REScO3 Substrates—Purveyors of Strain Engineering

The thermodynamic and crystallographic background for the development of substrate crystals that are suitable for the epitaxial deposition of biaxially strained functional perovskite layers is reviewed. In such strained layers the elastic energy delivers an additional contribution to the Gibbs free energy, which allows the tuning of physical properties and phase transition temperatures to desired values. For some oxide systems metastable phases can even be accessed. Rare‐earth scandates, REScO3, are well suited as substrate crystals because they combine mechanical and chemical stability in the epitaxy process with an adjustable range of pseudo‐cubic lattice parameters in the 3.95 to 4.02 Å range. To further tune the lattice parameters, chemical substitution for the RE or Sc is possible.


Introduction
In his seminal paper "On the equilibrium of heterogeneous substances," [1] J. W. Gibbs pointed out that in thermodynamic equilibrium either the entropy is maximum (for constant energy) or the energy is minimum (for constant entropy). These conditions are expressed nowadays as the condition that the Gibbs free energy must have a minimum. Usually this Gibbs energy is described by as a function of pressure p and temperature T. Here H is enthalpy, S is entropy, and U is internal energy. Gibbs recognized [1] ; however, other potential contributions to G, such as energies of DOI: 10.1002/crat.201900111 electric, magnetic, or elastic fields. The energetic contribution of a homogeneous elastic field is proportional to the biscalar product of elastic stress and elastic strain ɛ. Both are tensors of rank 2, and for strained bodies their product must be added as a contribution to G in Equation (1). [2] To obtain significant contributions to G, usually strains of order ɛ = 10 −3 to ≈3 × 10 −2 are required, which are not straightforward to be reached by external tension or compression, particularly in brittle materials. An elegant way to impart such strains is through strain engineering by the epitaxial growth of single crystalline layers on substrates that are "almost" (but not completely) lattice matched. For conventional semiconductors this method has become widely utilized to enhance mobility and make transistors with higher performance. [3] For ferroelectric materials (e.g., BaTiO 3 ) strain engineering can be employed to raise the phase transition of the unwanted paraelectric state by several hundred Kelvin. [4] Even more exciting is the prospect of using strain engineering to stabilize phases that are normally metastable, that is, to access the properties of phases that were previously "hidden." Examples include stabilizing a ferroelectric polymorph of SrTiO 3 (which in the absence of strain is not ferroelectric at any temperature) at room temperature, [16] stabilizing a simultaneously ferromagnetic and ferroelectric polymorph of EuTiO 3 (the strongest known multiferroic [5] ), and stabilizing an emergent polarization vortex phase exhibiting negative capacitance. [6] Substrate crystals for strain engineering should fulfil the following requirements: 1) The available substrate assortment should cover a wide range of lattice parameters to enable the growth of epitaxial thin films with a broad choice of strain states (sign and magnitude). 2) Substrates should be chemically and crystallographically compatible with the thin films to be grown upon them, during deposition and thereafter (this means that perovskites are prospective candidates, because many oxide thin films of interest are perovskites too). 3) Substrate crystals should not undergo disruptive phase transitions between their melting point and room temperature. 4) To pave the way for an integration of elements into siliconbased electronic devices, the corresponding substrate materials should be stable in contact with silicon.
Back in 1957 Geller reported room-temperature crystallographic data for a series of perovskite type compounds with orthorhombic www.advancedsciencenews.com www.crt-journal.org Figure 1. Left: One (cubic) unit cell of SrTiO 3 [12] contains 1 formula unit. Right: One (orthorhombic) unit cell of NdScO 3 [13] contains four formula units.
symmetry and chemical formula REScO 3 (RE = Gd, Nd, Pr, La, and Y). [7] Most of these compounds were shown to melt congruently at high temperature, T f > 2000°C. [8] Using a thermodynamic approach the oxides of a few chemical elements including scandium and most of the rare-earth (RE) elements were identified to fulfil condition 4. [9] Thus, it would appear that rare-earth scandates satisfy the important criteria for use as substrates for strain engineering of oxide perovskite thin films.
A significant share of the development of growth technologies for REScO 3 single crystals for substrate applications was done at the author's home institution, the Leibniz Institute for Crystal Growth in Berlin, Germany. This paper reviews these efforts and presents subsequent results on related substrate crystals that go beyond pure REScO 3 compounds to extend the range of lattice constants available for the strain engineering of perovskite thin films.

Pure REScO 3
For the rare-earth elements from RE = La to Tm, plus Y, compounds of the type REScO 3 are known to exist. Of these, high pressure synthesis is required to form phase-pure powders of ErScO 3 , TmScO 3 , and YScO 3 . [10] The rare-earth scandates melt congruently for RE = La…Dy, but HoScO 3 decomposes upon heating in the solid phase to a bixbyite type (Ho,Sc) 2 O 3 solid solution prior to melting. [8,11] The same is true for YScO 3 . [8,11] Because the other REScO 3 are in equilibrium with the melt, crystal growth by the Czochralski method seemed feasible-except for the very high melting LaScO 3 (T f = 2390°C [8] ) and CeScO 3 .
All REScO 3 crystallize in the GdFeO 3 structure type that is usually described as a distorted perovskite. Figure 1 compares the crystal structure of the ideal (cubic, space group Pm3m, #221) perovskite SrTiO 3 with that of NdScO 3 (orthorhombic, space group Pbnm, #62). In both structures the smaller cations Ti 4+ or Sc 3+ , respectively, are octahedrally coordinated to oxygen, but the octahedra are tilted and distorted in the scandates, resulting in lower symmetry. The lattice parameter in the c-direction is doubled, whereas a and b are scaled by a factor ≈ √ 2. Consequently, the REScO 3 unit cell contains four formula units.
In the setting of Figure 1 (right) the lattice parameters a and b are similar, and correspond to √ 2 times the "pseudocubic" lattice parameter a pc of the structure; c ≈2 a pc . [13] Even if a single parameter a pc is not able to describe the complex strain relations in an epitaxial layer, it is suitable for a first approximation. Other descriptions are the pseudocubic lattice parameters that span the (110) substrate surface, i.e., 1 2 √ a 2 + b 2 and 1 2 c [14] or a pc = 3 √ 1 4 abc. [15] In an alternative setting the space group #62 is Pnma, and one has a ≈ c, with the long axis being the b-axis.
The structural similarity of the REScO 3 makes them ideal substrates for the epitaxy of perovskite type oxides such as SrTiO 3 and BaTiO 3 . Haeni et al. [16] could deposit unrelaxed SrTiO 3 (100) layers on the (110) surface of DyScO 3 that are tensile strained by almost 1% for films as thick as 35 nm. [17] As the lattice mismatch between the thin film and the underlying substrate increases, the thickness of the unrelaxed film that can be formed decreases. For high strains (beyond roughly 3 × 10 −3 , depending upon the specific material and growth conditions), film relaxation occurs immediately through the introduction of defects, for example, dislocations or cracks. [21] Depending on the elastic stiffness of the layer, the elastic energy contribution to the Gibbs energy amounts to several 100 kJ mol −1 , which can shift, for example, ferroelectric transition temperatures by hundreds of Kelvins. In recent years REScO 3 substrates have become commercially available, for example, from CrysTec, Berlin, Germany. [18] The accessible pure REScO 3 compounds offer a pc ranging from 3.95 (DyScO 3 , [19] ) to 4.02 Å (PrScO 3 , [20] ). This range matches well with values that are desired for strain engineering of important ferroelectric and multiferroic materials like (Ba,Sr)TiO 3 or BiFeO 3 . [21] Nevertheless, two problems remain unsolved: 1) The accessible REScO 3 offer stepwise a pc "jumps" from REScO 3 to RE'ScO 3 as the RE constituent is changed. Additionally, PmScO 3 crystals were never grown because promethium is radioactive, which makes this material practically useless. Hence, a more gradual fine-tuning of a pc is desirable. 2) The "largest" REScO 3 with RE = La, Ce have not yet been grown with substrate quality, because their melting points are extremely high, that is, too high for iridium crucibles, thus precluding crystal growth by the Czochralski method. Isostructural LaLuO 3 with a pc = 4.17 Å, however, melts at 2120°C congruently and good crystals can be grown. [22]

Mixed Crystals (RE,RE′)ScO 3
Both of the issues mentioned above can possibly be solved by the growth of solid solutions between appropriate components. The first encouraging results were obtained with (Nd,Sm)ScO 3 (as a substitute for PmScO 3 ) and (Sm,Gd)ScO 3 (as a substitute for EuScO 3 ). [23] For both systems the rare-earth components are the next nearest neighbors in the RE series on either side of the RE of interest, with consequently very similar structural and thermodynamic properties.
As expected, both systems form solid solutions over the whole concentration range. Differential thermal analysis (DTA) studies www.advancedsciencenews.com www.crt-journal.org  Figure 4 of ref. [23]. Bottom: Hypothetical phase diagram that is obtained if T f (left) is lowered by 1000 K, and T f (right) is lowered by 1100 K. Table 1. Thermodynamic data for some perovskites. Accurate data for the upper 4 compounds are from FactSage databases. [25] The data for the REScO 3 compounds are estimated values from our own DTA measurements. [23,26] Formula Pbnm [14] SmScO 3 105 43 2440 Pbnm [14] GdScO 3 85 35 2423 Pbnm [14] DyScO 3 89 38 2370 Pbnm [14] of mixtures between pure REScO 3 reveal almost ideal behavior of the mixed crystal systems. Moreover, the two-phase liquid+solid field of both pseudobinary phase diagrams is extremely narrow, of the order of 2 K only (Figure 2 top). This is beneficial for crystal growth, because segregation during crystal growth is then marginal.
The almost ideal behavior of the REScO 3 -RE′ScO 3 mixed crystal systems is certainly mainly due to the high chemical similarity of the components, and due to the very high liquidus temperatures where excess energies become often smaller. The narrow liquid + solid phase fields that were found not only in the two systems mentioned above, but also for (Tb,Gd)ScO 3 and (Tb,Dy)ScO 3 [26] are very helpful for the growth of bulk crystals with low segregation. Nonetheless, the narrowness initially seemed a bit surprising, because in other systems with very similar components (e.g., alkali halides [24] ) the span of the two-phase field is often wider.
Reasons for this different behavior are found in the numerical data of Table 1 and in the Schröder-van-Laar equations that describe for ideal systems the solidus and liquidus curves as functions of the heats of fusion ΔH f and melting points T f of both components. Equation (4) shows that ΔH f is scaled by reciprocal temperature differences. Because ΔH f = ΔS f ·T f , the entropy change during melting is more relevant than the enthalpy change. Table 1 reports data for some perovskites with melting points between 1353 and 2491 K (1080 and 2218°C, respectively). The first four data rows are compiled from a reliable reference; for the last 4 substances the accuracy of the melting points are approx. ± 20 K from our own DTA measurements. The ΔH f values for the REScO 3 were estimated by comparing the melting peak areas with that of melting Al 2 O 3 (T f = 2327 K, ΔH f = 118.41 kJ mol −1 ). It turns out that the ΔS f are of the order of 33-43 J (mol·K) −1 for perovskites that have comparatively low symmetry at T f , but are significantly higher for the undistorted perovskites with cubic symmetry. Additionally it is known at least for the REScO 3 , that they show a significant degree of structural disorder, especially vacancies on the RE position. [14,19] The entropy gain during melting from the crystalline structure to the highly disordered melt is obviously less significant, if the crystal structure itself shows already some degree of disorder.
Another numerical effect influencing the width of the twophase field results from the difference of inverse temperatures in Equation (4): If at very high T both melting points are similar, this difference will always be small; but this difference scales ΔH f . From that stems a drastic effect which is demonstrated by the lower panel of Figure 2, where a hypothetical phase diagram was calculated with Equations (2)-(4) for identical ΔH f , but with T f that are lowered by 1000 K on the left side and by 1100 K on the right side; here the width of the two-phase field approaches a significant level of 30 K.

Related Perovskites
The (RE,RE′)ScO 3 solid solutions that were described in the previous section allow fine-tuning of the lattice constants only in the range of the accessible rare-earth scandates, which end for large lattice constants with PrScO 3 . Lanthanum lutetate LaLuO 3 is another isostructural orthorhombic perovskite with a very large lattice constant a pc = 4.17 Å and accessible T f = 2120°C. [22] It seemed interesting to check if LaLuO 3 forms solid solutions with LaScO 3 , and if substrate crystals can be grown.
The extremely high melting point of LaScO 3 (T f = 2390°C) is beyond the thermal limits that were accessible for DTA measurements, but for samples between pure LaLuO 3 and 59 mol% LaScO 3 melting peaks could be recorded. [29] The measurements proved that both end members form solid solutions in any arbitrary ratio, but in contrast to the (RE,RE')ScO 3 systems from the previous section, the two-phase field between solidus and liquidus is fairly wide (up to ≈50 K). This is an impediment to crystal growth because strong segregation must be expected. Fortunately, resulting from a small positive excess enthalpy of the solid www.advancedsciencenews.com www.crt-journal.org phase, an azeotrope exists at ≈8 mol% LaScO 3 , where the solidus and liquidus meet in one point. Consequently segregation vanishes at the azeotrope point and remains small in its vicinity. From a starting melt with 27.6% LaScO 3 a 75 mm long single crystal could be grown where the LaScO 3 content changed by segregation from 35.4% (top) to 31.1% (bottom). By X-ray diffraction studies of another crystal with 43% LaScO 3 a pseudocubic lattice constant a pc = 4.1224 Å was measured-nearly halfway between those of the end members. [30] The La(Lu,Sc)O 3 crystals mentioned above can only be grown in the region of large a pc , close to LaLuO 3 , and the gap towards the REScO 3 is wide. It seemed worthwhile to check if partial substitution of Sc 3+ by Lu 3+ could result in crystals with lattice parameters that are somewhat higher than for "pure" rare-earth scandates. Unfortunately, pure lutetates RELuO 3 are not stable for many RE. In initial DTA studies it was revealed that, in contrast to some reports in the literature, the perovskite NdLuO 3 is only a metastable phase. [31] Instead, a mixture of (Nd,Lu) 2 O 3 solid solutions of the cubic "C-phase" (bixbyite structure) and the monoclinic "B-phase" is found. Nevertheless, further studies of the ternary system Nd 2 O 3 -Lu 2 O 3 -Sc 2 O 3 revealed that a wide composition field around NdScO 3 exists, where the first phase to crystallize is the perovskite phase. This potential growth window can be used to access pseudo-cubic lattice parameters up to at least 4.031 Å. [32]

Conclusions
REScO 3 compounds with perovskite structures are useful substrates for the deposition of strained functional perovskite layers. "Strain engineering" of the film grown on such substrates introduces a third component to the Gibbs free energy G of the epitaxial layer: in addition to T·S and p·V, the biscalar product of stress and elastic strain is present which enables unpreceded physical effects, see Equation (1).
Rare-earth scandates are extraordinary well suited for such substrate applications for several reasons: 1) They show high structural similarity and chemical stability towards potential epitaxial layers. 2) Crystals of good quality and with technically relevant dimensions can be grown directly from the melt, because no structural phase transitions occur between the melting point and room temperature.
3) The lattice constants of the substrates and hence the biaxial elastic strain of the thin film can be tuned by an appropriate choice of the RE element.
This paper shows that the development of a growth technology for REScO 3 single crystal substrates for the effective strain engineering of perovskite thin films is far from finished. Instead, the range of accessible lattice parameters can be extended by alloying them on the RE site as well as on the Sc site, which allows the development of substrate crystals with lattice constants well beyond the limits of the "pure" REScO 3 series. Further progress is expected by increasing the number of alloying components.