Extremely long-lived magnetic excitations in supported Fe chains

We report on a theoretical study of the lifetime of the first excited state of spin chains made of an odd number of Fe atoms on Cu2N/Cu(100). Yan et al (Nat. Nanotech. 10, 40 (2015)) recently observed very long lifetimes in the case of Fe3 chains. We consider the decay of the first excited state induced by electron-hole pair creation in the substrate. For a finite magnetic field, the two lowest-lying states in the chain have a quasi-N\'eel state structure. Decay from one state to the other strongly depends on the degree of entanglement of the local spins in the chain. The entanglement in the chain accounts for the long lifetimes that increase exponentially with chain length. Despite their apparently very different properties, the behaviour of odd and even chains is governed by the same kind of phenomena, in particular entanglement effects. The present results account quite well for the lifetimes recently measured by Yan et al on Fe3


Introduction
The recent development of low-T, high resolution, scanning tunnelling microscopy (STM) allows detailed investigations on spin systems at surfaces 1,2,3,4,5,6 . The study of adsorbed spin systems made of only a few atoms was a very significant breakthrough in the context of miniaturization of electronic devices. The possibility to characterize and modify spin systems at the atomic level opened the way towards detailed analysis of these systems, of their behaviour and in particular of their quantum aspects (see a review in 7 ). The possibility to act on the spin variable of an adsorbate on a surface, i.e. to create local spin excitations, has been demonstrated; together with large magnetic anisotropies, this leads to the existence of well separated energy levels that could be used for logical devices. Indeed, atomic-scale logical devices have been assembled using atom manipulation techniques. A variety of magnetic nanostructures, and in particular of chains of magnetic atoms, have been built and analysed in STM experiments 8,9,10,11,12,13,14 that has triggered a series of associated or parallel theoretical studies 15,16,17,18,19,20,21 .
Atomic spin systems adsorbed on surfaces a priori obey quantum dynamics, if decoherence effects are not too strong 22,23,24,25 ; in addition, since adsorbed spin systems interact with the substrate, in particular with its thermal electron bath, magnetic excitations have finite lifetimes. For possible applications, it is of paramount importance to be able to determine the population lifetime of these spin excitations and their coherence time as well as to understand what the key parameters influencing these lifetimes are.
Few experiments have been devoted to the study of the lifetime of magnetic excitations in nano-structures at surfaces. We can mention the indirect measurement of population survival via STM-current saturation effects in the Mn/Cu 2 N system 12 , and the direct lifetime measurement via STM experiments with high time resolution in evennumbered chains of Fe on Cu 2 N/Cu(100) 13 , and in Fe 3 on Cu 2 N/Cu(100) 14 . Indeed, the different approaches were adapted to different time scales for the lifetime. In particular, the direct measurements, adapted to long time scales, revealed extremely long lifetimes, up to a few hours, for the Fe chains, that quickly increase in the even chain case as the chain length increases 14 . One can also mention measurements of excitation profiles 26,27,28 ; indeed a finite lifetime leads to broadening of the peaks in an excitation spectrum. However, there are other sources of broadening besides experimental effects (e.g. temperature, decoherence ...), so that extracting a lifetime from a peak width can be a difficult procedure.
On the theoretical side, besides some studies involving adjustable parameters 29,30 , only a few parameter-free studies have been reported including a perturbation calculation of electron scattering effects 31 , as well as excitation-width calculations 32,33 ; in both cases, a satisfying agreement with experimental data was reached.
Fe chains adsorbed on the Cu 2 N/Cu(100) surface present very appealing characteristics that make them choice systems for fundamental studies 12,13 . The nitride layer insulates the adsorbate from the metal substrate, thus limiting the flux of substrate electrons hitting the chain and reducing the decay rate of magnetic excitations. The chains can be described as a set of local spins interacting together (anti-ferromagnetic coupling, AFM) and with the substrate 12,13 . The magnetic anisotropy of each atomic spin is very large, so that the system is closer to an Ising chain than to a pure Heisenberg chain. As a consequence, the classical equilibrium state of an even Fe chain is a degenerate Néel state, with Fe atomic spins pointing parallel to the surface, in alternating directions along the chain. However, in a quantal view, the two Néel states are coupled (via exchange and/or transverse magnetic anisotropy), so that the ground state is a 50-50 mixture of the two Néel states. In the case of even chains, the detailed STM experimental study of Loth et al 13 showed that the two Néel states of the chain were observed and not the quantal ground state. It has been shown that inclusion of decoherence effects accounts for the observation of the classical Néel states 25,34 .
Spontaneous flip between the two Néel states in the even Fe chains was also observed 13 and accounted for 25 ; the flip rate quickly decreases with the chain length 13,25 , so that long chains yield a good model system for nano-magnetic memories 13 .
The case of odd Fe chains on Cu 2 N/Cu(100) is different, though it bears strong resemblances with the even Fe chain case. Indeed in both cases, the system is described by a set of local spins with a strong anisotropy and coupled by Heisenberg exchange. In a classical Ising model, the ground state is degenerate with two Néel-like states, with a non-vanishing spin projection and with Fe atomic spins pointing parallel to the surface, in alternating directions. When a magnetic field B is applied, the two Néel states of an odd chain split. The recent study by Yan et al 14 reported i) that a polarized magnetic STM tip very easily alters the system and ii) that for the Fe 3 chains, the upper Néel state has a very long lifetime, increasing roughly linearly with the applied magnetic field. It is the aim of the present work to compute the lifetime of the upper Néel state in a series of odd Fe chains on Cu 2 N using the method introduced by Novaes et al 31 . The decay of the magnetic excitation is induced by inelastic scattering of substrate electrons on the adsorbed chain, i.e. by electron-hole pair creation. This computation will lead to an analysis of the origin of the long lifetimes, as well as to a discussion of its dependence on the chain length.

2.a Model description of the Fe chains
The odd Fe chains on Cu 2 N/Cu(100) are described using the same model as for our earlier studies on spin chains 25,35,36 . It assumes that each Fe atom bears a local S=2 spin, with anisotropy terms, coupled by Heisenberg exchange. The effect of an applied B field is also included. The corresponding Hamiltonian reads: . . The projection of the total spin on the z-axis is almost equal to +2 and -2 for the two states.
Because of the existence of J and E terms, the two pure Néel states are coupled together via a high order indirect coupling, so that the result of the diagonalization of (1) does not yield exactly Néel states. When B is increased, the two states split almost linearly and the present study, as well as the earlier experimental study by Yan et al 14 , is devoted to the decay of , the upper quasi-Néel state, to the lower,

2.b Computation of the decay rate of the excited Néel state
The decay of the excited state is induced by inelastic scattering of a substrate electron by an atom of the chain that leads to the change of the magnetic state of the whole chain (1).
The lifetime τ of the excited state 〉 2 N | is the inverse of Γ , the decay rate from The decay corresponds to the super-elastic (final electron energy larger than its initial energy) scattering of substrate electrons by the adsorbate. The decay rate for a vanishing temperature can be expressed by the T transition matrix for electron scattering (j→f) by one of the atoms in the chain as (see a discussion of the decay rate by electron-hole pair creation in 39 and the application to spin decay in 7,31,40 ): states. The total energy is one can show that the decay rate reduces to :   , has a quasi-Néel state character, the lifetime appears to be roughly linear with B for all chains. In that range, the energy defect of the decay, δΩ, varies linearly with B (Zeeman term). In contrast, the probability entering in equation (3) depends on the spin entanglement in the chain, i.e. on the mixing between the two Néel states.

3.a Comparison between the different chains
Indirect high order terms in J and E are independent of B and since the energy difference between the two Néel states varies like B, this yield a mixing probability that varies like 1/B 2 .  The case of a single Fe adsorbate could appear as different from the other ones, although its lifetime fits well in the evolution observed on the chains. Indeed, entanglement of different atomic spins cannot be invoked in this case. As discussed in Hirjibehedin et al 2 , at B=0, the two lowest states correspond to an equal mixing of the S z =+2 and -2 of Fe with a small contribution from the S z =0 state. This structure is driven by the magnetic anisotropies D and E. When a B field is applied, the Zeeman term is able to uncouple the structure governed by the anisotropies and yields states that become more and more pure S z states as B increases.
At finite B, the transition between pure S z =+2 and -2 states is impossible via collision with an electron; however, since there is some correlation induced by the transverse anisotropy (the system state is a superposition of different S z states) a weak transition is possible, leading to the lifetime displayed in Fig.1 that increases as the applied field increases. The origin of long lifetimes in systems with large negative D terms, like single Fe atom, has already been discussed in 42 . We conclude that the role played by the J and E coupling terms in the chain case reduces to that of the E term alone in the Fe case; in all cases, decay proceeds via the correlations in the system: in the Fe case, the system state is not exactly a S z eigenstate and in the chain case, the system state is not exactly a single configuration of atomic spin states.

3.b Low B-field limit
The situation in the limit of vanishingly small B field is different from the one outlined above. It is illustrated in Figure 2, which shows the lifetime of the first excited state at a function of B on a double logarithmic scale to enhance the behaviour at extremely low B.
For B=0, the two Néel states are degenerated and coupled by indirect high order J and E terms (only the E term in the single Fe atom case). The two lowest eigenstates of Hamiltonian (1) are then equal mixtures of the two Néel states with small admixtures of higher lying states. The energy difference between the two low-lying states is very small and equal to twice V, the effective coupling between the two Néel states. The situation of the odd chains at B=0 is thus very close to that of the even chains 25,35 .
When a very small, finite, B field is applied, the Zeeman term splits the two quasi Néel states when it is able to overcome the V coupling and the system is in the situation described in the previous section. The lifetime of the first excited state is then constant at very small B and then switches to the linear behaviour discussed in the previous section (see Figure   2). The transition between the two regimes is similar in the different chains, although it appears in very different B ranges. Indeed, the switch occurs in the region where the Zeeman term is of the order of V, the effective coupling between the two Néel states, i.e. when it is large enough to decouple the two Néel states. The effective coupling V is due to higher and higher order indirect couplings as the chain length is increased and it decreases roughly exponentially with the length. As a consequence, the low B behaviour occurs at lower B field as the chain length is increased, as seen in Fig.2. Actually, for long chains, the low field regime might be difficult if not impossible to reach; any small stray magnetic field or more generally any small perturbation is able to uncouple the B=0 structure and leads to the regime discussed in the section 3a.
The effective coupling, V, can be obtained as half the energy splitting of the two eigenstates at B=0. It is shown in figure 3 as a function of the chain length. The present results (red squares) are seen to decrease roughly exponentially with the chain length, as expected for an indirect process involving higher order terms as N increases. These results were obtained with the set of magnetic parameters determined from a spectroscopic study by Yan et al for for the study on even chains) are also shown. It appears that the V coupling for odd and even chains with Bryant et al parameters follows the same exponential behaviour, so that again, odd and even chains, although they seem to behave qualitatively differently at first sight, actually exhibit the same kind of couplings. It also appears that the different sets of parameters lead to different results: they are very close for the short chains, e.g. for Fe 3 , but become significantly different for long chains; this is simply a consequence of the high order of the indirect coupling scheme that leads to the V coupling: the higher is the order, the more sensitive the V coupling is on small variations in the magnetic parameters. Similarly, for very long chains, the lifetime is sensitive to the precise choice of magnetic parameters.
One can stress that for a vanishing B and a long chain, one has two low lying eigenstates that interact with the underlying continuum of substrate electrons. Elastic collision of substrate electrons by the chain brings decoherence into the chain quantal system. Again, the situation is identical to that met in even chains (see a discussion in 25 ). In most realistic situations, the above computed lifetime for B=0 in long chains is not a meaningful quantity.
As for even chains 13 , except for extremely low T, an STM observation on long chains would not observe the system quantal ground state but would find an equal statistical population for the two Néel states and if the system is initially in one Néel states, it will relax slowly toward a classical equilibrium, with equal incoherent populations of the two Néel states 25 .

3.c Comparison with Yan et al data
Yan et al 14  or on an edge atom of the chain. They showed that this was due to the perturbation of the chain by the polarized tip that disappears when the tip is moved at larger distances from the chain. Only for a large tip-chain distance, can they measure the intrinsic lifetime. Figure 4 presents the two lifetimes measured by Yan et al ¡Error! Marcador no definido. on the central and edge atom of Fe 3 as a function of the applied B field for a fixed tip-chain distance as well as the 'asymptotic' intrinsic value at B=2T found for a large tip-chain distance. The calculated results (blue curve) correspond to an unperturbed chain and should then be compared to some average of the two sets of experimental data and/or to the 'asymptotic' point at 2T. The present results appear to reproduce the B-variation observed experimentally; however, they roughly underestimate the experimental data by a factor 2.8, i.e. the discrepancy is of the same order as the one found in our earlier study 31 on single Mn adsorbates on Cu 2 N/Cu(100).
Possible inaccuracies in the theoretical or experimental studies could account for this.
However, one can also stress the sensitivity of the present results to the magnetic parameters in Hamiltonian (1). To illustrate this sensitivity, Figure 4 also presents the results obtained with a 6.15 % change of the J, E and D parameters (E and J are lowered and D is increased, i.e. the changes bring the system closer to a pure Ising model with smaller entanglement in the chain and so a longer lifetime). The 6.15 % change is chosen to bring the computed results in agreement with the unperturbed experimental data at 2T (the three parameters, J, D and E, act in a similar way and agreement with experiment can also be found when modifying only one of the three parameters). This small change is to be compared with the accuracy of the spectroscopic fit reported by Yan et al 14 on the energies which amounts to around 5% for D and 10% for J. As a consequence, the discrepancy between theory and experiment seen in

Concluding summary
We The present results appear to reproduce reasonably well the experimental data of Yan et al on Fe 3 14 , allowing for a global factor of 2.8. The difference is attributed to an insufficient accuracy of the magnetic structure parameters of the chain. The very long lifetimes in the system are very sensitive to small variations in the magnetic structure parameters (exchange coupling and anisotropies). Computation of magnetic structure parameters have been performed via configuration interaction or DFT approaches for a variety of nanostructures at surfaces systems 15,18,19,20,43,44,45,46,47 . Further studies by ab initio approaches on the present system could be highly conclusive, despite the current difficulties in accurately computing magnetic structure parameters 48,49 .
It turns out that odd and even Fe chains on Cu 2 N/Cu(100) have very close magnetic structures, the basic interactions at play, in particular a weak entanglement of the local spins, being the same. In the absence of an applied magnetic field, both odd and even chains exhibit two closely spaced states, strongly influenced by decoherence. However, a small magnetic field is able to split the two states in the odd chain case, leading to the possibility of a very long lived state as observed by Yan et al 14 and computed here.