1D lattice model for binary growth and surface relaxation

Motivated by recent progress in producing metastable metallic binary alloy films and nanoclusters through atomic deposition techniques, we propose a simple, analytically solvable model of a growing Ising-type one-dimensional chain whose kinetics are determined by random addition of two atomic species and by compositional relaxation restricted to the surface. Solutions are presented for the influence of the external flux on both surface segregation and compositional short-range correlations in the bulk, which is frozen. Despite the simplicity of the model, these results can help to understand opposing trends in surface segregation and bulk short-range order, revealed in recent experiments and simulations for fcc-type nanoalloys.


Introduction
Growth of ultrathin films and nanoclusters by the molecular beam epitaxy (MBE) technique opens a way to generate non-equilibrium solid structures, which can display novel physical properties distinctly different from the equilibrium bulk properties [1]. To suppress equilibration, the substrate temperature is chosen low enough that atomic rearrangement processes in the bulk are frozen. The resulting bulk structure then exclusively depends on the kinetics at the growing surface, driven by the deposition flux and by atomic surface relaxation steps within the outermost 'active' layers. Multicomponent systems, grown under codeposition of two or more atomic species, offer the possibility to tune atomic ordering effects with the help of the growth parameters. For example, in MBE-grown metallic magnetic nanoalloys it has become possible to induce an anisotropic short range order that entails perpendicular magnetic anisotropy, a feature of great current interest from a technological viewpoint [2,3].
Theoretically, the relationship between the frozen atomic structure below the surface and the growth mechanism is only poorly understood, despite some studies in the recent past. The emergence of frozen short and long range atomic correlations in ordering alloys and their interplay with magnetic properties has been investigated recently by kinetic Monte Carlo simulations both for films [4] and nanoclusters [5]- [7]. Other recent studies of frozen-in correlations emerging from surface processes focused on the density correlation function below a surface that grows according to the non-conserved Kadar-Parisi-Zhang equation [8].
Here, we investigate a minimal model for unidirectional binary AB alloy growth. It consists of a linear atomic array that includes a surface relaxation mechanism by allowing AB-exchange between the two progressing outermost positions. This effect should mimic indirect and direct exchange processes between unlike atoms A and B in the outermost layers of realistic threedimensional systems. Pairwise interatomic interactions imply that the chosen AB-exchange rate depends on the occupation of the third position from the top. Formally, this model is equivalent to a growing sequence of two-state systems: after addition of one particle only a newly formed AB two-state system is allowed to relax in a way that depends on the frozen state formed in previous steps. This generates a kind of linear kinetic Ising model with competing dynamics [9], represented by the AB-exchange and the external flux. Stationary state properties and transient structures can be derived from exact recursion relations for three-point correlators. When applied to measurements, this model should allow one to elucidate opposing trends in surface segregation and short-range order, and the dependence of both features on the deposition fluxes F A and F B . Effects of this kind have been observed in kinetic Monte Carlo simulations of three-dimensional cluster and film growth, where they could be analyzed quantitatively within a specific energetic model for ordering fcc alloys [5,6].

Growth model
Consider a finite one-dimensional array of A and B atoms at positions m above the substrate; 1 m n. The array evolves in time according to the following elementary steps: 1. Growth at discrete times nτ ; n = 1, 2, . . . ; by adding an A or B atom with probability p A or p B = 1 − p A to the top of the array. 2. In between two successive growth events, unlike atoms in the first and second position from the top are allowed to interchange positions. Specifically, if the top atom is A and the second one is B, the exchange rate is denoted by w 1 , while for the reverse order we denote it by w 2 . This exchange process is described by a continuous time 2 × 2 master equation. 3. The rates w 1 and w 2 depend on the occupation of the third position from the top, such that the energetics of the system is incorporated via the detailed balance condition. The occupation of the third and lower positions are frozen. Let V αβ be the nearest neighbor (NN) interaction energy between species α and β (α, β = A or B) and α a surface potential that acts only on an α-atom in the top position. Then, where γ = A or B specifies the (frozen) occupation of the third position. For example, if γ = A, an AB-attraction will decrease the rate w 1 (A) but increase w 2 (A). Figure 1 illustrates the deposition and exchange processes described in this way. It is convenient to introduce the parameters The (negative) Ising interaction J generates short-range correlations, while h acts as a surface field. For example, for J > 0 and h > 0 AB-alternation and surface segregation of B-atoms will be favored. From (1) it follows that with 4 From the form of (7) (see below) it will become clear that the deposition time τ , which is related to the external fluxes by F A = p A /τ ; F B = p B /τ , enters the problem via the factors reflecting the competition between external flux and the internal dynamics. In fact, α describes the relaxation during the deposition time τ by exchange of two unlike atoms at first and second position from the surface, given that third position is occupied by an α-atom.

Kinetic equations
Let us introduce the three-point probabilities p (n) (α, β, γ ) that an array of length n 3 immediately after the addition of the nth atom (at time t = nτ ) contains atoms of species α, β and γ in the first, second and third position from the top, respectively. Similarly q (n) (α, β, γ ) is the corresponding probability immediately before the addition of the (n + 1)th atom. We now derive recurrence relations which provide complete information on the array evolution. First of all, since like atoms do not interchange, Interchange of unlike atoms leads to where U (t, γ ) is the (2 × 2) time evolution operator describing AB-exchange, with U (0; γ ) = 1 and w i ≡ w i (γ ).
Equations (6) and (7) provide the mapping of probabilities over one time interval τ . According to the deposition rule, matching with the subsequent time interval is expressed by where we have used the notation for the two-point probabilities of the two upper positions.
Physically, the most important properties of the chain are surface segregation and occupational correlations in the frozen 'bulk' below the growing surface. A measure of surface segregation in an array of length n is the probability 4 for the topmost position being occupied by an α-atom, right before deposition of the (n + 1)th atom. Joint probabilities for the frozen occupations at positions m and m − 1 and frozen singleparticle densities in arrays of length n m + 2 are, respectively, given by and In terms of the quantities Q (n) i these relations can be rewritten as 5 4 Alternatively, surface segregation can be measured by averaging the time-dependent occupational probability of the top position over one period τ .

Results
Let us first focus on the stationary solution of (10), which corresponds to an eigenvector of the matrix (11) with eigenvalue unity. For the stationary probabilities Q st i we readily find The main features of our model become already apparent by setting p A = p B = 1/2. Then, from equations (15) and (16) with To facilitate our further discussion, we adopt symmetrical rates w 1,2 (α) = νexp( E α /2k B T ), where E α is the difference in energy before and after the respective exchange jump and ν is some prefactor. Two special cases of (20) and (21) with J = 0 and h = 0, respectively, already illustrate the influence of the external flux on surface segregation and ordering, where Evidently, both quantities (23) and (24) diminish upon increasing the flux or decreasing τ . In the limit where deposition is much faster than exchange, τ → 0, we have λ → 1, corresponding to random incorporation of A and B into the array with q st (α) = 0; C st (α, β) = 0. Generally, the correlated occupation of a pair of NN sites is built by a succession of three interdependent equilibration steps within that pair and with the adjacent sites. Because of the associated constraints the frozen NN correlator (24) is smaller than the corresponding quantity in the fully equilibrated NN Ising chain even in the limit τ → ∞ or λ → 0.   The combined effect of the parameters J and h is displayed in figures 2 and 3. An increasing J impedes surface segregation, as seen from figure 2(a) for τ → ∞. Figure 2(b) shows the temperature-dependence of q st (B) for fixed J/ h = 4. Following recent work on a realistic, three-dimensional growth model [5,6] we assume here that ν is thermally activated, ν = ν 0 exp(−E t /k B T ), with E t a transition state energy and ν 0 an attempt frequency. It follows that for low temperatures, ντ 1, surface segregation becomes kinetically suppressed ( α ∼ 1), whereas at higher temperatures with α 1 results of the type as in figure 2(a) are recovered. Thus a non-monotonic temperature dependence of q st (B) emerges with a maximum at some τ -dependent blocking temperature T b (τ ) with respect to surface So far we considered only the stationary solution of (10), reached in the limit n → ∞. In practice, regarding structural properties of nanoclusters with a height of only a small number of atomic layers, one needs to consider the small-n, transient behavior of equation (10), where initial conditions Q (2) i are determined by the specific interactions between adatoms and the substrate. To illustrate possible transient effects, we evaluate equations (10), (17) and (16) by choosing random initial conditions Q (2) i = 1/4. As seen from figure 4(a), occupational profiles show a decay of p (m) (A) towards the stationary value p α = 1/2, and a corresponding increase in p (m) (B). Superimposed are oscillations that reflect short range correlations due to the effective AB-attraction. The reason for the A-atom excess and B-atom depletion in the transient zone getting buried as the system approaches stationary growth, is the surface 9 field h which drives the enrichment of B-atoms at the free surface. In fact, as expected from conservation of A-and B-particles, one can show that for large n the quantity q st (B) exactly balances the B-depletion in the transient zone. Figure 4

Conclusions
As a soluble model for binary alloy growth on a substrate we studied a linear atomic array with Ising-type interactions that grows by adding A-and B-atoms and includes surface relaxation, while the bulk remains frozen. With these ingredients it appears to be the simplest atomistic model which can describe structural properties emerging solely from processes at the growing surface. For stationary growth we find that bulk short range order gets suppressed when the surface field h is increased, a feature consistent with three-dimensional simulations for fcc nanoalloy growth [6], and that surface segregation diminishes upon increasing the ordering interaction J . Although part of these features are reminiscent of the competition between surface segregation and ordering in equilibrated alloys [10], their physical origin here is fundamentally different, because of the frozen bulk structure. Moreover, the transition of the surface from equilibrated to kinetically hindered states with decreasing T leads to a non-monotonic T -dependence in the segregation parameter q st (B) and the short-range order parameter C st (A, B), again consistent with those three-dimensional simulations. There, because of kinetic hindrance, a drop in the B-atom segregation and in the degree of AB-ordering with decreasing temperature was found, in analogy to figures 2(b) and 3(b), respectively. Thus it appears that our model can serve as a basis for understanding important aspects in the far-from equilibrium structural evolution of more realistic systems.