Characterization of micrometric and superficial residual stresses using high frequency surface acoustic waves generated by interdigital transducers Marc Duquennoy,a) Mohammadi Ouaftouh, Julien Deboucq, Jean-Etienne Lefebvre, Frederic Jenot, and Mohamed Ourak IEMN-DOAE, UMR CNRS 8520, Universit e de Valenciennes, 59313 Valenciennes, France
(Received 11 March 2013; revised 12 September 2013; accepted 30 September 2013) Controlling thin film deposition of materials and property gradients is a major challenge for the implementation of applications in microelectronics or glassmaking. It is essential to control the level of residual stress and thus important to have the right tools to characterize this stress in terms of scale and nature of the deposits. In this context, dispersion of ultrasound surface waves caused by the presence of a residual micrometric surface stress was studied in an amorphous medium for different superficial fields of residual stress. The design and implementation of SAW-IDT MEMS sensors enabled quasi-monochromatic Rayleigh-type surface waves to be generated and the dispersion phenomenon to be studied over a wide range of frequencies. The thicknesses of the stressed cortical zones as well as the level of stress were estimated with good accuracy using an inverse method. C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4826176] V PACS number(s): 43.35.Zc, 43.35.Pt, 43.35.Yb, 43.35.Cg [JBL]
I. INTRODUCTION
The study and control of residual mechanical stress are of increasing importance in numerous fields such as microelectronics and materials with property gradients with a view to meeting new functional requirements. The effects of these stresses can be harmful in certain applications (breakdown phenomenon1 and sometimes beneficial like, for example, to improve the transport properties in silicon2 or to mechanically reinforce glass.3 In this study, the influence of the superficial field of stress, the depths of which are in the order of tens of micrometers, on the propagation of surface acoustic waves (SAW) were studied theoretically and experimentally on chemically tempered sheets of glass. Based on the theory of acoustoelasticity, this study shows the influence of such a field of stress on the propagation of surface waves and in particular highlights dispersion phenomena. By studying this dispersion, it was possible to determine, by inversion, certain important characteristics such as the depth and level of superficial stress. But, in order to characterize thin depths (a few tens of lm) it is necessary to generate and detect ultrasonic waves over a broad frequency range up to 60 MHz. Various ultrasonic methods can be used for the nondestructive testing of these materials. Among the different types of ultrasonic waves that propagate near the surface of materials, Rayleigh-type surface acoustic waves are particularly interesting because the energy is concentrated within a layer under the surface of about one wavelength thick. They thus constitute a good candidate for the characterization of sheets or layer on substrate-type structures. To test the
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
4360
J. Acoust. Soc. Am. 134 (6), December 2013
Pages: 4360–4371
materials at different depths, different excitation frequencies were used to obtain the required wavelengths. Several techniques can be used to generate Rayleigh-type surface waves. Wedge sensors are traditionally used to generate surface waves, but above 10 MHz the losses and attenuations related to this sensor technology become too significant. Another interesting technique is laser-ultrasonics, which offers numerous advantages such as the possibility of noncontact generation and broadband generation.4,5 In recent years, several publications have demonstrated the relevance of this method for the characterization of thin films.6–8 However, depending on the nature of the materials, the suitability of this method of generation varies according to the penetration depth and/or fragility of the layers (problem of ablation). Finally, the acoustic signature is also an interesting technique enabling measurements to be carried out at very high frequency.9 However, it is essential to work in immersion, which in some cases is not feasible in terms of the integrity of the structure or device to be controlled. In this study, we designed and implemented surface acoustic waves InterDigital Transducer (SAW-IDT). This original solution is based on the development of an SAW-IDT MicroElectroMechanical Systems (MEMS), to generate quasi-monochromatic surface waves and obtain a rapid and accurate estimation of the phase velocity, key information for the characterization of the layers. Moreover, the use of SAW-IDT MEMS allowed HF surface waves to be generated over a broad frequency range.10 SAW-IDT MEMS are typically used in acoustoelectronic signal processing devices such as surface wave filters, oscillators, resonators, etc. Today, most SAW applications are in the field of telecommunications and the frequencies used are typically very high and can reach several gigahertz.11 IDT sensors are rarely used in non destructive testing (NDT) applications, but are usually used at frequencies of a few megahertz.12
0001-4966/2013/134(6)/4360/12/$30.00
C 2013 Acoustical Society of America V
II. PROPAGATION OF ACOUSTIC WAVES IN A STRESSED MEDIUM
Acoustoelasticity, i.e., the dependence of the acoustic wave velocity on the stress in an elastic material, is based on a continuum theory of small disturbances (ultrasonic waves) superimposed on an elastically stressed medium.13,14 The general theory of finite deformation was presented by Murnaghan13 and later in a treatise by Truesdell and Toupin.15 A modern theory of acoustoelasticity was presented by introducing third-order elastic constants into the constitutive equation.14,16,17 This formalism requires the deformation energy to be developed up to third order, making the theoretical developments very cumbersome. Different developments14,18–22 have all shown that it is possible to introduce the notion of Effective Elastic Constants (EEC). Through the introduction of EEC, it is possible to use a classic formalism (second order) in the case of stressed materials. A stressed material can thus be considered, via the EEC, as a stress free material presenting second order elastic constants different from the second order elastic constants of an unstressed material.23 To establish expressions for ultrasonic wave velocities in a stressed material, the type of wave and the mechanical behavior of the material need to be taken into account. Three deformation states are defined to distinguish the deformation process. The unstressed reference configuration (no stress and no strain) is called the “natural state.” The position of a material point is given by the position vector x whose “natural coordinates” are xI ðx1 ; x2 ; x3 Þ. The finite deformation (applied or residual) due to homogeneous pre-stress, in the elastic area, transforms the configuration to another state called the “initial state,” which is in static equilibrium. The position of this material point is then given by the position vector x whose x 1 ; x2 ; x3 Þ. Finally, when a “initial coordinates” are xI ð dynamic perturbation is superposed on the “initial state,” the configuration changes and takes the material to a third state called the “final state.” The position of the material point is then defined by the position vector x~ whose “final coorx 1 ; x~2 ; x~3 Þ. A common Cartesian frame (X1, dinates” are x~I ð~ X2, X3) is used to refer to the position of material points in the three states (Fig. 1). Thus, xI , xI , and x~I are the coordinates of the same material point in the natural, initial, and final states, respectively. The physical quantities which refer to the initial
state are denoted by the subscript “I.” The initial displacement (that corresponds to the transition from the natural state to the initial state) and the final displacement (that corresponds to the transition from the initial state to the final state) are denoted by u and u~, respectively. To establish the equations of motion of an acoustic wave in a stressed material, the following hypothesis must be assumed: The solid is hyper-elastic, all deformations are in the elastic area and uniform, the predeformation is static and the predeformations and rotations are small. The perturbation caused by the acoustic wave is assumed to be small with regard to the initial deformation; it can thus be written: k~ u k k u k.24 Equation (1) is the equation of motion with initial coordinates @ @ u~c @ 2 u~a ¼q 2 ðRabcd þ dac rbd Þ (1) @ xd @t @ xb with
Rabcd
@ um @ um ¼ Cabcd 1 þ Cabcdmn @ xm @ xn @ ua @ ub @ uc þ Cmbcd þ Camcd þ Cabmd @ xm @ xm @ xm @ ud þ Cabcm ; @ xm
(2)
where Cabcd and Cabcdef are second- and third-order elastic constants; q is the density defined in the deformed state; r is the Cauchy stress tensor, defined over an area of material elements in the deformed state. From this equation of motion, the expressions of acoustic wave velocities can be established. The coefficients Rabcd depend on the second- and third-order elastic constants and consequently on the symmetry of the material. The equation of motion [Eq. (1)] expressed with the initial coordinates is analogous to the equation of motion relative to an unstressed medium. The similarity that exists between these equations allows a stressed material to be considered as an unstressed material with EEC which take into account the disturbances linked to the presence of stress. This description considers the following formalism: CEabcd ¼ Cabcd þ dCabcd :
(3)
CEabcd is the EEC tensor. Cabcd are the second order elastic constants that correspond to the mechanical characteristics of the material free of all stress. dCabcd represent the variations due to the deformation terms. In initial coordinates, CEabcd ¼ Rabcd þ rbd dac and the presence of the term rbd translates the dependence of the bulk wave propagation velocity in relation to the field of stress. The expressions of these EEC are provided in Appendix A. In the case of an isotropic material undergoing a biaxial stress field, the expressions of these EEC are provided in Appendix B. III. DISPERSION PHENOMENA IN A SUPERFICIALLY STRESSED STRUCTURE A. Modeling of a sheet with a superficial stress
FIG. 1. Sketch of the three states of the material: Natural, initial, and final. J. Acoust. Soc. Am., Vol. 134, No. 6, December 2013
In numerous cases, fields of stress appear during the processing of materials, with sudden variations in temperature Duquennoy et al.: Characterization of residual stresses
4361
often being the cause of residual stress. However, chemical and mechanical causes also exist. In the context of amorphous materials such as glass, the fragility of the material can be limited using surface reinforcement by chemical or thermal tempering. The principle of thermal tempering of glass is the same as for steel in metallurgy: It consists in forming a compression layer on the surface. Another way of creating surface compression in glass consists in chemical tempering during which sodium ions are replaced by more “voluminous” potassium ions. This chemical reaction occurs by diffusion by immersing the glass in a solution of molten potassium nitrate. A rectangular stress profile, also called “U” profile, is thus obtained (Fig. 2). An important characteristic of this tempering is the introduction of a superficial field of stress on the surface.25–27 In practice, chemical tempering is carried out at approximately 450 C; the immersion time in potassium nitrate depends on the surface stress desired and is generally about 10 h. This tempering is much easier to control than thermal tempering and much higher levels of stress can be attained. The surface compression stress can be in the order of several hundred MegaPascals (approximately 700 MPa, for example, for chemical tempering of 3 h) and only affects a thickness of a few tens of micrometers. Chemical tempering more particularly enables the treatment of very fine sheets of glass (
(Received 11 March 2013; revised 12 September 2013; accepted 30 September 2013) Controlling thin film deposition of materials and property gradients is a major challenge for the implementation of applications in microelectronics or glassmaking. It is essential to control the level of residual stress and thus important to have the right tools to characterize this stress in terms of scale and nature of the deposits. In this context, dispersion of ultrasound surface waves caused by the presence of a residual micrometric surface stress was studied in an amorphous medium for different superficial fields of residual stress. The design and implementation of SAW-IDT MEMS sensors enabled quasi-monochromatic Rayleigh-type surface waves to be generated and the dispersion phenomenon to be studied over a wide range of frequencies. The thicknesses of the stressed cortical zones as well as the level of stress were estimated with good accuracy using an inverse method. C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4826176] V PACS number(s): 43.35.Zc, 43.35.Pt, 43.35.Yb, 43.35.Cg [JBL]
I. INTRODUCTION
The study and control of residual mechanical stress are of increasing importance in numerous fields such as microelectronics and materials with property gradients with a view to meeting new functional requirements. The effects of these stresses can be harmful in certain applications (breakdown phenomenon1 and sometimes beneficial like, for example, to improve the transport properties in silicon2 or to mechanically reinforce glass.3 In this study, the influence of the superficial field of stress, the depths of which are in the order of tens of micrometers, on the propagation of surface acoustic waves (SAW) were studied theoretically and experimentally on chemically tempered sheets of glass. Based on the theory of acoustoelasticity, this study shows the influence of such a field of stress on the propagation of surface waves and in particular highlights dispersion phenomena. By studying this dispersion, it was possible to determine, by inversion, certain important characteristics such as the depth and level of superficial stress. But, in order to characterize thin depths (a few tens of lm) it is necessary to generate and detect ultrasonic waves over a broad frequency range up to 60 MHz. Various ultrasonic methods can be used for the nondestructive testing of these materials. Among the different types of ultrasonic waves that propagate near the surface of materials, Rayleigh-type surface acoustic waves are particularly interesting because the energy is concentrated within a layer under the surface of about one wavelength thick. They thus constitute a good candidate for the characterization of sheets or layer on substrate-type structures. To test the
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
4360
J. Acoust. Soc. Am. 134 (6), December 2013
Pages: 4360–4371
materials at different depths, different excitation frequencies were used to obtain the required wavelengths. Several techniques can be used to generate Rayleigh-type surface waves. Wedge sensors are traditionally used to generate surface waves, but above 10 MHz the losses and attenuations related to this sensor technology become too significant. Another interesting technique is laser-ultrasonics, which offers numerous advantages such as the possibility of noncontact generation and broadband generation.4,5 In recent years, several publications have demonstrated the relevance of this method for the characterization of thin films.6–8 However, depending on the nature of the materials, the suitability of this method of generation varies according to the penetration depth and/or fragility of the layers (problem of ablation). Finally, the acoustic signature is also an interesting technique enabling measurements to be carried out at very high frequency.9 However, it is essential to work in immersion, which in some cases is not feasible in terms of the integrity of the structure or device to be controlled. In this study, we designed and implemented surface acoustic waves InterDigital Transducer (SAW-IDT). This original solution is based on the development of an SAW-IDT MicroElectroMechanical Systems (MEMS), to generate quasi-monochromatic surface waves and obtain a rapid and accurate estimation of the phase velocity, key information for the characterization of the layers. Moreover, the use of SAW-IDT MEMS allowed HF surface waves to be generated over a broad frequency range.10 SAW-IDT MEMS are typically used in acoustoelectronic signal processing devices such as surface wave filters, oscillators, resonators, etc. Today, most SAW applications are in the field of telecommunications and the frequencies used are typically very high and can reach several gigahertz.11 IDT sensors are rarely used in non destructive testing (NDT) applications, but are usually used at frequencies of a few megahertz.12
0001-4966/2013/134(6)/4360/12/$30.00
C 2013 Acoustical Society of America V
II. PROPAGATION OF ACOUSTIC WAVES IN A STRESSED MEDIUM
Acoustoelasticity, i.e., the dependence of the acoustic wave velocity on the stress in an elastic material, is based on a continuum theory of small disturbances (ultrasonic waves) superimposed on an elastically stressed medium.13,14 The general theory of finite deformation was presented by Murnaghan13 and later in a treatise by Truesdell and Toupin.15 A modern theory of acoustoelasticity was presented by introducing third-order elastic constants into the constitutive equation.14,16,17 This formalism requires the deformation energy to be developed up to third order, making the theoretical developments very cumbersome. Different developments14,18–22 have all shown that it is possible to introduce the notion of Effective Elastic Constants (EEC). Through the introduction of EEC, it is possible to use a classic formalism (second order) in the case of stressed materials. A stressed material can thus be considered, via the EEC, as a stress free material presenting second order elastic constants different from the second order elastic constants of an unstressed material.23 To establish expressions for ultrasonic wave velocities in a stressed material, the type of wave and the mechanical behavior of the material need to be taken into account. Three deformation states are defined to distinguish the deformation process. The unstressed reference configuration (no stress and no strain) is called the “natural state.” The position of a material point is given by the position vector x whose “natural coordinates” are xI ðx1 ; x2 ; x3 Þ. The finite deformation (applied or residual) due to homogeneous pre-stress, in the elastic area, transforms the configuration to another state called the “initial state,” which is in static equilibrium. The position of this material point is then given by the position vector x whose x 1 ; x2 ; x3 Þ. Finally, when a “initial coordinates” are xI ð dynamic perturbation is superposed on the “initial state,” the configuration changes and takes the material to a third state called the “final state.” The position of the material point is then defined by the position vector x~ whose “final coorx 1 ; x~2 ; x~3 Þ. A common Cartesian frame (X1, dinates” are x~I ð~ X2, X3) is used to refer to the position of material points in the three states (Fig. 1). Thus, xI , xI , and x~I are the coordinates of the same material point in the natural, initial, and final states, respectively. The physical quantities which refer to the initial
state are denoted by the subscript “I.” The initial displacement (that corresponds to the transition from the natural state to the initial state) and the final displacement (that corresponds to the transition from the initial state to the final state) are denoted by u and u~, respectively. To establish the equations of motion of an acoustic wave in a stressed material, the following hypothesis must be assumed: The solid is hyper-elastic, all deformations are in the elastic area and uniform, the predeformation is static and the predeformations and rotations are small. The perturbation caused by the acoustic wave is assumed to be small with regard to the initial deformation; it can thus be written: k~ u k k u k.24 Equation (1) is the equation of motion with initial coordinates @ @ u~c @ 2 u~a ¼q 2 ðRabcd þ dac rbd Þ (1) @ xd @t @ xb with
Rabcd
@ um @ um ¼ Cabcd 1 þ Cabcdmn @ xm @ xn @ ua @ ub @ uc þ Cmbcd þ Camcd þ Cabmd @ xm @ xm @ xm @ ud þ Cabcm ; @ xm
(2)
where Cabcd and Cabcdef are second- and third-order elastic constants; q is the density defined in the deformed state; r is the Cauchy stress tensor, defined over an area of material elements in the deformed state. From this equation of motion, the expressions of acoustic wave velocities can be established. The coefficients Rabcd depend on the second- and third-order elastic constants and consequently on the symmetry of the material. The equation of motion [Eq. (1)] expressed with the initial coordinates is analogous to the equation of motion relative to an unstressed medium. The similarity that exists between these equations allows a stressed material to be considered as an unstressed material with EEC which take into account the disturbances linked to the presence of stress. This description considers the following formalism: CEabcd ¼ Cabcd þ dCabcd :
(3)
CEabcd is the EEC tensor. Cabcd are the second order elastic constants that correspond to the mechanical characteristics of the material free of all stress. dCabcd represent the variations due to the deformation terms. In initial coordinates, CEabcd ¼ Rabcd þ rbd dac and the presence of the term rbd translates the dependence of the bulk wave propagation velocity in relation to the field of stress. The expressions of these EEC are provided in Appendix A. In the case of an isotropic material undergoing a biaxial stress field, the expressions of these EEC are provided in Appendix B. III. DISPERSION PHENOMENA IN A SUPERFICIALLY STRESSED STRUCTURE A. Modeling of a sheet with a superficial stress
FIG. 1. Sketch of the three states of the material: Natural, initial, and final. J. Acoust. Soc. Am., Vol. 134, No. 6, December 2013
In numerous cases, fields of stress appear during the processing of materials, with sudden variations in temperature Duquennoy et al.: Characterization of residual stresses
4361
often being the cause of residual stress. However, chemical and mechanical causes also exist. In the context of amorphous materials such as glass, the fragility of the material can be limited using surface reinforcement by chemical or thermal tempering. The principle of thermal tempering of glass is the same as for steel in metallurgy: It consists in forming a compression layer on the surface. Another way of creating surface compression in glass consists in chemical tempering during which sodium ions are replaced by more “voluminous” potassium ions. This chemical reaction occurs by diffusion by immersing the glass in a solution of molten potassium nitrate. A rectangular stress profile, also called “U” profile, is thus obtained (Fig. 2). An important characteristic of this tempering is the introduction of a superficial field of stress on the surface.25–27 In practice, chemical tempering is carried out at approximately 450 C; the immersion time in potassium nitrate depends on the surface stress desired and is generally about 10 h. This tempering is much easier to control than thermal tempering and much higher levels of stress can be attained. The surface compression stress can be in the order of several hundred MegaPascals (approximately 700 MPa, for example, for chemical tempering of 3 h) and only affects a thickness of a few tens of micrometers. Chemical tempering more particularly enables the treatment of very fine sheets of glass (
