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1 In the format provided by the authors and unedited. DOI: /NPHYS3965 Control of the millisecond spin lifetime of an electrically probed atom William Paul 1, Kai Yang 1,2, Susanne Baumann 1,3, Niklas Romming 4, Taeyoung Choi 1, Christopher P. Lutz 1, and Andreas J. Heinrich 1,5 1 IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, California 95120, USA. 2 School of Physical Sciences and Key Laboratory of Vacuum Physics, University of Chinese Academy of Sciences, Beijing , China. 3 Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland. 4 Department of Physics, University of Hamburg, Hamburg, Germany. 5 Physics Department, Ewha Womans University, Seoul, Republic of Korea (present address). Table of contents 1. Relaxation rates Relaxation by inelastic electron scattering Tunnel barrier transmission factors Quantum states and inelastic scattering probability Layer-dependence of quantum states and transition intensities Rates with applied bias voltage Temperature dependence Point contact conductance with different tips Lifetime on multi-layer MgO Inelastic electron scattering applied to multi-layer films Additional non-electronic rate Lifetime at B = 2 T Magnetic field orientation and T1 vs B saturation STM topography Thickness of MgO films Point contact conductance of Fe atoms on 1-4 ML MgO films STM topography of 1 and 2 ML MgO films dz/dv of 1 and 2 ML MgO films References NATURE PHYSICS 1

2 1. Relaxation rates The Fe atom under study is placed in close proximity to metallic electrodes that act as reservoirs of electrons available for inelastic collisions with the Fe spin. The total rate of transitions from state 1 to 0, = 1, is expressed as the sum of independent transition rates due to electron scattering and non-electronic processes : = 1 =. (S1) The main goal of this paper is to experimentally tune by adjusting the strength of the coupling to nearby electrodes; hence we focus on the first term, the electronic scattering rate. In the following sections, we will describe the quantum spin states, the transition intensities between them, the derivation of our model for the spin lifetime as a function of the tip-sample conductance - (a key experimental parameter), and the approximations we have made. 1.1 Relaxation by inelastic electron scattering Relaxation of the excited spin state can occur by = 0, 1 transitions when an electron from either the tip or sample scatters inelastically off the atom and returns to either the tip or sample electrode. The scattering electron may exchange both energy and angular momentum (changing the magnetic quantum number by = 1) or may only exchange energy ( = 0). This process creates an electron-hole pair in the tip or sample electrode, or split between both. These processes are described by inelastic electron tunneling and occur even for the case of zero bias voltage between the electrodes: The electron (from a filled state within energy of the Fermi energy) gains energy from the spin and can then enter an empty state above the Fermi energy. The rate of transitions between state 1 and 0 caused by inelastic electron scattering is given by the sum of rates over pairs of electrodes (s = sample, t = tip) = 1 = (S2) Each individual rate (generalized to electrodes m and n) can be expressed using the conventional model for inelastic tunneling as - = - ( ) 1 ( ) -. (S3) Here, we have approximated the integral over the Fermi-Dirac distributions ( ) as in the low temperature limit where. The zero-bias elastic conductance between electrodes m and n is -. Together, the factor - gives an attempt rate at which NATURE PHYSICS 2

3 electrons are available for scattering. A small fraction of this rate, described by the inelastic scattering probability, is the inelastic rate, i.e. the rate of the 1 0 transitions. The inelastic scattering probability contains the quantum mechanical transition intensity and the spin-polarized densities of states of the electrodes and is introduced in detail in Supplementary Section 1.3. In this work, we take the spin polarized densities of states to be constant in the small range of energies probed around the Fermi level. The model includes single scattering events only; higher order scattering processes are not considered here. Eqs. (S2) and (S3) can be combined to obtain the rate as a function of the inelastic scattering probabilities and the conductance between electrodes: = 1 = { }. (S4) In general, the value of - is specific to each pair of electrodes because it depends on their spin polarized densities of states. Following the discussion in Supplementary Section 1.3, we expect the inelastic scattering probabilities to be nearly independent of electrode pair (a variation of only ~10% is expected), hence we make the approximation that = - is constant for any pair of electrodes, and factor it from the sum: = 1 = { } (S5) The tip-tip conductance - can be re-written - - (derived in Section 1.2), and we take - = - to obtain the electronic contribution to the spin relaxation rate in the form it appears in the main text: = 1 = (S6) We note that the experimentally set - is established at a sample bias voltage of 10 mv, and thus does not correspond to the true zero-bias conductance as employed in Eq. (S3). We refer the reader to the conductance spectra of Figure S2, which show that the junction conductance changes by only a few percent in the region < 14 mv. An additional uncertainty of ~15% in - is expected due to the procedure of opening the constant-current feedback loop using either the ground-state (for long T 1 where two-state switching was visible) or using the timeaveraged magnetic state (for T 1 shorter than our detection bandwidth). 1.2 Tunnel barrier transmission factors We seek to understand the conductances in the model in terms of generic tunnel barrier properties; here we will express the tip-tip conductance - in terms of the tip-sample conductance - (an experimental parameter) and the sample-sample conductance - (a property of the insulating film that is constant for a given thickness). This can be done by NATURE PHYSICS 3

4 considering the transmission factors for the tip-side (vacuum) and sample-side (MgO) tunnel barriers as follows. The tip-sample conductance depends on the transmission through both tip- and sampleside tunnel barriers. Let us define that the probability of electron tunneling decays at rate over distance on the tip side of the atom, and at rate over distance on the sample side of the atom. These parameters are shown in the experimental schematic in Figure S1. We can express the tip-sample conductance as - = =, (S7) where we have defined the transmission factors for the tip- and sample- side of the atom to be = and = respectively. is the metallic conductance of the atom, which is expected to be of the order of the quantum of conductance = 2 = 77.5 S. reflects the number of open conductance channels and may be somewhat larger than if several conductance channels contribute. Figure S1: Schematic of transmission through tunneling barriers on tip- and sample- side of the Fe adatom on MgO. Similarly, we can write the sample-sample conductance as two passes through the sampleside tunnel barrier, - = ) =, (S8) and the tip-tip conductance as two passes through the tip-side tunnel barrier: - = ) =. (S9) Combining Eqs. (S7) through (S9) yields the desired expression for - needed to derive Eq. (S6) in the previous section: - = -. (S10) - We turn now to the sample conductance, which we experimentally determine by bringing the tip into point contact with the Fe atom. This conductance corresponds to a single pass through the MgO tunnel barrier, which is expressed by the transmission factor as = =, (S11) NATURE PHYSICS 4

5 Eq. (S8) and (S11) can be combined to obtain an expression for the sample-sample conductance - as a function of the single-pass sample conductance - =. (S12) We have summarized these conductance expressions in Table S1. sample-sample tip-sample tip-tip sample Table S1: Summary of conductance expressions - = = - = - = = - - = In the derivation of Eqs. (S6)-(S12), we have assumed that the metallic conductance that multiplies the transmission factors is constant, i.e. independent of the particular conductance pathway (tip-tip vs. sample-sample). We expect to lie in the range of a few, reflecting the number of open conduction channels. Variation depending on the chemical nature of the last atom on the tip, for example, is expected. We observe variations of 30 in for different tip terminations (Supplementary Sections 3 & 7.1). 1.3 Quantum states and inelastic scattering probability We now discuss the inelastic scattering probability - that describes the probability per available electron that the 1 0 spin transition takes place 1 3. In other words, this is the probability that an electron tunnels inelastically between electrodes m and n, and in so doing, absorbs energy from the excited Fe spin. We will see that most electrons interact elastically with the Fe spin the probability of inelastic scattering - is quite small. It will be shown that the inelastic scattering probability is roughly independent of electrode pair, and thus appears outside the brackets of Eq. (S5). The principal ingredients of are the quantum mechanical transition intensity between states 1 and 0 and the spin-polarized densities of states of the electrodes and, ( ) and ( ), which are assumed to be constant as a function of energy near the Fermi level. We write - for electrode pair - as - = 1 ( ) ( ),,. (S13) The summation is over all combinations of initial and final tunneling electron spin states. The transition intensity function is based on the exchange interaction between the Fe spin and the tunneling electron ( ):, = 0, 1,. (S14) NATURE PHYSICS 5

6 Here,, is the product state of the Fe spin = { 0, 1 } and the tunneling electron spin = {, } (expressed in the z basis). We can expand the sum in Eq. (S13) over all possible initial and final tunneling electron spin states (we now denote spin polarization of the densities of states in superscript for brevity): = 1 { 0, 1, + 0, 1, + 0, 1, + 0, 1, } (S15) The first two terms describe = 0 transitions, where the spin of the tunneling electron is conserved: no angular momentum is exchanged between the Fe spin and the tunneling electron (these transitions are made possible in the Fe spin due to the fact that they are not eigenstates of (fourth order mixing)). We will see shortly that = 0 transitions are the dominant inelastic transitions between the Fe ground states. The second two terms describe = 1 transitions in which the tunneling electron reverses its spin and exchanges angular momentum with the Fe spin. We can simplify the transition intensities by writing the exchange operator in terms of raising and lowering operators in the z basis as = + ( + ) = 0, 1, = 0, 1, , + 1, = 0 1 (S16) = 0, 1, = 0 1 (S17) = 0, 1, = 1 4 0, + 1, (S18) = 0, 1, = 1 4 0, + 1, (S19) and, which describe = 1 transitions that flip the spin of a tunneling electron, are precisely zero for the Spin Hamiltonian introduced in the main text, = + ( + ) +. This is because the 0 and 1 states have no amplitude in the basis differing by = 1. For the parameters = 4.7 me, = 41 ne, and = 2.6, at = 2 T, the eigenstates are: 0 = = = 2 (S20) 1 = = = 2 (S21) NATURE PHYSICS 6

7 The transition intensities for = 0 and = 1 transitions are summarized in Table S2 for several values of in-plane field, while keeping the out-of-plane field constant at 2 T. At zero lateral field, where the eigenvectors are as shown in Eqs. (S20)-(S21), the = 1 transition intensity is zero. In our experiment, there is a finite ~10 angle (Supplementary Section 5) between the applied field and the easy axis of magnetization, which corresponds to the in-plane field condition = 0.35 T. The effect of the lateral field is minor on the state mixing and the = 1 transition intensity remains seven orders of magnitude less than the = 0 transition intensity, yielding no measurable effect on T 1. For the extreme case of = = 2 T, = 1 transitions still underwhelm = 0 transitions by a several orders of magnitude. =2T = 0 T =2T = 0.35 T =2T = 2 T = 0 = = 1 = Table S2: Summary of = 0 and = 1 transitions between ground states of the Fe atom. Calculated with the effective spin amiltonian with parameters = 4.7 me, = 41 ne, and = 2.6. For finite, we take = 2 for the in-plane direction. For our Fe atom with strong easy axis anisotropy, we will approximate that = =. This simplifies the sum in Eq. (S15) to only the first two terms: (S22) Our next task is to investigate the spin-polarized densities of states and. We define the degree of spin polarization by the parameter ( 1 +1) such that the density of up electrons is = and the density of down electrons is = 1 2. A typical spin polarized ST tip has 0.3. For the non-spin-polarized Ag(001) substrate we take = 0. The spin polarized densities of states for tip and sample are thus: = 1+ 2 = 1 2 = 1 2 = 1 2. (S23) - for each pair of sample and tip electrodes, as they appear in Eq. (S4), are thus: - = = (S24) - = = = (S25) = (S26) NATURE PHYSICS 7

8 - = = (S27) Interestingly, it is only in the case of tip-tip tunneling that the tip spin polarization affects the probability of a = 0 transition occurring. For a maximally spin-polarized tip ( = 1) this leads to a factor of two enhancement compared to the non-spin-polarized ( = 0) case. In our experiments, this effect is expected to be less than ~10% for a typical spin-polarization of 0.3, hence we work under the approximation that = - = - = - = -, used in Eq. (S5). To summarize, transitions between ground states of the Fe atom are dominated by those that exchange only energy but not angular momentum with the tunneling electron ( = 0). The spin polarization of the tip introduces a minor correction to the inelastic scattering probability for tip-tip tunneling. For the spin of Fe on MgO, we can work under the assumption that Layer-dependence of quantum states and transition intensities In the extrapolation of our electronic de-excitation model to multiple layers of MgO (main text Figure 4b), we have assumed that the inelastic scattering probability is constant as a function of layer thickness. We expect this assumption to hold true if the bonding geometry of Fe, which sets up the crystal field and the resulting spin states, is independent of layer thickness. To check the validity of this assumption, we experimentally asses the dependence of the zero-field splitting of the spin states on MgO film thickness by inelastic electron tunneling spectroscopy (IETS). Variations in the zero-field splitting ( 0 2 and 1 3 energies) reflect changes in the Hamiltonian parameters as a function of film thickness. From these variations, we can estimate the change to the inelastic scattering probability 0 1. From the weak dependence of the zero-field splitting, we infer that varies by at most ~30% in moving from the 2 ML to 3 ML thick MgO film. Before we compare IETS spectra of Fe atoms on MgO films of different thickness, we must first look at the tip-atom distance dependence in the IETS spectra. When the STM tip is brought into close proximity with the Fe atom, forces acting on the atom distort the crystal field and thus the energies of the magnetic states. So, in order to directly compare the multilayer IETS data, we must position the tip at a similar tip-atom separation. The influence of tip proximity on IETS step energies is illustrated in Figure S2a for Fe on 2 ML MgO. We set the tip-atom distance by the tunneling current setpoint before opening the feedback loop to acquire the spectra. The conductance step at 14.5 mv, corresponding to the zero field splitting 3D (D is the out-of-plane anisotropy parameter), is roughly constant at large tip-atom distance (currents of pa at 10 mv). The step energy increases to 15.2 mev at closest approach (1.5 na). At 1.5 na, the tip is about 1 Å from point contact to the Fe atom (see I(z) traces in the main text Figure 4a). NATURE PHYSICS 8

9 To assess the dependence of the IETS step on MgO layer thickness, we took IETS spectra at comparable tip-atom distance for Fe on 2, 3, and 4 ML MgO films about 1 Å tip-atom separation. This was done by opening the feedback loop at currents of 1.5 na, 150 pa, and 17 pa, such that the tip-atom distance was about 1 Å for each measurement (see I(z), Figure S6). A much larger tip-atom gap would be difficult to realize due to the already low current condition for the 4 ML film (17 pa at 10 mv). Figure S2b shows that the conductance step decreases in energy by 2.3 mv as the MgO thickness is increased from 2 to 4 ML. The subtle changes in bonding geometry influence the crystal field of the Fe atom, thus the anisotropy parameters of our spin Hamiltonian are slightly modified. From the variation of the zero field splitting energy, we conclude that the anisotropy parameters change by ~ 15% from 2 ML MgO to 3 ML MgO, and ~2% from 3ML to 4 ML MgO. We will next estimate the change to the transition intensities given similar variations of the Hamiltonian parameters. Figure S2: (a) di/dv on Fe on 2ML MgO as a function of tip height (set by tunneling current as shown). The spin excitation step moves from 14.5 mv to 15.2 mv at closest approach to the atom (dark blue). At 1.5 na at 10 mv, the tip apex is about 1 Å from point contact with the Fe atom. (B = 0 T, T = 1.2, modulation voltage 400 V rms at 806 Hz) (b) di/dv on Fe on 2 to 4 ML MgO. Tip height is about 1 Å from point contact in all cases. Open loop conditions are 1.5 na at 10 mv (2 ML), 150 pa at 10 mv (3 ML), 17 pa at 10 mv (4 ML). (B = 0 T, T = 1.2, modulation voltage 250 V rms at 806 Hz) NATURE PHYSICS 9

10 Figure S3 Transition intensities for 0 (blue) and 1 (orange) transitions as a function of spin Hamiltonian parameters over a range of 15% of their nominal value (3 14 mev, 41 nev). (a) and (b) Variation of parameter for (a) Bz = 2 T, Bx = 0 T and (b) Bz = 2 T, Bx = 0.35 T. (c) and (d) Variation of C parameter for (c) Bz = 2 T, Bx = 0 T and (d) Bz = 2 T, Bx = 0.35 T. We vary the anisotropy parameters D and C by 15% (an amount comparable to the experimentally observed change in zero field splitting) and plot the calculated transition intensities for 0 and 1 transitions in Figure S3 (blue and orange points, respectively). The out-of-plane anisotropy parameter has no effect on the transition intensities, Figure S3a and b. The in-plane anisotropy parameter, which sets the degree of spin mixing between the 0 and 1 states, changes the transition intensity for 0 transitions by ~30% for a 15% variation in C, Figure S3c and d. These calculations were performed for a pure out-of-plane magnetic field, Figure S3a and c, as well as a 10 tilt with respect to the easy axis, Figure S3b and d. No effect of the magnetic field alignment is observed. In summary, we expect that the layer-dependent crystal field will lead to changes in the transition intensity between the ground states of the Fe atom, on the order of ~30% as we move from 2 ML to thicker films. Because the observed lifetime is more strongly limited for thicker layers by a non-electronic process (see discussion on elbow conductance in Supplementary Section 4), we do not take this variation into account. NATURE PHYSICS 10

11 1.5 Rates with applied bias voltage The majority of the discussion of this paper involves lifetime with zero applied bias between the tip and sample this corresponds experimentally to the conditions of a pumpprobe experiment, where the dynamical evolution of the spin is measured in the absence of applied tip-sample bias. The conditions are quite different for the two-state switching data presented in Fig. 1d of the main text in which there was 10 mv DC bias voltage applied to the sample electrode. The bias voltage enabled us to detect the spin state through the tunnel current, but the presence of these electrons also changed the dynamics and occupation of the spin states. Here we extend the zero-bias relaxation model introduced in Section 1.1 to consider the additional transition rates due to the tunneling electrons and observe the effect on the equilibrium occupation of the states. The relaxation rate due to electrons scattering between tip t and sample s electrodes was given by the sum of rates over the pairs Eq. (S2). We previously considered that the reverse (excitation) rate was negligible because we worked at zero bias and low temperature. Here, we will have to consider both relaxation and excitation rates, which we denote by the state transitions indicated in superscripts. The relaxation rate is, and the excitation rate is : (S28) (S29) Each rate component - accounts for electrons transitioning from filled states in electrode m to empty states in electrode n. We generalize our previous zero-bias description (Eq. (S3)) by accounting for the applied bias voltage between electrodes -. We assume the conventions that the inelastic transition energy is positive (as in Eq. (S3)) and the elementary charge e is also positive. For the relaxation (1 0) process, each rate component is given by - - ( ) 1 ( - ). (S30) Similarly, for the excitation (0 1) process, - - ( ) 1 ( - ). (S31) At the low-temperature limit (described earlier for zero bias by Eqs. (S3) (S6)), we can readily calculate the rate components found in Eq. (S28) & (S29). For the case of voltages less than the excitation energy ( ) the total relaxation rate is unaffected by the NATURE PHYSICS 11

12 applied voltage and reduces to the rate given by Eq. (S6), and the excitation rate remains zero. In the following we treat the case of low temperature ( ) and voltages exceeding the excitation energy ( ), which describes the conditions of the two-state switching of Fig. 1d. The relaxation rate with an applied sample bias = - = - is then given by ( + ) - + = ( + ) (S32) Note that one of the tunneling rate components is zero (either - =0, or - = 0) depending on the sign of V. In our example of tunnel-current driven two-state switching, this relaxation rate (out of the excited state) corresponds to the inverse of the lifetime of the excited state : 1 =. Similarly, the excitation rate due to the applied bias is 0+0+ ( ) - +0 = ( ) -. (S33) In our example of tunnel-current driven two-state switching, this excitation rate (for exciting the spin out of the ground state) corresponds to the inverse of the lifetime of the ground state: 1 =. We plot the ground state and excited state lifetimes in Figure S4 based on this derivation. The data points and black fit line correspond to the zero-bias lifetimes treated in the main text (Fig. 2c). To this, we have added the calculated and for an applied bias of = 10 mv. The two-state switching measurement, Fig. 1d, was conducted at - 10 S. At this conductance, we expect to measure an excited-state lifetime of the order 2.5 times shorter than in pump-probe measurements. This is in fair agreement with the 1.5 ms lifetime extracted from the two-state switching and the ~3 ms lifetime obtained from pump-probe measurements interpolated for = 2.5 T (Fig. 3). The driven transition rates also allow us to gain insight into the equilibrium population of the spin state as a function of tunnel current parameters. Limiting our discussion to just the two lowest energy states (valid for bias voltages less than the 14 mev spin excitation energy), we can write down the detailed balance condition for the occupation of states 0 and 1 =, (S34) where and are the occupations of states 0 and 1. The ratio of the state occupation as a function of bias voltage and tip-sample conductance - can then be obtained by inserting Eq. (S32) & (S33) into (S34): NATURE PHYSICS 12

13 = ( ) ( + ) - - (S35) Figure S4: (a) Spin lifetime at zero bias (black) fit to data from Fig. 2c ( = 2T). The zero bias model is extended to include tip-sample tunneling current due to a sample bias of = 10 mv. The calculated lifetime with applied bias is shown for the ground state (blue) and excited state (red) separately. (b) Tunnel-current driven occupation of the excited state calculated for = 10 mv and = 0. The tunnel-current induced spin heating is maximal at the lifetime shoulder, where the hot tip-sample electrons represent a maximal fraction of the impinging electrons. The occupation of the excited state = ( + )=1 (1+ ) is plotted in Figure S4b. Notably, spin heating is maximal at the position of the conductance elbow, where the tipsample tunneling is maximized compared to sample-sample or tip-tip scattering. We note that the occupation of the excited state is somewhat higher than seen in the histograms of Fig. 1d, and could be due to the presence of a non-electronic spin relaxation rate which was not taken into account in the present discussion (discussed in Fig. 4). Another source of discrepancy could be the relative strength of the conduction pathways which modify NATURE PHYSICS 13

14 the assumption that (Supplementary Section 1.2). In other words, the substrate-substrate tunneling conductance might not be as strong as estimated by our simple model. NATURE PHYSICS 14

15 2. Temperature dependence Our experiments were conducted at low temperature (0.6 or 1.2 Kelvin, as indicated in figure captions) where. Within experimental uncertainty, the spin lifetime was found to be independent of temperature in the range investigated. This is illustrated in Figure S5 for Fe on 3 ML MgO (also up to 4.2 K). Consequently, for these temperatures, occupied phonon states and hot electrons do not contribute to relaxation of the Fe spin state. The independence of temperature is implied in our approximation that the integral over filled and empty states in q. (S2) is equal to. The exact solution to the integral is (, ) = ( ) 1 ( ) = 1. (S36) At our lowest applied field of = 0.2 T (Figure 3 main text), we begin to deviate slightly from the approximation that. et at = 0.6 K, where = 2.4, the integral deviates from the zero temperature approximation by only 10%, hence the lifetime is mostly unaffected even at this low field. Despite the small difference, we included the effect of finite temperature (exact solution to the integral) in our calculation of shown in Figure 3 for completeness. Figure S5: Spin lifetime as a function of tip-sample conductance measured on Fe (atom H see inset topography) at temperatures of 0.6, 1.2, and 4.2 K. The spin lifetime of another Fe (atom I) at 1.2 K is also plotted, demonstrating that the measured lifetime is independent of the particular atom. STM topography taken at constant current (4 pa, 100 mv). Atoms appear elongated due to a blunt STM tip. 3 ML MgO at B = 5 T. NATURE PHYSICS 15

16 3. Point contact conductance with different tips The conductance measured at point contact with the Fe atom varied from tip to tip, presumably due to the detailed atomic structure and chemical nature of the last atom on the tip apex. In Figure S6, we compare I(z) spectra taken with two different tips. The dashed curves (tip 1) are reproduced from the main text, and the solid curves are data taken with a different STM tip apex (tip 2). Tip 1 was indented into the bare Ag substrate until a sharp apex was formed. Therefore, it is likely that tip 1 terminates with an Ag atom. Tip 2 was fabricated by transferring a Fe atom to the tip apex to form a spin polarized tip. It is likely that the Fe atom resides at or near the apex of tip 2. No changes to the tip were induced by the point contact experiments, and tip 2 remained spin polarized. Figure S6: I(z) traces acquired on Fe atoms on 2-4 ML MgO films at sample bias voltage of 10 mv. Curves obtained from two different STM tips are plotted: Data from tip 1 (dotted lines) are shown in the main text (Figure 4a). Tip 2 yields a comparable conductance at point contact as tip 1, but is systematically lower by ~30%. The same Fe atom was used for both 4 ML traces. All other Fe atoms were different. B = 6 T for tip 1, B = 6 T for tip 2. NATURE PHYSICS 16

17 4. Lifetime on multi-layer MgO 4.1 Inelastic electron scattering applied to multi-layer films In this section, we discuss the inelastic electron scattering model in more detail, in particular, the relationship between the substrate conductance and the model s prediction of the spin lifetime for multi-layer films. We start with the electronic relaxation model, Eq. (S6) and use Eq. (S12) to replace - by = 1 = (S37) The above expression is the foundation of the multi-layer model where the only thicknessdependent parameter is the conductance of the Mg layer. The inelastic scattering probability, the energy gap, and the metallic conductance are assumed to be independent of the layer thickness (Supplementary Section 1). The scattering probability and the energy gap depend on B. In this work, and are assumed to be independent of B. Figure S7 shows the calculated electron scattering limited lifetime for 1 4 ML insulating films following from Eq. (S37). For the sake of appreciating the general features of the lifetime as a function of insulating film thickness, we have chosen nominal values for that scale by a factor of 10 per monolayer (rather than the experimentally measured factor of ~9). We take = 10 and - = 10, where denotes the number of monolayers. These are explicitly tabulated in Table S3. insulator thickness (ML) sample conductance sample-sample conductance - = Table S3: Nominal conductance parameters used for calculations presented in Figure S7. The vertical axis in Figure S7 is the T 1 lifetime scaled to the prefactor of Eq. (S37), in other words, we take = 1. The hori ontal axis is the tip-sample conductance - in units of. At the right-most side of the plot, asterisks mark the metallic conductance value. For any given layer thickness, the maximal tip sample conductance is. When - =, the STM tip is in point-contact with the Fe atom; thus no higher - can be accessed and we plot the calculated T 1 only up to this point. This conductance value is indicated by the crosses and dashed vertical lines. The calculated lifetimes for each film thickness are shown by solid lines. For each layer, the lifetime evolves from a constant value at low tip-sample conductance to 1 - at high NATURE PHYSICS 17

18 tip-sample conductance (slope 2 on the log-log plot). An elbow occurs for each lifetime curve at - = - = -. The elbows are indicated by circles and correspond to the intersection of the substrate- and tip- limited lifetime curves. In the following discussion, we will refer to the elbow position as -. We note that an important relationship between the sample conductance and the lifetime elbow conductance - is visually illustrated by the logarithmic horizontal axis. From Eq. (S12), we have that is the geometric mean between the sample-sample conductance and the metallic contact conductance: = - G. (S38) This is graphically evinced by the equidistant spacing of the elbow - to the sample conductance ( +) and the sample conductance to the metallic contact conductance (+ ). We demonstrate this explicitly by the orange arrows for the 4 ML film. Figure S7: Using multi-layer insulating films to protect excited states from electron scattering. The lifetime is scaled to units of (i.e. prefactor of Eq. (S37) taken to be unity). The tip sample conductance is in units of, the metallic point contact conductance. NATURE PHYSICS 18

19 In the regime of sample-limited lifetime, - -, the lifetime depends only on samplesample scattering and is independent of the tip height. The lifetime here is proportional to that is, from layer to layer, a two-order of magnitude increase in T 1 is predicted by the electron scattering model. In the regime of tip-limited lifetime, - - where 1 -, the horizontal separation of the lifetime curves is determined by the values of each layer. This attribute is reflected in our experimental data in Figure 4 of the main text. The horizontal separation of the fitted curves is set by the experimentally determined substrate conductance for each MgO thickness. The excellent agreement of the experimental data to the model in the - - region supports our assumption that the inelastic scattering probability does not change significantly between thicknesses. If had varied significantly from layer to layer, each individual curve would be shifted vertically, and the horizontal separation in the - - region would not match the spacings, rendering the model fit poor. For the multi-layer fit, the inelastic scattering probability fit parameter multiplies all lifetime curves, and corresponds to a vertical shift of the whole group of curves on the log-log plot. The value of the metallic conductance fit parameter shifts all of the curves in the horizontal direction. In any case, lifetime curves for each thickness are fixed relative to each other by the substrate conductances ; the free fit parameters correspond only to translations of the whole group. 4.2 Additional non-electronic rate Experimentally, we find that the measured T 1 follows the electronic relaxation model very well for - -. The observed 1 - provides strong evidence for relaxation due to inelastic electron tunneling that requires two passes through the tip-side tunnel barrier (Eqs. (S8) (S10)). owever, at low -, we do not find that the intrinsic (plateau) lifetime follows instead, T 1 saturates at a lower value. We account for this effect in our fitting by adding a non-electronic relaxation rate (here assumed to be independent of MgO thickness) to the electronic relaxation rate. This additional constant rate, as per Eq. (S1), sets an upper limit to T 1. The experimental data suggests that the spin lifetime of Fe on 2 ML is not limited by the non-electronic rate because the plateau lifetime is substantially shorter than that of Fe on 3 or 4 ML. This argument does not directly exclude the possibility that depends strongly on film thickness; however, the value of the metallic contact conductance (fit parameter) and the position of the lifetime elbow contribute additional evidence that the lifetime of Fe on 2 ML of MgO is limited by electron scattering, as we now discuss. To see this, we first observe that in the case with large, the measured T 1 elbow moves to higher conductance. This is illustrated in Figure S8 where the position of the elbow moves to the right of the electronic-scattering only elbow, which occurs at -. This means that if T 1 of Fe on 2 ML were severely limited by a non-electronic rate, - would occur at much lower conductance than the measured elbow. In order to explain a lower conductance for -, we would need a larger value of the metallic contact conductance for NATURE PHYSICS 19

20 a given substrate conductance. This is because of the relationship - =. The fit parameter was found to be 4.9 ± 0.6 for the data in the main text, and 3.9 ± 1.4 in an additional data set (Supplementary Section 4.3). These values are already approaching the maximum conductance achievable for fully open channels of a 3d metal atom 4 ( 5 ), so there is little room for to increase. For example, in order to explain a non-electronic rate 10 greater than the electronic rate for Fe on 2 ML of MgO, would be unphysically large, at more than 12. We therefore conclude that T 1 of Fe on 2 ML is mostly limited by scattering of substrate electrons. Figure S8: Effect of a non-electronic relaxation rate (here 10 the rate of electronic-relaxation) on the measured elbow position. The measured elbow moves to higher conductance in the presence of an additional relaxation rate. 4.3 Lifetime at B = 2 T We present another set of data for the lifetime dependence on MgO layer thickness, this time taken on different atoms at B = 2 T, which shows the same qualitative features as the data taken at B = 5 T in the main text. This data is shown in Figure S9 and is fit with the same procedure as described in the main text, using the same point contact conductance data to set - for each film thickness. s fit parameters, we find = (1.4 ± 0.3) 10 and = 3.9 ± 1.4, and non-electronic rate = 30 ± 10 Hz. NATURE PHYSICS 20

21 Figure S9: lifetime as a function of tip-sample conductance for 2, 3, 4 ML MgO films at B = 2 T. Error bars determined by exponential fits to pump-probe data. Solid lines are a simultaneous fit of all data points to the model 1. Dashed lines use the resulting fit parameters to calculate the lifetime 1 due to electronic relaxation alone. The individual atoms we investigated in the above layer-dependent experiment are identified in Figure S10. Figure S10: Topographic STM image of individually adsorbed Fe atoms on a multi-layer MgO film on Ag(001). The particular atoms studied for the layer-dependent T1 at B = 2 T are circled. (2 0.1 V). NATURE PHYSICS 21

22 5. Magnetic field orientation and T1 vs B saturation Our magnetic field is applied mostly along the easy axis of magnetization, i.e. the out-ofplane direction. The angle between the surface normal of the sample and the applied field is ~10. This finite angle is intended to provide a 7.5 tilt to the sample with respect to the dewar so that adatoms can be deposited onto its surface. An additional 7.5 is present due to the misalignment of the solenoid with respect to the dewar. The two contributions of 7.5 are in different directions and produce a net angle of 10.3 between the applied field and the surface normal. Thus, the applied field produces an out-of-plane field 0.98, and an in-plane field The lateral component of the magnetic field may play a role in the observed plateau of for magnetic fields above 2.5 T (Figure 3). The lateral field creates spin state mixing, and thus is expected to yield an increased transition rate. However, the effective spin Hamiltonian description of the quantum states does not reproduce the increase in transition intensity required to reproduce the saturation (Supplementary Sections 1.3 and 1.4). We suggest that a more complete model of the atomic states might be needed to reproduce this behavior. An alternative possibility is that the plateau may be due to non-electronic relaxation processes, such as relaxation by creation of phonons. We note that the non-electronic rate increases as the field is increased. (The non-electronic rate r ne grows from 30 Hz at 2 T (Fig. S9) to 98 Hz at 5T (Fig. 4)). These rates are of the magnitude needed to produce a crossover from electronic to non-electronic relaxation in this field range, so we believe this is a plausible source of the plateau. NATURE PHYSICS 22

23 6. STM topography The topographic STM image shown in Figure 1a of the main text is rendered with a lighting mode which helps in the visualization of step edges and adatoms. However, this lighting mode removes the terrace height information. A greyscale rendering of the topography is shown in Figure S11. Figure S11: Topographic STM image of individual Fe atoms adsorbed on a multi-layer MgO film on Ag(001) ( V). Same data as Figure 1a of the main text. NATURE PHYSICS 23

24 7. Thickness of MgO films The MgO films used for our spin lifetime studies have a thickness of 2, 3, and 4 monolayers (ML). Single monolayer MgO films occur very infrequently and appear as slight elevations in the low-bias STM topography at the edges of the more common 2 ML patches. Note that in previous work from our group 5 8, films that were identified as 1 ML were in fact the more common 2 ML type that we have now identified. Recent work shows that growth conditions can determine the prevalence of 1 ML regions, with isolated 2 ML regions occurring readily 9. Figure S12 presents an overview of the features of Fe on 1 and 2 ML films, and in the sections that follow, we address the characterization of the film thickness. Figure S12: (a) STM topography of 1 and 2 ML films of MgO on Ag(001). Atoms on 1 and 2 ML patches of MgO that are circled by dotted lines are analyzed in (b-e). A tall metallic mound, created by indenting the tip into Ag, is indicated by poke. (b and c) Topographic STM images of Fe on 1 and 2 ML MgO. Dashed lines indicate the location of profile cuts shown in (e). (d) di/dv spectra taken on 1 and 2 ML atoms. The inelastic spin excitations (dashed lines) increase in energy from ~14 mv for Fe on 2ML to ~16 mv for Fe on 1 ML MgO. Spectra are normalized to the zero-bias conductance and offset for clarity. (1 ML mv; 2 ML mv) (e) Cross-section profiles of the Fe atoms shown in (b) and (c). (Imaging parameters indicated in figure, T = 1.2 K, B = 4.8 T) NATURE PHYSICS 24

25 Figure S12a shows a STM image taken at 10 mv of a sample with mostly 2 ML MgO film, regions of bare Ag, and small areas of 1 ML MgO film. Films of 1 and 2 ML appear co-planar in non-contact AFM measurements taken at constant height (not shown). Figure S12b and c show larger images of the Fe atoms that are circled in Figure S12a. These adatoms are identified as Fe by their spin excitation spectra, Figure S12d. We note that a ~2 mev shift to higher energy is observed for Fe on 1 ML. Figure S12e shows cross sections of the atoms as indicated by dashed lines in the topographic images (Figure S12b and c). 7.1 Point contact conductance of Fe atoms on 1-4 ML MgO films The thickness of the 1 ML MgO film is most noticeably revealed by the measured conductance at point contact with the Fe atom. We have plotted the I(z) data from the main text (Fe on 2-4 ML of MgO) as dashed lines in Figure S13a. In solid lines we have added I(z) data taken with a different tip on the atoms studied in Figure S12. It is clear that the 1 ML film extends the trend of a factor of ~9 change in conductance per monolayer; we therefore conclude that these regions of the film are thinner by one atomic layer than the 2 ML areas (which were in previous works identified as single layer 5 8 ). Shown in Figure S13b is the conductance at point contact (maximum conductance in the I(z) traces of Figure S13a) plotted as a function of layer thickness. Extrapolating to zero film thickness, we find a conductance of 1.5. This conductance is very close to which is strong evidence that the films identified as 1 ML are in fact of monolayer thickness. Figure S13: (a) Tunnel current as a function of tip-atom distance for Fe atoms on 1-4 monolayer MgO films on Ag(001) at a sample bias of 10 mv. Distance zero is defined as the position of point contact to the Fe atom (maximum conductance). Dotted lines correspond to data from the main text Figure 4a. Solid lines are data taken on 1 and 2 ML films and are measured with a new tip apex and different atoms. (b) Maximal conductance from I(z) curves as a function of MgO film thickness. NATURE PHYSICS 25

26 7.2 STM topography of 1 and 2 ML MgO films In this section, we present topographic STM data of the 1 and 2 ML MgO films taken at different bias voltages for comparison with published data 8. Figure S14: (a) STM topography of 1 and 2 ML MgO films on Ag(001). (b) Topographic profiles along lines indicated in (a). (c) Same region as (a) taken at bias voltages of 3 V (left) and 2 V (right). (d) Difference of 3 V and 2 V topography. (e) Profiles of difference topography along lines indicated in (d). (STM setpoint indicated in figure). NATURE PHYSICS 26

27 At low bias voltage, the energy of the tunneling electrons is well within the band gap of MgO. Figure S14a shows the topography taken at 0.1 V. The contrast in STM topography is not straightforward due to the buried interface structure of the MgO and the convolution of topographic and electronic effects. We will not interpret the topography in any detail here, other than to say at this low bias, the MgO can be considered as a tunneling barrier with a modified barrier height compared to vacuum tunneling. Figure S14b shows cross-sections of the topography as shown by the dashed lines in Figure S14a, where we reference height relative to the Ag terrace. A technique to determine the thickness of MgO films has been proposed by Baumann et al. 8 in which the topography of STM images taken at 3 V and 2 V are subtracted. The topographic difference is due to the difference in the integrated density of states in this voltage window, which is the energy range of layer-dependent electronic states of the MgO. The obtained contrast is thus an indicator of film thickness. In the measurements shown in Figure S14c-e, very little height contrast between 1 and 2 ML films is obtained when subtracting topographic images taken at 3 V and 2 V. 7.3 dz/dv of 1 and 2 ML MgO films We measured dz/dv spectra on 1 and 2 ML films of MgO to investigate their electronic properties. The spectra, plotted in Figure S15, are taken under closed loop conditions, where we sweep the bias voltage from 0.1 to 3.5 V and numerically differentiate the z(v) data. The 2 ML spectrum (blue) shows the presence of the MgO-Ag interface state (1.8 V) and the MgO state (2.4 V, attributed to either a surface state or its conduction band edge), in agreement with previously obtained results 8. The presence of both interface and MgO states, at ~2 and ~2.5 V respectively, is a common feature of films of thickness 2 ML and above. The electronic structure of the 1 ML film is markedly different, with only one peak at 2.9 V. Figure S15: dz/dv spectra on 1 and 2 ML films of MgO on Ag(001), taken by numerical differentiation of the z(v) data acquired under constant current feedback at I = 100 pa. NATURE PHYSICS 27

28 We note that the thickness determination technique based on differencing STM topographic data taken at 3 V and 2 V results in an unfortunate cancellation of the layerdependent spectral features. The integral between 2 V and 3 V is remarkably similar yielding only a few pm in height contrast (Figure S14e). We expect that differencing topographic images acquired at 2.75 V and 2 V would be a better choice, as it would maintain good resolution of the thickness of multi-layer films while enhancing the contrast between films of 1 and 2 ML. NATURE PHYSICS 28

29 8. References 1. Loth, S. et al. Controlling the state of quantum spins with electric currents. Nat. Phys. 6, (2010). 2. Lorente, N. & Gauyacq, J.-P. Efficient Spin Transitions in Inelastic Electron Tunneling Spectroscopy. Phys. Rev. Lett. 103, (2009). 3. Fernández-Rossier, J. Theory of Single-Spin Inelastic Tunneling Spectroscopy. Phys. Rev. Lett. 102, (2009). 4. Agraït, N., Levy Yeyati, A. & van Ruitenbeek, J. M. Quantum properties of atomic-sized conductors. Phys. Rep. 377, (2003). 5. Rau, I. G. et al. Reaching the magnetic anisotropy limit of a 3d metal atom. Science 344, (2014). 6. Baumann, S. et al. Origin of Perpendicular Magnetic Anisotropy and Large Orbital Moment in Fe Atoms on MgO. Phys. Rev. Lett. 115, (2015). 7. Baumann, S. et al. Electron paramagnetic resonance of individual atoms on a surface. Science 350, (2015). 8. Baumann, S., Rau, I. G., Loth, S., Lutz, C. P. & Heinrich, A. J. Measuring the threedimensional structure of ultrathin insulating films at the atomic scale. ACS Nano 8, (2014). 9. Pal, J. et al. Morphology of Monolayer MgO Films on Ag(100): Switching from Corrugated Islands to Extended Flat Terraces. Phys. Rev. Lett. 112, (2014). NATURE PHYSICS 29