Abstract
Recent experimental data demonstrate emerging magnetic order in platinum atomically thin nanowires. Furthermore, an unusual form of magnetic anisotropy – colossal magnetic anisotropy (CMA) – was earlier predicted to exist in atomically thin platinum nanowires. Using spin dynamics simulations based on firstprinciples calculations, we here explore the spin dynamics of atomically thin platinum wires to reveal the spin relaxation signature of colossal magnetic anisotropy, comparing it with other types of anisotropy such as uniaxial magnetic anisotropy (UMA). We find that the CMA alters the spin relaxation process distinctly and, most importantly, causes a large speedup of the magnetic relaxation compared to uniaxial magnetic anisotropy. The magnetic behavior of the nanowire exhibiting CMA should be possible to identify experimentally at the nanosecond time scale for temperatures below 5 K. This timescale is accessible in e.g., soft xray free electron laser experiments.
Introduction
Late 4d and 5d transition metals such as palladium and platinum are paramagnetic in the bulk, but at the same time exhibit enhanced magnetic susceptibility. Thus, perturbations such as reduced dimensionality may result in emerging magnetism in these metals. The magnetic state might only exist at very low temperatures, or have other features making it difficult to observe experimentally. Recently, Strigl et al.^{1} demonstrated emerging magnetic order in platinum atomic contacts and chains by measuring the magnetoconductance. Here, we take an alternative route and address the time evolution of magnetic order in platinum nanowires. The underlying idea is that the unusual anisotropy predicted to exist in these systems, where the magnetic moments of the wires depend strongly on the angle of deviation from the easyaxis^{2}, could affect the dynamics in such a way that it the dynamical behavior could function as a measurable signature for the emergent magnetism and its associated colossal magnetic anisotropy.
Understanding spin relaxation and longrange order in lowdimensional systems are questions of fundamental interest. Recently, they have also become core technological issues in the quest for eversmaller nanosized magnetismbased information storage systems. Generally, as the dimensionality of a system is reduced, fluctuations become larger and more important and the tendency toward magnetic ordering decreases. According to the MerminWagner theorem^{3}, infinite 1D chains with sufficiently short range magnetic interactions should spontaneously break up into segments with different spin orientation. This in turn implies that longrange order would be impossible in these systems. However, these early spinlattice models assume the absence of kinetic barriers as well as anisotropies. Kinetic barriers hinder thermally induced transitions between available magnetic configurations and may result in longlived stable states creating ordered magnetic structures below a certain threshold temperature. Thus, by introducing such barriers one might hope to build 1D magnetic systems with longrange magnetic order and even zerodimensional magnetic systems with the capability to store information on a macroscopic time scale^{3,4,5}.
Magnetic anisotropy energy can be exploited to introduce the barriers in the system, which is realized in practice by growing 1D systems on a substrate or by using magnetic species with substantial orbital moments^{6}. 1D nanowires may exhibit many different types of magnetic arrangements depending on the exchange coupling between the spins, the atomic geometry, the shape of the nanowire and the size and type of anisotropy. Recent experimental studies of chains of Fe atoms on a Cu_{2}N substrate showed evidence of both ferromagnetic^{7} and antiferromagnetic^{8} ordering at low temperatures, depending on the relative positioning of the atoms. Gambardella et al.^{6} observed both long and shortrange magnetic order in Co chains arranged on a Pt(997) surface, with a blocking temperature of 15 K for the longrange order. In meandered Fe nanowires grown on Au(788), Shiraki et al.^{9} confirmed the theoretical expectation^{10,11} that the average size of the ferromagnetic domains in the nanowire decreases exponentially with temperature. Strigl et al.^{1} showed experimentally that even nanowires of platinum, which is paramagnetic in bulk, demonstrate signatures of local magnetic order.
Theoretical investigations on the Heisenberg model for quantum and classical spin chains date long back, with early works concentrating on the timeindependent properties. In the limit of an infinite number of spins, it was possible to obtain analytical results for the thermal equilibrium observables, with the isotropic Heisenberg chain displaying an exponential decay of the timeindependent spinspin correlation and absence of longrange ordering^{3,12,13}. Numerical calculations based on transfer matrix formalism, augmented by approximate analytical calculations, enabled investigations on how the spinspincorrelation, the susceptibility and the specific heat depend on external magnetic field^{14} or on anisotropy^{15}, respectively.
The effect of temperature and magnetic anisotropy have been the subject of previous theoretical studies of spin dynamics in spin chains^{16,17,18,19} where the relaxation dynamics was found to depend substantially on the description of the magnetic anisotropy. Recently, analytical solutions were obtained for the dynamics of intrinsic localized modes, in particular in form of onespin, twospin and threespin excitations, of an anisotropic Heisenberg ferromagnetic spin chain in the presence of an external magnetic field^{20}.
In this work, we explore the spin dynamics in a platinum atomic wire (see Fig. 1) using atomistic spin dynamics simulations where the interactions have been calculated from first principles^{21}. Specifically, we analyze how the dynamics is altered when we include an energy barrier against relaxation in the form of magnetic anisotropy. In this context, colossal magnetoanisotropy (CMA)^{2} – a new type of magnetic anisotropy where the magnetic moments become zero for large enough angles between the wire and the magnetic moment – is of special interest.
We have employed atomistic spin dynamics (ASD) simulations^{22} as implemented by Skubic et al.^{21}. In brief, the ASD method is based on solving the equations of motion for the atomic moments, m_{i}, as expressed by the LandauLifshitzGilbert (LLG) equation
The time evolution described by the LLG equation comes about through a combination of precessional motion around the quantization axis and dissipation. The dissipative part was originally introduced phenomenologically. It is however intrinsic and can be derived by calculating the time evolution of the spin observable in the presence of the full spinorbital coupling^{23}. The gyromagnetic ratio is denoted by γ, B_{i} is the effective magnetic field on atom i and b_{i} is a stochastic magnetic field with a Gaussian distribution, the magnitude of which is related to the temperature. The Gilbert damping parameter is denoted by α. We have used the semiimplicit solver by Mentink et al.^{24} to treat the time evolution in the LLG equations. The effective magnetic field is formally defined as the functional derivative of the Gibbs free energy of the magnetization and is taken as in our simulations, where is the hamiltionian of the system. The Hamiltonian we consider consists of two terms – describing Heisenberg exchange and magnetic anisotropy, respectively. The Heisenberg Hamiltonian is given by , where J_{ij} is the strength of the exchange interaction between the moments on site i and site j. Magnetic anisotropy is modeled in two different ways, i.e., in the form of uniaxial anisotropy (UMA) and CMA. In both cases, the anisotropy axis, e_{K}, is chosen to be along the nanowire axis. The UMA is introduced as , with K being the strength of the anisotropy along e_{K} and being the unit vector pointing in the direction of ith magnetic moment. The CMA, in turn, is treated as a combination of a modified uniaxial anisotropy energy term and a dependence of the magnitude of the magnetic moments as a function of the angle of deviation ϕ_{i} of the moments from the nanowire axis. Both effects have in this work been modeled in the spin dynamics simulations by parametrization of the firstprinciples results reported by Smogunov et al.^{2} who found an anisotropy energy of 1.8 meV and reported a monotonic decrease in the magnitude of magnetic moments of platinum atoms, from 0.4 μ_{B} for zero deviation angle to zero for ϕ ≈ 45°. It is found that the modified uniaxial anisotropy can actually be quite well described by , where is given by
to be contrasted with the cos^{2}(ϕ_{i}) behavior of the UMA. The expressions above for the anisotropy energies present a simple way of including the effects of anisotropy on the dynamics in this case and will provide us with an understanding of the effect of CMA on the spin dynamics of platinum wires. We note however that with nonconstant magnitudes of the magnetic moments – the case in CMA – the equation of motion itself will in principle be modified. First steps in this direction were recently taken in connection to modeling of longitudinal and transversal fluctuations of magnetic moments in bcc Fe^{25}. Here we employ a simple rescaling scheme where for each time step, the magnitude m_{i} of each magnetic moment m_{i} is determined by its deviation from the nanowire axis. In particular, we use the data provided by Smogunov et al. (see Fig. 2a in ref. 2) to parametrize the magnitude of magnetic vectors as a function of the deviation angle. In addition, we also perform, for comparison, simulations for the corresponding system with no magnetic anisotropy (NMA), where no anisotropy term is added to the Hamiltonian. The NMA systems are thus modelled in the same way that the UMA systems are modelled with the exception that the anisotropy constant is set to zero.
The interatomic exchange interactions J_{ij} entering the spin Hamiltonian have been calculated from first principles by means of the “frozen magnon” approximation^{26} – i.e. by inverse Fourier transform of the qdependence on the total energy E(q) for a large number of spinspirals with wave vector q. The spin spirals were calculated with a full potential linearised augmentedplane wave (FPLAPW) method^{27} using the magnetic force theorem^{28}, starting from the ferromagnetic ground state.
From our frozen magnon approach we have extracted values for the eight nearest neighbour exchange couplings. The exchange interactions are found to be strongly dominated by the nearestneighbour interaction J_{1} with a strength of 0.49 mRy/. In our formulation, the exchange energy contribution to the Hamiltonian depends not only on relative orientation of magnetic vectors, but also scales with their length, while the exchange interaction strengths themselves are kept constant during the simulations. For both types of anisotropy modeled, the anisotropy constant K was set to K = 0.13 mRy^{2}.
The Gilbert damping parameter α in Eq. (1) defines the energy and angular momentum transfer between the spins and the bath, e.g. electrons, lattice, and the reduced dimensionality can affect the value of α, as exemplified in Steiauf and Fähnle^{29}. It is expected that α varies with systems having different magnetocrystalline anisotropy types since both effects depend strongly on the spinorbit coupling. Although the damping parameter can be calculated from first principles^{23,30,31,32,33,34}, we used the same α value for the chains with UMA, CMA and NMA so as to isolate the effect of the anisotropy mechanism on the dynamic response and varied it over an order of magnitude (from α = 0.01 to α = 0.1) in order to investigate the effect of dissipation in further detail. We find that changing the value of the damping parameter acts essentially as a time rescaling, and thus affects the behavior of the dynamics in a very simple way. This result is in agreement with NéelBrown relaxation theory for magnetic systems with axial symmetry, where the relaxation time is inversely proportional to the damping parameter^{10,11}. Unless otherwise stated, we have used α = 0.05 in our simulations for UMA, CMA, and NMA systems.
The time evolution of the average magnetization of ensembles of 1000 atom long platinum wires with UMA, CMA and NMA is shown in Fig. 2. Such a large number of atoms has been chosen in order to get good statistics and clear, smooth curves. However, as long as the chain length is larger than the correlation length, variation of the chain length does not change the results significantly.
The relaxation times for the CMA wires (blue curves) are significantly shorter than for the UMA wires (red curves) over the whole temperature range studied (3–15 K). However, the relaxation mechanisms in both cases appear to be similar and involve two steps as an inflection point can be seen on all curves corresponding to the CMA and UMA wires. The first step is associated with the smallangle precession around the anisotropy axis and establishment of local equilibrium, while the second step involves nucleation of reversedmagnetization domains. The first relaxation step is very rapid for both CMA and UMA and occurs on a subpicosecond timescale for all studied temperatures. The second step is slower, and, therefore, it is the timescale of domain nucleation that defines the relaxation time in UMA and CMA wires. For both UMA and CMA, the duration of the second relaxation step is strongly temperature dependent. For example, for temperatures below 7 K (data not shown) the spin flip relaxation is not even noticeable for the wires with UMA, during the entire simulation time of 1 ns. In contrast, at 9 K one can clearly see how the spinflips, on a time scale of about 1 ns, contribute significantly to the total decrease of the average magnetic moment and destruction of longrange order.
The NMA wires (green curves) do not show the twostep decrease in the average magnetization and the temperature dependence of the relaxation time is not as pronounced as in the other two cases which is a sign of the fact that the relaxation process is fundamentally different in this case compared to when anisotropy is present in the system. In the absence of anisotropy, the concept of spin flip is not suitable since excitations of an isotropic Heisenberg system have the form of collective spin wave formation.
In the low temperature regime, the CMA wires need longer times to relax than the anisotropyfree wires (see the upper panel of Fig. 2). On the other hand, the relaxation time changes more with temperature for the CMA case, compared to the NMA case. As a consequence, there is a crossover temperature around 5 K (see the middle panel of Fig. 2) above which the CMA wires relax faster than the NMA wires.
The relaxation times as a function of inverse temperature for UMA, CMA and NMA wires are shown in Fig. 3. Here we have considered chain lengths of 100 atoms since they give indistinguishable changes compered with the 1000 atom chains used in Fig. 2 but requires less computational effort.
Here, we have defined the relaxation time as the time it takes for the average magnetization to reach 1/e of its maximum (i.e. initial) value. It is seen that for the wires with UMA and CMA, the temperature dependence of relaxation time, τ_{r}, follows the Arrhenius law
implying thermal activation as a mechanism of the relaxation process. In Eq. (2), E_{a} is interpreted as an activation energy, v_{0} is the attempt frequency, T is the absolute temperature, and k_{B} is the Boltzmann constant.
If anisotropy is present, thermal magnetic relaxation in the nanowire involves nucleation of domains with the reversed magnetization. Each nucleation event requires overcoming an energy barrier, E_{a}, and the time scale is defined by Eq. (2) in the highbarrier limit. This mechanism is similar to the NéelBrown relaxation scenario for an ensemble of noninteracting spins with the activation energy defined by the magnetic anisotropy of each spin^{10,11}. In magnetic nanowires, the activation energy, E_{a}, as well as the preexponential factor, v_{0}, are affected by exchange interaction between atomic moments and, in particular, by the anisotropy type (see Fig. 3), as explained below. If the anisotropy is removed, the relaxation behavior of the wire cannot be fitted successfully to the Arrhenius formula, which is a sign of a fundamentally different relaxation mechanism, as explained earlier.
It seems clear that our simulated results agree very well with the Arrhenius law not only for the UMA case but also for the CMA wires. This is an interesting result in itself considering that in the CMA case, the potential landscape is altered as a function of the magnetic moment rotation. However, the activation energies are different. A leastsquares fit of the spindynamics data shows that the activation energy for the UMA wires, 0.51 mRy, is more than three times larger than for the CMA wires, for which the value of 0.15 mRy is found. This result is consistent with the much more rapid relaxation in the CMA case compared to the UMA case.
In order to shed light on the microscopic mechanism of spin relaxation in UMA and CMA wires and gain a better understanding of why the effective activation energy is significantly lower for the latter, we present an illustrative visualization of the relaxation process using colorcoded spin mapping of individual trajectories of the atomic moments, see Fig. 4. Here all wires start from the ferromagnetic ground state.
We now go through the maps starting with the uppermost row, i.e. the UMA case. At 9 K (the leftmost panel), only a few short sections of flipped spins – which also relax back to the unflipped state after a quite short time, on the order of tens of ps – can be observed during the entire simulation time of 1 ns. As the temperature is increased, the number of streaks with flipped spins increases (see middle and rightmost maps in the top row), and the flipped regions have a much longer lifetimes, as only a few regions can be seen relaxing back to the unflipped state. The initial streak width remains roughly constant over the entire simulation time but as more and more regions lump together, the flipped regions become wider, forming a clear domain structure.
In contrast, the CMA wires do not exhibit the wide domain formation as the one observed for the uniaxial anisotropy case. The spin map for the CMA case at 3 K, shown in the leftmost column in the middle row in Fig. 4, resembles partly the maps for the UMA case with the difference that the flipped domains are much narrower and with the existence of clear sharp green/yellow lines, signifying atoms where the local moments have vanished. As the temperature increases, more and more atoms lose their moments but it is also seen that due to the thermal fluctuations, a moment with a magnitude close to zero might flip towards the anisotropy axis, regaining the magnetic moment in the process. Even at 5 K (middle panel, middle row) there is no visible long range order despite the very short simulation time of 50 ps. At higher temperatures the wire becomes even more disordered with life times of small domains in the subpicosecond range.
The very narrow stripes in the middle row of Fig. 4 indicate that the minimum size of a stable reversedmagnetization domain in the CMA wires is significantly smaller than in the UMA wires. This seems to be the main reason for the lower activation energy in the CMA wires, because the energy cost for the critical domain nucleation is, to a first approximation, proportional to the domain size. Furthermore, Fig. 4 demonstrates that even atomically thin domains represent relatively longlived metastable states in the CMA wires. This is expected since the magnitude of the magnetic moments decreases quickly as they rotate away from the easy axis, thus lowering the effective exchange interaction and making spins less connected to each other in the CMA wires. Direct calculations of minimum energy paths for the magnetization switching in the UMA wires using the geodesic nudged elastic band (GNEB) method^{35} show that the minimum stable domain size is 3 spins with corresponding activation energy of 0.64 mRy, which is in a good agreement with Arrhenius fits to the spin dynamics data (see Fig. 3). The activation energy given by the minimum energy path is the one predicted by the rate theories, in particular, harmonic transition state theory (HTST)^{36} or Kramers’ theory^{37} (for adaptions of these rate theories to magnetic systems, see refs 38 and 39). These rate theories assume an establishment of thermal equilibrium locally at an energy minimum before transition to another minimum takes place as well as a specific form of dynamics at the top of the energy barrier: HTST neglects all multiple recrossing events, while Kramers’ theory assumes Langevin dynamics in a parabolic potential. These assumptions, which are satisfied only approximately, as well as finite number of switching events in spin dynamics calculations contribute to a slight disagreement between the theoretical prediction and simulation result.
Although the GNEB method accounts for the change in the magnitude of magnetic moments, it is problematic to apply it to the CMA wires because there are large regions in the configuration space where magnetic moments vanish and, therefore, the energy landscape is completely flat. Any initially defined switching path inevitably passes through such regions and gets stuck because the energy gradient guiding the optimization towards the minimum energy path vanishes there. Adaption of the GNEB method to systems with flat regions in the energy landscape is an important methodological problem which will be addressed in a future study. However, a single spinflip scenario for the magnetization reversal in CMA wires is supported by the fact that the activation energy derived from the spin dynamics simulations agrees very well with the anisotropy energy given by the anisotropy constant K.
Finally, turning to the wires lacking anisotropy (third row in Fig. 4) it is clear that timestable domains are not formed in the absence of anisotropy. Long range orde is not apparent even at 3 K. The lack of anisotropy also introduces an oscillatory behavior of the magnetism in the evolution of the different domains, adding to the disorder.
In conclusion, we find that the CMA wires relax much faster than the wires with UMA, and we attribute this to the decreasing magnitude of the magnetic moments: the decrease in effective exchange interactions makes the size of a critical reversedmagnetization domain smaller, thus lowering the energy cost for the domain nucleation. We also find that for both these types of anisotropy, the spin relaxation times can be described quite well with the NéelBrown model of magnetic relaxation. According to our relaxationtime calculations, the magnetic behavior of the CMA wire should be possible to resolve experimentally at the nanosecond time scale for temperatures below 1 K. This time scale should be accessible for soft xray free electron lasers^{40} and even pumpprobe xray transition microscopy^{41} even though the lateral resolution needed might prove very difficult to achieve.
Additional Information
How to cite this article: Bergman, A. et al. Spin relaxation signature of colossal magnetic anisotropy in platinum atomic chains. Sci. Rep. 6, 36872; doi: 10.1038/srep36872 (2016).
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Acknowledgements
We acknowledge financial support from Vetenskapsrådet (VR), The Royal Swedish Academy of Sciences (KVA), the Knut and Alice Wallenberg Foundation (KAW), Swedish Energy Agency (STEM), Swedish Foundation for Strategic Research (SSF), Carl Tryggers Stiftelse (CTS), eSSENCE, Icelandic Research Fund, and Göran Gustafssons Stiftelse (GGS). The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Center (NSC), Linköping University, the PDC Center for High Performance Computing (PDCHPC), KTH, and the High Performance Computing Center North (HPC2N), Umeå University.
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Affiliations
Department of Physics and Astronomy, Materials Theory Division, Uppsala University, Box 516, SE75120 Uppsala, Sweden
 Anders Bergman
 & Anna Delin
Department of Materials and Nano Physics, School of Information and Communication Technology, KTH Royal Institute of Technology, Electrum 229, SE16440 Kista, Sweden
 Johan Hellsvik
 , Pavel F. Bessarab
 & Anna Delin
Science Institute, University of Iceland, VRIII, 107 Reykjavík, Iceland
 Pavel F. Bessarab
Swedish eScience Research Center (SeRC), KTH Royal Institute of Technology, SE10044 Stockholm, Sweden
 Anna Delin
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Contributions
A.D. and A.B. initially designed the project; A.B. and P.F.B. performed the calculations; all authors contributed to analysing the data and writing the paper.
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The authors declare no competing financial interests.
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Correspondence to Anna Delin.
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Journal of Nanoparticle Research (2018)
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