Theory of triplet-triplet annihilation in optically detected magnetic resonance

Triplet-triplet annihilation allows two low-energy photons to be upconverted into a single high-energy photon. By essentially engineering the solar spectrum, this allows solar cells to be made more efficient and even exceed the Shockley-Quiesser limit. Unfortunately, optimizing the reaction pathway is difficult, especially with limited access to the microscopic time scales and states involved in the process. Optical measurements can provide detailed information: triplet-triplet annihilation is intrinsically spin dependent and exhibits substantial magnetoluminescence in the presence of a static magnetic field. Pulsed optically detected magnetic resonance is especially suitable, since it combines high spin sensitivity with coherent manipulation. In this paper, we develop a time-domain theory of triplet-triplet annihilation for complexes with arbitrary spin-spin coupling. We identify unique “Rabi fingerprints” for each coupling regime and show that this can be used to characterize the microscopic Hamiltonian.

Figure 1

General description of the time-domain dynamics of triplet-triplet complexes. Singlet excitons are excited by incident light. At a later time they can decay into triplet excitons. Some of these will form spatially bound pairs that can either dissociate back into two independent triplet excitons or annihilate and form an excited singlet exciton. The rate of these reactions depend on whether the triplet-triplet complex is in the quintet (spin-2), triplet (spin-1), or singlet (spin-0) manifold. Prior to an electronic transition occurring, spin mixing can change the overall spin content $\left(S\right)$ or spin projection $\left(m_s\right)$ of the complexes. Spin transitions can occur incoherently through interactions with the local spin environment or coherently through the absorption or emission of microwave radiation. Driving these transitions changes the overall probability of recombination and leads to a macroscopic change in sample luminosity.

Figure 2

Spin transitions and energy levels of triplet-triplet complexes with no (a), intermediate (b), and strong (c) spin-spin coupling. In the uncoupled regimes there are two independent triplet excitons with allowed spin transitions $|T_{+}\rangle \leftrightarrow |T_0\rangle$ and $|T_0\rangle \leftrightarrow |T_{-}\rangle$ with Rabi frequencies of $\sqrt{2}\gamma B_1$. Each exciton has three energy levels which are split by the dipolar and Zeeman coupling. The intermediate coupling regime is characterized by up to 16 possible spin transitions with Rabi frequencies between 0 and $\sqrt{6}\gamma B_1$. In the strongly coupled regime, distinct singlet, triplet, and quintet states are formed. The strongly allowed transitions are shown in black, and nominally forbidden transitions are shown by dashed lines: these produce arbitrarily slow transitions in the biexciton limit.

Figure 3

The singlet content of the $|5\rangle$ (blue) and $|7\rangle$ (red) basis states as a function of the spin mixing angle $\varphi$, which is a measure of spin-spin coupling. All other states have zero singlet content and are not shown. In the weak-coupling limit, $|5\rangle$ has $\frac{1}{3}$ singlet content and $|7\rangle$ has $\frac{2}{3}$ singlet content. In the strong-coupling limit (large mixing angle) $|5\rangle$ converges to the quintet state and $|7\rangle$ converges to a singlet state. The maximum mixing angle is given by ${\varphi }_{\max }=\frac{1}{2}{tan}^{-1}\left(2\sqrt{2}\right)$. Analytic expressions for $|5\rangle$ and $|7\rangle$ as a function of $\varphi$ can be found in the Appendix.

Figure 4

Analytically derived transition frequencies as a function of the spin mixing angle, with the maximum mixing angle given by ${\varphi }_{\max }=\frac{1}{2}{tan}^{-1}\left(2\sqrt{2}\right)$. The visible transitions that induce a change in the overall singlet content are shown in (a), and invisible transitions that do not induce a change in singlet content are shown in (b). Due to the symmetries of the Hamiltonian, the transition frequencies occur in pairs, so there are up to 16 components. In the weak-coupling limit Rabi frequencies of $\gamma B_1$ and $\sqrt{2}\gamma B_1$ are seen. These diverge from the expected $\sqrt{2}\gamma B_1$ exciton transitions since the hyperfine coupling has been neglected in our analytic treatment. In the strong-coupling limit only six transitions remain, with Rabi frequencies of $\sqrt{2}\gamma B_1,2\gamma B_1$, and $\sqrt{6}\gamma B_1$. These correspond to transitions within the triplet and quintet subspaces. The transitions occur in pairs and are labeled on the right-hand side (see online version for color). See the text for the explicit basis states.

Figure 5

Analytic solutions for the ensemble spin dynamics in the uncoupled (a) and strongly coupled (b) regimes. In the uncoupled regime the triplet excitons nutate independently with a Rabi frequency of $\sqrt{2}\gamma B_1$, producing a Pake doublet with a total width $2D$. The cross section is shown in blue. Multiple components occur in the strong-coupled regime. The triplet states will nutate with a Rabi frequency of $\sqrt{2}\gamma B_1$ and form a Pake doublet with a total width of $2D$, and the quintet states nutate with Rabi frequencies of $2\gamma B_1$ or $\sqrt{6}\gamma B_1$ and form Pake doublets with widths of $2D$ and $6D$, respectively. Since these transitions involve no first-order change in the singlet content, they will not produce a change in the luminosity. The strength of each component has been normalized such that each frequency component has a constant area.