Theory of exciton-polaron complexes in pulsed electrically detected magnetic resonance

Several microscopic pathways have been proposed to explain the large magnetic effects observed in organic semiconductors; however, it is difficult to identify and characterize the microscopic process which actually influences the overall magnetic field response in a particular instance. Pulsed electrically detected magnetic resonance provides an ideal platform for this task as it intrinsically monitors the charge carriers of interest and provides dynamical information which is inaccessible through conventional magnetoconductance measurements. Here we develop a general time-domain theory to describe the spin-dependent recombination of exciton-polaron complexes following the coherent manipulation of paramagnetic centers through electron paramagnetic resonance. A general Hamiltonian is treated, and it is shown that the transition frequencies and resonance positions of the exciton-polaron complex can be used to estimate interspecies coupling. This work also provides a general formalism for analyzing multipulse experiments which can be used to extract relaxation and transport rates.

Figure 1

(a) General picture of the triplet exciton-polaron quenching mechanism. Exciton-polaron complexes are formed spin independently due to the reorganization energy associated with a shared nuclear distortion. A complex may spin independently dissociate back in a separate exciton and polaron or recombine, possibly due to weak exciton-polaron exchange. Intersystem crossing will mix the doublet and quartet states prior to recombination or dissociation. (b) A diagramatic representation of the Hamiltonian for an exciton-polaron complex with arbitrary coupling. The exciton will have internal exchange $\left(J_{ex}\right)$ and dipolar $\left(D_{ex}\right)$ coupling, and there may also be exchange $\left(J_{ex,p}\right)$ and dipolar $\left(D_{ex,p}\right)$ coupling between the exciton and the polaron.

Figure 2

Behavior of an exciton-polaron complex depends strongly on the strength of the exciton-polaron coupling. Solid lines represent strongly driven transitions and dashed lines represent transitions which become forbidden in the weak or strong coupling limits. (a) In the weak coupling regime the basis states are close to the product states and the transitions are the $|\uparrow \rangle \leftrightarrow |\downarrow \rangle$ polaron transition, with a Rabi frequency of $\gamma B_1$, and the $|T_{+}\rangle \leftrightarrow |T_0\rangle ,|T_0\rangle \leftrightarrow |T_{-}\rangle$ exciton transitions with a Rabi frequency of $\sqrt{2}\gamma B_1$. The $|\downarrow T_{+}\rangle \leftrightarrow |\uparrow T_{-}\rangle$ transition becomes weakly allowed in the presence of nonzero coupling. (b) In the intermediate coupling regime the basis states are neither the product nor quartet-doublet states and there are up to eight visible transitions with Rabi frequencies between zero and $\sqrt{3}\gamma B_1$. (c) In the strongly coupled regime quartets and doublets are formed. Strong driving occurs within the doublet and quartet manifolds and driving between the manifolds becomes arbitrarily slow, leading to a vanishing signal. The Rabi frequencies of these transitions are $\sqrt{3}\gamma B_1$ and $2\gamma B_1$.

Figure 3

Transition frequencies of an exciton-polaron complex as a function of the mixing angle. In the uncoupled regime $\left(\theta =0\right)$ the polaron will nutate at a frequency of $\gamma B_1$ and the exciton will nutate at a frequency of $\sqrt{2}\gamma B_1$. Weak coupling splits the exciton and polaron resonances and permits the $\downarrow T_{+}\leftrightarrow \uparrow T_{-}$ transition to be weakly driven. In the strong coupling limit pure quartet and doublet states are formed. Quartet mixing will occur at frequencies of $2\gamma B_1 \left(Q_{1/2}\leftrightarrow Q_{-1/2}\right)$ and $\sqrt{3}\gamma B_1 \left(Q_{\pm 3/2}\leftrightarrow Q_{\pm 1/2}\right)$. Driving can also occur within the doublet manifold at a frequency of $\gamma B_1$. Driving between the doublet and quartet states becomes arbitrarily slow in the strong coupling limit. Transition frequencies are calculated by using Eq. (45).

Figure 4

Transition frequencies of a bipolaron counterion complex, which is treated by introducing a negative mixing angle. At an angle of ${\theta }_1=\frac{{cos}^{-1}\sqrt{\frac{2}{3}}-{cos}^{-1}\sqrt{\frac{1}{3}}}{2}$ the bipolaron is broken apart by the strong Coulombic attraction from the counterion. If the coupling is increased further an exciton-polaron complex is formed for a mixing angle of ${\theta }_2={cos}^{-1}\sqrt{\frac{2}{3}}-{cos}^{-1}\sqrt{\frac{1}{3}}$. From mixing angles of ${\theta }_2$ to ${\theta }_3={cos}^{-1}\sqrt{\frac{1}{3}}$ the transition frequencies of the exciton-polaron complex evolve in an identical manner to the positive mixing angle regime. Transition frequencies are calculated from Eq. (45).