## Abstract

The future of nanoscale spin-based technologies hinges on a fundamental understanding and dynamic control of atomic-scale magnets. The role of the substrate conduction electrons on the dynamics of supported atomic magnets is still a question of interest lacking experimental insight. We characterized the temperature-dependent dynamical response of artificially constructed magnets, composed of a few exchange-coupled atomic spins adsorbed on a metallic substrate, to spin-polarized currents driven and read out by a magnetic scanning tunneling microscope tip. The dynamics, reflected by two-state spin noise, is quantified by a model that considers the interplay between quantum tunneling and sequential spin transitions driven by electron spin-flip processes and accounts for an observed spin-transfer torque effect.

For magnetic storage technology (*1*), where magnets represent bits of information (*2*, *3*), effective manipulation of the magnetization without a magnetic field is of crucial importance. All-electrical manipulation offers technological advantages, such as highly localized bit control free of moving parts and compatibility with standard semiconductor technology (*4*, *5*). Integral to current-induced switching is the spin-transfer torque (*4*–*6*) (STT) effect, where angular momentum transfer from a flux of incident spin-polarized electrons can exert “torque” on the magnetization, causing it to switch in a preferential direction (*7*). STT can efficiently reverse the magnetization of magnetic layers (*8*, *9*), drive domain walls (*10*) and vortices, and be combined with giant magneto-resistance technology to generate high-frequency electrical oscillators (*11*, *12*). The ultimate goal of such spin-based technologies would be the total electrical control of an atomic “bit.” Factors complicating the achievement of this goal include the presence of a substrate, which can dramatically modify both the magnetic anisotropy and the moment at the single atom level (*13*), and spin quantization, which can play an enhanced role in the magnetization dynamics, for example, via quantum tunneling (*14*).

For small magnets that are sufficiently decoupled from an electron bath, such as molecular magnets (*15*), STT can be ascribed to asymmetric spin pumping induced by spin-polarized transport electrons that favorably excite particular spin states, via transfer of angular momentum, over or through an anisotropy barrier (*16*, *17*). A generalized Anderson model taking into account the coupling of the electron baths to the quantum spin adequately describes the dynamics of such isolated spins (*16*–*19*). For small magnets that are directly adsorbed on a metallic substrate where the moment strongly couples to the substrate conduction electrons (*20*, *21*), it is still questionable whether such a model is applicable and what the exact role is of the substrate conduction electrons on the spin dynamics of the magnet (*22*).

By using spin-polarized scanning tunneling microscopy (SP-STM), we demonstrate that it is possible to characterize the real-time current-driven dynamics of tailored magnets composed of a few direct-exchange-coupled Fe atoms on a metallic copper single-crystal surface. STM topographs of single Fe atoms on the surface and subsequent assembly of a five-atom Fe magnet by tip-induced atomic manipulation (*23*, *24*) are shown in Fig. 1, A and B. Density functional theory (DFT) (*24*) revealed two stable geometric configurations, “pyramid” (Fig. 1C) and “flat” (Fig. 1D), of such five-atom Fe magnets on the copper surface. The total magnetic moment of each configuration, including orbital contributions, is *m _{J}* ≈ 15.2μ

_{B}(where μ

_{B}is a Bohr magneton) and

*m*≈ 14.7μ

_{J}_{B}, respectively, which averages to ~3μ

_{B}per atom, close to the magnetic moment of a single Fe adatom (~3.5μ

_{B}) (

*20*,

*25*). The large total magnetic moment results from a strong ferromagnetic exchange coupling between the constituent Fe atoms to the total angular momentum (

*26*). We thus treat the magnet as a single total angular momentum,

*J*(

*21*), which is related to the total magnetic moment via

*m*=

_{J}*g*μ

_{B}

*J*{where

*g*is the

*g*factor [assumed

*g*≈ 2 (

*20*)]}; we will refer to the total angular momentum as the spin for short. A considerable out-of-plane magnetic anisotropy (

*24*) leads to a schematic level diagram of the spin states (Fig. 1E).

At modest current values using a magnetically coated tip that has out-of-plane spin sensitivity (*24*), spin-based telegraph noise (*27*) can be read out on top of the assembled magnet in constant-current mode (Fig. 1F). It represents full reversal of the magnetization between two degenerate ground states, labeled |±15/2〉 in Fig. 1E, aligned parallel/antiparallel to the surface normal. We refer to these up and down states as “1” and “0.” Although all ±|*J*_{z}| states contribute to the spin dynamics, the limited time resolution of our experiment allows us to resolve the signal from the magnet only in one of the two degenerate ground states, because they are the longest lived. Therefore, it is possible to extract the occupational lifetimes τ^{±} of the 1 and 0 states as well as the mean lifetime τ* = (τ^{+} + τ^{−})/2 from the measured spin noise, for a given tunneling current *I*_{t}, bias voltage *V*_{s}, out-of-plane magnetic field *B*, and temperature *T* (*24*). The histogram of the two-state noise (Fig. 1, G and H), at *B* = 0 T, shows a considerable asymmetry in the state-resolved occupational lifetimes τ^{±}, indicating a preferential state. This preferential state is reversed when the bias polarity, and hence the spin-polarized current direction, is reversed. The polarity-dependent asymmetry in the state of the magnet, in the absence of a magnetic field, results from a considerable STT induced by the spin-polarized tunneling electrons.

The mean switching frequency (ν* = 1/τ*) has a linear dependence on *I*_{t} (Fig. 2A), indicating that the total number of tunneling electrons *q* (= *I*_{t} × τ*) needed to reverse the magnetic state is independent of the current (Fig. 2B); for a five-atom magnet, ~5 × 10^{9} incident tunneling electrons are needed to reverse the magnetic state. Similar behavior is observed as the magnet size is increased from five to six atoms. The fact that *q* is independent of *I*_{t} is a hallmark of an inelastic spin excitation (*17*, *20*, *21*, *28*, *29*), where each tunneling electron has a fixed probability of exciting the spin eigenstates of the magnet (*30*). Also the observed asymmetry between the state-dependent lifetimes (τ^{±}) (*B *= 0), resulting from STT, is independent of the magnitude of *I*_{t} (Fig. 2, A and B, histograms). If Joule heating was important here, increasing *I*_{t} would reduce this asymmetry because heating would symmetrically reverse the magnetization, like seen for larger-scale Fe islands (*27*). Therefore, Joule heating can be ignored in the chosen range of both *I*_{t} and *V*_{s} in our experiment. This is most likely the result of the comparatively small bias voltage *V*_{s} used here, which is too small to strongly couple the spin to phonons (*31*).

The dependence of the lifetime τ* on the applied bias *V*_{s} (Fig. 2C) for several selected five-atom magnets exhibits a stronger than exponential increase in τ* as the energy of the tunneling electrons (*eV*_{s}) is lowered. Both current- and voltage-dependent studies of τ* suggest that, in the absence of current or for very small tunneling electron energy, these magnets remain stable for extremely long times. The exact current and energy dependence of τ* depends intricately on the spin-dependent energy landscape of the magnet (Fig. 1E), which is determined by the magnetic anisotropy and the total spin of the system.

In order to quantify the dynamics and link the experimentally observed lifetimes to the magnetic properties of the system, we used the following quantum impurity model (*24*). The spin eigenstates |α〉 of the magnet in an out-of-plane magnetic field *B* are derived from a “giant spin” Hamiltonian *15*), where *D* and *E* are the axial and transverse magnetic anisotropy, respectively, which determine the preferential orientation of the spin. A master equation (*24*) characterizes the dynamics by considering all possible transition probabilities between the various spin eigenstates |α〉 (Fig. 1E) induced both by sequential transitions over the anisotropy barrier [driven by spin-flip scattering of tunneling electrons (t*e*^{−}) and substrate conduction electrons (s*e*^{−})] and by quantum tunneling (QT) through the anisotropy barrier.

Model calculations (Fig. 2C) show excellent agreement with the experimental data for magnetic anisotropy values of *D* ≈ –0.1 meV and *E* ≈ 0.02 meV, confirming the uniaxial out-of plane anisotropy with a weak in-plane anisotropy predicted by ab initio methods (fig. S3). Because *E* is small, *V*) is very sensitive to small variations in the anisotropy, as seen experimentally for different magnets. Moreover, as *V*_{s} is decreased, τ* dramatically increases, spanning three decades. This divergent behavior of τ* is seen as the energy approaches the calculated first transition energy, ∆_{01} (Fig. 2C, dashed lines), for a given magnet. No switching was observed below |*V*_{s}| < 2 mV for currents *I*_{t} ≤ 3 nA up to maximum observation times of 2 hours.

To further investigate the role of QT and spin-flip scattering of (t*e*^{−}) and (s*e*^{−}) on the dynamics of the magnet, we varied the temperature. To remove the current dependency, we plotted the switching charge *q*(*T*) rather than the lifetime (Fig. 3). The required charge *q* to switch a five-atom magnet decreases as the temperature increases, reflecting the decrease of τ*(*T*) (Fig. 3A). The slope is rather small for the low-temperature range *T* ≤ 1.5 K, but *q* falls off more strongly for *T* ≥ 1.5 K, resulting in a shoulder in *q*(*T*) at *T* ≈ 1.5 K. For *T* ≥ 2 K, an exponential tail indicates Arrhenius behavior, similar to what was observed for larger-scale Fe islands (*27*). The Arrhenius behavior for *T* ≥ 2 K is evident from the log plot in Fig. 3A (inset), whereas a plateaulike regime appears for *T* ≤ 1.5 K. Such plateaus have been attributed to QT of the magnetization through the anisotropy barrier (*19*); however, our results show a deviation from a constant transition rate in the low-temperature regime, indicating that QT alone cannot account for the observed temperature dependence of τ*.

In order to link the temperature dependence of *q*(*T*) to the different spin-transition processes, we used the quantum impurity model with the *D* and *E* values extracted from the fits in Fig. 2C and fit the *T* dependence of *q* for different *I*_{t} (*24*) (solid lines in Fig. 3A), yielding good agreement with experiment. On closer inspection, the model reveals that this temperature dependence results from the following two spin transition mechanisms that contribute to the switching of the magnet in addition to t*e*^{−}-driven sequential transitions: (i) sequential transitions of the magnet induced by spin-flip scattering of thermally excited substrate electron-hole pairs generated by the broadening of the Fermi function. This process, which is analogous to “damping” (*32*), dominates at higher temperatures. (ii) Nonequilibrium QT resulting from transverse anisotropy (*E* ≠ 0). Note that the equilibrium QT, which fully reverses the magnetic state from one ground state to the other without any additional spin-flip processes (*14*), is blocked because for a half-integer spin the states across the barrier, which have the same |*J _{z}*|, have zero overlap. However, in nonequilibrium, a two-step process where a t

*e*

^{−}or s

*e*

^{−}electron changes

*J*can lead to QT because there is a finite overlap of states across the barrier for particular values of ∆

_{z}*J*≠ 0 resulting from

_{z}*E*≠ 0 (Fig. 1E). The most dominant QT channels responsible for reversal are closer to the top of the barrier, because the energy barrier is smaller for these states than for states near the bottom of the barrier. Therefore, QT processes are mainly preceded by sequential spin transitions that drive the system into higher excited states, resulting in a weak temperature dependence of the whole QT process. The temperature-dependent interplay between the two transition effects (i) and (ii) leads to a shoulder in

*q*(

*T*); at higher temperature, the well-known Arrhenius behavior is recovered. Moreover, the shape of the shoulder depends on the size of the transverse anisotropy

*E*(fig. S2). Thus, the shoulder serves as a delineation of the temperature range where QT plays a prominent role in the magnetization dynamics. Therefore, the underlying mechanism of how the magnetization is fully reversed can be understood within the quantum impurity model when accounting for both substrate conduction electrons and nonequilibrium-induced QT.

We explored the effects of an external out-of-plane magnetic field, *B*, on the observed dynamics. With increasing *B*, the asymmetry in the spin noise increases; that is, the lifetime τ^{+} of one of the ground states |α_{+}〉 becomes larger at the expense of the lifetime τ^{−} of the other ground state |α_{−}〉 (Fig. 4, A and B). This is a result of the Zeeman energy, *g*μ_{B}*B **B*; τ^{+} indeed shows a monotonic increase (and τ^{−} a monotonic decrease) with increasing *B*. The balance τ^{+} = τ^{−} occurs at a nonzero magnetic field, |*B*_{STT}| ≈ 0.1 T. The reason for the observed asymmetry at *B* = 0 T is a result of the spin polarization of the tunneling current created by the magnetic tip. The spin-polarized current favors t*e*^{−} sequential transitions over the barrier in one particular direction as opposed to the other (*17*), analogous to STT. As a proof of the STT effect, we reversed the direction of *I*_{t} by changing the bias polarity, resulting in the expected sign change of *B*_{STT} (Fig. 4B).

The observed asymmetry between the lifetimes τ^{+}(*B*) and τ^{−}(*B*) can be further quantified by the nonequilibrium magnetization *M*(*B*) = [τ^{+}(*B*) − τ^{−}(*B*)]/[τ^{+}(*B*) + τ^{−}(*B*)] (Fig. 4C), which saturates toward ±1 at a comparatively large magnetic field *B* > 0.5 T for the five-atom magnet. *M*(*B*) calculated from the quantum impurity model assuming the anisotropy values extracted from the voltage-dependent τ* (Fig. 2C) is in good agreement with experiments for both bias polarities. The spin polarization of the tip, assumed to be about 10% in the calculation, accounts for the observed polarity-dependent STT effect, *B*_{STT}, of the nonequilibrium magnetization curve. The nonzero transverse anisotropy is responsible for the large magnetic field needed to saturate the magnetization as compared with the single Fe atom limit (*20*). The presence of finite *E* in combination with a large spin (*J*) leads to a line shape of *M*(*B*) that is nearly independent of *T* for temperatures below the Arrhenius regime (Fig. 4D), in contrast to atoms that exhibit no transverse anisotropy (*13*).

The observed STT behavior persists for larger magnets, with the overall saturation field dropping as a result of the increased total spin as manifested by the sharper line shape in *M*(*B*) (Fig. 4C). As the size of the magnet increases, similar behavior is seen in *q*(*I*_{t}), but the overall value of *q* needed to reverse the larger magnet increases (Fig. 2, A and B). The general shape of *q*(*T*) (Fig. 3B) persists for larger magnets, proving that the underlying mechanisms that govern the dynamics remain the same. Whereas *D* and *E* change as the size of the magnet increases, thus affecting τ*(*I*_{t},*V*_{s}), the strong increase in *q* needed to switch the magnet as the magnet gets larger is most likely a consequence of the increased spin of the magnet. Increasing the spin results in an increased number of sequential transitions needed to reverse the magnetization, requiring a higher total charge to reverse the magnetic state.

Although the relaxation of the magnets studied here is far from purely quantum, namely the quantum phase is destroyed, it is surprising that, for such a strongly hybridized spin coupled to an electron bath, quantum effects are indeed necessary to fully describe the dynamics of the system. The strength of the STT studied here is determined solely by the total spin polarization of the tip independent of the total current (*24*). This can be seen in the nearly constant asymmetry in Fig. 2, A and B, and fig. S4 regardless of the mean switching frequency ν*, illustrating the quantum nature of STT in atomic-scale magnets (*16*). Our work brings to light fundamental processes of interest for future magnetic memory devices that are scaled to atomic dimensions.

## Supplementary Materials

www.sciencemag.org/cgi/content/full/339/6115/55/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S4

## References and Notes

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**Acknowledgments:**Financial support from the European Research Council (ERC) Advanced Grant “FURORE”; Bundesministerium für Bildung und Forschung; the Deutsche Forschungsgemeinschaft via the SFB668, SPP 1285 (B.B.), FOR1346, and the Graduiertenkolleg 1286 “Functional Metal-Semiconductor Hybrid Systems”; the Cluster of Excellence “Nanospintronics” funded by the Forschungs-und Wissenschaftsstiftung Hamburg; and HGF-YIG Programme VH-NG-717 (Functional nanoscale structure probe and simulation laboratory–Funsilab, S.L.) is gratefully acknowledged.