We report the results of the combined experimental and theoretical studies of the low-lying exciton states in crystalline copper phthalocyanine. We derive the eigen energy spectrum for the two lowest intramolecular Frenkel excitons coupled to the intermolecular charge transfer exciton state and compare it with temperature dependent optical absorption spectra measured experimentally, to obtain the parameters of the Frenkel-charge-transfer exciton intermixing. The two Frenkel exciton states are spaced apart by 0.26 eV, and the charge transfer exciton state is 50 meV above the lowest Frenkel exciton. Both Frenkel excitons are strongly mixed with the charge transfer exciton, showing the coupling constant 0.17 eV which agrees with earlier experimental measurements. These results can be used for the proper interpretation of the physical properties of crystalline phthalocyanines.

Over the years, intensive research on organic molecular solids has been motivated by practical as well as fundamental interest.^{1–3} Practical interest is prompted by the need to develop sustainable efficient molecular thin films for applications in modern optoelectronics. Fundamental interest comes from the need to understand the complex nature of intrinsic quasiparticle excitations—electronic, vibronic, excitonic, etc., as well as their intercorrelation effects—to be able to control the physical properties of the entire class of organic molecular solids.

Phthalocyanines are one of the most stable organic materials used in technological applications such as industrial pigments, dyes, gas sensors, and fuel cells.^{4–7} Their semiconducting properties are attractive for optoelectronics applications such as field-effect transistors,^{8} light-emitting diodes,^{9–11} photovoltaic cells,^{11–13} single-molecule devices,^{14} and spintronic devices.^{15} Copper phthalocyanine (CuPc) molecular crystals we discuss in this Letter are used in many optical applications due to their strong absorption in the visible spectral range and high temperature stability. Their low-lying optical excitations have been studied by different techniques,^{16–19} with no consensus reached so far, however, as to what extent intermolecular charge transfer (CT) states affect the excitation signal. The CT exciton state here consists of a pair of opposite sign charge carriers at neighboring sites of the quasi-one-dimensional (quasi-1D) crystal formed by 1D face-to-face stacks of *π*-coupled planar organic molecules whose 1D lattice period is much less than the shortest inter-stack distance. CT excitations are considered to be an important intermediate state in the creation of free carriers by light absorption.^{1–3} Fundamental understanding of their physical nature is important both for applications such as solar energy conversion and for the development of organic optoelectronics in general.

Here we report our results of the combined experimental and theoretical studies of the low-lying exciton states in *α*-Herringbone crystalline CuPc thin films. CuPc (95%) obtained from Sigma-Aldrich was used to grow thin films on glass substrates to a thickness of 100 nm using thermal evaporation at room temperature. CuPc grown at room *T* on glass substrates forms the *α*-Herringbone crystal structures of [100]-orientation.^{7} Our films were ∼50–100 nm crystallites oriented randomly in the substrate plane with all molecules stacked at 25° (*α*-phase) with respect to the substrate normal. Samples were placed in the vacuum cryostat and cooled to *T* in the range 77–295 K. Absorption Spectroscopy was completed with a Varian Cary 50 spectrophotometer, and we show that the Frenkel–CT exciton intermixing model explains unambiguously the spectral features we observe.

For CuPc in the vapor-phase and in solution, the visible range absorption spectra are known to be dominated by a single absorption line and a satellite positioned (depending on particular sample) at about 1.82 eV and 2.03 eV, respectively.^{20–23} Figure 1(a) shows our spectra measured for *α*-Herringbone CuPc molecular crystals. The absorption profile (*Q*-band) is seen to consist of two main features at about 1.8 eV and 2.0 eV, which both exhibit shoulders at 1.7 eV and 2.16 eV, respectively. The entire absorption pattern, when decomposed into the four Gaussians, is seen not only to reduce its peak intensity but also to shrink as *T* increases. Such a drastic difference between the molecular crystal and single CuPc molecule absorption indicates a significant intermolecular coupling and possible Frenkel–CT exciton hybridization (intermixing) to lead to a more complicated absorption behavior after crystallization.^{17,18} Previous theoretical work on organic crystals has indeed predicted that intramolecular Frenkel excitons can couple to CT excitons between adjacent molecules,^{24–27} showing crystal absorption features similar to those we observe for our CuPc thin films—features that most likely represent properties of the periodic 1D lattices of organic molecules in general.

In what follows, we develop an analytical theory for the intra-intermolecular exciton intermixing in periodic 1D chains of organic molecules with two isolated low-lying Frenkel exciton states (typical of CuPc and other transition metal phthalocyanine molecules^{20–23}). We use the exact Hamiltonian diagonalization procedure described in Ref. 26 to obtain the eigen energy spectrum for the two intramolecular Frenkel excitons coupled to the intermolecular CT exciton in the periodic 1D lattice of *N* molecules. The total Hamiltonian of this system can be written in the momentum (*k*) space as follows

Here

is the Hamiltonian for the Frenkel excitons created by the photon absorption on a single molecule, and

describes their hopping between neighboring sites on the 1D periodic molecular chain. The operators $Bkv\u2020$ and *B _{kv}* are the

*k*-space Fourier transforms of the operators $Bnv\u2020$ and

*B*which create and annihilate, respectively, a neutral excitation (Frenkel exciton) of the vibrational level

_{nv}*v*of the

*first*electronically excited molecular state of the molecule at the lattice site

*n*. In our case

*v*= 0, 1 to represent the first and second intramolecular Frenkel excitons, respectively. As the distance between the molecular planes within the 1D stacks of crystallized molecules is much less than the other lattice constants of molecular thin films, the strongest intermolecular interactions occur within the 1D molecular stacks. This justifies the 1D molecular lattice approximation we use. The parameters $\Delta Fv$ and $Lkvv\u2032=\u2211m(\u22600)eikmM0mvv\u2032$ represent the on-site energy of an intramolecular Frenkel exciton of the molecular vibrational level

*v*and its transfer probability amplitude between the vibrational levels

*v*and $v\u2032$ of the molecules on the 1D lattice, respectively. The latter can be evaluated using the nearest neighbor approximation to result in $Lkvv\u2032\u22482Msvsv\u2032\u2009cos\u2009k$, where

*M*is the nearest-neighbor exciton transfer integral and $sv=\u3008\chi n1v|\chi n00\u3009$ is the vibronic overlap factor between the ground state and the vibrational level

*v*of the

*first*electronically excited molecular state of the molecule at the site

*n*. The quantity

*s*is directly related to the Frank-Condon factor $F0v=|\u3008\chi 1v|\chi 00\u3009|2=g2ve\u2212g2/v!\u2009$, where

_{v}*g*is the exciton-phonon coupling constant. In such a formulation, one assumes the strong exciton-phonon coupling (i.e.,

*g*≈ 1), whereby the 1D lattice configurations are neglected in which the Frenkel exciton and an intramolecular optical phonon are located at different lattice sites,

^{26}resulting in

*s*

_{0}≈

*s*

_{1}≈ 0.6 in our case.

The Hamiltonian

describes the nearest-neighbor intermolecular ground-state CT exciton with the on-site energy Δ_{CT}. Here, the operators $C\u0303kg\u200a\u2020\u2009(C\u0303ku\u200a\u2020)$ and $C\u0303kg\u2009(C\u0303ku)$ are the symmetric (antisymmetric) combinations of the *k*-space Fourier transforms of the operators $Cn,\xb11\u200a\u2020$ and $Cn,\xb11$. The latter create and annihilate a localized nearest-neighbor exciton with the hole and the electron located at the lattice sites *n* and *n* ± 1, respectively. Possible CT state hopping as well as excited CT states are neglected. The intermolecular CT exciton can be visualized as a reversible tunnel transfer of an electron (or a hole) from the excited molecule at the lattice site *n* to its nearest neighbor. In the *k*-space, such a process is described by the Frenkel–CT exciton coupling Hamiltonian of the form

where $\u03f5k=\u03f5+2\u2009cos2(k/2)+\u03f5\u22122\u2009sin2(k/2),\u2009\u03f5\xb1=\u03f5e\xb1\u03f5h$, with *ϵ _{e}* and

*ϵ*representing the nearest-neighbor electron and hole tunnel transfer integrals, respectively.

_{h}The total Hamiltonian (1)–(5) is a quadratic form, and so it can be diagonalized exactly to give the eigen energies and new eigen states formed due to the Frenkel–CT exciton coupling and intermixing. In our case of *v* = 0, 1, this can even be done analytically by means of the unitary canonical transformation to new operators

In these new operators, we assume for simplicity that only one Frenkel exciton is excited in the system at a time (the case in most of the practical experiments under low and moderate laser excitation densities), which mixes up with the CT exciton and *not* with another Frenkel exciton. The subscript *v* refers to a particular one Frenkel exciton (either *v* = 0, or *v* = 1) and the subscript *μ* refers to the *two* types of the particles involved in the process of mixing to give the *same* number of the mixed quasiparticle states, accordingly. The unknown mixing coefficients $uv\mu \u2217(k)$ and $cv\u2217(k)$ are constrained to fulfill the standard bosonic commutation relations as follows: $[\xi v\mu (k),\xi v\u2032\mu \u2032\u2020(k\u2032)]=\delta kk\u2032\delta vv\u2032\delta \mu \mu \u2032,\u2009[\xi v\mu (k),\xi v\u2032\mu \u2032(k\u2032)]=0$.

where *E*_{0} is the energy constant, and $\u210f\omega v\mu (k)$ are the eigen energies given by solutions to the operator identity

Plugging the Hamiltonian (1)–(5) into this identity yields the following set of simultaneous algebraic equations

with $Dkvv\u2032=\Delta Fv\u2009\delta vv\u2032+2Msvsv\u2032\u2009cos\u2009k$, in which negligible off-diagonal terms must be dropped for consistency with Eq. (6). This yields four solutions to the set (9) as follows

to represent the eigen energy spectrum of the Frenkel–CT exciton coupled system.

From comparison of Eq. (10) with the measured absorption spectra in Fig. 1(a), one can estimate the relative positioning of the Frenkel and CT exciton states and how strong they are coupled in the *α*-Herringbone CuPc crystal structure. To do so, we set *k* = 0 in Eq. (10) and divide it by Δ_{CT}, to obtain the dimensionless expression

In this equation, $\u210f\omega v(\xb1)=\u210f\omega v(\xb1)(0)=Ev(\xb1),\u2009EF1=D000$ is the lowest Frenkel exciton (*F*1) excitation energy, $\Delta =D011\u2212D000=EF2\u2212EF1$ with *E _{F}*

_{2}being the second lowest Frenkel exciton (

*F*2) excitation energy,

*s*=

*s*

_{0}=

*s*

_{1}= 0.6 as was outlined above, and

*ϵ*=

*ϵ*

_{0}is the constant to represent the intermolecular Frenkel–CT exciton coupling strength. The graphical representation of this equation is shown in Fig. 1(b) to allow one identify those transition energies in the coupled Frenkel–CT exciton system which are related to the four absorption peaks seen in Fig. 1(a). These transition energies are marked as

*x*

_{1},

*x*

_{2},

*x*

_{3}, and

*x*

_{4}both in Figs. 1(a) and 1(b). They can be written in terms of Eqs. (10) and (11) as $x1=\u210f\omega 0(+)\u2212\u210f\omega 0(\u2212),\u2009x2=\u210f\omega 1(+)\u2212\u210f\omega 1(\u2212),\u2009x3=\u210f\omega 1(+)\u2212\u210f\omega 0(+)$, and $x4=\u210f\omega 1(\u2212)\u2212\u210f\omega 0(\u2212)$, where

*x*

_{1}= 2.0 − 1.7 = 0.3 eV,

*x*

_{2}= 2.16 − 1.8 = 0.36 eV,

*x*

_{3}= 2.16 − 2.0 = 0.16 eV, and

*x*

_{4}= 1.8 − 1.7 = 0.1 eV as it can be seen from the data in Fig. 1(a). Additionally, Eqs. (10) and (11) provide the following constraint equations:

Here, the first equation in the first line gives Δ = 0.26 eV. The second one can be used to test the appropriateness of our data analysis, yielding for data in Fig. 1(a) the true equality 0.16 eV– 0.1 eV = 0.36 eV– 0.3 eV just as the model prescribes. The remaining unknowns, *E _{F}*

_{2}, Δ

_{CT}, and

*ϵ*, can be obtained from the set of three independent equations to include the second and third line constraint equations (12) along with one of Eq. (10), e.g., that corresponding to the lowest eigen energy peak at 1.7 eV in Fig. 1(a). This results in

*E*

_{F}_{1}= 1.82 eV,

*E*

_{F}_{2}= 2.08 eV, Δ

_{CT}= 1.88 eV, and

*ϵ*= 0.17 eV. Note that

*E*

_{F}_{1}and

*E*

_{F}_{2}thus obtained represent the lowest intramolecular excitation energies in the CuPc molecule. They are in a good agreement with the CuPc absorption peak positions in the vapor-phase and in solution,

^{20–23}thereby providing another proof of the appropriateness of our analysis and of the model itself. The intermolecular coupling constant

*ϵ*represents the energy needed to transfer a bound excited electron (or a hole) to a neighboring molecular site, and so it can also be interpreted as the

*lower*bound for the electron-hole charge separation energy. The value of

*ϵ*we obtained agrees with the charge separation energy 0.6 ± 0.4 eV reported for CuPc molecular films earlier.

^{16}

A diagram of the Frenkel–CT energy level splitting and intermixing can now be plotted as a function of varied *ϵ* using Eq. (11) [or Eq. (10) taken at *k* = 0] with the parameters obtained. The diagram is shown in Fig. 2. The vertical red dotted line indicates the actual value of *ϵ* obtained. The four energies on the intersection with the red dotted line agree well with the low *T* spectral peak positions in Fig. 1(a). Their respective eigen states represent the coupled Frenkel–CT exciton system. The shrinkage of the absorption pattern with increasing *T*, which can be seen as a contraction of the Gaussian decomposition in Fig. 1(a), can now be understood as being due to an intermolecular coupling decrease as *T* increases—most likely due to the averaging over the phonon vibrations whose role in the lattice increases with *T*. This shifts the vertical red dotted line in Fig. 2 to the left, to shrink the entire energy level pattern just the way Fig. 1(a) shows.

We would also like to comment on the *Q*-band absorption profile Gaussian decomposition we used in our data analysis here. It was first demonstrated theoretically by Toyozawa for bulk crystalline semiconductors^{28} that if the exciton-phonon coupling is weak (*g* ≪ 1) and the exciton effective mass is small (Wannier-type exciton), then the exciton absorption band is of a narrow Lorentzian shape at not too low and not too high temperatures, with the half-value width proportional to *T* because of the *k* = 0 exciton level broadening due to the weak exciton-phonon scattering. If, on the other hand, the exciton effective mass is large (Frenkel-type exciton), or the exciton-phonon coupling is strong (*g* ≳ 1), or the temperature is high, then the exciton absorption band profile is of a broad Gaussian shape with the weakly *T*-dependent (∼*T*^{1/2}) half-value width. This is also valid for the exciton absorption line shape dependence on crystal imperfections if one replaces *T* by the density of lattice imperfections.^{28} Very similar absorption profile peculiarities were also reported for CT-excitons in bulk molecular crystals,^{29} for Wannier excitons trapped in quasi-2D semiconductor quantum dots of varied lateral size,^{30} and for the positronium atom (light “exciton isotope”) in alkali halide crystals.^{31–34} These features can be qualitatively understood from the general Heisenberg uncertainty principle, whereby the decrease in the exciton spatial localization volume—either due to the self-trapping, or due to the trapping by lattice imperfections, or because this is an excitation localized on a molecule in the molecular lattice of an organic crystal like the Frenkel excitons in our case—results in the increase of the exciton translational momentum uncertainty to yield a broad (Gaussian) absorption band profile. Spatial delocalization, on the contrary, generally decreases the exciton momentum uncertainty to result in a relatively narrow (Lorentzian) exciton absorption line shape. The Gaussian decomposition of the *Q*-band absorption profile is appropriate for our analysis here since our model assumes the strong exciton-phonon coupling [*g* ≈ 1, see the discussion following Eq. (3)]. The correctness of this choice is supported by the very weak *T*-dependence of the individual Gaussian peaks in Fig. 1(a), in agreement with the general Gaussian exciton absorption line shape behavior outlined above, no matter whether the homogeneous broadening (phonons), or the inhomogeneous broadening (imperfections), or both of them contribute the most to the *Q*-band absorption profile.

To summarize, we develop the Frenkel–CT exciton intermixing theory to understand the absorption spectra measured for *α*-Herringbone crystalline CuPc films at *T* = 77–295 K. Two low-lying Frenkel exciton states are shown to be strongly mixed with the CT exciton state. Determined are the relative arrangement of their excitation energies and the intermolecular coupling constant. The *T*-dependence of the absorption pattern is explained. These results can be used for the proper interpretation of the physical properties of crystalline phthalocyanines.

Experimental and part of the theoretical component (A.P.) of this work are funded by the UNC-GA Research Opportunity Initiative grant. I.V.B. acknowledges the DOE grant support (DE-SC0007117).