Abstract
Electron and hole spins in organic lightemitting diodes constitute prototypical twolevel systems for the exploration of the ultrastrongdrive regime of lightmatter interactions. Floquet solutions to the timedependent Hamiltonian of pairs of electron and hole spins reveal that, under nonperturbative resonant drive, when spinRabi frequencies become comparable to the Larmor frequencies, hybrid lightmatter states emerge that enable dipoleforbidden multiquantum transitions at integer and fractional gfactors. To probe these phenomena experimentally, we develop an electrically detected magneticresonance experiment supporting oscillating driving fields comparable in amplitude to the static field defining the Zeeman splitting; and an organic semiconductor characterized by minimal local hyperfine fields allowing the nonperturbative lightmatter interactions to be resolved. The experimental confirmation of the predicted Floquet states under strongdrive conditions demonstrates the presence of hybrid lightmatter spin excitations at room temperature. These dressed states are insensitive to power broadening, display BlochSiegertlike shifts, and are suggestive of long spin coherence times, implying potential applicability for quantum sensing.
Introduction
A spin in a magnetic field is a perfect discretelevel quantum system, for which resonant electromagnetic radiation can drive coherent propagation in a wellcontrolled perturbative fashion following Rabi’s theory^{1}. Such propagation is used for magnetic resonance applications in spectroscopy and imaging, where the thermalequilibrium population of spin states is altered under resonance, inducing an effective magnetization change of nuclear or electronic spins^{2}. Experiments are generally performed under the condition that the Zeeman splitting between spin states induced by a static magnetic field is much larger in energy, or frequency, than the Rabi frequency^{1}. This weakdrive limit of resonant pumping is described by perturbation theory. Magnetic resonance can also be probed by observables secondary to magnetization, such as the permutation symmetry of pairs of spins of electronic excitations. Such spindependent transitions are reflected in luminescence or conductivity in atomic, molecular, or solidstate systems^{3,4} in optically or electrically detected magnetic resonance (ODMR, EDMR) spectroscopies^{5}. Color centers in crystals, like atomic vacancies in silicon carbide or diamond, are widely used in ODMRbased quantum metrology and quantuminformation processing^{6}, but are limited in one regard: as dipolar and exchange coupling in the spin pair is strong, substantial level splitting arises at zero external field, posing a lower limit on resonance frequency. Such a limitation does not exist for weakly coupled spin½ chargecarrier pairs which form, for example, by electron transfer in molecular donoracceptor complexes, where they account for a range of magneticfield effects^{7}.
A particularly versatile way to study weakly coupled spin½ pairs is offered by organic lightemitting diodes (OLEDs), which generate light from the recombination of electrically injected electrons and holes that bind in pairs of singlet or triplet permutation symmetry^{8}. As the formation rates of intramolecular excitons from intermolecular electronhole carrier pairs differ for singlet and triplet spin permutations, and singlet pairs tend to have shorter lifetimes than triplets, an increase of singlet content in the electronhole pair depletes the reservoir of available carriers and leads directly to a change in conductivity^{9,10,11}. Because carrier spins interact with local hyperfine fields, which originate from unresolved hyperfine coupling between chargecarrier spins and the nuclear spins of the ubiquitous protons^{12}, carrier migration through the active OLED layer gives rise to spin precession and, ultimately, mixing of singlet and triplet carrierpair configurations. As a result, OLEDs can exhibit magnetoresistance on nanotesla scales at room temperature^{5}. A static magnetic field tends to partially suppress this hyperfine spin mixing, an effect that is reversed under magnetic resonance conditions, which give rise to distinct resonances in the magnetoresistance functionality^{13}.
It is important to stress that such studies are only possible in materials characterized by very weak spinorbit coupling^{14}, such as organic semiconductors, and are not generic to electronhole recombination in LEDs as a whole. In an inorganic LED, for example, spin mixing occurs by spinorbit coupling in addition to the fact that recombination does usually not occur into tightly bound, and hence thermally stable, excitonic species. Light generation in inorganic LEDs, in contrast to OLEDs, is therefore primarily not spin dependent. As such, the subsequent discussion strictly only applies to OLEDs comprising materials of weak spinorbit coupling, i.e., of low atomicorder number. The appeal of experimenting with spins in OLEDs is that their coherence time, i.e., the transverse spinrelaxation time T_{2}, is only weakly dependent on temperature^{15}. This independence is a direct consequence of the lack of spinorbit coupling, which leaves the carrier dynamics in the local hyperfine fields as the coherencelimiting effect^{15}.
In principle, magnetic resonances can be resolved down to very small frequencies of a few megahertz, limited only by the overall strength of the hyperfine interaction^{5,16}. One advantage of EDMR in OLEDs over conventional EPR in radicalpairbased spin½ systems^{17} is that the sample volume can be made almost arbitrarily small. It is therefore possible to achieve high levels of homogeneity both in terms of the static field B_{0}, which defines the Zeeman splitting, and the amplitude of the resonant driving field, B_{1}, whereas at the same time penetrating the nonperturbative regime of ultrastrong drive where B_{1} becomes comparable to B_{0}, so that the Rabi frequency approaches the Larmor frequency^{18}. Such an ultrastrongdrive regime is of great interest in contemporary condensedmatter physics, although embodiments thereof have proven very challenging to find^{19,20} and mostly arise in the form of ultrastrong coupling of twolevel systems in resonant optical cavities^{21,22,23}.
The breakdown of the perturbative regime of OLED EDMR has previously been identified under drive conditions of \(B_1 \approx 0.1B_0\), where conventional power broadening gives way to a variety of strongdrive effects^{24}. Once the drivingfield strength exceeds the inhomogeneous broadening of the individual spins of the pair induced by the hyperfine fields, the spins become indistinguishable with respect to the radiation and a new set of spinpair eigenstates is formed^{25}. The resonant field locks the spin pairs into the triplet configuration, in analogy to the formation of a subradiant state in Dicke’s description of electromagnetic coupling of noninteracting twolevel systems^{26,27}. The spinDicke state manifests itself in EDMR by the appearance of a particular inverted resonance feature^{25,28}. Experimentally, it is challenging to access this regime of strong and ultrastrong drive for the simple reason that very high oscillating magnetic field strengths have to be generated in close proximity to the OLED. Using either coils^{24} or a monolithic microwire integrated in the OLED structure^{18}, we were previously able to probe the strongdrive regime of up to \(B_1 \approx 0.3B_0\). We indeed observed an inversion of the resonance signal in the device current along with spectral narrowing occurring owing to the spinDicke effect^{18,24}. Larger oscillating field strengths were not accessible with these earlier experimental setups. In addition, the magnitude of the B_{0} field necessary to clearly resolve the resonance was previously limited by the inhomogeneous broadening of the resonance spectrum due to the hyperfine fields arising from the omnipresent protons^{16}. Besides limiting spectral resolution, this inhomogeneous broadening also constrains the effective coupling strength of the spin states to the driving field^{25}. Qualitatively speaking, the disorder increases quantummechanical distinguishability of the individual spins with respect to the driving field, lowering the overall degree of coherence of the incident radiation with the spin ensemble that can be achieved under resonant drive^{25}.
To date, there has been no formulation of the theoretical expectation of the nature of spindependent transitions of two weakly coupled spin½ carriers, electron and hole, under ultrastrong resonant drive. Although a Floquet formalism to treat this problem has been put forward previously, this was only pursued in the weakdrive limit^{29}.
In this work, we begin by setting out a strategy for computing such transitions in the ultrastrongdrive regime using the periodic timedependent spin Hamiltonian in the Floquet formalism. The numerically rather detailed calculations can be condensed into a diagrammatic representation of the resulting hybrid lightmatter states, the spin states of the OLED dressed by the driving electromagnetic field. This approach gives us a complete computation of the EDMR magnetic resonance spectrum of an OLED under the condition of ultrastrong drive, with B_{1} exceeding B_{0}. With substantial experimental improvements both in terms of the material used and with regard to the monolithic OLEDmicrowire structure we succeed in experimentally identifying the main predicted Floquet spin states of the OLED under ultrastrong drive conditions. These states are manifested as magneticdipoleforbidden multiplequantum transitions and resonances at fractional gfactors.
Results
Spin transitions in OLEDs under nonperturbative resonant drive
Theoretical treatment of the spin dynamics in a strongly driven electronhole pair beyond the perturbative regime is extremely challenging and has not been considered in detail previously. We approach this problem using quantummechanical Floquet theory^{30}. The electronhole spinpair Hamiltonian, H(t), is time dependent owing to the presence of a timedependent (sinusoidal) driving field. Besides the drive, we incorporate in H(t) the electron and hole spin coupling to the external field B_{0} and the effective local internal hyperfine fields, as well as the isotropic exchange and dipolar interactions between the electron and hole spins. Note that, given the exceedingly weak spinorbit coupling of the organic semiconductor material used in the experiments discussed in the following, the spin Hamiltonian is perfectly defined without an explicit spinorbit term, using effective Landé gfactors instead. We direct the interested reader to a recent joint experimental and theoretical examination of the influence of spinorbit coupling in these materials^{14}. Although all these interactions are time independent, the local hyperfine fields and the dipolar interaction are taken to be random and different for different configurations. Further details of the statistical distribution of spin pairs and the numerical approach to account for this are discussed in Supplementary Notes 1.5 and 1.6.
The timeperiodic character of the driving field allows us to write the Fourier decomposition of the Hamiltonian as \(H_{\alpha \beta }\left( t \right) = \mathop {\sum }\nolimits_n H_{\alpha \beta }^{(n)}e^{in\omega t}\), where ω is the angular frequency of the incident radiation. Here, and in the following, the Greek indices α and β run over the four spinpair states, the three triplets \(T_ + ,\;T_0,\;T_ \) and the singlet S, which are chosen as the spin Hilbert space basis for the subsequent discussion. The Floquet dressed states \(\left. {\alpha ,n} \right\rangle\) with photon number \(n = 0,\; \pm \!1,\; \pm \!2,\; \ldots\) are defined as the product of the spin Hilbert space basis state α and the Fourierspace basis state n〉. The dressed states form an orthonormal basis in an infinitedimensional Floquet Hilbert space. A change of n modifies the dressed state while retaining the same spinpair component. The timeindependent, infinitedimensional Floquet Hamiltonian H_{F} is then defined in matrix representation as
The Floquet approach takes advantage of the fact that H_{F} is time independent and Hermitian, thus possessing stationary eigenvectors, \({\psi _{\alpha ,n}} \rangle\), and real eigenvalues, \(\varepsilon _{\alpha ,n}\). The Floquet Hamiltonian has a periodic structure so that a shift of the indices by the same integer l changes only the diagonal matrix elements
rendering periodicity of the eigenvectors, \(\langle {\alpha ,n + l{\mathrm{}}\psi _{\alpha ,m + l}} \rangle = \langle {\alpha ,n{\mathrm{}}\psi _{\alpha ,m}} \rangle\), and eigenvalues, \(\varepsilon _{\alpha ,n + l} = \varepsilon _{\alpha ,n} + l\omega\).
The spindensity matrix of an ensemble of spin pairs, ρ, satisfies the stochastic Liouville equation and is used to compute the steadystate singlet content of the spin pair, i.e., the observable responsible for the experimentally measured conductivity of the OLED. Ultimately, this observable can be expressed by the trace of the steadystate spindensity matrix, \({\mathrm{tr}}\tilde \rho _0\), where the tilde indicates the steadystate solution to the stochastic Liouville equation, which is explicitly time dependent because of the timeperiodic Hamiltonian, with the index 0 referring to the timeindependent Fourier component thereof. As described in detail in Supplementary Note 1.2, from this solution, we can express \({\mathrm{tr}}\tilde \rho _0\) through the following phenomenological parameters: the rate of spinpair generation, G; the spinpair dissociation rate, r_{d}; the singlet and triplet recombination rates, r_{S} and r_{T}; and the spinlattice relaxation time, T_{sl}. For convenience, we introduce a characteristic total decay rate of the pair, \(w_d = r_d + r_T + 1/T_{{\mathrm{sl}}}\), and define the difference in singlet and triplet recombination rates \(k_r = r_S  r_T\), which determines the overall magneticfield response of the conductivity. Using \({\psi _{\alpha ,0}} \rangle\) as the eigenvectors of the Floquet Hamiltonian and \({\Pi}_S\) as the projection operator onto the dressed singlet subspace, \({\Pi}_S = \mathop {\sum }\nolimits_n \left. {S,n} \right\rangle \left\langle {S,n} \right.\), we arrive at an expression for the steadystate spindensity matrix as
Equation (3) is derived perturbatively, to leading order in the small difference of singlet and triplet recombination rates k_{r}. Note that \(k_r \ll \left D \right,\;\left J \right\), where D and J are the average dipolar and exchange interactions between electron and hole spins within the pairs. Further details of this approximation are discussed in Supplementary Note 1.4.
The steadystate device current measured in an experiment is a function of the static and oscillating magnetic fields, which give rise to a change \({\Delta} I\left( {B_0,B_1} \right) = I\left( {B_0,B_1} \right)  I\left( {B_0,0} \right)\) of the spindependent OLED current, i.e., the magnetic resonance. This steadystate spindependent device current probed in experiments corresponds to the average of \({\mathrm{tr}}\tilde \rho _0\) over all the random distributions of local hyperfine fields and dipolar couplings, i.e. the average over spatial orientations of electronic spin pairs with respect to the applied magnetic field,
where I_{0} is a spinindependent factor that is determined by the conductivity of the OLED.
We utilize Eqs. (3) and (4) for the numerical calculation of the EDMR signal \({\Delta} I\left( {B_0,\;B_1} \right)\). The Floquet Hamiltonian H_{F} is determined from Eq. (1), for a randomly generated configuration of local hyperfine fields and dipolar coupling. To approximately calculate the infinitedimensional eigenvectors \(\psi _{\alpha ,0}\) of H_{F} entering in Eq. (3), we truncate the infinitedimensional Floquet Hamiltonian and the corresponding Hilbert space as described in Supplementary Note 1.3. The truncation procedure consists of restricting the photon numbers by some integer value, N_{0}, or equivalently restricting the indices m and n in Eq. (1) to run between −N_{0} and N_{0}, where N_{0} is determined by the requirements on the accuracy of the numerical procedure. The eigenvectors of the truncated Floquet Hamiltonian are used to evaluate \({\mathrm{tr}}\tilde \rho _0\) from Eq. (3). The procedure is repeated multiple times and the fielddependent current is found as the average of the numerical values computed for each random configuration according to Eq. (4).
Diagrammatic representation of hybrid lightmatter states
In order to build up an intuitive understanding of the resonant transitions of the electronhole spin pair emerging under strong and ultrastrong resonance drive and to differentiate the Floquet states responsible for the specific transitions, we develop a diagrammatic representation^{31} of the dressed states \(\left. {\alpha ,n} \right\rangle\). For simplicity, we describe the transitions in terms of effective gfactors of resonances. Obviously, this does not relate to the gfactor of the resonant spins—these are all nearly free electrons—but to the ratio between the driving field oscillation frequency and the frequency corresponding to the Zeeman splitting induced by the static magnetic field B_{0}. Figure 1a illustrates the fourfold basis of the Floquet states. A resonant transition involving the creation or annihilation of a photon, shown in Fig. 1b, gives rise to an effective resonance in the overall singlet content of the pair at a gfactor g ≈ 2. This resonance arises from a superposition of transitions depicted by an infinite number of diagrams of increasing order. Specific examples of such diagrams describing the g ≈ 1 twophoton resonance owing to simultaneous annihilation or creation of two photons, shown in Fig. 1c, are given in Fig. 1e. The vortices of the diagrams mark transitions in spin state and occur either by photons, marked as dots, or due to magnetic scattering interactions such as hyperfine or dipolar spinspin coupling (crosses). The entire singlet content of the spin ensemble in the steady state is computed by approximating the infinite summation over all states (see Supplementary Note 1.4). A further group of transitions (Fig. 1d) involves a change of the magnetic quantum number by \({\Delta} m = \pm \!2\), corresponding to a halffield resonance at g ≈ 4. Since the steadystate singlet content is given by a summation over all spin permutations, resonances at fractional gfactors will also arise as combinations of the different processes.
Even though multiphoton transitions in twolevel spin systems have been discussed in the context of electronspin resonance spectroscopy^{32,33,34}, these are not analogous to twophoton absorption of electricdipole transitions. As each photon carries angular momentum \(\ell = 1\), a \({\Delta} \ell = 2\) electricdipole transition requires a final electronic orbital with different angular momentum. A spin½ system can only undergo \({\Delta} \ell = 1\) transitions, however, implying that a magneticdipole transition by absorption of two identical photons is impossible. Instead, twophoton magneticdipole transitions arise with a combination of different photons^{34} whose magneticfield components have transversal and longitudinal orientation with regard to the magnetic field B_{0}.
Computation of the EDMR spectrum as a function of driving strength
In Fig. 2a the calculated change of spindependent current \({\Delta} I(B_0,\;B_1)\) as a function of Zeeman splitting B_{0} and driving field amplitude B_{1} for an incident radiation frequency of 85 MHz is plotted (see Supplementary Note 1.5 for numerical details). The width of the fundamental resonance g ≈ 2 at low driving fields is defined by the expectation value of the local hyperfine field experienced by electron or hole spins, \({\Delta} B_{{\mathrm{hyp}}}\)^{25,28}, as described in Supplementary Note 5. The effective gfactors of the resonant species are indicated on the lefthand field ordinate. Six distinct resonances are seen, with the amplitudes and positions of these depending on B_{1}. The six features are assigned the Floquetstate transitions marked in the diagram of spinpair eigenstates in Fig. 2b. Here, states \(\left. {2} \right\rangle\) and \(\left. {3} \right\rangle\) depend on the particular nature of the intrapair interaction, i.e., dipolar and exchange coupling (see Supplementary Note 1.1). Three resonances arise from fullfield \({\Delta} m = \pm\! 1\) transitions, and three from halffield \({\Delta} m = \pm \!2\) transitions. Feature (i) is owing to the onephoton spin½ resonance and initially undergoes power broadening, before splitting due to the ACZeeman effect and subsequently inverting in amplitude owing to the spinDicke effect^{24}. The g ≈ 1 feature (ii) results from twophoton transitions, and the g ≈ 2/3 feature (iii) from threephoton absorption. Resonances also occur between pure triplet states and result from transitions involving either one (iv) (g ≈ 4), three (v) (g ≈ 4/3), or five (vi) (g ≈ 4/5) photons. Features (i–iii) clearly move towards lower B_{0} values with increasing B_{1}, which is a manifestation of the BlochSiegert shift (BSS), discussed in more detail below and in Supplementary Note 6. The BSS of the g ≈ 2 resonance results in merging into a zerofield resonance at high B_{1}. The resonances appear selfsimilar, with the ACZeeman and spinDicke effects apparent in features (i–iii). Crucially, the inversion of ΔI at the onset of the spinDicke effect coincides with the formation of a new spin basis of the electromagnetically dressed state^{24}. In this hybrid lightmatter state, power broadening of the resonant transition is absent since the electromagnetic field is implicit in the state’s wavefunction^{25}, allowing the BSS to be resolved clearly.
Experimental probing of hybrid lightmatter Floquet states in OLED EDMR
Identifying these Floquet states experimentally in the spindependent transitions that control the magnetoresistance of OLEDs poses two challenges. First, the effective hyperfine fields must be sufficiently small, such that the different resonances do not overlap spectrally. Hyperfine coupling broadens the resonant magneticdipole transition inhomogeneously, making individual microscopic spins distinguishable in terms of their resonance energy. This disorder determines the threshold field for the onset of spin collectivity^{24}, when each individual spin becomes indistinguishable with respect to the driving field. Previous studies of the condition of strong magneticresonant drive of OLEDs employed either conventional hydrogenated organic semiconductors^{18,24}, or partially deuterated materials with reduced hyperfine coupling strengths^{24}. To maximize the resolution of the experiment, we synthesized a perdeuterated conjugated polymer, poly[2(2ethylhexyloxyd_{17})5methoxyd_{3}1,4phenylenevinylened_{4}] (dMEHPPV), with 97% of the protons replaced by deuterons^{35}. Second, OLEDs have to be designed with integrated microwires which generate the oscillating field^{18}. The smaller the OLED pixel relative to the microwire, which narrows down to a width of 150 μm, the lower the inhomogeneity in both static and oscillating magnetic fields—at the cost of EDMR signaltonoise ratio. The alternating current passed through the microwire generates heat, requiring careful optimization of electrical and thermal conductivity of the monolithic OLEDmicrowire device^{18}.
Figure 3 plots the change ΔI of the steadystate DC forward current I_{0} = 500 nA under 85 MHz RF radiation, as a function of B_{0} and the square root of the power P of the applied radiation. The dominant Floquet spin state transitions identified in the calculation in Fig. 2 are resolved in the experiment and labeled correspondingly: the powerbroadened g ≈ 2 spin resonance which inverts its sign in the spinDicke regime with subsequent BSS (i); the twophoton transition and the corresponding BSS of the g ≈ 1 resonance (ii); and the halffield resonance at g ≈ 4 (iv). In principle, \(\sqrt P \propto B_1\), although heating of the microwire at high powers will change the microwire impedance and therefore increase the uncertainty on the abscissa scale in Fig. 3 (see Supplementary Note 4). From an analysis of the experimentally observed power broadening given in Supplementary Fig. 2, we estimate that B_{1} fields of up to 3 mT are reached, consistent with the universal scaling of EDMR amplitude dependence on B_{1} with hyperfine field strength as shown in Supplementary Fig. 6. Note that, because the measurements were actually carried out before the calculations were completed, unfortunately, we did not probe the B_{0} field region above 7.5 mT, as there was no reason to anticipate additional features appearing there. We will leave experimental exploration of this interesting field region for future studies.
A simple intuitive rationalization of the intriguing twophoton transition can be formulated. Twophoton transitions between the up and down state of a spin½ species are dipole forbidden. In analogy to light waves, the resonant radiation can be described in the photon picture. A photon has an angular momentum quantum number of 1 and can assume one of the two projection states of either \(m = + 1\) or \( 1\) relative to the direction of propagation. These socalled \(\sigma ^ +\) and \(\sigma ^ \) photons can be associated with the right and lefthand circularly polarized fields. In a σphoton state, where the direction of propagation is along the quantization axis, the AC magneticfield component B_{1} is perpendicular to B_{0}. The absence of a photon state with m = 0 can be traced back to a photon having zero mass. However, a linear AC field B_{1} in the direction of quantization, \(B_1\parallel B_0\)_{,} is referred to as a πphoton propagating perpendicular to the quantization axis. These unconventional πphotons feature in atomic spectroscopy but are also discussed within a semiclassical picture of magnetic resonance^{2,34}, and can be thought of as photons of total angular momentum zero. Twophoton transitions are only possible with a combination of σ and πphotons. As the excitation scheme employed here would appear to only involve σtype photons, it is not immediately obvious how angular momentum is conserved to give rise to the strong twophoton resonances observed in theory and experiment.
Figure 4a provides an intuitive picture for the emergence of the twophoton transition. Transversal oscillating magneticfield components are orthogonal to B_{0}, but the latter is superimposed with local static isotropic hyperfine fields. This superposition leads to the effective static magnetic field, B_{s}, tilted from the z axis by a small but finite angle. Thus, the superposition of B_{0} and the local hyperfine fields results in a total static field B_{s} parallel to one component of the oscillating magnetic field, which is sufficient to enable the \(\sigma  \pi\) twophoton transition conserving angular momentum. To test this picture, in a separate experimental setup described in the Methods section, we probed the angular dependence of the g ≈ 1 resonance. The complete quantumstatistical simulation of the effect using the Floquet ansatz described above predicts a \(\sin 2\alpha\) dependence, in agreement with the geometric arguments forward above. The characteristic features of the prediction from theory are matched by the experimental data as shown in Fig. 4b. To allow the angle to be swept, these experiments were performed at a very low B_{0} of 780 μT corresponding to a resonance frequency of 22 MHz.
Twophoton resonances arise both under parallel and perpendicular excitation. This observation can be explained by the illustration in Fig. 4a: photons of both longitudinal (π) and transversal (σ) polarizations exist even for parallel or perpendicular field configurations. The lower B_{0}, the greater the contributions of the transversal hyperfine fields and the influence of dipolar coupling of spins within the spin pairs, which enable the resonances under fully parallel and perpendicular excitation. We note that a slight asymmetry in the angular dependency can be attributed to additional resonant contributions of the second harmonic of the RF signal generated by the RF amplifier. This second harmonic can be removed by filtering the incident RF radiation using a lowpass filter. Alternatively, we replicate this asymmetry in the calculation in Fig. 4b by introducing a second harmonic resonant contribution amounting to \({\sim} 0.01\;B_1\). The additional slight asymmetry in the measurement between, e.g., 45° and 135° is also observed for the fundamental resonance (not shown). This deviation presumably arises from the fact that the RF radiation generated by the stripline is not perfectly linearly polarized.
Discussion
We conclude that the Floquet ansatz presented here to solve the timedependent Hamiltonian of a twolevel system in the ultrastrongdrive regime provides both an accurate and surprisingly intuitive representation of the complex spin excitations arising under these conditions. These include the spinDicke state, which manifests a strong BSS, along with dressed lightmatter states, which support dipoleforbidden multiplequantum transitions. Spindependent recombination rates between paramagnetic chargecarrier states in OLEDs offer a remarkably versatile testbed to probe this regime of ultrastrong coupling, visualizing directly the predicted multiplequantum transitions and fractional gfactor resonances—to our knowledge, better than other known quantum systems at room temperature. The identification of the BSS at room temperature is particularly clear in these solidstate devices compared to any other spectroscopic probe of this phenomenon outside of nuclear magnetic resonance^{36,37,38}. Similarly, direct signatures of hybrid lightmatter Floquet states are observed here to influence spindependent OLED currents, much like the Floquet states reported in photoelectron spectroscopy of topological materials^{39}.
The theoretical results of Fig. 2 provide motivation to extend the phase space of the experiment to even higher ratios of \(B_1/B_0\). Although experimentally challenging, this is not entirely unfeasible, and may, for example, be conceivable by using superconducting stripline resonators^{9}. It would be particularly exciting to experimentally identify the spinDicke state, which is reflected by the inversion and narrowing of the resonance, in the threephoton and twophoton transitions (iii) and (ii) predicted in Fig. 2. The latter is especially interesting as the BSS appears to induce a zerofield resonance, i.e., a resonance feature for \(B_0 \to 0\;{\mathrm{mT}}\) and \(B_1 \approx 3.7\;{\mathrm{mT}}\).
Although we refrain at present from speculating too much on possible technological applications of our study, we do note that a clear feature in theory and experiment is the demonstration that power broadening is suppressed when the spinDicke state is formed: the onephoton resonance feature (i) powerbroadens linearly with driving field strength B_{1} when B_{1} is below the Dicke regime, whereas after inversion of the resonance sign (marked in red in Figs. 2 and 3) broadening of the inverted feature with a further increase in power is minimal. This absence of power broadening implies that the hybrid lightmatter state, the spinDicke state^{25}, appears to be protected against power broadening precisely because it is a hybrid state. The consequence of this protection should be a dramatically enhanced coherence time, which in turn may turn out to be useful in quantum magnetometry applications. Measuring this increased coherence time in pulsed experiments^{40} at very high B_{1} could potentially confirm this hypothesis. Such experiments would require coherent spin manipulation by RF pulses, which are shorter in duration than the RF wave period at the very low Larmor frequencies used here, i.e., subcycle magneticresonant coherent control, which is technically feasible with modern pulse synthesizers.
Finally, the work presented here also suggests exciting challenges for materials chemists. The linewidth of the resonances in the calculated spectra of Fig. 2 is determined primarily from the finite inhomogeneous broadening arising from the residual hyperfine field strengths of the deuterated conjugated polymer. By careful design of materials, it may be possible to reduce hyperfine coupling even further, for example, by extending the πelectron system in polycyclic aromatic hydrocarbons to lower the interaction between electronic and nuclear magnetic moments. With such advances, it will be possible to resolve resonances at even lower B_{0} fields and therefore reach even higher effective ratios of B_{1} to B_{0}.
Methods
Details of the computational procedure, including the modifications made to the conventional spinpair model of spindependent recombination in an OLED^{28,41}, are given in Supplementary Note 1.5 along with a list of the model parameters used.
Device fabrication
In order to establish and detect ultrastrong electronspin magnetic resonance drive conditions, we developed a new monolithic OLED device structure with an integrated RF microwire. These small circular (57 µm diameter) singlepixel OLEDs allow for bipolar chargecarrier injection using electron and hole injector layers on top of an electrically and thermally separated thinfilm wire capable of producing RF magnetic fields B_{1} when connected to an RF source. The wire itself runs across the substrate and is shrunk down to 150 μm width beneath the OLED pixel. Details of the layer sequence necessary to achieve electrical and thermal decoupling are given in ref. ^{18}. Further details can be found in Supplementary Note 2. The active polymer, dMEHPPV, was dissolved in toluene at a concentration of 4.5 g/L at a temperature of 50–70 °C and deposited by spincasting at 800 rpm, as described in ref. ^{35}. The layer was sandwiched as a 100 nm thin film between a Ca/Al stack for electron injection and a TiO_{2} (3 nm)/Au (80 nm)/Ti (10 nm) layer covered with PEDOT:PSS (Ossila Al 4083) for hole injection. The structure is comparable to that used in previous studies of the strongdrive regime, where a different πconjugated polymer with much larger resonance linewidths was used^{24}. For the study presented here, the activelayer material, the layer stack, and the pixel device geometry were redesigned and optimized in order to allow for larger B_{1}/B_{0} ratios to be reached. The monolithic singlepixel OLED device was made following a preparation and deposition protocol as illustrated in Jamali et al.^{18} with the following changes: (i) the active polymer layer where the electronhole pair recombination takes place was nominally 100 nm thick, consisting of spincoated perdeuterated dMEHPPV; (ii) the sample template preparation included placing the silicon wafer in a dry thermal oxidation furnace at 1000 °C for 78 min, producing a 50 nm SiO_{2} layer on top of the Si wafer for insulation and better adhesion of the subsequent layer (amorphous SiN); and (iii) an entirely different lateral layout of the templates (cf. Supplementary Fig. 1a) was used compared to that described in Jamali et al.^{18}, providing larger separation between the electrical contacts of the thinfilm wire RF source and the device electrodes in order to avoid crosstalk at high RF powers and minimize heating of the OLED during the room temperature measurements. A photograph of the pixel device template is shown in Supplementary Fig. 1a with the pixel located at the image center. An example of the currentvoltage characteristics of the device at room temperature is also given in Supplementary Fig. 1b.
Ultrastrongdrive EDMR spectroscopy
The OLED was subjected to a static magnetic field B_{0} and irradiated with RF radiation with inplane amplitude B_{1} using an Agilent N5181A frequency generator and an ENI 510 L RF power amplifier as illustrated in Supplementary Fig. 1a. At the same time, a steadystate electric current was induced with a V = 2.8 V bias using a Keithley voltage source. A Stanford Research (SR570) current amplifier was used to detect changes \(\delta I\) of the steadystate forward current of I_{0} = 500 nA with and without the RF radiation applied. Measurements of \({\Delta} I(B_0,\;\sqrt P )\) as a function of B_{0} and the square root of the applied microwave power P (which is proportional to B_{1}) were then obtained, as shown in Fig. 3. Further details of the measurement procedure and the determination of the \(\sqrt P\) to B_{1} conversion factor through power broadening are given in the Supplementary Notes 3–7.
Angledependent EDMR spectroscopy of the twophoton transition
The angledependence measurements of the twophoton resonance were performed in a separate lowfield setup with different OLED samples, described in detail in refs. ^{5,42}. These OLEDs were much larger, with a pixel area of 3.5 mm^{2}. RF excitation generated by an Anritsu MG3740A signal generator and amplified by a HUBERT A 1020 RF amplifier was applied to the sample through a coplanar stripline designed to match a 50 Ω impedance. In contrast to the measurements on the monolithic OLEDmicrowire structures, these experiments were performed under constantcurrent conditions with \(I = 250\;{\upmu {\mathrm{A}}}\) applied by a Keithley 238 sourcemeasure unit, so that the EDMR signal corresponds to a voltage measurement. Lockin detection with a Stanford Research Systems SR830 DSP lockin amplifier and 100% modulation of the RF amplitude at a frequency of 232 Hz was employed. We used a 3D array of Helmholtz coils (Ferronato BH3003A), each supplied by a CAEN ELS EasyDriver 0520. This allowed us to perform sweeps of B_{0} from −2 to +2 mT with an arbitrary orientation of the field vector. Owing to the limited magneticfield range of the 3D Helmholtzcoil set, we resorted to a lower excitation frequency of 22 MHz with \(B_1 \approx 200\;\upmu{\mathrm{T}}\).
Data availability
The raw data that support the plots within this paper and the other findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The Fortran code used for the numerical simulations discussed in this paper is available from the corresponding authors upon reasonable request.
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Acknowledgements
This work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award #DESC0000909. The synthesis of the perdeuterated monomer took place at the Australian National Deuteration Facility, which is partly funded by NCRIS, an Australian Government initiative. P.L.B. is an Australian Research Council Laureate Fellow and the synthesis work was supported in part by this Fellowship (FL160100067). J.M.L. and S.B. acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—ProjectID 314695032—SFB 1277.
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S.J. designed the highpower microwire OLED structure and performed magnetic resonance spectroscopy, with help from H.M., A.N., and H.P. V.V.M. developed the Floquet simulation code and performed all calculations discussed in the paper. T.G., S.B., and S.M. performed additional supporting experiments at low fields. P.L.B. led the synthesis of the perdeuterated conjugated polymer with support from D.M.S., A.E.L., and T.A.D. J.M.L. and C.B. conceived and supervised the project, and wrote the paper with input from all authors.
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Jamali, S., Mkhitaryan, V.V., Malissa, H. et al. Floquet spin states in OLEDs. Nat Commun 12, 465 (2021). https://doi.org/10.1038/s41467020201486
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