/Coherent electric field manipulation of Fe3+ spins in PbTiO3 (via Qpute.com)

## Abstract

Magnetoelectrics, materials that exhibit coupling between magnetic and electric degrees of freedom, not only offer a rich environment for studying the fundamental materials physics of spin-charge coupling but also present opportunities for future information technology paradigms. We present results of electric field manipulation of spins in a ferroelectric medium using dilute ferric ion–doped lead titanate as a model system. Combining first-principles calculations and electron paramagnetic resonance (EPR), we show that the ferric ion spins are preferentially aligned perpendicular to the ferroelectric polar axis, which we can manipulate using an electric field. We also demonstrate coherent control of the phase of spin superpositions by applying electric field pulses during time-resolved EPR measurements. Our results suggest a new pathway toward the manipulation of spins for quantum and classical spintronics.

## INTRODUCTION

Manipulation of magnetism by an electric field instead of a magnetic field is driving substantial research activity in condensed matter physics, motivated not only by the intriguing fundamental materials physics but also by the potential applications in future low-power spintronics (18). On the basis of insulating multiferroic heterostructures, one is able to switch the magnetization by an electric field rather than electric current, thus providing a pathway to significantly reduce energy consumption (1). In the past decade, numerous pathways for electric field control of magnetism have been envisaged, and several interesting physical mechanisms have also been revealed, including interfacial strain/stress coupling (9, 10), charge mediation (1113), exchange bias or coupling at ferromagnetic-antiferromagnetic interfaces (1417), manipulation of state of 3d orbitals (18, 19), etc.

However, all these approaches are mainly focused on the manipulation of the magnetization in materials with macroscopic long-range magnetic order (i.e., ferro/ferri/antiferromagnets). As a step toward understanding the fundamental limits of these magnetoelectric coupling phenomena, one could ask the question: Is it possible to manipulate isolated spins, as opposed to long-range ordered states, using an electric field? In addition to presenting an intriguing fundamental science challenge with the potential to affect quantum computing, it would be valuable to manipulate spin-spin interactions and the spins of individual atoms or ions (20). Recently, electric field control of individual molecular magnets and tuning of magnetic exchange in a molecular system were achieved (2123). These results show the fascinating potential for spin-electric coupling (24), which belongs to the family of magnetoelectric effects.

Complex oxides that include transition metal or rare earth ions can have charge, spin, orbital, and lattice degrees of freedom that interact and give high tunability at a microscopic level for desirable macroscopic properties. For example, the strong polarization and lattice control in perovskite oxides have enabled the strain coupling of ferroelectricity and multiferroic orders, which is generating emerging logic and storage approaches (1). The role of aliovalent cationic impurities, such as Fe3+ in PbTiO3, has been studied extensively from the perspective of how the impurities interact with ferroelectric domain walls and, thus, influence the switching and degradation mechanisms (25, 26). In particular, electron paramagnetic resonance (EPR) has been extensively used to probe the electronic structure of these impurities. However, there has been no work focused on understanding the spin state and manipulating the spins in these impurities using an electric field.

With this as the background, we demonstrate coherent electric field manipulation of spins doped into a prototypical ferroelectric, PbTiO3 in which Fe3+ (high spin S = 5/2, 3d5 electronic structure) is inserted into the Ti4+ site (27, 28). We found that the spins in the Fe3+ ions are oriented normal to the dz2 orbital direction as a consequence of spin-orbit coupling within the Fe3+ ion, while the orbitals are coupled to the tetragonal polar lattice distortion that is characteristic of this ferroelectric. We demonstrate through both ab initio calculations and EPR-based experiments that the spins can be switched in a nonvolatile fashion and coherently manipulated with an electric field applied to the ferroelectric layer. The results described in this work are representative of a large class of materials in which the polar order and the accompanying coupling to the spins can be systematically tuned and manipulated, thus opening a new door to explore pathways for full electric field control of spins in individual ions. Although our work is focused on exploring the limits of magnetoelectric coupling down to single ions, this work has possible implications for potential quantum computing and spintronics applications.

## RESULTS

Ab initio calculations were performed within the density functional theory (DFT) framework to get insight into the interaction between electric polarization and spin behavior. Fe3+ doping was modeled with 1 Ti4+ of 27 replaced by Fe3+ (details of the calculations are provided in Materials and Methods). The spin axis was varied relative to the ferroelectric dipole as depicted in Fig. 1A, allowing a comparison of energies for different spin directions. We considered two cases: first, a direct replacement of a Ti4+ with a Fe3+ and, second, a representative Fe-O defect complex to account for the valence difference. For the first case, we find that in the polar state with the tetragonal lattice distortion in the [001] direction, the lowest-energy spin direction is perpendicular to the ferroelectric dipole. The easy plane is perpendicular to the tetragonal distortion, with a calculated magnetocrystalline anisotropy energy (MCAE) of 530 μeV between the parallel and perpendicular directions, in reasonable agreement with the experimental value of 720 μeV (extracted from EPR measurements; see below). Varying the spin axis within the (001) plane gives an energy difference of only 5 μeV between the [100] and [110] directions, thus resulting in a spin easy plane. The presence of the FeTi/O-vacancy defect complex reduces the MCAE to ~300 μeV, but the spin easy plane remains in the plane perpendicular to the ferroelectric dipole (Fig. 1B).

We then studied the influence of ferroelectric dipole rotation (through the application of an electric field) on the magnetic anisotropy. In the nonpolar, cubic structure, the spin axis was degenerate as it was varied in the (001) plane (Fig. 1B). In this case, the high-spin Fe 3d5 orbital momentum is quenched in the Oh crystal field, resulting in no preferred spin axis. With a tensile strain along the c axis to vary the magnitude of the ferroelectric dipole, the displacement of Fe along the c axis results in a spin easy plane in the ab plane. This ferroelectric distortion breaks the parity symmetry of orbitals orthogonal to this displacement, inducing an orbital moment and results in a spin easy plane perpendicular to the ferroelectric distortion. Although there is C4 rotational symmetry around the polar axis, the charge density (shown in fig. S1A) is rather isotropic, resulting in effectively degenerate and isotropic spin axes in-plane. Therefore, we find that the MCAE is proportional to the tetragonality as shown in Fig. 1C. In switching the ferroelectric dipole from [001] to [010], the structure passes through an intermediate monoclinic phase in which the dipole aligns along the [011] direction (as described in the Materials and Methods). This intermediate monoclinic structure has an MCAE of 164 μeV with a spin easy axis along the [100] direction, confirming that the spin direction still remains perpendicular to the ferroelectric dipole. Thus, we conclude that the combination of spin-orbit coupling and the distorted crystal field provided by the symmetry-breaking ferroelectric displacement determines the magnetocrystalline anisotropy.

We performed experiments on weakly Fe3+-doped [20 to 50 parts per million (ppm)] PbTiO3 single crystals using EPR spectroscopy as shown in Fig. 2. A typical EPR spectrum is shown in Fig. 2A (right); the magnetic field is applied parallel to [100], i.e., perpendicular to the c axis. A single strong line is visible, and there are several weaker subsidiary lines. We can assign these to transitions between mS states within the S = 5/2 multiplet, where mS represents good quantum numbers for a magnetic field applied along the c axis. The strong line corresponds to the −1/2 ↔ 1/2 transition; a somewhat weaker line corresponds to the −3/2 ↔ 3/2 “forbidden” transition. We assign the observed spectrum to isolated Fe3+ ions, which replace Ti4+ ions with charge compensation far away from the paramagnetic ion. This is based on the fact that the tetragonal crystal field constant of this ion approaches zero when the temperature increases to the cubic-tetragonal phase transition, i.e., it follows well the c/a-1 ratio (27). In contrast, the FeTi/O-vacancy center, which may also occur in PbTiO3:Fe (28), exhibits a tetragonal constant that is much less sensitive to temperature as the vacancy creates a strong tetragonal distortion in vicinity of a paramagnetic ion, even in the cubic phase (29).

The large tetragonal crystal field of the PbTiO3 lattice (27, 28) mixes states within the S = 5/2 multiplet when the magnetic field deviates from the tetragonal axis, so the states involved in the transitions are not pure mS = ±1/2 and ± 3/2. This, in turn, leads to a strong dependence of the magnetic field position of the resonances as a function of the orientation of the magnetic field with respect to the crystal axes. This is demonstrated by the angular dependences of the resonance fields measured under rotation of the magnetic field, H, from the [001] to the [100] direction (Fig. 2A, left).

The effective g factor for the −1/2 ↔ 1/2 transition varies between geff = 5.970 for Hc (angle θ = 90°, corresponding to an EPR transition at Hres = 1123 Oe) and g = 2.0056 (Hres = 3342.5 Oe) for H // c (angle θ = 0°). Thus, the H-field position of the EPR transition offers us a probe of the local orientation of the ferroelectric polarization; conversely, by manipulating the ferroelectric polarization, we have control over the Fe3+ spin Hamiltonian. In particular, the strongest line in the Fe3+ spectrum corresponds to spins located in the domains with c // [001]. The other two lines (two orders weaker in intensity: blue and green lines in Fig. 2A, left) arise from spins in 90° domains with c axes in the (001) plane (i.e., a minority fraction of 90° domains in the crystal).

The angle dependence of the EPR spectra allows us to experimentally determine the spin anisotropy, as parameterized by the Hamiltonian (Eq. 1 in Materials and Methods) whose dominant anisotropy term is of the form −DSz2. The solid lines in Fig. 2A (left) are fits to the data yielding D = 120 μeV, i.e., an easy plane. This corresponds to a zero–magnetic field splitting between the mS = ±5/2 and mS = ±1/2 doublets of 720 μeV, which corresponds favorably with the calculations of the MCAE reported above.

This anisotropy experienced by the spins is imposed by the local ferroelectric order. It allows us to control the anisotropy axis by manipulating the ferroelectric polarization direction via application of an external electric field that exceeds the coercive field for 90° ferroelastic switching. We demonstrate this (see Fig. 2B) by recording a sequence of EPR spectra with the magnetic field parallel to [001]. The upper trace is for the pristine sample for which the c axis is along [001]. The middle trace shows the spectrum following application of an electric field of 28 kV/cm along [100] to switch the c axis perpendicular to the magnetic field. The process is fully controllable, as demonstrated by the lower trace, which follows application of the electric field of 28 kV/cm along [001] to restore the original electric polarization axis. The electric polarization in each state is stable in time, so this process amounts to nonvolatile controlled configuration of the spin Hamiltonian using an electric field.

Note that, here, we used weakly Fe-doped crystals in which the Fe3+ spins may be considered as isolated noninteracting spins under the experimental conditions we studied. For a given concentration of Fe, spin-spin interactions (predominantly dipolar for low concentrations and with an exchange contribution at sufficiently high concentrations) with an energy scale J will give rise to spin correlations below temperatures of order J/kB (where kB is the Boltzmann’s constant), which is in the microkelvin range for the samples we study here. We expect any such correlations to be sensitive to the ferroelectric order.

To determine the transverse spin relaxation (or phase memory) time T2, we performed EPR spin echo measurements (details in Materials and Methods and fig. S2) on the single crystal sample. Figure 3A shows spin coherence times as a function of temperature for several of the electron spin resonance transitions. These are derived from exponential fits, Aecho(τ) ∝ exp. (−2τ/T2) to the Hahn echo decays. A typical Hahn echo decay is shown in the inset, demonstrating a simple exponential dependence of the echo amplitude on the delay time τ; the fast modulations of the echo arise from the interaction of Fe3+ spins with 207Pb and 47,49Ti nuclear spins [electron spin echo envelope modulation (ESEEM)].

When H is parallel to the c axis, the π/2 microwave pulse generates a coherence between ±1/2 states whose energy separation is determined almost entirely by gz and the phase memory time is as long as 2 μs at 12 K. However, when the magnetic field is oriented perpendicular to the c axis and the anisotropy terms in the spin Hamiltonian mix states within the S = 5/2 multiplet, the transition energy depends also on these anisotropy energies. Fluctuations in these terms will contribute to phase relaxation, explaining why the phase memory time is significantly shorter for this H-field orientation. Overall, the coherence time in our system (reaching ~9 μs for the high-field, +1/2 ↔ −3/2 transition at low temperatures) is of the same order of magnitude as some of the currently proposed spin qubits (30, 31). The anisotropy in T2 illustrates that the fluctuations driving phase relaxation are also highly dependent on H-field orientation. The presence of ESEEM in the echo decay indicates that the magnetic nuclei 207Pb and 47,49Ti may also be involved in the phase relaxation and that the coherence is likely to be enhanced by isotopic purification of the PbTiO3. Figure 3B shows Rabi oscillations, demonstrating that coherent manipulation of these spin states can be achieved via microwave pulse control.

The fact that the spin-bearing Fe3+ ions lie in an environment that lacks inversion symmetry provides an alternative means of coherently controlling the quantum spin states via the linear electric field effect (32). We investigate this interaction by inserting an electric field pulse into a standard Hahn echo EPR sequence (Fig. 4B and fig. S3). The effect for E//c axis is shown in Fig. 4C. The low-field −1/2 ↔ +1/2 transition (32) shows little modification with the application of an electric field because it only weakly depends on crystal field modification. By contrast, a pronounced modulation of the spin echo transient is observed for the high-field +1/2 ↔ −3/2 transition (Fig. 4A) when the electric field pulse is applied. The integrated in-phase echo shows a cosine-shaped oscillation pattern superimposed onto a quasi-exponential decay as tE increases from zero to tE = τ (i.e., 4 μs), while the quadrature component of the echo exhibits a sine dependence on tE, also overlaid onto a quasi-exponential decay. This oscillation is due to an E field–induced change in the EPR resonance frequency via electric field modulation of the axial magnetocrystalline anisotropy, leading to a phase shift in the echo signal (32, 33). The decay in the echo amplitude is due to the inhomogeneity in the electric field across the crystal, resulting from the irregularity crystal shape and the high relative permittivity for PbTiO3. The echo subsequently recovers to its original value when tE = 2τ (i.e., 8 μs), as the phase shift induced by the electric field during the first period of free evolution (between the π/2 and π pulses) is reversed by the π pulse and progressively canceled. This confirms that the interaction between the spins and the electric field is coherent.

## DISCUSSION

A linear dependence of the spin echo intensity on the electric field is expected because of the lack of inversion symmetry in the local coordination environment for Fe3+. This is confirmed in Fig. 4D, which shows a linear dependence of the change in the EPR resonance frequency and the amplitude of the applied electric field. Furthermore, the difference between the −1/2 ↔ +1/2 and +1/2 ↔ −3/2 transitions indicates that the axial anisotropy parameter, D, is sensitive to the applied E field, while the Lande g factor is insensitive to this stimulus. This is further confirmed by measuring the electric field effect for the 1/2 ↔ −3/2 and −3/2 ↔ +3/2 (Hc) transitions (fig. S4). A similar dependence has been reported for Mn2+ in ZnO (33). However, note that using a ferroelectric as the host medium, the spin-electric coupling coefficient is three times larger than that of ZnO:Mn2+ and significantly larger than for paramagnetic dopants in silicon (see Materials and Methods and fig. S5) (34). The ferroelectric medium provides a mechanism for dynamically configuring the anisotropy for a given spin using local electrostatic gates. This raises the possibility of a field-programmable gate array, in which an array of bits can be dynamically configured in the hardware (schematically proposed in fig. S6). There are proposed quantum computing schemes that exploit heterogeneous qubits (35, 36); magnetic defects in ferroelectric hosts could offer a physical context in which these ideas may be explored. Our experimental observations may stimulate theoretical developments leveraging the spin configurability that we have demonstrated.

In summary, we present the first observations of electric field–dependent magnetocrystalline anisotropy and coherent spin manipulation in ferroelectric Fe-doped PbTiO3. EPR spin echo measurements reveal spin coherence lifetime of the order of a few microseconds. We believe that this can be significantly enhanced by appropriate design of the ferroelectric material (e.g., its tetragonality, the chemical species, isotopic purity, etc.). More broadly, a wide spectrum of ferroelectric materials, spanning inorganic oxides to polymeric systems such as polyvinylidene fluoride (37), presents an attractive platform to explore electric field control of magnetic properties for quantum or classical spintronics applications.

## MATERIALS AND METHODS

### DFT calculations

DFT calculations were carried out on 3 × 3 × 3 supercells of PbTiO3 (135 atoms) including one Fe ion on a Ti site with one electron added to correctly compensate the Fe3+ charge state (FeTi). The Fe-Fe distance of 11.625 Å ensured that the dopant ions were sufficiently isolated from each other. The Fe ion/O-vacancy defect complex was considered with a doubly ionized vacancy (VO··) on the site of the apical oxygen with the shortest Fe─O bond length. The Vienna Ab initio Simulation Package (3840) was used with projector augmented wave pseudo-potentials (41, 42) including Pb d,s,p; Ti s,p,d; Fe d,s; and O s,p electrons as valence. A plane wave cutoff energy of 750 eV was used with a 2 × 2 × 2 Gamma-centered k-point grid, which converged the total energy to 1 meV per formula unit. The PBEsol functional was used which gave PbTiO3 lattice parameters within 1% of experimental values. Atomic positions were relaxed within spin-polarized calculations until the residual forces on each atom were less than 0.01 eV/Å. For optimization of the doped structure with tetragonal ferroelectric displacement, the volume and cell shape were held fixed to those of undoped PbTiO3, while for the cubic structure, the symmetry was also constrained. To generate the monoclinic structure, starting from the PbTiO3 cubic unit cell, Ti was displaced along the [011] direction, and the ion positions, cell volume and shape were allowed to relax within symmetry constraints until the forces on each atom were less than 0.001 eV/Å. From this structure, a 3 × 3 × 3 supercell was constructed, and 1 Ti of 27 was replaced with an Fe ion. The ionic positions were again optimized with the dopant present to a force convergence of 0.02 eV/Å, with symmetry constrained and supercell volume and shape fixed.

The magnetocrystalline anisotropy was calculated by including spin-orbit coupling self consistently and varying the spin quantization axes. An effective Hubbard term Ueff = UJ = 4 eV was added to the Fe d orbitals within the Dudarev approach (43). The effect of the exchange-correction (J) parameter was tested for J = 0 to 1.8 eV with U = 4 eV and was found to increase the MCAE (fig. S1) but did not change the spin easy axis and thus did not affect the magnetic ground state.

### Sample preparation

The PbTiO3 single crystal used in pulsed EPR measurements was grown by a flux method using the B2O3-PbO melt containing stoichiometric amounts of PbCO3 and TiO2. The crystal had a majority of 180° domains oriented parallel to the crystal c axis perpendicular to the main face, and the content of 90° domains was less than 1%. Fe3+ was present in the crystal as accidental impurity at a concentration of 20 to 50 ppm. The Fe3+ concentration was estimated from Fe3+ EPR intensity in the usual way by comparing the EPR intensity in our crystals with the EPR intensity of Cr3+ in a MgO reference sample with known concentration of Cr3+. The typical crystal size in measurements was about 2 mm by 3 mm by 0.35 mm. For experiments with 90° switching of the polarization, the crystal dimensions were 1.2 mm by 0.7 mm by 0.5 mm, with the surfaces parallel to crystallographic (100) planes. To apply electric fields at two perpendicular directions, [100] and [001], electrodes were applied to the corresponding crystal surfaces using flexible silver paste (SPI-paint). The crystal with electrodes was also electrically insulated by picein compound, allowing application of electric fields up to 30 kV/cm.

### EPR analysis

The EPR spectra for PbTiO3 single crystal were carried out using Bruker E580 spectrometer. The following spectrometer parameters were used: a microwave frequency of 9.34 to 9.44 GHz, a temperature range of 3.5 to 296 K, a field modulation amplitude of 1.5 G at 100 kHz, and a microwave power of 0.2 mW. Pulsed EPR measurements to obtain spin memory time were also carried out using a Bruker E580 spectrometer and standard Bruker MD5 resonator with the resonance frequency of 9.75 GHz. Spin echo decay as a function of the pulse separation τ between π/2 and π microwave pulses was recorded in the Hahn echo pulse sequence π/2-τ-π-τ-echo (44). The length of π/2 pulse was 16 ns and τ varied from 200 to 20,000 ns with an increment of 4 ns.

EPR spectra were interpreted using spin Hamiltonian of the general form for Fe3+ ion (3d5, S = 5/2) in the tetragonal symmetry

$H=13DO20+1180FO40+1120a(O40+5O44)+GzHzSz+GxHxSx+GyHySy$

(1)with Gi = βgi and the usual definitions of the

$Olm$

operators. Here, D and F parameters correspond to axial (tetragonal) crystal fields of the second and fourth degree, respectively, with the z axis chosen along the direction of the axial crystal field, i.e., along the tetragonal c axis. a is the cubic crystal field splitting parameter. The zero-field splitting parameters determined for bulk crystal at both the room temperature (T = 296 K) and low temperatures (T < 20 K). At the room temperature, D = 0.9650 cm−1, a = 0.01 cm−1, and F = −0.0108 cm−1. Below 20 K, D = 1.18 cm−1, a = 0.056 cm−1, and F = −0.094 cm−1. The g factor is almost isotropic: gz = 2.0056 and gx,y = 2.009. The temperature dependences of the zero-field splitting parameters are consistent with those reported in (27, 28).

### Electric field dependent EPR analysis

The zero-field splitting of the Fe3+ energy levels, i.e., the parameter D in Eq. 1, can also depend on the external electric field, as we have shown by direct measurement of electric field influence on the zero-field splitting parameter. Measurements were performed for PbTiO3 crystal with c domain structure at different temperatures by applying an electric field parallel to the c axis. The magnetic field H was applied in two directions: (i) 15° away from the c axis to measure the spectral line of the high-field +1/2↔ −3/2 transitions (Fig. 4A) and (ii) perpendicular to the c axis to measure the −3/2↔ +3/2 transition (fig. S4C). All these transitions are sensitive to small modifications in the D parameter (fig. S2, E and F). For example, fig. S5 shows the EPR line under the application of an electric field in the range +30.3 to −24.2 kV/cm. The spectral lines linearly shift in the electric field. Note that value of the negative electric field was restricted to −24 kV/cm as at higher fields the electric polarization starts to switch to the opposite direction, leading to a reverse of the EPR line shift as well. Using the data of fig. S5, the spin-electric coupling coefficient was calculated from the relation: D(E) = D(E = 0) + κE, where κ = 10 Hz V−1 m and D(E = 0) = 32.740 GHz. This is a relatively large value, compared, for instance, with the equivalent parameter in ZnO:Mn2+, which is three times smaller (33). We note that the spin-electric coupling observed in Fig. 4 (κ = 1.7 Hz V−1 m) is weaker than this by a factor of approximately 6. This is because, in the experiment reported in Fig. 4, the electrodes applying the electric field pulses were insulated from the PbTiO3 crystal; in this configuration, because of the large relative permittivity, which for PbTiO3 is εr ≈ 50 (45), the electric field leads to a polarization surface charge that significantly screens the field in the bulk. In addition, this, combined with the unevenness in the thickness of the crystal, caused the inhomogeneity in the electric field and the rapid decay in the echo signal shown in Fig. 4C. In contrast, the data in fig. S5 were obtained with the electrodes in direct contact with the PbTiO3 crystal surface, thus suppressing the screening effect of the surface charge.

### Comparison with other solid-state quantum systems

The coherent spin-electric field coupling raises the possibility of quantum spin control with electric fields in PbTiO3. Here, we discuss the electric field required for nontrivial spin operations using both the nonresonant (7) and resonant (2) approaches.

We first consider the nonresonant approach for local spins control. Kane proposed an A-gate (7) in which a local electric field is applied to shift the resonance frequency of a spin out of/into the band width of a global microwave radiation pulse, hence achieving selective manipulation for identical spin qubits. We consider this approach for the −3/2 ↔ +3/2 transition of Fe3+ in PbTiO3 with the magnetic field applied perpendicular c axis. The electric field measurement (fig. S5A) shows that the transition frequency, f34, is tuned by the externally applied field such that δf34/E = 4 Hz V−1 m, where δf34 = δf34(E) − δf34(E = 0) is the change in the EPR frequency. For a microwave pulse of 100 ns (~ T2/100), a DC electric field of 25 kV/cm is sufficient to tune the spin on/off resonance with the global microwave radiation. Such an electric field is easy to access as demonstrated experimentally in our work. Note that the duration of the electric field would be extremely short (100 ns). Therefore, it would not change the local polarization axis as in the experiments shown in Fig. 2.

We now discuss the feasibility of generating a superposition of eigenstates with a resonant high-frequency electric field. An oscillating electric field at the frequency of fE leads to a time dependent anisotropy D(t) = D(E = 0) + κEcos(2πft). This time-dependent

term can excite a spin transition when the resonance condition, fE = f34, is fulfilled. The Rabi rate of this electric field–driven transition is given by

$fRabi=∣κE⟨ms=−3/2∣Ŝz2∣ms=+3/2⟩∣$

(2)where ∣ms = − 3/2⟩ and ∣ms = + 3/2⟩ correspond to the wave functions for the states.

Figure S5B shows the transition rate in Eq. 2 as a function of the angle between the H field and the c axis at 8000 Oe. A maximum Rabi rate of fRabi = 0.194κE = 1.94 Hz V−1 m is found when the field is applied 70° away from the c axis. This implies that an oscillating electric field of 50 kV/cm would give rise to a Rabi frequency of 10 MHz, already 100 times faster than the decoherence rate of the −3/2 ↔ +3/2 EPR transition.

Acknowledgments: Funding: The work at UC Berkeley was supported in part by the Center for Probabilistic Spin Logic for Low-Energy Boolean and Non-Boolean Computing (CAPSL), one of the Nanoelectronic Computing Research (nCORE) Centers as task 2759.002, and a Semiconductor Research Corporation (SRC) program sponsored by the NSF through CCF 1739635. Partial support from the U.S. Department of Energy (DOE) EFRC on Quantum Coherence is acknowledged. The materials theory/calculation work (to K.I. and S.M.G.) was supported by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under DOE contract no. DE-AC02-05CH11231. E.S. acknowledges support from the U.S.-Irish Fulbright Commission and work supported by the Air Force Office of Scientific Research under award number FA9550-18-1-0480. Computational resources were provided by the National Energy Research Scientific Computing Center and the Molecular Foundry, DOE Office of Science User Facilities supported by the Office of Science, DOE under contract no. DE-AC02-05CH11231. The work performed at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the DOE under the same contract. V.V.L. thanks the support from the Ministry of Education, Youth, and Sports of Czech Republic under project no. SOLID21-CZ.02.1.01/0.0/0.0/16_019/0000760. A.A. acknowledges support from the Engineering and Physical Sciences Research Council (EP/P000479/1), the QuantERA European Project SUMO, and the European Union’s Horizon 2020 research and innovation program under grant agreement numbers 863098 (SPRING) and 862893 (FATMOLS). R.C. thanks the Division of Chemical Sciences, Geosciences, and Biosciences of the DOE (contract no. DE-AC02-05CH11231). Author contributions: R.R., A.A., V.V.L., S.M.G., S.D., and W.H. designed the experiments. J.L. and V.V.L. performed EPR and spin echo measurement for the single crystals, along with the detailed analyses. K.I., E.S., and S.M.G. performed DFT calculations. R.R., S.M.G., W.H., J.L., S.D., K.I., A.A., and V.V.L. analyzed the data and cowrote the manuscript. R.R., S.M.G., and A.A. supervised the research. All authors contributed to the discussions and manuscript preparation. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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