Stretchable form of single crystal silicon for high performance electronics on rubber substrates

Abstract

The present invention provides stretchable, and optionally printable, semiconductors and electronic circuits capable of providing good performance when stretched, compressed, flexed or otherwise deformed. Stretchable semiconductors and electronic circuits of the present invention preferred for some applications are flexible, in addition to being stretchable, and thus are capable of significant elongation, flexing, bending or other deformation along one or more axes. Further, stretchable semiconductors and electronic circuits of the present invention may be adapted to a wide range of device configurations to provide fully flexible electronic and optoelectronic devices.

Claims

1. A stretchable semiconductor element comprising: a flexible substrate having a supporting surface; a semiconductor structure having a curved internal surface, wherein at least a portion of said curved internal surface is bonded to said supporting surface of said flexible substrate; and an encapsulating layer or coating that at least partially encapsulates the semiconductor structure; wherein said curved internal surface is continuously bonded to said supporting surface at substantially all points along said curved internal surface.

2. The stretchable semiconductor element of claim 1 wherein said semiconductor structure is a bent semiconductor structure.

3. The stretchable semiconductor element of claim 2 wherein said bent semiconductor structure has a wave-shaped, wrinkled, coiled or buckled conformation.

4. The stretchable semiconductor element of claim 2 wherein said bent semiconductor structure is under strain.

5. The stretchable semiconductor element of claim 2 wherein said bent semiconductor structure is under strain selected over the range of about 1% to about 30%.

6. The stretchable semiconductor element of claim 2 wherein said bent semiconductor structure has a conformation comprising a periodic wave that extends at least a portion of the length of said structure.

7. The stretchable semiconductor element of claim 6 wherein said bent semiconductor structure has a sine wave conformation having a periodicity selected from the range of about 1 micron and 100 microns and an amplitude selected from the range of about 50 nanometers and about 5 microns.

8. The stretchable semiconductor element of claim 2 wherein said bent semiconductor structure has a conformation comprising a plurality of buckles that extend along the length of said structure.

9. The stretchable semiconductor element of claim 2 wherein said bent semiconductor structure has a conformation that varies spatially in one-dimension or two dimensions, wherein said internal surface has a contour profile that varies spatially in one-dimension or two dimensions.

10. The stretchable semiconductor element of claim 1 wherein said curved internal surface has at least one convex region, at least one concave region or a combination of at least one convex region and at least one concave region.

11. The stretchable semiconductor element of claim 1 wherein said curved internal surface has a contour profile comprising a periodic wave or an aperiodic wave.

12. The stretchable semiconductor element of claim 1 wherein said semiconductor structure comprises a ribbon, wire, strip, disc, or platelet.

13. The stretchable semiconductor element of claim 1 wherein said semiconductor structure has a thickness selected over the range of about 50 nanometers to about 50 microns.

14. The stretchable semiconductor element of claim 1 wherein said flexible substrate comprises a polymer.

15. The stretchable semiconductor element of claim 1 wherein said semiconductor structure is a single crystalline inorganic semiconductor material.

16. The stretchable semiconductor element of claim 1 wherein said semiconductor structure comprises a material selected from the group consisting of: Si, Ge, SiC, AIP, AlAs, AlSb, GaP, GaAs, GaSb, InP, InAs, GaSb, InP, InAs, InSb, ZnO, ZnSe, ZnTe, CdS, CdSe, ZnSe, ZnTe, CdS, CdSe, CdTe, HgS, PbS, PbSe, PbTe, AlGaAs, AllnAs, AllnP, GaAsP, GalnAs, GaInP, AlGaAsSb, AlGaInP, and GalnAsP, carbon nanotubes, graphene and GaN.

17. A stretchable electronic circuit comprising: a flexible substrate having a supporting surface; an electronic circuit having a curved internal surface, wherein at least a portion of said curved internal surface is bonded to said supporting surface of said flexible substrate; an encapsulating layer or coating that at least partially encapsulates the electronic circuit; and wherein the electronic circuit comprises a stretchable semiconductor element according to claim 1.

18. The stretchable electronic circuit of claim 17, wherein said encapsulating layer or coating is a polymer.

19. A stretchable semiconductor element comprising: a flexible substrate having a supporting surface; a semiconductor structure having a curved internal surface, wherein at least a portion of said curved internal surface is bonded to said supporting surface of said flexible substrate; an encapsulating layer or coating that at least partially encapsulates the semiconductor structure; and wherein said semiconductor structure is a bent semiconductor structure and said bent semiconductor structure is under a strain selected over the range of 1% to 30%.

20. A stretchable semiconductor element comprising: a flexible substrate having a supporting surface; a semiconductor structure having a curved internal surface, wherein at least a portion of said curved internal surface is bonded to said supporting surface of said flexible substrate; an encapsulating layer or coating that at least partially encapsulates the semiconductor structure; wherein said semiconductor structure is a bent semiconductor structure having a conformation comprising a periodic wave that extends at least a portion of the length of said bent semiconductor structure; and wherein said periodic wave has a periodicity selected from the range of 1 micron and 100 microns and an amplitude selected from the range of 50 nanometers and 5 microns.

21. A stretchable semiconductor element comprising: a flexible substrate having a supporting surface; a semiconductor structure having a curved internal surface, wherein at least a portion of said curved internal surface is bonded to said supporting surface of said flexible substrate; and an encapsulating layer or coating that at least partially encapsulates the semiconductor structure; wherein said curved internal surface is discontinuously bonded to said supporting surface at selected points along said curved internal surface.

22. A stretchable semiconductor element comprising: a flexible substrate having a supporting surface; a semiconductor structure having a curved internal surface, wherein at least a portion of said curved internal surface is bonded to said supporting surface of said flexible substrate; an encapsulating layer or coating that at least partially encapsulates the semiconductor structure; wherein said semiconductor structure is a bent semiconductor structure; and said bent semiconductor structure has a conformation comprising a plurality of buckles that extend along the length of said structure.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 provides an atomic force micrograph showing a stretchable semiconductor structure of the present invention.

(2) FIG. 2 shows an atomic force micrograph providing an expanded view of a semiconductor structure having curved internal surface.

(3) FIG. 3 shows an atomic force micrograph of an array of stretchable semiconductor structures of the present invention.

(4) FIG. 4 shows optical micrographs of stretchable semiconductor structures of the present invention.

(5) FIG. 5 shows an atomic force micrograph of a stretchable semiconductor structure of the present invention having a semiconductor structure bonded to a flexible substrate having a three dimensional relief pattern on its supporting surface.

(6) FIG. 6 shows a flow diagram illustrating an exemplary method of making a stretchable semiconductor element of the present invention.

(7) FIG. 7 shows an image of an array of longitudinal aligned stretchable semiconductors structures having wave-shaped curved internal surfaces supported by a flexible rubber substrate.

(8) FIG. 8 shows a cross sectional image of a stretchable semiconductor structure of the present invention, wherein printable semiconductor structures 776 are supported by flexible substrate 777. As shown in FIG. 8, printable semiconductor structures 776 have internal surfaces having a contour profile of a periodic wave.

(9) FIG. 9A shows a process flow diagram illustrating an exemplary method of making an array of stretchable thin film transistors. FIG. 9B shows optical micrographs of an array of stretchable thin film transistors in relaxed and stretched configurations.

(10) FIG. 10: Schematic illustration of the process for building stretchable single crystal silicon devices on elastomeric substrates. The first step (top frame) involves fabrication of thin (thicknesses between 20 and 320 nm) elements of single crystal silicon or complete integrated devices (i.e. transistors, diodes, etc.) by conventional lithographic processing followed by etching of the top silicon and SiO.sub.2 layers of a silicon-on-insulator (SOI) wafer. After these procedures, the ribbon structures are supported by but not bonded to the underlying wafer (top frame). Contacting a pre-strained elastomeric substrate (poly(dimethylsiloxane) PDMS—stretched by dL) to the ribbons leads to bonding between these materials (middle frame). Peeling back the PDMS, with the ribbons bonded on its surface, and then releasing the pre-strain causes the PDMS to relax back to its unstrained state (unstressed length, L). This relaxation leads to the spontaneous formation of well controlled, highly periodic, stretchable ‘wave-shaped’ structures in the ribbons (bottom frame).

(11) FIG. 11: (A) Optical images of a large scale aligned array of wavy single crystal silicon ribbons (widths=20 μm; spacings=20 μm; thicknesses=100 nm) on PDMS. (B) Angled view scanning electron micrograph of four wavy silicon ribbons from the array shown in (A). The wavelengths and amplitudes of the wave structures are highly uniform across the array. (C) Surface height (top frame) and wavenumber of the Si Raman peak (bottom frame) as a function of position along a wavy Si ribbon on PDMS, measured by atomic force and Raman microscopy, respectively. The lines represent sinusoidal fits to the data. (D) Amplitudes (top frame) and wavelengths (bottom frame) of wavy silicon ribbons as a function of the thickness of the silicon, all for a given level of prestrain in the PDMS. The lines correspond to calculations, without any fitting parameters.

(12) FIG. 12: Buckling wavelength as a function of temperature. The slight decrease in wavelength with increasing temperature is due to thermal shrinkage of the PDMS, which leads to shorter wavelengths for samples prepared at higher temperatures.

(13) FIG. 13: Peak silicon strain as a function of silicon thickness, for a prestrain value of ˜0.9%. The red symbols correspond to bending strains computed using wavelengths and amplitudes extracted based on equations that describe the buckling process. The black symbols correspond to similar calculations but using wavelengths and amplitudes measured by AFM.

(14) FIG. 14: (A) Atomic force micrographs (AFM; left frames) and relief profiles (right frames; the lines are the sinusoidal fit to experimental data) of wavy single crystal silicon ribbons (width=20 μm; thickness=100 nm) on PDMS substrates. The top, middle, and bottom parts correspond to configurations when the PDMS is strained, along the ribbon lengths, by −7% (compression), 0% (unperturbed) and 4.7% (stretching), respectively. (B) Average amplitudes (black) and changes in wavelength (red) of wavy silicon ribbons as a function of strain applied to the PDMS substrate (top frame). For the wavelength measurements, different substrates are used for tension (circles) and compression (squares). Peak silicon strains as a function of applied strain (bottom frame). The lines in these graphs represent calculations, without any free fitting parameters.

(15) FIG. 15: Top view AFM image of wavy silicon ribbons on PDMS, and line cut evaluated at an angle relative to the long dimension of the ribbons.

(16) FIG. 16: Silicon ribbon strain as a function of applied strain. The red symbols correspond to strains computed by numerical integration of the contour length, using wavelengths and amplitudes extracted using equations that describe the buckling process. The black symbols correspond to strains measured from the ratio of surface to horizontal distance in AFM surface profile along the wavy Si ribbons.

(17) FIG. 17: (A) Optical images of a stretchable single crystal silicon pn diode on a PDMS substrate at −11% (top), 0% (middle) and 11% (bottom) applied strains. The aluminum regions correspond to thin (20 nm) Al electrodes; the pink and green regions correspond to n (boron) and p (phosphorous) doped areas of the silicon. (B) Current density as a function of bias voltage for stretchable silicon pn diodes, measured at various levels of applied strain. The curves labeled ‘light’ and ‘dark’ correspond to devices exposed to or shielded from ambient light, respectively. The solid curves show modeling results. (C) Current-voltage characteristics of a stretchable Schottky barrier silicon MOSFET, measured at −9.9%, 0% and 9.9% applied strains (The gate voltage varies from 0V to −5V with a 1V step).

(18) FIG. 18: Peak silicon strain as a function of applied strain. The blue line is based on an accordion bellows model, and the black is an approximation for small strain which is also consistent with buckling mechanics.

(19) FIG. 19: Electrical measurements of wavy pn diodes, evaluated in three different devices before (before cycle) and after (after cycle) ˜100 cycles of compressing (to ˜5% applied strain), stretching (to ˜15% applied strain) and releasing for three different devices (#1, #2 and #3). The data indicate no systematic changes in device properties. The level of observed variation is comparable to that associated with repeated probing of a single device without changing the applied strain, and is likely due to slight differences in probe contacts.

(20) FIG. 20: Optical images (top frames) of a wavy silicon Schottky barrier MOSFET in its unperturbed state (middle) and in compression (top) and tension (bottom). Schematic illustration (bottom frame) of the device.

(21) FIG. 21: Transfer curves measured in a ‘wavy’ silicon Schottky barrier MOSFET at different applied strains.

(22) FIG. 22: Schematic illustration of steps for generating ‘buckled’ and ‘wavy’ GaAs ribbons on PDMS substrates. The left bottom frame shows the deposition of thin SiO.sub.2 on the surfaces of the ribbons to promote strong bonding to the PDMS. This bonding leads to the formation of the wavy geometry shown in the right middle frame. Weak, van der Waals bonding (and moderate to high levels of prestrain) leads to the buckled geometry, as shown in the right top frame.

(23) FIG. 23: Images of wavy GaAs ribbons on a PDMS substrate, as formed with a prestrain of ˜1.9% generated through thermal expansion. Optical (A), SEM (B), three-dimensional AFM (C) and top-view AFM (D) images of the same sample. The SEM image is obtained by tilting the sample stage at the angle of 45° between sample surface and detection direction. (Spots on the ribbons might be residues from the sacrificial AlAs layers.) (E, F) Surface height profiles plotted along the lines in blue and green as shown in (D), respectively.

(24) FIG. 24: (A) Optical micrographs of wavy GaAs ribbons formed with a prestrain of 7.8%, strongly bonded to the PDMS, collected at different applied strains. The blue bars on the left and right highlight certain peaks in the structure; the variation in the distance between these bars indicate the dependence of the wavelength on applied strain. (B) Change in wavelength as a function of applied strain for the wavy GaAs ribbons shown in (A), plotted in black; similar data for a system of sample (A) after embedding in PDMS, plotted in red.

(25) FIG. 25: Images of GaAs ribbons integrated with ohmic (source and drain) and Schottky (gate) contacts to form complete MESFETs. (A) Optical micrographs of wavy ribbons formed using a prestrain of 1.9% and strong bonding to the PDMS, showing the formation of periodic waves only in the sections without electrodes (grey). (B) Optical and (C) SEM images of buckled ribbons formed with a prestrain of ˜7% and weak bonding to the PDMS. (D) Optical image of two buckled devices shown in (B) after they were stretched to be flat. (E) A set of optical images of an individual ribbon device shown in (B) with different external applied strains (i.e., compressing strain of 5.83%, no applied strain, and stretching strain of 5.83% from top to bottom) after it was embedded in PDMS.

(26) FIG. 26: (A) Optical images of a GaAs ribbon MESFET on a PDMS stamp with different strains built in the PDMS substrate. The prestrain applied to the PDMS stamp was 4.7% before the devices were transferred onto its surface. (B) Comparison of I-V curves for the device shown in (A) before and after the system was applied 4.7% stretching strain.

(27) FIGS. 27A-C provides images at different degrees of magnification of a stretchable semiconductor of the present invention exhibiting stretchability in two dimensions.

(28) FIGS. 28A-C provide images of three different structural conformations of stretchable semiconductors of the present invention exhibiting stretchability in two dimensions.

(29) FIGS. 29A-D provide images of stretchable semiconductors of the present invention prepared by prestraining the elastic substrate via thermal expansion.

(30) FIG. 30 shows optical images of stretchable semiconductors exhibiting stretchability in two dimensions under varying stretching and compression conditions.

(31) FIG. 31A shows an optical image of stretchable semiconductors exhibiting stretchability in two dimensions fabricated via prestraining an elastic substrate via thermal expansion. FIGS. 31B and 31 C provide experimental results relating to the mechanical properties of the stretchable semiconductors shown in FIG. 31A.

DETAILED DESCRIPTION OF THE INVENTION

(32) Referring to the drawings, like numerals indicate like elements and the same number appearing in more than one drawing refers to the same element. In addition, hereinafter, the following definitions apply:

(33) “Printable” relates to materials, structures, device components and/or integrated functional devices that are capable of transfer, assembly, patterning, organizing and/or integrating onto or into substrates. In one embodiment of the present invention, printable materials, elements, device components and devices are capable of transfer, assembly, patterning, organizing and/or integrating onto or into substrates via solution printing or dry transfer contact printing.

(34) “Printable semiconductor elements” of the present invention comprise semiconductor structures that are able to be assembled and/or integrated onto substrate surfaces, for example using by dry transfer contact printing and/or solution printing methods. In one embodiment, printable semiconductor elements of the present invention are unitary single crystalline, polycrystalline or microcrystalline inorganic semiconductor structures. In this context of this description, a unitary structure is a monolithic element having features that are mechanically connected. Semiconductor elements of the present invention may be undoped or doped, may have a selected spatial distribution of dopants and may be doped with a plurality of different dopant materials, including P and N type dopants. The present invention includes microstructured printable semiconductor elements having at least one cross sectional dimension greater than or equal to about 1 micron and nanostructured printable semiconductor elements having at least one cross sectional dimension less than or equal to about 1 micron. Printable semiconductor elements useful in many applications comprises elements derived from “top down” processing of high purity bulk materials, such as high purity crystalline semiconductor wafers generated using conventional high temperature processing techniques. In one embodiment, printable semiconductor elements of the present invention comprise composite structures having a semiconductor operational connected to at least one additional device component or structure, such as a conducting layer, dielectric layer, electrode, additional semiconductor structure or any combination of these. In one embodiment, printable semiconductor elements of the present invention comprise stretchable semiconductor elements and/or heterogeneous semiconductor elements.

(35) “Cross sectional dimension” refers to the dimensions of a cross section of device, device component or material. Cross sectional dimensions include width, thickness, radius, and diameter. For example, semiconductor elements having a ribbon shape are characterized by a length and two cross sectional dimensions; thickness and width. For example, printable semiconductor elements having a cylindrical shape are characterized by a length and the cross sectional dimension diameter (alternatively radius).

(36) “Supported by a substrate” refers to a structure that is present at least partially on a substrate surface or present at least partially on one or more intermediate structures positioned between the structure and the substrate surface. The term “supported by a substrate” may also refer to structures partially or fully embedded in a substrate.

(37) “Solution printing” is intended to refer to processes whereby one or more structures, such as printable semiconductor elements, are dispersed into a carrier medium and delivered in a concerted manner to selected regions of a substrate surface. In an exemplary solution printing method, delivery of structures to selected regions of a substrate surface is achieved by methods that are independent of the morphology and/or physical characteristics of the substrate surface undergoing patterning. Solution printing methods useable in the present invention include, but are not limited to, ink jet printing, thermal transfer printing, and capillary action printing.

(38) “Substantially longitudinally oriented” refers to an orientation such that the longitudinal axes of a population of elements, such as printable semiconductor elements, are oriented substantially parallel to a selected alignment axis. In the context of this definition, substantially parallel to a selected axis refers to an orientation within 10 degrees of an absolutely parallel orientation, more preferably within 5 degrees of an absolutely parallel orientation.

(39) “Stretchable” refers to the ability of a material, structure, device or device component to be strained without undergoing fracture. In an exemplary embodiment, a stretchable material, structure, device or device component may undergo strain larger than about 0.5% without fracturing, preferably for some applications strain larger than about 1% without fracturing and more preferably for some applications strain larger than about 3% without fracturing.

(40) The terms “flexible” and “bendable” are used synonymously in the present description and refer to the ability of a material, structure, device or device component to be deformed into a curved shape without undergoing a transformation that introduces significant strain, such as strain characterizing the failure point of a material, structure, device or device component. In an exemplary embodiment, a flexible material, structure, device or device component may be deformed into a curved shape without introducing strain larger than or equal to about 5%, preferably for some applications larger than or equal to about 1%, and more preferably for some applications larger than or equal to about 0.5%.

(41) The term “buckle” refers to a physical deformation that occurs when a thin element, structure and/or device responds to a compressive strain by bending in a direction out of the plane of the element, structure and/or device. The present invention includes stretchable semiconductors, devices and components having one or more surfaces with a contour profile comprising one or more buckles.

(42) “Semiconductor” refers to any material that is a material that is an insulator at a very low temperature, but which has a appreciable electrical conductivity at a temperatures of about 300 Kelvin. In the present description, use of the term semiconductor is intended to be consistent with use of this term in the art of microelectronics and electronic devices. Semiconductors useful in the present invention may comprise element semiconductors, such as silicon, germanium and diamond, and compound semiconductors, such as group IV compound semiconductors such as SiC and SiGe, group III-V semiconductors such as AlSb, AlAs, Aln, AlP, BN, GaSb, GaAs, GaN, GaP, InSb, InAs, InN, and InP, group III-V ternary semiconductors alloys such as Al.sub.xGa.sub.1-xAs, group II-VI semiconductors such as CsSe, CdS, CdTe, ZnO, ZnSe, ZnS, and ZnTe, group I-VII semiconductors CuCl, group IV-VI semiconductors such as PbS, PbTe and SnS, layer semiconductors such as PbI.sub.2, MoS.sub.2 and GaSe, oxide semiconductors such as CuO and Cu.sub.2O. The term semiconductor includes intrinsic semiconductors and extrinsic semiconductors that are doped with one or more selected materials, including semiconductor having p-type doping materials and n-type doping materials, to provide beneficial electronic properties useful for a given application or device. The term semiconductor includes composite materials comprising a mixture of semiconductors and/or dopants. Specific semiconductor materials useful for in some applications of the present invention include, but are not limited to, Si, Ge, SiC, AlP, AlAs, AlSb, GaN, GaP, GaAs, GaSb, InP, InAs, GaSb, InP, InAs, InSb, ZnO, ZnSe, ZnTe, CdS, CdSe, ZnSe, ZnTe, CdS, CdSe, CdTe, HgS, PbS, PbSe, PbTe, AlGaAs, AlInAs, AlInP, GaAsP, GaInAs, GaInP, AlGaAsSb, AlGaInP, and GaInAsP. Porous silicon semiconductor materials are useful for applications of the present invention in the field of sensors and light emitting materials, such as light emitting diodes (LEDs) and solid state lasers. Impurities of semiconductor materials are atoms, elements, ions and/or molecules other than the semiconductor material(s) themselves or any dopants provided to the semiconductor material. Impurities are undesirable materials present in semiconductor materials which may negatively impact the electronic properties of semiconductor materials, and include but are not limited to oxygen, carbon, and metals including heavy metals. Heavy metal impurities include, but are not limited to, the group of elements between copper and lead on the periodic table, calcium, sodium, and all ions, compounds and/or complexes thereof.

(43) “Plastic” refers to any synthetic or naturally occurring material or combination of materials that can be molded or shaped, generally when heated, and hardened into a desired shape. Exemplary plastics useful in the devices and methods of the present invention include, but are not limited to, polymers, resins and cellulose derivatives. In the present description, the term plastic is intended to include composite plastic materials comprising one or more plastics with one or more additives, such as structural enhancers, fillers, fibers, plasticizers, stabilizers or additives which may provide desired chemical or physical properties.

(44) “Dielectric” and “dielectric material” are used synonymously in the present description and refer to a substance that is highly resistant to flow of electric current. Useful dielectric materials include, but are not limited to, SiO.sub.2, Ta.sub.2O.sub.5, TiO.sub.2, ZrO.sub.2, Y.sub.2O.sub.3, SiN.sub.4, STO, BST, PLZT, PMN, and PZT.

(45) “Polymer” refers to a molecule comprising a plurality of repeating chemical groups, typically referred to as monomers. Polymers are often characterized by high molecular masses. Polymers useable in the present invention may be organic polymers or inorganic polymers and may be in amorphous, semi-amorphous, crystalline or partially crystalline states. Polymers may comprise monomers having the same chemical composition or may comprise a plurality of monomers having different chemical compositions, such as a copolymer. Cross linked polymers having linked monomer chains are particularly useful for some applications of the present invention. Polymers useable in the methods, devices and device components of the present invention include, but are not limited to, plastics, elastomers, thermoplastic elastomers, elastoplastics, thermostats, thermoplastics and acrylates. Exemplary polymers include, but are not limited to, acetal polymers, biodegradable polymers, cellulosic polymers, fluoropolymers, nylons, polyacrylonitrile polymers, polyimide-imide polymers, polyimides, polyarylates, polybenzimidazole, polybutylene, polycarbonate, polyesters, polyetherimide, polyethylene, polyethylene copolymers and modified polyethylenes, polyketones, poly(methyl methacrylate, polymethylpentene, polyphenylene oxides and polyphenylene sulfides, polyphthalamide, polypropylene, polyurethanes, styrenic resins, sulphone based resins, vinyl-based resins or any combinations of these.

(46) “Elastomer” refers to a polymeric material which can be stretched or deformed and return to its original shape without substantial permanent deformation. Elastomers commonly undergo substantially elastic deformations. Elastic substrates useful in the present invention comprise, at least in part, one or more elastomers. Exemplary elastomers useful in the present invention may comprise, polymers, copolymers, composite materials or mixtures of polymers and copolymers. Elastomeric layer refers to a layer comprising at least one elastomer. Elastomeric layers may also include dopants and other non-elastomeric materials. Elastomers useful in the present invention may include, but are not limited to, thermoplastic elastomers, styrenic materials, olefenic materials, polyolefin, polyurethane thermoplastic elastomers, polyamides, synthetic rubbers, PDMS, polybutadiene, polyisobutylene, poly(styrene-butadiene-styrene), polyurethanes, polychloroprene and silicones.

(47) “Good electronic performance” and “high performance” are used synonymously in the present description and refer to devices and device components have electronic characteristics, such as field effect mobilities, threshold voltages and on-off ratios, providing a desired functionality, such as electronic signal switching and/or amplification. Exemplary transferable, and optionally printable, semiconductor elements of the present invention exhibiting good electronic performance may have intrinsic field effect mobilities greater than or equal 100 cm.sup.2 V.sup.−1 s.sup.−1, preferably for some applications greater than or equal to about 300 cm.sup.2 V.sup.−1 s.sup.−1. Exemplary transistors of the present invention exhibiting good electronic performance may have device field effect mobilities great than or equal to about 100 cm.sup.2 V.sup.−1 s.sup.−1, preferably for some applications greater than or equal to about 300 cm.sup.2 V.sup.−1 s.sup.−1, and more preferably for some applications greater than or equal to about 800 cm.sup.2 V.sup.−1 s.sup.−1. Exemplary transistors of the present invention exhibiting good electronic performance may have threshold voltages less than about 5 volts and/or on-off ratios greater than about 1×10.sup.4.

(48) “Large area” refers to an area, such as the area of a receiving surface of a substrate used for device fabrication, greater than or equal to about 36 inches squared.

(49) “Device field effect mobility” refers to the field effect mobility of an electronic device, such as a transistor, as computed using output current data corresponding to the electronic device.

(50) “Young's modulus” is a mechanical property of a material, device or layer which refers to the ratio of stress to strain for a given substance. Young's modulus may be provided by the expression;

(51) $\begin{array}{cc}E=\frac{\left(\mathrm{stress}\right)}{\left(\mathrm{strain}\right)}=\left(\frac{L_0}{\Delta L}\times \frac{F}{A}\right);& \left(\mathrm{II}\right)\end{array}$
wherein E is Young's modulus, L.sub.0 is the equilibrium length, ΔL is the length change under the applied stress, F is the force applied and A is the area over which the force is applied. Young's modulus may also be expressed in terms of Lame constants via the equation:

(52) $\begin{array}{cc}E=\frac{\mu \left(3\lambda +2\mu \right)}{\lambda +\mu };& \left(\mathrm{III}\right)\end{array}$
wherein λ and μ are Lame constants. High Young's modulus (or “high modulus”) and low Young's modulus (or “low modulus”) are relative descriptors of the magnitude of Young's modulus in a give material, layer or device. In the present invention, a High Young's modulus is larger than a low Young's modulus, preferably about 10 times larger for some applications, more preferably about 100 times larger for other applications and even more preferably about 1000 times larger for yet other applications.

(53) In the following description, numerous specific details of the devices, device components and methods of the present invention are set forth in order to provide a thorough explanation of the precise nature of the invention. It will be apparent, however, to those of skill in the art that the invention can be practiced without these specific details.

(54) The present invention provides stretchable semiconductors and electronic circuits capable of providing good performance when stretched, compressed flexed or otherwise deformed. Further, stretchable semiconductors and electronic circuits of the present invention may be adapted to a wide range of device configurations to provide fully flexible electronic and optoelectronic devices.

(55) FIG. 1 provides an atomic force micrograph showing a stretchable semiconductor structure of the present invention. The stretchable semiconductor element 700 comprises a flexible substrate 705, such as a polymer and/or elastic substrate, having a supporting surface 710 and a bent semiconductor structure 715 having a curved internal surface 720. In this embodiment, at least a portion of the curved internal surface 720 of bent semiconductor structure 715 is bonded to the supporting surface 710 of the flexible substrate 705. The curved internal surface 720 may be bonded supporting surface 710 at selected points along internal surface 720 or at substantially all points along internal surface 720. The exemplary semiconductor structure illustrated in FIG. 1 comprises a bent ribbon of single crystalline silicon having a width equal to about 100 microns and a thickness equal to about 100 nanometers. The flexible substrate illustrated in FIG. 1 is a PDMS substrate having a thickness of about 1 millimeter. Curved internal surface 720 has a bent structure comprising a substantially periodic wave extending along the length of the ribbon. As shown in FIG. 1, the amplitude of the wave is about 500 nanometers and the peak spacing is approximately 20 microns. FIG. 2 shows an atomic force micrograph providing an expanded view of a bent semiconductor structure 715 having curved internal surface 720. FIG. 3 shows an atomic force micrograph of an array of stretchable semiconductor structures of the present invention. Analysis of the atomic force micrograph in FIG. 3 shows that the bent semiconductor structures are compressed by about 0.27%. FIG. 4 shows optical micrographs of stretchable semiconductor structures of the present invention.

(56) The contour profile of curved surface 720 allows the bent semiconductor structure 715 to be expanded or compressed along deformation axis 730 without undergoing substantial mechanical strain. This contour profile may also allow the semiconductor structure to be bent, flexed or deformed in directions other than along deformation axis 730 without significant mechanical damage or loss of performance induced by strain. Curved surfaces of semiconductor structures of the present invention may have any contour profile providing good mechanical properties, such as stretchabilty, flexibility and/or bendability, and/or good electronic performance, such as exhibiting good field effect mobilities when flexed, stretched or deformed. Exemplary contour profiles may be characterized by a plurality of convex and/or concave regions, and by a wide variety of wave forms including sine waves, Gaussian waves, Aries functions, square waves, Lorentzian waves, periodic waves, aperiodic waves or any combinations of these. Wave forms useable in the present invention may vary with respect to two or three physical dimensions.

(57) FIG. 5 shows an atomic force micrograph of a stretchable semiconductor structure of the present invention having a bent semiconductor structure 715 bonded to a flexible substrate 705 having a three dimensional relief pattern on its supporting surface 710. The three-dimensional relief pattern comprises recessed region 750 and relief features 760. As shown in FIG. 5, bent semiconductor structure 715 is bound to supporting surface 710 in recessed region 750 and on relief features 760.

(58) FIG. 6 shows a flow diagram illustrating an exemplary method of making a stretchable semiconductor structure of the present invention. In the exemplary method, a prestrained elastic substrate in an expanded state is provided. Prestraining can be achieved by any means known in the art including, but not limited to, roll pressing and/or prebending the elastic substrate. Prestraining may also be achieved via thermal means, for example, by thermal expansion induced by raising the temperature of the elastic substrate. An advantage of prestraining via thermal means is that expansion along a plurality of different axes, such as orthogonal axes, is achievable.

(59) An exemplary elastic substrate useable in this method of the present invention is a PDMS substrate having a thickness equal to about 1 millimeter. The elastic substrate may be prestrained by expansion along a single axis or by expansion along a plurality of axes. As shown in FIG. 6, at least a portion of the internal surface of a printable semiconductor element is bonded to the external surface of the prestrained elastic substrate in an expanded state. Bonding may be achieved by covalent bonding between the internal surface of the semiconductor surface, by van der Waals forces, by using adhesive or any combinations of these. In an exemplary embodiment wherein the elastic substrate is PDMS, the supporting surface of the PDMS substrate is chemically modified such that is has a plurality of hydroxyl groups extending from its surface to facilitate covalent bonding with a silicon semiconductor structure. Referring back to FIG. 6, after binding the prestrained elastic substrate and semiconductor structure, the elastic substrate is allowed to relax at least partially to a relaxed state. In this embodiment, relaxation of the elastic substrate bends the internal surface of said semiconductor structure, thereby generating a semiconductor element having a curved internal surface.

(60) As shown in FIG. 6, the fabrication method may optionally include a second transfer step and, optional bonding step, wherein the transferable semiconductor element 715 having a curved internal surface 720 is transferred from the elastic substrate to another substrate, preferably a flexible substrate, such as a polymer substrate. This second transfer step may be achieved by bringing an exposed surface of the semiconductor structure 715 having a curved internal surface 720 in contact with a receiving surface of the other substrate that binds to the exposed surface of the semiconductor structure 715. Bonding to the other substrate may be accomplished by any means capable of maintaining, at least partially, the bent structure of the semiconductor element, including covalent bonds, bonding via van der Waals forces, dipole-dipole interactions, London forces and/or hydrogen bonding. The present invention also includes the use of adhesives layers, coatings and/or thin films provided between an exposed surface of the transferable semiconductor structure and the receiving surface.

(61) Stretchable semiconductor elements of the present invention may be effectively integrated into a large number functional devices and device components, such as transistors, diodes, lasers, MEMS, NEMS, LEDS and OLEDS. Stretchable semiconductor elements of the present invention have certain functional advantages over conventional rigid inorganic semiconductors. First, stretchable semiconductor elements may be flexible, and thus, less susceptible to structural damage induced by flexing, bending and/or deformation than conventional rigid inorganic semiconductors. Second, as a bent semiconductor structure may be in a slightly mechanically strained state to provide a curved internal surface, stretchable semiconductor elements of the present invention may exhibit higher intrinsic field effect mobilities than conventional unstrained inorganic semiconductors. Finally, stretchable semiconductor elements are likely to provide good thermal properties because they are capable of expanding and contracting freely upon device temperature cycling.

(62) FIG. 7 shows an image of an array of longitudinal aligned stretchable semiconductors having a wavy conformation As shown in FIG. 7, the semiconductor ribbons are provided in a periodic wave conformation and are supported by a single flexible rubber substrate.

(63) FIG. 8 shows a cross sectional image of a stretchable semiconductor element of the present invention, wherein semiconductor structures 776 are supported by flexible substrate 777. As shown in FIG. 8, semiconductor structures 776 have internal surfaces having a contour profile of a periodic wave. As also shown in FIG. 8, the periodic wave conformation extends through the entire cross sectional dimension of the semiconductor structures 776.

(64) The present invention also provides stretchable electronic circuits, devices and device arrays capable of good performance when stretched, flexed or deformed. Similar to the stretchable semiconductor elements described above, the present invention provides stretchable circuits and electronic devices comprising a flexible substrate having a supporting surface in contact with a device, device array or circuit having a curved internal surface, such as a curved internal surface exhibiting a wave structure. In this structural arrangement, at least a portion of the curved internal surface of the device, device array or circuit structure is bonded to the supporting surface of the flexible substrate. The device, device array or circuit of this aspect of the present invention is a multicomponent element comprising a plurality of integrated device components, such as semiconductors, dielectrics, electrodes, doped semiconductors and conductors. In an exemplary embodiment, flexible circuits, devices and device arrays having a net thickness less than about 10 microns comprise a plurality of integrated device components at least a portion of which have a periodic wave curved structure.

(65) In a useful embodiment of the present invention, a free standing electronic circuit or device comprising a plurality of interconnected components is provided. An internal surface of the electronic circuit or device is contacted and at least partially bonded to a prestrained elastic substrate in an expanded state. Prestraining can be achieved by any means known in the art including, but not limited to, roll pressing and/or prebending the elastic substrate, and the elastic substrate may be prestrained by expansion along a single axis or by expansion along a plurality of axes. Bonding may be achieved directly by covalent bonding or van der Waals forces between at least a portion of the internal surface of the electronic circuit or device and the prestrained elastic substrate, or by using adhesive or an intermediate bonding layer. After binding the prestrained elastic substrate and the electronic circuit or device, the elastic substrate is allowed to relax at least partially to a relaxed state, which bends the internal surface of the semiconductor structure. Bending of the internal surface of the electronic circuit or device generates a curved internal surface which in some useful embodiments has a periodic or aperiodic wave configuration. The present invention includes embodiments wherein all the components comprising the electronic device or circuit are present in a periodic or aperiodic wave configuration.

(66) Periodic or aperiodic wave configurations of stretchable electronic circuits, devices and device arrays allow them to conform to stretch or bent configurations without generating large strains on individual components of the circuits or devices. This aspect of the present invention provides useful electronic behavior of stretchable electronic circuits, devices and device arrays when present in bent, stretched or deformed states. The period of periodic wave configurations formed by the present methods may vary with (i) the net thickness of the collection of integrated components comprising the circuit or device and (ii) the mechanical properties, such as Young's modulus and flexural rigidity, of the materials comprising integrated device components.

(67) FIG. 9A shows a process flow diagram illustrating an exemplary method of making an array of stretchable thin film transistors. As shown in FIG. 9A, an array of free standing printable thin film transistors is provided using the techniques of the present invention. The array of thin film transistors is transferred to a PDMS substrate via dry transfer contact printing methods in a manner which exposes internal surfaces of the transistors. The exposed internal surfaces are next contacted with a room temperature cured prestrained PDMS layer present in an expanded state. Subsequent full curing of the prestrained PDMS layer bonds the internal surfaces of the transistors to the prestrained PDMS layer. The prestrained PDMS layer is allowed to cool and assume an at least partially relaxed state. Relaxation of PDMS layers introduces a periodic wave structure to the transistors in the array, thereby making them stretchable. The inset in FIG. 9A provides an atomic force micrograph of a array of stretchable thin film transistors made by the present methods. The atomic force micrograph shows the periodic wave structure that provides for good electronic performance in stretched or deformed states.

(68) FIG. 9B shows provides optical micrographs of an array of stretchable thin film transistors in relaxed and stretched configurations. Stretching the array in a manner generating a net strain of about 20% on the array did not fracture or damage the thin film transistors. The transition from a relax configuration to a strain configuration was observed to be a reversible process. FIG. 9B also provides a plot of drain current verse drain voltage for several potentials applied to the gate electrode showing that the stretchable thin film transistors exhibit good performance in both relaxed and stretched configurations.

Example 1: A Stretchable Form of Single Crystal Silicon for High Performance Electronics on Rubber Substrates

(69) We have produced a stretchable form of silicon that consists of sub-micrometer single crystal elements structured into shapes with microscale periodic, wave-like geometries. When supported by an elastomeric substrate, this ‘wavy’ silicon can be reversibly stretched and compressed to large strains without damaging the silicon. The amplitudes and periods of the waves change to accommodate these deformations, thereby avoiding significant strains in the silicon itself. Dielectrics, patterns of dopants, electrodes and other elements directly integrated with the silicon yield fully formed, high performance ‘wavy’ metal oxide semiconductor field effect transistors, pn diodes and other devices for electronic circuits that can be stretched or compressed to similarly large levels of strain.

(70) Progress in electronics is driven mainly by efforts to increase circuit operating speeds and integration densities, to reduce their power consumption and, for display systems, to enable large area coverage. A more recent direction seeks to develop methods and materials that enable high performance circuits to be formed on unconventional substrates with unusual form factors: flexible plastic substrates for paperlike displays and optical scanners, spherically curved supports for focal plane arrays and conformable skins for integrated robotic sensors. Many electronic materials can provide good bendability when prepared in thin film form and placed on thin substrate sheets or near neutral mechanical planes in substrate laminates. In those cases, the strains experienced by the active materials during bending can remain well below typical levels required to induce fracture (˜1%). Full stretchability, a much more challenging characteristic, is required for devices that can flex, stretch or reach extreme levels of bending as they are operated or for those that can be conformally wrapped around supports with complex, curvilinear shapes. In these systems, strains at the circuit level can exceed the fracture limits of nearly all known electronic materials, especially those that are well developed for established applications. This problem can be circumvented, to some extent, with circuits that use stretchable conducting wires to interconnect electronic components (e.g. transistors) supported by rigid, isolated islands. Promising results can be obtained with this strategy, although it is best suited to applications that can be achieved with active electronics at relatively low coverages. We report a different approach, in which stretchability is achieved directly in thin films of high quality single crystal silicon that have micron scale periodic, ‘wave’-like geometries. These structures accommodate large compressive and tensile strains through changes in the wave amplitudes and wavelengths rather than through potentially destructive strains in the materials themselves. Integrating such stretchable ‘wavy’ silicon elements with dielectrics, patterns of dopants and thin metal films leads to high performance, stretchable electronic devices.

(71) FIG. 10 presents a fabrication sequence for wavy single crystal silicon ribbons on elastomeric (i.e. rubber) substrates. The first step (top frame) involves photolithography to define a resist layer on a silicon-on-insulator (SOI) wafer, followed by etching to remove the exposed parts of the top silicon. Removing the resist with acetone and then etching the buried SiO.sub.2 layer with concentrated hydrofluoric acid releases the ribbons from the underlying silicon substrate. The ends of the ribbons connect to the wafer to prevent them from washing away in the etchant. The widths (5˜50 μm) and lengths (˜15 mm) of the resist lines define the dimensions of the ribbons. The thickness of the top silicon (20˜320 nm) on the SOI wafers defines the ribbon thicknesses. In the next step (middle frame), a flat elastomeric substrate (poly(dimethylsiloxane), PDMS; 1˜3 mm thick) is elastically stretched and then brought into conformal contact with the ribbons. Peeling the PDMS away lifts the ribbons off of the wafer and leaves them adhered to the PDMS surface. Releasing the strain in the PDMS (i.e. the prestrain) leads to surface deformations that cause well-defined waves to form in the silicon and the PDMS surface. (FIGS. 11A and 11B) The relief profiles are sinusoidal (top frame, FIG. 11C) with periodicities between 5 and 50 μm and amplitudes between 100 nm and 1.5 μm, depending on the thickness of the silicon and the magnitude of prestrain in the PDMS. For a given system, the periods and amplitudes of the waves are uniform to within ˜5%, over large areas (˜cm.sup.2). The flat morphology of the PDMS between the ribbons and the absence of correlated phases in waves of adjacent ribbons suggest that the ribbons are not strongly coupled mechanically. FIG. 11C (bottom frame) shows micro Raman measurements of the Si peak, measured as a function of distance along one of the wavy ribbons. The results provide insights into the stress distributions.

(72) The behavior in this static wavy configuration is consistent with non-linear analysis of the initial buckled geometry in a uniform, thin high modulus layer on a semi-infinite low modulus support:

(73) $\begin{array}{cc}{\lambda }_0=\frac{\pi h}{\sqrt{{\mathrm{.Math.}}_c}},A_0=h\sqrt{\frac{{\mathrm{.Math.}}_{\mathrm{pre}}}{{\mathrm{.Math.}}_c}-1}\mathrm{where}{\mathrm{.Math.}}_c={0.52\left[\frac{E_{\mathrm{PDMS}}\left(1-v_{\mathrm{Si}}^2\right)}{E_{\mathrm{Si}}\left(1-v_{\mathrm{PDMS}}^2\right)}\right]}^{2/3}& \left(1\right)\end{array}$
is the critical strain for buckling, ε.sub.pre is the level of prestrain, λ.sub.0 is the wavelength and A.sub.0 is the amplitude. The Poisson ratio is ν, the Young's modulus is E, and the subscripts refer to properties of the Si or PDMS. The thickness of the silicon is h. This treatment captures many features of the as-fabricated wavy structures. FIG. 11D shows, for example, that when the prestrain value is fixed (˜0.9% for these data), the wavelengths and amplitudes both depend linearly on the Si thickness. The wavelengths do not depend on the level of prestrain (FIG. 12). Furthermore, calculations that use literature values for the mechanical properties of the Si and PDMS (E.sub.Si=130 GPa, E.sub.PDMS=2 MPa, ν.sub.Si=0.27, ν.sub.PDMS=0.48) yield amplitudes and wavelengths that are within ˜10% (maximum deviation) of the measured values. The “ribbon strain” is computed from the ratio of the effective lengths of the ribbons (as determined from the wavelength) to their actual lengths (as determined from surface distances measured by AFM), and yield values that are approximately equal to the prestrain in the PDMS, for prestrains up to ˜3.5%. The peak (i.e. maximum) strains in the silicon itself, which we refer to as silicon strains, are estimated from the ribbon thicknesses and radii of curvature at the extrema of the waves according to κh/2, where κ is the curvature, in regimes of strain where the waves exist and where the critical strain (˜0.03% for the cases examined here) is small compared to the peak strains associated with bending. For the data of FIG. 11, the peak silicon strains are ˜0.36(±0.08)%, which is more than a factor of two smaller than the ribbon strains. This silicon strain is the same for all ribbon thicknesses, for a given prestrain (FIG. 13). The resulting mechanical advantage, in which the peak silicon strain is substantially less than the ribbon strain, is critically important for achieving stretchability. We note that buckled thin films have also been observed in metals and dielectrics evaporated or spin cast onto PDMS (in contrast to preformed, transferred single crystal elements and devices, as described herein).

(74) The dynamic response of the wavy structures to compressive and tensile strains applied to the elastomeric substrate after fabrication is of primary importance for stretchable electronic devices. To reveal the mechanics of this process, we measure the geometries of wavy Si ribbons by AFM as force is applied to the PDMS to compress or stretch it parallel to the long dimension of the ribbons. This force creates strains both along the ribbons and perpendicular to them, due to the Poisson effect. The perpendicular strains lead primarily to deformations of the PDMS in the regions between the ribbons. The strains along the ribbons, on the other hand, are accommodated by changes in the structure of the waves. The three-dimensional height images and surface profiles in FIG. 14A present representative compressed, unperturbed and stretched states (collected from slightly different locations on the sample). In these and other cases, the ribbons maintain their sinusoidal (lines in the right hand frames of FIG. 14A) shapes during deformation, in which approximately half of the wave structure lies beneath the unperturbed position of the PDMS surface as defined by the regions between the ribbons (FIG. 15). FIG. 14B shows the wavelength and amplitude for compressive (negative) and tensile (positive) applied strains relative to the unperturbed state (zero). The data correspond to averaged AFM measurements collected from a large number (>50) of ribbons per point. The applied strains are determined from the measured end-to-end dimensional changes of the PDMS substrate. Direct surface measurements by AFM as well as contour integrals evaluated from the sinusoidal wave shapes show that the applied strains are equal to the ribbons strains (FIG. 16) for the cases examined here. (The small amplitude (<5 nm) waves that persist at tensile strains larger than the prestrain minus the critical strain might result from slight slippage of the Si during the initial buckling process. The computed peak silicon strains and ribbon strains in this small (or zero) amplitude regime underestimate the actual values.) Interestingly, the results indicate two physically different responses of the wavy ribbons to applied strain. In tension, the waves evolve in a non-intuitive way: the wavelength does not change appreciably with applied strain, consistent with post-buckling mechanics. Instead, changes in amplitude accommodate the strain. In this regime, the silicon strain decreases as the PDMS is stretched; it reaches ˜0% when the applied strain equals the prestrain. By contrast, in compression, the wavelengths decrease and amplitudes increase with increasing applied strain. This mechanical response is similar to that of an accordion bellows, which is qualitatively different than the behavior in tension. During compression, the silicon strain increases with the applied strain, due to the decreasing radii of curvature at the wave peaks and troughs. The rates of increase and magnitudes of the silicon strains are, however, both much lower than the ribbon strains, as shown in FIG. 14B. This mechanics enables stretchability.

(75) The full response in regimes of strain consistent with the wavy geometries can be quantitatively described by equations that give the dependence of the wavelength λ on its value in the initial buckled state, λ.sub.0, and the applied strain ε.sub.applied according to:

(76) $\begin{array}{cc}\lambda =\{\begin{array}{cc}{\lambda }_0& \mathrm{for}\mathrm{tension}\\ {\lambda }_0\left(1+{\mathrm{.Math.}}_{\mathrm{applied}}\right)& \mathrm{for}\mathrm{compression}\end{array},& \left(2\right)\end{array}$
This tension/compression asymmetry can arise, for example, from slight, reversible separations between the PDMS and the raised regions of Si, formed during compression. For this case, as well as for systems that do not exhibit this asymmetric behavior, the wave amplitude A, for both tension and compression, is given by a single expression, valid for modest strains (<10-15%):

(77) $\begin{array}{cc}A=\sqrt{A_0^2-h^2\frac{{\mathrm{.Math.}}_{\mathrm{applied}}}{{\mathrm{.Math.}}_c}}=h\sqrt{\frac{{\mathrm{.Math.}}_{\mathrm{pre}}-{\mathrm{.Math.}}_{\mathrm{applied}}}{{\mathrm{.Math.}}_c}-1},& \left(3\right)\end{array}$
where A.sub.0 is the value corresponding to the initial buckled state. These expressions yield quantitative agreement with the experiments without any parameter fitting, as shown in FIG. 14A. When the waviness, which accommodates the tensile/compressive strains, remains, the peak silicon strain is dominated by the bending term and is given by (33)

(78) $\begin{array}{cc}{\mathrm{.Math.}}_{\mathrm{Si}}^{\mathrm{peak}}=2{\mathrm{.Math.}}_c\sqrt{\frac{{\mathrm{.Math.}}_{\mathrm{pre}}-{\mathrm{.Math.}}_{\mathrm{applied}}}{{\mathrm{.Math.}}_c}-1},& \left(4\right)\end{array}$
which agrees well with the strain measured from curvature, in FIG. 14B. (See also FIG. 18). Such an analytic expression is useful to define the range of applied strain that the system can sustain without fracturing the silicon. For a prestrain of 0.9%, this range is −27% to 2.9%, if we assume that the silicon failure strain is ˜2% (for either compression or tension). Controlling the level of prestrain allows this range of strains (i.e. nearly 30%) to balance desired degrees of compressive and tensile deformability. For example, a prestrain of 3.5% (the maximum that we examined) yields a range of −24% to 5.5%. We note that such calculations assume that the applied strain equals the ribbons strain, even at extreme levels of deformation. Experimentally, we find that these estimates are often exceeded due to the ability of the PDMS beyond the ends of the ribbons and between the ribbons to accommodate strains so that the applied strain is not completely transferred to the ribbons.

(79) We have created functional, stretchable devices by including at the beginning of the fabrication sequence (FIG. 10, top frame) additional steps to define patterns of dopants in the silicon, thin metal contacts and dielectric layers using conventional processing techniques. Two and three terminal devices, diodes and transistors respectively, fabricated in this manner provide basic building blocks for circuits with advanced functionality. A dual transfer process in which the integrated ribbon devices are first lifted off of the SOI onto an undeformed PDMS slab, and then to a prestrained PDMS substrate created wavy devices with metal contacts exposed for probing. FIGS. 17A and 17B show optical images and electrical responses of a stretchable pn-junction diode for various levels of strain applied to the PDMS. We observe no systematic variation in the electrical properties of the devices with stretching or compressing, to within the scatter of the data. The deviation in the curves is due mainly to variations in the quality of probe contacts. These pn-junction diodes can be used as a photodetector (at reverse-biased state) or as photovoltaic devices, in addition to normal rectifying devices. The photocurrent density is ˜35 mA/cm.sup.2 at a reverse bias voltage of ˜−1V. At forward bias, the short-circuit current density and open-circuit voltage are ˜17 mA/cm.sup.2 and 0.2V, respectively, which yields a fill factor of 0.3. The shape of the response is consistent with modeling (solid curves in FIG. 17B). The device properties do not change significantly, even after ˜100 cycles of compressing, stretching and releasing (FIG. 19). FIG. 17C shows current-voltage characteristics of a stretchable, wavy silicon Schottky barrier metal oxide semiconductor field effect transistor (MOSFET) formed with procedures similar to those used for the pn diode, and with an integrated thin layer (40 nm) of thermal SiO.sub.2 as a gate dielectric (33). The device parameters extracted from electrical measurements of this wavy transistor—linear regime mobility ˜100 cm.sup.2/Vs (likely contact limited), threshold voltage ˜−3 V—are comparable those of devices formed on the SOI wafers using the same processing conditions. (FIGS. 20 and 21). As with the pn diodes, these wavy transistors can be reversibly stretched and compressed to large levels of strain without damaging the devices or significantly altering the electrical properties. In both the diodes and the transistors, deformations in the PDMS beyond the ends of the devices lead to device (ribbon) strains that are smaller than the applied strains. The overall stretchability results from the combined effects of device stretchability and these types of PDMS deformations. At compressive strains larger than those examined here, the PDMS tended to bend in a manner that made probing difficult. At larger tensile strains, the ribbons either fractured, or slipped and remained intact, depending on the silicon thickness, the ribbon lengths and the strength of bonding between the silicon and PDMS.

(80) These stretchable silicon MOSFETs and pn diodes represent only two of the many classes of ‘wavy’ electronic devices that can be formed. Completed circuit sheets or thin silicon plates can also be structured into uniaxial or biaxial stretchable wavy geometries. Besides the unique mechanical characteristics of wavy devices, the coupling of strain to electronic properties, which occurs in many semiconductors, provides opportunities to design device structures that exploit mechanically tunable, periodic variations in strain to achieve unusual electronic responses.

Materials and Methods

(81) Sample Preparation:

(82) The silicon-on-insulator (SOI) wafers consist of Si (thicknesses of 20, 50, 100, 205, 290 or 320 nm) on SiO.sub.2 (thicknesses of 145 nm, 145 nm, 200 nm, 400 nm, 400 nm or 1 μm) on Si substrates (Soitec, Inc.). In one case, we use an SOI wafer of Si (thickness of ˜2.5 μm) and SiO.sub.2 (thickness of ˜1.5 μm) on Si (Shin-Etsu). In all cases, the top Si layer has a resistivity between 5˜20 Ωcm, doped with boron (p-type) or phosphorous (n-type). The top Si of these SOI wafers is patterned with photolithoresist (AZ 5214 photoresist, Karl Suss MJB-3 contact mask aligner) and reactive ion etched (RIE) to define the Si ribbons (5˜50 μm wide, 15 mm long) (PlasmaTherm RIE, SF6 40 sccm, 50 mTorr, 100 W). The SiO.sub.2 layer is removed by undercut etching in HF (49%); the etching time is mainly dependent on the width of the Si ribbons. The lateral etch rate is typically 2˜3 μm/min. Slabs of poly(dimethylsiloxane) (PDMS) elastomer (Sylgard 184, Dow Corning) are prepared by mixing base and curing agent in a 10:1 weight ratio and curing at 70° C. for >2 hrs or at room temperature for >12 hrs.

(83) These flat slabs of PDMS (thicknesses of 1˜3 mm) are brought into conformal contact with the Si on the etched SOI wafer to generate the wavy structures. Any method that creates controlled expansion of the PDMS prior to this contact followed by contraction after removal from the wafer, can be used. We examine three different techniques. In the first, mechanical rolling of PDMS after contacting the SOI substrate created the prestrains. Although the wavy structures could be made in this manner, they tended to have non-uniform wave periods and amplitudes. In the second, heating the PDMS (coefficient of thermal expansion=3.1*10.sup.−4 K.sup.−1) to temperatures of between 30° C. and 180° C. before contact and then cooling it after removal from the SOI, generated wavy Si structures with excellent uniformity over large areas, in a highly reproducible fashion. With this method, we control the prestrain level in PDMS very accurately by changing the temperature (FIG. 12). The third method uses PDMS stretched with mechanical stages before contact with the SOI and then physically released after removal. Like the thermal approach, this method enabled good uniformity and reproducibility, but is more difficult to finely tune the pre-strain level, compared to the thermal method.

(84) For the devices such as pn junction diodes and transistors, electron beam evaporated (Temescal BJD1800) and photolithographically patterned (through etching or liftoff) metal layers (Al, Cr, Au) serve as contacts and gate electrodes. Spin-on-dopants (SOD) (B-75X, Honeywell, USA for p-type; P509, Filmtronics, USA for n-type) are used to dope the silicon ribbons. The SOD materials are first spin-coated (4000 rpm, 20 s) onto pre-patterned SOI wafers. A silicon dioxide layer (300 nm) prepared by plasma-enhanced chemical vapor deposition (PECVD) (PlasmaTherm) is used as a mask for the SOD. After heating at 950° C. for 10 sec, both the SOD and masking layer on SOI wafer are etched away using 6:1 buffered oxide etchant (BOE). For the transistor devices, thermally grown (1100° C., 10˜20 min. dry oxidation with high purity oxygen flow in furnace to thicknesses between 25 nm and 45 nm) silicon dioxide provide the gate dielectric. After completing all device processing steps on the SOI substrate, the Si ribbons (typically 50 μm wide, 15 mm long) with integrated device structures are covered by photoresist (AZ5214 or Shipley S1818) to protect the device layer during HF etching of the underlying SiO.sub.2. After removing the photoresist layer by oxygen plasma, a flat PDMS (70° C., >4 hrs) slab without any prestrain is used to remove the ribbon devices from the SOI substrate, in a flat geometry. A slab of partially cured PDMS (>12 hrs at room temperature after mixing the base and curing agent) is then contacted to the Si ribbon devices on the fully cured PDMS slab. Completing the curing of the partially cured PDMS (by heating at 70° C.), followed by removal of this slab, transfer the devices from the first PDMS slab to this new PDMS substrate. The shrinkage associated with cooling down to room temperature creates a prestrain such that removal and release creates the wavy devices, with electrodes exposed for probing.

(85) Measurements:

(86) Atomic force microscopy (AFM) (DI-3100, Veeco) is used to measure the wave properties (wavelength, amplitude) precisely. From the acquired images, the sectional profiles along the wavy Si are measured and analyzed statistically. A home built stretching stage was used, together with the AFM and a semiconductor parameter analyzer (Agilent, 5155C) to measure the mechanical and electrical responses of wavy Si/PDMS. Raman measurements are performed with Jobin Yvon HR 800 spectrometric analyzer using the 632.8 nm light from a He—Ne laser. The Raman spectrum is measured at 1 μm intervals along the wavy Si, with a focus adjusted at each position along the lengths of the structures to maximize the signal. The measured spectrum is fitted by Lorentzian functions to locate the peak wavenumber. Due to the slight dependence of the peak wavenumbers on the focal position of the microscope, the Raman results only provide qualitative insights into the stress distributions.

(87) Calculation of Contour Length, Ribbon Strain and Silicon Strain:

(88) The experimental results show that, for the range of materials and geometries explored here, the shape of wavy Si can be accurately represented with simple sine functions, i.e., y=A sin(kx) (k=2π/λ). The contour length is then calculated as L=∫.sub.0.sup.λ√{square root over (1+y′.sup.2)}dx. The ribbon strain of wavy Si is calculated using

(89) $\mathrm{.Math.}=\frac{.Math.\lambda -L.Math.}{\lambda }.$
The peak silicon strains occur at peaks and troughs of the waves, and are calculated using

(90) ${\mathrm{.Math.}}_{\mathrm{Si}}^{\mathrm{peak}}=\frac{h}{2R_c},$
where h is the Si thickness, and R.sub.c is the radius of curvature at peak or trough, which is given by

(91) $R_c=\frac{1}{y^{″}}{.Math.}_{x=\pm \left[\frac{\left(2n-1\right)\pi }{2k}\right]}$
where n is an integer and y″ is the second derivative of y with respect to x. Using the sine function approximation to the actual shape, the silicon peak strain is given by

(92) 0${\mathrm{.Math.}}_{\mathrm{Si}}^{\mathrm{peak}}=\frac{2{\pi }^2\mathrm{Ah}}{{\lambda }^2}.$
FIG. 12 shows the wavelength as a function of temperature used to create the prestrain. As shown by FIG. 13, the peak strain is independent of Si thickness h due to the linear dependence of the wave amplitude and wavelength on thickness (A˜h, λ˜h). FIG. 15 shows that wavy structures involve nearly equal upward and downward displacements relative to the level of the PDMS surface between the ribbons. The silicon ribbon strain is equal to the applied strain for the systems examined here (FIG. 16).

(93) An accordion bellows model: When the silicon can separate from the PDMS in compression, the system is governed by accordion bellows mechanics, rather than by buckling mechanics. In the bellows case, the wavelength for compressive applied strains (ε.sub.applied) is λ.sub.0(1+ε.sub.applied) where λ is the wavelength in the unstrained configuration, as described by Eq. (2). Since the contour length of the silicon ribbon is approximately the same, prior to and after compressive strain, we can use the following relation to determine the wave amplitude A.

(94) ${\int }_0^{{\lambda }_0}\sqrt{1+{\left[{\left(A_0\sin \frac{2\pi }{{\lambda }_0}x\right)}^{\prime }\right]}^2}\mathrm{dx}={\int }_0^{{\lambda }_0\left(1+{\mathrm{.Math.}}_{\mathrm{applied}}\right)}\sqrt{1+{\left[{\left(A\sin \frac{2\pi }{{\lambda }_0\left(1+{\mathrm{.Math.}}_{\mathrm{applied}}\right)}x\right)}^{\prime }\right]}^2}\mathrm{dx}$
This equation has the asymptotic solution

(95) $A=\sqrt{1+{\mathrm{.Math.}}_{\mathrm{applied}}}\sqrt{A_0^2-h^2\frac{{\mathrm{.Math.}}_{\mathrm{applied}}}{{\mathrm{.Math.}}_c}}\mathrm{for}\frac{A_0}{{\lambda }_0}1.$
At small compressive strain, this equation reduces to Eq. (3), which applies also to the case where separation of the Si from the PDMS is not possible and the system follows buckling mechanics. The peak silicon strain is given by

(96) ${\mathrm{.Math.}}_{\mathrm{Si}}^{\mathrm{peak}}=\frac{2{\mathrm{.Math.}}_c}{{\left(1+{\mathrm{.Math.}}_{\mathrm{applied}}\right)}^{3/2}}\sqrt{\frac{{\mathrm{.Math.}}_{\mathrm{pre}}-{\mathrm{.Math.}}_{\mathrm{applied}}}{{\mathrm{.Math.}}_c}-1}.$
For modest compressive strains, this expression is approximately the same as Eq. (4). The peak silicon strains, like the wave amplitudes, have similar functional forms for both the bellows and the buckling models, in the limit of modest applied strains. FIG. 18 shows the peak strain computed according to the expression above, and according to Eq. (4).

(97) Device Characterization:

(98) A semiconductor parameter analyzer (Agilent, 5155C) and a conventional probing station are used for the electrical characterization of the wavy pn junction diodes and transistors. The light response of pn-diode is measured under an illumination intensity of ˜1 W/cm.sup.2, as measured by an optical power meter (Ophir Optronics, Inc., Laser Power Meter AN/2). We use mechanical stages to measure the devices during and after stretching and compressing. As a means to explore the reversibility of the process, we measure three different pn diodes before and after ˜100 cycles of compressing (to ˜5% strains), stretching (to ˜15% strains) and releasing, in ambient light. FIG. 19 shows the results. FIGS. 20 and 21 show images, schematic illustrations and device measurements from the wavy transistors.

Example 2: Buckled and Wavy Ribbons of GaAs for High-Performance Electronics on Elastomeric Substrates

(99) Single crystalline GaAs ribbons with thicknesses in the submicron range and well-defined, ‘wavy’ and/or ‘buckled’ geometries are fabricated. The resulting structures, on the surface of or embedded in an elastomeric substrate, exhibit reversible stretchability and compressibility to strains >10%, more than ten times larger than that of GaAs itself. By integrating ohmic and Schottky contacts on these structured GaAs ribbons, high-performance stretchable electronic devices (e.g., metal-semiconductor field-effect transistors) can be achieved. This kind of electronic system can be used alone or in combination with similarly designed silicon, dielectric and/or metal materials to form circuits for applications that demand high frequency operation together with stretchability, extreme flexibility or ability to conform to surfaces with complex curvilinear shapes.

(100) Performance capabilities in traditional microelectronics are measured mainly in terms of speed, power efficiency and level of integration. Progress in other, more recent forms of electronics, is driven instead by the ability to achieve integration on unconventional substrates (e.g., low-cost plastics, foils, paper) or to cover large-areas. For example, new forms of X-ray medical diagnosis are achievable with large-area imagers that conformally wrap around the body to digitally image desired tissues. Lightweight, wall-size displays or sensors that can be deployed onto a variety of surfaces and surface shapes provide new technologies for architectural design. Various materials including small organic molecules, polymers, amorphous silicon, polycrystalline silicon,.sup.]single crystalline silicon nanowires and microstructured ribbons have been explored to serve as semiconductor channels for the type of thin film electronics that might support these and other applications. These materials enable transistors with mobilities that span a wide range (i.e., from 10.sup.−5 to 500 cm.sup.2/V.Math.s), and in mechanically bendable thin film formats on flexible substrates. Applications with demanding high speed operations, such as large-aperture interferometric synthetic aperture radar (InSAR) and radio frequency (RF) surveillance systems, require semiconductors with much higher mobilities, such as GaAs, or InP, etc. The fragility of single crystalline compound semiconductors creates a number of fabrication challenges that must be overcome in order to fabricate high-speed, flexible transistors with them. We establish a practical approach to build metal-semiconductor field-effect transistors (MESFETs) on plastic substrates by using printed GaAs wire arrays created from high-quality bulk wafers. These devices exhibit excellent mechanical flexibility and f.sub.T's that approach 2 GHz, even in moderately scaled devices (e.g. micron gate lengths). This example demonstrates GaAs ribbon based MESFETs (as opposed to wire devices) designed with special geometries that provide not only bendability, but mechanical stretchability to levels of strain (˜10%) that significantly exceed the intrinsic yield points of the GaAs itself (˜2%). The resulting type of stretchable high performance electronic systems can provide extremely high levels of bendability and the capacity to integrate conformally with curvilinear surfaces. This GaAs system example extends our described ‘wavy’ silicon in four important ways: (i) it demonstrates stretchability in GaAs, a material that is, in practical terms, much more mechanically fragile than Si, (ii) it introduces a new ‘buckled’ geometry that can be used for stretchability together with or independently of the previously described ‘wavy’ configuration, (iii) it achieves a new class of stretchable device (i.e. the MESFET), and (iv) it demonstrates stretching over a larger range and with greater symmetry in compression/tension than that achieved in silicon.

(101) FIG. 22 illustrates steps for fabricating stretchable GaAs ribbons on an elastomeric substrate made of poly(dimethylsiloxane) (PDMS). The ribbons are generated from a high-quality bulk wafer of GaAs with multiple epitaxial layers. The wafer is prepared by growing a 200-nm thick AlAs layer on a (100) semi-insulating GaAs (SI-GaAs) wafer, followed by sequential deposition of a SI-GaAs layer with thickness of 150 nm and Si-doped n-type GaAs layer with thickness of 120 nm and carrier concentration of 4×10.sup.17 cm.sup.−3. A pattern of photoresist lines defined parallel to the (0 | |) crystalline orientation serves as masks for chemical etching of the epilayers (including both GaAs and AlAs). Anisotropic etching with an aqueous etchant of H.sub.3PO.sub.4 and H.sub.2O.sub.2 isolated these top layers into individual bars with lengths and orientations defined by the photoresist, and with side walls that form acute angles relative to the wafer surface. Removing the photoresist after the anisotropic etching and then soaking the wafer in an ethanol solution of HF (2:1 in volume between ethanol and 49% aqueous HF) removes the AlAs layer and released ribbons of GaAs (n-GaAs/SI-GaAs). The use of ethanol, instead of water, for this step reduces cracking that can occur in the fragile ribbons due to the action of capillary forces during drying. The lower surface tension of ethanol compared to water also minimizes drying-induced disorder in the spatial layout of the GaAs ribbons. In the next step, the wafer with released GaAs ribbons is contacted to the surface of a prestretched flat slab of PDMS, with the ribbons aligned along the direction of stretching. In this case, van der Waals forces dominate the interaction between the PDMS and the GaAs. For cases that require stronger interaction strength, we deposit a thin layer of SiO.sub.2 onto the GaAs, and expose the PDMS to ultraviolet induced ozone (i.e., product of oxygen in air) immediately prior to contact. The ozone creates —Si—OH groups on the surface of PDMS, which react with the surface of the SiO.sub.2 upon contact to form bridging siloxane −Si—O—Si— bonds. The deposited SiO.sub.2 is discontinuous at the edges of each ribbon because of the geometry of their sidewalls. For both the weak and strong bonding procedures, peeling the PDMS from the mother wafer transfers all the ribbons to the surface of the PDMS. Relaxing the prestrain in the PDMS led to the spontaneous formation of large scale buckles and/or sinusoidal wavy structures along the ribbons. The geometry of the ribbons depends strongly on the prestrain (defined by ΔL/L) applied to the stamp, the interaction between the PDMS and ribbons, and the flexural rigidity of the ribbons. For the ribbons investigated here, small prestrains (<2%) create highly sinusoidal ‘waves’ with relatively small wavelengths and amplitudes (FIG. 22 right, middle frame), for both the strong and weak interaction cases. These geometries in GaAs are similar to those reported for Si. Higher prestrains (e.g., up to ˜15%) can be applied to create similar type of waves where there is a strong bonding interaction between the ribbons and the substrate. A different type of geometry, comprising of aperiodic ‘buckles’ with relatively large amplitudes and widths, formed in the case of weak interaction strengths and large prestrains (FIG. 22 right, top frame). In addition, our results show that both kinds of structures—buckles and waves—can coexist in a single ribbon whose flexural rigidity varies along its length (due, for example, to thickness variations associated with device structures).

(102) FIG. 23 shows several micrographs of wavy GaAs ribbons with thickness of 270 nm (including both n-GaAs and SI-GaAs layers) and widths of 100 μm (all of the ribbons discussed in this example have widths of 100 μm) formed with strong bonding between PDMS (thickness of ˜5 mm) and ribbons. The fabrication followed the procedures for strong bonding, using 2-nm Ti and 28-nm SiO.sub.2 layers on the GaAs. A biaxial prestrain of ˜1.9% (calculated from the thermal response of PDMS) is created in the PDMS by thermal expansion (heating to 90° C. in an oven) immediately prior to and during bonding. This heating also accelerates the formation of interfacial siloxane bonds. Cooling the PDMS to room temperature (˜27° C.) after transferring the GaAs ribbons released the prestrain. Frames A, B and C of FIG. 23 show images collected with an optical microscope, scanning electron microscope (SEM) and atomic force microscope (AFM), respectively, from the same sample. The images show the formation of periodic, wavy structures in the GaAs ribbons. The waves are quantitatively analyzed by evaluating linecuts (FIGS. 23E and 23F) from AFM image (FIG. 23D). The contour parallel to the longitudinal direction of the ribbon clearly shows a periodic, wavy profile, consistent with a computed fit to a sine wave (dashed line of FIG. 23E). This result agrees with nonlinear analysis of the initial buckled geometry in a uniform, thin, high-modulus layer on a semi-infinite low-modulus support. The peak-to-peak amplitude and wavelength associated with this function are determined to be 2.56 and 35.0 μm, respectively. The strains computed from the ratio of the horizontal distances between the adjacent two peaks on the stamp (i.e., the wavelength) to their actual contour lengths between the peaks (i.e., the surface distances measured by AFM), which we refer to as ribbon strains, yield values (i.e., 1.3%) that are smaller than the prestrain in the PDMS. This difference might be attributed to low shear modulus of PDMS and island effects related to the length of GaAs ribbons being shorter than the length of PDMS substrate. The surface strain of GaAs ribbons at peaks and troughs, which we refer to as maximum GaAs strains, is estimated from the ribbon thicknesses and radii of curvature at the peaks or troughs of the waves according to κh/2, where κ is the curvature. In this evaluation, the direct contribution of strains in the PDMS stamp to the GaAs are ignored because the PDMS can be treated as a semi-infinite support whose modulus is low compared to that of GaAs (Young's Modulus of GaAs: 85.5 GPa versus that of PDMS: 2 MPa). For the data of FIG. 23E, the maximum GaAs strains are ˜0.62%, which is more than a factor of two smaller than the ribbon strain (i.e., 1.3%). This mechanical advantage provides stretchability in the GaAs ribbons, with physics similar to that found for wavy Si.

(103) As shown in FIG. 23F, the peak and trough regions of the ribbons are higher and lower than the contour level (right portion of the green curve) of the surface of pristine PDMS (i.e., the areas without ribbons), respectively. The result suggests that the PDMS under the GaAs adopts a wavy profile as a result of the upward and downward forces imparted to the PDMS by the GaAs ribbons in the peaks and troughs, respectively. The precise geometry of the PDMS near the peaks of the waves is difficult to evaluate directly. We suspect that in addition to the upward deformation there is also a lateral necking caused by the Poisson effect. The wavy ribbons on the PDMS stamp can be stretched and compressed by applying strains to the PDMS (so-called applied strain denoted as positive for stretching and negative for compression, respectively). The insets of FIGS. 23A and 23B show images of the ribbons that deformed to their original, flat geometry when a relatively small stretching strain (i.e., ˜1.5%) was applied. Further stretching transfers more tensile strain to the flat GaAs ribbons, resulting in their breakage of ribbons when this excess strain reaches the failure strain of GaAs. Compressive strains applied to the substrate reduce the wavelengths and increase the amplitudes of the wavy ribbons. Failure in compression occurs when the bending strains at peaks (and troughs) exceed the failure strains. This variation of wavelength with strain is consistent with previous observations in silicon, and is different than the predictions of wavelength invariance derived from ideal models.

(104) The stretchability of wavy GaAs ribbons can be improved by increasing the prestrain applied to PDMS through the use of a mechanical stage (as opposed to thermal expansion). For example, transferring GaAs ribbons with SiO.sub.2 layers onto the surface of a PDMS stamp with prestrain of 7.8% generated wavy ribbons without any observable cracking in the GaAs (FIG. 24A). The bending strains at the peaks in this case are estimated to be ˜1.2%, which is lower than the failure strain (i.e., ˜2%) of GaAs. Similar to the low prestrain case, the wavy ribbons behave like an accordion when the system is stretched and compressed: the wavelengths and amplitudes change to accommodate the applied strain. As shown in FIG. 24A, the wavelengths increase with tensile strain until the ribbons become flat and decrease with compressive strain until the ribbons break. These deformations are completely reversible, and do not involve any measurable slipping of the GaAs on the PDMS. The wavelength changes linearly with applied strains in both compression and tension (see, the black lines and symbols in FIG. 24B), in contrast to the mildly asymmetric behavior observed in Si ribbons with weak bonding and much lower prestrains.

(105) In practical applications, it might be useful to encapsulate the GaAs ribbons and devices in a way that maintains their stretchability. As a simple demonstration of one possibility, we cast and cured PDMS pre-polymers on samples such as the one shown in FIG. 24A to embed the ribbons in PDMS. The embedded systems exhibit similar mechanical behavior to the unembedded ones, i.e., stretching the system increases wavelength and compressing the system decreases wavelength (the red lines and symbols in FIG. 24B). Shrinkage due to curing the second layer of PDMS generated some moderate amount of additional strain (˜1%). This strain resulted in a slight decrease in the wavelength of the wavy ribbons, thereby expanding slightly the range of stretchability. FIG. 24B shows the difference. Overall, the systems generated with prestrain of ˜7.8% can be stretched or compressed to strains of up to ˜10% without inducing any observable breakage in the GaAs.

(106) The wavy GaAs ribbons on PDMS substrates can be used to fabricate high-performance electronic devices, such as MESFETs, the electrodes of which are formed through metallization and processing on the wafer, before transfer to PDMS. These metal layers change the flexural rigidity of the ribbons in a spatially dependent manner. FIG. 25A shows GaAs ribbons integrated with ohmic stripes (source and drain electrodes) and Schottky contacts (gate electrodes) after transfer to a PDMS substrate with prestrain of ˜1.9%. The ohmic contacts consisted of metal stacks including AuGe (70 nm)/Ni (10 nm)/Au (70 nm) formed on the original wafers through lithographically defined masks along with sequential annealing of the wafers at elevated temperature (i.e., 450° C. for 1 min) in a quartz tube with flowing N.sub.2. These ohmic segments have lengths of 500 μm. The distances between two adjacent ohmic contacts are 500 μm (i.e., channel length). Schottky contacts with lengths of 240 μm (i.e., gate length) are generated by directly depositing 75-nm Cr layer and 75-nm Au layer through electron-beam evaporation against the photolithographically designed mask. The electrodes have widths equal to the GaAs ribbons, i.e., 100 μm; their relatively large sizes facilitate probing. The dimensions of electrodes and semiconductor channels can be significantly decreased to achieve enhanced device performance. As shown in FIG. 25A, these stretchable GaAs MESFETS exhibit, shortrange, periodic waves only in the regions without electrodes. The absence of waves in the thicker regions might be attributed to their enhanced flexural rigidity due mainly to the additional thickness associated with the metals. Periodic waves can be initiated in the thicker regions by using prestrains larger than ˜3%. In these cases, however, the ribbons tend to break at the edges of the metal electrodes due to critical flaws and/or high peak strains near these edges. This failure mode limits the stretchability.

(107) To circumvent this limitation, we reduced the strength of interaction between the MESFETs and the PDMS by eliminating the siloxane bonding. For such samples, prestrain >3% generated large, aperiodic buckles with relatively large widths and amplitudes due to physical detachment of the ribbons from PDMS surface. FIG. 25B presents this type of system, as prepared with a prestrain of ˜7%, in which the big buckles form in the thinner regions of the devices. The detachment seems to extend slightly to the thicker sections with ohmic stripes, as indicated by the vertical lines. The contrast variation along ribbons is attributed to reflections and refraction associated passage of light through the curved GaAs segments. The SEM image (FIG. 25C) clearly shows the formation of arc-shape buckles and flat, unperturbed PDMS. These buckles display asymmetric profiles (as indicated by the red curves) with tails to the sides with ohmic contacts. This asymmetry might be attributed to the unequal lengths (500 μm versus 240 μm) of ohmic stripes and Schottky contacts for individual transistors. This kind of buckled MESFET can be stretched to its original flat status (FIG. 25D) with applied stretching strains between ˜6% and ˜7%. However, compressing the system shown in FIG. 25B leads to continuous detachment of ribbons from PDMS surface to form larger buckles because of weak bonding. Embedding such devices in PDMS, according to the previously described procedures, eliminates this type of uncontrolled behavior. FIG. 25B shows such a system, in which the liquid PDMS precursor fills the gaps underneath the buckles. The fully surrounding PDMS confines the ribbons and prevents them from sliding and detaching. The embedded devices can be reversibly stretched and compressed to strains up to ˜6% without breaking the ribbons. It is notable that when the embedded system was compressed by ˜5.83% (top frame of FIG. 25E), periodic, small waves formed in the regions with metal electrodes as well as new ripples in the buckled regions. The formation of these new small waves, in combination with the large buckles, enhances the compressibility. Stretching the system forces the buckled regions to compress and stretch the PDMS in a manner that enables some flattening of these buckles, thereby elongating the projected lengths of the ribbons (bottom frame of FIG. 25E). These results suggest that embedded devices with big buckles, which is a geometry distinct from the waves, represent a promising method to achieve stretchability and compressibility that can be used in combination with or separately from the wavy approach.

(108) The performance of buckled devices can be evaluated by directly probing the current flow from source to drain. FIG. 26A shows GaAs-ribbon devices fabricated on a wafer, picked up using a flat PDMS stamp and transfer printed onto a PDMS substrate with a prestrain of 4.7%. In this configuration, the metal electrodes are exposed to air for electrical probing. After the prestretched PDMS is relaxed to a strain of 3.4%, periodic small waves formed in the thin regions of the MESFET (FIG. 26A: second frame from the top). When the prestretched PDMS stamp is fully relaxed, the small waves in each segment of pure GaAs coalesced into an individual big buckle (FIG. 26A: third frame from the top). The buckled devices can be stretched to their flat status with applied stretching strain of 4.7% (FIG. 26A: bottom frame). The IV curves of the same device with applied strains of 0.0% (FIG. 26A: third frame from the top) and 4.7% (FIG. 26A: bottom frame) are plotted in FIG. 26B with red and black colors, respectively. The results indicate that the current flow from source to drain of buckled MESFETs on PDMS substrates can be well modulated with the voltages applied to the gate and that the applied stretching strain generates only a minor effect on device performance.

(109) In summary, this example discloses an approach to form ‘buckled’ and ‘wavy’ GaAs ribbons on and embedded in PDMS elastomeric substrates. The geometrical configurations of these ribbons depend on the level of prestrain used in the fabrication, the interaction strength between the PDMS and ribbons, and on the thicknesses and types of materials used. Buckled and wavy ribbons of GaAs multilayer stacks and fully formed MESFET devices show large levels of compressibility/stretchability, due to the ability of their geometries to adjust in a manner that can accommodate applied strains without transferring those strains to the materials themselves. Successful realization of large levels of mechanical stretchability (and, as a result, other attractive mechanical characteristics such as extreme bendability) in an intrinsically fragile material like GaAs provides similar strategies applicable to a wide range of other materials classes.

(110) The thermally-induced prestrain is ascribed to thermal expansion of PDMS stamp, which has the bulk linear coefficient of thermal expansion of α.sub.L=3.1×10.sup.−4 μm/μm/° C. On the other hand, the coefficient of thermal expansion for GaAs is only 5.73×10.sup.−6 μm/μm/° C. Therefore, the prestrain on PDMS (relatively GaAs ribbons) for the sample that was prepared at 90° C. and cooled down to 27° C. is determined according to Δα.sub.L×ΔT=(3.1×10.sup.−4−5.73×10.sup.−6)×(90−27)=1.9%.

(111) Methods: GaAs wafers with customer-designed epitaxial layers are purchased from IQE Inc., Bethlehem, Pa. The lithographic processes employed AZ photoresist, i.e., AZ 5214 and AZ nLOF 2020 for positive and negative imaging, respectively. The GaAs wafers with photoresist mask patterns are anisotropically etched in the etchant (4 mL H.sub.3PO.sub.4 (85 wt %), 52 mL H.sub.2O.sub.2 (30 wt %), and 48 mL deionized water) that is cooled in the ice-water bath. The AlAs layers are dissolved with a diluted HF solution (Fisher® Chemicals) in ethanol (1:2 in volume). The samples with released ribbons on mother wafers are dried in a fume hood. The dried samples are placed in the chamber of electron-beam evaporator (Temescal FC-1800) and coated with sequential layers of 2-nm Ti and 28-nm SiO.sub.2. The metals for the MESFET devices are deposited by electron-beam evaporation before removal of AlAs layers. PDMS stamp with thickness of ˜5 mm is prepared by pouring the mixture of low-modulus PDMS (A:B=1:10, Sylgard 184, Dow Corning) onto a piece of silicon wafer pre-modified with monolayer of (tridecafluoro-1,1,2,2-tetrahydrooctyl)-1-trichlorosilane, followed by baking at 65° C. for 4 hrs. In order to generate strong bonding, the stamps are exposed to UV light for 5 minutes. In the transfer process, the stamps are stretched through thermal expansion (in oven) and/or mechanical forces. The wafers with released ribbons are then laminated on the surfaces of the stretched PDMS stamps and left in contact at elevated temperatures (dependent on the required prestrains) for 5 minutes. The mother wafers are peeled from the stamps and all the ribbons are transferred to stamps. The prestrains applied to the stamps are released through cooling down to room temperature and/or removing the mechanical forces, resulting in the formation of wavy profiles along the ribbons. In the mechanical evaluations, we use a specially designed stage to stretch as well as compress the PDMS stamps with ‘wavy’ and ‘buckled’ GaAs ribbons.

Example 3: Two-Dimensional Stretchable Semiconductors

(112) The present invention provides stretchable semiconductors and stretchable electronic devices capable of stretching, compressing and/or flexing in more than one direction, including directions oriented orthogonal to each other. Stretchable semiconductors and stretchable electronic devices of this aspect of the present invention exhibit good mechanical and electronic properties and/or device performance when stretched and/or compressed in more than one direction.

(113) FIGS. 27A-C provides images at different degrees of magnification of a stretchable silicon semiconductor of the present invention exhibiting stretchability in two dimensions. The stretchable semiconductor shown in FIG. 27A-B was prepared by pre-straining an elastic substrate via thermal expansion.

(114) FIGS. 28A-C provide images of three different structural conformations of stretchable semiconductors of the present invention exhibiting stretchability in two dimensions. As shown, the semiconductor structures in FIG. 28A exhibit an edge line wavy conformation, the semiconductor structures in FIG. 28B exhibit a Herringbone wavy conformation and the semiconductor structures in FIG. 28C exhibit a random wavy conformation.

(115) FIGS. 29A-D provide images of stretchable semiconductors of the present invention fabricated via prestraining an elastic substrate via thermal expansion.

(116) FIG. 30 shows optical images of stretchable semiconductors exhibiting stretchability in two dimensions prepared via prestraining an elastic substrate via thermal expansion. FIG. 30 shows images corresponding to a variety of stretching and compression conditions.

(117) FIG. 31A shows an optical image of stretchable semiconductors exhibiting stretchability in two dimensions fabricated via prestraining an elastic substrate via thermal expansion. FIGS. 31B and 31 C provide experimental results relating to the mechanical properties of the stretchable semiconductors shown in FIG. 31A.

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STATEMENTS REGARDING INCORPORATION BY REFERENCE AND VARIATIONS

(119) The following references relate to self assembly techniques which may be used in methods of the present invention to transfer, assembly and interconnect printable semiconductor elements via contact printing and/or solution printing techniques and are incorporated by reference in their entireties herein: (1) “Guided molecular self-assembly: a review of recent efforts”, Jiyun C Huie Smart Mater. Struct. (2003) 12, 264-271; (2) “Large-Scale Hierarchical Organization of Nanowire Arrays for Integrated Nanosystems”, Whang, D.; Jin, S.; Wu, Y.; Lieber, C. M. Nano Lett. (2003) 3(9), 1255-1259; (3) “Directed Assembly of One-Dimensional Nanostructures into Functional Networks”, Yu Huang, Xiangfeng Duan, Qingqiao Wei, and Charles M. Lieber, Science (2001) 291, 630-633; and (4) “Electric-field assisted assembly and alignment of metallic nanowires”, Peter A. Smith et al., Appl. Phys. Lett. (2000) 77(9), 1399-1401.

(120) All references throughout this application, for example patent documents including issued or granted patents or equivalents; patent application publications; unpublished patent applications; and non-patent literature documents or other source material; are hereby incorporated by reference herein in their entireties, as though individually incorporated by reference, to the extent each reference is at least partially not inconsistent with the disclosure in this application (for example, a reference that is partially inconsistent is incorporated by reference except for the partially inconsistent portion of the reference).

(121) Any appendix or appendices hereto are incorporated by reference as part of the specification and/or drawings.

(122) Where the terms “comprise”, “comprises”, “comprised”, or “comprising” are used herein, they are to be interpreted as specifying the presence of the stated features, integers, steps, or components referred to, but not to preclude the presence or addition of one or more other feature, integer, step, component, or group thereof. Separate embodiments of the invention are also intended to be encompassed wherein the terms “comprising” or “comprise(s)” or “comprised” are optionally replaced with the terms, analogous in grammar, e.g.; “consisting/consist(s)” or “consisting essentially of/consist(s) essentially of” to thereby describe further embodiments that are not necessarily coextensive.

(123) The invention has been described with reference to various specific and preferred embodiments and techniques. However, it should be understood that many variations and modifications may be made while remaining within the spirit and scope of the invention. It will be apparent to one of ordinary skill in the art that compositions, methods, devices, device elements, materials, procedures and techniques other than those specifically described herein can be applied to the practice of the invention as broadly disclosed herein without resort to undue experimentation. All art-known functional equivalents of compositions, methods, devices, device elements, materials, procedures and techniques described herein are intended to be encompassed by this invention. Whenever a range is disclosed, all subranges and individual values are intended to be encompassed as if separately set forth. This invention is not to be limited by the embodiments disclosed, including any shown in the drawings or exemplified in the specification, which are given by way of example or illustration and not of limitation. The scope of the invention shall be limited only by the claims.