The strain game

For at least 400 years, humans have studied the effects of pressure (hydrostatic strain) on the properties of materials.^{Reference Lindh1} In the 1950s, it was shown that biaxial strain, where a film is clamped to a substrate but free in the out-of-plane direction, can alter the transition temperatures of superconductors^{Reference Lock2} (T _c) and ferroelectrics (T _C).^{Reference Forsbergh3}

What has changed in recent years is the magnitude of the biaxial strain that can be imparted. Bulk ferroic oxides are brittle and will crack under moderate strains, typically 0.1%. One way around this limitation is the approach of bulk crystal chemists, to apply “chemical pressure” through isovalent cation substitution. A disadvantage of such a bulk approach, however, is the introduction of disorder and potentially unwanted local distortions. Epitaxial strain, the trick of the thin-film alchemist, provides a potentially disorder-free route to large biaxial strain and has been used to greatly enhance the mobility of transistors^{Reference Nguyen, Brown, Thompson and Jelloian4,Reference Welser, Hoyt and Gibbons5} (see the article by Bedell et al. in this issue), increase catalytic activity (see the article by Yildiz in this issue), alter band structure^{Reference Shockley and Bardeen6} (see the article by Yu et al. in this issue), and significantly increase superconducting,^{Reference Sato and Naito7,Reference Bozovic, Logvenov, Belca, Narimbetov and Sveklo8} ferromagnetic,^{Reference Beach, Borchers, Matheny, Erwin, Salamon, Everitt, Pettit, Rhyne and Flynn9–Reference Fuchs, Arac, Pinta, Schuppler, Schneider and Löhneysen11} and ferroelectric^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12–Reference Lee, Fang, Vlahos, Ke, Jung, Kourkoutis, Kim, Ryan, Heeg, Roeckerath, Goian, Bernhagen, Uecker, Hammel, Rabe, Kamba, Schubert, Freeland, Muller, Fennie, Schiffer, Gopalan, Johnston-Halperin and Schlom16} transition temperatures. This approach, which we refer to as the “strain game,” is illustrated in Figure 1 for elastically strained films of oxides with the perovskite structure.

Figure 1.

How the “strain game” is used to impart biaxial elastic strain on an oxide film via epitaxy. (a) The crystal structure of an unstrained oxide, in this case an oxide with the perovskite structure, in its unstrained state. When deposited on a substrate with a larger or smaller spacing, the energetic preference of the deposited atoms to follow the template of the underlying substrate (epitaxy) can result in the film beginning in either (b) biaxial compression or (c) biaxial tension. (d) A number line showing the pseudotetragonal or pseudocubic a-axis lattice constants in angstroms of some perovskites and perovskite-related phases of interest, including ferroics (above the number line) and of some of the perovskite substrates that are available commercially (below the number line). LSAT, (LaAlO₃)_0.29—(SrAl_1/2Ta_1/2O₃)_0.71; NSAT, (NdAlO₃)_0.39—(SrAl_1/2Ta_1/2O₃)_0.61; SAGT, Sr_1.04Al_0.12Ga_0.35Ta_0.50O₃. Photographs of exemplary single crystals used as substrates, each with diameter, d. (d) An updated version of the plot in Reference 24.

Strains of about ±3% are common in epitaxial oxide films today,^{Reference Adamo, Ke, Wang, Xin, Heeg, Hawley, Zander, Schubert, Schiffer, Muller, Maritato and Schlom17–Reference Chen, Luo, Huang, Qi, Yang, You, Hu, Wu, Wang, Gao, Sritharan and Chen20} with the record to date being a whopping 6.6% compressive strain achieved in thin BiFeO₃ films grown on (110) YAlO₃.^{Reference Zeches, Rossell, Zhang, Hatt, He, Yang, Kumar, Wang, Melville, Adamo, Sheng, Chu, Ihlefeld, Erni, Ederer, Gopalan, Chen, Schlom, Spaldin, Martin and Ramesh21–Reference Martin and Schlom24} These strains are an order of magnitude higher than where these materials would crack in bulk.^{Reference Griffith25–Reference Freund and Suresh27}

Strained SrTiO₃ and the importance of suitable substrates

The strain game for ferroics was ignited by the demonstration that an oxide that normally is not ferroelectric at any temperature can be made ferroelectric at room temperature through the application of biaxial strain.^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12} Such a gigantic shift in properties and T _C had never before been clearly seen in any ferroic system; nonetheless, this achievement was the experimental realization of what had been predicted years earlier by theory.^{Reference Pertsev, Tagantsev and Setter28,Reference Pertsev, Tagantsev and Setter29} Figure 2 shows the strain phase diagram of (001)_p SrTiO₃ calculated by thermodynamic analysis,^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12,Reference Schlom, Chen, Eom, Rabe, Streiffer and Triscone14,Reference Warusawithana, Cen, Sleasman, Woicik, Li, Kourkoutis, Klug, Li, Ryan, Wang, Bedzyk, Muller, Chen, Levy and Schlom15,Reference Pertsev, Tagantsev and Setter28–Reference Devonshire30} where the p subscript indicates pseudocubic Miller indices. These predictions imply that a biaxial tensile strain on the order of 1% will shift the T _C of SrTiO₃ to the vicinity of room temperature.^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12,Reference Pertsev, Tagantsev and Setter28–Reference Li, Choudhury, Haeni, Biegalski, Vasudevarao, Sharan, Ma, Levy, Gopalan, Trolier-McKinstry, Schlom, Jia and Chen32} Although many researchers had grown SrTiO₃ films on substrates with different spacings, the lattice mismatches were so large and the films so thick that the films were no longer elastically strained.

Figure 2.

(a) The strain phase diagram of (001)_p-oriented SrTiO₃ calculated assuming a single-domain state for all structural and ferroelectric phases. The letters T and O used in the phase notations indicate tetragonal and orthorhombic crystallographic symmetries, respectively, under a constraint. The paraelectric, ferroelectric, and ferroelastic (or antiferrodistortive structural) natures of the phases are revealed by the superscripts P, F, and S, respectively. The components of the polarization vector P corresponding to the phases (along the crystallographic directions of pseudocubic SrTiO₃) are indicated within the parentheses following the phase notation. The dashed line corresponds to (001)_p SrTiO₃ commensurately strained to (001) Si. At room temperature, it is strained in biaxial compression by ∼1.7%. This strain decreases slightly with temperature due to the larger thermal expansion coefficient of SrTiO₃ than silicon. (b) Plan-view and (c) cross-sectional annular dark field (ADF) scanning transmission electron microscopy (STEM) images of a nominally 2 unit-cell-thick (0.8 nm) epitaxial (001)_p SrTiO₃ film grown on (001) Si and capped with amorphous silicon. (d) Cross-sectional ADF-STEM images of a nominally 5 unit-cell-thick (2.0 nm) commensurate epitaxial (001)_p SrTiO₃ film grown on (001) Si. The image shows an average film thickness of 6.5 unit cells, suggesting nonuniform SrTiO₃ coverage. In the area marked with an arrow, more than one SrTiO₃ grain is imaged in projection. (a) Adapted from References 14 and 15 and has only a minor difference from that presented in References 28 and 29. (b) and (c) Reprinted with permission from Reference 78. © 2008 American Physical Society. (d) Reprinted with permission from Reference 15. © 2009 AAAS.

Fully commensurate, elastically strained epitaxial films have the advantage that high densities of threading dislocations (e.g., the ∼10¹¹ dislocations cm^–2 observed, for example, in partially relaxed (Ba_xSr_1–x)TiO₃ films)^{Reference Canedy, Li, Alpay, Salamanca-Riba, Roytburd and Ramesh33,Reference Misirlioglu, Vasiliev, Aindow, Alpay and Ramesh34} are avoided. Strain fields around dislocations locally alter the properties of a film, making its ferroelectric properties inhomogeneous and often degraded.^{Reference Chu, Szafraniak, Scholz, Harnagea, Hesse, Alexe and Gösele35–Reference Nagarajan, Jia, Kohlstedt, Waser, Misirlioglu, Alpay and Ramesh37} To achieve highly strained ferroic films and keep them free of such threading dislocations, one needs to keep them thin, typically not more than a factor of five beyond the Matthews-Blakeslee critical thickness, beyond which it becomes energetically favorable (though typically constrained by kinetics) for a film to relax by the introduction of dislocations.^{Reference Freund and Suresh27,Reference Biegalski, Fong, Eastman, Fuoss, Streiffer, Heeg, Schubert, Tian, Nelson, Pan, Hawley, Bernhagen, Reiche, Uecker, Trolier-McKinstry and Schlom38} Thickness-dependent studies, involving the growth of a ferroic on just one substrate material to study the effect of strain in partially relaxed films, are not as easy to interpret as experiments utilizing commensurate films grown on several different substrate materials covering a range of lattice spacings. In the former, the strains are inhomogeneous, and the high concentration of threading dislocations can obfuscate intrinsic strain effects.

Exploring the strain predictions in Figure 2a was greatly simplified by the development of new substrates with a broad range of spacings to impart a desired strain state into the overlying SrTiO₃ film. These substrates have the same structure as SrTiO₃—the perovskite structure—but different lattice spacings. The number of perovskite single crystals that are available commercially as large substrates (with surfaces at least 10 mm × 10 mm in size) has nearly doubled in the last decade due to the work of the present authors.^{Reference Uecker, Wilke, Schlom, Velickov, Reiche, Polity, Bernhagen and Rossberg39–Reference Uecker, Klimm, Bertram, Bernhagen, Schulze-Jonack, Brützam, Kwasniewski, Gesing and Schlom41} Today, various single crystal perovskite and perovskite-related substrates are commercially available (see Figure 1d), including LuAlO₃,^{Reference Lempicki, Randles, Wisniewski, Balcerzyk, Brecher and Wojtowicz42,Reference Petrosyan, Shirinyan, Pedrini, Durjardin, Ovanesyan, Manucharyan, Butaeva and Derzyan43} YAlO₃,^{Reference Asano, Kubo, Michikami, Satoh and Konaka44} LaSrAlO₄,^{Reference Brown, Pendrick, Kalokitis and Chai45} NdAlO₃,^{Reference Miyazawa, Toshima and Morita46} LaAlO₃,^{Reference Berkstresser, Valentino and Brandle47,Reference Berkstresser, Valentino and Brandle48} LaSrGaO₄,^{Reference Hontsu, Ishii, Kawai and Kawai49} (NdAlO₃)_0.39—(SrAl_1/2Ta_1/2O₃)_0.61 (NSAT),^{Reference Mateika, Kohler, Laudan and Volkel50} NdGaO₃,^{Reference Simon, Platt, Lee, Lee, Daly, Wire, Luine and Urbanik51,Reference Berkstresser, Valentino and Brandle52} (LaAlO₃)_0.29—(SrAl_1/2Ta_1/2O₃)_0.71 (LSAT),^{Reference Mateika, Kohler, Laudan and Volkel50,Reference Chakoumakos, Schlom, Urbanik and Luine53} LaGaO₃,^{Reference Sandstrom, Giess, Gallagher, Segmüller, Cooper, Chisholm, Gupta, Shinde and Laibowitz54} SrTiO₃,^{Reference Merker55–Reference Scheel, Bednorz and Dill58} Sr_1.04Al_0.12Ga_0.35Ta_0.50O₃ (SAGT), DyScO₃,^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12,Reference Uecker, Wilke, Schlom, Velickov, Reiche, Polity, Bernhagen and Rossberg39} TbScO₃,^{Reference Uecker, Velickov, Klimm, Bertram, Bernhagen, Rabe, Albrecht, Fornari and Schlom40} GdScO₃,^{Reference Choi, Biegalski, Li, Sharan, Schubert, Uecker, Reiche, Chen, Pan, Gopalan, Chen, Schlom and Eom13,Reference Uecker, Wilke, Schlom, Velickov, Reiche, Polity, Bernhagen and Rossberg39,Reference Lim, Kriventsov, Jackson, Haeni, Schlom, Balbashov, Uecker, Reiche, Freeouf and Lucovsky59} EuScO₃, SmScO₃,^{Reference Uecker, Wilke, Schlom, Velickov, Reiche, Polity, Bernhagen and Rossberg39,Reference Soukiassian, Tian, Vaithyanathan, Haeni, Chen, Xi, Schlom, Tenne, Sun, Pan, Choi, Eom, Li, Jia, Constantin, Feenstra, Bernhagen, Reiche and Uecker60} KTaO₃,^{Reference Feenstra, Boatner, Budai, Christen, Galloway and Poker61} NdScO₃,^{Reference Uecker, Wilke, Schlom, Velickov, Reiche, Polity, Bernhagen and Rossberg39,Reference Yang, He, Suresha, Kuo, Peng, Haislmaier, Motyka, Sheng, Adamo, Lin, Hu, Chang, Tjeng, Arenholz, Podraza, Bernhagen, Uecker, Schlom, Gopalan, Chen, Chen, Ramesh and Chu62} PrScO₃,^{Reference Coh, Heeg, Haeni, Biegalski, Lettieri, Edge, O’Brien, Bernhagen, Reiche, Uecker, Trolier-McKinstry, Schlom and Vanderbilt63} and LaLuO₃;^{Reference Ovanesyan, Petrosyan, Shirinyan, Pedrini and Zhang64} many of these are produced with structural perfection rivaling that of conventional semiconductors. The perfection of the substrate, the best of which are grown by the Czochralski method (which is not applicable to most ferroic oxides because they do not melt congruently), can be passed on to the film via epitaxy. This has led to the growth of strained epitaxial films of the ferroics SrTiO₃,^{Reference Biegalski, Fong, Eastman, Fuoss, Streiffer, Heeg, Schubert, Tian, Nelson, Pan, Hawley, Bernhagen, Reiche, Uecker, Trolier-McKinstry and Schlom38,Reference Biegalski, Jia, Schlom, Trolier-McKinstry, Streiffer, Sherman, Uecker and Reiche65} BaTiO₃,^{Reference Schlom, Chen, Eom, Rabe, Streiffer and Triscone14} BiFeO₃,^{Reference Ihlefeld, Tian, Liu, Doolittle, Bernhagen, Reiche, Uecker, Ramesh and Schlom66} BiMnO₃,^{Reference Lee, Ke, Misra, Ihlefeld, Xu, Mei, Heeg, Roeckerath, Schubert, Liu, Musfeldt, Schiffer and Schlom67} and EuTiO₃,^{Reference Lee, Fang, Vlahos, Ke, Jung, Kourkoutis, Kim, Ryan, Heeg, Roeckerath, Goian, Bernhagen, Uecker, Hammel, Rabe, Kamba, Schubert, Freeland, Muller, Fennie, Schiffer, Gopalan, Johnston-Halperin and Schlom16} with rocking curve full width at half maximum values ≤11 arcsec (0.003°)—identical to those of the commercial substrates upon which they are grown and significantly narrower (indicative of higher structural perfection) than the most perfect bulk single crystals of these same materials.

Using these new perovskite substrates, predictions of the SrTiO₃ strain phase diagram shown in Figure 2 were assessed. Not only was it found possible to transmute SrTiO₃ into a room temperature ferroic,^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12,Reference Irvin, Levy, Haeni and Schlom68,Reference Ma, Levy, Biegalski, Trolier-McKinstry and Schlom69} but the experimentally determined point group,^{Reference Antons, Neaton, Rabe and Vanderbilt31,Reference Li, Choudhury, Haeni, Biegalski, Vasudevarao, Sharan, Ma, Levy, Gopalan, Trolier-McKinstry, Schlom, Jia and Chen32,Reference Vasudevarao, Kumar, Tian, Haeni, Li, Eklund, Jia, Uecker, Reiche, Rabe, Chen, Schlom and Gopalan70–Reference Denev, Kumar, Biegalski, Jang, Folkman, Vasudevarao, Han, Reaney, Trolier-McKinstry, Eom, Schlom and Gopalan72} direction and magnitude of spontaneous polarization (P _S),^{Reference Antons, Neaton, Rabe and Vanderbilt31,Reference Li, Choudhury, Haeni, Biegalski, Vasudevarao, Sharan, Ma, Levy, Gopalan, Trolier-McKinstry, Schlom, Jia and Chen32,Reference Biegalski, Jia, Schlom, Trolier-McKinstry, Streiffer, Sherman, Uecker and Reiche65,Reference Vasudevarao, Kumar, Tian, Haeni, Li, Eklund, Jia, Uecker, Reiche, Rabe, Chen, Schlom and Gopalan70–Reference Biegalski, Vlahos, Sheng, Li, Bernhagen, Reiche, Uecker, Streiffer, Chen, Gopalan, Schlom and Trolier-McKinstry73} observed shifts in T _C^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12,Reference Vasudevarao, Kumar, Tian, Haeni, Li, Eklund, Jia, Uecker, Reiche, Rabe, Chen, Schlom and Gopalan70,Reference Vasudevarao, Denev, Biegalski, Li, Chen, Trolier-McKinstry, Schlom and Gopalan71} and soft mode frequency with biaxial strain,^{Reference Nuzhnyy, Petzelt, Kamba, Kužel, Kadlec, Bovtun, Kempa, Schubert, Brooks and Schlom74} and existence of a transition to a simultaneously ferroelectric and ferroelastic phase at lower temperatures^{Reference Vasudevarao, Kumar, Tian, Haeni, Li, Eklund, Jia, Uecker, Reiche, Rabe, Chen, Schlom and Gopalan70–Reference Denev, Kumar, Biegalski, Jang, Folkman, Vasudevarao, Han, Reaney, Trolier-McKinstry, Eom, Schlom and Gopalan72,Reference Nuzhnyy, Petzelt, Kamba, Kužel, Kadlec, Bovtun, Kempa, Schubert, Brooks and Schlom74} were all in accord with theory. Not all of the experimental observations, however, were in agreement with theory. For example, it was observed that strained SrTiO₃ films exhibit a significant frequency dependence to their dielectric response.^{Reference Biegalski, Jia, Schlom, Trolier-McKinstry, Streiffer, Sherman, Uecker and Reiche65,Reference Biegalski, Vlahos, Sheng, Li, Bernhagen, Reiche, Uecker, Streiffer, Chen, Gopalan, Schlom and Trolier-McKinstry73} This relaxor ferroelectric behavior is due to defects. On account of the strain, the SrTiO₃ matrix is highly polarizable and can be easily polarized by defect dipoles that arise from non-stoichiometry in the SrTiO₃ film. Based on how the properties of strained SrTiO₃ films vary with non-stoichiometry, strained, perfectly stoichiometric SrTiO₃ films are not expected to show relaxor behavior.^{Reference Lee, Skoromets, Biegalski, Lei, Haislmaier, Bernhagen, Uecker, Xi, Gopalan, Martí, Kamba, Kužel and Schlom75}

To make it possible for strain-enabled or strain-enhanced functionalities to be exploited in mainstream device architectures, it is desirable to play the strain game on substrates relevant to the semiconductor industry. One such example is the integration of commensurately strained SrTiO₃ films with silicon.^{Reference Warusawithana, Cen, Sleasman, Woicik, Li, Kourkoutis, Klug, Li, Ryan, Wang, Bedzyk, Muller, Chen, Levy and Schlom15} The lattice mismatch between (001)_p SrTiO₃ and (001) Si is 1.7%, as indicated by the dashed line on the left side of the strain phase diagram in Figure 2a. From theory (Figure 2a), such a film would be expected to be ferroelectric with a T _C near room temperature. This integration is, however, rather challenging due to (1) the high reactivity of silicon with many elements and their oxides,^{Reference Hubbard and Schlom76,Reference Schlom, Guha and Datta77} and (2) the thermodynamic driving force for a pristine silicon surface to oxidize and form an amorphous native oxide (SiO₂), which blocks epitaxy, under the oxidizing conditions typically used for the growth of oxide thin films. Despite these impediments, thin commensurate SrTiO₃ films have been grown directly on silicon without discernible intermediate layers and free of reaction, and are found to be ferroelectric at room temperature.^{Reference Warusawithana, Cen, Sleasman, Woicik, Li, Kourkoutis, Klug, Li, Ryan, Wang, Bedzyk, Muller, Chen, Levy and Schlom15}

Planar-view and cross-sectional annular dark field (ADF) scanning transmission electron micrographs (STEM) of epitaxial (001)_p SrTiO₃ films grown on and commensurately strained to (001) Si are shown in Figure 2b–d. Although the interface between SrTiO₃ and silicon is seen to be abrupt and free of reaction, these images reveal additional challenges to the growth of SrTiO₃ on silicon. First is the propensity of SrTiO₃ to nucleate as islands and not wet the surface of the silicon substrate.^{Reference Kourkoutis, Hellberg, Vaithyanathan, Li, Parker, Andersen, Schlom and Muller78} Even using kinetically limited growth conditions,^{Reference Warusawithana, Cen, Sleasman, Woicik, Li, Kourkoutis, Klug, Li, Ryan, Wang, Bedzyk, Muller, Chen, Levy and Schlom15,Reference Li, Hu, Wei, Yu, Zhang, Droopad, Demkov, Edwards, Moore, Ooms, Kulik and Fejes79} it takes multiple unit cells of growth before the SrTiO₃ islands coalesce.^{Reference Kourkoutis, Hellberg, Vaithyanathan, Li, Parker, Andersen, Schlom and Muller78} Second are frequent out-of-phase boundaries in the SrTiO₃ film (see Figure 2d) resulting from the step height of (001) Si (0.14 nm) not matching the step height of (001)_p SrTiO₃ (0.39 nm). Out-of-phase boundaries can form during the coalescence of SrTiO₃ islands that have nucleated on different (001) Si terraces. The arrow in Figure 2d marks an area where the SrTiO₃ islands are out-of-phase with each other.

For the growth of high-quality films of ferroic oxides with a desired strain state, not only are appropriate substrates of exceptionally high perfection needed, but also methods to prepare them with smooth, well-ordered surfaces with a specific chemical termination on which epitaxial growth can be reliably initiated. For example, chemical-mechanically polished (001) SrTiO₃ substrates display a mixture of SrO- and TiO₂-terminated surfaces. A combination of chemical etching and annealing has been shown to yield SrTiO₃ substrates with known surface termination. Kawasaki et a1.^{Reference Kawasaki, Takahashi, Maeda, Tsuchiya, Shinohara, Ishiyama, Yonezawa, Yoshimoto and Koinuma80} showed that an NH₄F-buffered HF solution with controlled pH enables etching of the more basic SrO layer and leaves a completely TiO₂ terminated surface on the substrate. This method of preparing a TiO₂-terminated (001) SrTiO₃ surface has been further perfected by Koster et a1.^{Reference Koster, Kropman, Rijnders, Blank and Rogalla81} SrO-terminated (001) SrTiO₃ substrates can also be prepared.^{Reference Schrott, Misewich, Copel, Abraham and Zhang82} A means to prepare low defect surfaces with controlled termination has since been developed for (110) SrTiO₃,^{Reference Biswas, Rossen, Yang, Siemons, Jung, Yang, Ramesh and Jeong83} (111) SrTiO₃,^{Reference Biswas, Rossen, Yang, Siemons, Jung, Yang, Ramesh and Jeong83,Reference Chang, Park and Kim84} (001)_p LaAlO₃,^{Reference Blok, Wan, Koster, Blank and Rijnders85,Reference Ohnishi, Takahashi, Nakamura, Kawasaki, Yoshimoto and Koinuma86} (111)_p LaAlO₃,^{Reference Blok, Wan, Koster, Blank and Rijnders85} (110) NdGaO₃,^{Reference Ohnishi, Takahashi, Nakamura, Kawasaki, Yoshimoto and Koinuma86} (001)_p LSAT,^{Reference Ohnishi, Takahashi, Nakamura, Kawasaki, Yoshimoto and Koinuma86,Reference Ngai, Schwendemann, Walker, Segal, Walker, Altman and Ahn87} (110) DyScO₃,^{Reference Kleibeuker, Koster, Siemons, Dubbink, Kuiper, Blok, Yang, Ravichandran, Ramesh, ten Elshof, Blank and Rijnders88,Reference Kleibeuker, Kuiper, Harkema, Blank, Koster, Rijnders, Tinnemans, Vlieg, Rossen, Siemons, Portale, Ravichandran, Szepieniec and Ramesh89} (110) TbScO₃,^{Reference Kleibeuker, Koster, Siemons, Dubbink, Kuiper, Blok, Yang, Ravichandran, Ramesh, ten Elshof, Blank and Rijnders88} (110) GdScO₃,^{Reference Kleibeuker, Koster, Siemons, Dubbink, Kuiper, Blok, Yang, Ravichandran, Ramesh, ten Elshof, Blank and Rijnders88} (110) EuScO₃,^{Reference Kleibeuker, Koster, Siemons, Dubbink, Kuiper, Blok, Yang, Ravichandran, Ramesh, ten Elshof, Blank and Rijnders88} (110) SmScO₃,^{Reference Kleibeuker, Koster, Siemons, Dubbink, Kuiper, Blok, Yang, Ravichandran, Ramesh, ten Elshof, Blank and Rijnders88} (001) KTaO₃,^{Reference Bae, Sigman, Norton and Boatner90} (110) NdScO₃,^{Reference Kleibeuker, Koster, Siemons, Dubbink, Kuiper, Blok, Yang, Ravichandran, Ramesh, ten Elshof, Blank and Rijnders88} and (110) PrScO₃^{Reference Kleibeuker, Koster, Siemons, Dubbink, Kuiper, Blok, Yang, Ravichandran, Ramesh, ten Elshof, Blank and Rijnders88} substrates.

The strain game is capable of enhancing the properties of a multitude of ferroelectric systems. Shifts in ferroelectric T _C of roughly 300 K per percent biaxial strain, quite comparable to those predicted^{Reference Pertsev, Tagantsev and Setter28–Reference Li, Choudhury, Haeni, Biegalski, Vasudevarao, Sharan, Ma, Levy, Gopalan, Trolier-McKinstry, Schlom, Jia and Chen32} and observed^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12,Reference Irvin, Levy, Haeni and Schlom68–Reference Nuzhnyy, Petzelt, Kamba, Kužel, Kadlec, Bovtun, Kempa, Schubert, Brooks and Schlom74} for SrTiO₃, were first predicted by theory and subsequently verified by experiments on biaxially strained BaTiO₃^{Reference Choi, Biegalski, Li, Sharan, Schubert, Uecker, Reiche, Chen, Pan, Gopalan, Chen, Schlom and Eom13,Reference Schlom, Chen, Eom, Rabe, Streiffer and Triscone14,Reference Pertsev, Zembilgotov and Tagantsev91–Reference Tenne, Turner, Schmidt, Biegalski, Li, Chen, Soukiassian, Trolier-McKinstry, Schlom, Xi, Fong, Fuoss, Eastman, Stephenson, Thompson and Streiffer94} and PbTiO₃^{Reference Schlom, Chen, Eom, Rabe, Streiffer and Triscone14,Reference Pertsev, Zembilgotov and Tagantsev91,Reference Pertsev and Koukhar95–Reference Fong, Stephenson, Streiffer, Eastman, Auciello, Fuoss and Thompson100} films. Strain effects of comparable magnitude have also been observed in strained (Ba,Sr)TiO₃ films^{Reference Abe, Yanase, Sano, Izuha, Fukushima and Kawakubo101,Reference Yanase, Abe, Fukushima and Kawakubo102} and in strained-layer superlattices: KTaO₃/KNbO₃,^{Reference Specht, Christen, Norton and Boatner103} SrTiO₃/SrZrO₃,^{Reference Christen, Knauss and Harshavardhan104} SrTiO₃/BaZrO₃,^{Reference Christen, Specht, Silliman and Harshavardhan105} PbTiO₃/SrTiO₃,^{Reference Dawber, Lichtensteiger, Cantoni, Veithen, Ghosez, Johnston, Rabe and Triscone106,Reference Bousquet, Dawber, Stucki, Lichtensteiger, Hermet, Gariglio, Triscone and Ghosez107} BaTiO₃/SrTiO₃,^{Reference Neaton and Rabe108–Reference Li, Hu, Tenne, Soukiassian, Schlom, Chen, Xi, Choi, Eom, Saxena, Lookman and Jia111} and CaTiO₃/SrTiO₃/BaTiO₃.^{Reference Sai, Meyer and Vanderbilt112–Reference Lee, Christen, Chisholm, Rouleau and Lowndes114}

The success of theory in predicting the effect of strain on a multitude of ferroelectrics, together with advances in the ability to customize the structure and strain of oxide heterostructures at the atomic-layer level, has enabled a new era: ferroelectric oxides by design.^{Reference Schlom, Chen, Eom, Rabe, Streiffer and Triscone14} The appropriate theoretical methods to design strain-enhanced ferroelectrics depend on the material and whether or not domains need to be taken into account. First-principles methods are good for new materials where the coefficients of the Landau–Devonshire free energy expansion,^{Reference Devonshire30} a Taylor expansion of the free energy of a material in powers of its order parameter (polarization for typical ferroelectrics), are unknown. Due to the relatively small number of atoms that can be included in such calculations, however, the calculations are limited to single-domain materials. Measurements on materials can yield the coefficients needed in Landau–Devonshire thermodynamic analysis to calculate the effect of strain in the absence of domains.^{Reference Devonshire30}

Phase-field simulations, which also require coefficients obtained from either experiment or first-principles calculations, can be used to take domains into account.^{Reference Li, Hu, Liu and Chen97,Reference Li, Hu, Liu and Chen98} An example is (BaTiO₃)_n/(SrTiO₃)_m strained-layer superlattices, where n and m refer to the thickness, in unit cells, of the (001)_p BaTiO₃ and (001)_p SrTiO₃ layers, respectively. Despite the 2.3% lattice mismatch between the (001)_p BaTiO₃ and (001)_p SrTiO₃ layers, such superlattices can be commensurately strained. In Figure 3, the structural characterization of a commensurate (BaTiO₃)_n/(SrTiO₃)_m superlattice with n = 1 and m = 30 is shown. The macroscopic regularity of this superlattice, which was grown by molecular beam epitaxy (MBE), is demonstrated by the presence and sharpness of all of the superlattice reflections in its x-ray diffraction pattern (see Figure 3b).^{Reference Soukiassian, Tian, Vaithyanathan, Haeni, Chen, Xi, Schlom, Tenne, Sun, Pan, Choi, Eom, Li, Jia, Constantin, Feenstra, Bernhagen, Reiche and Uecker60,Reference Ravichandran, Yadav, Cheaito, Rossen, Soukiassian, Suresha, Duda, Foley, Lee, Zhu, Lichtenberger, Moore, Muller, Schlom, Hopkins, Majumdar, Ramesh and Zurbuchen119} Despite the BaTiO₃ layer being just a single unit cell thick (0.4 nm) and well-separated from neighboring BaTiO₃ layers, ultraviolet Raman measurements show that it is still ferroelectric.^{Reference Soukiassian, Tian, Vaithyanathan, Haeni, Chen, Xi, Schlom, Tenne, Sun, Pan, Choi, Eom, Li, Jia, Constantin, Feenstra, Bernhagen, Reiche and Uecker60}

Figure 3.

Structural characterization of a [(BaTiO₃)₁/(SrTiO₃)₃₀]₂₀ strained-layer superlattice grown on (001) SrTiO₃. (a) Atomic-resolution scanning transmission electron microscopy-electron energy loss spectroscopy (STEM-EELS) image (left) revealing the presence of atomically sharp interfaces with minimal intermixing. The STEM-EELS spectroscopic image at the left shows barium in purple and titanium in green. Distortions in the STEM-EELS map are not structural, but are artifacts from sample drift during data acquisition. Schematics of the crystal structures contained in the superlattice are shown on the right. (b) θ-2θ x-ray diffraction pattern of the same superlattice showing all expected satellite peaks and having a superlattice periodicity, c, of 12.1 nm. (a) Reprinted with permission from Reference 119. © 2013 Nature Publishing Group.

The ability to compare theory and experiment has motivated refinements in theory, including attention to not only mechanical (strain) and electrical boundary conditions (whether the ferroelectric is bounded by conducting electrodes or insulating layers), but also to unequal biaxial strain^{Reference Zembilgotov, Pertsev, Böttger and Waser115,Reference Sheng, Li, Zhang, Choudhury, Jia, Gopalan, Schlom, Liu and Chen116} and the ability of a ferroelectric film to break up into multiple domains.^{Reference Li, Hu, Liu and Chen97,Reference Li, Hu, Liu and Chen98} For strained-layer superlattices of BaTiO₃/SrTiO₃, it was shown that quantitative agreement between the predicted and observed T _C for superlattices with a wide range of periodicities only occurred if calculations in which the possibility of multiple domains was considered.^{Reference Tenne, Bruchhausen, Lanzillotti-Kimura, Fainstein, Katiyar, Cantarero, Soukiassian, Vaithyanathan, Haeni, Tian, Schlom, Choi, Kim, Eom, Sun, Pan, Li, Chen, Jia, Nakhmanson, Rabe and Xi109,Reference Li, Hu, Tenne, Soukiassian, Schlom, Xi, Choi, Eom, Saxena, Lookman, Jia and Chen110} For some BaTiO₃/SrTiO₃ superlattices, such three-dimensional phase-field calculations indicated that the low energy configuration was a multiple-domain state, which allowed the polarization in the (001)_p SrTiO₃ layers to drop considerably when the (001)_p BaTiO₃ layer was thinner than the (001)_p SrTiO₃ layer, resulting in a significant increase in T _C compared to the single-domain state.^{Reference Tenne, Bruchhausen, Lanzillotti-Kimura, Fainstein, Katiyar, Cantarero, Soukiassian, Vaithyanathan, Haeni, Tian, Schlom, Choi, Kim, Eom, Sun, Pan, Li, Chen, Jia, Nakhmanson, Rabe and Xi109,Reference Li, Hu, Tenne, Soukiassian, Schlom, Xi, Choi, Eom, Saxena, Lookman, Jia and Chen110} The domains anticipated to be present by theory, in order to quantitatively explain the observed T _C values, have recently been observed in BaTiO₃/SrTiO₃ strained-layer superlattices.^{Reference Kathan-Galipeau, Wu, Li, Chen, Soukiassian, Xi, Schlom and Bonnell117,Reference Kathan-Galipeau, Wu, Li, Chen, Soukiassian, Zhu, Muller, Xi, Schlom and Bonnell118}

Strain engineering of multiferroics

Emboldened by these successes, the strain game has more recently turned to enhancing materials containing multiple ferroic order parameters (i.e., multiferroics such as BiFeO₃) or to create new multiferroics from materials that are on the verge of being ferroic (e.g., EuTiO₃). Illustrative examples are described in the sections that follow.

Strained BiFeO₃—Morphing a room-temperature multiferroic

Bismuth ferrite, BiFeO₃, is one of the few materials that is simultaneously ferroelectric and magnetically ordered (antiferromagnetically in the case of BiFeO₃) at room temperature.^{Reference Wang, Neaton, Zheng, Nagarajan, Ogale, Liu, Viehland, Vaithyanathan, Schlom, Waghmare, Spaldin, Rabe, Wuttig and Ramesh120–Reference Wang, Zhao, Wang, Gai, Balke, Chi, Lee, Tian, Zhu, Cheng, Keavney, Yi, Ward, Snijders, Christen, Wu, Shen and Xu124} All of the other room-temperature multiferroics are, however, metastable. These include the high-pressure phase BiCoO₃,^{Reference Belik, Iikubo, Kodama, Igawa, Shamoto, Niitaka, Azuma, Shimakawa, Takano, Izumi and Takayama-Muromachi122} strain-stabilized ScFeO₃ with the corundum structure,^{Reference Li, Adem, McMitchell, Xu, Thomas, Warren, Giap, Niu, Wan, Palgrave, Schiffmann, Cora, Slater, Burnett, Cain, Abakumov, van Tendeloo, Thomas, Rosseinsky and Claridge123} and the hexagonal polymorph of LuFeO₃ that has been stabilized via epitaxy.^{Reference Wang, Zhao, Wang, Gai, Balke, Chi, Lee, Tian, Zhu, Cheng, Keavney, Yi, Ward, Snijders, Christen, Wu, Shen and Xu124} In its unstrained state, BiFeO₃ has the highest remnant polarization of any known ferroelectric.^{Reference Wang, Neaton, Zheng, Nagarajan, Ogale, Liu, Viehland, Vaithyanathan, Schlom, Waghmare, Spaldin, Rabe, Wuttig and Ramesh120,Reference Ramesh and Spaldin121,Reference Li, Wang, Wuttig, Ramesh, Wang, Ruette, Pyatakov, Zvezdin and Viehland125–Reference Dho, Qi, Kim, MacManus-Driscoll and Blamire127} BiFeO₃ also exhibits several polymorphs that are relatively close in energy to each other. Further, in BiFeO₃, the four fundamental degrees of freedom—electronic spin, charge, orbital, and lattice—are highly interactive. As a consequence, its ground state exhibits a strong sensitivity to temperature, pressure, electric fields, and magnetic fields. These qualities make BiFeO₃ unique and a natural candidate to tweak using the strain game.

Elastic strain, imparted via epitaxy, is able to distort and eventually change the structure and energy of the ground state of BiFeO₃. In its unstrained state, BiFeO₃ is rhombohedral.^{Reference Kubel and Schmid128} The evolution of the structure of BiFeO₃ can be understood from ab initio calculations, Figure 4a. As an in-plane biaxial compressive strain is imposed via the substrate, the rhombohedral (R) structure becomes progressively monoclinic (and perhaps even triclinic) until a critical strain of ∼4.5% is reached. For substrates that impose a larger strain (such as (110) YAlO₃, which imposes a hefty 6.6% biaxial compressive strain on commensurate BiFeO₃ thin films grown upon it^21–24), the structure changes into a “super-tetragonal” (T) state (or a monoclinic derivative thereof) with a distinct jump in the ratio of its pseudotetragonal lattice constants c/a.^{Reference Zeches, Rossell, Zhang, Hatt, He, Yang, Kumar, Wang, Melville, Adamo, Sheng, Chu, Ihlefeld, Erni, Ederer, Gopalan, Chen, Schlom, Spaldin, Martin and Ramesh21} Such an isostructural monoclinic-to-monoclinic phase transition—in which the symmetry does not change, but the coordination chemistry changes dramatically—has also been observed in other materials.^{Reference Hatt, Spaldin and Ederer129–Reference Haines, Léger and Schulte132} The overlapping, roughly parabolic free energy versus strain curves of the R- and T-phases can be seen in Figure 4a. It is the change in ground state with strain from the R phase to the T phase that enables the huge (6.6%) biaxial strains to be achieved in BiFeO₃ films. In more typical oxide systems, the free energy versus strain landscape limits the growth of epitaxial perovskite films under common growth conditions to about 3% strain.

Figure 4.

(a) Ab initio calculations of the energy and structure (ratio of pseudotetragonal lattice constants c/a) of the ground state of BiFeO₃ as a function of bi-axial, in-plane compressive strain on various substrates: YAO, YAlO₃; LAO, LaAlO₃; STO, SrTiO₃; LSAT, (LaAlO₃)_0.29—(SrAl_1/2Ta_1/2O₃)_0.71; DSO, DyScO₃. (b) Atomic force microscopy image of a partially relaxed, ∼70-nm-thick BiFeO₃ film grown on a LaAlO₃ substrate that exhibits a characteristic tetragonal and rhombohedral (T+R) mixed phase nanostructure. (c) High-resolution transmission electron microscopy image of the T+R mixed phase that illustrates the commensurate nature of the interface, with the complete absence of misfit dislocations. (d) X-ray magnetic circular dichroism photoemission electron microscopy (XMCD-PEEM) image obtained using the iron absorption edge showing that the highly constrained R-phase shows enhanced ferromagnetism compared to the bulk or the T-phase. Enhanced magnetic contrast is given from the ratio of PEEM images taken with left and right circularly polarized x-rays at the same location. Black and white contrasts indicate magnetic moments pointing parallel and antiparallel to the incident x-rays. (a) Reprinted with permission from Reference 21. © 2009 AAAS. (b) Adapted from Reference 21. (c) Reprinted with permission from Reference 21. © 2009 AAAS. (d) Reprinted with permission from Reference 134. © 2011 Nature Publishing Group.

Partial relaxation of the epitaxial constraint by increasing the film thickness leads to the formation of a mixed-phase nanostructure that exhibits the coexistence of both the R- and T-phases, as illustrated in the atomic force microscope (AFM) image in Figure 4b. This mixed-phase nanostructure is fascinating from many perspectives. First, high-resolution electron microscopy shows that the interface between these two phases is essentially commensurate, Figure 4c. This is important because it means that movement of this interface should be possible simply by the application of an electric field, as is indeed the case.^{Reference Zhang, Xiang, He, Seidel, Zeches, Yu, Yang, Wang, Chu, Martin, Minor and Ramesh133} Second, and perhaps more importantly, the highly distorted R-phase in this ensemble, shows significantly enhanced ferromagnetism. This can be discerned from the x-ray magnetic circular dichroism-photoemission electron microscopy (XMCD-PEEM) image in Figure 4d. The R-phase appears in either bright or dark stripe-like contrast in such PEEM images, corresponding to the thin slivers being magnetized either along the x-ray polarization direction or anti-parallel to it.

A rough estimate of the magnetic moment of this highly strained R-phase (from the PEEM images as well as from superconducting quantum interference device magnetometry measurements) gives a local moment of the order of 25–35 emu/cc.^{Reference He, Chu, Heron, Yang, Liang, Kuo, Lin, Yu, Liang, Zeches, Kuo, Juang, Chen, Arenholz, Scholl and Ramesh134} It is noteworthy that the canted moment of the R-phase (∼6–8 emu/cc) is not observable by the XMCD technique due to the small magnitude of the moment. This enhanced magnetic moment in the highly strained R-phase disappears around 150°C.^{Reference He, Chu, Heron, Yang, Liang, Kuo, Lin, Yu, Liang, Zeches, Kuo, Juang, Chen, Arenholz, Scholl and Ramesh134} Application of an electric field converts this mixed (R+T) phase into the T-phase, and the enhanced magnetic moment disappears; reversal of the electric field brings the mixed phase back accompanied by the magnetic moment in the distorted R-phase.^{Reference He, Chu, Heron, Yang, Liang, Kuo, Lin, Yu, Liang, Zeches, Kuo, Juang, Chen, Arenholz, Scholl and Ramesh134}

There has been limited work on the tensile side of the BiFeO₃ strain phase diagram. It was, however, predicted, using phase-field calculations, that an orthorhombic (O) phase of BiFeO₃ should exist under sufficient tensile strain,^{Reference Zeches, Rossell, Zhang, Hatt, He, Yang, Kumar, Wang, Melville, Adamo, Sheng, Chu, Ihlefeld, Erni, Ederer, Gopalan, Chen, Schlom, Spaldin, Martin and Ramesh21} and recent work has shown that an O-phase can indeed be stabilized.^{Reference Yang, He, Suresha, Kuo, Peng, Haislmaier, Motyka, Sheng, Adamo, Lin, Hu, Chang, Tjeng, Arenholz, Podraza, Bernhagen, Uecker, Schlom, Gopalan, Chen, Chen, Ramesh and Chu135} These observations on biaxially strained BiFeO₃ films raise several questions. First, what is the magnetic ground state of the various strained BiFeO₃ phases (e.g., do they have enhanced canting or exhibit spin glass behavior?). Second, given that spin-orbit coupling is the source of the canted moment in the bulk of BiFeO₃, can this enhanced moment be explained based on the strain and confinement imposed on the R-phase? Finally, what is the state of the Dzyaloshinskii–Moriya vector, the antisymmetric microscopic coupling between two localized magnetic moments, in such a strained system?

Strained EuTiO₃—Transforming a boring dielectric into the world’s strongest ferroelectric ferromagnet

The strain game involving EuTiO₃ is another tale in which theory led the way to a remarkable strain-enabled discovery. The idea behind this new route to ferroelectric ferromagnets is that appropriate magnetically ordered insulators that are neither ferroelectric nor ferromagnetic, of which there are many, can be transmuted into ferroelectric ferromagnets. Fennie and Rabe predicted^{Reference Fennie and Rabe136} that EuTiO₃, a normally boring paraelectric and antiferromagnetic insulator (in its unstrained bulk state), could be transformed using strain into the strongest known multiferroic with a spontaneous polarization and spontaneous magnetization each 100× superior to the reigning multiferroic it displaced, Ni₃B₇O₁₃I.^{Reference Rivera and Schmid137,Reference von Wartburg138}

The physics behind this discovery makes use of spin-lattice coupling as an additional parameter to influence the soft mode of an insulator on the verge of a ferroelectric transition.^{Reference Birol and Fennie139} The soft mode is the lowest frequency transverse optical phonon, which as it goes to zero results in the phase transition from a paraelectric to a ferroelectric. Appropriate materials for this (1) have a ground state that in the absence of strain is antiferromagnetic and paraelectric, (2) are on the brink of a ferroelectric transition, and (3) exhibit large spin-lattice coupling manifested by a significant decrease in permittivity as the material is cooled through its Néel temperature.^{Reference Fennie and Rabe136} EuTiO₃ meets these criteria and has much in common with SrTiO₃ except that EuTiO₃ magnetically orders at 5 K due to the existence of localized 4f moments on the Eu²⁺ site.^{Reference McGuire, Shafer, Joenk, Alperin and Pickart140,Reference Chien, DeBenedetti, De and Barros141} Similar to SrTiO₃, strain can be used to soften the soft mode and drive it to a ferroelectric instability. In contrast to SrTiO₃, which is diamagnetic, the permittivity of bulk EuTiO₃ is strongly coupled with its magnetism, showing an abrupt decrease in permittivity at the onset of the antiferromagnetic Eu²⁺ ordering.^{Reference Katsufuji and Takagi142} This indicates that the soft mode frequency hardens when the spins order antiferromagnetically; conversely, it will soften if the spins order ferromagnetically. This extra interaction provides the coupling favoring a simultaneously ferroelectric and ferromagnetic ground state under sufficient strain in EuTiO₃.

Although testing this prediction seems straightforward, the groups who first tested it ran into an unforeseen complication: no matter what substrate they deposited EuTiO₃ on, it was ferromagnetic. With its identical lattice constant (both are 3.905 Å at room temperature), SrTiO₃ is an obvious substrate for the growth of unstrained epitaxial EuTiO₃ films. Surprisingly, as-grown EuTiO_3–δ thin films synthesized by pulsed-laser deposition (PLD) on (001) SrTiO₃ substrates exhibit expanded out-of-plane spacings (0.4% to 2% longer than bulk EuTiO₃)^{Reference Wang, Fleet, Brock, Dale and Suzuki143–Reference Fujita, Wakasugi, Murai, Zong and Tanaka146} and are ferromagnetic with a Curie temperature of about 5 K.^{Reference Kugimiya, Fujita, Tanaka and Hirao144,Reference Chae, Chang, Kim, Lee, Choi and Jung145}

Could the observed expanded lattice spacings in EuTiO_3–δ thin films be due to oxygen vacancies? The effect of oxygen deficiency on lattice constant has been studied in EuTiO_3–δ bulk samples down to the EuTiO_2.5 limit of the perovskite EuTiO_3–δ structure, and negligible (<0.5%) variation in the cubic lattice constant was found.^{Reference Schafer147,Reference McCarthy, White and Roy148} Oxygen vacancies alone are thus insufficient to explain the 2% variation in out-of-plane lattice spacings observed in epitaxial EuTiO_3–δ films grown on (001) SrTiO₃ by PLD.^{Reference Kugimiya, Fujita, Tanaka and Hirao144–Reference Fujita, Wakasugi, Murai, Zong and Tanaka146}

One possible explanation is that the ferromagnetism observed in epitaxial EuTiO₃ films prepared by PLD on SrTiO₃ arises from extrinsic effects, masking the intrinsic properties of EuTiO₃ thin films. Extrinsic effects are known to occur in thin films, particularly for deposition methods involving energetic species, which can induce defects. Another factor favoring defect introduction is the relatively low growth temperatures common for oxide thin-film growth, enabling defects to be frozen in. For example, some epitaxial SrTiO₃ films grown on SrTiO₃ substrates by PLD have been reported to be ferroelectric,^{Reference Kim, Kim, Kim, Noh, Choi, Park and Yoon149} in striking contrast to the intrinsic nature of unstrained SrTiO₃, which is not ferroelectric at any temperature.^{Reference Müller and Burkard150} Homoepitaxial SrTiO₃ films grown by PLD are also known to exhibit lattice spacings that deviate significantly from the SrTiO₃ substrates they are grown on,^{Reference Tarsa, Hachfeld, Quinlan, Speck and Eddy151–Reference Ohnishi, Shibuya, Yamamoto and Lippmaa153} although bulk SrTiO_3–δ (in either single crystal or polycrystalline form) exhibits negligible variation in its cubic lattice constant up to the SrTiO_2.5 limit^{Reference Kestigian, Dickinson and Ward154,Reference Tenne, Gonenli, Soukiassian, Schlom, Nakhmanson, Rabe and Xi155} of the perovskite SrTiO_3–δ structure. The sensitivity of EuTiO₃ that made it an appropriate material to transmute via strain into a multiferroic also makes it quite sensitive to defects. To overcome this issue and examine the intrinsic effect of strain on EuTiO₃, a more delicate deposition technique was needed.

In contrast to PLD, homoepitaxial SrTiO₃ films grown by MBE^{Reference Brooks, Fitting Kourkoutis, Heeg, Schubert, Muller and Schlom156} show bulk behavior and none of the unusual effects reported in homoepitaxial SrTiO₃ films grown by PLD.^{Reference Kim, Kim, Kim, Noh, Choi, Park and Yoon149,Reference Tarsa, Hachfeld, Quinlan, Speck and Eddy151–Reference Ohnishi, Shibuya, Yamamoto and Lippmaa153} Indeed unstrained, stoichiometric EuTiO₃ thin films grown by MBE on (001) SrTiO₃ have the same lattice constant as bulk EuTiO₃ and are antiferromagnetic.^{Reference Lee, Ke, Podraza, Fitting Kourkoutis, Heeg, Roeckerath, Freeland, Fennie, Schubert, Muller, Schiffer and Schlom157} Seeing that MBE can produce EuTiO₃ films with intrinsic properties in their unstrained state, MBE was used to test Fennie and Rabe’s strained EuTiO₃ predictions.^{Reference Fennie and Rabe136} Commensurate EuTiO₃ films were grown on three substrates: (001) LSAT, (001) SrTiO₃, and (110) DyScO₃ to impart –0.9%, 0%, and +1.1% biaxial strain, respectively. Using scanning transmission electron microscopy with electron energy loss spectroscopy (STEM-EELS), the oxidation state of the film constituents and the abruptness of the interface between the film and substrate was checked with atomic resolution and chemical specificity (Figure 5a). Experimental measurements utilizing second harmonic generation (SHG) and magneto-optic Kerr effect (MOKE) confirmed that the EuTiO₃ grown on (110) DyScO₃ was simultaneously ferroelectric (Figure 5b) and ferromagnetic (Figure 5c), while on the other substrates it was not, in agreement with theory^{Reference Fennie and Rabe136} and resulting in the strongest multiferroic material known today.^{Reference Lee, Fang, Vlahos, Ke, Jung, Kourkoutis, Kim, Ryan, Heeg, Roeckerath, Goian, Bernhagen, Uecker, Hammel, Rabe, Kamba, Schubert, Freeland, Muller, Fennie, Schiffer, Gopalan, Johnston-Halperin and Schlom16}

Figure 5.

(a) Elemental maps of europium and dysprosium observed by scanning transmission electron microscopy-electron energy loss spectroscopy on a 22-nm-thick commensurate EuTiO₃/DyScO₃ film, confirming an abrupt EuTiO₃/DyScO₃ interface with the correct oxidation states. Observation of (b) ferroelectricity by second harmonic generation and (c) ferromagnetism by magneto-optical Kerr effect in this same strained EuTiO₃ grown on (110) DyScO₃, confirming predictions that under sufficient biaxial strain, EuTiO₃ becomes multiferroic. Control samples with zero (EuTiO₃/SrTiO₃) or opposite (EuTiO₃/LSAT)) strain are consistent with the theoretically predicted strain phase diagram for EuTiO₃. LSAT, (LaAlO₃)_0.29—(SrAl_1/2Ta_1/2O₃)_0.71. Reprinted with permission from Reference 16. © 2010 Nature Publishing Group.

There are many other exciting predictions that remain to be verified of even stronger and higher temperature ferroelectric ferromagnets in strained SrMnO₃^{Reference Lee and Rabe158} and EuO,^{Reference Bousquet, Spaldin and Ghosez159} as well as the prediction that an electric field on the order of 10⁵ V cm^–1 can be used to turn on ferromagnetism in EuTiO₃ when it is poised on the verge of such a phase transition via strain.^{Reference Fennie and Rabe136} It was recently shown that through the application of an electric field, the antiferromagnetic ground state of EuTiO₃, strained to be close to where it would have a ferromagnetic ground state (but still on the antiferromagnetic side), can be electrically tuned to the verge of the ferromagnetic state.^{Reference Ryan, Kim, Birol, Thompson, Lee, Ke, Normile, Karapetrova, Schiffer, Brown, Fennie and Schlom160} Turning on magnetism in a material by applying an electric field to it remains an open challenge. Such an important milestone would be a key advance to the field of ferroics, both scientifically and technologically.

Electronics has flourished because of the ability to route voltages with ease and on extremely small scales. If magnetism could be similarly controlled and routed, it would impact memory devices, spin valves, and many other spintronics devices and make numerous hybrid devices possible.

Strained Sr_n+1Ti_nO_3n+1—Creating a tunable dielectric with record performance

Strain has also been used to create a new family of tunable microwave dielectrics, which due to their low dielectric loss have a figure of merit at room temperature that rivals those of all such known materials.^{Reference Lee, Orloff, Birol, Zhu, Goian, Rocas, Haislmaier, Vlahos, Mundy, Kourkoutis, Nie, Biegalski, Zhang, Bernhagen, Benedek, Kim, Brock, Uecker, Xi, Gopalan, Nuzhnyy, Kamba, Muller, Takeuchi, Booth, Fennie and Schlom161} In contrast to standard (textbook) dielectrics, whose dielectric displacement (D) as a function of applied electric field (E) can be described by the linear equation

(1)$$D = {\varepsilon _0}\;K\,E,$$

where ε₀ is the permittivity of free space and K is the dielectric constant of the material, a tunable dielectric has a highly nonlinear relationship between D and E. The nonlinearity results in the effective dielectric constant of the material (D/ε₀E) behaving not as a constant, but changing greatly with E; changes of tens of percent in the dielectric “constant” are common in tunable dielectrics at high E. This nonlinearity can be described by adding higher order terms to Equation 1. In tensor form, this more general relationship is

(2)$${D_i} = {\varepsilon _0}\,{K_{ij}}\,{E_j} + {\varepsilon _{ijk}}\,{E_j}\,{E_k} + {\varepsilon _{ijkl}}\,{E_j}\,{E_k}\,{E_l} + \ldots ,$$

where ε_ijk and ε_ijkℓ are higher order permittivity coefficients.

Highly nonlinear dielectrics, including the most extensively studied material with such properties, Ba_xSr_1–xTiO₃,^{Reference Vendik162–Reference Tagantsev, Sherman, Astafiev, Venkatesh and Setter165} are found in displacive ferroelectric systems at temperatures just above the paraelectric-to-ferroelectric transition temperature.^{Reference Vendik162,Reference Tagantsev, Sherman, Astafiev, Venkatesh and Setter165} These tunable dielectrics are normally used in their paraelectric state to avoid the dielectric losses that would occur in the ferroelectric state due to the motion of domain walls. Although thin films provide an excellent geometry for the application of high E at low applied voltages, Ba_xSr_1–xTiO₃ films unfortunately suffer significant dielectric losses arising from defects; the dielectric loss in today’s best Ba_xSr_1–xTiO₃ tunable dielectrics films at GHz frequencies is about an order of magnitude worse than the best Ba_xSr_1–xTiO₃ in bulk form.^{Reference Tagantsev, Sherman, Astafiev, Venkatesh and Setter165}

A new approach to tunable microwave dielectrics is to take a system with exceptionally low loss and introduce a ferroelectric instability into it using strain. The appeal of this approach is that it can be applied to systems that are known to have exceptionally low dielectric loss, particularly in thin-film form. Such systems do not need to have a ferroelectric instability in their unstrained form; strain can be used to help impart the ferroelectric instability. This relaxed design constraint greatly increases the number of eligible systems that could yield improved performance over the best of today’s known tunable dielectrics.

This approach has been applied to Sr_n+1Ti_nO_3n+1 phases—where (SrO)₂ crystallographic shear^{Reference Andersson and Wadsley166,Reference Anderson, Browne, Cheetham, Vondreel, Hutchison, Lincoln, Bevan and Straehle167} planes provide an alternative to point-defect formation for accommodating non-stoichiometry.^{Reference Tilley168,Reference Tilley169} These phases are known to have low dielectric loss,^{Reference Nakamura, Sun, Shan, Inaguma, Itoh, Kim, Sohn, Ikeda, Kitamura and Konagaya170,Reference Wise, Reaney, Lee, Price, Iddles and Cannell171} even in thin-film form.^{Reference Orloff, Tian, Fennie, Lee, Gu, Mateu, Xi, Rabe, Schlom, Takeuchi and Booth172} In their unstrained state, these phases are centrosymmetric, and thus lack a ferroelectric instability.

The strain game in combination with control of the distance between the (SrO)₂ planes in Sr_n+1Ti_nO_3n+1 phases can, however, induce a ferroelectric instability.^{Reference Lee, Orloff, Birol, Zhu, Goian, Rocas, Haislmaier, Vlahos, Mundy, Kourkoutis, Nie, Biegalski, Zhang, Bernhagen, Benedek, Kim, Brock, Uecker, Xi, Gopalan, Nuzhnyy, Kamba, Muller, Takeuchi, Booth, Fennie and Schlom161,Reference Birol, Benedek and Fennie173} The emergence of this ferroelectric instability in Sr_n+1Ti_nO_3n+1 phases commensurately strained in biaxial tension to (110) DyScO₃ substrates (about 1% biaxial tension) can be seen in Figure 6a. For n ≥ 3, a ferroelectric instability is evident from the double-well energy potential of the ion displacements, and this double-well becomes deeper with increasing n. A deeper well corresponds to the ferroelectric instability occurring at higher temperature until, by n = 6, it is just below room temperature.^{Reference Lee, Orloff, Birol, Zhu, Goian, Rocas, Haislmaier, Vlahos, Mundy, Kourkoutis, Nie, Biegalski, Zhang, Bernhagen, Benedek, Kim, Brock, Uecker, Xi, Gopalan, Nuzhnyy, Kamba, Muller, Takeuchi, Booth, Fennie and Schlom161}

Figure 6.

(a) First-principles calculations showing how the index n of Sr_n+1Ti_nO_3n+1 phases strained commensurately to (110) DyScO₃ substrates can be used to control the local ferroelectric instability. Plotted is the energy (per n) with respect to the nonpolar state of Sr_n+1Ti_nO_3n+1 phases for polar distortions in the (001) plane. (b) The real and imaginary parts of the in-plane dielectric constant (K ₁₁) of an n = 6 Sr_n+1Ti_nO_3n+1 film grown on (110) DyScO₃ as a function of frequency. The inset shows the film loss tangent on a linear frequency scale in the gigahertz frequency regime along with a linear fit. (c) Bright-field scanning transmission electron microscopy image of an n = 6 Sr_n+1Ti_nO_3n+1 film grown on (110) DyScO₃. Sr_n+1Ti_nO_3n+1 phases can readily accommodate nonstoichiometry by the formation of (SrO)₂ double layers, the bright horizontal and vertical running layers in this image. Reprinted with permission from Reference 161. © 2013 Nature Publishing Group.

The low loss of this tunable dielectric at high frequencies is evident from Figure 6b, where both the real and imaginary parts of the in-plane dielectric constant K ₁₁ are shown over a frequency range spanning more than eight orders of magnitude and up to 125 GHz.^{Reference Lee, Orloff, Birol, Zhu, Goian, Rocas, Haislmaier, Vlahos, Mundy, Kourkoutis, Nie, Biegalski, Zhang, Bernhagen, Benedek, Kim, Brock, Uecker, Xi, Gopalan, Nuzhnyy, Kamba, Muller, Takeuchi, Booth, Fennie and Schlom161} Only at these highest frequencies can the dielectric loss be seen. This low loss yields unparalleled performance at room temperature for these new tunable dielectrics.^{Reference Gevorgian and Kollberg164} The reason for the low dielectric loss, far lower than today’s best Ba_xSr_1–xTiO₃ tunable dielectrics at GHz frequencies, is believed to be related to the propensity of Sr_n+1Ti_nO_3n+1 phases to form (SrO)₂ planar defects in response to local stoichiometry deviations rather than point defects.^{Reference Tilley168,Reference Tilley169} These (SrO)₂ double layers show up as bright layers in the STEM image of an n = 6 Sr_n+1Ti_nO_3n+1 film in Figure 6c. This is in contrast to the high point-defect concentrations that perovskite films like SrTiO₃ can incorporate in thin-film form;^{Reference Lee, Skoromets, Biegalski, Lei, Haislmaier, Bernhagen, Uecker, Xi, Gopalan, Martí, Kamba, Kužel and Schlom75,Reference Brooks, Fitting Kourkoutis, Heeg, Schubert, Muller and Schlom156} such point defects could be the reason that Ba_xSr_1–xTiO₃ tunable dielectric films have significantly higher dielectric losses than bulk Ba_xSr_1–xTiO₃.^{Reference Tagantsev, Sherman, Astafiev, Venkatesh and Setter165}

Other ways oxides can react to strain

As the examples described in this article show, the strain game can be a powerful control parameter for enhancing the properties of ferroic oxides. It is important to consider, however, other ways in which a complex oxide might react to strain. Should there be an energetically more favorable route to accommodate the imposed strain, the system might take it. Examples include the possibility that the imposed strain will lead to changes in film composition, microstructure, or crystal structure (e.g., ordering [or disordering] of cations, anions, or their vacancies,^{Reference Klenov, Donner, Foran and Stemmer174,Reference Donner, Chen, Liu, Jacobson, Lee, Gadre and Morgan175} atom clustering, the stabilization of other polymorphs, or rotations of the oxygen coordination polyhedra^{Reference Rondinelli and Spaldin176–Reference Choi, Kwon, Jeen, Hamann-Borrero, Radi, Macke, Sutarto, He, Sawatzky, Hinkov, Kim and Lee178}) rather than simply dilating or compressing the spacings between the atoms in the unstrained structure equally. Octahedral rotation patterns from underlying layers and substrates can be imparted into films over distances of several nanometers and can thus complicate a simple biaxial strain picture. An example is the creation of T-phase BiFeO₃ at relatively low biaxial strain (–1.4%, where the BiFeO₃ should be monoclinic), due to the competition between different octahedral rotation patterns in the film and in the substrate.^{Reference Yang, Schlepütz, Adamo, Schlom and Clarke179} Such chemical and structural changes may also cause dramatic changes in properties, for better or worse, but it is important to distinguish the underlying causes of the changes in properties.

Advances in electron microscopy have been particularly beneficial in identifying these underlying causes (see the Hÿtch and Minor article in this issue). With respect to ferroics, recent advances enabling polarization mapping on the atomic scale,^{Reference Jia, Nagarajan, He, Houben, Zhao, Ramesh, Urban and Waser180–Reference Jia, Urban, Alexe, Hesse and Vrejoiu183} octahedral rotation imaging,^{Reference Borisevich, Chang, Huijben, Oxley, Okamoto, Niranjan, Burton, Tsymbal, Chu, Yu, Ramesh, Kalinin and Pennycook184} and improved STEM-EELS analysis^{Reference Muller, Kourkoutis, Murfitt, Song, Hwang, Silcox, Dellby and Krivanek185} are invaluable in separating true strain effects from changes in composition, structure, and microstructure that can occur to mitigate against a strain effect. These techniques have the potential to visualize closure domains in ferroelectrics,^{Reference Jia, Mi, Urban, Vrejoiu, Alexe and Hesse182,Reference Jia, Urban, Alexe, Hesse and Vrejoiu183} usual domain walls with mixed Bloch–Néel–Ising character in strained ferroelectric superlattices,^{Reference Lee, Behera, Wu, Xu, Li, Sinnott, Phillpot, Chen and Gopalan186} and the exciting possibility of a dipole-dipole coupling analog of exchange-spring magnets^{Reference Kneller and Hawig187,Reference Fullerton, Jiang and Bader188} recently predicted to occur in strained ferroelectric systems.^{Reference Wu, Ma, Li, Gopalan and Chen189}

Other intriguing predictions, including using strain to poise a material (e.g., SrCoO₃^{Reference Lee and Rabe190}) on the brink of a metal-insulator transition, which could then be turned on or off through the application of an electric or magnetic field or even a small additional strain, are relevant to emerging piezotronic devices.^{Reference Newns, Elmegreen, Liu and Martyna191,Reference Newns, Elmegreen, Hu Liu and Martyna192}

Outlook

Elastic strain engineering of ferroics with the perovskite structure has been the most pursued to date. Yet there are many other fascinating oxides that either have a ferroic ground state or are borderline ferroic materials that might be enticed into a ferroic ground state with strain. Such non-perovskite structural families that are relevant include chrysoberyl Cr₂BeO₄,^{Reference Newnham, Kramer, Schulze and Cross193} pyrochlore Ho₂Ti₂O₇,^{Reference Harris, Bramwell, McMorrow, Zeiske and Godfrey194} YMnO₃,^{Reference Fennie and Rabe195,Reference Choi, Horibe, Yi, Choi, Wu and Cheong196} MnWO₄,^{Reference Arkenbout, Palstra, Siegrist and Kimura197} delafossite CuFeO₂,^{Reference Ye, Ren, Huang, Fernandez-Baca, Dai, Lynn and Kimura198} CuO,^{Reference Kimura, Sekio, Nakamura, Siegrist and Ramirez199} and hexaferrite Sr₃Co₂Fe₂₄O₄₁.^{Reference Kitagawa, Hiraoka, Honda, Ishikura, Nakamura and Kimura200} These systems are generally ignored by the thin-film community and have been the focus of the single crystal ferroic community. We believe the issue is the lack of suitable substrates for these latter structures; removing this roadblock would unleash a huge opportunity in strain engineering for the future. Thin films also have the technological advantage of lower switching voltages and the ability to integrate them into more sophisticated heterostructures, as are relevant for devices.

Imagine the opportunities for strain engineering that substrates for the non-perovskite systems would bring. Substrates for YMnO₃ would enable more variants of hexagonal manganite ferroics to be constructed. These variants include not only known materials, but metastable polymorphs (e.g., LuFeO₃ that is isostructural to YMnO₃ rather than its stable centrosymmetric perovskite form)^{Reference Wang, Zhao, Wang, Gai, Balke, Chi, Lee, Tian, Zhu, Cheng, Keavney, Yi, Ward, Snijders, Christen, Wu, Shen and Xu124,Reference Bossak, Graboy, Gorbenko, Kaul, Kartavtseva, Svetchnikov and Zandbergen201,Reference Wang, Zhao, Wang, Gai, Balke, Chi, Lee, Tian, Zhu, Cheng, Keavney, Yi, Ward, Snijders, Christen, Wu, Shen and Xu202} by utilizing lattice misfit strain energies and interfacial energies to favor the desired metastable phase over the equilibrium phase (epitaxial stabilization),^{Reference Machlin, Chaudhari, Machlin and Rowland203–Reference Gorbenko, Samoilenkov, Graboy and Kaul206} or the prospect of interfacial multiferroicity that has been predicted to emerge in superlattices between centrosymmetric components.^{Reference Rondinelli and Fennie207,Reference Mulder, Benedek, Rondinelli and Fennie208}

Similarly, substrates with the LiNbO₃ structure would enable the growth of the LiNbO₃-polymorph of FeTiO₃ and related multiferroics.^{Reference Fennie209,Reference Varga, Kumar, Vlahos, Denev, Park, Hong, Sanehira, Wang, Fennie, Streiffer, Ke, Schiffer, Gopalan and Mitchell210} A range of appropriate substrates, like the range of substrates available for perovskites shown in Figure 1d, for each ferroic system of interest would allow the thin-film tricks of strain engineering,^{Reference Haeni, Irvin, Chang, Uecker, Reiche, Li, Choudhury, Tian, Hawley, Craigo, Tagantsev, Pan, Streiffer, Chen, Kirchoefer, Levy and Schlom12–Reference Lee, Fang, Vlahos, Ke, Jung, Kourkoutis, Kim, Ryan, Heeg, Roeckerath, Goian, Bernhagen, Uecker, Hammel, Rabe, Kamba, Schubert, Freeland, Muller, Fennie, Schiffer, Gopalan, Johnston-Halperin and Schlom16} epitaxial stabilization,^{Reference Machlin, Chaudhari, Machlin and Rowland203–Reference Gorbenko, Samoilenkov, Graboy and Kaul206} dimensional confinement,^{Reference Lee, Orloff, Birol, Zhu, Goian, Rocas, Haislmaier, Vlahos, Mundy, Kourkoutis, Nie, Biegalski, Zhang, Bernhagen, Benedek, Kim, Brock, Uecker, Xi, Gopalan, Nuzhnyy, Kamba, Muller, Takeuchi, Booth, Fennie and Schlom161,Reference Birol, Benedek and Fennie173} polarization engineering,^{Reference Sai, Meyer and Vanderbilt112,Reference Singh-Bhalla, Bell, Ravichandran, Siemons, Hikita, Salahuddin, Hebard, Hwang and Ramesh211} and superlattice formation^{Reference Soukiassian, Tian, Vaithyanathan, Haeni, Chen, Xi, Schlom, Tenne, Sun, Pan, Choi, Eom, Li, Jia, Constantin, Feenstra, Bernhagen, Reiche and Uecker60,Reference Specht, Christen, Norton and Boatner103–Reference Lee, Christen, Chisholm, Rouleau and Lowndes114,Reference Kathan-Galipeau, Wu, Li, Chen, Soukiassian, Xi, Schlom and Bonnell117–Reference Ravichandran, Yadav, Cheaito, Rossen, Soukiassian, Suresha, Duda, Foley, Lee, Zhu, Lichtenberger, Moore, Muller, Schlom, Hopkins, Majumdar, Ramesh and Zurbuchen119,Reference Rondinelli and Fennie207,Reference Mulder, Benedek, Rondinelli and Fennie208} to be freely applied to a much larger set of ferroic building blocks. Strain engineering of ferroic oxides is in its infancy and considering its brief, but vibrant past, a brilliant future awaits.