Two models are employed to classify excitons — the small 1. Phase Transition from Dielectric radius Frenkel exciton model, and the large radius Wannier– to Conducting State Mott exciton model [1]. The charge-transfer exciton (CTE) Consider the CTEs on a single D-A interface. We occupies an intermediate place in the classification based on assume that they are aligned normal to the interface plane, the exciton internal structure [2]. The lowest energy CTE resulting in mutual repulsion. For example, if the static usually extends over two nearest neighbor molecules. In a CTE dipole moment is equal to 20 D and the distance CTE, the electron is localized on the acceptor and the hole, between them is 5 (the lattice constant at the interface), the on the donor. Such localization in organic crystals is usually repulsion energy is near 1 eV. If the distance between CTEs stable because the energy of electron-hole attraction is large increases to 10, the repulsion energy decreases to 0.1 eV.

compared to the corresponding widths of the conduction It is important that these energies are of the order of their and valence bands and due to the strong tendency of the separation B from the lowest conduction band (B < 0.5eV, CTE to undergo self-trapping [3]. Due to the separation see [8]). Thus, at high CTE concentrations we can expect of the electron and the hole in a CTE, the static dipole that repulsion energy populates the higher energy states with moment created by positive and negative ions can assume free carriers, thus producing photoconductivity even at very low temperature [5c]. In [5c] the photoconductivity was values as large as 10-25D. This feature determines the considered under the assumption that the time required for most characteristic properties of the CTE. For example, a phase transition to the conducting state is smaller than the CTEs contribute to the large second order nonlinear the CTE life time. Such a phase transition was obtained by polarizability 2 due to their large dipole moment [4]. As it minimizing the total energy of the CTEs and dissociated was recently shown [5], the same feature can be responsible excitations (free carriers). Consider for simplicity a 2D for a new type of photovoltaic effect in asymmetric D-A array of self trapped CTEs at T = 0. The energy of CTEs superlattices [5a], for unusual intensity dependences of (concentration n1) and the energy of dissociated e–h pairs nonlinear polarizabilities of D-A superlattices [5b], and also (concentration n2), can be calculated by assuming that total for phase transitions to conducting states in the system number of excitations, determined by the optical pumping of CTEs [5c]. In all of these cases, it was assumed that CTEs intensity, is constant: n1 + n2 = n. The energy of the CTE at D-A interfaces between alternating layers of donors and array is therefore E1 = n1+Eint, where Eint is the total acceptops are the lowest energy electronic excited states.

repulsion energy. This energy can be estimated using the These states are usually populated after lattice relaxation average distance between CTEs,. In the case of dipoles p from higher energy Frenkel-type electronic or vibronic states.

ApIn this work, we investigate the transition from a dielectric V =, (1) to a conducting state in a system of CTEs at a D-A interface (see also [6]). The realistic possibility to consider where A is a geometric constant depending on the CTE such organic crystalline structures appeared recently due to distribution in the interface plane. For example, for a square the progress in the development of the organic molecular lattice A 10. Since the CTE concentration by definition beam deposition and other related techniques [7]. This is n1 = 1/2, the electrostatic energy of the interaction opens a wide range of possibilities for creating new types between the dipole moments is: Eint = Vn1/2 = Ap2n5/2/2.

of ordered organic multilayer structures including ordered We can approximate the energy of the dissociated pairs as interfaces. E2 =(+B)n2, where the kinetic energy of the free carriers 782 V.M. Agranovich, S.A. Kiselev, Z.G. Soos, S.R. Forrest CTE to disappear. First, recombination occurs because of the finite lifetime of the CTE,. The second mechanism is dissociation. The CTE exciton dissociates when, due to the dipole-dipole interaction, the energy of the particular exciton exceeds some threshold. If there are n1 CTEs occupying the D-A interface, the electrostatic energy of the jth exciton in the electric field of the other excitons surrounding this site is npVi =, (j = i). (4) ri j j=The ith CT exciton dissociates when the repulsion energy, Vi, is larger than the energy B. The electrostatic potential energy of the exciton strongly increases when few CTEs occupy the nearest neighbor lattice sites. If this occurs, one or more CTEs will dissociate. Such a mechanism should Figure 1. The number of the charge-transfer excitons (n1) at the donor-acceptor interface and the number of the dissociated excitons result in correlations between exciton positions, and ordering (n2) as the functions of the total number of excitations [according of the system of non-mobile CT excitons can be expected.

to the simplified analytical model, Ed. (4)].

Such a spatial ordering suggests the existence of a critical pump light intensity above which there is an onset of photoconductivity. Thus, just above this threshold, we expect an onset of cold photoconductivity. In simulations, we neglect has been neglected (due to the self-trapping and narrow the process of recombination of free carriers which can result electronic bands, see above). Near the threshold, where the in the creation of CTEs. Near the threshold, where the consentration n2 n1 and also n1 1, we can neglect concentration of free carriers is small, the contribution of the interaction of the free carriers with the CTEs. The total this process to the number of CTEs will be small and can be energy of the system then can be written as neglected. However, even at higher concentration, the effect of free carrier recombination can be reduced by applying Ap2(n - n2)5/E = E1(n1)+E2(n2) =n+ + Bn2. (2) the electric field along the interface. This field will separate electrons and holes and thus will create the photocurrent Minimizing the above expression with respect to n2 gives which has to be measured. Computer simulations were performed for a two-dimensional square lattice containing 2/4B 600 600 sites. Under continuous pumping of the sample n2 = n -. (3) 5Apwith a constant intensity, the CTEs are generated in the process described above. In order to avoid the influence of It is clear from Eq. 3 that n2 is positive at boundary conditions, we simulate the evolution of only the n > ncr =[4B/(5Ap2)]2/3 (see Fig. 1). The appearance of central part of the lattice. This square, the central sublattice, free carriers at n > ncr is considered to be a phase transition consists of 200 200 D-A sites, N = 40.000. Next, we from the dielectric to conducting state. This transition replicate the central sublattice by adding 8 square sublattices corresponds to photoconductivity at low temperature (i. e., surrounding the central one. That is, the exciton positions to cold photoconductivity) and is due to long range dipolecalculated for the central 200 200 sites square lattice dipole interactions between CTEs. In this consideration, we is reflected via a mirror symmetry operation to the other neglected the randomness in the CTE distribution and did surrounding 8 squares. To simulate the time evolution, we not consider the establishment of steady state in ensembles, run the system through equally spaced time steps separated which is dependent on the pump intensity and the CTE by the interval t. The value of t = /50 is chosen lifetime. These effects are explicitly taken into account in to be much shorter than the CTE lifetime. We start the our computer simulations [6].

simulations when there are no CTEs at the interface. Under the influence of the pumping the excitons begin to appear.

2. Model for Numerical Simulations After the time, the number of CT excitons occupying the lattice reaches the steady state value. From this time and Results on, the necessary statistical information is collected. The The D-A sites are arranged is a square lattice. The D-A time evolution of the system is simulated as follows. At interface is uniformly irradiated with a time-independent every time step a few CTEs (depending on the pumping source of intensity, I. Only one CTE can be generated at intensity, I) are created at randomly choosed positions at any site. As long as the CT exciton is generated it will stay the central sublattice. Then we go over the central sublattice at the lattice site and it cannot move to other D-A sites sites and check every D-A molecule. With some probability because of self-trapping. There are two mechanisms for the the exciton at this site can recombine, as explained above. It Физика твердого тела, 1999, том 41, вып. Phase Transition to the Conducting State in a System of Charge-Transfer Excitons... the same, the concentration of CTEs which corresponds to intersection of linear S2 asymptote with the n1 curve.

In Fig. 2, the value of n1 is approximately 200 and thus the corresponding critical dimensionless concentration Ccr = 200/40000 = 0.5%. The curve in Fig. 1 corresponds to the value M = Ba3/p2 = 0.01. It follows from the analytical theory that for the same M the critical concentration Ccr = (4Ba3/5Ap2)2/3 = (4M/5A)2/3 = 0.85%.

Thus, a random CTE distribution decreases the critical concentration for the transition to conducting state. This effect could be expected, because, for random distribution, in contrast to analytical model of ordered CTEs, the small distances between CTEs are allowed even at low CTEs concentration. In both approaches the critical concentration strongly depends on the values of B, p and a. For example, Figure 2. The steady state number of the charge-transfer excitons for B = 0.2eV, p = 20D, and a =5, corresponding — (n1) occupying the donor-acceptor interface and the number of to M = 0.1, the analytical model yields Ccr = 4%, while the dissociated pairs production (S2) as the functions of the pump the computer simulations Ccr = 2.5%. For M = 0.05, intensity.

Ccr = 2.6% (analytical approach) or 1.5% (computer simulations), and so on. Thus, for random CTE distribution at the D-A interface, the critical concentration is almost twice as small as analytical model predicts for ordered also can dissociate if its electrostatic energy is high enough.

CTEs with infinite lifetime. The dissociation prevents the The rules for these events to happen at one particular D-A creation of clusters of the closely placed CT excitons and, site are: 1. If the site is empty, the charge-transfer exciton can especially, it prohibits CTEs from occupying adjacent sites be created with the probability Pc = It/N. 2. If a chargeat the D-A interface. It cuts off the high energy tail of transfer exciton already occupies this site, it can recombinate the CTEs energy distribution function. The peak of energy with the probability Pr = 1 - exp(-t/ ). Next, during distribution increases (more details see in [6]) with the the same time step, we calculate the energy of every CTE number of CTEs and so does the width of the distribution.

in the electrostatic field produced by the dipole moments of It is interesting to note that the position of the peak at the all other excitons. The energy of the ith CT exciton can be repulsion energy distribution (corresponding to the energy found using the Eq. (4). If this energy is greater than the of highest probability) varies with the steady state number dissociation threshold, B, the CT exciton dissociates. Finally, of CTEs approximately in the way which the theoretical we recalculate the energies of the CT excitons that remain at the D-A interface.

All results reported below are collected after steady state is achieved. Fig. 2 shows the dependence of the number of CTEs (n1) on the value S which is the product of generation intensity of the CTEs I and the CTE’s lifetime : S = I.

The steady state number of dissociated pairs is determined by their own lifetime, but we do not estimate here the concentration of carriers and conductivity. Nevertheless, on the Fig. 2 we plotted the value S2 which is equal to the number of dissociations which take place at given S in steady state during time. We find a qualitative agreement with the analytical theory in that the CTEs populate the D-A interface only until some utmost concentration be achieved. Further increase of the pump S results mainly in the dissociation of CTEs into electron-hole pairs. When the number of the CTEs reaches the saturation density, the number of dissociations S2 increases linearly with S. It is interesting to compare the critical concentration of CTEs derived from analytical model (see Section 2) with the results of computer simulations. Following qualitatively the results of analytical model (Fig. 1), we can take as a critical concentration of CTEs the concentration corresponding to Figure 3. Position of the energy distribution peak as a function of saturation of CTEs at the interface or, what is nearly the number of the CT excitons, n1.

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