6th NDT in Progress Lamb waves in an anisotropic plate of a single crystal silicon wafer

Transcription

1 6th NDT in Progress 2011 International Workshop of NDT Experts, Prague, Oct 2011 Lamb waves in an anisotropic plate of a single crystal silicon wafer Young-Kyu PARK 1, Young H. KIM 1 1 Applied Acoustics Laboratory, Korea Science Academy of KAIST; Busan, Korea Phone: , Fax: ; yhkim627@kaist.ac.kr Abstract Lamb waves are dispersive in a plate and elastic properties of anisotropic materials are dependent on the crystal orientation. It is expected that the wave propagation on a thin anisotropic plate is complex due to dispersion of Lamb waves which will be dependent on the propagation direction. Therefore it is essential to understand elastic waves in an anisotropic plate. In the present work, a silicon wafer of 100 mm diameter and 0.5 mm thick with [100] direction was interrogated with leaky Lamb wave (LLW). Lamb waves were generated in a silicon wafer with the variation in incidence angles. Edge reflected LLWs were detected by the same transducer in a pulseecho set-up. In order to clarify the propagation characteristics in anisotropic plates, 2-dimensional angular profile has been proposed in the presentation. In order to predict dispersion curves of Lamb waves, longitudinal and transverse wave velocities were calculated from the static elastic constants. As results, it was found that 2- dimensional angular profile represents the crystal structures of plates. It was also found that there is unexpected peak in angular profile due to anisotropy of the silicon wafer. Measures dispersion curves in a silicon wafer were qualitatively similar to calculated ones. Keywords: Silicon single crystal wafer, leaky lamb wave, dispersion, anisotropy 1. Introduction Lamb waves are dispersive in a plate due to boundary conditions. There are numerous modes depending on the wave structures in a plate. The phase and group velocities in a plate are also dependent on the thickness of the plate and the frequency of ultrasound. When ultrasound propagates along the anisotropic plates, complicated phenomena will be involved. Elastic properties of anisotropic materials are dependent on the crystal orientations. Silicon is one of well known materials and various researches on this material have been carried out [1]. Elastic properties of silicon wafer have been interrogated by laser based Lamb wave propagation [2]. However, it is limited to the velocities at fixed frequency. In addition, group velocity was measured whereas phase velocity is mentioned in theory. Leaky Lamb wave has been employed to evaluate various plates [3]. This technique has an advantage to change phase velocity by oblique incident of ultrasound. In the present work, edge reflected leaky Lamb wave was employed to evaluate a crystal silicon wafer of <100> orientation. The phase velocities of Lamb waves were determined by changing incident angle of ultrasound. Propagation characteristics depending on the crystal orientation was investigated by rotating specimen. A new diagram so called 2-dimensional angular profile has been proposed in order to represent anisotropic Lamb wave propagation. <111> wafer and aluminium plate were also interrogated by leaky Lamb wave in order to compare the anisotropic characteristics. The wave speeds in certain propagation direction were assumed as same as those of isotropic plate of which elastic moduli are as same as those of static moduli silicon in given direction, and dispersion curves were calculated. From the incident angle and frequency components of leaky Lamb waves, experimental dispersion curves were obtained and compared to the calculated one.

2 2. Theoretical Backgrounds 2.1 Elastic constants of silicon wafer Tensor formalism is required in order to account for anisotropy of silicon single crystal. The general relationship between stress and strain is where is the second order stiffness tensor, and are strain and stress tensor, respectively. Since silicon single crystal is cubic crystal, there are only 3 constants,, and, and stiffness tensor is given by To obtain Young s modulus in [100] direction, set all other stresses to zero and solve for /. This gives The Poisson s ratio can similarly be obtained as.. The Young s modulus in any direction can be obtained by calculating the stiffness matrix in rotated coordinates. Young s modulus, Poisson s ratio and shear modulus obtained by a computer program [4] are shown in Fig. 1. Figure 1. (a) Young s modulus, (b) Poisson s ratio and (c) shear modulus obtained by a program. The unit of moduli is Pa [4].

3 2.2 Leaky Lamb wave Fig. 2 shows schematic diagram of leaky Lamb wave. When ultrasound is obliquely incidence on a plate in water, the Lamb wave can be generated and propagates along the plate and reflected at the edge of the plate. Some energy of them leaks into water and produces reflected and transmitted beam. The condition for phase matching is satisfying the Snell's law. When Lamb waves propagate along the plate with the incidence angle,, the velocity of incidence wave is equal to the wave speed in water,, which is taken as 1,500 m/s in the present work. The velocity of refracted wave in the plate is the phase velocity of Lamb wave,, and the refraction angle is 90. The Snell's law gives This equation implies that the phase velocity of Lamb wave generated in the plate can be determined from the incidence angle. Figure 2. Schematic diagram of generation and detection of leaky Lamb wave. 3. Experiments 3.1 Single crystal Silicon wafer P type silicon wafer of <100> orientation, 100 mm diameter and 0.54 mm thick was used as one of single crystal plate. N type silicon wafer of <111> orientation, 150 mm diameter and 2 mm thick as well as aluminium plates as similar dimensions as silicon wafers were also used for the comparison of the results. Data for <100> wafer were intensively analyzed and others were qualitatively analyzed. It is well known that doping in silicon does not affect elastic properties of silicon, elastic constants of silicon wafer in calculation were assumed to be as same as pure silicon single crystal. 3.2 Ultrasonic system The ultrasonic transducer of 0.5 inch diameter and 5 MHz center frequency was used for the generation and detection of leaky Lamb wave in pulse-echo setup. The rf waveforms were

4 captured by 12 bit digitizer with 200 MHz sampling rate. The incident angle was changed from 0 to 30 in 0.5 step, and specimen was rotated 360 in 1 step. 4. Results and discussion 4.1 RF waveforms Fig. 3 shows typical rf waveforms of LLW for the <100> silicon wafer. It is clear that different rf waveform is observed not only for different incident angle but also for different propagation direction. In the θ=10 and θ=15 of incident angle, there are not so much differences for different propagation direction. However, dramatic differences were found in the θ=22 of incident angle. New type of leaky Lamb wave can be observed (especially at the φ=15 ) and the position and magnitude are different for different propagation direction. Therefore ultrasonic propagation in silicon wafer is anisotropic and dispersive. Figure 3. Typical waveforms of leaky Lamb wave signals. φ is propagation angle from <100> axis and θ is incident angle. The vertical full scale is 1 V peak-to-peak, and the horizontal full scale is 40 µs for all waveforms. 4.1 Angular profiles Figure 4 shows angular profiles, which is the magnitude of leaky Lamb wave as a function of incident angle, for different propagation direction. For φ=0 and 45, there are two major peaks near 12 and 22, whereas for φ=15 and 30, there are one more major peaks between 12 and 22 and the peak position is moved with the change of propagation direction. That

5 means the propagation in the direction of off-principal axis is more complicate than that in principal axis. It may be caused by the direction mismatch of phase and group velocities. Peak Amplitude (V) φ = 0 φ=15 φ=30 φ= Incident Angle (Deg) Figure 4. Angular profile as a function of propagation direction in <100> wafer D Angular profile In order to show the angular profile dependence on the propagation direction 2-dimensional angular profile was proposed in the present work. 2D angular profile, (, is a polar representation in grey scale. The distance from origin, direction and brightness are corresponding to the incident angle (θ), propagation direction (φ) and magnitude of leaky Lamb wave (,. Figure 5. 2D angular profile,, diagram of <100> wafer obtained by 5 MHz transducer.

6 Figure 5 shows the 2D angular profile obtained from the <100> wafer. It shows 3 major peaks as similar to Fig. 4. The first peak is observed near 12. The second peak which appears as a slant line is moving according to the propagation direction. It is closer to the third peak on the 0 and closer to the first peek in the 45. Third peak is observed near 22. The pattern of the first peak is a regular tetragonal shape, which is corresponding to the crystal structure of <100> wafer. This diagram seems to move down, and it may be cause by tilting of wafer rotation, and incident angle may have offset value which is changed by rotation of specimen D Angular profiles for various plates Figure 6 shows 2D angular profiles obtained from an aluminium plate of mm thick and an <111> silicon wafer of 2.0 mm thick and 150 mm diameter. The patterns of two profiles are corresponding to isotropic and hexagonal structures, respectively. It implies that 2D angular profiles can be used as a tool to characterize the texture of specimen. Figure 6. 2D angular profiles obtained from (a) aluminium plate of 1 mm thick and (b) <111> silicon wafer of 2.0 mm thick and 150 mm diameter. 4.4 Dispersion curves of leaky Lamb wave Dispersion curves can be obtained from the longitudinal and transverse wave speeds. Determination of wave modes and speeds in anisotropic medium is complicated. There are more than two wave modes and sometimes it is hard to define wave modes such as longitudinal and transverse. However, longitudinal and transverse wave speeds were determined from the static Young s modulus and shear modulus as a preliminary analysis. And Lamb wave dispersion curves were obtained by the method as same as isotropic case. The [110] was taken as propagation direction, and Young s modulus and two set of shear moduli were calculated. The larger value of shear modulus is corresponding to extremely small value of Poisson s ratio, and slower wave speed of transverse wave was taken. Since the wafer is thin, the number of wave modes is limited, and dispersion curves of A0, S0, A1 and S1 modes were calculated. From the incident angle and frequency components of leaky Lamb waves, experimental dispersion curves were obtained and compared to the calculated one.

7 Fig. 7 shows dispersion curves for [110] direction. Calculated and measured dispersion diagram shows qualitative agreement, however some differences were found. One of them is that measured phase velocities of S0 mode at low frequency and Rayleigh-like wave mode look like much faster than calculated one. It may be caused by either the offset of incident angle or the wafer was tilted from rotating plane. Moreover S0 mode seems to be double lines, it may be caused by double valued Poisson s ratio in [100] direction. Figure 7 Dispersion curves for the [110] direction in <100> wafer. Measured value is shown in grey level and calculated one is in dot point. 5. Conclusion As conclusion, the crystal orientation can be determined by leaky Lamb waves. However, more quantitative analysis of anisotropy is strongly required for more accurate understanding Lamb wave propagation in silicon wafer. More study with higher frequency will be helpful to investigate much more wave modes in anisotropic plates. References 1. M A Hopcroft, 'What is the Young s modulus of silicon?', J Mircoelectromechanical System, Vol 19, No 2, pp , April P A Kotidis, 'Laser ultrasonics-based material analysis system and method utilizing Lamb modes', US Patent 5,956,143, September 21, Y H Kim, S -J Song, S D Kwon, Y -M Cheoung, H -K Jung, 'Determination of ultrasonic wave velocities and phase velocity dispersion curves of an Inconel 600 plate using resonant ultrasonic spectroscopy and leaky Lamb waves', Ultrasonics, Vol 42, No1-9, pp , April 2004) 4. V Kaajakari, Silicon as an anisotropic mechanical material - a tutorial,