Mise en œuvre de deux méthodes interférométriques pour la caractérisation mécanique des films minces par l'essai de gonflement. Applications au cas du silicium monocristallin

Abstract

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. the good agreement between theory and experiments allows to valid the apparatus. 1.

Fabrication of Buckling Free Ultrathin Silicon Membranes by Direct Bonding with Thermal Difference

“…As for the coefficient β, C, and C*, they are related to form factors of the membrane and are, respectively, equal to 1.26.10 À3 , 26.12, and 13.63 for a squared membrane. 19 The weight is obtained assuming a pressure P uniformly applied to the membrane. Equation 1 has been applied to our case to determine the deflection of a silicon membrane as a function of the stress of the membrane.…”

“…However, the membrane thickness is very small compared to its lateral dimensions, so that its bending stiffness is no longer sufficient to counteract the deformation induced by its weight. On the basis of the theory of plates and shells developed by Bonnotte et al , Constancias et al have recently proposed a method for determining the minimum compensation stress required to achieve a negligible deflection of a membrane with a thickness lower than one micron. An application of this method to the case of a squared monocrystalline silicon membrane with a square side results in the following equation: with: where h is the calculated deflection, σ the stress and t the membrane thickness; g is the gravitational constant, E Si the Young modulus, ν Si is the Poisson’s ratio and ρ Si the density of silicon.…”

“…An application of this method to the case of a squared monocrystalline silicon membrane with a square side results in the following equation: with: where h is the calculated deflection, σ the stress and t the membrane thickness; g is the gravitational constant, E Si the Young modulus, ν Si is the Poisson’s ratio and ρ Si the density of silicon. As for the coefficient β, C , and C *, they are related to form factors of the membrane and are, respectively, equal to 1.26.10 –3 , 26.12, and 13.63 for a squared membrane . The weight is obtained assuming a pressure P uniformly applied to the membrane.…”

Fabrication of Buckling Free Ultrathin Silicon Membranes by Direct Bonding with Thermal Difference

“…As for the coefficient β, C, and C*, they are related to form factors of the membrane and are, respectively, equal to 1.26.10 À3 , 26.12, and 13.63 for a squared membrane. 19 The weight is obtained assuming a pressure P uniformly applied to the membrane. Equation 1 has been applied to our case to determine the deflection of a silicon membrane as a function of the stress of the membrane.…”

“…However, the membrane thickness is very small compared to its lateral dimensions, so that its bending stiffness is no longer sufficient to counteract the deformation induced by its weight. On the basis of the theory of plates and shells developed by Bonnotte et al , Constancias et al have recently proposed a method for determining the minimum compensation stress required to achieve a negligible deflection of a membrane with a thickness lower than one micron. An application of this method to the case of a squared monocrystalline silicon membrane with a square side results in the following equation: with: where h is the calculated deflection, σ the stress and t the membrane thickness; g is the gravitational constant, E Si the Young modulus, ν Si is the Poisson’s ratio and ρ Si the density of silicon.…”

“…An application of this method to the case of a squared monocrystalline silicon membrane with a square side results in the following equation: with: where h is the calculated deflection, σ the stress and t the membrane thickness; g is the gravitational constant, E Si the Young modulus, ν Si is the Poisson’s ratio and ρ Si the density of silicon. As for the coefficient β, C , and C *, they are related to form factors of the membrane and are, respectively, equal to 1.26.10 –3 , 26.12, and 13.63 for a squared membrane . The weight is obtained assuming a pressure P uniformly applied to the membrane.…”

<title>Automated interferometric system for bulge and blister test measurements of micromachined membranes</title>

“…Where C 1 , C 2 , C 3 are coefficients (2<Ci<10) which depends on the membrane shape and for C 2 on the Poisson ratio ν 7,8,10,12,21,22 . σ and Ε are respectively the film stress and Young's modulus.…”

Low-temperature thin-film indium bonding for reliable wafer-level hermetic MEMS packaging

“…The membrane deflects inward if an under pressure is present within the cavity compared to the surrounding environment, which is the case when the wafers are bonded in vacuum. If the deflections are small compared to the size of the membrane, they can be analytically related to the pressure difference between the cavity and the ambient pressure by using the formula from Bonnotte et al [16], assuming zero residual stress in the membrane:…”

Fabrication of Buckling Free Ultrathin Silicon Membranes by Direct Bonding with Thermal Difference

“…As for the coefficient β, C, and C*, they are related to form factors of the membrane and are, respectively, equal to 1.26.10 À3 , 26.12, and 13.63 for a squared membrane. 19 The weight is obtained assuming a pressure P uniformly applied to the membrane. Equation 1 has been applied to our case to determine the deflection of a silicon membrane as a function of the stress of the membrane.…”

“…However, the membrane thickness is very small compared to its lateral dimensions, so that its bending stiffness is no longer sufficient to counteract the deformation induced by its weight. On the basis of the theory of plates and shells developed by Bonnotte et al , Constancias et al have recently proposed a method for determining the minimum compensation stress required to achieve a negligible deflection of a membrane with a thickness lower than one micron. An application of this method to the case of a squared monocrystalline silicon membrane with a square side results in the following equation: with: where h is the calculated deflection, σ the stress and t the membrane thickness; g is the gravitational constant, E Si the Young modulus, ν Si is the Poisson’s ratio and ρ Si the density of silicon.…”

“…An application of this method to the case of a squared monocrystalline silicon membrane with a square side results in the following equation: with: where h is the calculated deflection, σ the stress and t the membrane thickness; g is the gravitational constant, E Si the Young modulus, ν Si is the Poisson’s ratio and ρ Si the density of silicon. As for the coefficient β, C , and C *, they are related to form factors of the membrane and are, respectively, equal to 1.26.10 –3 , 26.12, and 13.63 for a squared membrane . The weight is obtained assuming a pressure P uniformly applied to the membrane.…”

<title>Automated interferometric system for bulge and blister test measurements of micromachined membranes</title>

“…Where C 1 , C 2 , C 3 are coefficients (2<Ci<10) which depends on the membrane shape and for C 2 on the Poisson ratio ν 7,8,10,12,21,22 . σ and Ε are respectively the film stress and Young's modulus.…”

Low-temperature thin-film indium bonding for reliable wafer-level hermetic MEMS packaging

“…The membrane deflects inward if an under pressure is present within the cavity compared to the surrounding environment, which is the case when the wafers are bonded in vacuum. If the deflections are small compared to the size of the membrane, they can be analytically related to the pressure difference between the cavity and the ambient pressure by using the formula from Bonnotte et al [16], assuming zero residual stress in the membrane:…”

Fabrication of Buckling Free Ultrathin Silicon Membranes by Direct Bonding with Thermal Difference

“…As for the coefficient β, C, and C*, they are related to form factors of the membrane and are, respectively, equal to 1.26.10 À3 , 26.12, and 13.63 for a squared membrane. 19 The weight is obtained assuming a pressure P uniformly applied to the membrane. Equation 1 has been applied to our case to determine the deflection of a silicon membrane as a function of the stress of the membrane.…”

“…However, the membrane thickness is very small compared to its lateral dimensions, so that its bending stiffness is no longer sufficient to counteract the deformation induced by its weight. On the basis of the theory of plates and shells developed by Bonnotte et al , Constancias et al have recently proposed a method for determining the minimum compensation stress required to achieve a negligible deflection of a membrane with a thickness lower than one micron. An application of this method to the case of a squared monocrystalline silicon membrane with a square side results in the following equation: with: where h is the calculated deflection, σ the stress and t the membrane thickness; g is the gravitational constant, E Si the Young modulus, ν Si is the Poisson’s ratio and ρ Si the density of silicon.…”

“…An application of this method to the case of a squared monocrystalline silicon membrane with a square side results in the following equation: with: where h is the calculated deflection, σ the stress and t the membrane thickness; g is the gravitational constant, E Si the Young modulus, ν Si is the Poisson’s ratio and ρ Si the density of silicon. As for the coefficient β, C , and C *, they are related to form factors of the membrane and are, respectively, equal to 1.26.10 –3 , 26.12, and 13.63 for a squared membrane . The weight is obtained assuming a pressure P uniformly applied to the membrane.…”

<title>Automated interferometric system for bulge and blister test measurements of micromachined membranes</title>

“…Where C 1 , C 2 , C 3 are coefficients (2<Ci<10) which depends on the membrane shape and for C 2 on the Poisson ratio ν 7,8,10,12,21,22 . σ and Ε are respectively the film stress and Young's modulus.…”

Low-temperature thin-film indium bonding for reliable wafer-level hermetic MEMS packaging

“…The membrane deflects inward if an under pressure is present within the cavity compared to the surrounding environment, which is the case when the wafers are bonded in vacuum. If the deflections are small compared to the size of the membrane, they can be analytically related to the pressure difference between the cavity and the ambient pressure by using the formula from Bonnotte et al [16], assuming zero residual stress in the membrane:…”