distribution of hopping states broadened and new localized states for charge carriers are formed; it results in the reduction of charge mobility [17][18][19].
Experimentally measured conductivity is a macroscopic phenomenon and reflects on-stack mobility plus intra-stack hopping mobility. Intra-stack transport mainly contributes to the whole resistivity -i.e. the main charge-transport obstruction is between stacks.
From the charge carrier mobility point of view phtalocyanines seem to be promising materials -in devices with vacuum evaporated phthalocyanine thin films, the values reached from 10 -5 to 10 -4 cm 2 V -1 s -1 , when phthalocyanine was evaporated on hot substrates the value increased up to 10 -2 cm 2 V -1 s -1 [15][16][17][18][19][20][21][22]. Preliminary results obtained on sulphonated phthalocyanines, i.e., materials containing both electronic and electrolytic segments, seem to be very promising. The charge carrier mobility determined from the dependences of the source-drain current vs. the source-drain voltage of the OFET was surprisingly high. In the case of NiPc(SO 3 Na) 4 , the field-effect mobility was 0.02 cm 2 V -1 s -1 [12].
Phthalocyanines can be organized in columns at a supramolecular level, giving rise to conducting properties. The cofacial stacking of metallophthalocyanines enables electron delocalization along the main axis of the column through π-π orbital overlapping. Metallophthalocyanines generally crystallize in an inclined stacked insulating arrangements called or -modifications that do not allow an appropriate overlap of π-orbitals and hence no formation of a conduction band. Only in few cases stacked arrangements are found, being the most representative the nonplanar cone-shaped phthalocyaninatolead (II) (PbPc) in its monoclinic modification.
In the last few years, phthalocyanines ( Fig. 1) are being intensively studied as targets for optical switching and limiting devices, organic field effect transistors, sensors, light-emitting devices, low band gap molecular solar cells, optical information recording media, photosensitizers for photodynamic therapy, and nonlinear optical materials, among others. Phthalocyanines will burst also in a very near future into the nanotechnology field.
Phthalocyanine is an intensely blue-green coloured macrocyclic compound. Phthalocyanines form coordination complexes with most elements of the periodic table. Phthalocyanines are structurally related to other macrocyclic pigments, especially the porphyrins. Four pyrrolelike subunits are linked to form a 16-membered ring. The pyrrole-like rings within H 2 Pc are closely related to isoindole. Both porphyrins and phthalocyanines function as planar tetradentate dianionic ligands that bind metals through four inwardly projecting nitrogen centers. Such complexes are formally derivatives of Pc 2-, the conjugate base of H 2 Pc. Many derivatives of the parent phthalocyanine are known, where either carbon atoms of the macrocycle are exchanged for nitrogen atoms or where the hydrogen atoms of the ring are substituted by functional groups like halogens, hydroxy, amino, alkyl, aryl, thiol, alkoxy, nitro, etc.
Due to phthalocyanine stack formation and the presence of ionic groups charge transport consists of an electronic feature through the stack, charge hopping among the stacks and ionic type of the transport. In order to model the physical properties of H 2 Pc layer, quantum chemical calculations on H 2 Pc dimers and tetramers were performed at DFT level. The  and  types of polymorph of H 2 Pc dimer structures were optimized using several density functionals. The optimizations lead to geometry with almost parallel orientation of H 2 Pc planes. The optimised conformer structures are depicted in Fig. 2, the structural parameters calculated for different functional and basis sets are listed in Table 1. Calculated geometry of H 2 Pc dimer reasonably well represents the local part of the experimental structure of phthalocyanine films [15][16][17][18][19] Calculated stabilization energies listed in Table 1 are typical for  - stacking interactions and indicate that the polymorph  is more stable than the -modification; the calculated difference of stabilization energies is 1.5 kcal/mol for the functional MPW1B95. Table 1 shows that the withdrawal of electron from polymorph leads to geometry with more closely lying interplanar arrangements of monomeric subunits characterized by the distances a = 3.44 Å and b = 3.67 Å. Optimized geometry of anionic form is characterized by similar interplane distance 3.45 Å and larger parameter b = 3.80 Å. Cationic form of  polymorph is characterized by the distances a = 3.29 Å and b = 5.26 Å, anionic form of this polymorph by distances a = 3.33 Å and b = 3.68 Å. All following single point calculations were done at optimized geometries.
On the basis of the dimer calculations, the geometry optimization of H 2 Pc tetramer was done by MPW1B95/6-31G* calculations on both polymorphs. The middle part of the optimized tetramer structure interprets well the H 2 Pc crystal structure (the calculated interplane distance is 3.39 Å [15]).
The optical spectroscopy was used for the characterization of the phthalocyanine layer. Figure 3 shows the experimental spectrum in the region of the phtalocyanine Q band. The comparison of characteristic features of this spectrum with the previously measured spectra  At room temperature, the charge carrier transport can be described by a hopping within the Marcus theory. Subsequently, the rate expression of a self-exchange process is provided by the expression [ where  is the reduced Planck constant, k is the Boltzmann constant, T is the temperature an + is the reorganisation energy and t + is the electronic coupling matrix element (the measure of charge transport probability -electron transfer integral) for the charge transfer.
In order to estimate the electronic coupling between individual monomeric units, electron transfer integrals were calculated for and crystallographic modifications of the phthalocyanine dimer in HF approximation.
For the configuration with doubly occupied HOMO on the first monomer and singly occupied HOMO of the second one, calculation gives electron transfer integral (for otimized geometry MPW1B95/6-31G*) values for non-relaxed cation radical 11.7 meV and 5.7 meV (hole transfer) for the and modification, respectively. It is supposed that in the course of electron or hole hopping, the atoms can relax in order to reach the minimum of energy at the potential surface. Transfer integral for optimized cation radical crystal form with relaxed atoms (UMPW1B95/6-31G*) is 18.7 meV. The calculations indicate that electron mobility is smaller than hole mobility because optimized geometry of anionic form is characterized by larger interplane parameter b (see Tab H RP is the interaction energy between reactant and product states, S RP is the overlap between the reactant and product states and H RR is electronic energy of the reaktant state andH PP is electronic energy of the product state. All these terms were obtained via the direct coupling of localised monomer orbitals. On the molecular level, three factors, the electronic coupling (transfer integral t + ) between the individual parts of the molecule, the reorganisation energy λ + during charge transport and the effective length of hole transfer L, are usually considered to be important for charge transport in organic materials. The reorganization energy + consists of the sum of 1 + and 2 + , λ + = λ 1 + + λ 2 + , where the deformation energy of the system was calculated as the difference between the vertical and cationic state: λ 1 + = E + (Q N ) -E + (Q + ) and λ 2 + = E + (Q + ) -E + (Q N ). Here, E + (Q N ) is the total electronic energy of the cationic state in the neutral geometry, E + (Q + ) is the total energy of the cationic state in the cationic state geometry, E N (Q + ) is the total energy of the neutral state in the cationic state geometry and E N (Q N ) is the total energy of the neutral state in the neutral geometry. λ 1 + is frequently called as deformation energy of the system.
This makes it possible to evaluate the drift mobility of the charge carriers using the Einstein relation = eD/kT. It should also be noted that the calculated charge mobility mentioned here represents the zero electric field approximation value.
To explain the differences between the charge carrier mobility in sulphonated and nonsulphonated Ni phthalocyanines (NiPc), quantum chemical calculations were performed on the DFT (density functional theory) level for H 2 Pc, NiPc and NiPc(SO 3 Na) x (x = 1, 2) dimers and their cationic and anionic forms. Quantum chemical modeling was found to be very useful -we were able to see states of various molecular systems from new and comprehensive perspectives.
The optimized structure of the [Ni(Pc(SO 3 Na) 2 )] 2 is depicted in Fig.4 4 and approaching to the experimental value of 1.8 eV [13]. Fig. 9 shows how the variation of functional influences the energies of frontier orbitals.
The rate constant k ET of the electron transfer between molecular orbitals was expressed -see (1). The electronic coupling between the individual monomeric units V RP , the electrontransfer integral, can be calculated by several procedures [8][9][10][11][12][13]. Koopman's approximation, where the electronic coupling is estimated as half of the corresponding orbital energies, was used for the matrix-element estimation [14][15][16][17][18]. This method has already been used in the case of interacting metal-containing dimmers [22].   Fig. 10. The electron-transfer integral value varies strongly with the intermolecular separation, nevertheless the ratio between V RP calculated for SO 3 Na substituted and unsubstituted systems is close to this ratio obtained from equilibrium intermolecular separation.
Both water molecules surrounding or the application of an electric field can cause the dissociation of the ionic Na-SO 3 bond. In order to explain the effect of the Na + dissociation, calculations were performed on the model dimeric systems, where Na + ions were stepwise removed from the NiPc(SO 3 Na) 2 dimer . The stabilization energy of the system with one Na + removed (13.2 kcal mol -1 ) is even larger than that of [NiPc(SO 3 Na) 2 ] 2 . From Table 2, it can be seen that [Ni 2 Pc 2 (SO 3 Na) 3 (SO 3 )] -is the most stable and thus the best organized of all the structures investigated. The density of the states plot depicted in Fig. 11 shows how the withdrawal of one Na + influences the electronic structure. Projected Densities of States reported in this chapter were broadened using discrete molecular orbital levels -the broadening parameter used in our PDOS calculations (the full width at half maximum) was 0.3 eV.  In comparison with the unperturbed system, the set of HOMOs is no longer separated from the lower-lying occupied orbitals and contains contributions from the SO 3 -group. It can be supposed that after Na + ion dissociation the whole layer contains mobile Na + and fixed [NiPc(SO 3 Na)(SO 3 )] -ions surrounded by water molecules. In this case, the highest molecular orbitals, which contain contributions from the SO 3 -group and are close to the lower-lying occupied orbitals, allow the whole system to contain electronic states that are more extended and therefore more conductive in character.
Due to the withdrawal of the remaining Na + ions, the stabilization energies decrease. In the case of NiPc(SO 3 Na) 3.3 the charge carrier mobility (0.02 cm 2 V -1 s -1 ) is lower than for NiPc(SO 3 Na) 1.5 (1.08 cm 2 V -1 s -1 ) -see Tab. 3.
From calculating spatial models it was figured out that due to the sterical communication hindrance caused by third and fourth SO 3 Na groups the NiPc(SO 3 Na) 3.3 dimers are less stable. This fact results in reduced organized layer structure with reduced charge mobility.
Using DFT calculations, several aspects that influence the charge-carrier mobility in the case of the materials were compared: groups, it can be assumed that layers with (SO 3 Na) groups are organized more regularly -resulting in better - interaction and consequently better charge carrier mobility than in the case of nonsubstituted ones. The withdrawal of one Na + ion leads to a further stabilization of the structure and thus a higher probability of charge-carrier transport. The dissociations of more than one Na + ion strongly destabilize the layer structure. The probability of charge transfer through sulphonated Ni-phthalocyanine dimers with totally removed Na + ions should, therefore, be lower than in the case of standard sulphonated Ni phthalocyanines dimers. 3. The system is strongly affected by an external electric field and water molecules in the vicinity, which can cause the dissociation of the ionic Na-SO 3 bond. After the Na-SO 3 bond's dissociation, the whole layer contains mobile Na + and fixed [Ni 2 Pc 2 (SO 3 Na) 3 (SO 3 )]anions which allows the mobile holes to delocalize and increase the probability of hole transfer through the layer structure. We found that the highest molecular orbitals in [Ni 2 Pc 2 (SO 3 Na) 3 (SO 3 )] -contain new electronic contributions to the density of the states from the SO 3 -group, which should lead to an increase in the hole mobility and thus conductivity.
Other aspects which can increase charge carrier mobility in of the OFET structures based on sulphonated phthalocyanine derivatives should be mentioned: 1. Better resistivity to oxygen and thus the stability of the system in atmospheric air. 2. The interface SiO 2 /[NiPc(SO 3 Na) 2 ] x is between two hydrophilic groups and thus adheres materials enabling the formation of a molecular layer on the SiO 2 surface with proper and regular molecular orientation which increases the conductivity .
The objective of this chapter was to present results of DFT studies of class of organic materials with relatively high carrier mobility -phthalocyanines. It was shown how DFT can be used to calculate/model some parameters that influence charge carrier mobility on essentially level [18]. The calculations indicate that electron mobility in H 2 Pc is smaller than hole mobility because optimized geometry of anionic form is characterized by larger interplane parameter.
In the case of [NiPc(SO 3 Na)(SO 3 )] -, the contribution from the SO 3 anions to the whole HOMO orbital system enables the mobile holes to be more delocalized; thus the change charge transport (hopping) mechanism becomes more ballistic (less scattered and more effective). The withdrawal of one Na + ion leads to a further stabilization of the structure and thus a higher probability of charge-carrier transport.
 Larger stabilization energies of the systems with (SO 3 Na) groups -layers with (SO 3 Na) groups are organized more regularly -resulting in better π-π interaction and consequently better charge carrier mobility than in the case of nonsubstituted ones.  The withdrawal of one Na+ ion leads to a further stabilization of the structure and thus a higher probability of charge-carrier transport.  The dissociations of more than one Na+ ion strongly destabilize the layer structure. The probability of charge transfer through sulphonated Ni-phthalocyanine dimers with totally removed Na+ ions is lower than in the case of standard sulphonated Ni phthalocyanines dimers.