Magnetic behavior of one- and two-dimensional nanostructures stabilized by surface-state electrons: a kinetic Monte Carlo study

Recent experiments have demonstrated that it is possible to create macroscopic-ordered one- and two-dimensional nanostructures on (111) noble metal surfaces exploiting long-range substrate-mediated interaction. Here, we report on the systematic theoretical studies of magnetic properties of these atomic structures in the externally applied magnetic field. The spin dynamics is investigated by means of the kinetic Monte Carlo method based on transition-state theory. For the characteristic values of (i) magnetic anisotropy energy of adatoms and (ii) exchange coupling between adatoms in the considered class of nanostructures, we reveal the hysteresis-like behavior at low temperatures (typically at 1–3 K).


Introduction
Modern nanoscience manifests strong interest in magnetic nanostructures on surfaces. It is believed that such systems can be of great importance for the development of advanced atomicscale magnetic devices [1,2]. Magnetic nanostructures on metal substrates attract significant attention due to the enhanced magnetic moments of low-coordinated adatoms [3]- [5] and the large magnetic anisotropy energy (MAE) peculiar to a magnetic unit [6]- [9]. When the amount of thermal energy, which causes fluctuation of the magnetic moment, is enough to overcome the MAE barrier, the spin orientation is random. This takes place at the so-called blocking temperature [10,11]. In real magnetic ensembles, interaction between individual spins (exchange interaction, dipole-dipole interaction, and indirect exchange interaction through the substrate) could stabilize ferromagnetic (FM) order, leading to higher blocking temperatures [10]- [12]. Up to now, numerous experimental and theoretical studies have revealed a lot of nanoscale systems exhibiting hysteresis-like (i.e. FM) behavior: quasi-1D (one-dimensional) stripes [13]- [17], monatomic wires [10], [17]- [21], pillars [22], nanodots [23]- [27], nanoscale clusters and islands [28]- [30].
Recent studies on low-temperature self-assembly of metal adatoms on (111) metal substrates have demonstrated the existence of a new class of 1D and 2D systems: nanostructures of individual adatoms stabilized by the long-range substrate-mediated interaction (LRI) [31]- [33]. It is well known that conduction electrons on a (111) noble metal surface form a 2D nearly free electron gas confined in the vicinity of the top layer. Scattering of the surface states at adatoms (or defects) leads to the standing wave patterns of local density of states [34]- [42] and to indirect oscillating interaction between adsorbates [31]- [33]. On surfaces supporting the surface-state electrons the LRI decays as 1/r 2 (where r is the interatomic distance) and oscillates with a period of λ F /2 (where λ F is a surface-state Fermi wavelength) [43].
If two adatoms are separated by the first nearest neighbor (NN) distance r 0 (figure 1) the interaction is determined only by the short-range chemical bonding, and it is strongly attractive. At larger interatomic distances (>5 Å) the interplay between adatoms is caused by the elastic interaction [44,45] and the substrate-mediated LRI. The schematic description of the 'adatom-adatom' LRI is presented in figure 1. There is a repulsive barrier A; its magnitude is between 10 and 60 meV, depending on the sort of interacting adatoms and the type of surface [41], [46]- [52]. At low temperatures (<50 K) this repulsive barrier inhibits agglomeration of two adatoms into a dimer, decreasing the probability of cluster formation. At larger separations (see figure 1) there is an attractive minimum B; its magnitude is a few meV [41,46,47], [52]- [54]. Typically, this minimum is located at separations which are 5-10 times larger than the NN distance r 0 . For instance, on Cu(111) the position of the minimum B is at 12 Å [41,46,47,54], while on Ag(111) it is at 30 Å [52,53]. There is yet another repulsive barrier C, whose magnitude is usually less than 1 meV. According to [33], at larger separations r , E(r ) ∼ sin(α 1 r + α 2 )/r 2 . The positions of the minima and maxima of the LRI potential (figure 1) are determined by the type of surface and scattering properties of adatoms.
(figure 2(b)), molecular templates [61] or reconstructed substrates [62] for self-organization of 1D atomic structures. The separation between adatoms in all these nanostructures corresponds to the position of the local minimum B in the LRI potential ( figure 1). While the details of the growth of 1D and 2D nanostructures stabilized by surface-state electrons are well understood to date, the knowledge of their magnetic properties has still not been gained. The goal of this paper is to perform theoretical studies of magnetic properties of the 1D and 2D nanostructures stabilized by indirect LRI between adatoms on (111) metal surfaces. We demonstrate that in the externally applied magnetic field at low temperatures (typically at 1-3 K) such systems could exhibit hysteresis-like (i.e. FM) behavior promoted by the indirect exchange coupling between adatoms through the substrate. We reveal how the coercive field depends on the magnitude of exchange interaction between adatoms and on the MAE of individual adatoms within a nanostructure. We study the dependence of the magnetization response on the size of a system (number of adatoms in a nanostructure). As a method of investigation, we employ the kinetic Monte Carlo (kMC) model for simulation of spin dynamics [19]. The remainder of the paper is organized as follows. In section 2, we give the details of the model that was used to simulate the spin dynamics. In section 3, we present the results of our calculations and discuss them.

The model
In this section, we describe the model that was utilized to simulate spin dynamics and we discuss the parameters of the investigated systems.
Two different systems are considered. The first system is a 1D finite chain of spins (figure 3(a)), which represents a monatomic wire of magnetic adatoms stabilized by the LRI in the vicinity of a descending step (figure 2(b)) [60] or inside a molecular trench [61]. Since the average distance between the wires is several times larger than the interatomic distance within a wire [60,61], the effect of interchain exchange interaction on the magnetization response can be neglected. The second system under study is a 2D hexagonal-ordered array of spins (figure 3(b)), which represents magnetic adatoms assembled in a hexagonal nanoisland [46,47] or superlattice [52,53,57,58] (if the number of adatoms tends to infinity (figure 2(a))). For both systems the Hamiltonian of the classical Heisenberg model with an on-site anisotropy and external magnetic field B can be written as [19] where s i is the normalized spin value at site i, s z,i is the Z -component of the vector s i (see the direction definition in figure 3), the summation i, j is taken over all neighboring pairs of spins i and j, s i s j is a scalar product of spins at sites i and j. J is the exchange coupling constant (J < 0 for FM and J > 0 for antiferromagnetic (AFM) interaction), K is the MAE and µ is 5 the magnetic moment of an adatom. Individual spin is considered as a classical variable. In our study we concentrate on the FM-coupled spins only. The parameters of the Hamiltonian (1) are taken from our ab initio calculations. Density function theory reveals that the typical magnitude of the surface-state-mediated exchange interaction between magnetic adatoms is about 0.1-0.5 meV [54], when the 'adatom-adatom' separation corresponds to the position of the local minimum B in the LRI potential (figure 1). Depending on the sort of adatoms, this interaction can be either FM or AFM 5 . Within our study we consider that |J | is in the interval from 0 to 1 meV (J < 0), which implies a reasonable range of possible values of exchange coupling between the neighboring adatoms ( figure 3). The magnitude of the MAE K of an adatom in a chain (figure 3(a)) and in a 2D hexagonalordered array (figure 3(b)) is very close to the MAE of a single adatom on a clean substrate, since the separation between spins is a few times larger than the NN distance r 0 . Typically, the magnitude of K of a magnetic adatom on a metal surface is about 1.0 meV [64]. However, in some cases this value can be strongly enhanced. For instance, MAE of an individual Co adatom on Pt(111) is 9.3 meV [6]. Thus, within our study we consider that K varies from 0.1 to 10 meV. The magnetic moment of an adatom is taken to be 3.2µ B (µ B ≈ 0.058 meV T −1 is the Bohr magneton), which corresponds to the magnetic moment of a single iron adatom on Cu(111) [65,66]. The easy magnetization axis is considered to be perpendicular to the surface plane, i.e. collinear to the direction of the vector Z (figure 3).
We analyze the magnetization response for nanostructures shown on figures 3(a) and (b) to the externally applied magnetic field applied along the easy magnetization axis. We start from a strong field −B 0 (−5 T), oriented along the −Z -direction, at which all spin magnetic moments are aligned along the −Z -direction. The field strength increases by an increment B = 0.001 T gradually to +B 0 (5 T), which corresponds to the situation when the magnetic field is oriented along the Z -direction. Then the field decreases back to −B 0 , so that a sweeping cycle is complete. We consider that the magnitude of the sweeping rate of the external magnetic field dB/dt is about 130 T s −1 [19]. We use free boundary conditions in our simulations. Spin-spin correlation in open 1D chains and 2D hexagonal arrays is drastically reduced at the border sites [67]. This significantly alters the magnetization reversal rate of the systems, leading to lower blocking temperatures than in the case of periodic boundary conditions. Diffusion barriers during a single spin flip must be taken into account in order to describe spin dynamics in the investigated systems. In order to do this, we employ the kMC model recently introduced by Li and Liu [19]. This model has been successfully applied for the simulations of the magnetization response of monatomic Co chains on a stepped Pt(997) surface at different temperatures [19]. An individual spin i takes any value between −1 and +1 (s i = −1/ + 1 correspond to the spin orientation along the −Z /Z -direction, respectively). If model parameters J and K satisfy the condition 2K > |h i | (where h i = ( j J i j s j + µB)s i ), the energy increment during spin rotation to the opposite direction (from s i = −1 to +1 or from s i = +1 to −1) yields a transition-state barrier [19]: The rate of the single spin flip is calculated then according to the Arrhenius law: where k B is the Boltzmann constant and prefactor v 0 is 10 9 Hz [10,19]. If the condition 2K |h i | is satisfied, there is no transition state barrier for the spin reversal process, and the Glauber method [68] is used to compute the exponential factor of the rates, with the prefactor to be the same.
If the total number of spins in a system is equal to N , then for every kMC step N different spin flips with the rates v 1 , v 2 , . . . , v N are possible. The time increment τ corresponding to one step of the kMC algorithm is computed using the formulae [63]: where U is a randomly distributed number in the interval (0, 1).

Results and discussion
We start this section with the investigation of the magnetization response for a 1D chain of spins ( figure 3(a)). It is interesting to study how the coercive field B c depends on the length of a chain. The results in figure 5 demonstrate a size effect: the temperature dependence of B c for chains consisting of N = 100 (black circles) and N = 10 spins (white circles). In both cases, 1D chains exhibit FM behavior up to about 1.5 K. One can see that the size effect is quite weak: increasing the length of the chain by ten times increases B c by less than 10%. The typical values of coercive field B c in the studied systems are of the same order as in 1D Co wires grown on Pt(997) [10,19]. Figure 6 demonstrates the dependence of coercive field B c for the chain of N = 100 spins on the MAE K and exchange constant J . The temperature of the system is T = 0.5 K. From figure 6 one can see the onset of ferromagnetism in the whole range of values J and K considered in our study. Increasing J at a fixed K leads to an increase of B c . Similarly, at a fixed J , larger values of B c are reached at larger K . We note that we cannot determine B c for K > 1.5 meV, since there are no spin flips within the half-period of oscillations of the externally applied magnetic field: spins keep their initial magnetization fixed along the −Z -direction 6 .
B c depends on the value of the sweeping rate dB/dt. Our systematic studies demonstrate that B c increases monotonically with increasing dB/dt at fixed N , J , K . Now we turn to the discussion of the magnetization response for a 2D hexagonal-ordered array of spins ( figure 3(b)). 6 The half-period of oscillations of the externally applied magnetic field T osc = 10 T/130 T s −1 ∼ 0.08 s. On the other hand, at K = 1.5 meV, the average time of a single spin flip can be estimated using the formula (2) (see text) t flip ∼ 1/v 0 exp(((K − 0.5µB 0 ) 2 )/(K k B T )) (µ = 3.2 * 0.058 meV T −1 , k B = 0.086 meV K −1 , v 0 = 10 9 Hz, T = 0.5 K, B 0 = 4.5 T) and is equal to t flip ∼ 0.08 s. At larger K , t flip > T osc , therefore no spin flips are possible.   Figure 9 shows the dependence of coercive field B c for a 2D array of N = 100 spins on the MAE K and exchange constant J . The temperature of the system is T = 0.5 K. The FM behavior is seen for the whole range of values J and K involved in our study.  It is of great importance to discuss the Kondo effect [69]- [73], which could arise in the considered class of the 1D and 2D systems (figure 2). In the Kondo effect, a localized magnetic moment of an adatom is screened below the Kondo temperature T K and forms a correlated electron system with the surrounding conduction electrons on the non-magnetic host [69]- [73]. Exchange interaction between adatoms can modify the Kondo temperature considerably [74]. Generally, if the magnetic ordering temperature T m of a system of the FM-coupled units is larger than T K , the observation of FM response is possible only in the temperature range (T K , T m ) [74,75]. Since the typical temperature of magnetic ordering T m in the considered class of nanostructures is a few K, we note that all our conclusions are applicable to the systems that exhibit very small Kondo temperatures. Our results are also valid for the case of an underscreened Kondo lattice, when the coexistence of FM order and Kondo behavior is possible [76].
It is also worth noting that at low temperatures quantum effects (a spin switching induced by quantum tunneling (QT)) could play an important role [9]. In the following we discuss a possible effect of QT on the onset of ferromagnetism of spin chains and hexagonal-ordered 2D spin arrays. We suggest that the frequency of spin flips of an isolated adatom in the presence of QT can be described using the formula: v = v 1 + v 0 exp(−K /k B T ). Here, v 1 is the rate of spin switching induced by QT (it is independent of the temperature 7 ) and v 0 exp(−K /k B T ) is the frequency of thermoactivated spin flips. If T is high enough, the second term dominates, and the frequency of spin flips follows the Arrhenius law. However, at low T , the term v 0 exp(−K /k B T ) tends exponentially to zero, and there is a range of temperatures where v 1 dominates, i.e. v v 1 .
Let us consider an individual spin in the absence of external magnetic field. The Hamiltonian of such a system is H = −K (s z ) 2 , independently of the absence or presence of QT.
as a new 'effective' prefactor instead of v 0 (see equation (3)) for the calculation of switching rate in the presence of QT in the framework of the applied kMC algorithm [19].
Let us now discuss possible values of v 1 . The upper limit of v 1 is set by the condition v 1 v 0 , which is derived from the fact that at high T (T → ∞) the frequency of thermoactivated spin flips v 0 is much larger than the frequency of spin flips promoted by QT. Therefore, v 1 10 9 Hz. The rough estimation of the lower limit of v 1 can be obtained from the experimental studies of Meier et al [9] on the magnetization response of individual Co adatoms placed on a Pt(111) surface. The S-like paramagnetic curves over Co adatoms on Pt(111) have been recorded at two different temperatures, 0.3 and 4.2 K [9]. It is well known that a Co adatom on Pt(111) has MAE K = 9.3 meV [6,9]. From the formula 1/t av ∼ v 0 exp(−K /k B T ) (where t av is the average time of a thermoactivated spin flip), one finds that t av = 10 147 s at T = 0.3 K and t av = 10 2 s at T = 4.2 K. At the same time, Meier et al [9] have reported that spin switching takes place much faster than the resolution time of the experiment (100 Hz). Hence the observed value of t av is 0.01 s. It has been suggested that either QT or current-induced magnetization switching could promote 'fast' spin flips of individual Co adatoms [9]. If we suggest that QT is responsible for the observed spin switching, we obtain the lower limit of v 1 10 2 Hz. As a result, we conclude that most probably v 1 is in the range between 10 4 and 10 7 Hz.
In order to take into account quantum effects, we have modified the kMC model described in section 2. The procedure of calculation of activation barriers remains unchanged; however, we employ prefactor v eff (see equation (5)) instead of v 0 to compute transition rates (see equation (3)). To illustrate the concept of v eff , firstly we have followed the experimental setup of Meier et al [9] on the magnetization response of individual Co adatoms on Pt(111) at the temperatures 0.3 and 4.2 K. The following values of parameters have been used in our calculations: K = 9.3 meV, µ = 3.8µ B , B 0 = 7.5 T and dB/dt = 0.01 T s −1 (slow sweeping rate). The results of kMC simulations are presented in figure 10. For the calculations we have considered that v 1 = 10 4 Hz. We have also tested other magnitudes of v 1 within the range from 10 4 to 10 7 Hz and we have found that both the resulting curves remain unchanged. From figure 10 one can see that the magnetization response at both temperatures is paramagnetic. The values of saturation fields are about 0.3 T at 0.3 K and about 5.0 T at 4.2 K, in agreement with the experimental observations [9].
We now turn to the effect of QT on the possible onset of ferromagnetism in 1D spin chains and 2D hexagonal-ordered spin arrays at low temperatures. Figure 11 demonstrates the magnetization response for a chain of N = 100 spins at T = 1.0 K. We have already found (figure 4(a)) that this system exhibits FM behavior with coercive field B c = 0.26 T (black circles in figure 11). White circles represent the magnetic signal calculated at v 1 = 10 4 Hz: FM response with B c = 0.18 T. In the case of v 1 = 10 5 Hz (blue circles), we find that B c = 0.07 T. Red circles demonstrate the magnetization response at v 1 = 10 6 Hz: hysteresis loop with B c = 0.02 T is observed. If v 1 = 10 7 Hz, then B c = 0.003 T (not shown in figure 11). Taking into account the results shown in figure 11, we conclude that QT could decrease the coercive field B c . However, it does not influence the onset of ferromagnetism in 1D spin arrays.
We have also investigated the effect of QT on the magnetic behavior of the 2D array of N = 100 spins at T = 1.2 K (figure 12). Black circles demonstrate the hysteresis loop obtained in the absence of QT (v 1 = 0): the system exhibits FM behavior with B c = 0.96 T (see also figure 7(a)). At v 1 = 10 4 Hz the magnetization response is almost the same as that in the absence of QT, i.e. at v 1 = 0. The magnitude of the coercive field in this case is 0.95 T (not shown in figure 12). White circles (figure 12) demonstrate the magnetic signal at v 1 = 10 5 Hz: hysteresis loop with B c = 0.89 T is observed. At v 1 = 10 6 Hz we find that B c = 0.76 T (blue circles), whereas at v 1 = 10 7 Hz B c is decreased to 0.64 T (red circles). Thus, similar to the 1D spin chains (see figure 11), the presence of QT does not affect the onset of ferromagnetism in 2D spin arrays.

Conclusion
Using the Heisenberg Hamiltonian, we have theoretically studied the magnetic properties of 1D and 2D spin systems with the parameters (exchange coupling constant and MAE), which are typical for atomic-scale structures stabilized by the substrate-mediated interaction. Our investigation has been performed by means of kinetic Monte Carlo simulations based on transition-state theory. We have shown that in the externally applied magnetic field at low temperatures (typically at 1-3 K), the considered class of spin systems exhibits hysteresislike behavior. We have found that in the 2D hexagonal-ordered spin arrays the magnetization response depends on the size of the system, whereas in the 1D case the size-effect is not strong. Our systematic studies have demonstrated that QT does not influence the possible onset of ferromagnetism.