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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
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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,1M. D. Rabasović,1D. M. Todorović,2S. Galović,3and S. E. Bialkowski4
1Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade-Zemun, Serbia
2Institute for Multidisciplinary Researches, University of Belgrade, P.O. Box 33, 11030 Belgrade, Serbia
3Institute of Nuclear Sciences “Vinca”, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia
4Department of Chemistry and Biochemistry, Utah State University, Logan, Utah 84322-0300, USA
(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 accom-
plished 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. C2015 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 op-
tical absorption.1–6For 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 Rosencweig-
Gersho (RG)1and elastic-bending (EB)2,9models with an
open-cell (OC) experimental setup,12 one can use photoacous-
tic 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.9On 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, optically-
induced free carrier generation contributes to the TD and TE
effects.17–19 Free carriers can also induce elastic stress result-
ing in plasmaelastic (PE) effects. Subsequently, the semicon-
ductor PA signal is due to a combination of TD, TE, and PE ef-
fects. PA spectroscopy may thus be used to determine thermal,
mechanical, optical, and electronic characteristics of semicon-
ductors. 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 interfer-
ence often mask the “true” PA signal. Noise and signal distor-
tion 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” frequency-
dependent PA signal target response is calculated using RG
and EB theories and the relationship between TD, TE, and PE
signal components.9Good 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 opti-
cal excitation of a homogeneous semiconductor sample with
thickness ls(Figure 1(a)). Excitation intensity is given by
0034-6748/2015/86(3)/035110/9/$30.00 86, 035110-1 ©2015 AIP Publishing LLC
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035110-2 Markushev et al. Rev. Sci. Instrum. 86, 035110 (2015)
FIG. 1. Transmission photoacoustic spectroscopy model: (a) thermal diffusion and (b) thermal and plasma elastic analysis.
I=I0exp (−jωt), where j is the imaginary unit, tis the time,
ω=2πf, and fis the modulation frequency. General assump-
tions 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 prop-
agates 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)=[b1exp (γz)+b2exp (−γz)+b3n(z)
+b4exp (βz)]exp (jωt),(2)
where n(z)is the free carrier concentration, βis the sample
absorption coefficient, and b1,b2,b3, and b4are the complex
constants depending on the sample characteristics.9,14 Also,
γ=jω/Ds, with the thermal diffusivity, Dz, given by Ds
=ks/(ρC)where ksis the heat conductivity, ρis the density,
and Cis the heat capacity.
Equation (2) allows one to calculate the temperature
Ts(ls,t)at the back side of the sample (z=ls). This tempera-
ture is responsible to the TD component of PA signal δpTD,1–4
δpTD (ω)=γgp0Dg
lcT0√ωT(ls,ω),(3)
where T0and p0are the gas temperature and pressure, respec-
tively, γgis the adiabatic ratio, Dgis the gas thermal diffusivity,
and lcis the PA cell length.
It is well known that reducing the thickness of the Si
sample increases the TD and TE components, with TD increas-
ing 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
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 coor-
dinate ϕ.δpTE (ω)and δpPE (ω)are calculated using4,8
δpm(ω)=γp0
V0R s
0
2πrUz(r,z)dr,(4)
where exp (jωt)is omitted for simplicity.9,14–19,22 Here, m
=TE,PE indicates the particular effect, V0and p0are the cell
volume and ambient pressure, ris the radial coordinate, zis the
longitudinal position, Rsis 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).(5)
The sample bending term is
U1z(r,z)=α6R2
s−r2
ls3MT,(6)
while thermal expansion term is given by
U2z(r,z)=α
1+ν
1−ν
z
ls
T(z)dz
−3ν
1−ν
4z2−ls2
ls3MT+
2(z+ls
2)
ls
NT
.
(7)
Here, αis the coefficient of linear expansion, lsis the sam-
ple thickness, νis Poisson’s ratio, and T(z)is the temperature
distribution along the z–axis (Figure 1(b)). MTand NTare
defined as
MT=
ls/2
−ls/2
zT (z)dzand NT=
ls/2
−ls/2
T(z)dz.(8)
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035110-3 Markushev et al. Rev. Sci. Instrum. 86, 035110 (2015)
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.
In the case of free carrier generation, PE effects are also
calculated using Eqs. (4)–(8). However, one uses the coeffi-
cient of electronic deformation dninstead 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 2illustrates 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 Iare the target of the signal
analysis. We assume that model is valid across the modulation
frequency range and valid for samples other than the particular
Si used here.22
APPARATUS AND NOISE ANALYSIS
The photoacoustic apparatus consists of five main compo-
nents: (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 Ωcm, n-
type, and ⟨100⟩oriented Si wafer. The bulk sample parameters
TABLE I. Sample and system parameters.
Si sample bulk parameters
Density ρ=2.33×103kg m−3Coefficient of carrier diffusion Dn=1.2×10−3
Optical reflectivity R=0.30 Recombination velocity of the front surface s1=2ms−1
Optical absorption coefficient β=5.00 ×105m−1Recombination velocity of the rear surface s2=24 m s−1
Thermal diffusivity Ds=1.00 ×10−4m2s−1Young’s modulus E=1.31 ×1011 N m−2
Excitation energy ε=1.88 eV Linear thermal expansion α=3.0×10−6K−1
Energy gap εg=1.11 eV Coefficient of electronic deformation dn=−9.0×10−31 m3
Lifetime of photogenerated
carriers
τ=6.0×10−6s
Calculated system parameters in the case of optical excitation power P0=1.5×10−3W
LF
Sound card eLF1 time constant τeLF1 =(1.1±0.1)×10−2s Detector eLF2 time constant τeLF2 =(6.3±0.3)×10−3s
Sound card eLF1 frequency feLF1 =(15±1) Hz Detector eLF2 frequency feLF2 =(25±1) Hz
HF
Detector aHF1 frequency faHF1 =(14.7±0.2)×103Hz Detector aHF1 dumping factor δeHF1 =(8±1)×10−2ωeHF1
Detector aHF2 frequency faHF2 =(9.4±0.2)×103Hz Detector aHF2 dumping factor δeHF2 =(4.5±0.5)×10−1ωeHF2
Detector eHF1 frequency feHF1 =(14.9±0.4)×103Hz Detector eHF2 frequency feHF2 =(9.5±0.4)×103Hz
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035110-4 Markushev et al. Rev. Sci. Instrum. 86, 035110 (2015)
FIG. 3. Schematic diagram of experimental setup used for solid state pho-
toacoustics.
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 micro-
phone 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.
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),(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 sig-
nals 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) ampli-
tude RN(f)and (b) phase ϕN(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 (1-
20) kHz. Noise in the LF range is dominated by flicker noise
(FN), having typical f−1amplitude 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 RN(f)and (b) phases ϕN(f)in our experimental setup, as a function of modulation frequency f. The
f−1amplitude dependence is recognized as the flicker noise (dashed line), while f+1amplitude dependence is recognized as the coherent signal deviation (doted
line). Noise clearly divides our analysis to LF and HF range.
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035110-5 Markushev et al. Rev. Sci. Instrum. 86, 035110 (2015)
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 af-
fects 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 accom-
panying 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 ac-
counted for using a first-order high-pass filter cascade system
transfer function
Hel
LF (ω)=Hel
LF1 (ω)·Hel
LF2 (jω)
=−ωτeLF1
(1+jω·τeLF1)·ωτeLF2
(1+jω·τeLF2),(12)
where τeLF1 =(2πfeLF1)−1and τeLF2 =(2πfeLF2)−1are 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 fil-
ters 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,
Hel
HF (ω)=Hel
HF1 (ω)·Hel
HF2 (ω)
=1
(1+jω·τeHF1)·1
(1+jω·τeHF2),(13)
where τeHF1 =(2πfeHF1)−1and τHFe2 =(2πfeHF2)−1are 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
is the microphone acoustic deviation Hac
HF (ω). This artifact
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 dia-
phragm 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)
Hac
HF (ω)=Hac
HF1 (ω)·Hac
HF2 (ω)=ωaHF12
ωaHF12+jδaHF1ω−ω2
·ωaHF22
ωaHF22+jδaHF2ω−ω2,(14)
where δk=ωk/Qkis the damping and Qkis the quality factor
(k =aHF1, aHF2, . . . ). The first term in Eq. (14) is usually
characterized as the microphone cutofffrequency, 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. 1–8in 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).
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035110-6 Markushev et al. Rev. Sci. Instrum. 86, 035110 (2015)
a typical PA apparatus. Any PA apparatus is an integration
of excitation source, sample, sample chamber, pressure trans-
ducer (microphone), and processing electronics. Although the
reason for performing PA measurement is to obtain informa-
tion on the sample, an integrated “holistic” approach to signal
analysis is required to optimize the signal information for
subsequent sample parameter estimation. Our experience sug-
gests that all PA measurements will be subject to coherent and
incoherent noise, high- and low-frequency roll-offcaused by
microphone and measurement electronics characteristics, and
distortion due to electronic impedance mismatch and micro-
phone 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 compo-
nents.
To facilitate apparatus modeling, we assume that these
effects can be modeled by typical system transfer function ele-
ments 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 Yexp(ω)to obtain Sexp(ω)using Eq. (9);
(2) Fit Sexp(ω)to obtain all microphone and lock-in elec-
tronics characteristics ωmn,δmn, and τmn (m =eLF, eHF
or aHF, n =1,2,. . . ) using Eqs. (10),(12)–(14);
(3) Correct Sexp(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 7depicts 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 =1.1×10−2s
and τeLF2 =6.3×10−3s. In terms of modulation frequency,
feLF1 =15 Hz and feLF2 =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 measure-
ments14–17,20–22 (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 electro-
acoustic parameters are found to be feHF1 =14.9×103Hz
(τeHF1 =10.7×10−6s),feHF2 =9.5×103Hz (τeHF2 =16.7
×10−6s),faHF1 =14.7×103Hz, δaHF1 =0.08 ·ωaHF1,faHF2
=9.4×103Hz, and δaHF2 =0.45 ·ωaHF2. The faHF1 deter-
mined 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.
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035110-7 Markushev et al. Rev. Sci. Instrum. 86, 035110 (2015)
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
for the influence of H(ω)=Hel
LF (ω)+Hel
HF (ω)+Hac
HF (ω):
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 2and 8, one can see that the PA ampli-
tude 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 displace-
ment component is miscalculated using a strictly U1z(r,z)
∼ls−3dependence in Eq. (6). They account for the discrep-
ancy using a semi-empirical U1z(r,z)∼ls−2.8dependence.
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 ftogether with fitted
(dashed line) and theoretical (solid line) signal. Arrows depicts the results of applied signal correction procedure.
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035110-8 Markushev et al. Rev. Sci. Instrum. 86, 035110 (2015)
in very thin Si samples, especially those with thicknesses
below 50 µm.23 In our opinion, a theoretical ls−3dependence
is not a problem for samples greater than 50 µm. Thinner
samples may lose the ideal U1z(r,z)∼ls−3bending charac-
teristic, perhaps due to an effective sample radius or temper-
ature 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 depen-
dent 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 pre-
sented here may also be thought of as a normalization proce-
dure. 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.
CONCLUSIONS
Photoacoustic measurements of a 85 µm Si plate over
a 20 Hz–20 kHz modulation frequency range show that PA
signals can be represented as a combination of thermal diffu-
sion and thermal and plasma elastic components. An accu-
rate photoacoustic signal was obtained in an open-cell experi-
mental apparatus using noise and signal deviation recognition
and removal procedures based on both a theoretical elastic-
bending model and recognition of processes giving rise to
signal distortion. We find coherent noise to be the dominant
interference at high frequencies. Its origin was found to be
the current optical excitation source modulation system. A
subtractive procedure was found to be sufficient to obtain an
interference-free signal.
Subsequent analysis reveals strong signal distortion. At
low frequencies, apparent amplitude distortion was found for
f<100 Hz and phase distortion for f<1000 Hz. This distor-
tion was attributed to an impedance mismatch between the
microphone and the sound card used for lock-in like signal
processing. Accounting for this effect in a system transfer
function allowed signal processing procedures that simulta-
neously correct signal amplitudes and phases. The correction
procedure allows estimation of instrument time constants for
the detector (25 Hz) and the sound-card (15 Hz).
Major system resonances were detected in the high fre-
quency range and attributed to electro-acoustic detector char-
acteristics. In the open-cell configuration, the electret micro-
phone acts as an acoustic low-pass filter. Two characteristic
frequencies were found. The first was the microphone cutoff
frequency (14.9 kHz). The second was attributed to the micro-
phone 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 accu-
rate representation of the theoretical PA amplitude and phase
signals for the 85 µm thick Si plate. Thermal diffusion, ther-
moelastic, 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).
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