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Nondestructive evaluation of solids and deposited films by thermal-wave interferometry
Applied Surface North-Holland
Science
69 (1993) 65-68
applied surface science
Nondestructive evaluation of solids and deposited films by thermal-wave interferometry E. Oesterschulze,
M. Stopka, M. Tochtrop-Mayr,
K. Masseli and R. Kassing
University of Kassel, Heinrich-Plettstrasse 40, D-3500 Kassel, Germany Received
2 June
1992; accepted
for publication
30 October
1992
In order to investigate heat diffusion processes in dielectric or metallic materials, a differential interferometer was constructed allowing phase-sensitive measurement of the thermal expansion of periodically excited solid surfaces. To get insight into the physical process of heat propagation, a three-dimensional model of temperature diffusion will be introduced. This theoretical model may be considered as a first step for the understanding of signal formation in case of investigating the homogeneity or adhesion properties of deposited films by means of thermal-wave interferometry. First measurements will be presented and thermal parameters evaluated using the model mentioned above.
1. Introduction The spatially resolved investigation of thermal parameters of materials is of technical interest, especially in the case of structures of small dimensions. The thermal behaviour of dielectric or metallic mirrors under laser illumination, the investigation of deposited films or the heat dissipation in microelectronic devices may serve as example. Thus a method for monitoring thermal parameters is desirable. For nondestructive and remote-sensing material investigation, photo-thermal measurement techniques have been established in recent years [l-3]. Thermal-wave interferometry is one of the most sensitive techniques; moreover, it is a noncontact measuring system [4,5]. In combination with a three-dimensional model of heat propagation in solids it is possible to extract thermal parameters.
2. Experimental
I
Y-Scan
arrangement
The instrumental set-up consists of three components which are shown in fig. 1. 0169.4332/93/$06.00
The sample surface is excited by a focussed Ar-ion laser with a maximum output power of about 1 W. The laser beam is periodically modulated by an acousto-optical modulator (AOM) up to frequencies of 1 MHz. A scanning system allows the computer-controlled variation of the distance between excitation source and the point of measurement of the interferometer as described below.
0 1993 - Elsevier
Science
main
Publishers
Fig.
1. Schematic
B.V. All rights reserved
experimental set-up interferometer.
of the thermal
wave
E. Oesterschulze et al. / Nondestructke
66
The thermal expansion induced by this laser excitation is detected by a differential interferometer. In this means linear polarized light of a 10 mW He/Ne laser is reflected by a polarizing beam splitter (PBS). The following quarter-wave plate (QW) yields circularly polarized light. Passing through a Rochon prisma (RP) the beam is spatially separated into two orthogonal polarized beams. After reflection at a dichroic filter (DF) both beams are focussed onto the sample, one acting as reference and the other as measurement beam. On their way back through the optical system the polarized beams are recombined by the Rochon prisma. Leaving the quarter-wave plate the beam passes the polarizing beam splitter, and the zero-order interference fringe occurring at the polarizer (PI is detected by a photodiode (PD). To increase the detection sensitivity, the measured interferometer signal is electronically fed back (FB) to the sample holder using a piezoelectric transducer (PZT) which counteracts incidental vibrations of the measurement system. This procedure reduces the influence of mechanical noise signals on the interferometer output which is given by:
Z(6) =
zmax +‘min z~ax - 'min
45T6
cos . (1) 1 A i I,,, and Zmin denote the minimum and maximum amplitudes of the interferogram, h is the He/Ne laser wavelength and 6 describes the total path difference between the polarized beams. The latter consists of the mechanical and optical static path differences 6, and the thermal-induced surface expansion 8,(t) = A cos(wt), A being the amplitude of the thermal expansion. Adjusting the static phase lag to 6, = 5~/2 by tilting the same holder by means of the computer-controlled piezo-transducer, and considering that A will be small in comparison to the wavelength, Z may be approximated to: Z(t)
=
2
+
zmax + 'min 2 +
2
I,,,
- Zmin 4rrA 2 h
evaluation of solids and deposited films
frequency w. Z(w) is phase-sensitively a lock-in amplifier.
3. Theory of heat propagation
(2) This is a linear dependence between the expansion amplitude A and the measured intensity I of
by
in solids
The heat diffusion process in solids may be described by the three-dimensional heat-conduction equation [6]:
-v’(KT(r,
t)) +PC
where T(r, t) is the conductivity, p the and S(r, t) are the suming an isotropic, temperature-independent and c the conduction 1 U(r, AT(r,t)-;
aT(r, t) at
=
S(r, t),
(3)
temperature, K the thermal density, c the specific heat source and sink terms. Ashomogeneous medium with thermal constants K equation reduces to: t)
at
=-p.
S(r,
t)
(4)
K
= K/PC denotes the thermal diffusivity. This differential equation must be applied to the experimental situation of fig. 2 showing a solid half-space in contact with a gas half-space. The solid surface is illuminated with a laser beam inducing a heat source with a Gaussian distribution which is periodically modulated and exponentially damped while propagating into the solid in z-direction. Assuming that no absorption takes place in the gas phase, the conduction equation can be written as:
a
z
Solid
cos(tit)
detected
detection beam
thermal bump
iz
reference beam
f
r-
Fig. 2. Schematic description of the experimental geometry of laser illumination of a solid half-space in contact with a gas half-space.
E. Oesterschulze et al. / Nondestructive evaluation of solids and deposited films
67
extended to the investigation of layer/ substrate systems as has been shown by Iravani and Wickramasinghe [71. Neglecting elastic deformations, the temperature distribution Ts(r, z, t) considering a solid as shown in fig. 2, leads to the simple linear thermal expansion expression: where the indices g and s denote gas and solid half-space, respectively, /3 is the absorption coefficient, R the reflection coefficient, u the waist of the Gaussian distributed laser beam (TEM, mode), I, the laser intensity, and w the modulation frequency. These differential equations must be treated considering the boundary conditions that both temperature and thermal flux must be continuous functions at the discontinuity (z = 0): T, I Z-O = 7; 1==a and
K
(7) This mathematical problem was solved using a special integral transform technique, the so-called Hankel transform of zero order. Applying the Hankel transform to the differential eqs. (6) and (5) and taking the boundary conditions into account, the resultant temperature field will be given by the Hankel back transform: z
Tg(r, 2) =
I”P
m
~1
s
X
dwJo(w)
h(r)
=Lm dzbT,(r,
with b denoting coefficient.
z, t), the linear
(10) thermal
expansion
4. Results First measurements were carried out to confirm the above assumptions that thermal parameters may show no measurable temperature dependence in the considered temperature region. Therefore, keeping the distance between excitation laser beam and the point of interferometer measurement constant, the output power of the Ar-ion laser (modulated with 1 kHz) was linearly varied. The detected expansion signal of a brass sample shows an excellent linear behaviour as a function of power as can be seen in fig. 3. Various metal substrates were investigated to measure the respective thermal diffusivity. As an example, the thermal diffusivity of a Cu sample was determined by measuring the phase lag be-
0
K,q, -
K,P
e4pr
e
-2u= /2
Ksqs + Kg9,
7
(8)
z > 0:
IOP
T~(T, z) = =/
s
m
dwJo(v)
0
e-Pr _ Kgqg-KJ3 c-4? Kg4, + KSqS where q& = p2 + iw/a, s are the quadratic forms of the complex propagation vector in the (p, z)space. Without explaining any mathematical details, this way of treating heat propagation in solids based on the Hankel-transform technique can be
Fig. 3. Amplitude U of the detector output of a brass sample excited with a frequency of 1 kHz as a function of Ar-ion laser output power at a fixed measurement location.
E. Oesterschulze et al. / Nondestructice el‘aluation
68
xa
of solidsand deposited
films
A more complicated situation arises with investigation of layer/substrate systems. Fig. 5 shows the phase-lag measurements of a layer/ substrate system consisting of a 80 nm Au film deposited on a BaKSO-glass substrate. The solid lines represent theoretical results calculated by means of the extended model mentioned above.
1
H 120 8 90-
30 t
5. Conclusion 0 I
0.0
0.5
1
1.0
/
1.5
1
2.0
I
25
i
i
3.0 r/CL
Fig. 4. Phase-lag signal @ of a Cu sample as a function of the distance between the excitation source and measurement point for various frequencies: (0) 1066 Hz, (0 J 31Y0 Hz, (A) = 5260 Hz, (+) = 7220 Hz, ( x ) = 9070 Hz. Distances were normalized with respect to the one-dimensional diffusion length k.
tween excitation signal and the thermal expansion of the point of the interferometer on the sample surface as a function of the distance between them. Fig. 4 shows the normalized phaselag dependence for various laser modulation frequencies for the Cu sample. The mean value of the thermal diffusivity of this Cu sample was evaluated to be (111.0 + 5.7) X lo-” m’/s, differing only slightly from the literature value of 117.0 X lo-’ m*/s.
It has been shown that thermal-wave interferometry is a powerful tool for noncontact and nondestructive material analysis. This technique allows the spatially resolved investigation of laser-induced thermal expansion of sample surfaces. In combination with a three-dimensional temperature diffusion model, a quantitative evaluation of the thermal diffusivity of various samples was possible.
Acknowledgement This work is supported by the Department of Research and Technology of the Federal Republic of Germany (BMFT Project 13N6008).
References 111F.A. McDonald, Can. J. Phys. 64 (1986) 1023. 121J.C. Murphy, L.C. Aamodt, F.G. Satkiewicz, R.B. Givens 60 I
Fig. 5. Phase-lag signal @ of a 80 nm Au film on a BaKSO-glass substrate as a function of frequency ((0) = 3030 Hz, ( n ) = 1880 Hz, (A ) = 965 Hz) in comparison with theoretical calculations (solid lines). Distances were normalized with respect to the one-dimensional diffusion length w.
and P.R. Zarriello, Johns Hopkins APL Tech. Dig. 7 (1986) 187. 131R.L. Thomas, L.D. Favro and P.K. Kuo, Can. J. Phys. 64 (19x6) 1234. and Pho[41Z. Sodnik and H.J. Tiziani, in: Photoacoustic tothermal Phenomena, Eds. P. Hess and J. Pelzl (Springer, Heidelberg, 1988) pp. 400-403. J. Stohr, M. Tochtrop and R. Kassing, in: [51V. Kurzmann, Acoustic, Thermal Wave and Optical Characterization of Materials, Proc. Eur. Mater. Res. Sot. Spring Meeting 1989, Strasbourg, 1989, Eds. G.M. Crean, M. Locatelh and J. McGilp, pp. 117-120. Oxford, [61H.S. Carslaw and J.C. Jaeger, 2nd ed. (Clarendon, 1959). J. Appl. Phys. 58 [71 M.V. Iravani and H.K. Wickramasinghe, (1985) 122.
Science
69 (1993) 65-68
applied surface science
Nondestructive evaluation of solids and deposited films by thermal-wave interferometry E. Oesterschulze,
M. Stopka, M. Tochtrop-Mayr,
K. Masseli and R. Kassing
University of Kassel, Heinrich-Plettstrasse 40, D-3500 Kassel, Germany Received
2 June
1992; accepted
for publication
30 October
1992
In order to investigate heat diffusion processes in dielectric or metallic materials, a differential interferometer was constructed allowing phase-sensitive measurement of the thermal expansion of periodically excited solid surfaces. To get insight into the physical process of heat propagation, a three-dimensional model of temperature diffusion will be introduced. This theoretical model may be considered as a first step for the understanding of signal formation in case of investigating the homogeneity or adhesion properties of deposited films by means of thermal-wave interferometry. First measurements will be presented and thermal parameters evaluated using the model mentioned above.
1. Introduction The spatially resolved investigation of thermal parameters of materials is of technical interest, especially in the case of structures of small dimensions. The thermal behaviour of dielectric or metallic mirrors under laser illumination, the investigation of deposited films or the heat dissipation in microelectronic devices may serve as example. Thus a method for monitoring thermal parameters is desirable. For nondestructive and remote-sensing material investigation, photo-thermal measurement techniques have been established in recent years [l-3]. Thermal-wave interferometry is one of the most sensitive techniques; moreover, it is a noncontact measuring system [4,5]. In combination with a three-dimensional model of heat propagation in solids it is possible to extract thermal parameters.
2. Experimental
I
Y-Scan
arrangement
The instrumental set-up consists of three components which are shown in fig. 1. 0169.4332/93/$06.00
The sample surface is excited by a focussed Ar-ion laser with a maximum output power of about 1 W. The laser beam is periodically modulated by an acousto-optical modulator (AOM) up to frequencies of 1 MHz. A scanning system allows the computer-controlled variation of the distance between excitation source and the point of measurement of the interferometer as described below.
0 1993 - Elsevier
Science
main
Publishers
Fig.
1. Schematic
B.V. All rights reserved
experimental set-up interferometer.
of the thermal
wave
E. Oesterschulze et al. / Nondestructke
66
The thermal expansion induced by this laser excitation is detected by a differential interferometer. In this means linear polarized light of a 10 mW He/Ne laser is reflected by a polarizing beam splitter (PBS). The following quarter-wave plate (QW) yields circularly polarized light. Passing through a Rochon prisma (RP) the beam is spatially separated into two orthogonal polarized beams. After reflection at a dichroic filter (DF) both beams are focussed onto the sample, one acting as reference and the other as measurement beam. On their way back through the optical system the polarized beams are recombined by the Rochon prisma. Leaving the quarter-wave plate the beam passes the polarizing beam splitter, and the zero-order interference fringe occurring at the polarizer (PI is detected by a photodiode (PD). To increase the detection sensitivity, the measured interferometer signal is electronically fed back (FB) to the sample holder using a piezoelectric transducer (PZT) which counteracts incidental vibrations of the measurement system. This procedure reduces the influence of mechanical noise signals on the interferometer output which is given by:
Z(6) =
zmax +‘min z~ax - 'min
45T6
cos . (1) 1 A i I,,, and Zmin denote the minimum and maximum amplitudes of the interferogram, h is the He/Ne laser wavelength and 6 describes the total path difference between the polarized beams. The latter consists of the mechanical and optical static path differences 6, and the thermal-induced surface expansion 8,(t) = A cos(wt), A being the amplitude of the thermal expansion. Adjusting the static phase lag to 6, = 5~/2 by tilting the same holder by means of the computer-controlled piezo-transducer, and considering that A will be small in comparison to the wavelength, Z may be approximated to: Z(t)
=
2
+
zmax + 'min 2 +
2
I,,,
- Zmin 4rrA 2 h
evaluation of solids and deposited films
frequency w. Z(w) is phase-sensitively a lock-in amplifier.
3. Theory of heat propagation
(2) This is a linear dependence between the expansion amplitude A and the measured intensity I of
by
in solids
The heat diffusion process in solids may be described by the three-dimensional heat-conduction equation [6]:
-v’(KT(r,
t)) +PC
where T(r, t) is the conductivity, p the and S(r, t) are the suming an isotropic, temperature-independent and c the conduction 1 U(r, AT(r,t)-;
aT(r, t) at
=
S(r, t),
(3)
temperature, K the thermal density, c the specific heat source and sink terms. Ashomogeneous medium with thermal constants K equation reduces to: t)
at
=-p.
S(r,
t)
(4)
K
= K/PC denotes the thermal diffusivity. This differential equation must be applied to the experimental situation of fig. 2 showing a solid half-space in contact with a gas half-space. The solid surface is illuminated with a laser beam inducing a heat source with a Gaussian distribution which is periodically modulated and exponentially damped while propagating into the solid in z-direction. Assuming that no absorption takes place in the gas phase, the conduction equation can be written as:
a
z
Solid
cos(tit)
detected
detection beam
thermal bump
iz
reference beam
f
r-
Fig. 2. Schematic description of the experimental geometry of laser illumination of a solid half-space in contact with a gas half-space.
E. Oesterschulze et al. / Nondestructive evaluation of solids and deposited films
67
extended to the investigation of layer/ substrate systems as has been shown by Iravani and Wickramasinghe [71. Neglecting elastic deformations, the temperature distribution Ts(r, z, t) considering a solid as shown in fig. 2, leads to the simple linear thermal expansion expression: where the indices g and s denote gas and solid half-space, respectively, /3 is the absorption coefficient, R the reflection coefficient, u the waist of the Gaussian distributed laser beam (TEM, mode), I, the laser intensity, and w the modulation frequency. These differential equations must be treated considering the boundary conditions that both temperature and thermal flux must be continuous functions at the discontinuity (z = 0): T, I Z-O = 7; 1==a and
K
(7) This mathematical problem was solved using a special integral transform technique, the so-called Hankel transform of zero order. Applying the Hankel transform to the differential eqs. (6) and (5) and taking the boundary conditions into account, the resultant temperature field will be given by the Hankel back transform: z
Tg(r, 2) =
I”P
m
~1
s
X
dwJo(w)
h(r)
=Lm dzbT,(r,
with b denoting coefficient.
z, t), the linear
(10) thermal
expansion
4. Results First measurements were carried out to confirm the above assumptions that thermal parameters may show no measurable temperature dependence in the considered temperature region. Therefore, keeping the distance between excitation laser beam and the point of interferometer measurement constant, the output power of the Ar-ion laser (modulated with 1 kHz) was linearly varied. The detected expansion signal of a brass sample shows an excellent linear behaviour as a function of power as can be seen in fig. 3. Various metal substrates were investigated to measure the respective thermal diffusivity. As an example, the thermal diffusivity of a Cu sample was determined by measuring the phase lag be-
0
K,q, -
K,P
e4pr
e
-2u= /2
Ksqs + Kg9,
7
(8)
z > 0:
IOP
T~(T, z) = =/
s
m
dwJo(v)
0
e-Pr _ Kgqg-KJ3 c-4? Kg4, + KSqS where q& = p2 + iw/a, s are the quadratic forms of the complex propagation vector in the (p, z)space. Without explaining any mathematical details, this way of treating heat propagation in solids based on the Hankel-transform technique can be
Fig. 3. Amplitude U of the detector output of a brass sample excited with a frequency of 1 kHz as a function of Ar-ion laser output power at a fixed measurement location.
E. Oesterschulze et al. / Nondestructice el‘aluation
68
xa
of solidsand deposited
films
A more complicated situation arises with investigation of layer/substrate systems. Fig. 5 shows the phase-lag measurements of a layer/ substrate system consisting of a 80 nm Au film deposited on a BaKSO-glass substrate. The solid lines represent theoretical results calculated by means of the extended model mentioned above.
1
H 120 8 90-
30 t
5. Conclusion 0 I
0.0
0.5
1
1.0
/
1.5
1
2.0
I
25
i
i
3.0 r/CL
Fig. 4. Phase-lag signal @ of a Cu sample as a function of the distance between the excitation source and measurement point for various frequencies: (0) 1066 Hz, (0 J 31Y0 Hz, (A) = 5260 Hz, (+) = 7220 Hz, ( x ) = 9070 Hz. Distances were normalized with respect to the one-dimensional diffusion length k.
tween excitation signal and the thermal expansion of the point of the interferometer on the sample surface as a function of the distance between them. Fig. 4 shows the normalized phaselag dependence for various laser modulation frequencies for the Cu sample. The mean value of the thermal diffusivity of this Cu sample was evaluated to be (111.0 + 5.7) X lo-” m’/s, differing only slightly from the literature value of 117.0 X lo-’ m*/s.
It has been shown that thermal-wave interferometry is a powerful tool for noncontact and nondestructive material analysis. This technique allows the spatially resolved investigation of laser-induced thermal expansion of sample surfaces. In combination with a three-dimensional temperature diffusion model, a quantitative evaluation of the thermal diffusivity of various samples was possible.
Acknowledgement This work is supported by the Department of Research and Technology of the Federal Republic of Germany (BMFT Project 13N6008).
References 111F.A. McDonald, Can. J. Phys. 64 (1986) 1023. 121J.C. Murphy, L.C. Aamodt, F.G. Satkiewicz, R.B. Givens 60 I
Fig. 5. Phase-lag signal @ of a 80 nm Au film on a BaKSO-glass substrate as a function of frequency ((0) = 3030 Hz, ( n ) = 1880 Hz, (A ) = 965 Hz) in comparison with theoretical calculations (solid lines). Distances were normalized with respect to the one-dimensional diffusion length w.
and P.R. Zarriello, Johns Hopkins APL Tech. Dig. 7 (1986) 187. 131R.L. Thomas, L.D. Favro and P.K. Kuo, Can. J. Phys. 64 (19x6) 1234. and Pho[41Z. Sodnik and H.J. Tiziani, in: Photoacoustic tothermal Phenomena, Eds. P. Hess and J. Pelzl (Springer, Heidelberg, 1988) pp. 400-403. J. Stohr, M. Tochtrop and R. Kassing, in: [51V. Kurzmann, Acoustic, Thermal Wave and Optical Characterization of Materials, Proc. Eur. Mater. Res. Sot. Spring Meeting 1989, Strasbourg, 1989, Eds. G.M. Crean, M. Locatelh and J. McGilp, pp. 117-120. Oxford, [61H.S. Carslaw and J.C. Jaeger, 2nd ed. (Clarendon, 1959). J. Appl. Phys. 58 [71 M.V. Iravani and H.K. Wickramasinghe, (1985) 122.