Accurate estimation of a phase diagram from a single STM image

We propose a new approach to constructing a phase diagram using the effective Hamiltonian derived only from a single real-space image produced by scanning tunneling microscopy (STM). Currently, there have been two main methods to construct phase diagrams in material science: ab initio calculations and CALPHAD with thermodynamic information obtained by experiments and/or theoretical calculations. Although the two methods have successfully revealed a number of unsettled phase diagrams, their results sometimes contradicted when it is difficult to construct an appropriate Hamiltonian that captures the characteristics of materials, e.g., for a system consisting of multiple-scale objects whose interactions are essential to the system’s characteristics. Meanwhile, the advantage of our approach over existing methods is that it can directly and uniquely determine the effective Hamiltonian without any thermodynamic information. The validity of our approach is demonstrated through an Mg–Zn–Y long-period stacking-ordered structure, which is a challenging system for existing methods, leading to contradictory results. Our result successfully reproduces the ordering tendency seen in STM images that previous theoretical study failed to reproduce and clarifies its previously unknown phase diagram. Thus, our approach can be used to clear up contradictions shown by existing methods.

In this study, we present a new approach to constructing a phase diagram by combining our recently proposed theory 16,17 with two-dimensional STM images. Since our approach directly determines underlying many-body interactions only from the geometrical information of any-sized objects at thermodynamic equilibrium within a given accuracy, we can clarify that underlying interactions play an essential role in stabilizing the measured microscopic structure. Thus, when contradictions occur between ab initio calculations and experiments, it is expected that our approach, which does not require any thermodynamic information, can bridge this gap. To demonstrate the validity of our approach, the stacking-fault interface of Mg-Zn-Y alloy, in which L1 2 -type Zn 6 Y 8 clusters are arranged, is a suitable target. We herein show the interface phase diagram of the 18R-type LPSO phases in Mg-Zn-Y alloy, compare our results with previous studies, and successfully reproduce the short-range-order (SRO) of clusters seen in the STM image, which the previous work failed to reproduce quantitatively.

Results and Discussion
On the right-hand side of Fig. 1, we show a typical STM image of the distribution of Zn-Y clusters in the stacking-fault interface (see Methods for sample preparation). Since the STM is only sensitive to the topmost surface atoms, one can directly determine the individual positions of the clusters 18 . Before the evaluation of the SRO, we corrected the distortion of STM images, which originates from thermal drift. In particular, we calculated the vectors between all clusters and selected those corresponding to the 6th-nearest-neighbor (6NN) distance. STM images were corrected by linear transformation such that 6NN vectors had directions and magnitudes that were as correct as possible. Once we eliminated the effects of thermal drift, we could measure the relative positions between any pairs of clusters. We counted the number of cluster-cluster, cluster-vacancy, and vacancy-vacancy pairs in the field of the STM image up to a distance of 25NN and evaluated SRO.
Energy coefficients of mNN pairs in Eq. (2), V m , as determined by SRO from STM images (see Methods) are shown in Fig. 2(a). We also shows a schematic illustration of some mNN pairs at the interface in Fig. 2(b). Since in 18R-type LPSO, the interplane interactions are relatively weak 19 , this method is valid for reproducing the characteristics of an STM image, ignoring interplane interactions and regarding the interface as an isolated two-dimensional system, i.e., all interactions in Fig. 2 are in-plane interactions.
In order to confirm validity of V m in Fig. 2(a), we investigated the pair-formation energy of a single mNN (m = 6, 7, 8) cluster-cluster pair with an L × L two-dimensional triangular lattice under a periodic boundary condition. The pair-formation energy of ΔE(m, L) is defined as Mg cluster Here, E Mg denotes the total energy of a configuration filled with Mg, and E cluster denotes total energy of a configuration consisting of a single cluster and Mg atoms on the rest lattice points in a considered cell. The definition of Eq. (1) corresponds with pair interactions used in the previous DFT work 14 and reflects actual mNN cluster-cluster pair interactions. In Fig. 3, ΔE(m, L) is shown for m = 6-8 and L = 5-40. Figure 3 shows that positive values of ΔE(6, 5), ΔE(7, 6) and ΔE(8, 7) imply that 6-8NN cluster-cluster interactions behave repulsively under very small simulation cells, i.e., very dense cluster compositions. This tendency is similar to the result obtained by the previous DFT work 14 , which estimated interactions using a simulation cell with fully arranged clusters. Meanwhile, under dilute cluster compositions where L ≥ 10, ΔE(6, L) becomes strongly negative unlike ΔE(7, L) or ΔE(8, L). Negative value of ΔE(6, 40) in Fig. 3, which means 6NN cluster-cluster pairs are favorable, seems to contradict with the previous DFT result 14 , which held that 6NN cluster-cluster pairs are unfavorable. However, since strong oscillation of ΔE(m, L) in L ≤ 10 means that clustercluster interactions depend on the cluster composition, we actually clarify that the most probable reason for this disagreement is originate from insufficient consideration of dependence on cluster compositions in the previous DFT work 14 . Therefore, the 6NN cluster-cluster interaction behaves attractively for dilute clusters.
Although the previous result 14 successfully showed qualitative landscapes of the radial distribution function of nanoclusters, it could not reproduce the quantitative microscopic ordering tendency seen in STM images (solid   Fig. 1). This is because the previous DFT work determined cluster-cluster interactions by dense-cluster simulation cells, whereas we found that interactions behave differently between dense and dilute cluster concentrations, as shown in Fig. 3. Now, let us confirm that our multiscale interactions in Fig. 2(a) can capture the characteristics of SRO of clusters at the interface. Figure 4 shows thermodynamically averaged SRO at equilibrium, as obtained by Monte Carlo simulation (see Methods for a more detailed procedure), and this SRO corresponds to the radial distribution function of clusters. The strong SRO of 6NN pair means that the number of Mg-Mg and cluster-cluster pairs are relatively larger than that of Mg-cluster pairs, i.e., clusters are arranged so as to be at each other's 6NN position. Our SRO landscape agrees well with the radial distribution functions obtained by previous DFT and experimental work, especially for the peaks in 6, 9, and 15NN pairs. For further discussion, we took a snapshot at the interface in the simulation and compared it with an STM image in Fig. 1. From the STM image, we can clearly confirm chain-like cluster ordering, unlike the previous DFT work, which showed a uniform arrangement of clusters. As above, our snapshot quantitatively captures the features of STM images and supports the validity of our multiscale interactions.
The Mg 1−x (Zn 6 Y 8 ) x interface phase diagram obtained by different three STM images is presented in Fig. 5(a), which shows the ordering tendency in the cluster-rich phase (Fig. 5(b)) through a first-order order-disorder phase transition. Since, the interplane interactions are relatively weak in 18R-type LPSO 19 , it is valid to regard the interface of LPSO as an isolated two-dimensional system. As a result, the closed circle in Fig. 5(a), showing the point where the STM image in Fig. 1(b) is obtained, indicates that Fig. 1(b) is not a single phase but a order-disorder phase coexistence state.
In summary, we suggested a new approach based on microscopic geometric information for constructing phase diagrams in cases where this is difficult for conventional simulation methods. Using STM images, we demonstrated our approach through the interface of 18R-type Mg-Y-Zn LPSO. We clarified the contradictory information concerning cluster arrangements presented by experiments and simulations and successfully reproduced an interface phase diagram consistent with that obtained by experiment. Our approach is expected to become a powerful tool for modeling isolated systems subjected to experimental observations such as STM.

Methods
Sample preparation for STM images. The directional solidification process was applied to a Mg 85 Zn 6 Y 9 ingot. The ingot was annealed at 773 K for 168 h and quenched with water. We confirmed the presence of a diffraction spot corresponding to 18H-type LPSO by TEM observation.
Samples for STM observation were prepared by cutting the alloy ingot, typically into a reed shape of 8 mm × 5 mm × 0.5 mm. After the introduction of the sample into the preparation chamber of the low-temperature ultrahigh-vacuum STM (Unisoku 1200), we cooled the sample with a liquid-nitrogen flow for 10-15 min. Finally, we cleaved the sample using a pushing rod in the UHV chamber. The cleaved sample was immediately transferred into a pre-cooled observation chamber. All STM observations were carried out at liquid-nitrogen temperature (up to 77 K).
STM itself has enough spatial resolution to resolve each atomic position on the surface. However, due to some noise and thermal drift, the positions of the clusters determined by our method have some uncertainty. To demonstrate the accuracy of our method, we plotted relative positions of the clusters measured from all the clusters in a single STM image in Fig. 6. As clearly seen, the relative positions of the clusters construct a lattice; which indicate that the accuracy of positions is good enough to estimate SRO. Generalized Ising model. In recent alloy studies with ab initio calculations, the generalized Ising model 20 (GIM) has often been used for constructing a coarse-grained Hamiltonian, which captures the characteristics of a system within a given accuracy. Since we are interested in how clusters would be arranged at the interface with  Mg solvent, for simplicity, it is natural to consider the clusters as coarse-grained atoms, i.e., Mg and the center of cluster as +1 and −1 Ising-spin variables, respectively. In GIM, the total energy of a given configuration, σ, is expanded by a set of complete and orthogonal basis functions, { } k ξ , with indices where V k denotes an energy coefficient and k is an index specified as a symmetrically nonequivalent figure such as 1NN and 2NN pairs. In particular for a binary system, ξ k corresponds to the average product of spin variables s α on k over all symmetrically equivalent k in an Ising-spin configuration where α denotes a site on a given lattice and N k denotes the number of k in the configuration. V k is defined as the inner product between total energy and ξ k Here 〈|〉 denotes inner product for configuration space, ρ k 0 denotes a normalized constant for the inner product, and the term in 〈〉 denotes over all possible configurations except for pair k (i.e., σ N−k ). Transformation from the second to the third line means that summation is taken in terms of possible configurations on pair k (i.e., σ k ) instead of a summation over all possible configurations of σ. This transformation explicitly shows us that when a given system includes many-body interactions should include the many-body interactions except for that of k. Therefore, even though in this study pair-wise ks are only considered, our method correctly treats essential many-body interactions.
Short-range-order in our simulation. We calculated the SROs of clusters using the Metropolis algorithm with a 100 × 100 two-dimensional triangular lattice under fixed composition. Note that, to explicitly consider the size of the nano-sized clusters without loss of validity, interactions of 1-3NN pairs are configured as having a very high so as not to overlap with each other.
Determination of many-body interactions from a single STM image. We first assume that the measured structure obtained by STM is in thermodynamic equilibrium. Then, the expectation value of the kth coordination of the structure at temperature T, ξ k (T), can be given in the framework of classical statistical mechanics as where summation is taken over all possible configurations σ, and Z denotes a partition function. Under these conditions, we have recently derived the relationship 16,17 between the structure, ik ik ave B where Q ave represents a linearly averaged configuration and S ik denotes an element of the covariance matrix for the configurational density of states in ξ ξ ( , ) i k two-dimensional space for a non-interacting system. Both of these quantities can therefore be known a priori without any information about energy or temperature. From Eq. (6), we can thus directly determine the energy coefficients defined in Eq. (4) from a measured structure using the equation In this study, the interface is regarded as a Mg-cluster pseudo-binary alloy. To construct the alloy phase diagram, a semi-grand-canonical (SGC) ensemble is more often used than grand-canonical or canonical ensemble. In an SGC ensemble, compositions can vary under the number of atoms, N, kept fixed, i.e., we handle the difference of chemical potentials for each constituent element, µ µ µ = − cluster M g in spite of handling compositions. The total energy, E SGC , is defined as E E Nx SGC µ = − , and the free energy, φ, is defined as φ = −k T Y ln B , with Boltzmann constant k B and the partition function in the SGC ensemble Y defined as: where W denotes density of states (DOS). Note that Helmholtz free energy F in the canonical ensemble is related to the free energy φ in the SGC ensemble through Legendre transformation: We can obtain x using partial differentiation by interpolating φ for each chemical potential: