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1, although the basic theoretical considerationsapply to all related systems in which buckling occurs. Here, thinribbons of single-crystal semiconductors derived from high-qualitywafer-based sources of material are chemically bonded to f lat,prestrained elastomeric substrates of poly(dimethylsiloxane)(PDMS) (10). Releasing the prestrain leads to compressive strainson the ribbons that generate the wavy layouts. Applying strains tothese structures and observing them with high-resolution micros-copy techniques reveals the nature of their deformations andresponses to strain.Structures like those in Fig. 1 are of interest for applications inelectronics. In particular, wavy electronic materials, such as thesingle-crystal inorganic semiconductors of Fig. 1 (10) or polycrys-talline films of evaporated metals (11–14, 18, 19), provide fullyreversiblemechanical stretchability in electronic interconnects (11–14, 19) or in the active devices themselves, including metal-oxidefield effect transistors (MOSFETs) (10),metal-semiconductor fieldeffect transistors (MESFETs) (35), p–n junction diodes (10), andSchottky diodes (36). Integrated electronics that use such compo-nents could be important for devices such as f lexible displays (37),eye-like digital cameras (38), comformable skin sensors (39),intelligent surgical gloves (40), and structural health monitoringdevices (41). Besides their potential role in these applications, wavysingle-crystal inorganic films provide valuable testbeds for exam-ining the basic mechanics of the buckling process and the mechan-ical response of the wavy structures to applied strains. Unlike therelated and more thoroughly studied cases of amorphous or poly-crystalline films, high-purity single-crystal materials can be de-signed with dimensions (i.e., thicknesses, widths, and lengths) andmechanical properties (i.e., Young’s modulus and Poisson ratio)Author contributions: H.J., D.-Y.K., and J.S. contributed equally to this work; H.J., D.-Y.K.,Y.H., and J.A.R. designed research; H.J., D.-Y.K., J.S., and Y.H. performed research; H.J.,D.-Y.K., J.S., Y.S., Y.H., and J.A.R. analyzed data; and H.J., D.-Y.K., Y.H., and J.A.R.wrote thepaper.that are extremely well controlled. Recent methods have beendeveloped to allow the integration of defect-free single-crystal filmswith elastomeric substrates (10). These techniques enable system-atic and repeatable studies of the bucklingmechanics, to a precisionthat was not possible in previously studied systems. This articlepresents results of experiments that demonstrate the fundamentalaspects of the buckling process. Theoretical reexamination of thisclassical problem leads to an analytical mechanics theory thatprovides a coherent and quantitatively accurate picture of themechanics, which has direct connections to simple mechanics of anaccordion bellows. Some implications of these findings on appli-cations of buckled systems in electronics are presented.Results and DiscussionFig. 2 shows optical, scanning electron, and atomic force micro-graphs of structures similar to those schematically illustrated in Fig.1. Silicon-on-insulator (SOI) substrates or epitaxial layers on bulkwafers provided sources of high-quality single-crystal films withsemiconductor device-grade levels of materials purity, uniformityin thickness (less than 3%), and mechanical properties. Manip-ulating the surface chemistry of the ribbons created from theselayers and the PDMS substrates enables covalent interfacial bondsto form between these two materials upon physical contact. Inparticular, silane coupling reactions between hydroxyl groups onthe native oxide surfaces of Si ribbons and UV/ozone activatedsurfaces of the PDMS lead to exceptionally strong adhesion (42)and intimate mechanical coupling, as illustrated in the images ofFig. 2. In fact, the failure modes under extreme strains are cohesivein the ribbons (i.e., the ribbons crack) or the PDMS (i.e., the PDMStears); adhesive failures at the interfaces are not observed. Wavystructures formed in this manner are highly sinusoidal (Fig. 2), withexcellent uniformity in the amplitudes (less than 5%) and wave-lengths (less than 3%) over large areas (up to 15 mm 15 mm).Precision mechanical stages provide accurate means for applyingstrain to these structures. In situ scanning electron, optical, andatomic force microscopy can be used to quantify the mechanicalresponses.Several mechanics models have been developed for buckling incomposite structures of this type. Unlike classical column bucklinganalyses (e.g., ref. 43 and 44) that focus on the buckling load, thesemodels describe results in terms of the wavelength and amplitudeof the wavy structures. These dimensions are important in theemerging applications mentioned previously. Such models, whoserange of applicability lies in the small deformation limit, all lead tothe following predictions. For a thin film of thickness h and elasticmodulus Ef on a prestretched substrate (prestrain, pre) of modulusEs, releasing the prestrain leads to purely sinusoidal displacementdistributions with wavelengths of (2–9, 45–47) 0 2 h E f3E s 1/3.††[1]This equation predicts that the wavelength depends only on the filmthickness and the film/substrate modulus ratio, and not on theprestrain pre. The amplitude for the buckling process is given by (8)A0 h pre c 1, [2]††E E/(1 v2) is the plane-strain modulus, and v is the Poisson ratio.Fig. 1.