Determining the energetics of vicinal perovskite oxide surfaces

The energetics of vicinal SrTiO$_3$(001) and DyScO$_3$(110), prototypical perovskite vicinal surfaces, has been studied using topographic atomic force microscopy imaging. The kink formation and strain relaxation energies are extracted from a statistical analysis of the step meandering. Both perovskite surfaces have very similar kink formation energies and exhibit a similar triangular step undulation. Our experiments suggest that the energetics of perovskite oxide surfaces is mainly governed by the local oxygen coordination.


I. INTRODUCTION
The perovskite oxides are a fascinating class of material, due to their wealth in available physical properties, such as superconductivity, ferromagnetism, ferro-and dielectricity.
Despite their abundance of functional properties, the ABO 3 perovskite platform shares a common crystal structure with similar lattice parameters. For application within oxide thin film devices, grown multilayer (perovskite) heterostructures typically contain at least one functional active layer which is then directly supported onto a substrate, or with a bottom electrode layer in between film and substrate. In these (ultra)thin film structures enormous strains might be beneficial, as they can result in unanticipated functional properties such as altering T c of ferromagnetic and superconducting materials [1][2][3][4] . However, uncontrolled strain relaxation might also result in destructive cracking [5][6][7] or threading dislocation cascades [8][9][10] within thin films. It is therefore of utmost importance to identify and quantify strain relaxation behavior within (ultra)thin films. As we will demonstrate furtheron, strain relaxation phenomena are easily identified at the thin film interface. Besides strain relaxation, the kink formation energy is an important parameter in thin film growth, facilitating nucleation during growth and thereby determining the resulting growth to be either of rough 3D or 2D stepflow character. The formation of steps usually introduces surface stress, influencing the step edge morphology. In view of its technological relevance, most of the step-related surface studies have been focused so far on silicon as a model system 11,12 .
In this paper, we describe a method to determine both, the strain relaxation energy together with the step edge formation energy by the use of Atomic Force Microscopy (AFM) as it can image the surface topography irrespective of its band gap. To our knowledge, this study is unique in the field of complex oxides as it is a quantitative study to extract energetic parameters using AFM where AFM studies reported in literature so far [13][14][15] have been of qualitative character. By imaging the meandering of step edges, which are determined by the step free energy, the kink formation energy can be determined 16 . As in mono-metal oxides the geometric structure at the surface is considered to be a continuation of the atomic arrangement of building blocks 17 , we assume in our analysis that for the more complex perovskite oxides the ABO 3 building block is the elementary unit to describe the surface energetics.
This paper is arranged as follows. First, we describe the experimental details with empha-sis on the surface preparation to ensure the surface under study is in thermal equilibrium.
Next, we explain the use of correlation functions to extract the kink energy (E kink ) and strain relaxation energy constant (C) from the observed step meandering in topographical AFM images. Finally, we discuss the use of this procedure on two prototypical perovskite surfaces, i.e. SrTiO 3 (001) and DyScO 3 (110). The selected two systems have a lattice mismatch of 1.0% at room temperature, however, are known to result upon epitaxial growth of the first onto the latter in films with a high degree of uniformity and structural perfection in comparison to typical films 18 indicative of their ability to accomodate strain well. We conclude by summarizing the observed foundings and argue how the reported implications are anticipated to be of generic character.

II. EXPERIMENTAL DETAILS
To quantify the strain relaxation energy, we image the step meandering of vicinal perovskite surfaces on standard commercial available 5×5 mm 2 samples of SrTiO 3 (001) and DyScO 3 (110) using topographical AFM images. SrTiO 3 (001) and DyScO 3 (110) samples are often used substrates in thin film growth studies, as they are chemically stable and are known to have a low defect density. In addition, these materials have a well-defined surface of alternating AO and BO 2 planes, which is neutral in the case of SrTiO 3 (001) and polar for DyScO 3 (110). Furthermore, SrTiO 3 (001) has a pseudo cubic lattice constant of a 0 =3.905Å whereas DyScO 3 (110) has a nearly square surface mesh with an inplane lattice spacing of a 0 =3.95Å in the perovskite oxide spectrum 19 . The assumption of a cubic unit cell for both was used throughout our analysis.
Prior to annealing, all samples were ultrasonically rinsed in acetone and ethanol both for 10 minutes. In order to achieve single terminated BO 2 surfaces, we applied chemical etching procedures as described in detail elsewhere 20,21 . The samples were then annealed in a tube furnace using an O 2 flow of 150 l/h at a temperature of 1273 K. The AFM topographic images were taken in ex-situ tapping mode (TM). The fast scan direction was aligned perpendicular to the surface steps in order to optimize image analysis and to prevent tip artifacts in the images.
For the statistical analysis described below, it is of utmost importance that the surface is prepared at thermal equilibrium. By varying the annealing time and temperature followed by AFM imaging, we study the evolution of the step-edges to find the conditions for which the surface is at equilibrium. Fig. 1(a) shows the polished as-received surface, exhibiting disordered step edges, without a well-defined step distribution. Upon 30 minutes of annealing as described above, the steps become visible, but still reveal vacancy islands in the terraces as well as protrusions along the step edges, indicative of its non-equilibrium state, see Fig. 1 Prolonged annealing up to 4 hours results in meandering of the surface steps and a welldefined and narrow terrace width distribution, see Fig. 1(c) and (f). No vacancy islands are present in the terraces. Typically, the average terrace width (< L >) and its standard distribution (σ) are measured to study the strength of the step-step interactions. To verify the terrace width, its distribution P(l) versus the terrace width l is plotted in Fig. 1 The data can nicely be described by the fitted Gaussian, neglecting the weak shoulders and the slight asymmetry in the distribution. An average terrace width < L > of 760 a 0 (corresponding to 300 nm) is found for the DyScO 3 (110) surface in Fig. 1(c). The standard distribution (σ) of this distribution, only 86 a 0 (corresponding to 34 nm), is relatively small and indicative for step-step interactions. This very narrow distribution of DyScO 3 (110) compared to silicon can be explained by the coherent behavior of the DyScO 3 (110) stepedges 11 . The shoulder on the right of the average in the distribution is most probably due to step-step repulsion.
Continuing annealing up to 12 hours, see Fig. 1(d), reveals no significant change of the surface or step edge correlation function analysis as described below, a clear signature of a surface at thermal equilibrium. The same procedure was used to prepare and verify the SrTiO 3 (001) surface at thermal equilibrium, see Fig. 1(e). Note that the length scale at which thermal equilibrium is reached scales as t α , where t is the time and α an exponent that depends on the exact mass transport mechanism [22][23][24] . An exponent α = 1 2 describes mass exchange between the step edge and adatoms on the terrace, whereas α = 1 4 describes mass transport along step edges 25 . The large correlation length determined from the experimental obtained data (λ ≈ 2100 a 0 , see below) suggests that a non-equilibrium step instability is very unlikely 26,27 . In order to extract the kink energy E kink and strain relaxation energy from the meandering step edges in the topographical AFM images, step edge analysis was performed on the AFM images by first applying a Gaussian filter to reduce noise. Subsequently, leveling of the height data was done by plane fitting the topography.
Step edges were then detected by applying the Canny filter 28 , a multi-stage algorithm to detect a wide range of edges. Fig. 2(a) represents a processed AFM topographical image of a DyScO 3 (110) at thermal equilibrium after 4 hours of annealing. The lines in Fig. 2(a) represent the step edges, detected by the Canny filter.
The step meandering of perovskite oxide step edges can be represented by a triangular undulation of amplitude A and wavelength λ as schematically shown in Fig. 2 where h(r) is the deviation measured in the direction perpendicular to the step edge at position r, h(0) = < h(r) > = 0 is the deviation measured in the direction perpendicular to the step edge at position 0, r is the position along the high symmetry direction parallel to the step edge, and a 0 the unit cell length parallel to the step edge. Now < k 2 > is the mean-square kink length, see Fig. 2(b). From the mean square kink length, the kink energy can be calculated as: where k is the kink length, k B the Boltzmann constant and T the sample temperature. The kink energy E kink is related to the nearest neighbour energy E n as: By substitution, the mean square kink length < k 2 > and kink energy E kink relation can be simplified to: where: Eqs. 1-5 reveal that the kink energy E kink can be extracted by measuring the slope < k 2 > of the correlation function for small enough r. The linear dependence in this range of the correlation function implies a random kink formation distribution.
In order to determine the strain energy, we fit the correlation function of the perovskite oxide step waviness by a triangular step undulation model. Note that the triangular step undulation is indicative of a strong energy relaxation mechanism along the step. There is a competition between two energy terms in minimizing the surface free energy, i.e. the energy cost to create additional step length versus the energy reduction by strain relaxation due to the triangular undulations along the step edge.
The total free energy of a simple square-wave like step with periodicity λ can be described as 29 : where γ is the free energy per unit area (in perovskite oxides, the free energy per unit area for a step-up and step-down domain are equal), F step is the step free energy per unit length, υ is the Poisson ratio, µ the bulk modulus and ∆σ the force monopole at the step edge.
The total free energy per unit area of a triangular step edge can be calculated as with the strain relaxation energy constant C as, and, with A the amplitude as shown in Fig. 2(b). Now, Eq. 7 can be rewritten as: The integral of Eq. 10 can be solved by using trigonometric identities and integration by substitution, such that  (2), see Eq. 11, the total free energy per unit area F of a triangular step edge is equal to: The critical periodicity λ c follows from the minimum free energy per unit area by calculating dF dλ =0 as: Besides the ln(2) term in the exponent, Eq. 13 is identical to the relation derived by Alerhand et al. 30 .
Now, by determining the critical periodicity λ c from the experimentally obtained topographic AFM images, together with the obtained E kink (extracted from the slope < k 2 > of the correlation function for small enough r), one can calculate the strain relaxation energy constant C.

IV. ENERGETICS OF SRTIO 3 (001) AND DYSCO 3 (110) SURFACES
In order to demonstrate how the energetics of vicinal perovskite surfaces can be studied by measuring its surface topography, we determine the E kink and strain relaxation for two prototypical perovskites i.e. SrTiO 3 (001) and DyScO 3 (110). For this, we performed the image analysis as described on Fig. 1(e) and (c) respectively, identifying the step edges by use of the described Canny filter. From these, we calculated the corresponding (average) correlation functions as depicted in Fig. 3 which show surprisingly both qualitative and quantitave similarities for SrTiO 3 (001) and DyScO 3 (110). In Fig. 3(a) the strong correlation along the step edges. The ratio between minima and maxima amplitude A, see Fig. 1(b), slightly varies from step to step, however, the periodicity remains constant for both material systems.
In Fig. 3(b), the correlation functions are plotted for small r ranging from 0 -200 a 0 . At short length scales, see Fig. 3(b), the correlation function starts to deviate from the linear fit (G(r) ∼ r) around r > 60 a 0 to a quadratic relation (G(r) ∼ r 2 ).
From the determined correlation function, the critical periodicity λ c for both surfaces is found to be equal to 2100 a 0 (corresponding to ≈ 830 nm). Applying this to Eq. 13 results in a ratio of Fstep C = E kink C = 4.13. Now, the kink formation energy can be determined from Fig. 3(b), a < k 2 >=2.1±0.1 2 0 is extracted assuming random kink formation on this short length scale, which results in E kink = 0.10 eV/a 0 using Eqs. 4 and 5. Note, that from this the nearest neighbour energy E n can be determined as E kink = 0.21 ± 0.01 eV for SrTiO 3 (001) and DyScO 3 (110), both assumed as a (pseudo-)cubic crystal structure 31 . Furthermore, the strain relaxation energy constant C can then be determined as 24 meV/a 0 for both surfaces.
This value is close to the reported value range of E n = 0.25 -0.6 eV, commonly used in Kinetic Monte Carlo simulations to simulate thin film growth 32 .
As mentioned before, the correlation along the surface steps depends on the average terrace width < L >. When the average terrace width becomes too small, the correlation function becomes less pronounced resulting in overestimating E kink . A small terrace However, in the case that step-step interactions are present, the step-edge morphology, correlation function and extracted energetic values can be influenced when terrace widths become so small that step edges start interacting. In Fig. 4 we demonstrate this, by studying DyScO 3 (110) samples with increasing vicinal cut. For an average terrace width of only < L >=92 nm, the step edge morphology is influenced, resulting in a skewed correlation function which in its turn results to an overestimated nearest neighbour energy.
For larger terrace widths of 300 nm and 170 nm, see respectively Fig. 4(a) and (b), a similar correlation function behavior is found on small length scales (0 -100 a 0 ). At larger length scales (1000 -3000 a 0 ) the correlation has less pronounced minima and maxima then for the DyScO 3 (110) surface having smaller terrace widths (< L > = 170 nm). The mean square kink length decreases with larger step-step repulsion as the entropy of steps decreases for small step-step separations. Typically, the influence of the step-step interaction potential on a wandering step increases with 1/L 211 . The decrease of the mean square kink length < k 2 > = 2.1 ± 0.1 a 2 0 towards lower values (we described < k 2 > = 1.6 ± 0.1 a 2 0 for < L >=92 nm here) results in a change of the found nearest neighbour energy of E n = 0.21 ± 0.01 eV towards 0.23 ± 0.01 eV.
The proposed model explains the quasi 1D character of a perovskite oxide step edge for terrace widths larger than 170 nm. Smaller terrace widths show a more 2D character caused by the step-step interaction, which requires a more sophisticated model. Both, the entropic step-step interaction 33 and step interaction energy models 30 used to describe the interaction of Si step-edges are not valid for DyScO 3 (110) step-edges. The models assume a meandering Si step-edge catched between two straight step-edges, while the triangular undulation of step-edges on a DyScO 3 (110) surface behaves coherent.

V. CONCLUSIONS
In conclusion, we demonstrated a method to determine both, the strain relaxation energy together with the step edge formation energy on a perovskite surface by measuring its vicinal surface topography by use of AFM. Remarkably, we find similar triangular step edge undulations for two different perovskite surfaces at thermal equilibrium. From these we determine an average step edge undulation periodicity of ≈ 2100 a 0 for the two different From the similar step meandering along the step edges of the perovskite oxides studied and described here, we suggest that the surface energetics might depend on the local oxygen coordination. According to this, the surface energetics should change with different alternating planes / oxygen coordination, for example in the case of SrTiO 3 (001)/AO -BO 2 , SrTiO 3 (110)/ABO -O 2 and SrTiO 3 (111)/AO 3 -B planes, which is supported by DFT calculations and difference in required annealing times 34 . The importance of the oxygen coordination is also supported by the specific terminating plane at the surface, since it has been shown that nanostructures can be grown selective on mixed terminated substrates 15,35 .