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Quantum confinement in linear molecular chains with strong mixing of Frenkel and charge-transfer excitons
21 January 2002
Physics Letters A 293 (2002) 83–92 www.elsevier.com/locate/pla
Quantum confinement in linear molecular chains with strong mixing of Frenkel and charge-transfer excitons Karin Schmidt Institut für Angewandte Photophysik, Technische Universität Dresden, 01062 Dresden, Germany Received 16 November 2001; accepted 3 December 2001 Communicated by V.M. Agranovich
Abstract The excitonic spectrum in quasi-one-dimensional crystals with strong orbital overlap between neighboring molecules is discussed. In such crystals, the energy difference between the lowest Frenkel exciton and charge-transfer (CT) exciton becomes small and their strong mixing determines the nature of the lowest energy states. In particular, the spectrum of surface states is studied in detail, i.e., states localized at the chain ends, which can appear simultaneously with excitonic bulk states. The contribution of both kinds of states to the absorption spectrum of molecular chains of arbitrary length is investigated. The surface states found allow the determination of a characteristic quantum length with which exciton quantum confinement effects as a consequence of the strong mixing can be expected. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Widely used optoelectronic devices, such as laser diodes and modulators [1], exploit effects which only occur in low-dimensional structures. In these artificial structures the properties of the materials are tuned apart from its typical bulk features by quantum confinement. While the investigation of nanostructures opens the way for new devices of tailored functionality, it primarily provides deeper insight in the nature of the states of the given system. Mechanisms which are responsible for finite size effects incorporate the basic properties of the system. Therefore, the formulation of conditions under which quantum confinement takes place differ from material to material.
E-mail address: [email protected] (K. Schmidt).
In inorganic semiconductors, which typically form covalently bound crystals, quantum confinement of excitonic states means that structures smaller than the exciton Bohr radius aB show properties which significantly differ from the behavior of bulk structures [2]. On the other hand, in the van der Waals crystals formed of small organic molecules, the lowest excited states consist of small radius excitons. The spectra of the most prominent and best investigated organic crystals, the polyacenes, are analyzed in terms of Frenkel excitons. Often, the approximation of nearest neighbor interaction serves as a standard tool, e.g., for Jaggregates [3]. However, within systems idealized in this way it is impossible to find a quantum length in the order of a few lattice constants. Only the consideration of additional effects allow the identification of a characteristic system size. For example, the inclusion of exciton–phonon coupling [4] leads to the self-trap-
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 8 4 1 - 6
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ping of excitons. The localization length of such selftrapped excitons serves as a quantum length and can adopt the role of aB in inorganic semiconductors. This Letter is devoted to a class of quasi-onedimensional materials which can be considered as a the first step towards a link between standard organic, such as anthracene and pentacene, and covalently bound crystals. In such quasi-1D crystals, the distance between molecules in one direction is much smaller than in the others. Typically, these conditions are met in crystals of planar aromatic molecules which are arranged in face-to-face stacks providing a strong overlap of the molecular wavefunctions. In a large number of perylene derivatives, this situation is realized, among them 3,4,9,10-perylenetetracarboxylic dianhydride (PTCDA) and N ,N -dimethyl-3,4,9,10perylenetetracarboximide (MePTCDI). The molecular planes within the one-dimensional stacks are separated by 3.40 Å in MePTCDI [5] and 3.37 Å in PTCDA [6,7]. This separation is small in comparison with other lattice constants and also small in comparison with the size of the molecules. As a result, the interactions of the π -electron systems within the stacks is very strong, compared with the interaction in the other directions. Besides Frenkel excitons, also charge-transfer (CT) excitons should be taken into account for the analysis. A CT configuration consists of a pair of charge carriers localized at different molecules which are nearest neighbors in the stack. Due to the strong interaction in the stack, one can expect that the qualitative difference between Frenkel and CT excitations becomes smaller, their energies approach each other and their strong mixing determines the nature of the lowest energy states [8,9]. During the last years intensive investigations were performed in order to determine the nature of its lowest energy electronic states. Experiments such as absorption [9], and, for example, electroabsorption [10] can reveal the character of the lowest excitations in relatively large systems. Systematic studies of thin films [11– 13] give additional information about small systems in particular. In order to investigate quantum confinement, the general behavior of the finite chain has to be understood in the framework of the model description. Here, the finite size effects of such a system with strong mixing of Frenkel and CT excitations are studied in the electronic absorption spectrum.
The discussion is restricted to ideal finite chains, in which a strong coupling of Frenkel and CT exciton states is allowed, but exciton–phonon coupling as well as any diagonal and off-diagonal disorder effects (see Section 2) are not regarded. The considered system is kept as simple as possible in order to establish a clear distinction from the influence other effects, such as a possible change of the excitation energies at the outermost molecules due to the absence of one nearest neighbor [14]. The Letter is divided into two parts: Section 2 provides a brief review of the description of an finite chain of arbitrary size [15] in comparison to an infinitely extended chain [9]. The considerations presented here will closely follow the detailed discussion given these previous papers. Thereby, the emphasis lies on the properties shared by the infinite and finite systems. In Section 3, this comparison will explain the behavior of the states dependent on the system size. It will be shown that one type of solution which is localized at the ends of the chain, the so called surface states, allow the introduction of a characteristic quantum length which implies the appearance of quantum confinement. The model bases on a parametric description allowing very general statements. With the appropriate choice of the parameters, a large class of materials can be addressed, e.g., the limiting case of pure Frenkel exciton description is included. Nevertheless, the theory is applied to MePTCDI and PTCDA in order to compare the final results with available experimental observations.
2. Mixing of Frenkel and charge-transfer excitons in a finite molecular chain For the discussion of the excitonic spectrum in a one-dimensional molecular crystal (with one molecule per unit cell) the following Hamiltonian has been used: Hˆ = Hˆ F + Hˆ C + Hˆ FC , EF Bˆ n+ Bˆ n + Mnn Bˆ n+ Bˆ n , Hˆ F = n
Hˆ C =
nσ
nn + ˆ ECT Cˆ nσ Cnσ ,
K. Schmidt / Physics Letters A 293 (2002) 83–92
Hˆ FC =
e Bˆ n+ Cˆ n,+1 + Bˆ n+ Cˆ n,−1 n
chain an eigenstate takes the form
+ h Bˆ n+ Cˆ n+1,−1 + Bˆ n+ Cˆ n−1,+1
+ h.c.
85
|ψ = (1)
The operator Bˆ n+ (Bˆ n ) describes the creation (annihilation) of a molecular excitation at lattice site n, with n = 1, . . . , N , and N being the number of molecules. At each site only one electronically excited molecular state is assumed. Then EF is the on-site energy of a Frenkel exciton and Mnn is the hopping integral for molecular excitation transfer from molecule n to molecule n . In the summation in Hˆ F the terms with n = n are omitted. The Hamiltonian Hˆ F describes the Frenkel excitons in Heitler–London approximation. Besides the Frenkel excitons, nearest neighbor charge-transfer excitons are additionally included. A localized CT exciton with the hole at lattice site n and the electron at lattice site n + σ (σ = −1, +1) is + (C ˆ nσ ). Simcreated (annihilated) by the operator Cˆ nσ ilar to Frenkel excitons, only the vibrational ground state is considered for the CT excitons with ECT as their on-site energy. Hopping of CT states will not be considered. The mixing of Frenkel and CT excitons arises due to the last part Hˆ FC of the Hamiltonian. Here, the nonlocal transformation of a CT state into a Frenkel state at the lattice site of either hole or electron is allowed. The relevant transfer integrals e ( h ) can be visualized as transfer of an electron (hole) from the excited molecule n to its nearest neighbor. Without introducing a new parameter, the theory of the infinite chain [9] can be extended to finite chains. The treatment of the finite [15] in comparison with the infinite system differs only in the choice of the boundary conditions. In idealized chains, which shall be considered here, the on-site energies EF , ECT for each molecule and the coupling matrix elements M, e , h between the molecules are the same, even at the outermost molecules. Only the CT excitations from the outermost molecules to the one missing neighbor are omitted. In general, each eigenstate is a superposition of Frenkel excitations and CT excitations, described by the coefficients un and vn+1 , vn−1 , respectively. With the help of the creation operators of (1), in the finite
N
+ N−1 un Bˆ n+ 0 + vn,+1 Cˆ n,+1 0
n=1
+
n=1
N
+ vn,−1 Cˆ n,−1 0.
(2)
n=2
Here, v1,−1 and vN,+1 are absent, because the corresponding CT states are missing in a finite chain. The contribution of each eigenstate |ψ i to the optical absorption spectrum of a molecular chain is provided by the oscillator strength F i , which is primarily determined by the length of the transition dipole moment pi squared: Fi =
2me e2 h2 ¯
2 E i pi .
(3)
In order to evaluate pi , it is assumed that the length of the chain is small in comparison with the optical wavelength corresponding to the energy of electronic excitation. In this case, the effects of retardations (polaritonic effects) can be neglected. Additionally, the change of the optical electric field along the chain can be neglected in the calculation of the exciton transition dipole moment. In this approximation, the total transition dipole moment from the ground |0 to the excited crystal state |ψ i is pi = pF
N
uin + p+CT
n=1
N−1 n=1
i vn,+1 + p−CT
N n=2
i vn,−1 ,
(4)
where pF and p+CT , p−CT are the molecular transition dipole moment and transition dipole moments for ±CT states, respectively. The intrinsic CT transition dipole moment p±CT is (at least) an order of magnitude smaller than pF [9]. As a consequence, the absolute value of pi is mainly given by the Frenkel part
N
uin pF ,
n=1
even if the Frenkel coefficients uin are rather small. Thus, the CT contribution is omitted in the following.
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K. Schmidt / Physics Letters A 293 (2002) 83–92
The quantity N 2 (pi )2 i i f ∼ F 2= un (p )
N −1 . (6) 2 In the case of periodic boundary conditions (infinite chain) κ is purely imaginary (Re(κ) = 0) and directly corresponds to the quasi-momentum k using the assignment k = Im(κ) (k real). By applying the boundary conditions of the finite system, κ is not a continuous parameter anymore, and the imaginary parts k := Im(κ) are discrete values (cf. [15, Eq. (28)]). Two cases have to be carefully distinguished: For purely imaginary solutions of κ (Re(κ) = 0, κ = ik), the function un becomes proportional to cos nk. The boundary conditions determine the set of allowed κ, but the relation determining E(k) (cf. [15, Eq. (14)] with [9, Eq. (25)]) exactly takes the form known from the infinite system: (E − EF ) + 2M cos k (E − ECT ) − 2 e2 + h2 + 2 e h cos k = 0. (7) un = A cosh nκ,
(5)
n=1
provides a measure of the oscillator strength of a state |ψ i in relation to the molecular excitation with the transition dipole moment pF . The classification of solutions (2) is carried out in comparison to the infinite chain: In case of periodic boundary conditions [9] three different excitonic bands are formed, derived from the local excitations EF and the two possible CT excitations ECT . Each state |ψ is characterized by a quantum number k, the quasi-momentum. The dispersion (energetic alignment and curvature) E(k) of the bands as well as the composition of the states in terms of Frenkel and CT-type excitations can be easily studied as a function of the parameters used in the Hamiltonian. Due to the inversion symmetry of the system, one band is exclusively composed of CT excitations and shows no dispersion in the Brillouin zone [9]. The two remaining bands consist of mixed Frenkel–CT states and show a dispersion which is determined by the parameters EF , ECT , M, e , h [9]. In the following the solutions of the finite chain are discussed. Since the system has a center of inversion, all eigenstates (2) are either symmetric or antisymmetric. It follows immediately from relation (4) that the transition dipole moment pi differs from zero only for symmetric solutions. For convenience, below the molecules are enumerated with n = 0, ±1, ±2, . . ., ±(N − 1)/2 (N odd), starting from the center of the chain. For even N , all results of the analysis are qualitatively similar, and for N 1 they exactly coincide [15]. By further analysis of the structure of (2) it can be shown that each coefficient vn,±1 can be expressed in terms of un (see [15]); except the case of pure CT states. There, the Frenkel excitations do not contribute at all (un = 0 for all n). Such CT states are degenerate at the energy ECT and clearly correspond to the nonmixing CT band of the infinite chain. For absorption, states with f i = 0 (Eq. (5)), i.e., symmetrical states mixed of Frenkel and CT excitations are relevant. Below, only those states are discussed, each characterized by its Frenkel coefficients un . Using a complex parameter κ, with κ = Re(κ) + ik, the coefficients un can be expressed as
n = 0, ±1, ±2, . . ., ±
Therefore, the states are expected to be aligned on the bands of the corresponding infinite chain. Their exact matching motivates to call the solutions bulk states. As second case, one has to consider states, which are parameterized with a κ with Re(κ) = 0. This type of solution does not correspond to a solution already known from the infinite chain. Below, such type of states will be called surface states, because they are similar to surface states of quasi-particles in 3D crystals. This identification bases on their characteristic features which appear in long molecular chains (N 1): The real part of κ is responsible for an exponential decay e|n| Re(κ) from the outermost molecules into the inner chain: 1 un ∼ cosh nκ ≈ ei|n|k e|n| Re(κ) , 2 N −1 N −3 |n| = (8) , ,.... 2 2 Without loss out generality, Re(κ) was set positive in Eq. (8). In most cases, a nonvanishing real part of κ is accompanied with an imaginary part k = 0 or k = π , causing either a monotonous, or an oscillating decay:
k = 0, nonoscillating, 1|n| |n| Re(κ) un ∼ e k = π, oscillating. (−1)|n| (9) In both cases the decay is exponential which localizes the states at the outermost molecules. Since the CT
K. Schmidt / Physics Letters A 293 (2002) 83–92
Fig. 1. Surface states of a long chain (N = 41) are visualized with the distribution of the coefficients un as a function of the lattice site n. The left curve () represents the surface state with the nonoscillating exponential decay, the right curve () shows the state with oscillating decay. For comparison, also the bulk state with lowest k is given. The exact values for the un were obtained after full calculation with the parameters M = h = 0.2 eV, e = 0, EF = ECT .
coefficients vn±1 are directly connected to the Frenkel coefficients un (cf. [15, Eqs. (4), (5)] ), the localization can be equivalently observed by using the vn±1 . Since the states have to obey inversion symmetry, i.e., they are either symmetric or antisymmetric, they are delocalized over the whole chain. Nevertheless, the main contribution of the un is concentrated at the ends of the chain (|n| = (N − 1)/2, (N − 3)/2, . . .). In very long chains (N 1), each state can be represented by two tails localized at the chain end, which have no overlap in the inner chain. This behavior is summarized in Fig. 1. Due to the symmetry of these states, only the left or the right tail, respectively, of the solutions are displayed. For comparison, also the bulk state with lowest k = Im(κ) given. This localization at the ends of the chain finally gives reason for calling these states surface states. They are fully analogous to Tamm states for electrons [16]. It is important to note that such states arise even in ideal chains in which the on-site energies at the boundaries are not adjusted to the surrounding. Independent from the type of decay, each surface state is located energetically between the bands of the infinite chain. Both the symmetric and the antisymmetric state become degenerated in long chains, since both involve the same number of molecules. As visualized in Fig. 1, the right-hand side of the antisymmetric state is the mirror image of the r.h.s. of the correspond-
87
Fig. 2. States in a finite MePTCDI chain for different chain lengths N . Left: Energy as a function of k = Im(κ). The solid lines represent the bands of the corresponding infinite chain. States with pure CT character are not shown, but are aligned on the nondisperging CT band at ECT = 2.15 eV. Right: Oscillator strength represented by the relative transition dipole moment squared according to Eq. (5). For comparison, also the intensities of the dipole-allowed band states are shown, represented by |ui (k = 0)|2 (scaled with factor 500, symbol ui (k) according to ui (k) from [9]).
ing symmetric state. If surface states appear at all, or what type of surface states can be expected, is exclusively determined by the parameter set [15]. As an example, the situation in the crystals of MePTCDI and PTCDA is displayed in Figs. 2 and 3. Although the materials look very similar, as might be concluded from the applied parameters, the nature of their surface states is completely different. For the crystals MePTCDI and PTCDA a full set of parameters which are required for Hamiltonian (1) is available. The parameters, collected in Table 1, have been obtained from experiments [18] to explain, e.g., the polarization dependence of the crystal absorption spectra. The resulting surface state energies are preliminary, because the accuracy of the fitting parameters is limited (see [9,18]). Fig. 2 summarizes the situation in MePTCDI. From the three excitonic bands of the infinite chain, two bands consist of mixed Frenkel and CT excitons and therefore show a dispersion (Fig. 2, left). In an infinite chain, only states with k = 0 can be probed by light. Only the two states with k = 0 which belong to a band with Frenkel exciton admixture, according to the assumption made for the oscillator strength (Eq. (5)), possess oscillator strength (Fig. 2, right). As it can be nicely seen, the bulk states calculated for finite chains of different length N exactly correspond to a
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K. Schmidt / Physics Letters A 293 (2002) 83–92
Fig. 3. States in a finite PTCDA chain for different chain lengths N . Left: Energy as a function of k = Im(κ). The solid lines represent the bands of the corresponding infinite chain. States with pure CT character are not shown, but are aligned on the nondisperging CT band at ECT = 2.27 eV. Right: Oscillator strength represented by the relative transition dipole moment squared according to Eq. (5). For comparison, also the intensities of the dipole-allowed band states are shown, represented by |ui (k = 0)|2 (scaled with factor 500, symbol ui (k) according to ui (k) from [9]).
Table 1 Fitting parameters which were obtained in [18]; the values of e , h , and M listed here already contain the vibronic overlap factor. Due a new evaluation, the parameters h and e differs from the parameters given in [15]
MePTCDI PTCDA
EF (eV)
ECT (eV)
S0
e (eV)
h (eV)
M (eV)
2.23 2.34
2.15 2.27
0.643 0.622
−0.047 −0.006
−0.017 −0.054
0.044 0.041
band state. Within each band, the main contribution to the absorption spectrum is provided by the particular bulk state which wavevector k is closest to zero. Since this smallest k becomes larger with decreasing chain length N , its associated state ‘travels’ into the band and its energy is shifted by an amount determined by the dispersion of the underlying band. In MePTCDI, the two surface states found, a pair of a symmetric and an antisymmetric state, are of the oscillating type. Hence, its wavevector k = Im(κ) = π corresponds to the boundary of the Brillouin zone (Fig. 2). In the limit of long chains, this surface states appear degenerate at ES = 2.14 eV. Independent from the length N of the chains discussed in Fig. 2, the oscillator strength of the symmetric surface state is constant. This observation has to be attributed to the strong localization at the outermost molecules. Only the two outermost molecules
on the left- and on the right-hand side, respectively, possess a nonvanishing un and therefore the surface states of a chain of N = 5 molecules are equivalent to the surface states in a chain with N = 17 molecules (see Fig. 2). The consideration of PTCDA (Fig. 3) yield a bulk state behavior which is similar to MePTCDI. In this case, however, the surface states which occur belong to the nonoscillating type. As a consequence, its associated wavevector is exactly zero. Since its energetic position at ES = 2.24 eV as well as its oscillator strength are not affected by a chain length varied between 5 and 21 molecules, the localization is again very strong. Hence, the situation displayed in Fig. 3 corresponds to the limit of long chains, even for N = 5. Remarkably, the oscillator strength of the symmetric surface state in PTCDA is considerably larger in comparison with its counterpart in MePTCDI. As the visualization in Fig. 1 clearly shows, this originates directly from the different nature of the surface states, because in the oscillating surface state (MePTCDI) the contributions un partially compensate each other.
3. Quantum confinement In this section, the dependence of the energy and oscillator strength of both the bulk and the surface states on the chain length N are investigated. The behavior, found in particular for small systems, can be attributed to quantum confinement as soon as the system provides an intrinsic length, i.e., a length uniquely defined by the parameter set EF , ECT , M, e , h . The aim of the next paragraphs is to identify such an intrinsic quantum length. The band structure in the first Brillouin zone of the corresponding infinite chain turns out to be a useful tool to study the N -dependence of bulk states. From the expression for the oscillator strength f i (Eq. (5)) it can be expected that the states with a permitted wavevector k closest to k = 0 will carry the largest oscillator strengths (similar to analogous states in molecular chains with Frenkel excitons) which scale almost linearly with the system size N . The larger the chain, the closer the smallest allowed wavevector k gets to zero which is the limit given by the infinite chain. The energy of the states closest to the center of the first Brillouin zone is shifted due to the shift of their
K. Schmidt / Physics Letters A 293 (2002) 83–92
89
Fig. 4. Comparison of the energy and oscillator strength of the surface states and the lower band bulk state with highest oscillator strength as a function of the chain length N . The lower part shows the energetic positions of the surface states (, ) and the bulk state (•) (open symbols represent odd surface states); in the upper part the intensities of the corresponding states are compared. Left: MePTCDI, with the oscillating surface state (); Right: PTCDA, with the nonoscillating surface state (). The oscillator strength of the bulk states scale almost linearly with N , whereas the oscillator strength remains constant down to N = 5. Below N 5 the antisymmetric and the symmetric surface states suspend their degeneracy and the oscillator strength of the latter becomes dependent on N .
wavevectors towards higher k (cf. Fig. 2). The direction and the amount of this shift is determined by the dispersion of the bands. The contribution of the surface states in the absorption spectrum differs from the bulk states. For long chains, the surface states can be represented by tails located at the left- or right-hand side of the chain, respectively (Fig. 1). Only these tails can contribute to the transition dipole moment. Such tails can be characterized by an effective size L := Re(κ)−1 , reflecting how many sites possess a nonvanishing un (Fig. 1). Since (Re κ) is exclusively determined by the parameter set (see previous section), the characteristic quantum length under question can be assigned with the effective size L of the surface state and has to be interpreted as the radius of the mixed Frenkel–CT exciton. As long as chains longer than twice the extension of such a tail are considered, the oscillator strength of the symmetric state will be independent from the actual number of molecules N . For small molecular chains (or stacks) the tails of the surface states from both ends of the chain will overlap. As soon as structures smaller than 2L are considered, quantum confinement effects are observed. Only if the quantum length is too small (less than two lattice constants), a discussion of quantum confinement is not reasonable anymore.
A variety of parameter constellations yield an effective size of the surface states which is sufficiently large. If, for example, EF and ECT are resonant, L can exceed 4 lattice constants. Fig. 1 shows the result of a calculation using parameter chosen as EF = ECT = 0, M = 0.2 eV, h = 0.2 eV, and e = 0, which yield an effective size of 5 lattice constants. At both sides of the chain, at least 7 molecules possess a nonvanishing Frenkel coefficient un . Hence, in chains containing less than ≈12 molecules the oscillator strength of the symmetric surface state exhibits a pronounced dependence on the number of molecules within the chain. It is important to note that quantum confinement not only affects the surface states, but simultaneously forces the bulk states to change their properties. Nevertheless, the quantum length L is determined by means of the surface states which arise due to the absence of CT excitations at the outermost molecules. In Fig. 4 we display the N -dependence of the energies and the oscillator strength of the surface states and the lowest allowed bulk state for N 3, calculated for MePTCDI and PTCDA. For larger N (N 5), the surface states are degenerate and the oscillator strength of the symmetric surface state remains constant. Hence, the surface states are strongly localized and accordingly, the effective size is smaller than two lattice constants for
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K. Schmidt / Physics Letters A 293 (2002) 83–92
both crystals: L ≈ 1.58 for MePTCDI and L ≈ 1.95 for PTCDA, respectively. So far, only a few experiments are available for MePTCDI and PTCDA which investigate the dependence of the absorption on the film thickness. In experiments done by So et al. [11] and Leonhardt et al. [12], very thin films of PTCDA with thicknesses in the order of a few lattice constants were investigated. It was found that the lowest absorption maximum was shifted ≈200 cm−1 to higher energies when the layer thickness was reduced from more than 60 to 3 lattice constants. For MePTCDI, the corresponding experiment was done by Beckers et al. [13] who investigated thin films grown on quartz glass. Similar to PTCDA, the lowest transmission minimum was shifted ≈400 cm−1 to higher energies when the layer thickness was reduced from more than 140 to less than 5 lattice constants. Besides finite size effects, also the film morphology and the particular substrate chosen may alter the absorption spectrum. Hence, several attempts were already made to explain this phenomenon, for PTCDA [12,14] and for MePTCDI [13], respectively, each starting from a completely different point of view. Considering surface states in finite chains due to Frenkel–CT exciton coupling offers another possible point of view [15]. In [14] also surface states were taken into account, but the mechanism presented there is different from what shall be discussed now. Structural investigations of thin films of MePTCDI and PTCDA [9,19] revealed that during the growth the (102) plane of these materials lies almost parallel to the substrate plane. After comparison with the crystal structure [5–7], the molecular planes are aligned within the (102) plane, allowing the quasi-one-dimensional face-to-face formation of the molecules perpendicular to the substrate. Therefore, the film thickness can be directly related to the chain length N . The experimental findings cannot be discussed using quantum confinement in the sense introduced above, simply because the effective size L < 2 found is too small. Alternatively, in the framework of the model the situation can be explained as follows: The low energy part of the absorption spectra consists of the surface state and the lower bulk state which are well separated in energy. At higher thicknesses (N > 5) the bulk state governs the spectrum in the region under question: Its oscillator strength scales almost linearly with N . In contrast, in the surface state
only the outermost molecules contribute to the absorption. Hence its oscillator strength is rather small and remains constant (Fig. 4). Below N ≈ 5, the energy of the nearest lowest bulk state shows a considerable shift, because its corresponding k belongs to the region of the lower band with steepest dispersion (compare Fig. 4 with Fig. 2 for MePTCDI and Fig. 3 for PTCDA). Simultaneously, however, the qualitative difference between bulk and surface state almost vanishes. The tails of the symmetric surface state start to overlap. Thereby, all molecules are allowed to take part in the absorption and consequently, the oscillator strength of the surface states becomes sensitive to N . Such a surface state might provide now the leading contribution to the absorption spectrum. Due to the energetic separation between the bulk states and the surface state the increasing importance of the latter induces a peak shift in the absorption spectrum. In conclusion, the experimentally observed peak shift in the absorption spectrum cannot be attributed to the energy shift of the lower bulk state. Within the model presented in this Letter, this shift corresponds to the energy difference between the nearest lower bulk state in the limit of long chains and the symmetric surface state in a very short chain (N = 3). This is an extension to the estimation given in [15], where the surface state energy was taken in the limit of long chains. For both crystals, the calculation predicts a shift of the lowest peak towards higher energies (see Table 2). The amount of ≈160 cm−1 for MePTCDI and ≈80 cm−1 for PTCDA, respectively, is in the same order of magnitude as the experimentally observed shifts. However, due to the limited accuracy of the fitting parameters taken from [9,18], this qualitative agreement might be fortuitous. To perform a better comparison of experimentally obtained data with predictions using a model as presented here, experiments are required which allow a clear separation of finite size effects from effects originating, e.g., from morphology. Therefore, the combination of absorption measurements with structural characterization incorporating precise determination of the film thickness is of crucial importance. To ensure further that the chain length can be assigned with the film thickness, very smooth films, i.e., films with a narrow thickness distribution, are needed. Therefore, the best way to realize this requirements would be the investigation of highly ordered films grown in a layer-
K. Schmidt / Physics Letters A 293 (2002) 83–92
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Table 2 Comparison of the energies of the symmetric surface state in the shortest chain (N = 3) and the nearest lower bulk state of a large chain (N = 17), and the corresponding oscillator strength ratios of surface and bulk state. The second column from the right shows the resulting shift ∆Esb of the surface state in comparison with the bulk. Experimental data (right column) support the order of the amount as well as the direction of the theoretically predicted shift N =3 MePTCDI PTCDA
N = 17
Es (eV)
(|ps |/|pb |)2
Eb (eV)
(|ps |/|pb |)2
2.13 2.25
1.86 34.36
2.11 2.24
0.15 0.29
by-layer fashion on well defined substrates. This can be archived by the epitaxial growth of ultrathin organic films on surfaces of single crystalline substrates. Such investigations are currently under way in our institute [17].
4. Concluding remarks The eigenstates within a finite molecular chain with strong mixing of Frenkel and CT excitons were studied as function of the system size. Only pure electronic excitations have been considered, whereas the vibronic structure of the Frenkel [9] and the CT excitons as well as exciton–phonon coupling were not taken into account. Additionally, the molecular chain has been treated in an idealized picture: All sites in the chain are treated in the same fashion, except the CT excitations at the outermost molecules to the missing neighbors were omitted. In ideal chains which lowest energy excited states are described exclusively by Frenkel excitons, all solutions are correlated to a band state of the corresponding infinite chain. Now, the admixture of CT excitons implies the possibility of surface state formation. These surface states provide the key to quantum confinement even in idealized systems. In long chains, the surface states have almost no influence the absorption spectrum, because they are localized at the ends of the chain. In contrast, they gain importance for small chains. The typical size of these states, which is determined completely by the applied parameter set, provides an intrinsic quantum length being characteristic for the given system. In ideal chains, this quantum length corresponds to the radius of a coupled Frenkel–CT exciton. If this radius is sufficiently large, quantum confinement effects take
∆Esb (eV)
Exp. (eV)
0.02 0.01
0.05 [13] 0.03 [11,12]
place as soon as the systems becomes smaller than this effective size. Even if the localization of the surface states is very strong, the surface state in small chains involves all molecules and behaves similar to bulk states. Then, the oscillator strength of the surface state can contribute significantly to the absorption spectrum. As shown above, this situation was realized for MePTCDI and PTCDA, where the effective surface state size was found to be rather small (L < 2). While reducing the system size, a surface state close to a bulk state with nonvanishing oscillator strength dominated the latter, causing a peak shift within the absorption spectrum. In order to provide a reasonable description of the experimental data, of course, several concepts must be combined. Then, the energies of bulk and surface states will be rearranged and additional surface states might appear. For instance, the exciton phonon coupling has to be incorporated. As another important effect, one should take into account that the situation of a molecule at the very end of a chain differs from a bulk molecule [20], because it has to adjust to the absence of its nearest neighbor. Furthermore, in real experiments the investigated finite molecular chains are attached to some substrate, e.g., after organic molecular beam deposition [19] or placed in the neighborhood of other molecular substances [10]. Such influences might change the on-site energies of the outermost molecules as well as their coupling to the inner of the chain [20].
Acknowledgements The author thanks V.M. Agranovich for many useful suggestions and discussion of the results. Financial support has been provided by the Deutsche
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Forschungsgemeinschaft (Graduiertenkolleg “Struktur- und Korrelationseffekte in Festkörpern”).
References [1] S.M. Sze, Physics of Semiconductor Devices, 2nd ed., Wiley, New York, 1981. [2] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, Physics and Material Properties, Springer, Berlin, 1996, Chapter 9. [3] H. Fidder, J. Knoester, D.A. Wiersma, J. Chem. Phys. 95 (1991) 7880. [4] V.M. Agranovich, A.M. Kamchatnov, Chem. Phys. 245 (1999) 175. [5] E. Hädicke, F. Graser, Acta Cryst. C 42 (1986) 189. [6] A.J. Lovinger, S.R. Forrest, M.L. Kaplan, P.H. Schmidt, T. Venkatesan, J. Appl. Phys. 55 (1984) 476; A.J. Lovinger, S.R. Forrest, M.L. Kaplan, P.H. Schmidt, T. Venkatesan, Bull. Am. Phys. Soc. 28 (1983) 363. [7] M.L. Kaplan, C.S. Day, A.J. Lovinger, P.H. Schmidt, S.R. Forrest, Full set of crystal structure data, private communication (1994). [8] M.H. Hennessy, Z.G. Soos, R.A. Pascal Jr., A. Girlando, Chem. Phys. 245 (1999) 199.
[9] M. Hoffmann, K. Schmidt, T. Fritz, T. Hasche, V.M. Agranovich, K. Leo, Chem. Phys. 258 (2000) 73. [10] E.I. Haskal, Z. Shen, P.E. Burrows, S.R. Forrest, Phys. Rev. B 51 (1995) 4449. [11] F.F. So, S.R. Forrest, Y.Q. Shi, W.H. Steier, Appl. Phys. Lett. 56 (1990) 674. [12] M. Leonhardt, O. Mager, H. Port, Chem. Phys. Lett. 313 (1999) 24. [13] U. Beckers, O. Stenzel, S. Wilbrandt, U. Falke, C.V. Borczyskowski, J. Phys. Condens. Matter 10 (1998) 1721. [14] V.M. Agranovich, R.D. Atanasov, G.P. Bassani, Chem. Phys. Lett. 199 (1992) 621. [15] V.M. Agranovich, K. Schmidt, K. Leo, Chem. Phys. Lett. 325 (2000) 308; V.M. Agranovich, K. Schmidt, K. Leo, Chem. Phys. Lett. 336 (2001) 536. [16] I. Tamm, Phys. Z. Sowjetunion 1 (1932) 733. [17] R. Nitzsche, Diploma thesis, Technische Universität Dresden (2001). [18] M. Hoffmann, Ph.D. thesis, Technische Universität Dresden (2000). [19] S.R. Forrest, Chem. Rev. 97 (1997) 1793. [20] W. Müller, R. Schiller, W. Nolting, Eur. Phys. J. B 16 (2000) 705.
Physics Letters A 293 (2002) 83–92 www.elsevier.com/locate/pla
Quantum confinement in linear molecular chains with strong mixing of Frenkel and charge-transfer excitons Karin Schmidt Institut für Angewandte Photophysik, Technische Universität Dresden, 01062 Dresden, Germany Received 16 November 2001; accepted 3 December 2001 Communicated by V.M. Agranovich
Abstract The excitonic spectrum in quasi-one-dimensional crystals with strong orbital overlap between neighboring molecules is discussed. In such crystals, the energy difference between the lowest Frenkel exciton and charge-transfer (CT) exciton becomes small and their strong mixing determines the nature of the lowest energy states. In particular, the spectrum of surface states is studied in detail, i.e., states localized at the chain ends, which can appear simultaneously with excitonic bulk states. The contribution of both kinds of states to the absorption spectrum of molecular chains of arbitrary length is investigated. The surface states found allow the determination of a characteristic quantum length with which exciton quantum confinement effects as a consequence of the strong mixing can be expected. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Widely used optoelectronic devices, such as laser diodes and modulators [1], exploit effects which only occur in low-dimensional structures. In these artificial structures the properties of the materials are tuned apart from its typical bulk features by quantum confinement. While the investigation of nanostructures opens the way for new devices of tailored functionality, it primarily provides deeper insight in the nature of the states of the given system. Mechanisms which are responsible for finite size effects incorporate the basic properties of the system. Therefore, the formulation of conditions under which quantum confinement takes place differ from material to material.
E-mail address: [email protected] (K. Schmidt).
In inorganic semiconductors, which typically form covalently bound crystals, quantum confinement of excitonic states means that structures smaller than the exciton Bohr radius aB show properties which significantly differ from the behavior of bulk structures [2]. On the other hand, in the van der Waals crystals formed of small organic molecules, the lowest excited states consist of small radius excitons. The spectra of the most prominent and best investigated organic crystals, the polyacenes, are analyzed in terms of Frenkel excitons. Often, the approximation of nearest neighbor interaction serves as a standard tool, e.g., for Jaggregates [3]. However, within systems idealized in this way it is impossible to find a quantum length in the order of a few lattice constants. Only the consideration of additional effects allow the identification of a characteristic system size. For example, the inclusion of exciton–phonon coupling [4] leads to the self-trap-
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 8 4 1 - 6
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ping of excitons. The localization length of such selftrapped excitons serves as a quantum length and can adopt the role of aB in inorganic semiconductors. This Letter is devoted to a class of quasi-onedimensional materials which can be considered as a the first step towards a link between standard organic, such as anthracene and pentacene, and covalently bound crystals. In such quasi-1D crystals, the distance between molecules in one direction is much smaller than in the others. Typically, these conditions are met in crystals of planar aromatic molecules which are arranged in face-to-face stacks providing a strong overlap of the molecular wavefunctions. In a large number of perylene derivatives, this situation is realized, among them 3,4,9,10-perylenetetracarboxylic dianhydride (PTCDA) and N ,N -dimethyl-3,4,9,10perylenetetracarboximide (MePTCDI). The molecular planes within the one-dimensional stacks are separated by 3.40 Å in MePTCDI [5] and 3.37 Å in PTCDA [6,7]. This separation is small in comparison with other lattice constants and also small in comparison with the size of the molecules. As a result, the interactions of the π -electron systems within the stacks is very strong, compared with the interaction in the other directions. Besides Frenkel excitons, also charge-transfer (CT) excitons should be taken into account for the analysis. A CT configuration consists of a pair of charge carriers localized at different molecules which are nearest neighbors in the stack. Due to the strong interaction in the stack, one can expect that the qualitative difference between Frenkel and CT excitations becomes smaller, their energies approach each other and their strong mixing determines the nature of the lowest energy states [8,9]. During the last years intensive investigations were performed in order to determine the nature of its lowest energy electronic states. Experiments such as absorption [9], and, for example, electroabsorption [10] can reveal the character of the lowest excitations in relatively large systems. Systematic studies of thin films [11– 13] give additional information about small systems in particular. In order to investigate quantum confinement, the general behavior of the finite chain has to be understood in the framework of the model description. Here, the finite size effects of such a system with strong mixing of Frenkel and CT excitations are studied in the electronic absorption spectrum.
The discussion is restricted to ideal finite chains, in which a strong coupling of Frenkel and CT exciton states is allowed, but exciton–phonon coupling as well as any diagonal and off-diagonal disorder effects (see Section 2) are not regarded. The considered system is kept as simple as possible in order to establish a clear distinction from the influence other effects, such as a possible change of the excitation energies at the outermost molecules due to the absence of one nearest neighbor [14]. The Letter is divided into two parts: Section 2 provides a brief review of the description of an finite chain of arbitrary size [15] in comparison to an infinitely extended chain [9]. The considerations presented here will closely follow the detailed discussion given these previous papers. Thereby, the emphasis lies on the properties shared by the infinite and finite systems. In Section 3, this comparison will explain the behavior of the states dependent on the system size. It will be shown that one type of solution which is localized at the ends of the chain, the so called surface states, allow the introduction of a characteristic quantum length which implies the appearance of quantum confinement. The model bases on a parametric description allowing very general statements. With the appropriate choice of the parameters, a large class of materials can be addressed, e.g., the limiting case of pure Frenkel exciton description is included. Nevertheless, the theory is applied to MePTCDI and PTCDA in order to compare the final results with available experimental observations.
2. Mixing of Frenkel and charge-transfer excitons in a finite molecular chain For the discussion of the excitonic spectrum in a one-dimensional molecular crystal (with one molecule per unit cell) the following Hamiltonian has been used: Hˆ = Hˆ F + Hˆ C + Hˆ FC , EF Bˆ n+ Bˆ n + Mnn Bˆ n+ Bˆ n , Hˆ F = n
Hˆ C =
nσ
nn + ˆ ECT Cˆ nσ Cnσ ,
K. Schmidt / Physics Letters A 293 (2002) 83–92
Hˆ FC =
e Bˆ n+ Cˆ n,+1 + Bˆ n+ Cˆ n,−1 n
chain an eigenstate takes the form
+ h Bˆ n+ Cˆ n+1,−1 + Bˆ n+ Cˆ n−1,+1
+ h.c.
85
|ψ = (1)
The operator Bˆ n+ (Bˆ n ) describes the creation (annihilation) of a molecular excitation at lattice site n, with n = 1, . . . , N , and N being the number of molecules. At each site only one electronically excited molecular state is assumed. Then EF is the on-site energy of a Frenkel exciton and Mnn is the hopping integral for molecular excitation transfer from molecule n to molecule n . In the summation in Hˆ F the terms with n = n are omitted. The Hamiltonian Hˆ F describes the Frenkel excitons in Heitler–London approximation. Besides the Frenkel excitons, nearest neighbor charge-transfer excitons are additionally included. A localized CT exciton with the hole at lattice site n and the electron at lattice site n + σ (σ = −1, +1) is + (C ˆ nσ ). Simcreated (annihilated) by the operator Cˆ nσ ilar to Frenkel excitons, only the vibrational ground state is considered for the CT excitons with ECT as their on-site energy. Hopping of CT states will not be considered. The mixing of Frenkel and CT excitons arises due to the last part Hˆ FC of the Hamiltonian. Here, the nonlocal transformation of a CT state into a Frenkel state at the lattice site of either hole or electron is allowed. The relevant transfer integrals e ( h ) can be visualized as transfer of an electron (hole) from the excited molecule n to its nearest neighbor. Without introducing a new parameter, the theory of the infinite chain [9] can be extended to finite chains. The treatment of the finite [15] in comparison with the infinite system differs only in the choice of the boundary conditions. In idealized chains, which shall be considered here, the on-site energies EF , ECT for each molecule and the coupling matrix elements M, e , h between the molecules are the same, even at the outermost molecules. Only the CT excitations from the outermost molecules to the one missing neighbor are omitted. In general, each eigenstate is a superposition of Frenkel excitations and CT excitations, described by the coefficients un and vn+1 , vn−1 , respectively. With the help of the creation operators of (1), in the finite
N
+ N−1 un Bˆ n+ 0 + vn,+1 Cˆ n,+1 0
n=1
+
n=1
N
+ vn,−1 Cˆ n,−1 0.
(2)
n=2
Here, v1,−1 and vN,+1 are absent, because the corresponding CT states are missing in a finite chain. The contribution of each eigenstate |ψ i to the optical absorption spectrum of a molecular chain is provided by the oscillator strength F i , which is primarily determined by the length of the transition dipole moment pi squared: Fi =
2me e2 h2 ¯
2 E i pi .
(3)
In order to evaluate pi , it is assumed that the length of the chain is small in comparison with the optical wavelength corresponding to the energy of electronic excitation. In this case, the effects of retardations (polaritonic effects) can be neglected. Additionally, the change of the optical electric field along the chain can be neglected in the calculation of the exciton transition dipole moment. In this approximation, the total transition dipole moment from the ground |0 to the excited crystal state |ψ i is pi = pF
N
uin + p+CT
n=1
N−1 n=1
i vn,+1 + p−CT
N n=2
i vn,−1 ,
(4)
where pF and p+CT , p−CT are the molecular transition dipole moment and transition dipole moments for ±CT states, respectively. The intrinsic CT transition dipole moment p±CT is (at least) an order of magnitude smaller than pF [9]. As a consequence, the absolute value of pi is mainly given by the Frenkel part
N
uin pF ,
n=1
even if the Frenkel coefficients uin are rather small. Thus, the CT contribution is omitted in the following.
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The quantity N 2 (pi )2 i i f ∼ F 2= un (p )
N −1 . (6) 2 In the case of periodic boundary conditions (infinite chain) κ is purely imaginary (Re(κ) = 0) and directly corresponds to the quasi-momentum k using the assignment k = Im(κ) (k real). By applying the boundary conditions of the finite system, κ is not a continuous parameter anymore, and the imaginary parts k := Im(κ) are discrete values (cf. [15, Eq. (28)]). Two cases have to be carefully distinguished: For purely imaginary solutions of κ (Re(κ) = 0, κ = ik), the function un becomes proportional to cos nk. The boundary conditions determine the set of allowed κ, but the relation determining E(k) (cf. [15, Eq. (14)] with [9, Eq. (25)]) exactly takes the form known from the infinite system: (E − EF ) + 2M cos k (E − ECT ) − 2 e2 + h2 + 2 e h cos k = 0. (7) un = A cosh nκ,
(5)
n=1
provides a measure of the oscillator strength of a state |ψ i in relation to the molecular excitation with the transition dipole moment pF . The classification of solutions (2) is carried out in comparison to the infinite chain: In case of periodic boundary conditions [9] three different excitonic bands are formed, derived from the local excitations EF and the two possible CT excitations ECT . Each state |ψ is characterized by a quantum number k, the quasi-momentum. The dispersion (energetic alignment and curvature) E(k) of the bands as well as the composition of the states in terms of Frenkel and CT-type excitations can be easily studied as a function of the parameters used in the Hamiltonian. Due to the inversion symmetry of the system, one band is exclusively composed of CT excitations and shows no dispersion in the Brillouin zone [9]. The two remaining bands consist of mixed Frenkel–CT states and show a dispersion which is determined by the parameters EF , ECT , M, e , h [9]. In the following the solutions of the finite chain are discussed. Since the system has a center of inversion, all eigenstates (2) are either symmetric or antisymmetric. It follows immediately from relation (4) that the transition dipole moment pi differs from zero only for symmetric solutions. For convenience, below the molecules are enumerated with n = 0, ±1, ±2, . . ., ±(N − 1)/2 (N odd), starting from the center of the chain. For even N , all results of the analysis are qualitatively similar, and for N 1 they exactly coincide [15]. By further analysis of the structure of (2) it can be shown that each coefficient vn,±1 can be expressed in terms of un (see [15]); except the case of pure CT states. There, the Frenkel excitations do not contribute at all (un = 0 for all n). Such CT states are degenerate at the energy ECT and clearly correspond to the nonmixing CT band of the infinite chain. For absorption, states with f i = 0 (Eq. (5)), i.e., symmetrical states mixed of Frenkel and CT excitations are relevant. Below, only those states are discussed, each characterized by its Frenkel coefficients un . Using a complex parameter κ, with κ = Re(κ) + ik, the coefficients un can be expressed as
n = 0, ±1, ±2, . . ., ±
Therefore, the states are expected to be aligned on the bands of the corresponding infinite chain. Their exact matching motivates to call the solutions bulk states. As second case, one has to consider states, which are parameterized with a κ with Re(κ) = 0. This type of solution does not correspond to a solution already known from the infinite chain. Below, such type of states will be called surface states, because they are similar to surface states of quasi-particles in 3D crystals. This identification bases on their characteristic features which appear in long molecular chains (N 1): The real part of κ is responsible for an exponential decay e|n| Re(κ) from the outermost molecules into the inner chain: 1 un ∼ cosh nκ ≈ ei|n|k e|n| Re(κ) , 2 N −1 N −3 |n| = (8) , ,.... 2 2 Without loss out generality, Re(κ) was set positive in Eq. (8). In most cases, a nonvanishing real part of κ is accompanied with an imaginary part k = 0 or k = π , causing either a monotonous, or an oscillating decay:
k = 0, nonoscillating, 1|n| |n| Re(κ) un ∼ e k = π, oscillating. (−1)|n| (9) In both cases the decay is exponential which localizes the states at the outermost molecules. Since the CT
K. Schmidt / Physics Letters A 293 (2002) 83–92
Fig. 1. Surface states of a long chain (N = 41) are visualized with the distribution of the coefficients un as a function of the lattice site n. The left curve () represents the surface state with the nonoscillating exponential decay, the right curve () shows the state with oscillating decay. For comparison, also the bulk state with lowest k is given. The exact values for the un were obtained after full calculation with the parameters M = h = 0.2 eV, e = 0, EF = ECT .
coefficients vn±1 are directly connected to the Frenkel coefficients un (cf. [15, Eqs. (4), (5)] ), the localization can be equivalently observed by using the vn±1 . Since the states have to obey inversion symmetry, i.e., they are either symmetric or antisymmetric, they are delocalized over the whole chain. Nevertheless, the main contribution of the un is concentrated at the ends of the chain (|n| = (N − 1)/2, (N − 3)/2, . . .). In very long chains (N 1), each state can be represented by two tails localized at the chain end, which have no overlap in the inner chain. This behavior is summarized in Fig. 1. Due to the symmetry of these states, only the left or the right tail, respectively, of the solutions are displayed. For comparison, also the bulk state with lowest k = Im(κ) given. This localization at the ends of the chain finally gives reason for calling these states surface states. They are fully analogous to Tamm states for electrons [16]. It is important to note that such states arise even in ideal chains in which the on-site energies at the boundaries are not adjusted to the surrounding. Independent from the type of decay, each surface state is located energetically between the bands of the infinite chain. Both the symmetric and the antisymmetric state become degenerated in long chains, since both involve the same number of molecules. As visualized in Fig. 1, the right-hand side of the antisymmetric state is the mirror image of the r.h.s. of the correspond-
87
Fig. 2. States in a finite MePTCDI chain for different chain lengths N . Left: Energy as a function of k = Im(κ). The solid lines represent the bands of the corresponding infinite chain. States with pure CT character are not shown, but are aligned on the nondisperging CT band at ECT = 2.15 eV. Right: Oscillator strength represented by the relative transition dipole moment squared according to Eq. (5). For comparison, also the intensities of the dipole-allowed band states are shown, represented by |ui (k = 0)|2 (scaled with factor 500, symbol ui (k) according to ui (k) from [9]).
ing symmetric state. If surface states appear at all, or what type of surface states can be expected, is exclusively determined by the parameter set [15]. As an example, the situation in the crystals of MePTCDI and PTCDA is displayed in Figs. 2 and 3. Although the materials look very similar, as might be concluded from the applied parameters, the nature of their surface states is completely different. For the crystals MePTCDI and PTCDA a full set of parameters which are required for Hamiltonian (1) is available. The parameters, collected in Table 1, have been obtained from experiments [18] to explain, e.g., the polarization dependence of the crystal absorption spectra. The resulting surface state energies are preliminary, because the accuracy of the fitting parameters is limited (see [9,18]). Fig. 2 summarizes the situation in MePTCDI. From the three excitonic bands of the infinite chain, two bands consist of mixed Frenkel and CT excitons and therefore show a dispersion (Fig. 2, left). In an infinite chain, only states with k = 0 can be probed by light. Only the two states with k = 0 which belong to a band with Frenkel exciton admixture, according to the assumption made for the oscillator strength (Eq. (5)), possess oscillator strength (Fig. 2, right). As it can be nicely seen, the bulk states calculated for finite chains of different length N exactly correspond to a
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K. Schmidt / Physics Letters A 293 (2002) 83–92
Fig. 3. States in a finite PTCDA chain for different chain lengths N . Left: Energy as a function of k = Im(κ). The solid lines represent the bands of the corresponding infinite chain. States with pure CT character are not shown, but are aligned on the nondisperging CT band at ECT = 2.27 eV. Right: Oscillator strength represented by the relative transition dipole moment squared according to Eq. (5). For comparison, also the intensities of the dipole-allowed band states are shown, represented by |ui (k = 0)|2 (scaled with factor 500, symbol ui (k) according to ui (k) from [9]).
Table 1 Fitting parameters which were obtained in [18]; the values of e , h , and M listed here already contain the vibronic overlap factor. Due a new evaluation, the parameters h and e differs from the parameters given in [15]
MePTCDI PTCDA
EF (eV)
ECT (eV)
S0
e (eV)
h (eV)
M (eV)
2.23 2.34
2.15 2.27
0.643 0.622
−0.047 −0.006
−0.017 −0.054
0.044 0.041
band state. Within each band, the main contribution to the absorption spectrum is provided by the particular bulk state which wavevector k is closest to zero. Since this smallest k becomes larger with decreasing chain length N , its associated state ‘travels’ into the band and its energy is shifted by an amount determined by the dispersion of the underlying band. In MePTCDI, the two surface states found, a pair of a symmetric and an antisymmetric state, are of the oscillating type. Hence, its wavevector k = Im(κ) = π corresponds to the boundary of the Brillouin zone (Fig. 2). In the limit of long chains, this surface states appear degenerate at ES = 2.14 eV. Independent from the length N of the chains discussed in Fig. 2, the oscillator strength of the symmetric surface state is constant. This observation has to be attributed to the strong localization at the outermost molecules. Only the two outermost molecules
on the left- and on the right-hand side, respectively, possess a nonvanishing un and therefore the surface states of a chain of N = 5 molecules are equivalent to the surface states in a chain with N = 17 molecules (see Fig. 2). The consideration of PTCDA (Fig. 3) yield a bulk state behavior which is similar to MePTCDI. In this case, however, the surface states which occur belong to the nonoscillating type. As a consequence, its associated wavevector is exactly zero. Since its energetic position at ES = 2.24 eV as well as its oscillator strength are not affected by a chain length varied between 5 and 21 molecules, the localization is again very strong. Hence, the situation displayed in Fig. 3 corresponds to the limit of long chains, even for N = 5. Remarkably, the oscillator strength of the symmetric surface state in PTCDA is considerably larger in comparison with its counterpart in MePTCDI. As the visualization in Fig. 1 clearly shows, this originates directly from the different nature of the surface states, because in the oscillating surface state (MePTCDI) the contributions un partially compensate each other.
3. Quantum confinement In this section, the dependence of the energy and oscillator strength of both the bulk and the surface states on the chain length N are investigated. The behavior, found in particular for small systems, can be attributed to quantum confinement as soon as the system provides an intrinsic length, i.e., a length uniquely defined by the parameter set EF , ECT , M, e , h . The aim of the next paragraphs is to identify such an intrinsic quantum length. The band structure in the first Brillouin zone of the corresponding infinite chain turns out to be a useful tool to study the N -dependence of bulk states. From the expression for the oscillator strength f i (Eq. (5)) it can be expected that the states with a permitted wavevector k closest to k = 0 will carry the largest oscillator strengths (similar to analogous states in molecular chains with Frenkel excitons) which scale almost linearly with the system size N . The larger the chain, the closer the smallest allowed wavevector k gets to zero which is the limit given by the infinite chain. The energy of the states closest to the center of the first Brillouin zone is shifted due to the shift of their
K. Schmidt / Physics Letters A 293 (2002) 83–92
89
Fig. 4. Comparison of the energy and oscillator strength of the surface states and the lower band bulk state with highest oscillator strength as a function of the chain length N . The lower part shows the energetic positions of the surface states (, ) and the bulk state (•) (open symbols represent odd surface states); in the upper part the intensities of the corresponding states are compared. Left: MePTCDI, with the oscillating surface state (); Right: PTCDA, with the nonoscillating surface state (). The oscillator strength of the bulk states scale almost linearly with N , whereas the oscillator strength remains constant down to N = 5. Below N 5 the antisymmetric and the symmetric surface states suspend their degeneracy and the oscillator strength of the latter becomes dependent on N .
wavevectors towards higher k (cf. Fig. 2). The direction and the amount of this shift is determined by the dispersion of the bands. The contribution of the surface states in the absorption spectrum differs from the bulk states. For long chains, the surface states can be represented by tails located at the left- or right-hand side of the chain, respectively (Fig. 1). Only these tails can contribute to the transition dipole moment. Such tails can be characterized by an effective size L := Re(κ)−1 , reflecting how many sites possess a nonvanishing un (Fig. 1). Since (Re κ) is exclusively determined by the parameter set (see previous section), the characteristic quantum length under question can be assigned with the effective size L of the surface state and has to be interpreted as the radius of the mixed Frenkel–CT exciton. As long as chains longer than twice the extension of such a tail are considered, the oscillator strength of the symmetric state will be independent from the actual number of molecules N . For small molecular chains (or stacks) the tails of the surface states from both ends of the chain will overlap. As soon as structures smaller than 2L are considered, quantum confinement effects are observed. Only if the quantum length is too small (less than two lattice constants), a discussion of quantum confinement is not reasonable anymore.
A variety of parameter constellations yield an effective size of the surface states which is sufficiently large. If, for example, EF and ECT are resonant, L can exceed 4 lattice constants. Fig. 1 shows the result of a calculation using parameter chosen as EF = ECT = 0, M = 0.2 eV, h = 0.2 eV, and e = 0, which yield an effective size of 5 lattice constants. At both sides of the chain, at least 7 molecules possess a nonvanishing Frenkel coefficient un . Hence, in chains containing less than ≈12 molecules the oscillator strength of the symmetric surface state exhibits a pronounced dependence on the number of molecules within the chain. It is important to note that quantum confinement not only affects the surface states, but simultaneously forces the bulk states to change their properties. Nevertheless, the quantum length L is determined by means of the surface states which arise due to the absence of CT excitations at the outermost molecules. In Fig. 4 we display the N -dependence of the energies and the oscillator strength of the surface states and the lowest allowed bulk state for N 3, calculated for MePTCDI and PTCDA. For larger N (N 5), the surface states are degenerate and the oscillator strength of the symmetric surface state remains constant. Hence, the surface states are strongly localized and accordingly, the effective size is smaller than two lattice constants for
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both crystals: L ≈ 1.58 for MePTCDI and L ≈ 1.95 for PTCDA, respectively. So far, only a few experiments are available for MePTCDI and PTCDA which investigate the dependence of the absorption on the film thickness. In experiments done by So et al. [11] and Leonhardt et al. [12], very thin films of PTCDA with thicknesses in the order of a few lattice constants were investigated. It was found that the lowest absorption maximum was shifted ≈200 cm−1 to higher energies when the layer thickness was reduced from more than 60 to 3 lattice constants. For MePTCDI, the corresponding experiment was done by Beckers et al. [13] who investigated thin films grown on quartz glass. Similar to PTCDA, the lowest transmission minimum was shifted ≈400 cm−1 to higher energies when the layer thickness was reduced from more than 140 to less than 5 lattice constants. Besides finite size effects, also the film morphology and the particular substrate chosen may alter the absorption spectrum. Hence, several attempts were already made to explain this phenomenon, for PTCDA [12,14] and for MePTCDI [13], respectively, each starting from a completely different point of view. Considering surface states in finite chains due to Frenkel–CT exciton coupling offers another possible point of view [15]. In [14] also surface states were taken into account, but the mechanism presented there is different from what shall be discussed now. Structural investigations of thin films of MePTCDI and PTCDA [9,19] revealed that during the growth the (102) plane of these materials lies almost parallel to the substrate plane. After comparison with the crystal structure [5–7], the molecular planes are aligned within the (102) plane, allowing the quasi-one-dimensional face-to-face formation of the molecules perpendicular to the substrate. Therefore, the film thickness can be directly related to the chain length N . The experimental findings cannot be discussed using quantum confinement in the sense introduced above, simply because the effective size L < 2 found is too small. Alternatively, in the framework of the model the situation can be explained as follows: The low energy part of the absorption spectra consists of the surface state and the lower bulk state which are well separated in energy. At higher thicknesses (N > 5) the bulk state governs the spectrum in the region under question: Its oscillator strength scales almost linearly with N . In contrast, in the surface state
only the outermost molecules contribute to the absorption. Hence its oscillator strength is rather small and remains constant (Fig. 4). Below N ≈ 5, the energy of the nearest lowest bulk state shows a considerable shift, because its corresponding k belongs to the region of the lower band with steepest dispersion (compare Fig. 4 with Fig. 2 for MePTCDI and Fig. 3 for PTCDA). Simultaneously, however, the qualitative difference between bulk and surface state almost vanishes. The tails of the symmetric surface state start to overlap. Thereby, all molecules are allowed to take part in the absorption and consequently, the oscillator strength of the surface states becomes sensitive to N . Such a surface state might provide now the leading contribution to the absorption spectrum. Due to the energetic separation between the bulk states and the surface state the increasing importance of the latter induces a peak shift in the absorption spectrum. In conclusion, the experimentally observed peak shift in the absorption spectrum cannot be attributed to the energy shift of the lower bulk state. Within the model presented in this Letter, this shift corresponds to the energy difference between the nearest lower bulk state in the limit of long chains and the symmetric surface state in a very short chain (N = 3). This is an extension to the estimation given in [15], where the surface state energy was taken in the limit of long chains. For both crystals, the calculation predicts a shift of the lowest peak towards higher energies (see Table 2). The amount of ≈160 cm−1 for MePTCDI and ≈80 cm−1 for PTCDA, respectively, is in the same order of magnitude as the experimentally observed shifts. However, due to the limited accuracy of the fitting parameters taken from [9,18], this qualitative agreement might be fortuitous. To perform a better comparison of experimentally obtained data with predictions using a model as presented here, experiments are required which allow a clear separation of finite size effects from effects originating, e.g., from morphology. Therefore, the combination of absorption measurements with structural characterization incorporating precise determination of the film thickness is of crucial importance. To ensure further that the chain length can be assigned with the film thickness, very smooth films, i.e., films with a narrow thickness distribution, are needed. Therefore, the best way to realize this requirements would be the investigation of highly ordered films grown in a layer-
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Table 2 Comparison of the energies of the symmetric surface state in the shortest chain (N = 3) and the nearest lower bulk state of a large chain (N = 17), and the corresponding oscillator strength ratios of surface and bulk state. The second column from the right shows the resulting shift ∆Esb of the surface state in comparison with the bulk. Experimental data (right column) support the order of the amount as well as the direction of the theoretically predicted shift N =3 MePTCDI PTCDA
N = 17
Es (eV)
(|ps |/|pb |)2
Eb (eV)
(|ps |/|pb |)2
2.13 2.25
1.86 34.36
2.11 2.24
0.15 0.29
by-layer fashion on well defined substrates. This can be archived by the epitaxial growth of ultrathin organic films on surfaces of single crystalline substrates. Such investigations are currently under way in our institute [17].
4. Concluding remarks The eigenstates within a finite molecular chain with strong mixing of Frenkel and CT excitons were studied as function of the system size. Only pure electronic excitations have been considered, whereas the vibronic structure of the Frenkel [9] and the CT excitons as well as exciton–phonon coupling were not taken into account. Additionally, the molecular chain has been treated in an idealized picture: All sites in the chain are treated in the same fashion, except the CT excitations at the outermost molecules to the missing neighbors were omitted. In ideal chains which lowest energy excited states are described exclusively by Frenkel excitons, all solutions are correlated to a band state of the corresponding infinite chain. Now, the admixture of CT excitons implies the possibility of surface state formation. These surface states provide the key to quantum confinement even in idealized systems. In long chains, the surface states have almost no influence the absorption spectrum, because they are localized at the ends of the chain. In contrast, they gain importance for small chains. The typical size of these states, which is determined completely by the applied parameter set, provides an intrinsic quantum length being characteristic for the given system. In ideal chains, this quantum length corresponds to the radius of a coupled Frenkel–CT exciton. If this radius is sufficiently large, quantum confinement effects take
∆Esb (eV)
Exp. (eV)
0.02 0.01
0.05 [13] 0.03 [11,12]
place as soon as the systems becomes smaller than this effective size. Even if the localization of the surface states is very strong, the surface state in small chains involves all molecules and behaves similar to bulk states. Then, the oscillator strength of the surface state can contribute significantly to the absorption spectrum. As shown above, this situation was realized for MePTCDI and PTCDA, where the effective surface state size was found to be rather small (L < 2). While reducing the system size, a surface state close to a bulk state with nonvanishing oscillator strength dominated the latter, causing a peak shift within the absorption spectrum. In order to provide a reasonable description of the experimental data, of course, several concepts must be combined. Then, the energies of bulk and surface states will be rearranged and additional surface states might appear. For instance, the exciton phonon coupling has to be incorporated. As another important effect, one should take into account that the situation of a molecule at the very end of a chain differs from a bulk molecule [20], because it has to adjust to the absence of its nearest neighbor. Furthermore, in real experiments the investigated finite molecular chains are attached to some substrate, e.g., after organic molecular beam deposition [19] or placed in the neighborhood of other molecular substances [10]. Such influences might change the on-site energies of the outermost molecules as well as their coupling to the inner of the chain [20].
Acknowledgements The author thanks V.M. Agranovich for many useful suggestions and discussion of the results. Financial support has been provided by the Deutsche
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Forschungsgemeinschaft (Graduiertenkolleg “Struktur- und Korrelationseffekte in Festkörpern”).
References [1] S.M. Sze, Physics of Semiconductor Devices, 2nd ed., Wiley, New York, 1981. [2] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, Physics and Material Properties, Springer, Berlin, 1996, Chapter 9. [3] H. Fidder, J. Knoester, D.A. Wiersma, J. Chem. Phys. 95 (1991) 7880. [4] V.M. Agranovich, A.M. Kamchatnov, Chem. Phys. 245 (1999) 175. [5] E. Hädicke, F. Graser, Acta Cryst. C 42 (1986) 189. [6] A.J. Lovinger, S.R. Forrest, M.L. Kaplan, P.H. Schmidt, T. Venkatesan, J. Appl. Phys. 55 (1984) 476; A.J. Lovinger, S.R. Forrest, M.L. Kaplan, P.H. Schmidt, T. Venkatesan, Bull. Am. Phys. Soc. 28 (1983) 363. [7] M.L. Kaplan, C.S. Day, A.J. Lovinger, P.H. Schmidt, S.R. Forrest, Full set of crystal structure data, private communication (1994). [8] M.H. Hennessy, Z.G. Soos, R.A. Pascal Jr., A. Girlando, Chem. Phys. 245 (1999) 199.
[9] M. Hoffmann, K. Schmidt, T. Fritz, T. Hasche, V.M. Agranovich, K. Leo, Chem. Phys. 258 (2000) 73. [10] E.I. Haskal, Z. Shen, P.E. Burrows, S.R. Forrest, Phys. Rev. B 51 (1995) 4449. [11] F.F. So, S.R. Forrest, Y.Q. Shi, W.H. Steier, Appl. Phys. Lett. 56 (1990) 674. [12] M. Leonhardt, O. Mager, H. Port, Chem. Phys. Lett. 313 (1999) 24. [13] U. Beckers, O. Stenzel, S. Wilbrandt, U. Falke, C.V. Borczyskowski, J. Phys. Condens. Matter 10 (1998) 1721. [14] V.M. Agranovich, R.D. Atanasov, G.P. Bassani, Chem. Phys. Lett. 199 (1992) 621. [15] V.M. Agranovich, K. Schmidt, K. Leo, Chem. Phys. Lett. 325 (2000) 308; V.M. Agranovich, K. Schmidt, K. Leo, Chem. Phys. Lett. 336 (2001) 536. [16] I. Tamm, Phys. Z. Sowjetunion 1 (1932) 733. [17] R. Nitzsche, Diploma thesis, Technische Universität Dresden (2001). [18] M. Hoffmann, Ph.D. thesis, Technische Universität Dresden (2000). [19] S.R. Forrest, Chem. Rev. 97 (1997) 1793. [20] W. Müller, R. Schiller, W. Nolting, Eur. Phys. J. B 16 (2000) 705.