Si Micropillars Research Articles

In-Situ Microscale Compression Testing of Li-Si Micropillars

Silicon is considered as an ideal anode material for high-capacity Li-ion batteries (LIB) thanks to its high theoretical capacity, but the commercial viability is severely hampered by considerable volume expansion of Si upon lithiation that can lead to mechanical failure (e.g., cracks and delamination). Hence, understanding the mechanical behaviors of the Li-Si system is of high interest, but the experimental studies have been limited to thin film electrodes by means of wafer curvature measurements or nanoindentation. In this work, we for the first time present in-situ microscale compression testing of Li-Si micropillars and extract the key mechanical properties (e.g., Young’s modulus, yield strength, and fracture energy) from the resulting stress-strain curves. Prior to testing, Si micropillars of various sizes/shapes and oriented at different crystal directions were lithiated/delithiated at room temperature. We then performed the microscale compression testing on these lithiated Si micropillars using a flat-head indenter in environmental scanning electron microscope (ESEM) while obtaining live video images for data-image correlation. The results suggest that the measured elastic moduli of the Li-Si system agree with the values obtained from the standard nanoindentation experiments but with much smaller variance. The precise nature of these experiments allows us to observe the mechanical properties dependent on the crystal orientation of Si and lithiation rate. In addition, various failure behaviors of Li-Si under compressive stresses will be presented, which is difficult to be obtained through the existing experimental techniques.

Competitive Growth of Ge Quantum Dots on a Si Micropillar with Pits for a Precisely Site-Controlled QDs/Microdisk System.

Semiconductor quantum dots (QDs)/microdisks promise a unique system for comprehensive studies on cavity quantum electrodynamics and great potential for on-chip integrated light sources. Here, we report on a strategy for precisely site-controlled Ge QDs in SiGe microdisks via self-assembly growth of QDs on a micropillar with deterministic pits and subsequent etching. The competitive growth of QDs in pits and at the periphery of the micropillar is disclosed. By adjusting the growth temperature and Ge deposition, as well as the pit profiles, QDs can exclusively grow in pits that are exactly located at the field antinodes of the corresponding cavity mode of the microdisk. The inherent mechanism of the mandatory addressability of QDs is revealed in terms of growth kinetics based on the non-uniform surface chemical potential around the top of the micropillar with pits. Our results demonstrate a promising approach to scalable and deterministic QDs/microdisks with strong light-matter interaction desired for fundamental research and technological applications.

Electrochemical Redox Cycling Behavior of Gold Nanoring Electrodes Microfabricated on a Silicon Micropillar.

We report the microfabrication and characterization of concentric gold nanoring electrodes (Au NREs), which were fabricated by patterning two gold nanoelectrodes on the same silicon (Si) micropillar tip. Au NREs of 165 ± 10 nm in width were micropatterned on a 6.5 ± 0.2 µm diameter 80 ± 0.5 µm height Si micropillar with an intervening ~ 100 nm thick hafnium oxide insulating layer between the two nanoelectrodes. Excellent cylindricality of the micropillar with vertical sidewalls as well as a completely intact layer of a concentric Au NRE including the entire micropillar perimeter has been achieved as observed via scanning electron microscopy and energy dispersive spectroscopy data. The electrochemical behavior of the Au NREs was characterized by steady-state cyclic voltammetry and electrochemical impedance spectroscopy. The applicability of Au NREs to electrochemical sensing was demonstrated by redox cycling with the ferro/ferricyanide redox couple. The redox cycling amplified the currents by 1.63-fold with a collection efficiency of > 90% on a single collection cycle. The proposed micro-nanofabrication approach with further optimization studies shows great promise for the creation and expansion of concentric 3D NRE arrays with controllable width and nanometer spacing for electroanalytical research and applications such as single-cell analysis and advanced biological and neurochemical sensing.

Effects of excess electrons/holes on fracture toughness of single-crystal Si

This study demonstrates that bond strength can be enhanced by injecting excess electrons or holes into a material by electron beam irradiation. To determine the effect of excess electrons/holes on the interatomic bond strength, fracture toughness tests were performed on single-crystal Si micropillars under various electron-beam irradiation conditions. The fracture toughness under electron beam irradiation was 4%–11% higher than that under non-irradiated conditions. In particular, an increase in strength was large in tests performed under hole-injection conditions. Furthermore, in first-principles calculations of the tensile strength of excess electrons/hole-doped Si, the ideal tensile strength monotonically decreased with an injection in excess electrons and increased monotonically with the injection of holes. This is qualitatively consistent with the experimental result that the fracture toughness increases under hole-injection conditions.

Electric current-assisted manipulation of liquid metals using a stylus at micro-and nano-scales

A novel methodology, based on wetting and electromigration, for transporting liquid metal, over long distances, at micro-and nano-scale using a stylus is reported. The mechanism is analogous to a dropper that uses ‘suction and release’ actions to ‘collect and dispense’ liquid. In our methodology, a stylus coated with a thin metal film acts like the dropper that collects liquid metal from a reservoir upon application of an electric current, holds the liquid metal via wetting while carrying the liquid metal over large distances away from the reservoir and drops it on the target location by reversing the direction of electric current. Essentially, the working principle of the technique relies on the directionality of electromigration force and adhesive force due to wetting. The working of the technique is demonstrated by using an Au-coated Si micropillar as the stylus, liquid Ga as the liquid metal to be transported, and a Kleindiek-based position micro-manipulator to traverse the stylus from the liquid reservoir to the target location. For demonstrating the potential applications, the technique is utilized for closing a micro-gap by dispensing a minuscule amount of liquid Ga and conformally coating the desired segment of the patterned thin films with liquid Ga. This study confirms the promising potential of the developed technique for reversible, controlled manipulation of liquid metal at small length scales.

The Radon transform as a tool for 3D reciprocal-space mapping of epitaxial microcrystals.

This work presents a new approach suitable for mapping reciprocal space in three dimensions with standard laboratory equipment and a typical X-ray diffraction setup. The method is based on symmetric and coplanar high-resolution X-ray diffraction, ideally realized using 2D X-ray pixel detectors. The processing of experimental data exploits the Radon transform commonly used in medical and materials science. It is shown that this technique can also be used for diffraction mapping in reciprocal space even if a highly collimated beam is not available. The application of the method is demonstrated for various types of epitaxial microcrystals on Si substrates. These comprise partially fused SiGe microcrystals that are tens of micrometres high, multiple-quantum-well structures grown on SiGe microcrystals and pyramid-shaped GaAs/Ge microcrystals on top of Si micropillars.

Fabrication of the ZnO nanowires/TiO2 nanowires/Si micropillars structures for the gas sensor application

It is the first time that Zinc Oxide (ZnO) nanowires and Titanium Dioxide (TiO2) nanowires structures are fabricated on the Silicon (Si) micropillars surface to form ZnO nanowires/TiO2 nanowires/Si micropillars structure successfully. And this structure is applied to the gas sensor. The Si wafer with countless micropillars is used as substrates, which are fabricated by Cesium Chloride (CsCl) self-assembly lithography and dry etching. The TiO2 nanowires and ZnO nanowires are prepared by hydrothermal reaction separately. From the SEM images, 60 min ZnO nanowires growth and 3 h TiO2 nanowires growth on 600 nm average diameter and 3.2 µm height Si micropillars is the most suitable morphology for the gas sensor application. From the XRD curves, both the ZnO and TiO2 nanowires on the Si micropillars surface are good crystallinity. Compared with the ZnO nanowires/Si micropillars structure, the ZnO nanowires/TiO2 nanowires/Si micropillars can increase the response to ethanol no matter the concentration. For 500 ppm ethanol, the response from ZnO nanowires/TiO2 nanowires/Si micropillars is about 76%, which is much higher than the 44% of that from ZnO nanowires/Si micropillars.

Growth of thick [1 1 1]-oriented 3C-SiC films on T-shaped Si micropillars

In this paper we report the morphology and the microstructural properties of thick [1 1 1]-oriented 3C-SiC films epitaxially grown on T-shaped Si micropillars. This compliant substrate was designed to release the stress developed in 3C-SiC grown on Si, due to the lattice mismatch and the different thermal expansion coefficients between 3C-SiC and Si. In this way it was possible to have 3C-SiC films as thick as 10 and 16 μm, with small bowing and no cracks. Our study relies on the use of an Electron Microscopy approach and elucidates the structure of the crystallographic defects across the 3C-SiC film, such as stacking faults (SFs), nano-twins and grain boundaries (GBs). After examination of the morphological and structural characteristics of the Si micropillar array, we analyzed the crystallographic properties of the thin 3C-SiC deposit on the Si micropillar sidewalls, since it may have an impact on the upper film. To assess the crystal quality of the 3C-SiC film, we quantified the SF density at the 3C-SiC surface, even estimating semi-quantitatively the depletion of SFs at much larger thickness. Hence, we analyzed the regions where 3C-SiC microcrystals grown on neighboring Si micropillars coalesce and form a continuous layer. We found that the coalescence between adjacent 3C-SiC microcrystals produces twinned regions, which terminate with the formation of GBs. We noticed that SFs may annihilate at the GBs, leading to the SF reduction across the 3C-SiC film. Finally, we observed the closure of the GBs inside the 3C-SiC film, thus improving the crystal quality of its surface. This work endorses the use of Si micropillars as compliant substrate to grow thick [1 1 1]-oriented 3C-SiC films with good crystal quality and is preliminary to the exploitation of 3C-SiC for high performing microelectronic devices.

A numerical fitting routine for frequency-domain thermoreflectance measurements of nanoscale material systems having arbitrary geometries

In this work, we develop a numerical fitting routine to extract multiple thermal parameters using frequency-domain thermoreflectance (FDTR) for materials having non-standard, non-semi-infinite geometries. The numerical fitting routine is predicated on either a 2D or 3D finite element analysis that permits the inclusion of non-semi-infinite boundary conditions, which cannot be considered in the analytical solution to the heat diffusion equation in the frequency domain. We validate the fitting routine by comparing it with the analytical solution to the heat diffusion equation used within the wider literature for FDTR and known values of thermal conductivity for semi-infinite substrates (SiO2, Al2O3, and Si). We then demonstrate its capacity to extract the thermal properties of Si when etched into micropillars that have radii on the order of the pump beam. Experimental measurements of Si micropillars with circular and square cross sections are provided and fit using the numerical fitting routine established as part of this work. Likewise, we show that the analytical solution is unsuitable for the extraction of thermal properties when the geometry deviates significantly from the standard semi-infinite case. This work is critical for measuring the thermal properties of materials having arbitrary geometries, including ultra-drawn glass fibers and laser gain media.

Vapor-Based Synthesis of ZIF-8 Thin Films and Gas Adsorption Properties

On-site and real-time measurement of gas concentrations are crucial for both the understanding and the monitoring of industrial and environmental processes. In recent years, there is an increasing need to develop portable multi-gas analysis tools allowing in situ detection of complex gas mixtures mainly due to safety, process and environmental considerations. A promising approach is based on the integration of the different parts of the analytical system (i.e. pre-concentrator, gas chromatography column, gravimetric sensors) in a silicon die by using standard microelectronic technologies. Each of these devices need to be coated by an appropriate functional layer.Metal Organic Frameworks (MOF), hybrid microporous crystalline materials with tuneable properties, are attractive for this type of application regarding their high specific surface area and chemical stability. However, these materials are usually synthetized via solvothermal techniques, which complicates the growth of continuous thin films and their integration in micro-devices. Recently, vapor phase-based routes were reported for the synthesis of MOF thin films [1-2]. These solvent-free synthesis methods hold great economic and environmental perspectives since they can not only avoid the use of solvents and metallic salts, but also reduce contamination and impurity incorporated into the crystals during the synthesis step. Moreover, vapour-phase process are usually preferred to obtain conformal coatings in the high aspect ratio features of devices and they present a good compatibility with the Si technology. These breakthroughs have paved the way for the use of MOF in micro- and nanotechnologies but much still remains to be done to control the growth and understand the potential and the limits of these growth methods.In this work, pinhole-free and crystalline zeolitic imidazolate framework-8 (ZIF-8) layers were grown on different types of substrates and devices that can be used for gas analysis. Si wafers, Quartz Crystal Microbalance, Si micro-pillars arrays and 3D foam were envisaged. First, ZnO films were deposited by Atomic Layer Deposition (ALD). Then, the formation of ZIF-8 was realized through a vapor-solid reaction with a vaporized linker (2-methylimidazole). Continuous ZIF-8 films were obtained and the impact of the process parameters on the MOF growth and on the material properties was studied. Additionally, the composition, roughness and structure of the films were studied by FTIR, EDX, XPS, XRD and AFM experiments. Finally, the as-synthesized films were thermally activated and the porosity was assessed using ellipsometric-porosimetry. The adsorption properties of the films were also investigated using gravimetric gas sensors. Several gases were tested (methanol, toluene, formaldehyde) in order to determine the benefit of ZIF-8 thin films in sensors. It is demonstrated that the synthesis approach based on an ALD deposition followed by a vapor-solid treatment in appropriate conditions is compatible with different kinds of substrates, which allows the use of ZIF-8 as sensitive layer in devices for gas sensing.[1] I. Stassen et al., Nat. Mat., 15, 304–310 (2016)[2] E. Ahvenniemi et al., Chem. Com., 52, 1139-1142 (2016)

Heterogeneous optoelectronic characteristics of Si micropillar arrays fabricated by metal-assisted chemical etching

Recent progress achieved in metal-assisted chemical etching (MACE) has enabled the production of high-quality micropillar arrays for various optoelectronic applications. Si micropillars produced by MACE often show a porous Si/SiOx shell on crystalline pillar cores introduced by local electrochemical reactions. In this paper, we report the distinct optoelectronic characteristics of the porous Si/SiOx shell correlated to their chemical compositions. Local photoluminescent (PL) images obtained with an immersion oil objective lens in confocal microscopy show a red emission peak (≈ 650 nm) along the perimeter of the pillars that is threefold stronger compared to their center. On the basis of our analysis, we find an unexpected PL increase (≈ 540 nm) at the oil/shell interface. We suggest that both PL enhancements are mainly attributed to the porous structures, a similar behavior observed in previous MACE studies. Surface potential maps simultaneously recorded with topography reveal a significantly high surface potential on the sidewalls of MACE-synthesized pillars (+ 0.5 V), which is restored to the level of planar Si control (− 0.5 V) after removing SiOx in hydrofluoric acid. These distinct optoelectronic characteristics of the Si/SiOx shell can be beneficial for various sensor architectures.

Depth-dependent EBIC microscopy of radial-junction Si micropillar arrays

Recent advances in fabrication have enabled radial-junction architectures for cost-effective and high-performance optoelectronic devices. Unlike a planar PN junction, a radial-junction geometry maximizes the optical interaction in the three-dimensional (3D) structures, while effectively extracting the generated carriers via the conformal PN junction. In this paper, we report characterizations of radial PN junctions that consist of p-type Si micropillars created by deep reactive-ion etching (DRIE) and an n-type layer formed by phosphorus gas diffusion. We use electron-beam induced current (EBIC) microscopy to access the 3D junction profile from the sidewall of the pillars. Our EBIC images reveal uniform PN junctions conformally constructed on the 3D pillar array. Based on Monte-Carlo simulations and EBIC modeling, we estimate local carrier separation/collection efficiency that reflects the quality of the PN junction. We find the EBIC efficiency of the pillar array increases with the incident electron beam energy, consistent with the EBIC behaviors observed in a high-quality planar PN junction. The magnitude of the EBIC efficiency of our pillar array is about 70% at 10 kV, slightly lower than that of the planar device (≈ 81%). We suggest that this reduction could be attributed to the unpassivated pillar surface and the unintended recombination centers in the pillar cores introduced during the DRIE processes. Our results support that the depth-dependent EBIC approach is ideally suitable for evaluating PN junctions formed on micro/nanostructured semiconductors with various geometry.

From Geometry to Activity: A Quantitative Analysis of WO3/Si Micropillar Arrays for Photoelectrochemical Water Splitting

AbstractThe photoelectrochemical (PEC) activity of microstructured electrodes remains low despite the highly enlarged surface area and enhanced light harvesting. To obtain a deeper understanding of the effect of 3D geometry on the PEC performance, well‐defined WO3/n‐Si and WO3/pn‐Si micropillar arrays are fabricated and subjected to a quantitative analysis of the relationship between the geometry of the micropillars (length, pitch) and their PEC activity. For WO3/n‐Si micropillars, it is found that the photocurrent increases for WO3/n‐Si pillars, but not in proportion to the increase in surface area that results from increased pillar length or reduced pillar pitch. Optical simulations show that a reduced pillar pitch results in areas of low light intensity due to a shadowing effect. For WO3/pn‐Si micropillar photoelectrodes, the p–n junction enhances the photocurrent density up to a factor of 4 at low applied bias potential (0.8 V vs RHE) compared to the WO3/n‐Si. However, the enhancement in photocurrent density increases first and then decreases with reduced pillar pitch, which scales with the photovoltage generated by the p–n junction. This is related to an increased dead layer of the p–n junction Si surface, which results in a decreased photovoltage even though the total surface area increases.

Variable-Wavelength Quick Scanning Nanofocused X-Ray Microscopy for In Situ Strain and Tilt Mapping.

Compression of micropillars is followed in situ by a quick nanofocused X-ray scanning microscopy technique combined with 3D reciprocal space mapping. Compared to other attempts using X-ray nanobeams, it avoids any motion or vibration that would lead to a destruction of the sample. The technique consists of scanning both the energy of the incident nanofocused X-ray beam and the in-plane translations of the focusing optics along the X-ray beam. Here, the approach by imaging the strain and lattice orientation of Si micropillars and their pedestals during in situ compression is demonstrated. Varying the energy of the incident beam instead of rocking the sample and mapping the focusing optics instead of moving the sample supplies a vibration-free measurement of the reciprocal space maps without removal of the mechanical load. The maps of strain and lattice orientation are in good agreement with the ones recorded by ordinary rocking-curve scans. Variable-wavelength quick scanning X-ray microscopy opens the route for in situ strain and tilt mapping toward more diverse and complex materials environments, especially where sample manipulation is difficult.

Enhanced metal assisted etching method for high aspect ratio microstructures: Applications in silicon micropillar array solar cells

A solar cell device, fabricated on high density array cylindrical pillars, enables photogenerated carrier collection in the radial direction, thus shortening the path length of the carriers reaching the junction. It also provides advantages over conventional planar junction solar cells, such as reduced surface reflectance and enhanced light trapping. In this study, highly ordered Si micropillars were fabricated by photolithography and metal assisted etching (MAE) methods. It is shown that the use of ethanol as a solvent during the etching process and increasing HF concentration in the MAE solution both improve the quality of the surfaces of the pillars. Micropillars with smooth sidewalls and a high aspect ratio were obtained in this way. Solar cells with a radial junction were then fabricated on these micropillars. Standard doping, SiO2/SiNx passivation, and metallisation steps were carried out for the fabrication of solar cells with different micropillar lengths. A significant decrease in reflectance values was observed as the micropillar length increased, as expected. Solar cell short circuit current density (Jsc) and efficiency (η) of the solar cells tended to increase with micropillar length up to 11.5 µm and then decrease due to increased surface recombination. The maximum efficiency achieved in this study is 17.26%.

Stretchable, Bifacial Si-Organic Hybrid Solar Cells by Vertical Array of Si Micropillars Embedded into Elastomeric Substrates.

Stretchable electronics has enabled many unforeseen applications in a variety of fields. Mechanical design concepts to achieve the stretchability without affecting the device functionality, however, are limited to few known practices, such as mechanical buckling, serpentine shape, or simple elastomeric composites. In this paper, we propose another mechanics design principle for high stretchability (>100%) based on the composite of vertical array of Si micropillars embedded into elastomer poly(dimethylsiloxane). The orthogonalization of active functional elements to applied strain direction enables highly stretchable electronic devices, where the applied strain is mostly absorbed into elastomer on interpillar space. On the other hand, the vertical pillars do not experience any noticeable strain at all. As a proof-of-concept demonstration, we fabricate stretchable Si-organic hybrid solar cells using such a design and the cell shows reasonable level of cell efficiency compared with planar counterparts. The cell can be stretched reversibly without any noticeable performance degradation. Furthermore, the cell can be operated in a bifacial mode by employing stretchable, transparent Ag nanowire-based electrodes. The mechanical design for stretchability demonstrated here would provide new opportunities for stretchable electronics.

Electrochemical behavior of a gold nanoring electrode microfabricated on a silicon micropillar

We report on the microfabrication and characterization of a gold nanoring electrode (Au NRE) patterned on top of a silicon (Si) micropillar. An NRE of 165 ± 10 nm in width was micropatterned on 4.6 ± 1 μm diameter × 17.5 ± 2.5 μm long Si micropillar with an intervening 50 nm thick hafnium oxide insulating layer. Scanning electron microscopy and energy dispersive spectroscopy data confirmed excellent cylindricality of the micropillar with vertical sidewalls and a completely intact layer of Au NRE encompassing the entire micropillar perimeter. The electrochemical behavior of the Au NRE was characterized by a steady-state cyclic voltammogram with extremely high signal-to-noise ratios of 2500 and charging currents as small as 1.5 ± 0.3 pA. A “semicircle spectrum” from the Nyquist plot indicates a kinetically-controlled voltammetric current response, which is unique to nanoscale electrodes. The variations in the exchange current values, which are dependent on the NRE area and standard rate constant, place a greater emphasis on the isotropic gold wet etching process that defined the geometry of the Au NRE. Circuit fitting of the electrochemical impedance spectra suggests that the NRE geometry can be varied from inlaid to nanotrench with depths controllable by the etching process. This resulted in differing stead-state voltammetric currents and interfacial properties in terms of charge transfer resistance, constant phase element and trench resistance values. The applicability of Au NREs to electrochemical sensing is demonstrated by detecting lead, a neurotoxin at 100 ppb levels. Also, by surface-modification with multi-walled carbon nanotubes, dopamine, a neurochemical implicated in various brain disorders, is detected at a sensitivity as low as 100 nM with 1000-fold selectivity versus common interferents. The flexibility of the microfabrication approach allows for the creation of multiple NREs of controllable width and nanometer spacing on a single micropillar. This capability and in particular where the NRE is micropatterned on three-dimensional microstructures as will be reported herein, facilitates unique electroanalytical capabilities such as intracellular electrochemistry and highly multiplexed detection for emerging biological applications.

Silicon microstructures through the production of silicon nanowires by metal-assisted chemical etching, used as sacrificial material

A simple, inexpensive and wafer-scale method to obtain Si microstructures is proposed. The method consists in a sequence of steps that include a selective metal-assisted chemical etching process to create regions of Si nanowires that are sacrificed in a post-etching process, leaving microstructures standing. As a proof of concept, Si micropillars with length of 7 µm and diameter of 1.4 µm were fabricated. The advantage of the proposed method is its simplicity, allowing the production of microstructures in a basic chemical laboratory.

Surface Defect Passivation of Silicon Micropillars

AbstractReactive ion etching (RIE) used to fabricate high‐aspect‐ratio (HAR) nano/microstructures is known to damage semiconductor surfaces which enhances surface recombination and limits the conversion efficiency of nanostructured solar cells. Here, defect passivation of ultrathin Al2O3‐coated Si micropillars (MPs) using different surface pretreatment steps is reported. Effects on interface state density are quantified by means of electrochemical impedance spectroscopy which is used to extract quantitative capacitance–voltage and conductance–voltage characteristics from HAR dielectric–semiconductor structures which would otherwise suffer from high gate leakage currents if tested using solid‐state metal–insulator–semiconductor structures. High‐temperature thermal oxidation to form a sacrificial oxide on RIE‐fabricated Si MPs, followed by atomic layer deposition of 4 nm thick Al2O3 after removal of the sacrificial layer produces an interface trap density (D it) as low as 1.5 × 1011 cm−2 eV−1 at the mid‐gap energy of silicon. However, a greatly reduced mid‐gap D it (2 × 1011 cm−2 eV−1) is possible even with a simple air annealing procedure having a maximum temperature of 400 °C.

Solving the critical thermal bowing in 3C-SiC/Si(111) by a tilting Si pillar architecture

The exceptionally large thermal strain in few-micrometers-thick 3C-SiC films on Si(111), causing severe wafer bending and cracking, is demonstrated to be elastically quenched by substrate patterning in finite arrays of Si micro-pillars, sufficiently large in aspect ratio to allow for lateral pillar tilting, both by simulations and by preliminary experiments. In suspended SiC patches, the mechanical problem is addressed by finite element method: both the strain relaxation and the wafer curvature are calculated at different pillar height, array size, and film thickness. Patches as large as required by power electronic devices (500–1000 μm in size) show a remarkable residual strain in the central area, unless the pillar aspect ratio is made sufficiently large to allow peripheral pillars to accommodate the full film retraction. A sublinear relationship between the pillar aspect ratio and the patch size, guaranteeing a minimal curvature radius, as required for wafer processing and micro-crack prevention, is shown to be valid for any heteroepitaxial system.