Longitudinal and transverse electron paramagnetic resonance in a scanning tunneling microscope

Recording EPR spectra with an STM

Our EPR-STM setup is depicted in Fig. 1A: A spin-polarized tip is positioned above a magnetic adatom adsorbed on a double layer of MgO on Ag(100) (10). A magnetic field splits the atomic energy levels by the Zeeman interaction, and a resonant rf excitation induces transitions between these split states. EPR spectra are acquired by sweeping the out-of-plane B_ext while keeping the frequency ω_rf/2π constant and detecting the rf-induced change in the dc tunneling current ∆I using a modulation scheme of the rf source, followed by a lock-in detection (10). Figure 1B shows typical constant frequency EPR spectra on Fe and TiH. Following (4), EPR is detected electrically through the tunneling magnetoresistance (TMR) of the tip-adatom junction. Because the conductance of the STM junction depends on the relative alignment of the magnetic moments of the tip and adatom, the EPR dynamics induces a change of both the dc and ac TMR. The dc TMR variation is caused by the time-averaged population change of the magnetic adatom states and is detected as a change of the dc tunneling current. The ac TMR originates from the rf conductance change and gives rise to an additional (homodyne) dc tunneling current via mixing with the rf voltage. For further experimental details, see Materials and Methods and (10).

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To discriminate between different EPR mechanisms, we first have to characterize the basic elements of an EPR-STM experiment, i.e., the rf excitation, the EPR species, and the magnetic tip. The rf excitation is provided by an antenna capacitively coupled to the STM tip and is well understood from a previous study (10). In addition, Fe and TiH adatoms are two well-known, yet magnetically distinct systems (1, 2, 10). However, the structure of the magnetic tip is completely unknown and requires further characterization. To this end, we record an extensive EPR dataset on the Fe and TiH adatoms as shown in Fig. 1A using the same magnetic microtip at a similar external magnetic field, resulting in 119 spectra in the EPR-STM parameter space (see Fig. 2). Note that spectra for different V_rf values are offset for better visibility in Fig. 2 (C to D). Without any further analysis, these spectra already reveal important characteristic features of EPR-STM: (i) The amplitude and width of the EPR signal of both magnetic adatoms grow with increasing rf voltage amplitude V_rf and decreasing standoff distance s, indicating that the excitation is stronger close to the tip. (ii) The external magnetic field at resonance changes by less than 20 mT with either s or V_rf, ruling out tip and bias-induced changes of the electronic ground state of the probed magnetic adatoms. (iii) The peak-to-peak amplitude is about twice as large for TiH as for Fe. (iv) The EPR spectra of TiH have opposite sign than those of Fe (note that the Fe spectra are inverted in Fig. 2, A and C), and (v) the EPR signal line shape of TiH has a stronger dependence on s than that of Fe. (vi) The EPR signal is mostly symmetric for Fe and more asymmetric for TiH, for which the asymmetry grows with increasing V_rf. The same general features are observed when changing the STM tip, for all the six different EPR-active microtips investigated in this study (see fig. S9). These observations indicate a different nature of EPR-STM for Fe and TiH, requiring a more detailed analysis of the recorded spectra to allow for an assessment of the EPR driving forces.

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Analysis of the EPR spectra and Rabi rate

For a quantitative analysis of the mechanisms that drive the EPR of Fe and TiH, we fit the spectra in Fig. 2 using a general model of the change in tunneling current flowing between a magnetic adatom and a spin-polarized tip in the presence of an rf bias. According to (4) and as summarized in section S2, the total rf-induced current is given by∆I=Ioff−aTMRIdcΩ2T1T21+∆ω2T22+Ω2T1T2(cos α+∆ωT2Vrf2ΩT1Vdcsin α)(1)where the first term is an offset that accounts for magnetic field–independent rectified rf currents due to STM junction conductance nonlinearities. The second term describes the TMR of the STM junction modulated by spin precession. It includes a term proportional to the time-averaged projection of the atomic magnetic moment on the tip magnetization (∼ cos α) and a homodyne contribution (∼ sin α), where α is the angle between the tip magnetization and the external magnetic field (see Fig. 1A). a_TMR is a parameter that describes the TMR amplitude, and I_dc is the dc set point current. Note that, for a vanishing V_dc, the ratio I_dc/V_dc that appears in the second term approaches the dc set point conductance of the tunneling junction. T₁ and T₂ are the longitudinal and transverse spin lifetimes, respectively, and ∆ω=2μ(Bext0−Bext)/ℏ is the detuning from the external magnetic field at which the resonance occurs Bext0, with ℏ as the reduced Planck constant and μ as the adatom magnetic moment. The latter is 1 μ_B for TiH (10) and 5.2 μ_B for Fe (27).

Equation 1 was initially derived for a S = 1/2 system such as TiH, but we show below that it is also valid for higher spin systems such as Fe if the Rabi frequency Ω of an effective two-level system is appropriately renormalized by the matrix element connecting the true initial and final magnetic states, as outlined in section S2. We also note that Eq. 1 neglects a possible spin torque initialization of the magnetic adatom spin (5), which would alter the EPR signal through a change of the off-resonant magnetic state population induced by inelastic spin-flip excitations by the tunneling electrons. The impact of this effect is minimized by measuring the EPR in a relatively narrow range of the dc bias voltage with constant polarity and using rf voltage amplitudes large compared to the inelastic spin-flip thresholds. Last, our model neglects the hyperfine interaction (11), which is justified because the investigated adatoms did not show an associated broadening or splitting of the EPR lines.

Given that all the spectra in Fig. 2 were acquired with the same tip, we perform a simultaneous fit of the entire set of EPR spectra based on the following assumptions: The magnetic moments of the probed adatoms are supposed to point on average along the out-of-plane external magnetic field B_ext (see Fig. 1A), which is justified for Fe owing to its large out-of-plane anisotropy (27) and for the isotropic TiH moment if B_ext is dominant with respect to in-plane components of the tip-induced magnetic field (2). We assume that the tunneling electrons are the main source of T₁ and T₂ events due to the large values of the dc and the rf current (see fig. S3A) as also previously observed (5, 28). Thus, we set T_1,2 = e/(r_1,2I), where r_1,2 is the probability that a single tunneling electron induces a T_1,2 event and I is the total current given by the sum of the dc and the root mean square rf current I_rf, which is obtained via the dc STM junction conductance (see Materials and Methods). We further account for a fixed increase in EPR spectral line width through a convolution of the EPR spectra with a 4-mT broad Gaussian. This broadening is caused by the atom-tracking scheme, in which the tip circles atop the magnetic adatom (3 mT; as deduced from typical magnetic field gradients) and by the finite B_ext sweep rate (1 mT) (10). With these assumptions, we fit all the EPR spectra with Eq. 1 using an adatom-independent value of α, adatom-specific parameters aTMRFe,Ti, r1Fe,Ti, and r2Fe,Ti, and adatom-specific local parameters B0Fe,Ti, IoffFe,Ti, and Ω^{Fe, Ti} that depend additionally on I_dc, V_dc, and V_rf (see Materials and Methods for further details). The best fit of all the 119 EPR spectra is found for α = (64 ± 2)°, aTMRFe=0.043−0.004+0.003, r1Fe=6−1+2∙10−9, r2Fe=0.99−0.24+0.33, aTMRTi=−0.70−0.05+0.04, r1Ti=0.032−0.003+0.003, and r2Ti=1.00−0.04+0.21 (see Fig. 1B, fig. S4, and section S3). Thus, for the relaxation times, we find that nearly every tunneling electron induces a T₂ event, whereas only a small fraction of them leads to a T₁ relaxation, in agreement with previous reports for Fe (5). For TiH, on the other hand, a difference in T₁ and T₂ can arise from the different probabilities for inelastic and elastic scattering events of the spin-polarized tunneling electrons with the adatom’s spin. Note that our model neglects relaxation mediated by phonons that play a minor role because of the relatively high tunneling currents and the thin MgO support (28). The opposite sign of a_TMR for Fe and TiH reflect the opposite polarities observed in the raw data (Figs. 1B and 2).

From the above fit parameters, we derive three important quantities, namely, the line width 1+Ω2T1T2/(πT2) (Fig. 3, A and B), the spectral amplitude a_TMRI_dcΩ²T₁T₂/(1 + Ω²T₁T₂) (Fig. 3, C and D), and the asymmetry T₂V_rf/(2ΩT₁V_dc) (fig. S6A). We observe that the line width grows almost linearly with the rf voltage amplitude V_rf at constant I_dc, which is a consequence of being in the strong-driving regime, i.e., Ω²T₁T₂ ⪆ 1. This is consistent with the saturated amplitude for Fe for all V_rf and with the saturating amplitudes for TiH at the two lowest values of I_dc (see Fig. 3, C and D). For TiH and the highest value of I_dc = 120 pA, that is for the smallest standoff distance s, we do not observe saturation of the amplitude at large V_rf. This finding might indicate a change of the TiH magnetization orientation due to an increased magnitude of the in-plane tip–magnetic field that is not included in our analysis. The asymmetry of the EPR signal of TiH (fig. S6B) grows linearly with V_rf and strongly depends on I_dc reflecting the intricate dependence of the Rabi rate on I_dc discussed below. Fe spectra show nearly symmetrical line shapes and accordingly have vanishing asymmetries (fig. S6A), which is consistent with previous studies (5) and can be understood by the long T₁, i.e., small r₁ of Fe compared to TiH. In essence, this difference arises because a tunneling electron can induce a direct transition in TiH, which corresponds to a spin excitation with ΔS = 1, whereas Fe has a large spin and orbital moment that cannot be directly excited by a single electron. The long T₁ of Fe suppresses the asymmetric EPR line shape originating from the homodyne component of Eq. 1. The shorter T₁ of TiH, on the other hand, gives an asymmetric line shape as also reported previously (2). Last, the experimental Rabi rate Ω is given in Fig. 3 (E and F) and ranges from about 100 MHz for TiH to about 1 MHz for Fe, consistent with the literature (12). This information allows us to perform a qualitative assessment of the different proposed EPR-STM mechanisms, as described below.

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Assessment of different EPR-STM mechanisms

We now contrast the observations summarized in Figs. 2 and 3 with the expectations for different excitation models of EPR-STM.

1) A Rabi rate Ω induced by the ac magnetic field originating from the rf tunneling current and the rf displacement current has been discarded previously by estimating the respective magnitudes (7, 10). In addition, we note that both contributions should not depend strongly on the standoff distance s, contrary to our measurements (see Fig. 2). Moreover, the rf magnetic field caused by the displacement current should depend monotonically on s, unlike what we observe for Fe in Fig. 3E. In addition, the displacement current should be proportional to the frequency ω_rf, which is not consistent with EPR measurements performed at different ω_rf values.

2) A spin torque–mediated EPR (24) is expected to be proportional to I_rf and independent of s. Such a mechanism is unlikely, given the strong dependence of Ω on s at constant dc set point current (see fig. S6, E and F).

3) A purely rf electric field–driven EPR-STM, in which rf-induced spin-polarized tunneling electrons couple via the exchange interaction to the adatom magnetic moment, has been proposed in (20). This coupling can be understood as a current-induced effective magnetic field driving the EPR. However, this mechanism can be discarded because it fails to explain EPR in half-integer spin systems such as TiH.

4) A change of the crystal field caused by adatom vibrations induced by the rf electric field (1, 22) should yield a Rabi rate that depends monotonically on s, unlike what is observed for Fe in Fig. 3E. Moreover, our multiplet calculations (see Fig. 4, Materials and Methods, section S6, and fig. S8) indicate that the crystal field operators yield vanishing EPR driving forces for Fe. Nevertheless, rf-induced variations of the crystal field could yield minor contributions to the Rabi rate in the case of TiH.

5) A modulation of the density of states by the precessing spin of the magnetic adatom mediated by spin-orbit coupling (19) can be ruled out because it should be observable even with a nonmagnetic tip. This is not observed experimentally and is inconsistent with the results presented in Fig. 5 (A and B), which show that the resonance field depends on the distance between the magnetic tip and the EPR species.

6) A modulation of the g-factor anisotropy of the EPR species by the vibrations induced by the rf electric field should lead to a Rabi rate that depends monotonically on s because the driving electric field is proportional to 1/s in a simple plate capacitor model (25). This is in contrast with our experimental findings for Ω shown in Fig. 3E.

7) In the PEC model (22), Ω is expected to be proportional to the conductance of the STM junction if the adatom-tip interaction is dominated by the exchange interaction (8). This prediction is partly inconsistent with our experimental Ω (see fig. S6, C and D), which might indicate an additional tip-adatom interaction such as dipolar coupling (see below). Apart from that, the PEC mechanism implies complex dependencies of Ω on the experimental parameters I_dc, V_dc, s, and V_rf (7) that require a quantitative evaluation.

8) A cotunneling mechanism (23) and an open quantum system approach (26) have been proposed to describe the excitation and detection of EPR, respectively. Testing these approaches requires a detailed knowledge of the wave functions of the tip and EPR species that is experimentally difficult to obtain. However, as we will discuss later, these approaches represent more general descriptions that include some of the other mechanisms.

On the basis of this analysis, we conclude that mechanisms (1 to 6) are not compatible with our experimental dataset. Further evaluation of (7) and of EPR-STM in general requires quantitative knowledge of the involved transition matrix elements and of the total magnetic field acting on the EPR species. We focus here on the most relevant magnetic moment operator mediating EPR (see below) but discuss further operators in section S6. Our analysis goes beyond an ideal S = 1/2 system because EPR encompasses a much larger variety of magnetic complexes with S > 1/2. It is thus important to determine what drives the EPR of Fe on MgO, which is known to have S = 2 (27), to reach a comprehensive understanding of EPR-STM.

Transition matrix elements of EPR-STM for atoms with S>1/2

To drive EPR, we consider a perturbative oscillating magnetic field B₁ acting on the magnetic adatom. The B₁ field interacts with the magnetic moment of the adatom μ̂=−μB(L̂+2Ŝ)/ℏ via the Zeeman interaction, and the corresponding interaction Hamiltonian reads H′=μB(L̂+2Ŝ)∙B1/ℏ, where Ŝ and L̂ are the spin and orbital angular momentum operators, respectively. In the derivation of Eq. 1 for single TiH adatoms, B₁ was assumed to be transverse to the static magnetic field B₀ inducing the Zeeman splitting of the adatom’s states, as in the standard two-level model of EPR [(4) and section S2]. This assumption, however, has not been tested in detail and (4) makes no predictions about the requirements on B₁ to drive EPR in Fe. For TiH on the bridge binding site (see Fig. 1A), we assume a nearly perfect physical S = 1/2 system due to the low binding site symmetry. Accordingly, the two lowest states ∣0⟩ and ∣1⟩ are the LS-basis states ∣M_L = 0, M_S = ± 1/2⟩ with quenched orbital moment, as reported previously (2, 10). Those states are the natural eigenstates of the L̂ and Ŝ operators and the interaction Hamiltonian becomes H′=2μBℏ(ŜxB1,x+ŜyB1,y+ŜzB1,z). Because Ŝz is diagonal in the ∣M_L = 0, M_S = ± 1/2⟩ basis, it is evident that only transverse B₁ fields can drive a transition between ∣0⟩ and ∣1⟩. For instance, for a transverse B₁ field along x, the off-diagonal matrix element that drives the transition is H01′=μBB1,x(t) because Ŝx=ℏ2σ̂x with σ̂x being the x component of the vector of Pauli matrices σ̂. Note that the transverse field oscillates in time proportionally to cosω_rft, i.e., B_{1, x}(t) = B_{1, x} cos ω_rft and that its amplitude relates to the Rabi rate according to ℏΩ = μ_BB_{1, x}.

The situation is more complex for Fe, which has a state multiplicity of 5 due to the effective spin S = 2, and the presence of strong orbital moments (see Fig. 4A). At zero magnetic field, the ground-state doublet is separated by about 14 meV from the next excited doublet (27). Therefore, only the two lowest states are thermally occupied in the range of temperature and magnetic field probed by our experiments. Transitions to higher doublets are too high in energy to be driven by the rf excitation. This renders Fe also an effective two-level system. Within this effective two-level system, we need to evaluate the interaction Hamiltonian H′ in the eigenstate basis ∣0⟩ and ∣1⟩, which are a general superposition of the ∣M_L, M_S⟩ basis wave functions. To describe the states ∣0⟩ and ∣1⟩, we use the wave functions obtained from a multiplet model that was successfully used to simultaneously describe the x-ray absorption spectral line shape and the low-energy excitations of Fe/MgO probed by STM (27). We find that the off-diagonal matrix elements H01′ are proportional to B_{1, z}, whereas the in-plane field components, B_{1, x} and B_{1, y}, yield vanishing matrix elements (see Materials and Methods and section S6 for more details). That is, only the z component of the magnetic moment operator yields an off-diagonal matrix element ⟨1∣L̂z+2Ŝz∣0⟩≠0. This is known in EPR spectroscopy to be the case for integer spins where longitudinal B₁ fields are used to drive the EPR transition (29). Further, the matrix element is strongly dependent on the Zeeman-splitting field B₀, as shown in Fig. 4B. Fe behaves as an integer-spin system, in which the levels are strongly intermixed by the crystal field and spin-orbit interaction (Fig. 4C). This leads to wave functions that are not eigenfunctions of the Zeeman Hamiltonian; thus, the composition of the states ∣0⟩ and ∣1⟩ changes with the external magnetic field. Setting again ℏΩ equals to the amplitude of H01′, we see that the Rabi rate Ω, besides being proportional to the z component of the B₁ field, is also proportional to the matrix element ⟨1∣L̂z+2Ŝz∣0⟩.

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Further, we describe the effective two-level system for Fe, not in terms of the magnetic moment operator, but by the two-level polarization operator P̂∝σ̂ (see section S6). Following this approach, we can model the two EPR species using the same model for the tunneling current while taking into account that the origin of the Rabi rate is different for the two species. We derive the Bloch equations in terms of the polarization vector P̂ with a driving term proportional to P̂x and with an effective driving field strength given by μB⟨1∣L̂z+2Ŝz∣0⟩B1,z/ℏ. Moreover, in the evaluation of the TMR for the read out of the EPR signal in the STM junction, we use the polarization vector P̂ instead of the physical magnetic moment of the system (see section S2) because the conductance of the STM junction should only depend on the occupation and coherence of the involved EPR states (26). This approach reflects the fact that the conductance of the STM junction depends on the nature of the magnetic adatom states and not only on the associated magnetic moment. Note that, for a real S = 1/2 system, the polarization operator is identical to the spin operator.

Thus, we obtain formally the same equation, Eq. 1, for the experimentally detected EPR signal for the two EPR species, Fe and TiH. However, the physical interpretation of the effective driving field component and strength that yield Ω differs for the two cases. In summary, our analysis shows that EPR in systems with S > 1/2 can be driven by STM, provided that longitudinal field gradients are nonzero. Note that the small in-plane magnetic field component of the tip produces negligible matrix elements for the in-plane magnetic moment operator as compared to its z component (see fig. S8).

Magnetic field acting on the adatoms

At the position of the EPR species, the total magnetic field is the sum of the external magnetic field B_ext and the tip-induced effective magnetic field B_eff. Quantitative analysis of the Rabi rate requires estimation of B_eff acting on the EPR species. Here, we determine B_eff by considering the measured resonance positions Bext0, i.e., the value of B_ext at resonance, as shown in Fig. 5 (A and B). The intrinsic resonance position in the absence of a tip-induced magnetic field is given by 2ℏω_rf/μ, which yields 247 mT for Fe at ωrfFe/2π=36 GHz and 286 mT for TiH at ωrfTi/2π=8 GHz. The measured EPR resonance position deviates from these values as a function of the standoff distance s. These deviations are caused by the finite B_eff produced by the tip. The upturn of Bext0 at s ≈ 420 pm for Fe indicates that the magnetic force changes from attractive to repulsive upon approaching this specific tip (Fig. 5A), which is unexpected if the interaction only contains an exchange contribution as determined in previous studies (2, 7, 8). This finding indicates that the tip magnetic field comprises two competing terms, which we assume to be an exchange field B_xc and an additional dipolar field B_dip. These two fields were shown to be present independently from one another for certain STM tips in (6) and were also discussed but not taken into account simultaneously in (4). We note that previous studies using an atomic force microscope with a magnetic tip (30) have shown that the exchange interaction might change sign depending on the overlap of the tip and the magnetic adatom wave functions. Here, however, we find that the dipolar field in addition to an exponentially decaying exchange field is sufficient to account for the observed change of B_eff without considering more complex exchange regimes. Given the cylindrical symmetry of the STM junction, it is sufficient to determine the x and z components of B_eff, which can be written as (22)Beff=(Bxc,x+Bdip,xBxc,z+Bdip,z)=((ae−s/λxc−bs3)sin α(ae−s/λxc+2bs3)cos α)(2)where a is the exchange parameter, b is the dipolar parameter, s is the tip standoff distance defined through point contact between the tip and the magnetic adatom (see Materials and Methods), and λ_xc is the exchange decay length. Note that we orient the coordinate system such that B_{eff, y} = dB_{eff, y}/ds = 0 along the z axis and for x = 0. We fit Bext0(s) using B_{eff, z} + 2ℏω_rf/μ (see Eq. 2) with a fixed α = 64° as determined above and find a good agreement between experiment and theory for λxcFe=(370±60) pm, a^Fe = ( − 0.6 ± 0.1) T, λxcTi=(170±20) pm, a^Ti = ( − 2.2 ± 0.1) T, and b = (0.2 ± 0.03)μ₀μ_B (see Fig. 5, A and B). The parameter b implies a tip magnetic moment of about 3 μ_B, which is reasonable, given that few Fe atoms form the tip apex. The values for a compare well with reports of the exchange field, ranging from 0.1 to 10 T for similar systems (4, 7, 31). For all six tips used within this work (see fig. S9), we find values of a < 0 independent of the adatom. The values for λ_xc are somewhat larger than reported values (4, 7, 31), but λ_xc is expected to strongly depend on the detailed atomic structure of the microtip. In this way, we completely characterize B_eff for this magnetic microtip, which is shown for Fe in Fig. 5C, assuming an isotropic exchange interaction. This allows us to derive the corresponding magnetic field gradients along z as shown in Fig. 5D containing substantial contributions from dipolar and exchange tip-adatom interactions at the same time.

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Quantitative evaluation of the PEC Rabi rate and comparison with experiment

Knowledge of the transition matrix elements and B_eff is essential to compute the Rabi rate expected for the PEC model [see section S2 and (22)], which is given byΩPECFe,TiH=∣μBℏ∆zωrfdBeffz,xds⟨1∣L̂+2Ŝ∣0⟩∣(3)

Here, different components of the magnetic field gradient drive Fe and TiH as discussed above. In more detail, the field that drives EPR is given by B1Fe=∆zωrfFedBeffFez/ds and B1TiH=∆zωrfTiHdBeffTiHx/ds, where ∆z_ωrf is the amplitude of the magnetic adatom displacement induced by the rf electric field between tip and adatom. To compute ∆z_ωrf, we calculate the structural response of the adatoms to a static electric field applied normal to the surface by means of density functional theory (DFT) (for details of the calculations see Materials and Methods). Because the adsorbate species Fe and TiH form a polar bond to the MgO substrate, an external electric field can displace the adatoms and vary the length of the bond to the surface. As the frequencies of the local vibrational modes of the adatoms lie at several terahertz [see Materials and Methods, fig. S7, and previous work (7)], we expect ∆z_ωrf to adiabatically adjust to the gigahertz electric fields, justifying our static approach in the calculations. As seen in Fig. 6 (A and B), the Fe and Ti adatoms are both displaced by about 0.5 pm/(V/nm) but in opposite directions. The opposite response appears as a 180° phase change in the driving terms and has no consequences for the measurements. Note that the inverted EPR spectra of Fe and TiH instead originate from the sign change in a_TMR. We observe that the displacement does not depend linearly on the electric field but follows a second-order polynomial (see also fig. S7). This result is rationalized by noting that the linear response is due to the coupling to permanent electric dipoles, whereas the second-order term arises from a coupling to induced electric dipoles that has (refers to “a coupling”) not been reported before (7). Last, ∆z_ωrf is obtained by considering only the terms oscillating at the fundamental frequency ω_rf derived from the second-order polynomial fit of the displacement (see Fig. 6, A and B), i.e., neglecting time-independent offsets and terms oscillating at 2ω_rf, and using the experimental electric field E = [V_dc + V_rf cos (ω_rft)]/s (see section S5).

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After these steps, we can lastly compute the PEC Rabi rate Ω_PEC using Eq. 3, as shown in Fig. 6 (F and G). We find that the calculated values of Ω_PEC match the experimental Rabi rate Ω reported in Fig. 3 (E and F) for TiH and only deviate by a factor of 2 for Fe. Notably, Ω_PEC describes the Ω(s) dependence adequately for Fe, i.e., changing from decreasing to slightly increasing at s ≈ 420 pm. This change in slope arises from the differences in the distance dependence of the exchange and dipolar interactions with the adatom. In addition, for TiH, the decreasing trend of Ω with s is reproduced correctly. Discrepancies in the magnitude are ascribed to an inaccurate determination of the electric field, which was shown to deviate from the plate capacitor model used here (32). Moreover, keeping the adatom magnetic moment fixed along z is especially critical for TiH at small standoff distances and can lead to errors. Last, including a finite phase between driving field and the precessing magnetic adatom spin, as well as a bias-dependent TMR and a spin-torque initialization (5) could further improve the agreement with the experiment.