Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain D. D. Markushev, M. D. Rabasović, D. M. Todorović, S. Galović, and S. E. Bialkowski Citation: Review of Scientific Instruments 86, 035110 (2015); doi: 10.1063/1.4914894 View online: http://dx.doi.org/10.1063/1.4914894 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/86/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of manipulating the signal-to-noise envelope power ratio on speech intelligibility J. Acoust. Soc. Am. 137, 1401 (2015); 10.1121/1.4908240 Photoacoustic Fourier transform infrared spectroscopy of nanoporous SiO x ∕ Si thin films with varying porosities J. Appl. Phys. 98, 114310 (2005); 10.1063/1.2138376 Signal-to-noise ratio improvement by sideband intermixing: Application to Doppler ultrasound vibrometry Rev. Sci. Instrum. 74, 4191 (2003); 10.1063/1.1599070 Erratum: “Differentiating between elastically bent rectangular beams and plates” [Appl. Phys. Lett. 80, 2284 (2002)] Appl. Phys. Lett. 80, 4871 (2002); 10.1063/1.1489489 Differentiating between elastically bent rectangular beams and plates Appl. Phys. Lett. 80, 2284 (2002); 10.1063/1.1459762
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REVIEW OF SCIENTIFIC INSTRUMENTS 86, 035110 (2015)
Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain D. D. Markushev,1 M. D. Rabasović,1 D. M. Todorović,2 S. Galović,3 and S. E. Bialkowski4
1
Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade-Zemun, Serbia Institute for Multidisciplinary Researches, University of Belgrade, P.O. Box 33, 11030 Belgrade, Serbia 3 Institute of Nuclear Sciences “Vinca”, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia 4 Department of Chemistry and Biochemistry, Utah State University, Logan, Utah 84322-0300, USA 2
(Received 3 September 2014; accepted 28 February 2015; published online 23 March 2015) Methods for photoacoustic signal measurement, rectification, and analysis for 85 µm thin Si samples in the 20-20 000 Hz modulation frequency range are presented. Methods for frequency-dependent amplitude and phase signal rectification in the presence of coherent and incoherent noise as well as distortion due to microphone characteristics are presented. Signal correction is accomplished using inverse system response functions deduced by comparing real to ideal signals for a sample with well-known bulk parameters and dimensions. The system response is a piece-wise construction, each component being due to a particular effect of the measurement system. Heat transfer and elastic effects are modeled using standard Rosencweig-Gersho and elastic-bending theories. Thermal diffusion, thermoelastic, and plasmaelastic signal components are calculated and compared to measurements. The differences between theory and experiment are used to detect and correct signal distortion and to determine detector and sound-card characteristics. Corrected signal analysis is found to faithfully reflect known sample parameters. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914894] INTRODUCTION
Photoacoustic (PA) spectroscopy of solids measures a pressure change resulting from sample heating due to optical absorption.1–6 For opaque solids, surface absorption is followed by heat transfer through the sample and coupling fluid. Solid heating also results in elastic bending and thermal expansion effects that may contribute to the total signal. The heat transfer is due to thermal diffusion (TD) while mechanical elastic effects are thermoelastic (TE) processes. In the end, all of these effects generate pressure changes within the sample cell and are detected as PA signal.7–10 Using RosencweigGersho (RG)1 and elastic-bending (EB)2,9 models with an open-cell (OC) experimental setup,12 one can use photoacoustic spectroscopy to determine thermal and elastic parameters describing solid samples.11–13 In the case of metals and insulators, only the TD and TE components exist, simplifying the analysis. But our ultimate goal is to characterize the thin films (primarily photocatalytic) layered on semiconductors.14–16 Therefore, we chose Si as a substrate having well defined thermal17 and perfect elastic characteristics.9 On the other hand, the choice of wavelengths used to illuminate the sample is limited by the photocatalytic film analysis needs, so the use of wavelengths that causes free carriers generation could not be avoided. In the case of Si, and semiconductors in general, opticallyinduced free carrier generation contributes to the TD and TE effects.17–19 Free carriers can also induce elastic stress resulting in plasmaelastic (PE) effects. Subsequently, the semiconductor PA signal is due to a combination of TD, TE, and PE effects. PA spectroscopy may thus be used to determine thermal, mechanical, optical, and electronic characteristics of semiconductors. With an apparatus adapted to one-layer samples, all of
these effects can be elucidated within a modulation frequency range from 20 Hz to 20 kHz by applying the EB model for data analysis.14,15,20,21 Unfortunately, noise, distortion and interference often mask the “true” PA signal. Noise and signal distortion analysis is subsequently an unavoidable task in accurate PA measurements. Recognition, quantitation, and correction for these instrumental effects allow one to obtain the “true” PA signal, clear of many types of interference and distortion. In this report, PA amplitude and phase signals are analyzed for noise and systematic distortion within the 20 Hz–20 kHz modulation frequency range. A 85 µm plate of well-characterized Si is used as the sample. Differences between “true” and experimental signals are identified and characterized as system noise and signal distortion originated from the instruments used to construct the apparatus. Signal distortion analysis is modeled by cascade high and low pass filters, and the corresponding system transfer functions are derived. Undistorted signals are obtained by applying the inverse system transfer functions. “True” frequencydependent PA signal target response is calculated using RG and EB theories and the relationship between TD, TE, and PE signal components.9 Good agreement is found between the processed PA signal and the theoretical model after accounting for instrumental response. The applicability of these signal models is shown to be credible across the frequency range for thin semiconductor samples.
THEORETICAL MODEL
The theoretical model used here is adopted for the optical excitation of a homogeneous semiconductor sample with thickness l s (Figure 1(a)). Excitation intensity is given by
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FIG. 1. Transmission photoacoustic spectroscopy model: (a) thermal diffusion and (b) thermal and plasma elastic analysis.
I = I0 exp (−jωt), where j is the imaginary unit, t is the time, ω = 2π f , and f is the modulation frequency. General assumptions are that the sample absorbs at least part of the incident optical energy and sample irradiation is uniform over the entire surface. As a consequence of the latter assumption, heat propagates only along the z-axes (Figure 1(a)). Thus, heat transfer is considered to be a 1D process. Basic model configuration in the case of transmission photoacoustic apparatus is presented in Figure 1(a). The total pressure change, i.e., photoacoustic signal, δp (ω) in the cell is a sum of those due to the TD, TE, and PE processes and is represented as δp (ω) = δpTD (ω) + δpTE (ω) + δpPE (ω) .
(1)
The temperature distribution Ts (z,t) can be calculated by9,14 Ts (z, ω) = [b1 exp (γz) + b2 exp (−γz) + b3n (z) + b4 exp (βz)] exp ( jωt) ,
important at higher frequencies and have a strong influence on the total PA signal generation. In the case of small vibrational amplitudes, the δpTE (ω) and δpPE (ω) calculations may use a cylindrical approximation (Figure 1(b)) that assumes the acoustic vibration modes are independent of the polar coordinate ϕ. δpTE (ω) and δpPE (ω) are calculated using4,8 γp0 Rs δpm (ω) = 2πrUz (r, z) dr, (4) V0 0 where exp ( jωt) is omitted for simplicity.9,14–19,22 Here, m = TE,PE indicates the particular effect, V0 and p0 are the cell volume and ambient pressure, r is the radial coordinate, z is the longitudinal position, Rs is the sample’s effective radius, and Uz (r, z) is the displacement function describing linear elastic bending and volume expansion. For TE effects, this function can be written in the form of a sum, Uz (r, z) = U1z (r, z) + U2z (r, z) .
(2)
where n (z) is the free carrier concentration, β is the sample absorption coefficient, and b1, b2, b3, and b4 are the complex constants depending on the sample characteristics.9,14 Also, γ = jω/D s , with the thermal diffusivity, Dz , given by D s = k s / (ρC) where k s is the heat conductivity, ρ is the density, and C is the heat capacity. Equation (2) allows one to calculate the temperature Ts (l s ,t) at the back side of the sample (z = l s ). This temperature is responsible to the TD component of PA signal δpTD,1–4 γg p0 Dg (3) δpTD (ω) = √ T (l s , ω) , l cT0 ω where T0 and p0 are the gas temperature and pressure, respectively, γg is the adiabatic ratio, Dg is the gas thermal diffusivity, and l c is the PA cell length. It is well known that reducing the thickness of the Si sample increases the TD and TE components, with TD increasing faster than TE.9,14–17 In the case of thin Si samples, TD effects usually dominate in the (20 Hz–20 kHz) frequency range. In the case of the thin Si semiconductor plate analyzed here (85 µm), the TD process is dominant for modulation frequencies lower than 1000 Hz. TE and PE effects become
(5)
The sample bending term is U1z (r, z) = α
6 Rs2 − r 2 l s3
MT ,
(6)
while thermal expansion term is given by z 1+ν T (z) dz U2z (r, z) = α 1 − ν l s ( 2 2 2 z+ 3ν 4z − l s − MT + 1 − ν ls l s3
ls 2
)
NT .
(7)
Here, α is the coefficient of linear expansion, l s is the sample thickness, ν is Poisson’s ratio, and T (z) is the temperature distribution along the z–axis (Figure 1(b)). MT and NT are defined as l s /2
l s /2 MT =
zT (z) dz and NT = −l s /2
T (z) dz. −l s /2
(8)
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FIG. 2. Theoretical (a) amplitude R( f ) and (b) phase ϕ( f ) of the PA signal (asterisks + line) as a function of the modulation frequency, f , together with their thermal diffusion (TD—solid line), thermal elastic (TE—dashed line), and plasma elastic (PE—dotted line) components.
frequency range and valid for samples other than the particular Si used here.22
In the case of free carrier generation, PE effects are also calculated using Eqs. (4)–(8). However, one uses the coefficient of electronic deformation d n instead of α, and the free carrier concentration n(z) instead of T(z). Details regarding Eqs. (4)–(8) can be found in Refs. 16 and 23. Figure 2 illustrates the theoretical PA signal (a) amplitude Rtheo ( f ) and (b) phase ϕtheo ( f ), and the corresponding TD, TE, and PE components. The calculations are performed for a 85 µm thick Si plate with a diameter of 0.7 cm. The bulk parameters are given in the Table I. This PA signal will be referred to as the “true” PA signal in our noise analysis and the bulk parameters listed in Table I are the target of the signal analysis. We assume that model is valid across the modulation
APPARATUS AND NOISE ANALYSIS
The photoacoustic apparatus consists of five main components: (1) sample, (2) light source, (3) cell, (4) detector, (5) electronic signal processing equipment. A schematic diagram illustrating the main components is depicted in Figure 3. A blank (6 in Figure 3) is added as an optical beam stop for noise measurements. All measurements are performed with a single Si thin plate sample, prepared from 3-5 Ω cm, ntype, and ⟨100⟩ oriented Si wafer. The bulk sample parameters
TABLE I. Sample and system parameters. Si sample bulk parameters Density Optical reflectivity Optical absorption coefficient Thermal diffusivity Excitation energy Energy gap Lifetime of photogenerated carriers
ρ = 2.33 × 103
−3
kg m
R = 0.30 β = 5.00 × 105 m−1 D s = 1.00 × 10−4 m2 s−1 ε = 1.88 eV εg = 1.11 eV τ = 6.0 × 10−6 s
Coefficient of carrier diffusion Recombination velocity of the front surface Recombination velocity of the rear surface Young’s modulus Linear thermal expansion Coefficient of electronic deformation
D n = 1.2 × 10−3 s 1 = 2 m s−1 s 2 = 24 m s−1 E = 1.31 × 1011 N m−2 α = 3.0 × 10−6 K−1 d n = −9.0 × 10−31 m3
Calculated system parameters in the case of optical excitation power P0 = 1.5×10−3 W LF Sound card eLF1 time constant Sound card eLF1 frequency
τeLF1 = (1.1 ± 0.1) × 10−2 s f eLF1 = (15 ± 1) Hz
Detector eLF2 time constant Detector eLF2 frequency
τeLF2 = (6.3 ± 0.3) × 10−3 s f eLF2 = (25 ± 1) Hz
HF Detector aHF1 frequency Detector aHF2 frequency Detector eHF1 frequency
f aHF1 = (14.7 ± 0.2) × 103 Hz f aHF2 = (9.4 ± 0.2) × 103 Hz f eHF1 = (14.9 ± 0.4) × 103 Hz
Detector aHF1 dumping factor Detector aHF2 dumping factor Detector eHF2 frequency
δeHF1 = (8 ± 1) × 10−2ωeHF1 δeHF2 = (4.5 ± 0.5) × 10−1ωeHF2 f eHF2 = (9.5 ± 0.4) × 103 Hz
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The measurement, Y (ω), can be expressed by the sum of signal and noise components, Y (ω) = S (ω) + N (ω) ,
(9)
where S (ω) is the PA signal and N (ω) is the noise. Here, PA signal S (ω) is defined by S (ω) = P (ω) H (ω) ,
FIG. 3. Schematic diagram of experimental setup used for solid state photoacoustics.
are given in Table I. Optical excitation is performed using a 660 nm laser diode modulated between 20 Hz and 20 kHz with a current modulation system.20 The irradiated sample surface is polished and the back is etched. We attempt to irradiate sample homogeneously in order to minimize the influence of the 3D effect. A Jin In Electronics Co. model ECM30 electret microphone is used to detect the photoacoustic signal. Electret microphones, in general, can produce nonlinear response at low frequencies. Such nonlinearity is primarily caused by high intensity signals and the subsequent nonlinear response of the electret material.24 Our experimental conditions produce low enough acoustic intensity that these nonlinear effects are minimal. The microphone uses battery power and the influence of electrical line (50 Hz) interference is significantly reduced. However, some line interference is observed at 50 Hz and 100 Hz using our experimental setup with blank beam block and microphone turned off. Data acquisition was performed using an Intel 82801Ib/ ir/ih hd PC audio controller. Software emulation of lock-in amplifier signal processing was used to extract the amplitudes and phases of the acoustic signal.
(10)
where H (ω) is the measurement system response and P(ω) represents the “true” PA signal (P (ω) ∼ δp (ω), Eq. (1)). Noise is a term generally used to designate all unwanted signals observed at the output of a system. Unwanted signals interfere and mask the true PA signal of interest. Several sources of noise are present in the PA measurements including noise sources in the detection and optical modulation systems. Noise is measured using the apparatus without sample irradiation. This is accomplished by placing a beam block, labeled “blank” in Fig. 3, between the light source and the sample. Typical measurements (dot + line) for noise (a) amplitude RN ( f ) and (b) phase ϕ N ( f ) are illustrated in Figure 4. Two frequency ranges can be distinguished: (1) low-frequency (LF) range (20-1000) Hz and (2) high frequency (HF) range (120) kHz. Noise in the LF range is dominated by flicker noise (FN), having typical f −1 amplitude dependence (dashed line— Figure 4(a)) and random phase (Figure 4(b)). The HF range is characterized by dominant crosstalk or interference, herein called “coherent signal deviation” (CSD), with typical f +1 amplitude dependence (dotted line—Figure 4(a)) and coherent phase (Figure 4(b)), both of which are driven by the current modulation system. The total noise N (ω) can be written in the form N (ω) = NFN (ω) + NCSD (ω) ,
(11)
where NFN (ω) is the flicker noise and NCSD (ω) is the coherent signal deviation. Flicker noise cannot be reduced but the coherent signal deviation can be subtracted.
FIG. 4. Measured noise (square + line) (a) amplitudes R N ( f ) and (b) phases ϕ N ( f ) in our experimental setup, as a function of modulation frequency f . The f −1 amplitude dependence is recognized as the flicker noise (dashed line), while f +1 amplitude dependence is recognized as the coherent signal deviation (doted line). Noise clearly divides our analysis to LF and HF range. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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FREQUENCY-DEPENDENT SIGNAL DEVIATION
Distortion alters the signal amplitude and phase to different degrees across the modulation frequency range. This distortion is incorporated into the system transfer function H(ω). Finite mechanical and electronic system response affects both amplitude and phase dependent signal components. System response functions are sought that produce appropriate relationship between amplitude and phase and describe a physically realistic system. Previous work evaluated different microphones, lock-ins and sound cards.14,16,20,23 These studies show that the system transfer function approach to PA signal recovery is appropriate in the 20 Hz–20 kHz modulation frequency range used in the present study. The frequency response of the microphone and accompanying electronics has band pass characteristics that depend on the design, manufacturing, and diaphragm properties. In the LF region, these can be modeled as high-pass filters with characteristic frequencies below 200 Hz. The high-pass filters significantly alter the PA signal amplitude at and below the characteristic frequency. The high-pass filter effects are accounted for using a first-order high-pass filter cascade system transfer function
el el el (ω) (ω) · HHF2 (ω) = HHF1 HHF 1 1 = · , (1 + jω · τeHF1) (1 + jω · τeHF2)
(13)
where τeHF1 = (2π f eHF1)−1 and τHFe2 = (2π f eHF2)−1 are the HF time constants of the used microphone and instruments. In both cases, the high- and low-pass characteristics alter the PA signal amplitudes R ( f ) and phases ϕ ( f ) as shown with a solid line in Figures 5(a) and 5(b). Theoretical signals without deviations, i.e., the “true” PA signals, are presented with dashed lines. Another apparent microphone distortion in the HF range ac (ω). This artifact is the microphone acoustic deviation HHF is due to air flow changes within the microphone volume. These changes are a function of how fast the microphone diaphragm can respond to pressure changes. Because the diaphragm is thin and flexible, it can bend in a number of ways, giving uneven frequency response. The system response can be described as acoustic low-pass filters25 and is represented by a cascade of second-order low-pass filters (Fig. 6) ac ac ac (ω) = HHF1 (ω) · HHF2 (ω) = HHF
ωaHF12
ωaHF12 + jδaHF1ω − ω2
2
el el el (ω) = HLF1 (ω) · HLF2 (jω) HLF ωτeLF2 ωτeLF1 · , (12) =− (1 + jω · τeLF1) (1 + jω · τeLF2)
where τeLF1 = (2π f eLF1)−1 and τeLF2 = (2π f eLF2)−1 are the time constants of the microphone and signal processing electronics, respectively. Although we find that two high-pass filters are appropriate, there is no reason why one or even more than two would not be appropriate for other apparatuses. A similar effect occurs in the HF region. The microphone and accompanying electronic instruments act as low-pass filters with characteristic frequencies between 6 kHz and 20 kHz. The low-pass filters are represented by a system transfer function representing a first-order low-pass filter cascade,
·
ωaHF2
2
ωaHF2 , + jδaHF2ω − ω2
(14)
where δk = ωk/Qk is the damping and Qk is the quality factor (k = aHF1, aHF2, . . . ). The first term in Eq. (14) is usually characterized as the microphone cutoff frequency, ωaHF1. The second term in the cascade is more subtle, exhibiting small amplitude but obvious phase influence. Among other factors, it depends on the geometry of the microphone body and has a characteristic frequency of ωaHF2.
RESULTS AND DISCUSSION
The main idea presented in this paper is to show that one can measure, correct, and analyze PA signals obtained with
FIG. 5. Theoretical PA signal (a) amplitudes R( f ) and (b) phases ϕ( f ) in frequency domain, calculated with Eqs. 1–8 in the case of 85 µm thick Si plate and bulk parameters given in Table I, obtained without (dashed line) and with electrical deviations included (solid line). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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a typical PA apparatus. Any PA apparatus is an integration of excitation source, sample, sample chamber, pressure transducer (microphone), and processing electronics. Although the reason for performing PA measurement is to obtain information on the sample, an integrated “holistic” approach to signal analysis is required to optimize the signal information for subsequent sample parameter estimation. Our experience suggests that all PA measurements will be subject to coherent and incoherent noise, high- and low-frequency roll-off caused by microphone and measurement electronics characteristics, and distortion due to electronic impedance mismatch and microphone frequency response.14–16,20,26,27 The degree to which signals are distorted is a function of the quality of components. But some distortion will exist with even the best of components. To facilitate apparatus modeling, we assume that these effects can be modeled by typical system transfer function elements based on known physics. The parameters of the system transfer functions are found that simultaneously characterize the real effects on signal amplitude and phase. Our general process for signal correction is given in the following steps: (1) Measure and subtract the coherent signal deviations NCSD(ω) from the Y exp(ω) to obtain S exp(ω) using Eq. (9); (2) Fit S exp(ω) to obtain all microphone and lock-in electronics characteristics ωmn, δmn, and τmn (m = eLF, eHF or aHF, n = 1,2,. . . ) using Eqs. (10), (12)–(14); (3) Correct S exp(ω) by removing all detected LF and HF deviations in H(ω), targeting the “true” PA signal in Eqs. (11) and (1); Our test system for this analysis is a thin Si plate sample with constant bulk values given in Table I. In this respect, the well-characterized Si, we use is a calibration sample used to deduce the apparatus constants. Figure 7 depicts the nascent signal of the 85 µm thick Si plate in (a) amplitude, Rexp ( f ), and (b) phase, ϕexp ( f ), (solid line) as a function of modulation frequency, f . The noise
amplitude and phase are measured using the blank beam block. The data confirm that flicker noise, NFN(ω), is a dominant but negligible (signal-to-noise ratio greater than 103) PA signal component in the LF range for this apparatus. On the other hand, the NCSD(ω) amplitude, having f +1.0 dependence, is very high and intercepts the PA signal within the (10–20) kHz range. Clearly, one must carefully measure this CSD interference and remove it from the total signal. The “clear” signal is obtained after subtractive removal, taking NCSD(ω) as a complex number. The “clear” signal amplitude, Rclear ( f ), and phase, ϕclear ( f ), are shown in comparison to the nascent data. The “clear” signal accounts for CSD, but not instrumental distortion. The experimental Rclear ( f ) and ϕclear ( f ) data (Figures 8(a) and 8(b)) are subsequently fit to Eqs. (10), (12)–(14). Following the fitting procedure determines the LF and HF system characteristics. Detector impedance mismatch with the sound card is the main cause of the PA signal deviation in the LF range. As expected, the deviation decreases the signal amplitude. Using Eq. (12), fitting yields the respective sound card and detector characteristic relaxation times of τeLF1 = 1.1 × 10−2 s and τeLF2 = 6.3 × 10−3 s. In terms of modulation frequency, f eLF1 = 15 Hz and f eLF2 = 25 Hz. This is in accordance with our expectation to have amplitude deviations between 1 Hz and 10 Hz for most of the lock-ins used for photoacoustic measurements14–17,20–22 (manuals can be found in Ref. 28), between 5 Hz and 20 Hz for most of the sound-cards,20 and from 20 Hz to 60 Hz for most of the electret microphones.14–17,20–22 Phase deviation is expected across the entire LF range. In the HF range, characteristic detector electroacoustic parameters are found to be f eHF1 = 14.9 × 103 Hz (τeHF1 = 10.7 × 10−6 s), f eHF2 = 9.5 × 103 Hz (τeHF2 = 16.7 × 10−6 s), f aHF1 = 14.7 × 103 Hz, δaHF1 = 0.08 · ωaHF1, f aHF2 = 9.4 × 103 Hz, and δaHF2 = 0.45 · ωaHF2. The f aHF1 determined by fitting approximates the microphone cutoff frequency. The solid line in Figures 8(a) and 8(b) shows that
FIG. 6. Theoretical (a) amplitudes R( f ) and (b) phases ϕ( f ) of PA signal in HF range, obtained for 85 µm thick Si plate with bulk parameters given in Table I, without (dashed line) and with (solid line) mentioned acoustic deviations. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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FIG. 7. Experimental (a) amplitudes R( f ) and (b) phases ϕ( f ) of the PA signal (solid line), FN and coherent signal deviation CSD measurements (square + line), together with the clear signal (asterisks + line).
the PA signal tends to Rtheo ( f ) and ϕtheo ( f ) upon accounting el el ac (ω) + HHF (ω) + HHF (ω): for the influence of H (ω) = HLF Rfit ( f ) → Rtheo ( f ) and ϕfit ( f ) → ϕtheo ( f ). The corrected signals match theoretical ones presented in Figure 2. All LF and HF system parameters calculated in this way with corresponding errors are presented in Table I. The TD, TE, and PE components of the total PA signal can be deduced based on the theoretical Rtheo ( f ) and ϕtheo ( f ). As clearly seen in Fig. 2, TD is dominant in the LF range. On the other hand, the TE processes, and in particular the elastic bending, become important above 10 kHz. A small PE effect contribution is also clearly observed. But one cannot estimate the free carrier contribution to the PE component from this alone. This PE component is due only to sample
expansion from elastic stress. More significant free carrier signal contributions are evident through the heat distribution along the sample, given in the terms of sample temperature, Eq. (2), for the TD and TE signal components in Eqs. (3)–(8). Comparing Figures 2 and 8, one can see that the PA amplitude follows the frequency-dependent response predicted by the model. Other researchers have concluded that the model used here is not valid, even in the LF range.13 They find a discrepancy between experiment and theory in the TD and TE mutual ratios and interpret this to indicate that the TE displacement component is miscalculated using a strictly U1z (r, z) ∼ l s −3 dependence in Eq. (6). They account for the discrepancy using a semi-empirical U1z (r, z) ∼ l s −2.8 dependence. In our experience, a similar discrepancy has been observed
FIG. 8. Clear experimental (a) amplitudes R( f ) and (b) phases ϕ( f ) of the PA signal (asterisks) as a function of the modulation frequency f together with fitted (dashed line) and theoretical (solid line) signal. Arrows depicts the results of applied signal correction procedure. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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in very thin Si samples, especially those with thicknesses below 50 µm.23 In our opinion, a theoretical l s −3 dependence is not a problem for samples greater than 50 µm. Thinner samples may lose the ideal U1z (r, z) ∼ l s −3 bending characteristic, perhaps due to an effective sample radius or temperature gradient change. Experimental protocols for clamping or mounting the sample on the microphone could also cause the change in TD and TE mutual ratio. Sample mounting could affect the slope and mutual ratio of the frequency dependent PA signal. But to conclude this, one needs reproducible measurements obtained by controlling the irradiation power, sample absorbance, and detector characteristics in addition to removing signal deviation as prescribed above. In any event, the signal correction procedure presented here may lead to better assessment of the validity of the theoretical model for the sample bending TE effect. We did not, however, observe this for our 85 µm thick Si sample. Finally, it is worth mentioning that the methodology presented here may also be thought of as a normalization procedure. In this case, the effective normalization is accomplished by determining the instrumental system response based on measurements of a sample with presupposed properties. Compared with the usual normalization techniques based on front- and back-side measurements, our procedure allows better use of the open photoacoustic cell (OPC) configuration resulting in larger PA signal amplitudes (smaller cell volume) and uses a simpler experimental apparatus.
frequency (14.9 kHz). The second was attributed to the microphone geometry and diaphragm properties. As a whole, the procedures appear to yield corrected signals that more accurately represent the sample physical characteristics. The processed signal appears to be an accurate representation of the theoretical PA amplitude and phase signals for the 85 µm thick Si plate. Thermal diffusion, thermoelastic, and plasmaelastic components calculated from the corrected signals are in agreement with the bulk parameters of this sample. This procedure should be valid for all solid samples and is suitable for multilayered samples composed of a substrate in the form of a thin plate, or a thin film as a coating, as long as frequency-dependent TD, TE, and PE effects are accounted for. ACKNOWLEDGMENTS
This work was supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia (Project Nos. ON171016 and III45005). 1A.
Rosencwaig and A. Gersho, “Theory of the photoacoustic effect with solids,” J. Appl. Phys. 47, 64-69 (1976). 2G. Roussel, F. Lepoutre, and L. Bertrand, “Influence of thermoelastic bending on photoacoustic experiments related to measurements of thermal diffusivity of metals,” J. Appl. Phys. 54, 2383-2391 (1983). 3F. McDonald and G. Westel, “Generalized theory of the photoacoustic effect,” J. Appl. Phys. 49, 2313-2322 (1978). 4A. C. Tam, “Application of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381-431 (1986). 5H. Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 16, 45-101 (1988). 6S. E. Bialkowski, “Photothermal spectroscopy methods for chemical analCONCLUSIONS ysis,” in Chemical Analysis: A Series of Monographs on Analytical ChemPhotoacoustic measurements of a 85 µm Si plate over istry and Its Applications, edited by J. D. Winefordner (John Wiley & Sons, Inc., New York, 1996), Vol. 134. a 20 Hz–20 kHz modulation frequency range show that PA 7M. Terazima, N. Hirota, S. E. Braslavsky, A. Mandelis, S. E. Bialkowski, G. signals can be represented as a combination of thermal diffuE. Diebold, R. S. D. Miller, D. Fournier, R. A. Palmer, and A. Tam, “Quantision and thermal and plasma elastic components. An accuties, terminology, and symbols in photothermal and related spectroscopies,” rate photoacoustic signal was obtained in an open-cell experiPure Appl. Chem. 76, 1083-1118 (2004). 8C. Haisch and R. Niessner, “Light and sound-photoacoustic spectroscopy,” mental apparatus using noise and signal deviation recognition Spectrosc. Eur. 14(5), 10-15 (2002). and removal procedures based on both a theoretical elastic9D. M. Todorovic and P. M. Nikolic, in Semiconductors and Electronic bending model and recognition of processes giving rise to Materials, edited by A. Mandelis and P. Hess (SPIE Optical Engineering Press, Belingham, Washington, 2000), Chap. 9, pp. 273-318. signal distortion. We find coherent noise to be the dominant 10L. F. Perondi and L. C. M. Miranda, “Minimal-volume photoacoustic cell interference at high frequencies. Its origin was found to be measurement of thermal diffusivity: Effect of thermoelastic sample bendthe current optical excitation source modulation system. A ing,” J. Appl. Phys. 62, 2955-2959 (1987). 11E. Marin, H. Vargas, P. Diaz, and I. Riech, “On the photoacoustic charsubtractive procedure was found to be sufficient to obtain an acterization of semiconductors: Influence of carrier recombination on the interference-free signal. thermodiffusion, thermoelastic and electronic strain signal generationmechSubsequent analysis reveals strong signal distortion. At anisms,” Phys. Status Solidi A 179, 387-402 (2000). low frequencies, apparent amplitude distortion was found for 12M. V. Marquerinit, N. Cellat, A. M. Mansanares, H. Vargas, and L. C. M. f < 100 Hz and phase distortion for f < 1000 Hz. This distorMiranda, “Open photoacoustic cell spectroscopy,” Meas. Sci. Technol. 2, 396401 (1991). tion was attributed to an impedance mismatch between the 13A. Somer, F. Camilotti, G. F. Costa, C. Bonardi, A. Novatski, A. V. C. microphone and the sound card used for lock-in like signal Andrade, A. V. C. Andrade, V. A. Kozlowski, Jr., and G. K. Cruz, “The processing. Accounting for this effect in a system transfer thermoelastic bending and thermal diffusion processes influence on photoafunction allowed signal processing procedures that simultacoustic signal generation using open photoacoustic cell technique,” J. Appl. Phys. 114, 063503 (2013). neously correct signal amplitudes and phases. The correction 14D. M. Todorovi´ c, D. D. Markushev, and M. D. Rabasovi´c, “Photoacoustic procedure allows estimation of instrument time constants for elastic bending in thin film—Substrate system,” J. Appl. Phys. 114, 213510 the detector (25 Hz) and the sound-card (15 Hz). (2013). 15D. M. Todorovi´ Major system resonances were detected in the high frec, M. D. Rabasovi´c, D. D. Markushev, and M. Sarajlic, “Photoacoustic elastic bending in thin film-substrate system: Experimental quency range and attributed to electro-acoustic detector chardetermination of the thin film parameters,” J. Appl. Phys. 116, 053506 acteristics. In the open-cell configuration, the electret micro(2014). 16D. M. Todorovi´ phone acts as an acoustic low-pass filter. Two characteristic c, M. D. Rabasovi´c, D. D. Markushev, M. Franko, and U. L. Štangar, “Study of TiO2 thin films on Si substrate by the photoacoustic frequencies were found. The first was the microphone cutoff This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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elastic bending method,” Sci. China: Phys., Mech. Astron. 56(7), 1285-1293 (2013). 17D. M. Todorovic, P. M. Nikolic, M. D. Dramicanin, D. G. Vasiljevic, and Z. D. Ristovski, “Photoacoustic frerquency heat-transmission technique: Thermal and carrier transport parameters measurement in silicon,” J. Appl. Phys. 78, 5750-5755 (1995). 18G. Gonzalez de la Cruz and Yu. G. Gurevic, “Electron and phonon thermal waves in semiconductors: An application to photothermal effects,” J. Appl. Phys. 80, 1726-1730 (1996). 19J. L. Pichardo-Molina and J. J. Alvarado-Gil, “Heat diffusion and thermoelastic vibration influence on the signal of an open photoacoustic cell for two layer system,” J. Appl. Phys. 95, 6450-6456 (2004). 20M. D. Rabasovi´ c, M. G. Nikoli´c, M. D. Drami´canin, M. Franko, and D. D. Markushev, “Low-cost, portable photoacoustic setup for solid samples,” Meas. Sci. Technol. 20, 095902 (2009). 21D. M. Todorovi´ c, “Photothermal elastic bending method in investigation of microelectromechanic structures,” Rev. Sci. Instrum. 74(1), 578-581 (2003). 22D. M. Todorovic, M. D. Rabasovic, D. D. Markushev, V. Jovic, and K. T. Radulovic, “Investigation of micromechanical structures by photoacoustic
Rev. Sci. Instrum. 86, 035110 (2015) elastic bending method,” Int. J. Thermophys. 33, 2222-2229 (2012). M. Todorovi´c, D. D. Markushev, M. D. Rabasovic, K. T. Radulovic, and V. Jovic, 28th International Conference on Microelectronics - Proceedings, MIEL 2012, in Photoacoustic elastic bending method: Study of the siliconmembranes (IEEE, 2012), pp. 169-172. 24M. V. Marquerinit, N. Cellat, A. M. Mansanarest, H. Vargas, and L. C. M. Miranda, “Open photoacoustic cell spectroscopy,” Meas. Sci. Technol. 2, 396-401 (1991). 25See http://www.moultonworld.pwp.blueyonder.co.uk/Lecture9_page.htm for information about the physics behind Eq. (14). 26S. Telenkov and A. Mandelis, “Signal-to-noise analysis of biomedical photoacoustic measurements in time and frequency domains,” Rev. Sci. Instrum. 81, 124901 (2010). 27B. Lashkari and A. Mandelis, “Comparison between pulsed laser and frequency-domain photoacoustic modalities: Signal-to-noise ratio, contrast, resolution, and maximum depth detectivity,” Rev. Sci. Instrum. 82, 094903 (2011). 28See http://www.signalrecovery.com/literature/manuals.aspx for different lock-in types, their manuals and characteristics. 23D.
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REVIEW OF SCIENTIFIC INSTRUMENTS 86, 035110 (2015)
Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain D. D. Markushev,1 M. D. Rabasović,1 D. M. Todorović,2 S. Galović,3 and S. E. Bialkowski4
1
Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade-Zemun, Serbia Institute for Multidisciplinary Researches, University of Belgrade, P.O. Box 33, 11030 Belgrade, Serbia 3 Institute of Nuclear Sciences “Vinca”, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia 4 Department of Chemistry and Biochemistry, Utah State University, Logan, Utah 84322-0300, USA 2
(Received 3 September 2014; accepted 28 February 2015; published online 23 March 2015) Methods for photoacoustic signal measurement, rectification, and analysis for 85 µm thin Si samples in the 20-20 000 Hz modulation frequency range are presented. Methods for frequency-dependent amplitude and phase signal rectification in the presence of coherent and incoherent noise as well as distortion due to microphone characteristics are presented. Signal correction is accomplished using inverse system response functions deduced by comparing real to ideal signals for a sample with well-known bulk parameters and dimensions. The system response is a piece-wise construction, each component being due to a particular effect of the measurement system. Heat transfer and elastic effects are modeled using standard Rosencweig-Gersho and elastic-bending theories. Thermal diffusion, thermoelastic, and plasmaelastic signal components are calculated and compared to measurements. The differences between theory and experiment are used to detect and correct signal distortion and to determine detector and sound-card characteristics. Corrected signal analysis is found to faithfully reflect known sample parameters. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914894] INTRODUCTION
Photoacoustic (PA) spectroscopy of solids measures a pressure change resulting from sample heating due to optical absorption.1–6 For opaque solids, surface absorption is followed by heat transfer through the sample and coupling fluid. Solid heating also results in elastic bending and thermal expansion effects that may contribute to the total signal. The heat transfer is due to thermal diffusion (TD) while mechanical elastic effects are thermoelastic (TE) processes. In the end, all of these effects generate pressure changes within the sample cell and are detected as PA signal.7–10 Using RosencweigGersho (RG)1 and elastic-bending (EB)2,9 models with an open-cell (OC) experimental setup,12 one can use photoacoustic spectroscopy to determine thermal and elastic parameters describing solid samples.11–13 In the case of metals and insulators, only the TD and TE components exist, simplifying the analysis. But our ultimate goal is to characterize the thin films (primarily photocatalytic) layered on semiconductors.14–16 Therefore, we chose Si as a substrate having well defined thermal17 and perfect elastic characteristics.9 On the other hand, the choice of wavelengths used to illuminate the sample is limited by the photocatalytic film analysis needs, so the use of wavelengths that causes free carriers generation could not be avoided. In the case of Si, and semiconductors in general, opticallyinduced free carrier generation contributes to the TD and TE effects.17–19 Free carriers can also induce elastic stress resulting in plasmaelastic (PE) effects. Subsequently, the semiconductor PA signal is due to a combination of TD, TE, and PE effects. PA spectroscopy may thus be used to determine thermal, mechanical, optical, and electronic characteristics of semiconductors. With an apparatus adapted to one-layer samples, all of
these effects can be elucidated within a modulation frequency range from 20 Hz to 20 kHz by applying the EB model for data analysis.14,15,20,21 Unfortunately, noise, distortion and interference often mask the “true” PA signal. Noise and signal distortion analysis is subsequently an unavoidable task in accurate PA measurements. Recognition, quantitation, and correction for these instrumental effects allow one to obtain the “true” PA signal, clear of many types of interference and distortion. In this report, PA amplitude and phase signals are analyzed for noise and systematic distortion within the 20 Hz–20 kHz modulation frequency range. A 85 µm plate of well-characterized Si is used as the sample. Differences between “true” and experimental signals are identified and characterized as system noise and signal distortion originated from the instruments used to construct the apparatus. Signal distortion analysis is modeled by cascade high and low pass filters, and the corresponding system transfer functions are derived. Undistorted signals are obtained by applying the inverse system transfer functions. “True” frequencydependent PA signal target response is calculated using RG and EB theories and the relationship between TD, TE, and PE signal components.9 Good agreement is found between the processed PA signal and the theoretical model after accounting for instrumental response. The applicability of these signal models is shown to be credible across the frequency range for thin semiconductor samples.
THEORETICAL MODEL
The theoretical model used here is adopted for the optical excitation of a homogeneous semiconductor sample with thickness l s (Figure 1(a)). Excitation intensity is given by
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FIG. 1. Transmission photoacoustic spectroscopy model: (a) thermal diffusion and (b) thermal and plasma elastic analysis.
I = I0 exp (−jωt), where j is the imaginary unit, t is the time, ω = 2π f , and f is the modulation frequency. General assumptions are that the sample absorbs at least part of the incident optical energy and sample irradiation is uniform over the entire surface. As a consequence of the latter assumption, heat propagates only along the z-axes (Figure 1(a)). Thus, heat transfer is considered to be a 1D process. Basic model configuration in the case of transmission photoacoustic apparatus is presented in Figure 1(a). The total pressure change, i.e., photoacoustic signal, δp (ω) in the cell is a sum of those due to the TD, TE, and PE processes and is represented as δp (ω) = δpTD (ω) + δpTE (ω) + δpPE (ω) .
(1)
The temperature distribution Ts (z,t) can be calculated by9,14 Ts (z, ω) = [b1 exp (γz) + b2 exp (−γz) + b3n (z) + b4 exp (βz)] exp ( jωt) ,
important at higher frequencies and have a strong influence on the total PA signal generation. In the case of small vibrational amplitudes, the δpTE (ω) and δpPE (ω) calculations may use a cylindrical approximation (Figure 1(b)) that assumes the acoustic vibration modes are independent of the polar coordinate ϕ. δpTE (ω) and δpPE (ω) are calculated using4,8 γp0 Rs δpm (ω) = 2πrUz (r, z) dr, (4) V0 0 where exp ( jωt) is omitted for simplicity.9,14–19,22 Here, m = TE,PE indicates the particular effect, V0 and p0 are the cell volume and ambient pressure, r is the radial coordinate, z is the longitudinal position, Rs is the sample’s effective radius, and Uz (r, z) is the displacement function describing linear elastic bending and volume expansion. For TE effects, this function can be written in the form of a sum, Uz (r, z) = U1z (r, z) + U2z (r, z) .
(2)
where n (z) is the free carrier concentration, β is the sample absorption coefficient, and b1, b2, b3, and b4 are the complex constants depending on the sample characteristics.9,14 Also, γ = jω/D s , with the thermal diffusivity, Dz , given by D s = k s / (ρC) where k s is the heat conductivity, ρ is the density, and C is the heat capacity. Equation (2) allows one to calculate the temperature Ts (l s ,t) at the back side of the sample (z = l s ). This temperature is responsible to the TD component of PA signal δpTD,1–4 γg p0 Dg (3) δpTD (ω) = √ T (l s , ω) , l cT0 ω where T0 and p0 are the gas temperature and pressure, respectively, γg is the adiabatic ratio, Dg is the gas thermal diffusivity, and l c is the PA cell length. It is well known that reducing the thickness of the Si sample increases the TD and TE components, with TD increasing faster than TE.9,14–17 In the case of thin Si samples, TD effects usually dominate in the (20 Hz–20 kHz) frequency range. In the case of the thin Si semiconductor plate analyzed here (85 µm), the TD process is dominant for modulation frequencies lower than 1000 Hz. TE and PE effects become
(5)
The sample bending term is U1z (r, z) = α
6 Rs2 − r 2 l s3
MT ,
(6)
while thermal expansion term is given by z 1+ν T (z) dz U2z (r, z) = α 1 − ν l s ( 2 2 2 z+ 3ν 4z − l s − MT + 1 − ν ls l s3
ls 2
)
NT .
(7)
Here, α is the coefficient of linear expansion, l s is the sample thickness, ν is Poisson’s ratio, and T (z) is the temperature distribution along the z–axis (Figure 1(b)). MT and NT are defined as l s /2
l s /2 MT =
zT (z) dz and NT = −l s /2
T (z) dz. −l s /2
(8)
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FIG. 2. Theoretical (a) amplitude R( f ) and (b) phase ϕ( f ) of the PA signal (asterisks + line) as a function of the modulation frequency, f , together with their thermal diffusion (TD—solid line), thermal elastic (TE—dashed line), and plasma elastic (PE—dotted line) components.
frequency range and valid for samples other than the particular Si used here.22
In the case of free carrier generation, PE effects are also calculated using Eqs. (4)–(8). However, one uses the coefficient of electronic deformation d n instead of α, and the free carrier concentration n(z) instead of T(z). Details regarding Eqs. (4)–(8) can be found in Refs. 16 and 23. Figure 2 illustrates the theoretical PA signal (a) amplitude Rtheo ( f ) and (b) phase ϕtheo ( f ), and the corresponding TD, TE, and PE components. The calculations are performed for a 85 µm thick Si plate with a diameter of 0.7 cm. The bulk parameters are given in the Table I. This PA signal will be referred to as the “true” PA signal in our noise analysis and the bulk parameters listed in Table I are the target of the signal analysis. We assume that model is valid across the modulation
APPARATUS AND NOISE ANALYSIS
The photoacoustic apparatus consists of five main components: (1) sample, (2) light source, (3) cell, (4) detector, (5) electronic signal processing equipment. A schematic diagram illustrating the main components is depicted in Figure 3. A blank (6 in Figure 3) is added as an optical beam stop for noise measurements. All measurements are performed with a single Si thin plate sample, prepared from 3-5 Ω cm, ntype, and ⟨100⟩ oriented Si wafer. The bulk sample parameters
TABLE I. Sample and system parameters. Si sample bulk parameters Density Optical reflectivity Optical absorption coefficient Thermal diffusivity Excitation energy Energy gap Lifetime of photogenerated carriers
ρ = 2.33 × 103
−3
kg m
R = 0.30 β = 5.00 × 105 m−1 D s = 1.00 × 10−4 m2 s−1 ε = 1.88 eV εg = 1.11 eV τ = 6.0 × 10−6 s
Coefficient of carrier diffusion Recombination velocity of the front surface Recombination velocity of the rear surface Young’s modulus Linear thermal expansion Coefficient of electronic deformation
D n = 1.2 × 10−3 s 1 = 2 m s−1 s 2 = 24 m s−1 E = 1.31 × 1011 N m−2 α = 3.0 × 10−6 K−1 d n = −9.0 × 10−31 m3
Calculated system parameters in the case of optical excitation power P0 = 1.5×10−3 W LF Sound card eLF1 time constant Sound card eLF1 frequency
τeLF1 = (1.1 ± 0.1) × 10−2 s f eLF1 = (15 ± 1) Hz
Detector eLF2 time constant Detector eLF2 frequency
τeLF2 = (6.3 ± 0.3) × 10−3 s f eLF2 = (25 ± 1) Hz
HF Detector aHF1 frequency Detector aHF2 frequency Detector eHF1 frequency
f aHF1 = (14.7 ± 0.2) × 103 Hz f aHF2 = (9.4 ± 0.2) × 103 Hz f eHF1 = (14.9 ± 0.4) × 103 Hz
Detector aHF1 dumping factor Detector aHF2 dumping factor Detector eHF2 frequency
δeHF1 = (8 ± 1) × 10−2ωeHF1 δeHF2 = (4.5 ± 0.5) × 10−1ωeHF2 f eHF2 = (9.5 ± 0.4) × 103 Hz
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The measurement, Y (ω), can be expressed by the sum of signal and noise components, Y (ω) = S (ω) + N (ω) ,
(9)
where S (ω) is the PA signal and N (ω) is the noise. Here, PA signal S (ω) is defined by S (ω) = P (ω) H (ω) ,
FIG. 3. Schematic diagram of experimental setup used for solid state photoacoustics.
are given in Table I. Optical excitation is performed using a 660 nm laser diode modulated between 20 Hz and 20 kHz with a current modulation system.20 The irradiated sample surface is polished and the back is etched. We attempt to irradiate sample homogeneously in order to minimize the influence of the 3D effect. A Jin In Electronics Co. model ECM30 electret microphone is used to detect the photoacoustic signal. Electret microphones, in general, can produce nonlinear response at low frequencies. Such nonlinearity is primarily caused by high intensity signals and the subsequent nonlinear response of the electret material.24 Our experimental conditions produce low enough acoustic intensity that these nonlinear effects are minimal. The microphone uses battery power and the influence of electrical line (50 Hz) interference is significantly reduced. However, some line interference is observed at 50 Hz and 100 Hz using our experimental setup with blank beam block and microphone turned off. Data acquisition was performed using an Intel 82801Ib/ ir/ih hd PC audio controller. Software emulation of lock-in amplifier signal processing was used to extract the amplitudes and phases of the acoustic signal.
(10)
where H (ω) is the measurement system response and P(ω) represents the “true” PA signal (P (ω) ∼ δp (ω), Eq. (1)). Noise is a term generally used to designate all unwanted signals observed at the output of a system. Unwanted signals interfere and mask the true PA signal of interest. Several sources of noise are present in the PA measurements including noise sources in the detection and optical modulation systems. Noise is measured using the apparatus without sample irradiation. This is accomplished by placing a beam block, labeled “blank” in Fig. 3, between the light source and the sample. Typical measurements (dot + line) for noise (a) amplitude RN ( f ) and (b) phase ϕ N ( f ) are illustrated in Figure 4. Two frequency ranges can be distinguished: (1) low-frequency (LF) range (20-1000) Hz and (2) high frequency (HF) range (120) kHz. Noise in the LF range is dominated by flicker noise (FN), having typical f −1 amplitude dependence (dashed line— Figure 4(a)) and random phase (Figure 4(b)). The HF range is characterized by dominant crosstalk or interference, herein called “coherent signal deviation” (CSD), with typical f +1 amplitude dependence (dotted line—Figure 4(a)) and coherent phase (Figure 4(b)), both of which are driven by the current modulation system. The total noise N (ω) can be written in the form N (ω) = NFN (ω) + NCSD (ω) ,
(11)
where NFN (ω) is the flicker noise and NCSD (ω) is the coherent signal deviation. Flicker noise cannot be reduced but the coherent signal deviation can be subtracted.
FIG. 4. Measured noise (square + line) (a) amplitudes R N ( f ) and (b) phases ϕ N ( f ) in our experimental setup, as a function of modulation frequency f . The f −1 amplitude dependence is recognized as the flicker noise (dashed line), while f +1 amplitude dependence is recognized as the coherent signal deviation (doted line). Noise clearly divides our analysis to LF and HF range. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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FREQUENCY-DEPENDENT SIGNAL DEVIATION
Distortion alters the signal amplitude and phase to different degrees across the modulation frequency range. This distortion is incorporated into the system transfer function H(ω). Finite mechanical and electronic system response affects both amplitude and phase dependent signal components. System response functions are sought that produce appropriate relationship between amplitude and phase and describe a physically realistic system. Previous work evaluated different microphones, lock-ins and sound cards.14,16,20,23 These studies show that the system transfer function approach to PA signal recovery is appropriate in the 20 Hz–20 kHz modulation frequency range used in the present study. The frequency response of the microphone and accompanying electronics has band pass characteristics that depend on the design, manufacturing, and diaphragm properties. In the LF region, these can be modeled as high-pass filters with characteristic frequencies below 200 Hz. The high-pass filters significantly alter the PA signal amplitude at and below the characteristic frequency. The high-pass filter effects are accounted for using a first-order high-pass filter cascade system transfer function
el el el (ω) (ω) · HHF2 (ω) = HHF1 HHF 1 1 = · , (1 + jω · τeHF1) (1 + jω · τeHF2)
(13)
where τeHF1 = (2π f eHF1)−1 and τHFe2 = (2π f eHF2)−1 are the HF time constants of the used microphone and instruments. In both cases, the high- and low-pass characteristics alter the PA signal amplitudes R ( f ) and phases ϕ ( f ) as shown with a solid line in Figures 5(a) and 5(b). Theoretical signals without deviations, i.e., the “true” PA signals, are presented with dashed lines. Another apparent microphone distortion in the HF range ac (ω). This artifact is the microphone acoustic deviation HHF is due to air flow changes within the microphone volume. These changes are a function of how fast the microphone diaphragm can respond to pressure changes. Because the diaphragm is thin and flexible, it can bend in a number of ways, giving uneven frequency response. The system response can be described as acoustic low-pass filters25 and is represented by a cascade of second-order low-pass filters (Fig. 6) ac ac ac (ω) = HHF1 (ω) · HHF2 (ω) = HHF
ωaHF12
ωaHF12 + jδaHF1ω − ω2
2
el el el (ω) = HLF1 (ω) · HLF2 (jω) HLF ωτeLF2 ωτeLF1 · , (12) =− (1 + jω · τeLF1) (1 + jω · τeLF2)
where τeLF1 = (2π f eLF1)−1 and τeLF2 = (2π f eLF2)−1 are the time constants of the microphone and signal processing electronics, respectively. Although we find that two high-pass filters are appropriate, there is no reason why one or even more than two would not be appropriate for other apparatuses. A similar effect occurs in the HF region. The microphone and accompanying electronic instruments act as low-pass filters with characteristic frequencies between 6 kHz and 20 kHz. The low-pass filters are represented by a system transfer function representing a first-order low-pass filter cascade,
·
ωaHF2
2
ωaHF2 , + jδaHF2ω − ω2
(14)
where δk = ωk/Qk is the damping and Qk is the quality factor (k = aHF1, aHF2, . . . ). The first term in Eq. (14) is usually characterized as the microphone cutoff frequency, ωaHF1. The second term in the cascade is more subtle, exhibiting small amplitude but obvious phase influence. Among other factors, it depends on the geometry of the microphone body and has a characteristic frequency of ωaHF2.
RESULTS AND DISCUSSION
The main idea presented in this paper is to show that one can measure, correct, and analyze PA signals obtained with
FIG. 5. Theoretical PA signal (a) amplitudes R( f ) and (b) phases ϕ( f ) in frequency domain, calculated with Eqs. 1–8 in the case of 85 µm thick Si plate and bulk parameters given in Table I, obtained without (dashed line) and with electrical deviations included (solid line). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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a typical PA apparatus. Any PA apparatus is an integration of excitation source, sample, sample chamber, pressure transducer (microphone), and processing electronics. Although the reason for performing PA measurement is to obtain information on the sample, an integrated “holistic” approach to signal analysis is required to optimize the signal information for subsequent sample parameter estimation. Our experience suggests that all PA measurements will be subject to coherent and incoherent noise, high- and low-frequency roll-off caused by microphone and measurement electronics characteristics, and distortion due to electronic impedance mismatch and microphone frequency response.14–16,20,26,27 The degree to which signals are distorted is a function of the quality of components. But some distortion will exist with even the best of components. To facilitate apparatus modeling, we assume that these effects can be modeled by typical system transfer function elements based on known physics. The parameters of the system transfer functions are found that simultaneously characterize the real effects on signal amplitude and phase. Our general process for signal correction is given in the following steps: (1) Measure and subtract the coherent signal deviations NCSD(ω) from the Y exp(ω) to obtain S exp(ω) using Eq. (9); (2) Fit S exp(ω) to obtain all microphone and lock-in electronics characteristics ωmn, δmn, and τmn (m = eLF, eHF or aHF, n = 1,2,. . . ) using Eqs. (10), (12)–(14); (3) Correct S exp(ω) by removing all detected LF and HF deviations in H(ω), targeting the “true” PA signal in Eqs. (11) and (1); Our test system for this analysis is a thin Si plate sample with constant bulk values given in Table I. In this respect, the well-characterized Si, we use is a calibration sample used to deduce the apparatus constants. Figure 7 depicts the nascent signal of the 85 µm thick Si plate in (a) amplitude, Rexp ( f ), and (b) phase, ϕexp ( f ), (solid line) as a function of modulation frequency, f . The noise
amplitude and phase are measured using the blank beam block. The data confirm that flicker noise, NFN(ω), is a dominant but negligible (signal-to-noise ratio greater than 103) PA signal component in the LF range for this apparatus. On the other hand, the NCSD(ω) amplitude, having f +1.0 dependence, is very high and intercepts the PA signal within the (10–20) kHz range. Clearly, one must carefully measure this CSD interference and remove it from the total signal. The “clear” signal is obtained after subtractive removal, taking NCSD(ω) as a complex number. The “clear” signal amplitude, Rclear ( f ), and phase, ϕclear ( f ), are shown in comparison to the nascent data. The “clear” signal accounts for CSD, but not instrumental distortion. The experimental Rclear ( f ) and ϕclear ( f ) data (Figures 8(a) and 8(b)) are subsequently fit to Eqs. (10), (12)–(14). Following the fitting procedure determines the LF and HF system characteristics. Detector impedance mismatch with the sound card is the main cause of the PA signal deviation in the LF range. As expected, the deviation decreases the signal amplitude. Using Eq. (12), fitting yields the respective sound card and detector characteristic relaxation times of τeLF1 = 1.1 × 10−2 s and τeLF2 = 6.3 × 10−3 s. In terms of modulation frequency, f eLF1 = 15 Hz and f eLF2 = 25 Hz. This is in accordance with our expectation to have amplitude deviations between 1 Hz and 10 Hz for most of the lock-ins used for photoacoustic measurements14–17,20–22 (manuals can be found in Ref. 28), between 5 Hz and 20 Hz for most of the sound-cards,20 and from 20 Hz to 60 Hz for most of the electret microphones.14–17,20–22 Phase deviation is expected across the entire LF range. In the HF range, characteristic detector electroacoustic parameters are found to be f eHF1 = 14.9 × 103 Hz (τeHF1 = 10.7 × 10−6 s), f eHF2 = 9.5 × 103 Hz (τeHF2 = 16.7 × 10−6 s), f aHF1 = 14.7 × 103 Hz, δaHF1 = 0.08 · ωaHF1, f aHF2 = 9.4 × 103 Hz, and δaHF2 = 0.45 · ωaHF2. The f aHF1 determined by fitting approximates the microphone cutoff frequency. The solid line in Figures 8(a) and 8(b) shows that
FIG. 6. Theoretical (a) amplitudes R( f ) and (b) phases ϕ( f ) of PA signal in HF range, obtained for 85 µm thick Si plate with bulk parameters given in Table I, without (dashed line) and with (solid line) mentioned acoustic deviations. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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FIG. 7. Experimental (a) amplitudes R( f ) and (b) phases ϕ( f ) of the PA signal (solid line), FN and coherent signal deviation CSD measurements (square + line), together with the clear signal (asterisks + line).
the PA signal tends to Rtheo ( f ) and ϕtheo ( f ) upon accounting el el ac (ω) + HHF (ω) + HHF (ω): for the influence of H (ω) = HLF Rfit ( f ) → Rtheo ( f ) and ϕfit ( f ) → ϕtheo ( f ). The corrected signals match theoretical ones presented in Figure 2. All LF and HF system parameters calculated in this way with corresponding errors are presented in Table I. The TD, TE, and PE components of the total PA signal can be deduced based on the theoretical Rtheo ( f ) and ϕtheo ( f ). As clearly seen in Fig. 2, TD is dominant in the LF range. On the other hand, the TE processes, and in particular the elastic bending, become important above 10 kHz. A small PE effect contribution is also clearly observed. But one cannot estimate the free carrier contribution to the PE component from this alone. This PE component is due only to sample
expansion from elastic stress. More significant free carrier signal contributions are evident through the heat distribution along the sample, given in the terms of sample temperature, Eq. (2), for the TD and TE signal components in Eqs. (3)–(8). Comparing Figures 2 and 8, one can see that the PA amplitude follows the frequency-dependent response predicted by the model. Other researchers have concluded that the model used here is not valid, even in the LF range.13 They find a discrepancy between experiment and theory in the TD and TE mutual ratios and interpret this to indicate that the TE displacement component is miscalculated using a strictly U1z (r, z) ∼ l s −3 dependence in Eq. (6). They account for the discrepancy using a semi-empirical U1z (r, z) ∼ l s −2.8 dependence. In our experience, a similar discrepancy has been observed
FIG. 8. Clear experimental (a) amplitudes R( f ) and (b) phases ϕ( f ) of the PA signal (asterisks) as a function of the modulation frequency f together with fitted (dashed line) and theoretical (solid line) signal. Arrows depicts the results of applied signal correction procedure. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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in very thin Si samples, especially those with thicknesses below 50 µm.23 In our opinion, a theoretical l s −3 dependence is not a problem for samples greater than 50 µm. Thinner samples may lose the ideal U1z (r, z) ∼ l s −3 bending characteristic, perhaps due to an effective sample radius or temperature gradient change. Experimental protocols for clamping or mounting the sample on the microphone could also cause the change in TD and TE mutual ratio. Sample mounting could affect the slope and mutual ratio of the frequency dependent PA signal. But to conclude this, one needs reproducible measurements obtained by controlling the irradiation power, sample absorbance, and detector characteristics in addition to removing signal deviation as prescribed above. In any event, the signal correction procedure presented here may lead to better assessment of the validity of the theoretical model for the sample bending TE effect. We did not, however, observe this for our 85 µm thick Si sample. Finally, it is worth mentioning that the methodology presented here may also be thought of as a normalization procedure. In this case, the effective normalization is accomplished by determining the instrumental system response based on measurements of a sample with presupposed properties. Compared with the usual normalization techniques based on front- and back-side measurements, our procedure allows better use of the open photoacoustic cell (OPC) configuration resulting in larger PA signal amplitudes (smaller cell volume) and uses a simpler experimental apparatus.
frequency (14.9 kHz). The second was attributed to the microphone geometry and diaphragm properties. As a whole, the procedures appear to yield corrected signals that more accurately represent the sample physical characteristics. The processed signal appears to be an accurate representation of the theoretical PA amplitude and phase signals for the 85 µm thick Si plate. Thermal diffusion, thermoelastic, and plasmaelastic components calculated from the corrected signals are in agreement with the bulk parameters of this sample. This procedure should be valid for all solid samples and is suitable for multilayered samples composed of a substrate in the form of a thin plate, or a thin film as a coating, as long as frequency-dependent TD, TE, and PE effects are accounted for. ACKNOWLEDGMENTS
This work was supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia (Project Nos. ON171016 and III45005). 1A.
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