Origin of space-separated charges in photoexcited organic heterojunctions on ultrafast time scales

We present a detailed investigation of ultrafast (subpicosecond) exciton dynamics in the lattice model of a donor/acceptor heterojunction. Exciton generation by means of a photoexcitation, exciton dissociation, and further charge separation are treated on equal footing. The experimentally observed presence of space-separated charges at $\lesssim 100\mathrm{fs}$ after the photoexcitation is usually attributed to ultrafast transitions from excitons in the donor to charge-transfer and charge-separated states. Here, we show, however, that the space-separated charges appearing on $\lesssim 100$-fs time scales are predominantly directly optically generated. Our theoretical insights into the ultrafast pump-probe spectroscopy challenge usual interpretations of pump-probe spectra in terms of ultrafast population transfer from donor excitons to space-separated charges.

Figure 1

(a) Illustration of the model Hamiltonian used in our study. (b) Active variables in the density matrix formalism and their interrelations in the resulting hierarchy of equations. The direction of a straight arrow indicates that in the equation for the variable at its start appears the variable at its end. Loops represent couplings to higher-order phonon-assisted density matrices which are truncated so that the particle number and energy of the free system are conserved.

Figure 2

(a) One-dimensional lattice model of a heterojunction. Various types of electronic couplings (in the donor, in the acceptor, and among them) are indicated. There is an energy offset between single-electron/hole levels in the donor and acceptor. (b) Band alignment produced by our model. (c) Energies of exciton states, in particular of donor excitons (black lines), CT (red lines), and CS (blue lines) states. Exciton wave function square moduli are shown for the lowest donor, CT, and CS state.

Figure 3

(a) Time dependence of the numbers of donor (XD), CT, and CS excitons. The inset shows the time dependence of these quantities normalized to the total exciton population in the system. (b) Probability that at time $t$ an electron is located at site $r$ as a function of $r$ for various values of $t$. In the legend, the probability that at instant $t$ an electron is located in the acceptor is given, while the inset shows its full time dependence. Dotted vertical lines indicate the end of the excitation.

Figure 4

Time dependence of the relative number of (a) donor and CT, (b) CS excitons, and (c) the probability $P_A^{\mathrm{e}}$ that an electron is in the acceptor, for different values of the transfer integrals $|J_{DA}^{\mathrm{c}}|=|J_{DA}^{\mathrm{v}}|=J_{DA}$ between the donor and the acceptor. Dotted vertical lines indicate the end of the excitation.

Figure 5

Time dependence of the relative number of (a) donor and CT, (b) CS excitons, and (c) the probability $P_A^{\mathrm{e}}$ that an electron is in the acceptor, for different values of the LUMO-LUMO energy offset $\mathrm{\Delta }{E}_{DA}^{\mathrm{c}}$. Dotted vertical lines indicate the end of the excitation.

Figure 6

Time dependence of the relative number of (a) donor and CT and (b) CS excitons, for different values of electronic coupling in the acceptor $J_A^{\mathrm{c}}$. (c) Squared moduli of dipole matrix elements (in arbitrary units) for direct generation of CS excitons from the ground state for different values of electronic coupling in the acceptor $J_A^{\mathrm{c}}$. Dotted vertical lines indicate the end of the excitation. Note that, globally, squared moduli of dipole matrix elements are largest for $|J_A^{\mathrm{c}}|=200$ meV (completely filled bars).

Figure 7

Time dependence of the relative number of (a) donor, (b) CT, (c) CS excitons, and (d) the probability $P_A^{\mathrm{e}}$ that an electron is in the acceptor, for different strengths of the carrier-phonon interaction. Dotted vertical lines indicate the end of the excitation.

Figure 8

(a) and (b) Time dependence of the relative number of space-separated (CT and CS) excitons for three different disorder realizations (r1, r2, and r3). (c) and (d) Time dependence of the probability that an electron is in the acceptor for three different disorder realizations. Time evolution of the respective quantities in ordered system is shown on each graph for comparison. Disorder is diagonal (affects only on-site energies), static, and Gaussian, standard deviations being $\sigma =50$ [(a) and (c)] and 100 meV [(b) and (d)].

Figure 9

Differential transmission signal $\mathrm{\Delta }{T}_{\mathrm{PIA}}$ [Eq. (31)] as a function of the pump-probe delay for (a) pump at 1500 meV (826 nm) and probe at 1000 meV (1240 nm) testing PIA dynamics from space-separated states, and (c) pump resonant with the lowest donor exciton (1210 meV, 1025 nm) and probe at 1130 meV (1096 nm) testing PIA dynamics from donor states. The inset of (c) shows the coherent exciton population $|y_{{\mathrm{XD}}_0}{|}^2$ of the lowest donor state ${\mathrm{XD}}_0$. (b) The same signal as in (a) at longer pump-probe delays ($>300\mathrm{fs}$). (d) $\overline{n}$ part of the signal shown in (c); the inset displays the incoherent exciton population ${\overline{n}}_{{\mathrm{XD}}_0}$ of the lowest donor state.