Surface roughening of heteroepitaxial films is typically attributed to competitive processes of strain energy and surface energy29,30. When the defect-mediated strain accommodation near the interface is incomplete, the residual internal strain tends to destabilise the flat film surface, thus resulting in undulated-surface films. As shown in Fig. 1, heteroepitaxial Ag(111) thin films grown on Cu(111) buffer layers were atomically flat, with occasional monoatomic-step edges (Supplementary Fig. 3). This implies that the lattice mismatch strain at the interface should be fully relaxed to its single-crystal state within as short a range as possible. This perfect strain relief is unusual when large inter-material lattice mismatches are considered (approximately 13%).
To explore the lattice strain relief in the Ag(111)/Cu(111) heterostructure, the atomic structure of the interface between the two materials was analysed using ADF-STEM imaging. The top panel of Fig. 3a shows the ADF-STEM image of the Ag(111)/Cu(111) heterostructure. Evidently, the interface at which the first Ag layer and the topmost Cu layer face each other (marked by dotted line) was atomically abrupt and structural-defect-free. As shown in the bottom panel of Fig. 3a, the stacking order of the Ag(111) planes was reversed in the order of B–A–C… on the forefront Cu(111), with a stacking order of …A–B–C. Thus, the C plane of Cu(111) and the B plane of Ag(111) faced each other at the interface. Given the large lattice mismatch between Ag(111) and Cu(111), in-plane Ag atoms in the first Ag(111) plane remarkably seemed to reside on the flat Cu(111) surface without displacement or intervention by misfit defects. Furthermore, the arrangement of Ag atoms starting from the second Ag(111) plane did not suffer from any structural distortions that could occur as a result of strain relaxation. The measured planar spacing values of Ag(111) and Cu(111) for the growth direction were 2.36 and 2.09 Å and for the in-plane directions were 1.67 and 1.48 Å, respectively, consistent with the values for their single-crystal structures (Figs. 1b, 1c, and Fig. 3a, top panel).
To examine the structural integrity of the grown Ag films, the projected atomic distances (PADs) of each atomic row across the interface were measured (Supplementary Fig. 4). The undistorted PADs of Ag–Ag (dAg−Ag) and Cu–Cu (dCu−Cu) atoms in the < 110>-orientated single-crystal structures are 2.51 and 2.23 Å, respectively. The histograms of the measured PADs for each atomic layer across the interface provide crucial information for understanding the unique strain-relaxation behaviour of the grown Ag films (Fig. 3b). First, the average PAD in the first Ag layer decreased by 0.09 Å and the PAD values substantially fluctuated in the ± 0.3 Å range. Owing to these characteristic in-plane atomic displacements, the mismatch strain was fully released within the first Ag monoatomic interface layer, and the consecutive Ag layers grew into a single-crystal structure without strain constraints. Thus, the first Ag monolayer acted as a perfect strain absorber. This ultimately thin strain-releasing layer could exclude the collective cooperation of displaced atoms forming a dislocation as usual for heteroepitaxial growth, thereby enabling defect-free growth of Ag films. Second, given the relatively large scatter of the Cu PADs in the topmost Cu(111) layer, the top Cu atoms were also slightly modulated to cooperate for the strain relief. Considering that the average Young’s modulus (63 GPa) of Ag(111) is much smaller than that of Cu(111) (102 GPa), no noticeable modulation of the Cu bond length would be observed in the topmost surface layer. However, for the Ag(111)/Cu(111) multilayer, Young’s modulus (87.5 GPa) of the Ag/Cu multilayer was approximately the average of those of Ag and Cu31. This suggests that the topmost Cu layer conjoined to the first Ag layer could be more ductile with reduced stiffness than the Cu layers inside the Cu film. Eventually, the misfit strain relief in the Ag(111)/Cu(111) heterostructure was, to some extent, assisted by the Cu atomic displacement in the topmost Cu surface layer. Finally, because the misfit strain was completely accommodated in the Ag–Cu interfacial layer, the Ag atomic displacements were no longer manifested immediately after the second Ag layer. Atomic-level crystallinity of the obtained Ag films was consistent with that of their bulk single crystals.
A close inspection of the in-plane atomic array of the first Ag layer revealed that most Ag atoms populated Ag lattice sites in an orderly manner, except for several Ag atoms that exhibited fuzzy contrast in ADF-STEM images (yellow ovals, Fig. 3c). This is questionable, considering the large scatter of in-plane Ag PADs (Fig. 3b). Diffusive and weak contrasts in ADF-STEM images appear when some of the atoms are displaced relative to their original atomic columns. The simulated ADF-STEM images of the atomic model with displaced Ag atoms and the intensity profile of the in-plane Ag atoms agreed well with experimental results (Fig. 3c, bottom graph). Taking this structural feature into account, we argue that some of the Ag atoms in the first Ag(111) monolayer were displaced in-plane from the original atomic columns. Multiple STEM observations of different TEM samples for PAD statistical analysis revealed that the in-plane displacements of Ag atoms occurred predominantly at every interval of the 7th or 8th atomic distance. The top panel in Fig. 3a shows one of the typically observed interfacial structures. After the site assignment by 2D Gaussian was fit to the ADF-STEM images (Fig. 3a, bottom panel), the occurrence of disordered sites was revealed to be repetitive, with a periodicity of seven or eight atoms. This observation implies that the Ag atomic displacements may follow an underlying crystallographic relationship between the Ag and Cu structures, which is related to long-range atomic distance mismatches.
In heterostructure combinations, EADM is crucial for successful heteroepitaxy when a substrate or buffer layer is atomically flat and grain boundary-free. Empirically, a film can be epitaxially grown on a substrate when the EADM between the film and substrate is less than 1%19,24. The EADM relation is expressed as df:ds ~ nf:ns, where df and ds are the atomic distances in the film and substrate, respectively, and nf and ns are the smallest nonreducible integers that closely match the ratio of the atomic distances, df/ds. In this relation, the value of the EADM is calculated as (nfdf – nsds)/nsds. For the case of the Ag(111)/Cu(111) heterostructure, the evaluated EADM is approximately 0.32% with the extended matching periodicity of 15 Ag atoms (15 × \({\text{d}}_{\text{Ag-Ag}}^{\left[\text{11}\stackrel{\text{-}}{\text{2}}\right]}\)) and 17 Cu atoms (17 × \({\text{d}}_{\text{Cu-Cu}}^{\left[\text{11}\stackrel{\text{-}}{\text{2}}\right]}\)) for the [11\(\stackrel{\text{-}}{\text{2}}\)] direction parallel to the interface, where \({\text{d}}_{\text{Ag-Ag}}^{\left[\text{11}\stackrel{\text{-}}{\text{2}}\right]}\) and \({\text{d}}_{\text{Cu-Cu}}^{\left[\text{11}\stackrel{\text{-}}{\text{2}}\right]}\) are 2.89 and 2.56 Å, respectively. In the plane-view model of the Ag(111)/Cu(111) heterostructure with this EADM relation (Fig. 3d), the crystallographic Moiré superlattice, defined by the extended lattice parameters of Sa (43.4 Å) and Sb (43.4 Å), is observed. Owing to the two-atom difference in the extended matching periodicity between the Ag and Cu atoms, the superlattice is constructed by four Moiré unit cells with different dimensions (M1, M2, M3 and M4 in Fig. 3d). In the superlattice, the matching points of the Ag and Cu atoms recur with two different periodicities of 7Ag:8Cu and 8Ag:9Cu, respectively (red balls in Fig. 3d), which can be better described by the cross-sectional atomic model (Fig. 3e). Given this EADM geometric relationship, the Ag-to-Cu matching points are expected to be unstable owing to the formation of the shortest bond length between the Ag–Cu atoms in the heterojunction. Thus, the Ag atom atop the Cu atom could be displaced toward one of the six valleys in the (111) plane to reduce the potential energy. In this case, horizontally elongated atomic columns could be formed by the displaced Ag atoms and repeated every seven or eight Ag atomic columns. The cross-sectional STEM image and corresponding image simulation results show these repeated in-plane atomic displacements (Figs. 3a, 3c) that appear as fuzzy and elongated contrasts along the interface. This unique structural feature clearly explains that the large scatter of interfacial Ag atomic distances is attributed to the periodical displacement of Ag atoms sitting atop Cu atoms, which acts as a major structural agent for the misfit strain accommodation. Thus, the formation of an atomically flat heterointerface is crucial for EADM-mediated strain-free growth of single-crystal metal films.