Probing long-range carrier-pair spin–spin interactions in a conjugated polymer by detuning of electrically detected spin beating

Weakly coupled electron spin pairs that experience weak spin–orbit interaction can control electronic transitions in molecular and solid-state systems. Known to determine radical pair reactions, they have been invoked to explain phenomena ranging from avian magnetoreception to spin-dependent charge-carrier recombination and transport. Spin pairs exhibit persistent spin coherence, allowing minute magnetic fields to perturb spin precession and thus recombination rates and photoreaction yields, giving rise to a range of magneto-optoelectronic effects in devices. Little is known, however, about interparticle magnetic interactions within such pairs. Here we present pulsed electrically detected electron spin resonance experiments on poly(styrene-sulfonate)-doped poly(3,4-ethylenedioxythiophene) (PEDOT:PSS) devices, which show how interparticle spin–spin interactions (magnetic-dipolar and spin-exchange) between charge-carrier spin pairs can be probed through the detuning of spin-Rabi oscillations. The deviation from uncoupled precession frequencies quantifies both the exchange (<30 neV) and dipolar (23.5±1.5 neV) interaction energies responsible for the pair's zero-field splitting, implying quantum mechanical entanglement of charge-carrier spins over distances of 2.1±0.1 nm.

The spin-pair process results in a double Gaussian profile, which is fit to this data with very good agreement. (c) The resonance lineshape at each acquisition time is then independently fit in the same manner, and the Gaussian center positions (g-factors) are given as a function of time. Taking the weighted mean of each g-factor allows the relative difference, Δg, between them to be determined, which provides a limit to the carrier pair's Larmor separation. The Rabi frequency detuning for the case of maximal exchange and zero dipolar coupling is simulated for isolated spin pairs within the ensemble distribution. Each spin pair in (a-c) is characterized by its Larmor separation, Δω, as labeled. Additionally, the average Larmor center of each pair is shifted along the detuning axis with respect to the ensemble average Larmor center. (d) The single spin-pair Rabi frequency detuning behavior was simulated for 121 cases of Δω, weighted according to probability of occurrence in the ensemble, and then added together to construct the ensemble average. The ensemble average has multiple features that are not observed by the experiment, notably the on-resonance frequency components at ~0.5B 1 and ~1.5B 1 . This combination of J and D can therefore be excluded as invalid. The same method is used for the inspection of the remaining J and D combinations, allowing strict bounds to be placed on these energies. Figure 7: Demonstration of the qualitative Rabi-frequency spectra placing bounds on acceptable exchange and dipolar combinations. Bounds on possible J and D combinations which can account for the measured Rabi detuning behavior. (a) The combination of spin-spin interaction energy constituents which leads to the experimentally observed offset, Δ, to the harmonic Rabi oscillation is reproduced. The first subsets of combinations that lie outside the range of acceptable values are marked by red arrows and crosses. A valid combination lying directly between these bounds is marked by a green arrow and circle. The corresponding on-resonance Rabi frequency components for each of these combinations is shown in (b-d). The combinations that lie outside the bounds of acceptable values (b,d) display significant splitting in their fundamental frequency component (i.e. around 1B 1 ), which is not observed experimentally. For the case lying within the range of acceptable values (c), the distribution of Rabi frequency components matches that of experiment very well. Note that due to the simplifications used for the simulation of the data in (b) through (d), only frequency components but not relative intensities can be directly compared to those observed in the experiment (see discussion in Supplementary Note 6).

Spin-Hamiltonian of charge carrier pairs
The spin-Hamiltonian for either unipolar or bipolar charge carrier pairs is represented by an expression where , is the gyromagnetic ratio and , is the spin operator for each charge, a and b, of the weakly coupled pair. 2 The first term describes each charge's Zeeman interaction, while the second one describes the isotropic exchange interaction between charges. The third determines the mutual dipolar coupling in the high-field limit, which applies to the conditions discussed here. Since this expression is independent of the charge polarities and is a material parameter, the entire spectrum of two-charge spin-½ interactions can be described by simply varying the magnitude of exchange, J, and dipolar, D, coupling strengths. 3,4 The four resulting energy eigenstates explicitly depend on J and D (e.g. one singlet and three triplet states for strongly bound excitons). As the eigenbasis states are defined, in part, by the magnitudes of J and D, the general expression for Rabi nutation frequency between eigenstates of the system is highly sensitive to these spin-coupling parameters.

Fine structure of detuning of Rabi oscillations in the presence of large hyperfine coupling
Small inhomogeneous broadening of the charge carriers' magnetic resonance line is crucial for resolving fine-structure in the detuned Rabi spectrum. Since hydrogen is an all abundant constituent of organic semiconductors, hyperfine fields at polaron sites have been found to exceed 1 mT. 1,5 Since these local fields are randomly distributed, they add Gaussian distributed random magnetic fields onto the externally applied magnetic field, resulting in a distribution of Larmor frequencies for charges within the spin pairs.
As the detuning term of Rabi's frequency formula depends on the Larmor frequency, , ,

Electrically detected Rabi nutation
The raw data (device current change) corresponding to the microwave pulse length dependence as a function of magnetic field displayed in Figure 3c of the main text is shown in Supplementary Figure 2.
The difference between the two plots is that the main text Figure 3 represents the raw data with an exponential decay function subtracted, which was fitted to the raw data for long pulse lengths. Since this procedure only introduces low-frequency harmonics to the data sets, it allows for a clearer visualization of the Rabi nutation in the main text Figure 3 without distorting the measured Rabi oscillation features around the fundamental and harmonic frequency components. However, as this procedure does distort the measured low-frequency oscillation components, the Fourier analysis displayed in the main text Figure   4a-c was therefore obtained from the raw data without this subtraction.

Extraction of Larmor separation from the EDMR spin-pair resonance
In the experiments reported, a (nanosecond-range) pulsed magnetic resonant microwave excitation induces a sudden transition between spin eigenstates, which subsequently changes the device current from a steady state to a value that represents the changed singlet-to-triplet ratio of the pair population. After the steady state returns to zero, the magnitude of the current change will evolve continuously. However, for any time t after the pulse, the dependence of the current change on the magnetic field will resemble the same functions, i.e. it will be described by the same resonance lines. Since the g-factor of each resonant carrier is fixed as a material parameter, the center position of each Gaussian profile also remains fixed (as does the width). In this case, the resonance profile obtained at each time step in the data acquisition can be extracted and fit individually with the two-Gaussian model. Supplementary Figure 3c shows the resulting g-factors obtained through this fitting process as a function of time following microwave excitation of the spin-system. Since each step in time constitutes an independent measurement of resonance lineshape, the weighted mean and associated error for each g-factor can be determined to very high precision. Even though the absolute g-factor for each carrier is resolution limited (to ±0.001) by calibration of the Hall sensor used to set the static magnetic field  0 , the relative difference between them, Δg, retains the precision of the weighted mean. With each carrier of the pair carrying spin-½, the only contribution to differences in Larmor frequency derives from Δg.

Supplementary Note 5 Calculating exchange and dipolar contributions to the Rabi harmonic frequency shift
The oscillation components caused by a magnetic resonantly driven two spin-½ system (the Rabioscillation of the two pair constituents as well as the beat components) are highly sensitive to the magnitudes of intra-pair exchange and dipolar coupling. In the weak-coupling limit (  0)

Calculation of interaction energy and limit on spin-pair distance
In the weak-coupling limit, the harmonic Rabi oscillation frequency lies at exactly the sum of the two fundamental oscillation frequencies. 2,6 When the harmonic deviates from this sum value, the magnitude of the deviation (main text Figure 5b) provides information on the spin-spin interaction strengths. 3,4 In order to quantify this difference the detuning spectrum must be measured. This spectrum is the Fourier

Supplementary Note 8 Intermediate pair-controlled spin-dependent transitions in organic semiconductors
The spin-dependent currents observed in this study reveal the coherent spin motion of pairs of weakly coupled charge carrier spin states with s=1/2, through recombination rates controlled by the spin pair's permutation symmetry which is determined by spin conservation. This mechanism, called polaron pair recombination, is very similar to spin-dependent radical pair reaction processes that are thought to influence avian magnetoreception. The term "weak" coupling in this context means that dipolar-and exchange interactions are smaller than the average Larmor-frequency differences within the pairs caused by the difference of random hyperfine fields to which the pair partners are exposed. The identification of the pair beating is the characteristic signature which reveals that spin-selection rules apply to pairs of s=1/2 which are subject to Pauli blockade, as discussed previously for polaron-pair recombination in the conjugated polymer MEH-PPV. 1,5 Spin-dependent recombination through so-called "intermediate pairs" of weakly coupled s=1/2 systems were first described and studied for recombination in inorganic semiconductors by Kaplan, Solomon, and Mott (KSM). 10 Essentially, by describing this mechanism but without explicitly mentioning this fact, KSM recognized that recombining charge carriers can behave in a very similar way to recombining radical pairs in photochemical reactions.
The crucial aspect of the mechanism described by KSM is that the electronic transitions governed by the usually spin-independent, yet for spin pair ensembles, this typically nevertheless implies that the dissociation rates depend on the spin state of the dissociating pair as the spin-selection rules that apply to the pair annihilation create an imbalance of singlet and triplet states. Because of this imbalance, a spinindependent dissociation probability will lead to spin-dependent dissociation rates, i.e. even if the pair dissociation process itself is independent of spin (singlet or triplet), the overall dissociation rate will be spin dependent.
Since the work by KSM, the intermediate pair mechanism has been implicated in spin-dependent transport of organic semiconductors in two qualitatively different ways: (i) in the form of "polaron pairs", a term that has been used in the literature exclusively for the description of oppositely charged, bipolar pairs of charge carriers which recombine spin dependently 12 ; and (ii) in the form of so-called "bipolarons", weakly spin-spin coupled pairs of equally charged carriers whose name pertains to the circumstance that these pairs are precursor states for doubly occupied states, bipolarons. 13 As discussed in the main text, the weakly coupled spin pairs reported for PEDOT:PSS appear to be polaron pairs rather than bipolarons. This conclusion may appear surprising given the fact that PEDOT:PSS is considered a unipolar conductor. However, the pair's charging state is irrelevant for the analysis of spin beating and the conclusions drawn about the measured intra-pair spin-interaction strength since the measurement applies equally to unipolar and bipolar pairs.