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Nonlinear acoustic crack detection in thermoelectric wafers
Mechanical Systems and Signal Processing 139 (2020) 106598
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Nonlinear acoustic crack detection in thermoelectric wafers J. Greenhall a,⇑, S. Grutzik b, A. Graham a, D.N. Sinha a, C. Pantea a a b
Acoustics and Sensors Team, Materials Physics and Applications Group (MPA-11), Los Alamos National Laboratory, Los Alamos, NM 87545, USA Component Science and Mechanics, Sandia National Laboratory, Albuquerque, NM 87123, USA
a r t i c l e
i n f o
Article history: Received 31 July 2019 Received in revised form 12 November 2019 Accepted 22 December 2019
Keywords: Nonlinear acoustics Nondestructive evaluation Acoustic crack detection Thermoelectric wafer testing
a b s t r a c t An acoustic technique for detecting cracks in bismuth telluride (Bi2Te3) thermoelectric wafers is presented. The technique is based on mechanically exciting the wafers with a low frequency pump signal that drives a crack to ‘‘breathe” by opening and closing periodically and a high frequency probe signal that interrogates the breathing crack. Interaction between the pump signal, the probe signal, and the crack leads to generation of acoustic nonlinearities in the wafer. In contrast to existing acoustic crack detection techniques we utilize standing waves within the wafers to facilitate simultaneous crack detection throughout the wafer, which reduces the total inspection time; we do not require uniform dimensions and material properties between wafers; and we do not affix the transmitter/ receiver to the wafers, which can lead to wafer damage. We present a technique that integrates multiple damage metrics to identify cracks within the wafers. We demonstrate the technique on 94 Bi2Te3 wafers, of which we correctly identify cracked wafers with 4.3% total error and we correctly identify crack-free wafers with 7.5% total error. This acoustic crack detection technique finds application in manufacturing of thermoelectric wafers and other materials, where identifying cracks early in the manufacturing process results in significant time and cost savings. Additionally, this acoustic crack detection technique has implications for other nondestructive inspection applications where cracks cannot be detected effectively through existing techniques. Published by Elsevier Ltd.
1. Introduction Thermoelectric materials such as bismuth telluride (Bi2Te3) have attracted significant attention due to their ability to convert thermal energy into electric energy or vice versa within a solid state device via the Peltier effect [1]. These materials have significant applications including cooling electronics at miniature scales [2], refrigeration [3], and thermal energy harvesting [4]. Bi2Te3 wafers tend to be highly brittle and they are susceptible to cracking during manufacturing, which would inhibit thermal-electric energy conversion. Thus, it is critical to detect cracks in Bi2Te3 to ensure the efficacy of the energy conversion devices. Ideally, cracks are identified when the Bi2Te3 material is in wafer form, prior to subsequent manufacturing processes to minimize wasted time/labor. Currently, there are a wide range of techniques for nondestructive identification of cracks utilizing optical microscopy [5,6], X-ray computed tomography (CT) [7,8], and ultrasound [9–16]. However, automated optical techniques based on image processing are sensitive to variations in surface texture and lighting and are limited to surface defects, manual optical techniques are costly and time-consuming, and X-ray CT requires prohibitively long measurement times. Current ultrasound crack detection techniques are categorized as either burst techniques, wherein ⇑ Corresponding author. E-mail address: [email protected] (J. Greenhall). https://doi.org/10.1016/j.ymssp.2019.106598 0888-3270/Published by Elsevier Ltd.
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J. Greenhall et al. / Mechanical Systems and Signal Processing 139 (2020) 106598
an ultrasound burst propagates through the material; or resonance techniques, wherein a standing ultrasound wave is established throughout the material. Burst techniques rely either on transmitting the burst into the material and measuring reflections [12] or radiation [13] from cracks, or they excite the crack and cause it to open and close, which generates nonlinear harmonics [9,11] and/or modulations [14]. Burst techniques are typically used on materials with dimensions significantly larger than the wavelength of the ultrasound to ensure that transmitted signal does not reflect from the edges of the material and interfere with the measured signal. Additionally, these techniques are sensitive to the placement of the transmitting and receiving ultrasound transducers with respect to the crack position and orientation, and typically require a large number of transmitter/receiver configurations to ensure that the cracks are properly interrogated. Conversely, resonance techniques utilize standing waves within the materials, which works well for material geometries on the order of the wavelength, and the standing wave sets up over the entire material simultaneously, which reduces sensitivity to the transmitter location. Current resonance techniques work by setting up a standing ultrasound wave into the wafer at a range of different frequencies to measure the resonance spectrum of the wafer. Cracks are then identified by contrasting the frequency spectrum to that of a baseline sample that is known to be crack-free [15]. The critical problem with this methodology is that it requires uniform geometry and material properties between all wafers. Alternatively, cracks can be detected by exploiting shifts in the frequency spectrum peaks at different excitation amplitudes [10]. But, this technique is limited by the fact that at sufficiently high amplitudes resonance peaks can shift due to nonlinearities within the material itself, which can be mistaken for cracks. Additionally, the techniques require measuring the frequency spectrum at multiple excitation amplitudes, which increases the overall measurement time. The current ultrasound crack detection techniques are also limited by the fact that they either assume a fixed rigid connection between the transmitter/receiver and the material specimen via an adhesive or vacuum [9–11,14,15], or they utilize a laser pulse to generate an ultrasound wave [12,13]. Obtaining a fixed rigid connection between a transmitter/receiver and the Bi2Te3 wafers via adhesive is not feasible due to the fact that removing the adhesive from the Bi2Te3 wafer can lead to wafer damage and due to the long adhesive curing time. Similarly, the stresses induced by laser pulse excitation can lead to Bi2Te3 wafer damage, which can adversely affect the Bi2Te3 material properties [17]. Finally, a fixed rigid connection cannot be achieved via vacuum due to the porous texture of the Bi2Te3 material. In contrast with existing ultrasound crack detection techniques, the objective of this work is to develop a fast, accurate technique for detecting cracks in Bi2Te3 wafers without the use of a fixed rigid connection between the transmitter/receiver and the wafer. To achieve this, we generate a resonating ultrasound wave within the Bi2Te3 wafer and quantify the vibration amplitude of the resonant wave and the amplitude of generated nonlinear harmonics and modulations. 2. Nonlinear crack behavior Fig. 1 shows a cross section schematic of a Bi2Te3 wafer (first resonant mode approximately 1.4 kHz) with a crack of initial opening a0 prior to excitation and opening aðx3 ; tÞ þ a0 during excitation that extends through the wafer thickness. Several models have been developed to model the nonlinear behavior of cracks. When modeling cracks in 1D, nonlinearities are introduced by augmenting the stress–strain relationship of the material. Van Den Abeele et al. include classical nonlinearities and nonlinear hysteretic effects [9]. Solodov et al. model a crack as a bilinear spring, with high stiffness when the crack faces are in contact and low stiffness when the crack faces are not in contact [11]. Alternatively, Aleshin et al. model cracks in 2D or 3D numerically by implementing nonlinear normal N and shear T force vectors acting on a small area A of each crack face [18,19]
N ¼ nAKHðaðx3 ; tÞ þ a0 Þ ðaðx3 ; tÞ þ a0 Þ;
ð1Þ
and
: T ¼ A l x; N N
ð2Þ
Here, n is the surface normal of the crack face, H(∙) is the Heaviside function, K ¼ dN=da is the stiffness of the closed crack ðaðx3 ; tÞ þ a0 6 0Þ, and m is the slip-stick friction coefficient, which is dependent on the velocity vector x_ between two contacting points on opposing crack faces. We excite the wafer with a single piezoelectric transducer driven by a two-tone signal (voltage) consisting of a high-amplitude, low-frequency xL sinusoidal ‘‘pump” signal added to a high-frequency xH , lowamplitude ‘‘probe” signal. The pump signal generates high-amplitude perturbations in the crack opening aðx3 ; tÞ and causes the crack to ‘‘breathe,” wherein it opens and closes periodically with frequency xL . When aðx3 ; tÞ þ a0 > 0 the crack opens
Fig. 1. Cracked Bi2Te3 wafer schematic with normal N and shear force T direction indicated.
J. Greenhall et al. / Mechanical Systems and Signal Processing 139 (2020) 106598
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Fig. 2. Frequency spectrum of a cracked (solid red) and crack-free Bi2Te3 wafer (dashed black) with a low-frequency xL pump and high-frequency xH probe signal.
and the normal and shear forces acting on the crack faces become zero and when aðx3 ; tÞ þ a0 < 0 the crack closes and the forces become nonzero. This crack breathing results in nonlinear effects including harmonic generation and modulation. Fig. 2 shows the theoretical frequency spectrum of the measured signal for a crack-free wafer (dashed black) and for a cracked wafer (solid red) using the nonlinear contact acoustic model for a 1D crack [10,11]. We observe that the measured signal for the crack-free wafer consists only of the pump xL and probe signals xH . For the cracked Bi2Te3 wafer we observe harmonic signals at frequencies of ðn þ 1ÞxL for n = 1, 2, 3. . . and modulated sideband signals at frequencies of xH nxL n = 1, 2, 3. . ., as well as the peaks at the pump and probe frequencies. Previous published techniques assume that the wavelength of the elastic waves is significantly smaller than the dimensions of the material specimen, which enables assuming purely propagating waves. Alternatively, in Bi2Te3 wafers with dimensions on the order of the elastic wavelength, we generate standing waves that form at specific frequencies corresponding to the resonance modes of the Bi2Te3 wafers. Generation of nonlinear harmonics and modulated sidebands is contingent on the crack opening perturbation amplitude being sufficiently large to periodically overcome the initial crack opening and periodically close the crack. The amplitude of the nonlinear harmonic and modulated sideband signals is dependent on the wafer spectrum of the Bi2Te3 wafer in the pump-band xL and probe-band xH , and the wafer spectrum over the range frequencies where the nonlinearities will be generated, i.e. ðn þ 1ÞxL for n = 1, 2, 3. . . for the harmonics and xH nxL n = 1, 2, 3. . . for the modulated sidebands.
Fig. 3. (a) Pump-band wafer spectrum of a cracked Bi2Te3 wafer with four resonant mode peaks labeled A-D. (b) Plot of the harmonic-band wafer spectrum (solid) and the measured nonlinear harmonic signal (dashed). (c) Optical microscope image of the cracked Bi2Te3 wafer and images of the displacement amplitudes for the four resonant modes indicated in Fig. 3(a).
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As a demonstrative example, we consider generation of the second harmonic 2xL for a pump signal of Frequencies xL =2p 2 ½8; 12 kHz. The pump signal at xL drives crack breathing, which generates the second harmonic at 2xL , thus the amplitude of the second harmonic signal at 2xL is dependent on the amplitude of the vibration at xL . Additionally, the second harmonic waves will propagate and interfere with one another. If the frequency of the second harmonic signal 2xL corresponds with a resonant mode of the Bi2Te3 wafer, then the interference will be constructive and the second harmonic signal will be amplified. Thus, the amplitude of the second harmonic signal at a particular frequency 2xL is dependent on the wafer spectrum at frequency xL and 2xL . Fig. 3 shows the experimentally measured frequency characteristics of a Bi2Te3 wafer (see Sec. 4 for experimental setup details). Fig. 3(a) shows the measured pump-band (xL ) wafer spectrum Y(x) averaged over the surface of a cracked Bi2Te3 wafer with four wafer modes (modes A-D) indicated. Fig. 3(b) shows the harmonic-band (2xL ) wafer spectrum Y(x) (solid) and the measured nonlinear harmonic signal Y(x) (dashed) that correspond to the pump-band wafer spectrum in Fig. 3(a). Here, the harmonic-band wafer spectrum is the natural (linear) response of the wafer within the harmonic frequency band; i.e. we excite and measure the response at the same frequency 2xL . Conversely, the nonlinear harmonic signal is the signal generated at frequency 2xL by exciting the wafer at frequency xL . From Fig. 3(c) we observe that for frequencies x where the pump-band spectrum has a peak near x and the harmonicband spectrum has a peak near 2x, there is a peak in the nonlinear harmonic at 2x. Conversely, peaks (e.g. Modes A and C) in the pump range spectrum that correspond to low values in the harmonic range spectrum and do not generate a nonlinear signal. We also observe peaks in the nonlinear harmonic at frequencies that correspond to peaks in the harmonic-band wafer spectrum but no peak in the pump-band wafer spectrum. Here, the amplitude of the nonlinear harmonic peak is dependent on the quality factor (Q-factor) of the harmonic-band peak; higher Q-factor leads to higher vibration amplitudes and less energy dissipation. Additionally, the amplitude of the crack opening perturbation aðx3 ; tÞ is dependent on the location, orientation, and shape of the crack. Fig. 3(c) shows an optical microscope image of the cracked wafer and the resonant modes A-D at frequencies x = 8.4 (mode A), 9.3, (mode B), 10.5 (mode C), and 10.9 kHz (mode D). At each resonant mode, we observe a standing wave, which consists of nodes (blue) and antinodes (yellow), where the vibration amplitude is minimum and maximum, respectively. We observe that modes A, B, and D produce vibration antinodes near the crack (boxed in red), whereas mode C produces a vibration node near the crack. This indicates that a pump frequency of xL = 8.4 kHz (mode A), 9.3 kHz (mode B) or 10.9 kHz (mode D) will induce crack breathing, but a pump frequency of xL = 10.5 kHz (mode C) will not. To ensure that we excite multiple (at least three) resonant modes and that the first two harmonic bands (2xL and 3xL Þ do þ for pump-band limits not overlap, we select pump signal that consists of a linear chirp over the range xL 2 x L ; xL þ xL = 8 kHz and xL = 12 kHz. 3. Damage quantification To identify cracks, we quantify three damage metrics from the measured vibration signal Y(x). Cracked wafers are subjected to a decrease in the mean vibration amplitude of the resonance modes due to a reduction in the quality factor of the Bi2Te3 wafer and hysteretic nonlinearities [10,20]. We quantify the mean resonance amplitude as
Mmean ¼ mean YðxÞ:
ð3Þ
x2½0;1Þ
Next, we quantify the harmonic amplitude metric as
R xþ L
Mharm ¼
xL
YðxÞðYð2xÞ þ Yð3xÞÞdx ; R xþ M mean xL YðxÞdx
ð4Þ
L
þ þ here, wafer resonance modes in the second and third harmonic-bands x 2 ½2x L ; 2xL and x 2 ½3xL ; 3xL are susceptible to excitation from outside sources such as bulk movement of the wafer, which may be amplified by wafer resonances in the harmonic-band. To filter out signals that are not generated by the nonlinearities in the crack, we multiply the harmonic specþ þ þ tra (YðxÞ for x 2 ½2x L ; 2xL and x 2 ½3xL ; 3xL ) by the pump excitation spectrum ðYðxÞ for x 2 ½xL ; xL ). As a result, the integral in Eq. (4) will attribute greater weight to the harmonic signals that correspond to resonances in the pump-band and it will reduce the weight of harmonic signals due to other sources. Additionally, we include the mean resonance amplitude Mmean in the denominator of Eq. (4) to amplify the harmonic metric for cracked wafers, which typically experience a reduction in the mean vibration amplitude of their resonance modes [10,16]. Finally, we quantify the modulated sideband amplitude metric as
R xH xL Mmod ¼
xH xþL
R x þxþ YðxH ÞYðxÞdx þ xHHþxL YðxH ÞYðxÞdx L R xþ M mean xL YðxÞdx
ð5Þ
L
Again in Eq. (5), we multiply the modulated sideband signal (YðxÞ for x 2 ½xH xþ and L ; xH xL þ x 2 ½xH þ xL ; xH þ xL ) by the probe excitation signal YðxH Þ and we include the mean resonance amplitude Mmean in the denominator to amplify the metrics for cracked wafers.
J. Greenhall et al. / Mechanical Systems and Signal Processing 139 (2020) 106598
Laser vibrometer
Foam support
Oscilloscope
5
PC
Bi2Te3 wafer Ruby
Ultrasound transducer
Function Motorized generator stage
Fig. 4. Schematic of the experimental acoustic crack detection apparatus.
4. Experimental setup Fig. 4 shows the experimental setup used to evaluate Bi2Te3 wafers. A PC-controlled function generator (Tektronix AFG 2340) produces the excitation signal (pump and probe), which is amplified (E&I 240L) and transmitted into the Bi2Te3 wafer. We then measure the vibrations along the surface of the Bi2Te3 wafer via laser Doppler vibrometer (Polytec OFV-5000 measuring out-of-plane velocity) and oscilloscope (Tektronix DPO 4350). To mitigate the effect of measurement location, we average the Bi2Te3 wafer vibration measurements over 25 points (5 5 grid with 1 mm spacing) around the center of the Bi2Te3 wafer using a two-axis motorized stage (2 Newmark NLS4 linear actuators) controlled by a motor controller (Newmark NSC-M2) to move the Bi2Te3 wafer between measurements. To achieve contact without an adhesive or vacuum, the Bi2Te3 wafer is balanced between a foam support and a hemispherical ruby that is affixed to the center of the transmitter (PZT-4, center frequency of 1.0 MHz). High-amplitude resonant waves in cracked or crack-free Bi2Te3 wafers can cause the wafer to ‘‘chatter,” which results in intermittent contact between the transmitter and the Bi2Te3 wafer and, thus, can generate acoustic nonlinearities similar to the cracks. As a result, we cannot accurately detect cracks using a single damage metric, and instead we utilize all three damage metrics, defined in Eqs. (3)–(5). We implement the acoustic crack detection technique on 94 Bi2Te3 wafers with approximate dimensions of 30.0 18.0 0.5 mm and either one or two triangular notches with approximate dimensions 2.2 2.2 mm removed from the corners. Manual optical microscopy inspection (Keyence VHX 6000, 50 magnification) of the wafers is used to determine the ‘‘true” damage state of each wafer; we identify 17 cracked wafers and 77 crack-free wafers. Using this optical technique, we identified defects as small as 13 mm, but we note the possibility of smaller crack openings not being identified. 5. Results and discussion Fig. 5(a) shows the acoustic metrics from Eqs. (3)–(5) for each of the Bi2Te3 wafers. The color of each dot identifies whether the wafer is cracked (red) or crack-free (green), based on optical microscopy. From Fig. 5(a), we observe that it
Fig. 5. (a) Acoustic damage metrics including the harmonic amplitude Mharm, the modulated sideband amplitude Mmod, and the mean signal amplitude Mmean metrics. We show optical microscope and X-ray computed tomography (CT) images for wafers where cracks were not detected (Fig. 5(b)) and cracks were detected (Fig. 5(c)).
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J. Greenhall et al. / Mechanical Systems and Signal Processing 139 (2020) 106598
is possible to place a hyperplane (blue line) that separates the cracked and crack-free wafers such that we correctly identify cracked wafers with a total error of 4.3% (4/94 Bi2Te3 wafers) and correctly identify the crack free wafers with a total error of 7.5% (7/94 Bi2Te3 wafers). Thus, we demonstrate that using the metrics in Eqs. (3)–(5) results in quasi-linearly separable Bi2Te3 wafer data points, which means that the cracks can be accurately distinguished via a single linear classifier (a plane). More complex classification techniques such as support vector machines [21], decision trees [22], or boosting [23] could be implemented to further reduce the classification error, but a full machine learning analysis is beyond the scope of this manuscript. To understand why cracks were not detected in some Bi2Te3 wafers, we image a subsample of Bi2Te3 wafers via optical microscopy and X-ray computed tomography (CT). Fig. 5(b) and (c) show the optical microscope image (right) and CT image of the crack cross section (left) for a cracked Bi2Te3 wafer that was misclassified acoustically (Fig. 5(b)) and a cracked Bi2Te3 wafer that was correctly classified acoustically (Fig. 5(c)). The larger average initial crack opening a0 = 73 mm (Fig. 5(b)) requires a higher crack opening perturbation amplitude |a| to close the crack and induce breathing, compared to the average initial crack of opening a0 = 48 mm in Fig. 5(c). This results in low harmonic and sideband modulation peaks, which inhibits crack detection. To induce breathing in a crack with large initial opening a0 , the excitation signal amplitude must be increased. However, sufficiently large excitation signal amplitude will also result in a periodic loss of contact between the ruby hemisphere and the Bi2Te3 wafer that induces nonlinear harmonic and sideband generation in both cracked and crack-free wafers, which can impede crack detection. As a result, the acoustic crack detection technique is more effective for detecting cracks with small initial crack openings as opposed to large initial crack openings. In practice, a crack with a large initial crack opening is easily identifiable optically with little to no magnification, whereas a crack with a small initial crack opening is more difficult to identify optically. Thus, the acoustic crack detection technique serves a critical role in identifying cracks with small initial crack opening. Additionally, we observe no crack growth due to the crack breathing, but we note that the breathing process has the potential to remove material from one or both crack faces. However, in practice, any cracking in the wafer typically results in disposal of the wafer, and, thus any subsequent damage to an already cracked thermoelectric wafer is inconsequential. 6. Conclusion In conclusion, we present a new technique for identifying cracks in thermoelectric wafers via excitation of acoustic nonlinearities. In contrast with existing crack detection techniques in the literature, the new technique does not require uniform shape and material between wafers and it does not require the transmitter/receiver to be affixed to the wafer via adhesive or vacuum, thus decreasing inspection time, reducing the likelihood of wafer damage, and facilitating measurement of porous materials. We demonstrate the new crack detection technique experimentally on 94 bismuth telluride (Bi2Te3) thermoelectric wafers and compare the results to the cracks measured via optical microscopy. Using a linear classifier, we demonstrate identification of the cracked Bi2Te3 wafers with 4.3% total error and the crack-free Bi2Te3 wafers with 7.5% total error, thus validating the new crack detection technique. This crack detection technique finds application in testing thermoelectric wafers, silicon wafers, piezoelectric wafers, etc., which can lead to significant savings in manufacturing time and cost. Additionally, the technique can be extended to other applications that require nondestructive crack identification, such as welding, concrete, and composite materials. CRediT authorship contribution statement J. Greenhall: Methodology, Data curation, Formal analysis, Investigation, Writing - original draft, Visualization. S. Grutzik: Methodology, Formal analysis. A. Graham: Funding acquisition, Resources. D.N. Sinha: Conceptualization, Project administration. C. Pantea: Conceptualization, Project administration, Writing - review & editing. Declaration of competing interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements We acknowledge Siddharth Komini Babu for performing the X-ray CT imaging of the cracked Bi2Te3 wafers. This work was funded by the National Nuclear Security Administration and the U.S. Department of Energy. References [1] S. Twaha, J. Zhu, Y. Yan, B. Li, A comprehensive review of thermoelectric technology: Materials, applications, modelling and performance improvement, Renew. Sustain. Energy Rev. 65 (2016) 698–726. [2] R. Chein, G. Huang, Thermoelectric cooler application in electronic cooling, Appl. Therm. Eng. 24 (14) (2004) 2207–2217. [3] J. Chen, Y. Zhou, H. Wang, J.T. Wang, Comparison of the optimal performance of single- and two-stage thermoelectric refrigeration systems, Appl. Energy 73 (3) (2002) 285–298.
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Nonlinear acoustic crack detection in thermoelectric wafers J. Greenhall a,⇑, S. Grutzik b, A. Graham a, D.N. Sinha a, C. Pantea a a b
Acoustics and Sensors Team, Materials Physics and Applications Group (MPA-11), Los Alamos National Laboratory, Los Alamos, NM 87545, USA Component Science and Mechanics, Sandia National Laboratory, Albuquerque, NM 87123, USA
a r t i c l e
i n f o
Article history: Received 31 July 2019 Received in revised form 12 November 2019 Accepted 22 December 2019
Keywords: Nonlinear acoustics Nondestructive evaluation Acoustic crack detection Thermoelectric wafer testing
a b s t r a c t An acoustic technique for detecting cracks in bismuth telluride (Bi2Te3) thermoelectric wafers is presented. The technique is based on mechanically exciting the wafers with a low frequency pump signal that drives a crack to ‘‘breathe” by opening and closing periodically and a high frequency probe signal that interrogates the breathing crack. Interaction between the pump signal, the probe signal, and the crack leads to generation of acoustic nonlinearities in the wafer. In contrast to existing acoustic crack detection techniques we utilize standing waves within the wafers to facilitate simultaneous crack detection throughout the wafer, which reduces the total inspection time; we do not require uniform dimensions and material properties between wafers; and we do not affix the transmitter/ receiver to the wafers, which can lead to wafer damage. We present a technique that integrates multiple damage metrics to identify cracks within the wafers. We demonstrate the technique on 94 Bi2Te3 wafers, of which we correctly identify cracked wafers with 4.3% total error and we correctly identify crack-free wafers with 7.5% total error. This acoustic crack detection technique finds application in manufacturing of thermoelectric wafers and other materials, where identifying cracks early in the manufacturing process results in significant time and cost savings. Additionally, this acoustic crack detection technique has implications for other nondestructive inspection applications where cracks cannot be detected effectively through existing techniques. Published by Elsevier Ltd.
1. Introduction Thermoelectric materials such as bismuth telluride (Bi2Te3) have attracted significant attention due to their ability to convert thermal energy into electric energy or vice versa within a solid state device via the Peltier effect [1]. These materials have significant applications including cooling electronics at miniature scales [2], refrigeration [3], and thermal energy harvesting [4]. Bi2Te3 wafers tend to be highly brittle and they are susceptible to cracking during manufacturing, which would inhibit thermal-electric energy conversion. Thus, it is critical to detect cracks in Bi2Te3 to ensure the efficacy of the energy conversion devices. Ideally, cracks are identified when the Bi2Te3 material is in wafer form, prior to subsequent manufacturing processes to minimize wasted time/labor. Currently, there are a wide range of techniques for nondestructive identification of cracks utilizing optical microscopy [5,6], X-ray computed tomography (CT) [7,8], and ultrasound [9–16]. However, automated optical techniques based on image processing are sensitive to variations in surface texture and lighting and are limited to surface defects, manual optical techniques are costly and time-consuming, and X-ray CT requires prohibitively long measurement times. Current ultrasound crack detection techniques are categorized as either burst techniques, wherein ⇑ Corresponding author. E-mail address: [email protected] (J. Greenhall). https://doi.org/10.1016/j.ymssp.2019.106598 0888-3270/Published by Elsevier Ltd.
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an ultrasound burst propagates through the material; or resonance techniques, wherein a standing ultrasound wave is established throughout the material. Burst techniques rely either on transmitting the burst into the material and measuring reflections [12] or radiation [13] from cracks, or they excite the crack and cause it to open and close, which generates nonlinear harmonics [9,11] and/or modulations [14]. Burst techniques are typically used on materials with dimensions significantly larger than the wavelength of the ultrasound to ensure that transmitted signal does not reflect from the edges of the material and interfere with the measured signal. Additionally, these techniques are sensitive to the placement of the transmitting and receiving ultrasound transducers with respect to the crack position and orientation, and typically require a large number of transmitter/receiver configurations to ensure that the cracks are properly interrogated. Conversely, resonance techniques utilize standing waves within the materials, which works well for material geometries on the order of the wavelength, and the standing wave sets up over the entire material simultaneously, which reduces sensitivity to the transmitter location. Current resonance techniques work by setting up a standing ultrasound wave into the wafer at a range of different frequencies to measure the resonance spectrum of the wafer. Cracks are then identified by contrasting the frequency spectrum to that of a baseline sample that is known to be crack-free [15]. The critical problem with this methodology is that it requires uniform geometry and material properties between all wafers. Alternatively, cracks can be detected by exploiting shifts in the frequency spectrum peaks at different excitation amplitudes [10]. But, this technique is limited by the fact that at sufficiently high amplitudes resonance peaks can shift due to nonlinearities within the material itself, which can be mistaken for cracks. Additionally, the techniques require measuring the frequency spectrum at multiple excitation amplitudes, which increases the overall measurement time. The current ultrasound crack detection techniques are also limited by the fact that they either assume a fixed rigid connection between the transmitter/receiver and the material specimen via an adhesive or vacuum [9–11,14,15], or they utilize a laser pulse to generate an ultrasound wave [12,13]. Obtaining a fixed rigid connection between a transmitter/receiver and the Bi2Te3 wafers via adhesive is not feasible due to the fact that removing the adhesive from the Bi2Te3 wafer can lead to wafer damage and due to the long adhesive curing time. Similarly, the stresses induced by laser pulse excitation can lead to Bi2Te3 wafer damage, which can adversely affect the Bi2Te3 material properties [17]. Finally, a fixed rigid connection cannot be achieved via vacuum due to the porous texture of the Bi2Te3 material. In contrast with existing ultrasound crack detection techniques, the objective of this work is to develop a fast, accurate technique for detecting cracks in Bi2Te3 wafers without the use of a fixed rigid connection between the transmitter/receiver and the wafer. To achieve this, we generate a resonating ultrasound wave within the Bi2Te3 wafer and quantify the vibration amplitude of the resonant wave and the amplitude of generated nonlinear harmonics and modulations. 2. Nonlinear crack behavior Fig. 1 shows a cross section schematic of a Bi2Te3 wafer (first resonant mode approximately 1.4 kHz) with a crack of initial opening a0 prior to excitation and opening aðx3 ; tÞ þ a0 during excitation that extends through the wafer thickness. Several models have been developed to model the nonlinear behavior of cracks. When modeling cracks in 1D, nonlinearities are introduced by augmenting the stress–strain relationship of the material. Van Den Abeele et al. include classical nonlinearities and nonlinear hysteretic effects [9]. Solodov et al. model a crack as a bilinear spring, with high stiffness when the crack faces are in contact and low stiffness when the crack faces are not in contact [11]. Alternatively, Aleshin et al. model cracks in 2D or 3D numerically by implementing nonlinear normal N and shear T force vectors acting on a small area A of each crack face [18,19]
N ¼ nAKHðaðx3 ; tÞ þ a0 Þ ðaðx3 ; tÞ þ a0 Þ;
ð1Þ
and
: T ¼ A l x; N N
ð2Þ
Here, n is the surface normal of the crack face, H(∙) is the Heaviside function, K ¼ dN=da is the stiffness of the closed crack ðaðx3 ; tÞ þ a0 6 0Þ, and m is the slip-stick friction coefficient, which is dependent on the velocity vector x_ between two contacting points on opposing crack faces. We excite the wafer with a single piezoelectric transducer driven by a two-tone signal (voltage) consisting of a high-amplitude, low-frequency xL sinusoidal ‘‘pump” signal added to a high-frequency xH , lowamplitude ‘‘probe” signal. The pump signal generates high-amplitude perturbations in the crack opening aðx3 ; tÞ and causes the crack to ‘‘breathe,” wherein it opens and closes periodically with frequency xL . When aðx3 ; tÞ þ a0 > 0 the crack opens
Fig. 1. Cracked Bi2Te3 wafer schematic with normal N and shear force T direction indicated.
J. Greenhall et al. / Mechanical Systems and Signal Processing 139 (2020) 106598
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Fig. 2. Frequency spectrum of a cracked (solid red) and crack-free Bi2Te3 wafer (dashed black) with a low-frequency xL pump and high-frequency xH probe signal.
and the normal and shear forces acting on the crack faces become zero and when aðx3 ; tÞ þ a0 < 0 the crack closes and the forces become nonzero. This crack breathing results in nonlinear effects including harmonic generation and modulation. Fig. 2 shows the theoretical frequency spectrum of the measured signal for a crack-free wafer (dashed black) and for a cracked wafer (solid red) using the nonlinear contact acoustic model for a 1D crack [10,11]. We observe that the measured signal for the crack-free wafer consists only of the pump xL and probe signals xH . For the cracked Bi2Te3 wafer we observe harmonic signals at frequencies of ðn þ 1ÞxL for n = 1, 2, 3. . . and modulated sideband signals at frequencies of xH nxL n = 1, 2, 3. . ., as well as the peaks at the pump and probe frequencies. Previous published techniques assume that the wavelength of the elastic waves is significantly smaller than the dimensions of the material specimen, which enables assuming purely propagating waves. Alternatively, in Bi2Te3 wafers with dimensions on the order of the elastic wavelength, we generate standing waves that form at specific frequencies corresponding to the resonance modes of the Bi2Te3 wafers. Generation of nonlinear harmonics and modulated sidebands is contingent on the crack opening perturbation amplitude being sufficiently large to periodically overcome the initial crack opening and periodically close the crack. The amplitude of the nonlinear harmonic and modulated sideband signals is dependent on the wafer spectrum of the Bi2Te3 wafer in the pump-band xL and probe-band xH , and the wafer spectrum over the range frequencies where the nonlinearities will be generated, i.e. ðn þ 1ÞxL for n = 1, 2, 3. . . for the harmonics and xH nxL n = 1, 2, 3. . . for the modulated sidebands.
Fig. 3. (a) Pump-band wafer spectrum of a cracked Bi2Te3 wafer with four resonant mode peaks labeled A-D. (b) Plot of the harmonic-band wafer spectrum (solid) and the measured nonlinear harmonic signal (dashed). (c) Optical microscope image of the cracked Bi2Te3 wafer and images of the displacement amplitudes for the four resonant modes indicated in Fig. 3(a).
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As a demonstrative example, we consider generation of the second harmonic 2xL for a pump signal of Frequencies xL =2p 2 ½8; 12 kHz. The pump signal at xL drives crack breathing, which generates the second harmonic at 2xL , thus the amplitude of the second harmonic signal at 2xL is dependent on the amplitude of the vibration at xL . Additionally, the second harmonic waves will propagate and interfere with one another. If the frequency of the second harmonic signal 2xL corresponds with a resonant mode of the Bi2Te3 wafer, then the interference will be constructive and the second harmonic signal will be amplified. Thus, the amplitude of the second harmonic signal at a particular frequency 2xL is dependent on the wafer spectrum at frequency xL and 2xL . Fig. 3 shows the experimentally measured frequency characteristics of a Bi2Te3 wafer (see Sec. 4 for experimental setup details). Fig. 3(a) shows the measured pump-band (xL ) wafer spectrum Y(x) averaged over the surface of a cracked Bi2Te3 wafer with four wafer modes (modes A-D) indicated. Fig. 3(b) shows the harmonic-band (2xL ) wafer spectrum Y(x) (solid) and the measured nonlinear harmonic signal Y(x) (dashed) that correspond to the pump-band wafer spectrum in Fig. 3(a). Here, the harmonic-band wafer spectrum is the natural (linear) response of the wafer within the harmonic frequency band; i.e. we excite and measure the response at the same frequency 2xL . Conversely, the nonlinear harmonic signal is the signal generated at frequency 2xL by exciting the wafer at frequency xL . From Fig. 3(c) we observe that for frequencies x where the pump-band spectrum has a peak near x and the harmonicband spectrum has a peak near 2x, there is a peak in the nonlinear harmonic at 2x. Conversely, peaks (e.g. Modes A and C) in the pump range spectrum that correspond to low values in the harmonic range spectrum and do not generate a nonlinear signal. We also observe peaks in the nonlinear harmonic at frequencies that correspond to peaks in the harmonic-band wafer spectrum but no peak in the pump-band wafer spectrum. Here, the amplitude of the nonlinear harmonic peak is dependent on the quality factor (Q-factor) of the harmonic-band peak; higher Q-factor leads to higher vibration amplitudes and less energy dissipation. Additionally, the amplitude of the crack opening perturbation aðx3 ; tÞ is dependent on the location, orientation, and shape of the crack. Fig. 3(c) shows an optical microscope image of the cracked wafer and the resonant modes A-D at frequencies x = 8.4 (mode A), 9.3, (mode B), 10.5 (mode C), and 10.9 kHz (mode D). At each resonant mode, we observe a standing wave, which consists of nodes (blue) and antinodes (yellow), where the vibration amplitude is minimum and maximum, respectively. We observe that modes A, B, and D produce vibration antinodes near the crack (boxed in red), whereas mode C produces a vibration node near the crack. This indicates that a pump frequency of xL = 8.4 kHz (mode A), 9.3 kHz (mode B) or 10.9 kHz (mode D) will induce crack breathing, but a pump frequency of xL = 10.5 kHz (mode C) will not. To ensure that we excite multiple (at least three) resonant modes and that the first two harmonic bands (2xL and 3xL Þ do þ for pump-band limits not overlap, we select pump signal that consists of a linear chirp over the range xL 2 x L ; xL þ xL = 8 kHz and xL = 12 kHz. 3. Damage quantification To identify cracks, we quantify three damage metrics from the measured vibration signal Y(x). Cracked wafers are subjected to a decrease in the mean vibration amplitude of the resonance modes due to a reduction in the quality factor of the Bi2Te3 wafer and hysteretic nonlinearities [10,20]. We quantify the mean resonance amplitude as
Mmean ¼ mean YðxÞ:
ð3Þ
x2½0;1Þ
Next, we quantify the harmonic amplitude metric as
R xþ L
Mharm ¼
xL
YðxÞðYð2xÞ þ Yð3xÞÞdx ; R xþ M mean xL YðxÞdx
ð4Þ
L
þ þ here, wafer resonance modes in the second and third harmonic-bands x 2 ½2x L ; 2xL and x 2 ½3xL ; 3xL are susceptible to excitation from outside sources such as bulk movement of the wafer, which may be amplified by wafer resonances in the harmonic-band. To filter out signals that are not generated by the nonlinearities in the crack, we multiply the harmonic specþ þ þ tra (YðxÞ for x 2 ½2x L ; 2xL and x 2 ½3xL ; 3xL ) by the pump excitation spectrum ðYðxÞ for x 2 ½xL ; xL ). As a result, the integral in Eq. (4) will attribute greater weight to the harmonic signals that correspond to resonances in the pump-band and it will reduce the weight of harmonic signals due to other sources. Additionally, we include the mean resonance amplitude Mmean in the denominator of Eq. (4) to amplify the harmonic metric for cracked wafers, which typically experience a reduction in the mean vibration amplitude of their resonance modes [10,16]. Finally, we quantify the modulated sideband amplitude metric as
R xH xL Mmod ¼
xH xþL
R x þxþ YðxH ÞYðxÞdx þ xHHþxL YðxH ÞYðxÞdx L R xþ M mean xL YðxÞdx
ð5Þ
L
Again in Eq. (5), we multiply the modulated sideband signal (YðxÞ for x 2 ½xH xþ and L ; xH xL þ x 2 ½xH þ xL ; xH þ xL ) by the probe excitation signal YðxH Þ and we include the mean resonance amplitude Mmean in the denominator to amplify the metrics for cracked wafers.
J. Greenhall et al. / Mechanical Systems and Signal Processing 139 (2020) 106598
Laser vibrometer
Foam support
Oscilloscope
5
PC
Bi2Te3 wafer Ruby
Ultrasound transducer
Function Motorized generator stage
Fig. 4. Schematic of the experimental acoustic crack detection apparatus.
4. Experimental setup Fig. 4 shows the experimental setup used to evaluate Bi2Te3 wafers. A PC-controlled function generator (Tektronix AFG 2340) produces the excitation signal (pump and probe), which is amplified (E&I 240L) and transmitted into the Bi2Te3 wafer. We then measure the vibrations along the surface of the Bi2Te3 wafer via laser Doppler vibrometer (Polytec OFV-5000 measuring out-of-plane velocity) and oscilloscope (Tektronix DPO 4350). To mitigate the effect of measurement location, we average the Bi2Te3 wafer vibration measurements over 25 points (5 5 grid with 1 mm spacing) around the center of the Bi2Te3 wafer using a two-axis motorized stage (2 Newmark NLS4 linear actuators) controlled by a motor controller (Newmark NSC-M2) to move the Bi2Te3 wafer between measurements. To achieve contact without an adhesive or vacuum, the Bi2Te3 wafer is balanced between a foam support and a hemispherical ruby that is affixed to the center of the transmitter (PZT-4, center frequency of 1.0 MHz). High-amplitude resonant waves in cracked or crack-free Bi2Te3 wafers can cause the wafer to ‘‘chatter,” which results in intermittent contact between the transmitter and the Bi2Te3 wafer and, thus, can generate acoustic nonlinearities similar to the cracks. As a result, we cannot accurately detect cracks using a single damage metric, and instead we utilize all three damage metrics, defined in Eqs. (3)–(5). We implement the acoustic crack detection technique on 94 Bi2Te3 wafers with approximate dimensions of 30.0 18.0 0.5 mm and either one or two triangular notches with approximate dimensions 2.2 2.2 mm removed from the corners. Manual optical microscopy inspection (Keyence VHX 6000, 50 magnification) of the wafers is used to determine the ‘‘true” damage state of each wafer; we identify 17 cracked wafers and 77 crack-free wafers. Using this optical technique, we identified defects as small as 13 mm, but we note the possibility of smaller crack openings not being identified. 5. Results and discussion Fig. 5(a) shows the acoustic metrics from Eqs. (3)–(5) for each of the Bi2Te3 wafers. The color of each dot identifies whether the wafer is cracked (red) or crack-free (green), based on optical microscopy. From Fig. 5(a), we observe that it
Fig. 5. (a) Acoustic damage metrics including the harmonic amplitude Mharm, the modulated sideband amplitude Mmod, and the mean signal amplitude Mmean metrics. We show optical microscope and X-ray computed tomography (CT) images for wafers where cracks were not detected (Fig. 5(b)) and cracks were detected (Fig. 5(c)).
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is possible to place a hyperplane (blue line) that separates the cracked and crack-free wafers such that we correctly identify cracked wafers with a total error of 4.3% (4/94 Bi2Te3 wafers) and correctly identify the crack free wafers with a total error of 7.5% (7/94 Bi2Te3 wafers). Thus, we demonstrate that using the metrics in Eqs. (3)–(5) results in quasi-linearly separable Bi2Te3 wafer data points, which means that the cracks can be accurately distinguished via a single linear classifier (a plane). More complex classification techniques such as support vector machines [21], decision trees [22], or boosting [23] could be implemented to further reduce the classification error, but a full machine learning analysis is beyond the scope of this manuscript. To understand why cracks were not detected in some Bi2Te3 wafers, we image a subsample of Bi2Te3 wafers via optical microscopy and X-ray computed tomography (CT). Fig. 5(b) and (c) show the optical microscope image (right) and CT image of the crack cross section (left) for a cracked Bi2Te3 wafer that was misclassified acoustically (Fig. 5(b)) and a cracked Bi2Te3 wafer that was correctly classified acoustically (Fig. 5(c)). The larger average initial crack opening a0 = 73 mm (Fig. 5(b)) requires a higher crack opening perturbation amplitude |a| to close the crack and induce breathing, compared to the average initial crack of opening a0 = 48 mm in Fig. 5(c). This results in low harmonic and sideband modulation peaks, which inhibits crack detection. To induce breathing in a crack with large initial opening a0 , the excitation signal amplitude must be increased. However, sufficiently large excitation signal amplitude will also result in a periodic loss of contact between the ruby hemisphere and the Bi2Te3 wafer that induces nonlinear harmonic and sideband generation in both cracked and crack-free wafers, which can impede crack detection. As a result, the acoustic crack detection technique is more effective for detecting cracks with small initial crack openings as opposed to large initial crack openings. In practice, a crack with a large initial crack opening is easily identifiable optically with little to no magnification, whereas a crack with a small initial crack opening is more difficult to identify optically. Thus, the acoustic crack detection technique serves a critical role in identifying cracks with small initial crack opening. Additionally, we observe no crack growth due to the crack breathing, but we note that the breathing process has the potential to remove material from one or both crack faces. However, in practice, any cracking in the wafer typically results in disposal of the wafer, and, thus any subsequent damage to an already cracked thermoelectric wafer is inconsequential. 6. Conclusion In conclusion, we present a new technique for identifying cracks in thermoelectric wafers via excitation of acoustic nonlinearities. In contrast with existing crack detection techniques in the literature, the new technique does not require uniform shape and material between wafers and it does not require the transmitter/receiver to be affixed to the wafer via adhesive or vacuum, thus decreasing inspection time, reducing the likelihood of wafer damage, and facilitating measurement of porous materials. We demonstrate the new crack detection technique experimentally on 94 bismuth telluride (Bi2Te3) thermoelectric wafers and compare the results to the cracks measured via optical microscopy. Using a linear classifier, we demonstrate identification of the cracked Bi2Te3 wafers with 4.3% total error and the crack-free Bi2Te3 wafers with 7.5% total error, thus validating the new crack detection technique. This crack detection technique finds application in testing thermoelectric wafers, silicon wafers, piezoelectric wafers, etc., which can lead to significant savings in manufacturing time and cost. Additionally, the technique can be extended to other applications that require nondestructive crack identification, such as welding, concrete, and composite materials. CRediT authorship contribution statement J. Greenhall: Methodology, Data curation, Formal analysis, Investigation, Writing - original draft, Visualization. S. Grutzik: Methodology, Formal analysis. A. Graham: Funding acquisition, Resources. D.N. Sinha: Conceptualization, Project administration. C. Pantea: Conceptualization, Project administration, Writing - review & editing. Declaration of competing interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements We acknowledge Siddharth Komini Babu for performing the X-ray CT imaging of the cracked Bi2Te3 wafers. This work was funded by the National Nuclear Security Administration and the U.S. Department of Energy. References [1] S. Twaha, J. Zhu, Y. Yan, B. Li, A comprehensive review of thermoelectric technology: Materials, applications, modelling and performance improvement, Renew. Sustain. Energy Rev. 65 (2016) 698–726. [2] R. Chein, G. Huang, Thermoelectric cooler application in electronic cooling, Appl. Therm. Eng. 24 (14) (2004) 2207–2217. [3] J. Chen, Y. Zhou, H. Wang, J.T. Wang, Comparison of the optimal performance of single- and two-stage thermoelectric refrigeration systems, Appl. Energy 73 (3) (2002) 285–298.
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