Strain relaxation of semiconductor membranes: insights from finite element modeling

Finite element (FE) models were employed to better understand the process used to fabricate elastically strain-relaxed in-place bonded semiconductor membranes for application as engineered substrates for semiconductor devices (Cohen et al 2005 Appl. Phys. Lett. 86 251902; Owen et al 2008 ECS Trans. 16 271; Owen et al 2009 Sci. Technol. 24 035011; Mooney et al 2014 Semicond. Sci. Technol. 29 075009; Salehzadeh et al 2014 Semicond. Sci. Technol. 29 085002). Initial structures consist of a compressively strained, square semiconductor membrane atop a sacrificial layer that is subsequently removed by etching in an hydrofluoric acid (HF) solution. Elastic relaxation of the compressive strain bends the free area of the membrane toward the substrate. But for the dimensions of most of the structures employed in the fabrication of in-place bonded membranes, the bending by elastic strain relaxation alone is not sufficient for the membrane corners to contact the substrate. FE models, therefore, confirm that an attractive force between the surface of the membrane and the surface of the substrate in the HF solution must act to hold the membrane in place during the wet etching process. FE models also confirm that in the etch solution the interface between the membrane and the substrate must be a sliding interface in order for the membrane to lie flat on the substrate once the sacrificial layer has been completely removed.


Introduction
Epitaxially grown heterostructures incorporating semiconductor alloy layers have long been employed to improve the characteristics of electronic and optoelectronic devices. Structures incorporating Al x Ga 1−x As layers having the entire range of alloy composition could be grown on GaAs Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
substrates because the lattice constants of the constituents, GaAs and AlAs, differ by only 0.1% [1]. Devices employing Al x Ga 1−x As/GaAs heterostructures include double heterostructure lasers [2] and modulation-doped field effect transistors [3]. In contrast, the lattice constant of the constituent materials is significantly different for other important semiconductor alloys. Two examples are In x Ga 1−x As, where the lattice constant difference between GaAs and InAs is ∼8% and Si 1−x Ge x where the lattice constant of Ge is ∼4% larger than that of Si. This lattice constant difference allows for the incorporation of alloy layers in which the band structure is modified by biaxial tensile or compressive strain. One example is the SiGe bipolar transistor where the compressively strained SiGe base region resulted in a unity-current-gain cutoff frequency, f T , of 113 GHz, more than a factor of 2 greater than that achieved for Si bipolar transistors [4]. Another is the InGaAs/GaAs laser [5].
The lattice constant mismatch between the alloy layer and the substrate limits the composition range of In x Ga 1−x As or Si 1−x Ge x alloys that can be grown epitaxially without the introduction of strain relieving defects that degrade device performance and reliability [6][7][8][9][10][11]. Therefore, the availability of only a few high-quality bulk semiconductor substrates, such as Si, GaAs, and InP, severely restricts applications of these semiconductor heterostructures. Various types of engineered substrates have been proposed to circumvent this limitation. Examples include very thick graded-composition Si 1−x Ge x buffer layers [12,13] or thinner Si 1−x Ge x buffer layers in which dislocation nucleation is initiated in a controlled manner by implantation of He ions [14,15]. In both these types of engineered substrates the strain relieving misfit dislocations are confined within the buffer layer and the density of threading arms running through the active device layers to the wafer surface is reported to be <10 7 cm −2 [12][13][14][15]. Prototype modulation doped field effect transistors with excellent performance fabricated on both types of buffer layers have been demonstrated [16][17][18]. Nevertheless, it is not known how low the threading dislocation density must be to avoid problems with various device fabrication processes and to achieve the very high device yield and reliability needed for ultra-largescale integration applications.
To eliminate dislocations entirely, epitaxial growth on thin, free-standing membranes such that strain relaxation occurs elastically, without dislocation formation, has been investigated [19][20][21][22][23]. Another approach involves the release of an epitaxially-grown device layer structure from the substrate on which it was grown, resulting in the elastic redistribution of the lattice mismatch strain in the layers of free-standing membrane [24].
A method that can be implemented over very large areas has been referred to as in-place bonding or solution bonding [25][26][27][28][29]. Here, the thin semiconductor membrane, consisting of a single layer or a series of layers, is grown epitaxially on a sacrificial layer. The membrane is first patterned and the sacrificial layer is then removed by wet etching in an HF solution, thus allowing lattice constant mismatch strain to be redistributed in the free membrane as it is separated from the substrate. These membrane features bond in place to the original substrate in the etch solution, thus avoiding contamination of the bonded interface by oxidation or by the various molecules that are adsorbed on surfaces when they are exposed to air [30]. The primary fabrication steps for Si/Si 0.8 Ge 0.2 /Si tri-layer membranes bonded to a Si substrate after removal of the SiO 2 sacrificial layer are shown schematically in figure 1 [25]. The initial structure was a silicon-on-insulator (SOI) wafer having epitaxially-grown Si 0.8 Ge 0.2 and Si cap layers as shown in figure 1(a). Since the Si/Si 0.8 Ge 0.2 /Si membrane is bonded in place on the Si substrate, it was inferred that the membrane must bend downward and adhere to the substrate as the sacrificial layer is removed by wet etching, as is indicated in figure 1(c). Room temperature bonding is not sufficiently  [25]. Reprinted from [25], with the permission of AIP Publishing. strong for most applications [30]; therefore, heat treatment is employed after the wafer is removed from the etch solution to form a strong covalent bond [25]. The registration error of bonded Si/SiGe/Si features with respect to their initial position defined by photolithography was found to be <30 nm, the same as for control wafers where the features were defined but the sacrificial oxide layer had not been removed [25]. A similar fabrication process was used for GaAs/In 0.08 Ga 0.92 As/GaAs membranes bonded to a GaAs substrate, where the sacrificial layer was either AlAs or Al 0.7 Ga 0.3 As [26][27][28][29]. Devices can subsequently be fabricated on the bonded features or they can be used as 'virtual substrates' for the growth of device layer structures.
Both the initial epitaxial layer structures of figure 1(a) and final bonded wafers of figure 1(d) were extensively characterized. High resolution x-ray diffraction (HRXRD) was used to measure the strain in the semiconductor layers [25][26][27][28] and atomic force microscopy was used to detect the surface steps resulting from misfit dislocations [25][26][27][28].
In some experiments Raman spectroscopy was also used to measure the strain in bonded features [25,28]. Transmission electron microscopy images showed the discontinuity of the crystal lattice at the bonded interface between the GaAs/In 0.08 Ga 0.92 As/GaAs membrane and the GaAs substrate [28]. The electrical resistance of the bonded interface between the GaAs/In 0.08 Ga 0.92 As/GaAs membrane and the GaAs substrate was found to be an order of magnitude lower than that reported for whole wafer bonded interfaces [29]. Unfortunately, no direct observations or measurements could be made during the wet etching process and questions remain. Specifically, what is the shape of the membrane as the sacrificial layer is removed and how is it that the bonded membrane is held in place on the substrate?
In this work finite element (FE) modeling was employed to investigate the membrane shape due to elastic redistribution of the lattice constant mismatch strain as the width of the sacrificial layer is reduced during the etching process. The FE models were validated by comparing the strain distribution in free-standing tri-layer Si/SiGe/Si membranes with that calculated using a force-balance analytical model [23], which was demonstrated to agree well with experimental results. Various membrane and supporting pedestal dimensions were explored. The FE models show that initially, as the sacrificial layer is etched away and the width of the supporting pedestal decreases, the free edges of Si 0.8 Ge 0.2 membranes bend downward towards the substrate at the edges of the pedestal. Then, as the pedestal becomes narrow enough that strain in the material on top of the pedestal is relaxed, the membrane flattens again, consistent with experimental observations of tri-layer membranes supported by a narrow pedestal [22,23]. Models of structures having the dimensions of the experimental in-place bonded structures demonstrate that the downward curvature of the membrane due to elastic strain relaxation alone would not bring the membrane into contact with the substrate during the etch process. Thus, the FE models confirm that, as reported in [25], an attraction between the membrane and the substrate acts during the wet etching process to hold the membrane in place once the sacrificial layer is completely removed. The nature of the interface between the membrane and the substrate, as indicated by the FE models, is also discussed.

FE models of Si 1−x Ge x /Si structures
FE modeling has been used previously to explore strain behavior in various semiconductor structures. Examples include the relaxation of lattice mismatch strain by undulations in thin Si 1−x Ge x films grown on Si substrates [31] and patterning induced strain redistribution in ultra-thin strained Si layers on SiO 2 [32]. In FE models the difference in lattice constant between the Si 1−x Ge x alloy and Si is established using a nominal coefficient of thermal expansion (CTE) for Si 1−x Ge x so that Si 1−x Ge x expands elastically with respect to Si when a heating step is implemented in the model [31]. The lattice constant of Si 1−x Ge x in Å is given by where x is the Ge fraction of the alloy [33]. The FE modeling software Abaqus/CAE V6.8-1 was used in this work [34]. Semiconductor layers were modeled as orthotropic materials, coherent epitaxial interfaces were modeled as tie interfaces, and linear elastic deformation of all materials was assumed. Models were constructed using the room temperature values for the materials constants of Si and Si 0.8 Ge 0.2 in table 1 [35]. Young's modulus and Poisson ratio used for thermal SiO 2 are 6.663 × 10 5 MPa and 0.17, respectively. The initial temperature was 0 K and the temperature after heating was 300 K, with the CTE of Si and SiO 2 set to zero and the CTE of Si 1−x Ge x set to achieve thermal expansion of the Si 1−x Ge x equivalent to the lattice constant difference between the alloy layer and Si at room temperature given by equation (1). The dimensions of the membrane layers are much smaller in the out-of-plane direction than in the in-plane direction and hence the mesh was typically set to be much finer in the out-of-plane direction in the membrane layers. To obtain accurate model results, the size of the mesh, in both the inplane and out-of-plane directions in all the different materials in the model was reduced until further reductions made a negligible different in the results.
Validation of these procedures was done by first modeling blocks of Si 1−x Ge x . The heating step in the model accounts for the larger lattice constant of Si 1−x Ge x compared to that of Si and therefore the expansion of the mesh when heat is applied in these simple models represents the difference between the lattice constant of Si 1−x Ge x and that of Si as calculated from equation (1). Next thin, large area Si 1−x Ge x layers on very thick Si substrates were modeled. A tie interface was implemented to model epitaxial semiconductor interfaces. For the diamond crystal structure, the mismatch strain in a pseudomorphic epitaxial layer is given by ε = (a l − a s ) /a s , where a l is the lattice constant of the layer and a s is the lattice constant of the substrate. The strain in the Si 1−x Ge x layer in the model structure, calculated from the distortion of the mesh, which in the FE models represents the in-plane and out-of-plane lattice constants of this crystalline material, was equal to the known lattice constant mismatch strain of a pseudomorphic Si 1−x Ge x layer of that alloy composition grown on a Si substrate.
FE model results were also compared with results from an analytical model. A force-balance analytical model of a free-standing tri-layer Si 1−x Ge x /Si/Si 1−x Ge x membrane was initially employed to predict elastic strain relaxation of the Si 1−x Ge x layer in structures consisting of a tri-layer Si 1−x Ge x /Si/Si 1−x Ge x membrane supported by a very narrow SiO 2 pedestal [23]. It was assumed that the narrow pedestal Schematic showing the force-balance model applied to free-standing epitaxial SiGe/Si/SiGe structures. The different lattice constants of the two materials results in stress at the coherent interfaces between the two materials [23]. Reprinted from [23], with the permission of AIP Publishing. acts as a point support for the membrane, thus having no effect on the strain distribution, and that strain relaxation effects at the edges of the membrane can be neglected. Strain in the Si 1−x Ge x layers determined by this analytical model is in good agreement with that measured in fabricated structures by HRXRD [23]. As well, strain in the Si 1−x Ge x layers and in the In 0.08 Ga 0.92 As layers of various tri-layer bonded membranes, determined from HRXRD measurements, agrees well with predictions by this analytical model, except when the thickness of the bonded membrane was <100 nm [25][26][27][28]. To further validate the FE analysis, free-standing tri-layer membrane structures were modeled and the results were compared with those calculated using this analytical model.
A schematic diagram of the force-balance model is shown in figure 2 [23]. Because the tri-layer membrane is symmetric, with the thin Si layer cladded by two Si 1−x Ge x layers of equal thickness, bending forces can be ignored. With the bending forces set to zero this tri-layer structure is equivalent to a bilayer structure consisting of a Si and a single SiGe layer. Strain in the Si and SiGe layers is given by ε (Si) = [S 11 (Si) + S 12 (Si)] P/t (Si) and ε (SiGe) = −[S 11 (SiGe) + S 12 (SiGe)]P/t (SiGe), where S 11 and S 12 are the crystal compliances, P is the force/unit length at the interface, and t is the total thickness of each material. Assuming coherent interfaces, ε (Si) = ε (SiGe) + ∆ε, where ∆ε is the lattice mismatch strain between the two materials. Substituting the values for ε (Si) and ε (SiGe) into the previous equation yields represents the biaxial moduli, E is Young's modulus, m is Poisson's ratio, and S 11 and S 12 are the crystal compliance values for the Si and Si 1−x Ge x films. For Si 1−x Ge x having low Ge fraction, the materials constants of are approximately equal to those of Si and equation (2) reduces to We see from equation (3) that strain in these free-standing membrane structures is shared between the two materials according to their relative thickness. The thicker the Si 1−x Ge x relative to the Si thickness, the smaller the compressive strain in Si 1−x Ge x and the greater the tensile strain in Si.
When a pseudomorphic Si 1−x Ge x layer is grown on a standard Si substrate, 100% of the lattice mismatch strain occurs as compressive strain in the Si 1−x Ge x layer. In contrast, when the thickness ratio t (SiGe) /t (Si) is 10, approximately 90% of the total mismatch strain occurs as tensile strain in the Si layer and only about 10% remains as compressive strain in the Si 1−x Ge x layers. In this example the Si 1−x Ge x layers are about 90% relaxed compared to a pseudomorphic Si 1−x Ge x layer of the same composition on a standard Si substrate. The solid curve of figure 3, calculated using equation (2), shows the strain relaxation of Si 1−x Ge x in the tri-layer membrane as a function of t (SiGe) /t (Si). Figure 4 shows a cross section view of the center region of the FE model of a free-standing tri-layer square membrane that was cut across the center parallel to the edges of the square. The membrane dimensions are 5 µm × 5 µm × 300 nm with t(Si 0.8 Ge 0.2) /t (Si) = 2 and the thermal expansion of Si 0.8 Ge 0.2 was set to allow for the lattice constant difference 0.8% between the two materials. In this model the mesh was established at T = 0 K with one mesh unit in the plane of the membrane equal to four mesh units perpendicular to the membrane plane in both materials. After the heating step, when the Si 0.8 Ge 0.2 layer has expanded corresponding to the larger lattice constant of Si 0.8 Ge 0.2 , the tie interface has forced the Si layer to expand in the in-plane direction with a corresponding reduction of the layer thickness. Therefore, in the Si layer four mesh units in the vertical direction are no longer equal to one mesh unit in the horizontal direction. This distortion of the mesh, which represents the change in the in-plane and out-ofplane lattice constants, shows the magnitude of the tensile or compressive strain in each layer. Looking at the red squares overlaid on the image in figure 4, we see that in the central Si layer four vertical mesh units are clearly compressed compared to the single horizontal mesh unit, showing that this layer is under tensile strain. In the two Si 0.8 Ge 0.2 layers the four mesh units in the out-of-plane direction are larger than the single mesh unit of the in-plane direction, indicating that those layers remain partially compressively strained. In this example the strain in the Si 0.8 Ge 0.2 layers has been reduced by 66% compared to the compressive strain in a thin epitaxial Si 0.8 Ge 0.2 layer of the same alloy composition on a thick Si substrate. The strain relaxation of the Si 1−x Ge x layers determined from a series of FE models agrees with the strain relaxation calculated by the analytical model, as is seen by the data points that overlap the solid curve in figure 3. The FE models show that in each layer the mesh is uniform across the membrane except at the membrane edges, indicating that the strain in each layer is uniform and that edge effects would contribute negligibly to the experiments results.

Results and discussion
The analytical model described above can be applied only to the membrane of the starting wafer shown in figure 1(a) and the in-place bonded membrane shown in figure 1(d). In contrast, FE models can be used to help understand the intermediate steps of the fabrication process. Specifically, FE models demonstrate how strain relaxation affects the shape of the membrane as the width of the sacrificial layer that supports it is reduced by wet etching. FE models also help understand the behavior of the membrane when it comes into contact with the substrate.

Membrane shape
A series of simple models consisting of 10 µm × 10 µm × 300 nm thick Si 0.8 Ge 0.2 membrane having a thermal expansion of 1% supported by a 1 µm thick SiO 2 pedestal of various widths on a 2.5 µm thick Si substrate are shown in figures 5 and 6. The SiO 2 /Si and SiO 2 /Si 1−x Ge x interfaces were modeled as tie interfaces, since bonded SOI wafers with covalently bonded interfaces after heat treatment had been employed for the experimental structures [22,23,25]. The mesh size in the models was reduced until further reductions gave a negligible difference in the results and the thickness of the Si substrate was increased until further increases gave a negligible difference. The thickness of the Si 0.8 Ge 0.2 membrane was chosen to be comparable to the thickness of published experimental in-place bonded membranes [25]. At the start of the wet etch process, the entire membrane is supported by the sacrificial layer and is flat, with the Si 0.8 Ge 0.2 layer fully compressively strained. As the sacrificial layer is removed and the width of the supporting pedestal decreases, relaxation of the compressive strain occurs by downward bending of the membrane at the edges of the pedestal. When the pedestal width is greater than ∼1 µm, the area of the Si 0.8 Ge 0.2 membrane lying on top of the pedestal remains compressively strained except at the edges of the pedestal. For these square membranes, the corners bend down closer to the substrate than the edge centers, as can be clearly seen in figure 6(a). Consistent with the tri-layer Si/Si 1−x Ge x /Si membranes supported by a narrow pedestal reported in [22,23] figures 5(d) and 6(b) both show that when the pedestal width is less than µm, the compressive strain over the entire area of the Si 0.8 Ge 0.2 layer is relaxed and the membrane is essentially flat. Note that the effects of gravity are not included in the FE models.
These model results showing strain relaxation of the membrane on top of the pedestal when the width of the interface between the pedestal and the film is less than 1 µm are consistent with x-ray microdiffraction measurements of epitaxial SiGe strips of various widths on Si(001) substrates [36]. When the width of a 240 nm thick Si 0.86 Ge 0.14 strip is 1 µm, the normal stress at the center of the strip is reduced to 15% of the blanket film value as the effects of strain relaxation at the edges of the strip overlap. Such edge effects must always be considered when characterizing strained nanoscale structures.
The distance below the top of the pedestal and the lower surface of both the corner of the membrane and the center of the membrane edge was measured to quantify the membrane curvature and determine the conditions for which the lower surface of the membrane would contact the upper surface of the Si substrate. This was accomplished by making cross sectional cuts of the models through the membrane center both parallel to the membrane edge and along the membrane diagonal, as indicated by the 0 • and 45 • lines on figure 6(b). Cross section cuts parallel to the membrane edge are shown in figure 5. The position of the bottom surface of the membrane corner and edge center relative to the top of the pedestal was determined from the change in the position of those mesh nodes after the heating step in the model was applied. As is shown in figure 7, the downward bending of the membrane is a maximum when the pedestal width is 4 µm and is negligible when the pedestal width is less than 1 µm. For this model structure, the maximum distance of the membrane corner below the     The membrane curvature was found to be essentially independent of the pedestal height. However, varying the membrane width or the lattice constant mismatch has a large effect on the maximum distance of the membrane edge center and corner below top of the pedestal. Figures 8 and 9 show that increasing either the width of the membrane or the lattice constant mismatch (thermal expansion) of the membrane significantly increases the maximum distance of the lower surface Tri-layer Si/Si 0.8 Ge 0.2 /Si membranes on SiO 2 pedestals on Si substrates were also modeled. For the same structure dimensions, the distance of the membrane above the Si substrate of the tri-layer membrane is greater that of a membrane consisting only of a Si 0.8 Ge 0.2 layer of the same composition and thickness. Since the Si 0.8 Ge 0.2 layer in the tri-layer membrane is only partially, not fully, relaxed, depending on the relative thickness of the two materials, the downward curvature of the tri-layer membrane would be equivalent to that of a single layer having a lower Ge fraction.
Wrinkling or buckling effects are not included in the FE models described above. The image of figure 10 is the 0 • cut of a structure showing the mesh on the top of a large area membrane of Si 0.8 Ge 0.2 that overhangs the pedestal by 2 µm. The mesh is square on the region of the membrane above the pedestal showing that this area, which is constrained by the pedestal, is under biaxial compressive strain. In contrast, the mesh on the overhanging edge, except at the membrane corners, is rectangular indicating the membrane has expanded outward from the pedestal edge, but remains constrained by the pedestal in the direction parallel to the membrane edge. Strain relaxation by wrinkling or buckling of undercut free edges of initially compressively strained films has been reported [37,38]. As well, similar wrinkling was observed at the edges of much larger area features on the mask used to fabricate the 5 µm × 5 µm SiGe/Si/SiGe tri-layer membrane structures reported in [22,23]. Sinusoidal wrinkling of the undercut edges on long stripes of polycrystalline and amorphous Si deposited on SiO 2 has been used to determine the initial strain in these Si films employed for the fabrication of micromechanical beams [37]. Sinusoidal wrinkling of overhanging Si 1−x Ge x epitaxial layers on Si substrates was characterized by a local wavelength (λ) and amplitude (A), which is a maximum at the edge of the free layer and decreases gradually toward the clamped edge that remains compressively strained [38]. Experimental results agree well with an analytical model based on Von Karman plate theory. For a long stripe, overhang length of 2 µm and initial mismatch strain of ∼0.8% (x = 0.20), the wavelength was calculated to be about 3.5 µm and the amplitude to be about 65 nm [38], comparable to the distance of the corner of the membrane below the top of the pedestal, as shown in figure 7 for a pedestal width of 6 µm, where the membrane overhangs the pedestal by 2 µm.
Strain relaxation by sinusoidal wrinkling of the overhanging membrane edge is expected in the initial stages of the etching process when the pedestal is relatively wide. Sinusoidal wrinkling would likely bring more of the membrane edge closer to the substrate, not only the corners as indicated in the FE models. However, sinusoidal wrinkling would also raise parts of the membrane edge farther above the initial position of the membrane, thus allowing the etch solution to flow underneath the membrane. Wrinkling effects become insignificant once the pedestal becomes narrow enough that the strain in the membrane on top of the pedestal relaxes. Note that no wrinkling of membranes supported by a pedestal <1 µm wide was reported [22,23,39]. Nor was wrinkling of the in-place solution bonded structures observed [25][26][27][28]. Therefore, wrinkling of the overhanging membrane edge does not significantly change what has been learned from the FE models, specifically, that the membrane would contact the Si surface by strain relaxation alone only in in the cases of a large area membrane and/or a thin sacrificial layer.

Interaction of the membrane with the substrate
Once the sacrificial layer is completely removed, the membrane lies flat on the substrate, held in-place place by Van der Walls forces when the sample is removed from the HF etch solution and dried. But the question remains: how is the membrane held in place during the wet etching process? The models presented here demonstrate that for the dimensions of the fabricated Si 1−x Ge x /Si in-place bonded structures reported in [1], the membrane corners would not contact the substrate due to strain relaxation alone. Therefore, an attractive force between the lower surface of the membrane and the upper surface of the substrate must act to bring the membrane into contact with the substrate and hold it in place during the etching process. For both the SiGe/Si and the InGaAs/GaAs materials systems the etchant employed was an HF solution, which renders the semiconductor surfaces hydrophobic [25][26][27][28][29]. And when a surfactant was added to the HF etch solution to reduce the hydrophobicity of the Si surfaces, small area Si/Si 1−x Ge x /Si tri-layer membrane features were not held in place on the substrate [25]. This experiment demonstrates that hydrophobic effects play an important role in keeping the membrane in place, presumably because water is repelled from both the membrane and substrate surfaces, thus pulling the membrane toward the substrate. Generally, hydrostatic forces are understood to be short range forces, acting over distances on the order of about 1 nm [40] and measurements of the distances over which hydrophobic effects occur are very difficult [41]. However, these FE models indicate that the attractive force holding the membrane in place on the substrate acts over much greater distances, i.e. over distances of as much as 50-100 nm.
To investigate the effects of an attractive force between the membrane and the substrate during the wet etching process, a series of FE models in which a uniform pressure was applied as a downward load acting either on the entire upper surface of the membrane or on a 200 nm wide band along the edge of the membrane were constructed. A uniform pressure in the range from 10 −4 N m −2 to 10 2 N m −2 was applied to the top surface of both Si and Si/Si 1−x Ge x /Si membranes of various dimensions. Whereas the applied pressure had a negligible effect on the area of the membrane on top of the SiO 2 pedestal, the free areas of the membrane were bent downward toward the Si substrate. When a fixed interface between the membrane and the substrate is implemented in the FE models, once the membrane corners contact the substrate surface they do not move and increasing the load further has little effect on the shape of the membrane. Except at the membrane corners, a gap always remains between the membrane and the substrate, inconsistent with the experiments showing that the bonded membranes lie flat on the substrate [25,28]. Alternatively, a frictionless interaction between the membrane and the substrate, which allows the membrane to slide along the substrate surface, was implemented. In this case, when the magnitude of the load was increased, the membrane tended to flatten at the corners and the area of contact between the membrane and the substrate increased. Including friction at the interface in the models was found to have a negligible effect on the shape of the membrane. These FE models confirm that, in order for the membrane to lie flat on the substrate once the sacrificial layer is completely removed, the interaction between the membrane and the substrate in the etch solution must be relatively weak.
As an example, figure 11 shows the results of a uniform downward pressure of 14 N m −2 applied to the top surface of a 5 µm × 5 µm × 300 nm thick Si membrane on an SiO 2 1 µm × 1 µm × 150 nm SiO 2 pedestal. The expected result that a smaller pressure is needed in order for the corners of a thinner membrane having the same area to contact the substrate was observed. Similarly, a smaller pressure is needed in order for the corners of a larger area membrane having the same thickness to contact the substrate. A comprehensive quantitative study of the response of membranes of various dimensions to an applied pressure is not included here, since an applied pressure is unlikely to be an accurate representation of the hydrophobic effect that was found to play a role in the experiments that motivated this work.
The FE models of structures having dimensions comparable to those of the reported in-place bonded structures demonstrate that an attractive force between the membrane and the substrate could bring the membrane corners into contact with the substrate as the sacrificial layer is removed. These models also confirm that in order for the bonded membrane to lie flat on the substrate, the interface between the membrane must be a sliding interface. Similarly, it has been found that in order for FE models of the strain in crystalline SiGe-on-insulator layers deposited locally on SOI wafers to agree with experimental results, the interfaces between the two materials must be decoupled [42]. Fundamental mechanisms operating at sliding interfaces are not well understood and require further study.

Conclusions
In this work FE models were employed to better understand the process used to fabricate strain relaxed in-place bonded semiconductor membranes for application as engineered substrates for semiconductor devices. Elastic strain relaxation of various square Si 1−x Ge x and Si/Si 1−x Ge x /Si membranes supported by a SiO 2 pedestal of varying widths was investigated. The FE models show that when the width of the sacrificial layer is greater than about 1 µm, the free area of the compressively strained membrane relaxes elastically by bending toward the Si substrate at the edges of the supporting SiO 2 pedestal. However, when the SiO 2 pedestal width is <1 µm, the strain in the membrane area on top of the pedestal is also relaxed and the membrane is essentially flat.
When the membrane area is sufficiently large and the sacrificial layer is sufficiently thin, the membrane would bend enough so that the corners would contact the substrate during the etch process. However, for the dimensions of most of the structures employed to fabricate in-place bonded membranes, the membrane corners would not touch the substrate due to elastic strain relaxation alone. The FE models therefore confirm that an attractive force between the lower surface of the membrane and the upper surface of the substrate must act to hold the membrane in place during the wet etching process. The FE models also confirm that, during the wet etching process, the interface between the membrane and the substrate must be a sliding interface in order for the membrane to lie flat on the substrate once the sacrificial layer has been completely removed.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.