Note: Descriptions are shown in the official language in which they were submitted.
CA 02305477 2000-04-17
FIELD OF INVENTION
The present invention relates to metrologic methodologies and instrumentation,
in
particular to laser-frequency domain infrared photothermal radiometry (PTR),
for measuring
electronic properties in industrial Si wafers, devices and other semiconductor
materials; and
metrologic methodologies for performing thermal-parameter depth profilometry
of intrinsic or
process-induced inhomogeneities in engineering materials. In particular, the
metrologic
application to measuring thermal diffusivity of layered solids, a~,
inhomogeneous thermal
diffusivity depth profiles, a(x), and electronic transport properties in
semiconductor wafers such
as: minority-carrier lifetime (i), surface recombination velocities (,S'),
diffusion coefficient (D)
and diffusion length (L).
BACKGROUND OF THE INVENTION
There are essentially two dynamic or time-dependent methods for measuring
thermal and
electronic properties of solids. Regarding thermal (or thermophysical)
properties, the first is the
periodic heat flow method (see for example L. Qian and P. Li, Appl. Opt. 29,
4241, 1990) and
the second one is the transient method (see W. P. Leung and C. A. Tam, J.
Appl. Phys. 56, 153,
1984), including the spectral analysis and cross-correlation (mufti-frequency)
method (S. Peralta,
S. C. Ellis, C. Christofides and A. Mandelis, J. Res. Non-Destructive Eval.,
3, 69, 1991).
In the periodic heat flow case, a solid sample is irradiated with a
harmonically modulated
laser beam thereby launching a thermal wave through the sample. The resulting
periodic
temperature profile at the front or back of the surface of the sample is
monitored at several
modulation frequencies f, also known as the frequency scan method. The
frequency dependent
thermal diffusion length p, is given by:
and is related to the phase-lag of the detected temperature wave with respect
to the heating
source and may be monitored using a lock-in amplifier.
In transient measurement techniques such as pulsed or mufti-frequency spectral
excitation, a sample is irradiated on one side with a laser pulse and the time
evolution of the
temperature is monitored and the rate of decay of the temperature is related
to thermal diffusivity
of the solid. Among the most common non-contact, non-destructive techniques
used for
characterizing electronic materials and semiconductor substrates are:
Photothermal radiometry
(PTR) [E. A. Ulmer and D. R. Frankl, Proc. IX-th Int. Conf. Physics
Semiconductors, Nauka,
1959, pp. 99-101; H. Nakamura, K. Tsubouchi, N. Mikoshiba and T. Fukuda, Jpn.
J. Appl. Phys.
24, L876 (1985); S. J. Sheard M. G. Somekh and T. Hiller, Mater. Sci. Eng. B5,
101, (1990); A.
Mandelis, R. Bleiss and F. Shimura, J. Appl. Phys. 74, 3431 (1993)];
laserlmicrowave
absorptiorzlreflection (LMR) [T. Warabisako, T. Saitoh, T. Motooka and T.
Tokuyama, Jpn. J.
Appl. Phys. Suppl. 22-1, 557 (1982); J. Waldemeyer, J. Appl. Phys. 63, 1977
(1988); Z. G. Ling
and P. K. Ajmera, J. Appl. Phys. 69, 519 (1991)]; infrared absorption (IA) [Y.
Mada, Jpn. J.
Appl. Phys. 18, 2171 (1979); F. Shimura, T. Okui and T. Kusama, J. Appl. Phys.
61, 7168
(1990); A. Buckzkowski, G. A. Rozgonyi and F. Shimura, Proc. MRS Spring Conf.
(1992)];
microwave photoconductance decay (,u PCD) [T. Tiegje, J. I. Haberman, R. W.
Francis and A.
2
CA 02305477 2000-04-17
K. Ghosh, J. Appl. Phys. 54, 2499 (1983)], or open circuit decay (OCYD [LJ.
Lehmann and H.
Foll, J. Electrochem. Soc. 135, 2831 (1988)]; surface photovoltage, SPV, [J.
Lagowski, P.
Edelman, M. Dexter, and W. B. Henley, Semicond. Sci. Thechnol. 7A, 185 (1992);
J. Lagowski,
V. Faifer, and P. Edelman, Electrocehm. Soc. Proc. 96-13, 512 (1995)]; laser
photomodulated
thermoreflectance, PMOR [A. Rocencwaig, in Photoacoustic and Thermal Wave
Phenomena in
Semiconductors, edited by A. Mandelis (Elsevier, New York, 1987), Chap. 5]. In
all of these
methods laser illumination is used to generate excess electron-hole pairs. The
resulting signal is
detected in the frequency-domain as a function of modulation frequency (in PTR
and PMOR) or
in the time domain as a transient signal (IA, LMR, p,-PCD, and OCVD).
SUMMARY OF THE INVENTION
The present invention consists of the development of a complete photothermal
radiometric instrumentation hardware and software metrologic system comprising
novel
combinations of signal generation and analysis techniques, computational
software, as well as
novel instrumental hardware configurations based on (but not confined to) the
physical principles
of laser infrared photothermal radiometry.
FREQUENCY SWEEP (CHIRP)PHOTOTHERMAL RADIOMETRY
The present invention provides a method of noncontact measurement of
electronic and
thermal transport properties in semiconductors such as thermal diffusivity
(a), minority carrier
lifetime (i), front and back surface recombination velocities (S), and
electronic carrier diffusion
length (L). In one aspect the present radiometric method comprises (a)
providing a sample of a
semiconductor wafer, including a scribeline between adjacent circuit devices;
(b) irradiating the
sample with an excitation source (laser); (c) generating a square-wave chirp
from a dual-channel
fast Fourier transform (FFT) analyzer to drive an acousto-optic modulator and
produce periodic
frequency sweeps (chirps) of the laser beam in the range including (but not
confined to) do to
100 kHz; (d) generating an instrumental transfer function, H(f), by fitting
the frequency-scan
data from a Si wafer with well-known electronic and thermal parameters to a
theoretical model
which uses these parameters, computing the necessary corrections to the
captured radiometric
amplitude and phase signal, and storing them in the FFT analyzer and in a
personal computer; (e)
fitting the obtained signal from arbitrary semiconductor samples to the same
theoretical model
of the photothermal radioemtric response, corrected for the instrumental
transfer function, by
using the multiparameter computational method (presented in this invention) to
obtain the
thermal and/or electronic parameters of these samples.
The present invention further provides for using the same chirp methodology as
described
above, for generating fast radiometric (or otherwise) frequency scans from
mufti-layered and
inhomogeneous materials, such as thermal barrier coatings and hardened steels,
in order to
measure the thermophysical properties (thermal diffusivity, a, and
conductivity, k) of mufti-layer
structures, and to reconstruct thermal diffusivity depth profiles, a(x), of
inherently or process-
related inhomogeneous structures.
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CA 02305477 2000-04-17
COMMON REJECTION MODE (DUAL PULSE)
The present invention also provides a general instrumental method for
detection of very
weak inhomogeneities among materials that are not possible to detect with
conventional signal
generation techniques. In one aspect the present method comprises (a)
providing a sample of the
material; (b) irradiating the sample with an optical or otherwise excitation
source of thermal
waves; (c) generating a real time periodic waveform consisting of two incident
pulses; (d)
detecting the signal (photothermal or otherwise) and feeding it to a lock-in
amplifier.
The present invention is by no means confined to thermal-wave signal
generation, but
encompasses all manner of modulated signals, such as acoustic, optical,
ultrasonic, X-rays and
any other signal generation method accessible to those skilled in the art.
MULTIPARAMETER COMPUTATIONAL RADIOMETRIC METHODOLOGY
In another aspect of the present invention a computational method for
determining a
unique set of thermal and electronic parameters of industrial semiconductor
(e.g. Si) wafers,
from frequency domain radiometric measurements, is also provided. This method
comprises the
steps of (a) providing a sample of the semiconductor; (b) irradiating the
sample with a periodic
optical (laser) or other free-carrier raising energy source generating a
blackbody radiation signal;
(c) detecting said radiometric signal and inputting said radiometric signal to
a lock-in amplifier
and storing the frequency scans in a personal computer; (d) applying the
multiparameter fitting
procedure using the electronic spread sheet coupled to a numerical function-
program.
DEPTHPROFILOMETRYAND ROUGHNESS ELIMINATIONALGORITHM
The present invention provides a method for reconstructing the thermal
diffusivity profile
of rough engineering materials by means of first eliminating roughness effects
from the
experimental data. In one aspect the method comprises of (a) providing a
sample of process-
related inhomogeneous material or mufti-layer structures; (b) irradiating the
sample with a
periodically excited source (laser); (c) detecting the photothermal
(radiometric or otherwise)
frequency sweep signal with a lock-in amplifier and storing the experimental
data in a personal
computer; (d) processing the experimental data with a heuristic approach to
roughness so as to
eliminate the effects of roughness; (e) applying to the processed data the
theoretical/computational model to reconstruct the thermal diffusivity
profile.
BRIEF DESCRIPTION OF THE DRAWINGS
The methods of the present invention will now be described by way of example
only, reference
being had to the accompanying drawings in which:
Fig. 1 illustrates an schematic diagram of one embodiment of an apparatus used
for measuring
thermal and electronic properties according to the methods of the present
inventions;
Lock-in Common Rejection Mode Figs
Fig. 2. (a) Optical excitation pulse train i(t); (b) radiometric repetitive
transient signal s(t) due to
i(t); (c)lock-in weighting function w(t).
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Fig. 3. Amplitude of the in-phase (IP) and quadrature {Q) component of the
lock-in analyzer
(LIA) radiometric response, as function of the separation between two pulses
for the zllT values
reported in the inset. The sample has been assumed homogeneous, i.e. Re~S(f)J
= - Im~S(~J and
z2/T = 25% , where S(~ is the system frequency response; T is the period; zl
is the duration of
pulse 1; and z2 is the duration of pulse 2.
Fig. 4. IP and Q components of the LIA response, as functions of the pulse
separation, 0, for
various arguments of S(~ reported in the inset. zllT and z2/T have been
assumed equal to 5 and
25 %, respectively.
Fig. 5. Dependence of the zero crossing values ~o,iP and Do,Q on the duration
of the first pulse for
the arg(S(~J values reported in the inset. z2/T has been assumed equal to 25%.
Fig. 6. Block diagram of the infrared laser photothermal radiometric system,
(a subset of the
apparatus shown in Fig. 1), used as a first embodiment of the pulse-separation
scan invention.
Fig. 7. Experimental IP- and Q-component data obtained on a Zr alloy sample
for the zllT values
reported in the inset. The modulation frequency was ~ 10 kHz and zz/T = 25%.
The solid lines
represent the theoretical fits calculated according to the theory (see section
with detailed
description of method below) assuming Re~S(fiJ = -Im~S(~J.
Fig. 8. Experimental Q-component zero crossing values obtained on the Zr alloy
sample withzl/T
= 2, 5, 7 and 10 %. The modulation frequency is 5 kHz while z2/T = 25%. The
shown
continuous line is theoretical (see section with detailed description of
method below).
Fig. 9. Q-magnitude pulse-separation scans obtained for zllT = S%, z2/T = 25%,
f = S00 Hz
and for the pump laser power values shown in the inset. Continuous lines are
theoretical fits (see
setion with detailed description of the method below) with Im~(S(fiJlRe~(S(~J
as the only
adjustable parameter.
Fig. 10. Conventional 50%-duty-cycle frequency-scan phase signals obtained for
the pump
power values reported in Fig. 9.
Fig. 11. Zoom of the data reported in Fig. 9 in the vicinity of the zero
crossing region. The solid
lines are fits calculated according to the theory described in detailed in
section below. The dotted
curves represent the theoretical data obtained for P = 150 mW, calculated for
arg~(S(~J values
reported in the inset.
Fig. 12. Microhardness depth profiles for two shot peened Zr-2.SNb alloy
samples. The inset
shows Almen intensities. The data corresponding to the N7 sample have been
smoothed.
Fig. 13(a). Experimental Q-component data obtained from the N7 Zr-2. SNb alloy
sample in Fig.
12 for the zllT values reported in the inset, and z2/T = 25%: (a} f = 500 Hz;
(b) f = 5 kHz. (b)
The near-zero crossing region for the Q-components of the two shot peened Zr-
2.SNb samples in
Fig. 12 and a Zr sample used as a reference. z~lT = 5%, z2/T = 25% and f = 500
Hz.
CA 02305477 2000-04-17
Fig. 14. The near-zero crossing region for the Q-components of the two shot
peened Zr-
2. SNb samples and the Zr reference. zllT = 5%, zzlT = 25% and f = 500 Hz.
Fig. 15. Experimental Q-component zero-crossing data obtained on the (a) CS ,
and (b) N7 shot
peened Zr-2.SNb alloy samples for the modulation frequencies reported in the
inset. Thezl/Tand
z2/T values are the same as in Fig. 13.
Fig. 16. Measured photothermal radiometric amplitude ratio, (a), and phase
difference, (b), for
the two shot peened Zr-2.SNb alloy samples. To aid the eye, the CS ratio has
been shifted
upward by +0.5. The relative amplitudes are consistent with the slopes of the
N7 and CS curves
in Fig. 13, which reveal a low photothermal amplitude response for the CS
sample.
Frequency-Sweep ("Chirp") Figures
Fig.l7. Schematic representation of photothermal radiometric apparatus
embodiment used for
simultaneous measurements of transient and frequency swept scans. M: mirror;
AOM:
Acousto-optic modulator; MCT: Mercury Cadmium Telluride detector; L: lens;
LIA: lock-in
amplifier; MC: mechanical chopper; FFT: fast Fourier transform. X(t) is the
chirp periodic
waveform launched by the FFT analyzer. Y(t) is the sample response to X(t).
H(fj is the output
spectral transfer function.
Fig. 18. Comparison of two radiometric signal transients of an unirradiated
(a) and irradiated (b)
spot of an n-type unoxidized Si wafer, and an unirradiated spot of a p-type Si
wafer with a 5000
~, oxide film (c). The parameters used for the fittings (solid lines) are
shown in this figure. The
horizontal bar on curve (a) indicates the duration of each set of frequency
swept measurements
(chirps). Arrows indicate the onset of each set of chirps
Fig. 19. Experimental PTR amplitude (a) and phase (b) responses of an n-Si
wafer. Curve (1)
corresponds to lock-in amplifier data obtained at steady state. Curve (2) is
the normalized
frequency-sweep (100 Hz- 100 kHz) transfer function, H(f), corresponding to
the first chirp
measurement (300 s) on the same wafer. Solid lines show the simultaneous best
fit using a finite-
thickness model 8. Parameters derived from the best fits are i) at steady
state: T = 110 p,s, DP =
cm2/s, S1 = 320 cm/s; ii) after 300 s of exposure: i = 110 ps, Dp = 10 cm2/s,
S1 = 830 cm/s
and L = 570 p,m. Inset: calculated surface recombination velocities of the n-
Si wafer at different
times for transients (a) and (b) of Fig. 2. The extremes of the solid line
denote the radiation turn-
off and turn-on times.
Computational Method Figures
Fig. 20 PTR signal amplitude (a) and phase (b) for lifetime simulations in
some Si samples.
Values uses for these simulations were: D~ Scmz/s, S1=130 cm/s, S2=1x104 cm/s,
a =0.116
cm2/s, CP 3x10-2° a.u, Ct=1 a.u.
Fig. 21 Linear relation between lifetime measurements and PTR signal amplitude
for a high-
resistivity p-Si wafer, evaluated at four different positions along the radial
direction.
Fig. 22 Amplitude (a) and phase (b) PTR images of a long-lifetime Si wafer,
probed from the
front (intact) surface and scanned over the coordinates of the back surface
mechanical defect site.
CA 02305477 2000-04-17
Fig. 23 Schematic representation of the horizontal furnace used for dry
isochronal oxidation
process.
Fig. 24 PTR signal amplitude (a) and phase (b) for front surface recombination
velocity
simulations in Si samples with long lifetime (i=1500 ps). Values for the
remaining simulation
parameters were: Dri Scm2/s, S2=1x104 cm/s, a =0.116 cm2/s, Cp 3x10-2°
a.u and Ct=1 a.u.
Fig. 25 PTR signal amplitude (a) and phase (b) for back surface recombination
velocity in Si
samples with long lifetime (i=1500 p.s). Values for the remaining simulations
parameter were:
Dri Scm2/s, S1=130 cm/s, a =0.116 cm2/s, Cp 3x10-2° a.u. and Ct=1
a.u.
Fig. 26 PTR signal amplitude (a) and phase for a Si sample with long-lifetime
wafer, before
(front 1 and back 1) and after (front 2 and back 2) back-surface damage,
respectively.
Fig. 27 Histogram for best-fit results for wafer 1 (door), 2 (source), 25
(door) and 26 (source)
processed in tube 15 (a) front surface recombination velocity (b) lifetime.
Fig. 28 p-PCD iron concentration (a, c) and lifetime measurement (b, d) for p-
silicon wafers:
sample 1 (BK101) and 2 (BK1003).
Fig. 29 shows the PTR signal for the six radial positions in sample 2
(BK1003).
Fig. 30 typical configuration of thermal annealing of industrial p-Si wafers
under an applied
electric field
Fig. 31 PTR amplitude (a) and phase (b) frequency scans for p-Si wafer.
Annealed under no
electric field condition. The solid lines represent the best fits using the
multiparameter
computational methodology presented in this invention.
Fig. 32 Experimental PTR amplitude (a) and phase (b) frequency responses
obtained from a
nonimplanted reference wafer and Si wafers implanted with P+ ions of 50 keV
energy to various
doses (ions/cm2): (1) 5x101°; (2) 1x1011; (3) 5x1011 ; (4) 1x1012; (5)
5x1012 ; (6) 1x1013 ; (7)
5x1013 ; (8) 1x1014 ; (9) 5x1014 ; (10) 1x1015 ; (11) 5x1015 ; (12) 1x1016.
Fig. 33 (a) Values of the minority carrier lifetime evaluate from the PTR
amplitude and phase
frequency responses as a function of implantation dose for implantation
energies of 50, 100, and
150 keV. (b) Experimental dependencies of PTR amplitude on implantation dose
for 50, 100 and
1 SO keV implantation energies taken at 10 kHz modulation frequency.
Fig. 34 a) Microscope photograph showing two different sizes of scribelines in
a patterned
wafer. b) Schematic representation of the cross-sectional geometry of the
wafers.
Fig. 35 Microscope photograph of a characteristic region located 2 cm away
from the wafer flat,
showing the topology of PTR line scans.
Fig. 36 PTR signal amplitude (a) and phase for six positions (three Si02 and
three poly-Si pads)
shown in Fig.35.
7
CA 02305477 2000-04-17
Fig. 37 PTR signal amplitude (a) and phase (b) thermolectronic images for
region A shown in
Fig. 34(a).
Depth profilometry/Roughness elimination algorithm figures
Fig. 38 Depth profilometric region under investigation.
Fig. 39 Frequency-domain photothermal radiometric instrumentation.
Fig. 40 Experimental data and forward theoretical fit of untreated AISI 8620
steels samples.
Fig. 41 Thermal diffusivity reconstruction of untreated AISI 8620 steels
samples.
Fig. 42 Simulation of roughness elimination method with 1.6pm roughness
thickness.
Fig. 43 Simulation of roughness elimination method with 7~m roughness
thickness.
Fig. 44 Simulation of roughness elimination method with 131tm roughness
thickness.
Fig. 45 Experimental elimination of roughness with 1.6p,m roughness thickness.
Fig. 46 Experimental elimination of roughness with 5.6pm roughness thickness.
Fig. 47 Experimental data of carburized samples at depths 0.02", 0.04" and
0.06" with two
levels of roughness.
Fig. 48 Elimination of roughness of carburized samples of figure 45.
Fig. 49 Thermal diffusivity reconstruction of carburized samples of figure 46.
Hardness profile
for each respective depth also shown.
Fig. 50 PTR amplitude (a) and phase (b) frequency response from a stainless
steel thermal
spray coating on a carbon steel substrate. The solid line is the Gaussian fit
to the high frequency
experimental data.
Fig. 51 PTR amplitude (a) and phase (b) frequency response from the stainless
steel thermal
spray coating on a carbon steel substrate of Fig. 48. The high frequency data
has been corrected
by the roughness elimination methodology.
Micro-weld application figuress
Fig. 52 CCD camera image of pins 1-to-4 of sample 8f2 (90 gf)examined using
PTR
Fig. 53 PTR phase (a) and amplitude (b) image at 10 kHz for pinl of sample 8f2
(bonding force was 90 gf)
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Fig. 54 PTR amplitude (a) and phase (b) frequency scans for pin 1-to-3 of
sample 8f1
(bonding force was 90 gf)
PRIOR ART
i) Conventional Photothermal Frequency Scan Method
The differences between the conventional frequency scan method and the common-
rejection mode method, and the frequency-sweep ("chirp") method are best
understood by
comparison of the various methods. The conventional frequency scan method will
be first
described followed by a description of the sweep frequency (Chirp), and the
common rejection
mode.
In a conventional photothermal radiometric embodiment, one dimensional
analysis of the
diffusion of the thermal wave generated inside a solid strip of thickness L by
a laser beam
modulated at angular frequency w, yields the following expression for the a.c.
temperature at the
irradiated surface:
T(~) = 1 orls . l + Rgs exp(-2a~SL) 1)
(
kscrs (1 + bgs ) 1- Rgs exp(-2~SL)
[see G. Busse and H. G. Walther, in Progress in Photoacoustic and Photothermal
Science and
Technology, edited by A. Mandelis, Vol. I, Chapter 5, p. 205, (Elsevier, New
York, 19991)],
where ks is the thermal conductivity of the sample; Io is the laser
irradiance; rls is the optical-to-
thermal energy conversion eWciency at the sample surface; and bgs is the
thermal coupling
coeff cient to the surrounding gas (air) given by:
b _ kg ~ ag (2)
sf _ ks ~ as
Here k~ is the thermal conductivity and a~ is the thermal diffusivity of
medium j with the
subscripts s and g referring to the sample and the gas, respectively. The
quantity Rgs given by:
R _ 1 _ bss (3)
gs 1 + bgs
is the thermal-wave reflection coefficient at the solid-gas interface and as
is a complex diffusion
coefficient given by:
a-s = (1 + 1) 2as (4)
9
CA 02305477 2000-04-17
It is assumed that the solid and air are in perfect thermal contact.
Expressions for the measured
quantities, phase and amplitude, can be derived from the real and imaginary
parts of Equation 1.
The measurements are made with respect to a thermally thick (L»p,) reference
sample where
the signal is given by:
T~ef (~) - kr~l(lrl, bgr )
The signal from the semi-infinite reference sample is used to compensate for
the instrumental
transfer function. For radiometric detection both T(w) and T~zr(c~)
expressions must be multiplied
by terms including surface emissivity, detector parameters, ambient
temperature, etc. This
constant multiplicative term, except for the sample dependent terms, is
cancelled out from the
normalized amplitude signal, Equation 1, divided by Equation S; and from the
normalized phase
signal in Equation (1), subtracted from the phase in Eq. (5).
By fitting the normalized experimental data (phase and amplitude) frequency
dependence to the
corresponding expressions derived from Equation l, the parameters Rgs and
L/(a)1'2 can be
calculated. Since the coupling medium is usually air [kg 0.026 Wrri 1K-1, ag
3.1x10-5 m2s2; A.
Rocencwaig, Photoacoustics and Photoacoustic Spectroscopy, Chem. Anal. Vol. 57
(J. Wiley &
Sons, New York, 1980), p. 96], the value of bgs«1. Therefore, R~ is almost
unity and its
sensitivity to ks is extremely small. That simplification renders L/(a)"2 the
only fitting parameter
for normalized phase data. In addition to L/(a)ln the normalized amplitude
data contain a
multiplicative factor due to any differences in the bulk thermal properties
and the surface finish
(e.g. roughness) and its possible differences between the sample and the
reference. This factor
may be cancelled out by setting the amplitude ratio to be unity at the high
frequency (thermally
thick) end, where the phase difference is expected to be zero. Setting the
amplitude ratio equal to
unity is possible because we are only interested in the shape of the
normalized curve, not in the
absolute magnitude. However there is a very real possibility that the surface
"finishes" will
overlap the lower frequency regions in either or both sampleand reference, in
which case the
"setting-to-unity" method will not work. For such cases a roughness
elimination algorithm from
the high-f "humps" developed by L. Nicolaides and A. Mandelis can be used (see
section below).
Since there exist extrema in the frequency curve of both amplitude and phase
(thermal-wave
interference), which are very sensitive to L/(a)"2, it is not necessary to fit
an entire frequency
range. These extrema could be used as a fast on-line measurement of small
variations in L or as
in an industrial environment.
ii) Conventional Photothermal Electronic Lifetime measurement methods.
For sometime now several laser-based photothermal techniques have been
developed to
monitor photoexited carrier kinetics and transport properties in
semiconductors, the advantage
over other, mainly electrical, methods being that electronic effects can thus
be monitored in a
non-contacting and non-destructive manner, therefore eliminating the need for
electrode
attachment [A. Rocencwaig, in Photoacorrstic and Thermal-Wave Phenomena in
Semiconductors, edited by A. Mandelis (Elsevier, New York, 1987); M. Wagner,
N. Winkler and
H. D. Geiler, Appl. Surf. Sci. 50, 373 (1991); A. Skumanich, D. Fournier, A.
C. Boccara and N.
to
CA 02305477 2000-04-17
M. Amer, Appl. Phys. Lett. 47, 402 (1985); A. Mandelis, A. A. Ward and K. T.
Lee, J. Appl.
Phys. 66, 5584 (1989)]. A distinct disadvantage of those photothermal
techniques, however, is
the fact that with either frequency-scanned detection [A. Rocencwaig, in
Photoacoustic and
Thermal-Wave Phenomena in Semiconductors, edited by A. Mandelis (Elsevier, New
York,
1987); ); A. Mandelis, A. A. Ward and K. T. Lee, J. Appl. Phys. 66, 5584
(1989)], both free
carrier (plasma)-wave and thermal wave responses from semiconductors are
strongly coupled
together [A. Mandelis and R. E. Wagner, Jpn. J. Appl. Phys. 35, 1786 (1996)].
As a result the
interpretation of the convoluted experimental data is usually complicated. The
task of
deconvoluting the two types of responses becomes cumbersome, and this renders
much of the
analysis qualitative. As an example, the photomodulated thermoreflectance
(PMOR) technique
[A. Rocencwaig, in Photoacoustic and Thermal-Wave Phenomena in Semiconductors,
edited by
A. Mandelis (Elsevier, New York, 1987), Chap. 5] produces signals ~R which
depend on both
the a.c. temperature of the laser-excited semiconductor surface OT(c~), and on
the
photogenerated electron-hole plasma wave ON(w),
( ) CaT ~~T (~) +CaN~~(~) (6)
Very tightly focused (~1 ~m2) pump beams can, in principle, lead to the
domination of PMOR
by the plasma response, yet this constraint almost invariably gives rise to
unwanted non-linear
phenomena, such as Auger recombination, which further complicate the
quantitative aspects of
the technique [R. E. Wagner and A. Mandelis, Semicond. Sci. Technol. 11, 289
(1996); and 300
(1996)]. Therefore, very tight laser focusing can be detrimental to the study
of electronic defects,
since the exceedingly high fluence may greatly perturb the experimental
behavior of an
electronic material.
Regarding laser infrared photothermal radiometry (PTR) of semiconductors, the
pulsed
(including spectral cross-correlation and impulse response) time-domain mode
may exhibit
severe overlap of free-carrier density and thermal effects [K. Cho and C.
Davis, IEEE J.
Quantum Electron. QE-25, 1112 ( 1989)] and non-optimized signal-to-noise
ratio, SNR [A.
Mandelis, Rev. Sci. Instrum. 65, 3309 (1994)]. Unlike the PMOR technique, it
has been shown
that the electronic (plasma-wave) component of the infrared emissivity PTR
signal fully
dominates the thermal-wave component in typical industrial Si wafers [A.
Mandelis, R. Bleiss
and F. Shimura, J. Appl. Phys. 74, 3431 (1993); A. Salnick, A. Mandelis, H.
Ruda and C. Jean, J.
Appl. Phys. 82, 1853 (1997)], thus making PTR the preferred method for
industrial
semiconductor metrologic technology development. In terms of physical
interpretation of
signals, the time-domain technique is considered preferable to the frequency-
domain counterpart
[S. J. Sheard M. G. Somekh and T. Hitler, Mater. Sci. Eng. B5, 101, (1990); Z.
H. Chen, R.
Bleiss, A. Mandelis and F. Shimura, J. Appl. Phys. 73, 5043 (1993)] due to the
inherent ability of
transient-response techniques to be interpretable in terms of simple system
time-delay constants.
The same information can be obtained, in principle, from the frequency-scanned
data; however,
this method requires the de-multiplexing of data over broad frequency ranges,
typical of the
existing relationship between Fourier tra-nsform pairs (i.e. time and
frequency domains).
Nevertheless, the superior frequency-domain SNR, which is achievable via lock-
in filtering and
demodulation, coupled with further improvements regarding either the
substantial acceleration of
the measurement process, or the SNR of the signal generation and processing
techniques
11
CA 02305477 2000-04-17
introduced in the present invention, renders the frequency-domain (FD) PTR
mode the
measurement method of choice for the development of novel industrial-level
semiconductor
metrologic technologies.
iii) Frequency-Swept Time-Delay-Domain (Chirp) Modulation in PTR Signal
Generation and Processing
In the frequency swept optical excitation mode, the temporal equivalent of a
single
ultrashort excitation pulse is generated over the duration of e.g. a 100-kHz
chirp and the sample
impulse-response (or cross-correlation) information contained in the output
signal spectral
response is recovered using (but not only confined to) the photothermal
correlation and spectral
analysis techniques described by A. Mandelis, IEEE Trans. Ultrasonics,
Ferroelectrics, Freq.
Control, UFFC-33, 590 (1986). With regard to photothermal (including PTR)
detection, the
main advantages of this technique for industrial instrumentation and
measurement system
development over other techniques such as the pulsed laser method, the
wideband random noise
correlation method and the point-by-point lock-in FD photothermal method are:
a) the much
accelerated speed of signal data acquisition to less than one minute over the
entire frequency
span dc-100 kHz for 1024 co-added and averaged frequency sweeps; and b) the
flexible signal
acquisition nature, capable of yielding the impulse response and/or the
transfer function of a
sample from the same set of data via instrumental, real-time, fast-Fourier
transformations, thus
potentially facilitating interpretation and parameter extraction in terms of
simple Green function
formalisms. The major disadvantage of the chirp method is the less-than-
optimal SNR owing to
the broadband nature of the data acquisition and noise content, compared to
the conventional
point-by-point lock-in filtering and demodulation technique.
iv) PTR Depth Profilometry for Rough Samples
Depth profilometry is an important inverse problem where the thermal
diffusivity profile
is inverted from experimental surface information. Thermal diffusivity is a
property that depends
on the microstructural properties of a material and thus can be used to
identify changes that take
place in a material as a result of surface modification processes, such as
laser processing, case
hardening and coating deposition. The benefits of this methodology for
processes such in the
heat treating and thermal spray industries are immense since it implies the
development of an on-
line non-destructive method for rapidly determining the metallurgical
properties of case treated
material and thermal spray coatings.
In inhomogeneous materials, the amplitude and phase signal channels carry
information
about any heat transport disruption or change below the surface, which must be
interpreted with
appropriate models, in order to yield reliable reconstructions of the
spatially variant thermal
diffusivity of the sample. One of the first theories of this kind of
inversions was described by
Vidberg et. al. [H.J. Vidberg, J. Jarrinen and D.O. Riska, Can. J. Phys. 64,
1178 (1986)]. This
model pertains to the thermal-wave surface signal obtained by measuring the
radial variation of
the surface temperature of a continuously inhomogeneous solid about a heated
point at a single
modulation frequency. Both thermal conductivity and heat capacity profiles
were reconstructed
using Pade approximants for the inversion of spatial Laplace transforms. There
are a number of
constraints which limit the applicability of this model. The most significant
ones are: (I) it is
12
CA 02305477 2000-04-17
only valid for a nonconventional experimental geometry; (2) the reconstructed
profiles are not
always numerically reliable; (3) the accuracy is limited to a depth
reconstruction on the order of
one thermal diffusion length; and (4) the reconstruction algorithm is
relatively complex and is
sensitive to the presence of small amounts of error. In an earlier publication
Jaarinen and
Luukkala [J. Jaarinen and M. Luukkala, J. Phys. (Paris) 44, C6-503 (1983)]
discussed a
numerical analysis of the same experimental geometry based on the solution of
the thermal-wave
equation at a single modulation frequency. The analysis uses a two-dimensional
finite difference
grid.
More recently, another major attempt [A. Mandelis, S.B. Peralta and J. Thoen,
J. Appl.
Phys. 70 , 1761 (1991)] was made to approach the thermal-wave inverse problem
more
rigorously and for more general geometries than the foregoing papers. In this
approach the well-
known Hamilton-Jacobi formalism from Classical Mechanics was introduced into
the thermal-
wave problem by treating the a.c. temperature field as a Thermal Harmonic
Oscillator (THO) [A.
Mandelis, J. Math. Phys. 26, 2676 (1985)] and inverting the amplitude and
phase of the
experimental data through matching to explicit theoretical expressions for a
semi-infinite solid
(or liquid). The first experimental inversions were obtained from the liquid
crystal
octylcyanobiphenyl (8CB) [A. Mandelis, E. Schoubs, S.B. Peralta and J. Thoen,
J. Appl. Phys.
70 , 1771 (1991)] using this method. Further inversions with semi-infinite
laser-processed solids
were reported later [T-C. Ma, M. Munidasa and A. Mandelis, J. Appl. Phys. 71,
6029 (1992), M.
Munidasa, T.C. Ma, A. Mandelis, S.K. Brown and L. Mannik, Mater. Sci. Eng.
A159, 111
(1992)]. An inversion procedure for a finite thickness problem has also been
reported based on
the same THO approach [A. Mandelis, J. Math. Phys. 26, 2676 (1985)]. More
recently, a newer
model [C. Glorieux, J. Fivez and J. Thoen, J. Appl. Phys. 73, 684 (1993))
motivated by the
approach described by Mandelis and co-workers [A. Mandelis, S.B. Peralta and
J. Thoen, J.
Appl. Phys. 70 , 1761 (1991); A. Mandelis, J. Math. Phys. 26, 2676 (1985); A.
Mandelis, E.
Schoubs, S.B. Peralta and J. Thoen, J. Appl. Phys. 70 , 1771 (1991); T-C. Ma,
M. Munidasa and
A. Mandelis, J. Appl. Phys. 71, 6029 (1992); M. Munidasa, T.C. Ma, A.
Mandelis, S.K. Brown
and L. Mannik, Mater. Sci. Eng. A159, 111 (1992); F. Funak, A. Mandelis and M.
Munidasa, J.
Phys.(Paris) IV, Colloque C7, 95 (1994)] was proposed, that assumed locally
constant or
linearly-dependent thermal conductivity on depth. In that work the solid was
divided up into a
virtual incremental discrete-layer system and in each layer forward and
reverse thermal-wave
equations were set up for constant conductivity and solved using computer-
based matrix
routines. The resulting equations were inverted for the depth-dependent
increments of the value
of the thermal conductivity using a commercially available nonlinear least-
squares fit routine. It
is well established that only true material discontinuities such as surfaces
and not virtual
incremental slices can generate reflected thermal waves. This raises questions
about the validity
and/or uniqueness of the inversions. Even if it is accurate for semi-infinite
solids, the theory
presents problems with the treatment of finite-thickness materials, as it
ignores the multiple inter-
reflections of the thermal wave between the two boundaries (surfaces) of the
material. Fivez and
Thoen reported yet another version [J. Fivez and J. Thoen, J. Appl. Phys. 75,
7696 {1994)] of the
foregoing inversion problem with a linear dependence of the local
(incremental) thermal
conductivity with depth. Explicit expressions were derived and matched with
experimental data
and the results of the inversions were in good agreement with those obtained
by the approach by
Ma et al. [T-C. Ma, M. Munidasa and A. Mandelis, J. Appl. Phys. 71, 6029
(1992)]. The major
shortcoming of this new approach is in its inability to treat semi-infinite
solids, as the explicit
formulas depend on the boundedness of the derived Bessel and Neumann
functions. Instead, it
13
CA 02305477 2000-04-17
requires flat profiles in the bulk of the material under investigation. This
is so because many of
the combinations of these functions utilized in this approach become infinite
in value as the
depth increases without bound. A recent theoretical approach by Lan et. al.
[T.T.N. Lan, U.
Seidel and H.G. Walther, J. Appl. Phys. 77, 4739 (1995)] combines the
approaches of both prior
papers[C. Glorieux, J. Fivez and J. Thoen, J. Appl. Phys. 73, 684 (1993), J.
Fivez and J. Thoen,
J. Appl. Phys. 75, 7696 (1994)]. Therefore, it has improved strengths, yet, it
is subject to some
combinations of their shortcomings: a flat profile of the thermal conductivity
at large distances
[T.T.N. Lan, U. Seidel, H.G. Walther, G. Goch and B. Schmitz J. Appl. Phys.
78, 4108 (1995)]
(i.e. at "infinity"), to induce boundedness, along with the lack of a
theoretical basis to treat
multiple thermal-wave reflections from the opposite surfaces of finitely-thick
samples. In a more
recent theoretical paper [J. Fivez and J. Thoen, J. Appl. Phys. 79, 2225
(1996)] Fivez and Thoen
presented a new analytical approach to the inverse problem which is valid for
semi-infinite solids
at sufficiently high frequencies, but shows significant deviations of
reconstructed thermophysical
profiles from the expected values at low frequencies (equivalent to large
depths in a sample).
Kolarov and Velinov [R. Kolarov and T. Velinov, J. Appl. Phys. 83 (4) (1998)]
developed a
method based on the Riccati first-order differential equation. The numerical
method presented
solved the general Riccati equation in real time. Recently, Walther and
Akeshin [H. G. Walther
and V. Aleshin, J. Appl. Phys. 8G (11) (1999)] developed a method which
combines laterally
scanned and frequency resolved measurements for the inspection of
inhomogeneous samples. A
lateral scan increases the ill-poseness of the problem since more degrees of
freedom are
introduced.
Mandelis et al. [A. Mandelis, F. Funak and M. Munidasa, J. Appl. Phys. 80
(10), 5570
(1996)] further formulated a complete generalized expression for the thermal-
wave field in an
inhomogeneous bounded solid. The method improved on the previously derived
formulas [A.
Mandelis, S.B. Peralta and J. Thoen, J. Appl. Phys. 70 , 1761 (1991); T-C. Ma,
M. Munidasa and
A. Mandelis, J. Appl. Phys. 71, 6029 (1992); F. Funak, A. Mandelis and M.
Munidasa, J.
Phys.(Paris) IV, Colloque C7, 95 (1994)] based on the THO approach by ensuring
proper
convergence to limiting cases. A successful application of the method was
further presented in
[M. Munidasa, F. Funak and A. Mandelis, J. Appl. Phys. 83 (S) 3495(1998)]. The
results were
promising but the material roughness response on the experimental data was
neglected.
In this invention a methodology based on the THO approach [A. Mandelis, J.
Math. Phys.
26, 2676 (1985)], for the thermal-wave field in a semi-infinite inhomogeneous
solid with a rough
layer is disclosed.
APPARATUS FOR NON-DESTRUCTIVELY MEASURING ELECTRONIC
PARAMETERS OF SEMICONDUCTORS AND THERMAL PARAMETERS AND
DEPTH PROFILES OF ENGINEERING MATERIALS
The novel complete instrumentation system (apparatus) using laser PTR as the
preferred
(but not sole) embodiment of the present invention for measuring the thermal
and electronic
transport parameters by means of the frequency scan, the frequency sweep
("chirp"), the
common-rejection-mode signal generation and processing method, as well as by
scanning wafer
imaging at a fixed frequency, will now be described.
A schematic diagram of the apparatus for measuring thermal and electronic
transport
properties of substrate or processed semiconductor wafers or chips is shown in
Fig. 1. A heating
laser 1 with modulated power up to a few watts and super-bandgap wavelength (<
1000 nm for
14
CA 02305477 2000-04-17
Si or <600 nm for very-near-surface probing) is directed onto the surface of a
sample 4 using
focusing optics 3. The radiation emitted by the surface of the sample 4 is
collected and focused
onto a detector 5 using a pair of reflecting objectives 3 (two off axis
parabolloidal mirrors or one
elliptical minor can also be used).
Detector 5 is a liquid-N2 cooled HgCdTe (EG & G Judson model J15D16-M204) with
an active
area of 1 mm2 or less and a spectrally sensitive range of 2-10 Vim. Other non-
cryogenic IR
detectors such as pyroelectric sensors or Golay cells can be substituted for
the LN2 detector, as
required. An anti-reflection (AR)-coated Ge window with a transmission
bandwidth of 2-13 ~tm
is mounted in front of detector 5 to block any visible radiation from the pump
laser 5. The pump
spot diameter on sample 4 is typically ca. 1-to-5 microns. The photothermal
signal, which is
proportional to the change of the IR radiation emitted from an area viewed by
detector 5, is
amplified by a preamplifier 6 (EG & G Judson model PA-101) before being sent
to a digital
lock-in amplifier 13 (e.g. Stanford Research Systems, Model SR 850). Lock-in
amplifier 13 is
interfaced with a computer 11 so that the frequency scan and data acquisition
and storage are
automated. The laser can also be modulated by the FFT dynamic signal analyzer
14 or the dual
pulse generator 12. The sample is mounted on an automated sample holder 8 with
X-Y scaning
capability for sample mapping applications through fast point-by-point image
construction.
It will be appreciated by those skilled in the art that numerous other
configurations for
repetitively heating samples and measuring the resulting photothermal
radiometric signal may be
used. For example IR detectors cooled by other means than liquid nitrogen or
modulated infrared
sensor arrays (CCD) for imaging purposes. The above example is meant to be non
limiting and
illustrative only.
DETAILED DESCRIPTION OF THE METHODS OF THE PRESENT INVENTION
i) Lock-in Common-Mode Rejection Method
a) Description of the method
Thermophysical properties are, in general, an indicator of the degree of
homogeneity of a
given sample because they are strongly affected by variations occurring in the
sample
microstructure. An introduction to thermal-wave non-destructive detection can
be found under
"Background of the Invention". Briefly, the common working principle of
conventional
photothermal techniques is based on the study of the periodic temperature
distribution, i.e. the
thermal wave, produced in a given sample as a result of heating due to an
intensity modulated
pump laser source impinging on the surface. Thermal waves inside a homogeneous
sample
diffuse over a characteristic distance, which is given by the thermal
diffusion length
p.(f) _ (a/~f)'~2, where a is the thermal diffusivity and f the modulation
frequency. By changing
the modulation frequency, the thermal wave propagates over different distances
and probes the
presence of thermal inhomogeneities located at various depths beneath the
surface. In fact,
thermal features inside the sample alter the heat transfer rate, thus
affecting the resulting surface
temperature distribution which is detected by various photothermal techniques.
Finally, by
analyzing the dependence of the photothermal signal from non-electronic
materials on the
modulation frequency, it is possible to derive some material parameters
(thermal diffusivity,
conductivity) and/or obtain information on inhomogeneities, such as position,
size, depth profile,
CA 02305477 2000-04-17
etc. This is the basic principle of all thermal wave inspection methods.
Conventional frequency
domain photothermal methods are basically single-ended techniques using an
intensity-
modulated heating beam intensity (either a 50% duty-cycle square wave, or a
sinusoidal wave),
and a lock-in amplifier (LTA) for signal processing. The limitations of single-
ended detection are
that, if the signal contributions from sample inhomogeneities are much smaller
than that from the
homogeneous bulk of the material (background signal), then they cannot be
easily detected. In a
single-ended technique the sensitivity of the experiment is determined by the
magnitude of the
background signal. Without further conditioning, the signal level is simply
too high to probe
variations of amplitude much smaller than this background. For the purposes of
this invention,
these very small variations will be called "contrast signals".
As discussed under "Prior Art", in order to obtain quantitative information
about the
sample properties, the photothermal signal must be normalized, i.e. compared
to that obtained
from a homogenous reference sample in order to account for the instrumental
transfer function.
Properly normalized signal amplitude ratios and phase differences must be
collected as a
function of the modulation frequency. This procedure introduces several
problems especially
when one intends to probe slightly inhomogeneous samples with theoretical
contrast signals
approaching the noise level of the experiment. In fact, the effect of
normalization is, in general,
to add some more noise to the measurement, thus resulting in poor SNR, which
usually masks
contrast signals. A strong noise reduction is required for these kinds of
applications and
conventional photothermal techniques do not compensate against slowly varying
drift
phenomena, which can occur during a measurement, because of their single-ended
nature [C.-H.
Wang and A. Mandelis, Rev. Sci. Instrum., (1999)]. All this despite the
advantage of the narrow-
bandwidth filtering action of the demodulating lock-in amplifier, since the
noise frequency
components within the filter bandwidth are not rejected and are still present
and become
enhanced during the normalization procedure.
The new lock-in common-mode-rejection demodulation scheme, introduced in this
invention, seems to be very promising for high-resolution thermal-wave non-
destructive material
evaluation (NDE) applications. If the sample is irradiated with a periodic
optical waveform
consisting of two pulses, then the LIA output is basically given by the
difference of the physical
response waveforms produced by each of the two pulses. This fact is of
fundamental importance
toward the improvement of low-dynamic range techniques, such as thermal-wave
NDE, in their
ability to detect relatively small signal variations from slightly different
materials. In practice,
the differential action has the effect of suppressing the signal baseline,
which leads to an
enhanced detectivity when compared to conventional single-ended techniques.
Thus, the
instrumental sensitivity is not compromised by the high-level signal baseline
and can easily
match the level of small signal variations introduced by slightly different
materials or by very
weak inhomogeinities in a given material. The principle of the invention can
be broadly applied
to any technique utilizing a lock-in analyzer demodulation scheme of periodic
signal waveforms.
In order to achieve a differential input with a single excitation source and
demodulation
instrument, a new periodic optical excitation waveform, Fig. 2(a), has been
designed, which
exploits advantages due to the built-in weighing-function waveform of the LIA
[G. L. Miller, J.
V. Ramirez and H. A. Robinson, J. Appl. Phys. 46, 2638 (1975); A. Mandelis,
Rev. Sci. Instrum.
65, 3309 (1994)]. As shown in Fig. 2(a), in a given period T the sample is
excited by two square-
wave pulses with center-to-center separation by a time interval 0. As a
consequence of the
asymmetric periodic excitation, the transient (photothermal) response s(t) of
the sample, Fig.
2(b), rises and decays twice during a period, also with a certain degree of
asymmetry. The in-
16
CA 02305477 2000-04-17
phase or quadrature component of the LIA response to the incoming signal s(t)
with a long
integration time constant may be written as [A. Mandelis, Rev. Sci. Instrum.
65, 3309 (1994)]
T T/2 T
y(t)_ Js(t)w(t)dt= ~(+)s(t)dt+ ~(-)s(t)dt (7)
0 o r/z
where w(t) is the square weighing function shown in Fig. 2(c) and assumed to
have a zero-delay
rising edge. Owing to the opposite signs of w(t) across the mid-period point
T/2 for zero phase
delay at t=0, the LIA acts like a real-time differential comparator whose
output level is a measure
of the degree of asymmetry of the two s(t) lineshapes in the two half periods.
Therefore, with this
waveform design, a differential input configuration is achieved, which
suppresses the signal
baseline and takes full advantage of the highly efficient noise suppression by
the LIA due to its
extremely narrow filtering.
The difference between analog and digital LIAs, which use square-wave and
synthesized
sine-wave reference signals, respectively, has been extensively treated
elsewhere [A. Mandelis,
Rev. Sci. Instrum. 65, 3309 (1994)]. The output is quantitatively the same for
the two types of
LIA, provided that a tracking filter is inserted into the input of the analog
version, in order to
reject the odd harmonics of the input signal.
b) Theory of output signal
In this section a theoretical description of the signal generation due to the
new waveform
of Fig. 2 will be given, providing analytical expressions for both the in-
phase (IP) and quadrature
(Q) components of the lock-in response. In particular, we are interested in
pointing out how the
signal output is influenced by the parameters of the composite optical
waveform (il, z2 and 0).
We will also demonstrate signal sensitivity to the response of the system
under investigation. In
the analysis, we assume that the system is excited with a repetitive waveform
consisting of two
square pulses within one period T having the same amplitude Io, durations il
and i2 , and
separation ~. For periodic waveforms it is convenient to consider the Fourier
series
representation of i(t)
+~
i(t) _ ~ck exp(~ Wit)
k=-ao
in complex form, or,
i(t) = 2 +~Cak cos(2T t)+bk sin(2'd't)~
k=I
with
17
CA 02305477 2000-04-17
_T
2
~x = ax 2~bx = T 1 (t) exp(- ~ ~ )dt = T j(T )
_T_
2
where I(f) is the Fourier transform of i(t) calculated over one period. By
applying the time shift
property to the two-square-pulse Fourier transform, it is easy to show that
~k~, 7ikT 2
sin( ) sin( )
ck = ~I° z~ T exP[-J2T~ k (T - ~)~} + {I° i2 T exp[-j2~c k (T +
~)~}
T ~k~', T 2 2 T ~ki2 T 2 2
T T
and, after some manipulation,
(11)
(-1)k I° { ( ~k0 )[ . ( ~kil ) . ( nki2 )~ j ( ~k~)[sin( ~kil ) _ sin(
~kz2 )~ ~
ck = ~k cos T sm T + sm T + sin T T T
(12)
The LIA monitors only the fundamental component of the harmonic signal, so we
can limit our
attention to the first term of the Fourier series, the coefficients of which
are given by
I ~0 ni ~i ~0 ~i 7zi
c, _ - ~ { cos( T )[sin( T' ) + sin( T2 )~ + J sm( T )[sin( T' ) - sin( T2 )]
} ( 13)
a, = 2 Re(c, ) = 2~° cos( T )[sin( T' ) + sin( T2 )] (14)
bl = -2 Im(ci ) = 2~° sin( T )[sin( T1 ) - sin( T2 )] (I S)
In order to calculate the LIA response to the excitation pulse train, we
introduce the system
frequency response S(~ = Re~S(~J + jlm~S(~J which can be unambiguously defined
for each
sample as the Fourier transform of the transient impulse response. In so
doing, the LIA output
may be written as:
Y(~°(fReLs(~l +Jlm~s(~l~(ai+jbi), (16)
which can be eventually decomposed into in-phase (IP) and quadrature (Q)
components given
by:
YIP = Re~Y(~J =
18
CA 02305477 2000-04-17
- 2I° {cos(~0)[S~( nTl ) + sin( nit )] Re[S(f )] + sin( n0 )[S~( nil ) -
sin( ~i2 )] ~[S(f )] }
T T T T T T
and
(17)
YQ - Iml ~'~J -
2I° ~ sue( ~~)[s~( ~Tl ) _ sin( ~i2 )] Re[S(f )] - cos(~c0)[sin( 7zil )
+ sin( nit )] ~[S(f )] }
T T T T T T
(I8)
It can be seen that, in order to obtain a true differential output, the pulse
widths must be different.
Otherwise, the effect of the new optical waveform is only to generate a signal
equivalent to that
obtained from the conventional frequency scan method, from which it differs
only by a
multiplicative (amplitude) factor. This is physically reasonable, because the
effect of two equal-
width pulses is the same in the two half periods, Fig. 2, and as a result it
does not reveal the
asymmetric behavior of the response s(t). If il is different from i2, then the
mixer - low-pass
filter action of the LIA mixes the IP- and Q-channel signals created by the
single-ended or by the
equal-width two-pulse waveform. It is most interesting that the demodulated
signal output
multiplication factors are the real and the imaginary part of the response
S(~. In fact, by choosing
suitable values of Z~, TZ, d , it is possible to balance the two terms of the
IP and Q components so
as to obtain zero magnitude for either the IP or the Q signal channel.
Figure 3 shows the theoretical behavior of the IP and Q channel outputs
obtained for
Re~S(~JIIm~S(~J = -1 ( as we are going to show in the next paragraph, in the
photothermal case
this condition corresponds to having a thermally homogeneous sample) as a
function of the
pulse separation 0 for different z~/T values. The plots clearly show the
existence of particular
pulse separation values 0°,lP and 0°,Q for which the IP or the Q
component is equal to zero.
Modifying the properties of the system leads to different values of the ratio
Re(S(~Jllm(S(fiJ,
thus shifting the output zero to a new position along the O axis provided that
zl and z2 are fixed.
The IP and Q loci of the zero crossing points can be derived from Eqs. (17)
and (18), and are
given by the following expressions:
7L'C 1 ?L'L 2
tan( ~~°°lP ) - ~Re[S(f )] sin( T ) + sin( T )~ (19)
T Im[S(f )] s~( ~i2 ) _ sin( nil )
T T
and
7L~C 1 ?L'L 2
tan( 7z0°.Q ) - ~ Im[S(f )] sm( T ) + sin( T ) ] (
T Re[S(f )] sin('-~' ) - sin( ~T2 )
T T
19
CA 02305477 2000-04-17
The existence of zeros in the outputs appears promising, because relatively
small variations in
the response of a physical system can be readily obtained from the position of
the zero on the D
axis for different values of il or i2. Moreover random fluctuations of the
signal amplitude, which
normally are not suppressed by LIA filtering [C.-H. Wang and A. Mandelis, Rev.
Sci. Instnum.,
(1999)], affect less the response of the experiment. Of course, an additional
noise suppression
factor in this technique is the constant, single-frequency bandwidth used for
the entire
measurement [J. Shen, A. Mandelis, and B. D. Aloysius, Int. J. Thermophys. 17,
1241 (1996)].
This substantially limits the noise output compared to the variable bandwidth
of a conventional
frequency- or time-scan [M. Munidasa and A. Mandelis, Rev. Sci. Instrum. 65,
2344 (1994)] and
is an instrumental feature commonly shared with the single pulsewidth-scan LIA
method [A.
Mandelis and M. Munidasa, U.S. Patent 5,667,300 (Sept. 16, 1997)]
In Figure 4 both the IP and Q amplitude, calculated according to Eqs. (17) and
(18), are
shown as functions of the pulse separation for z~lT = 5%, z~T = 25% and for
different
Im~S(~JIRe~S(~J ratios corresponding to the arg~S(f)J values reported in the
inset. The Q
component crosses the zero magnitude axis at lower values of 0 than the IP
component. This can
be understood in terms of the fact that the odd w(t) weighing function used at
the mixing stage to
obtain the Q component is the one actually shown in Fig. 2(c). The even
weighing function used
to obtain the IP component is shifted by Tl2, which implies an equivalent
shift of the zero
positions close to the upper edge of the dlT axis. Figure S shows the locus of
zeroes as a function
of tl for z~T = 25% and for the arg(S(~J values shown in the inset. It is seen
that the greater the
difference in value between pulsewidths zl and z2, the better the loci
positions are resolved. This
reflects, again, the fact that the use of two different pulse widths forces
the response to show a
measurably different behavior in the two half periods. These facts corroborate
the use of narrow
pulsewidths. Long pulsewidths limit the available 0 scan range and hence the
resolution of the
experiment. In some experimental situations, such as photothermal measurements
in
transmission across the thickness of a material, the magnitude of one of the
two LIA channels is
much greater than that of the other, zero-crossing, channel. This may
adversely affect the
effectiveness of the technique by forcing the LIA baseline to remain high for
both channels. In
those situations the baseline can be lowered and the effectiveness of the
technique can be
restored by e.g. using an appropriate lock-in analyzer, such as the EG&G
models 7220, 7260 or
7265 with the availability of a synchronous oscillator signal to produce an
offsetting voltage or
current signal into the differential input amplifier. The offsetting method is
described in the
EG&G Application Note AN 1001, "Input Offset Reduction zrsing the model
7265/7260/7220
Synchronozrs OscillatorlDemodulator Monitor Output" and must be used in
combination with
the lock-in common-mode rejection method as an integral part of the present
invention.
c) An application of the lock-in common-mode-rejection method using laser PTR
diagnostics.
In this application, measurements obtained on a homogeneous Zr alloy sample
will be
presented by way of example for the present invention. These measurements will
be further
compared with that obtained by irradiating the sample with the conventional
SO% duty-cycle
square wave, in order to compare their noise characteristics. Finally, some
preliminary
measurements on Zr-2.SNb shot-peeved samples will be presented as a case study
of weakly
inhomogeneous solids and for comparison with that obtained with the
conventional frequency
scan.
CA 02305477 2000-04-17
A simple PTR embodiment of the common-mode-rejection LIA methodology was
constructed. A schematic diagram of the experimental setup used to perform the
PTR
measurements is shown in Fig. 6 and it comprises a sub-set of the full system
shown in Fig. 1.
An Ar-ion laser (514 nm) from Coherent, model Innova 100, was used as a 250-mW
pump beam
with a 2-mm spot size impinging on the sample surface. The beam was intensity
modulated by
an acousto-optic modulator (AOM), the digital driver of which was connected to
a four-channel
delay digital generator (Stanford Research Model DG535). The digital delay
generator allows the
construction of the variable-width two-square-pulse waveform through
appropriate computer-
controlled software and is used to drive the AOM through the driver. The
emitted IR radiation
from the sample was collected and focused onto the detector using two Ag
coated off axis
paraboloidal minors. The PTR optical detection circuit was as described in
Fig. 1. The PTR
signal from the detector was pre-amplified (EG&G Judson Model PA 350) and fed
to an analog
LIA (EG&G Model 5210), which also provided the external triggering signal for
the digital
delay generator. A personal computer was used to control the modulation
waveform and to store
the LIA signal components.
Several experiments were performed using a crystalline Zr alloy "reference"
sample.
One experiment consisted of recording the PTR signal as a function of the two-
pulse separation
for different widths of the first pulse while the width of the second pulse
was kept fixed (z~T=
25%). The separation scan range was limited by the necessity to avoid the
overlapping of the two
pulses. In fact, linear conduction heat transfer theory relies on a linear
superposition of the effect
of each pulse, which further implies that the optical intensity should be two
times higher when
the pulses are driving the AOM in tandem. This condition is not fulfilled
under normal, single-
ended working conditions of the modulator. The aim of these measurements was
to measure the
instrumental time-delay shift that inevitably occurs between the reference and
the optical
excitation waveform due the finite risetime of the modulator and the
peripheral electronics. In
order to fit the data, the theoretical expressions for the IP and Q components
of the LIA have
been modified by inserting a time delay term d
YIP(f) _ - 2~° {cos(~(~,T d))[sin( T' ) + sin( T2 )j Re~S(f)~
(21)
+ sin( ~(~T d))(sin( T' ) - sin( T2 )~ ~~S(f)~ }
and
YQ (f ) = 2~° f sin( ~'(~T d) )fsin( T' ) - sin( TZ )l Re fs(f )~
(22)
- cos(~(~T d) )(sin( T' ) + sin( T2 )~ ImLs(f )~ }
The introduction of the delay term d shifts the crossing points for the IP and
Q channels, which,
according to Eqs.(21) and (22), must be modified as follows
21
CA 02305477 2000-04-17
1 2
7z Re[S(f )] sue( T ) + sin( T )
tan[ T (~o,iP + d)] _ ~ ~ S f ~z ~z ~ (23)
[ ( )] s~( T2 ) _ sin( T1 )
~c Im[S(f )] sin( T' ) + sin( T2 )
tan[ T (~o.Q + d)] _ {Re S f nT nz } (24)
[ ( )] s~( T1 ) - sin( T2 )
The experimental results have been fitted to the theoretical expressions (21)
and (22) by using d
as an adjustable parameter (fixed for a given repetition frequency), and
assuming Re[S(~] _ -
Im[S(~], which is theoretically consistent with the assumption of a
homogeneous (reference)
sample [See, for example, G. Busse and H. G. Walther inProgress in
Photothermal and
Photoacoustic Science and Technolo~y Vol. I: Principles and Perspectives of
Photothermal and
Photoacozrstic Phenomena, (A. Mandelis, Ed.,Elsevier, New York,1992), Chap. 5,
pp. 207 -
298]. It should be noted that, like the single-ended technique, the use of a
"reference" sample
here may be confined to calculating the one-point instrumental transfer-
function phase shift at
the given frequency. Furthermore, the position of the two zero-crossing signal
magnitude points
(one for the IP and one for the Q channel) can also be labeled as belonging to
a homogeneous
sample. Nevertheless, the former operation is not essential when only the
degree of departure
from homogeneity of a test sample is required. This method may thus
be used without signal normalization requirements to measure small relative
signal variations
between slightly different samples.
Measurements with the Zr alloy sample were performed at three modulation
frequencies
(0.5, S and 10 kHz). Typical experimental results are shown together with
their theoretical fits in
Fig. 7. We wish to point out the excellent agreement between theory and
experimental results,
which is indicative of the potential of the technique, in view of the very low
signal levels
encountered, especially at 10 kHz. This is the result of the efficient noise
suppression, in part due
to the common-mode rejection by the differential operation performed by the
LIA, and in part
due to the constant noise bandwidth of the fixed-frequency operation, as
discussed earlier on.
In Table I, the instrumental delays obtained for both the IP and Q components
are shown
for the various modulation frequencies of this application. It is noted that
for a given frequency
the delay values for the Q component are quite independent of z~ as they
should be, while those
for the IP component reveal a greater scatter. It is believed that this is due
to the fact that the
zeroes of the IP component are close to the upper edge of the 0 scan, Fig. 7,
and the fitting
procedure cannot afford the same quality as that for the Q component. For this
reason attention
will henceforth be limited only to the Q component of the signal.
In order to study the influence of the scatter in the delay data on the
performance of the
experiment, we inserted the average d value in Eq. (22) reported in the last
row of Table I, and
22
CA 02305477 2000-04-17
we fitted again all the data in order to find the Im((S(~JIRe(S(j~J ratio or,
equivalently, the ~o,Q
zero crossing positions. The tan((dog +d)~rITJ values obtained for the Q
component at 5 kHz as
a function of z~/T are reported in Fig. 8, together with the theoretical
interpolation given by Eq.
(10) with Im((S(~JIRe(S(~J as a parameter. The quality of the fits for the
remaining frequencies
(0.5 and 10 kHz) was very similar. It is concluded that the d value scatter
has a negligible effect
on the output response, which results in excellent agreement between the
experimental data and
the expected theoretical result for Zr (Im(S(~JIRe(S(~J = -1); see first row
of Table II. This
agreement means that, in general, the instrumental delay can be assumed
constant for a given
frequency, as long as distortions are not introduced in the optical waveform
shape. Moreover it
should be noted that this calibration is sample independent. This means that
for a given
experimental apparatus the delay values remain the same and it is not
necessary to repeat the
calibration.
Table I: Delay term dlT as a function of f and zllT obtained by fitting the
Il' and Q components
of the Zr response to Eqs. (1) and (2) assuming Re(S(fiJ = -Im(S(~J. r2/T was
equal to 25%.
Last row shows the average dvalue in each column.
T1/T f--500 f--5 f--10
Hz kHz kHz
IP Q IP Q IP Q
2 2.2 2.3 2.22 2.3 6.98 7.13
2.1 2.28 2.02 2.87 5.47 8.09
7 2.1 2.3 2.0 3.11 4.44 8.10
1.9 2.05 1.2 2.7 2.88 7.94
2.1 2.3 2.9 2.8 5 8
In order to evaluate the robustness of this new methodology, the same pulse
separation
scans have been performed for various pump laser powers. Conventional
frequency scans have
also been carried out in parallel under the same experimental conditions, in
order to compare the
relative SNR. In Figure 9 the Q component is reported as a function of pulse
separation. As can
be seen, even the data corresponding to the lowest power are in agreement with
the other sets
despite the very low magnitude (less than 2 ~V). The varying slopes of the
experimental data
about the zero crossing point are due to the corresponding S(~ amplitudes. The
zero crossing
points are coincident for all experimental laser powers, as expected from the
same sample, and
very good noise rejection is observed.
Table II: Im(S(~JIRe(S(~J ratio values for the three investigated samples (two
shot-
peened Zr-2.SNb alloys and the reference Zr sample), obtained by fitting the
zero
crossin oints to E . 18 .
SAMPL f=0.5 kHz f--5 kHz f--10 kHz
Zr -1.01 -0.99 -1.00
CS 0.928 -0.955 -0.996
23
CA 02305477 2000-04-17
The corresponding signal phase data, obtained by temporally varying the pump
intensity
as a SO% duty-cycle square wave, are reported in Fig. 10. The data
corresponding to the two
highest power values are in agreement, but those obtained at the lowest power
are increasingly
shifted with increasing modulation frequency. In the frequency range utilized
in the pulse-scan
measurements (f = S00 Hz), the phase shift is approx. -1.5°. In order
to give a comparison
between the two methodologies, Fig. 11 shows a zoom in the vicinity of the
zero crossing region
of the curves reported in Fig. 9. Here two additional curves are included,
showing the theoretical
interpolation of the data obtained for P = 1 SO mW, arbitrarily shifted by ~
1.5° with respect to
arg~(Sf)J = -45° (the semi-infinite photothermal case). It is evident
that the spread 02 among the
crossing points at all laser powers is much less than the phase-error
equivalent spread DI (1.5°
exhibited by the frequency scan). Once again, this confirms the superior noise
suppression
resulting from the lock-in differential action.
After the preliminary tests with the Zr alloy reference and the ensuing
calibration
procedure, experiments were performed with two Zr-2.SN6 shot-peened samples in
order to test
the sensitivity of the new instrumental methodology to minute thermomechanical
inhomogeneities and to compare the results with those obtained by means of the
conventional
50% duty-cycle frequency-scan PTR method. Shot peening [S.A.Meguid, ed.,
Surface
Engineering, (Elsevier Applied Science, New York, 1990)] is employed as an
effective
mechanical surface improvement method in metallic materials. This method
basically consists of
bombarding the metal surface with a large number of small spheres of steel,
glass or ceramic,
totally covering the surface with indentations. As a result, a thin surface
layer (on the order of
100 p.m) is plastically deformed. Plastic deformation causes strain hardening,
which improves
the fatigue life and corrosion resistance of the treated metal surface. In
selecting and controlling
shot peening parameters to optimize surface improvement, it is very important
to monitor the
effects caused by the shot peening process. These effects to-date are usually
evaluated by
destructive methods, such as Transmission Electron Microscopy (TEM).
The two examined samples were shot peened at Almen intensities CS and N7,
respectively. The microhardness profiles obtained by Vickers indentation tests
are shown in Fig.
12. The sample CS reveals quite a small variation (~ 10%) in the hardness
value over a depth
distance on the order of 100 pm, while the sample N7 exhibits an essentially
flat hardness
profile. Nevertheless, TEM examinations performed on this same sample have
indicated that shot
peening at N7 Almen intensity does affect the grain structure over a depth
lower than 60 p.m [K.
F. Amouzouvi, L. J. Clegg, R. C. Styles and J. E. Winegar, private
communication]. The
foregoing shot peening process was chosen to test the new technique because
its effects on the
thermophysical properties of metals are minuscule. For comparison,
photothermal depth
profilometry of hardened steels by heat treatment, generates a phase contrast
less than 5° even for
hardness variations of one order of magnitude [T.T.N.Lan and H.G.Walther, J.
Appl. Phys. 80,
5289 (1996)]. This suggests than a very small contrast signal should be
expected from shot
peened samples.
In Figure 13 the Q signals corresponding to various zl pulsewidths and fixed
zz are
reported as functions of the normalized pulse separation for the N7 sample and
for two different
modulation frequencies (500 Hz and 10 kHz). Figure 14 shows the shift of the
zero position due
to thermal response changes for both samples CS and N7 at 500 Hz. The results
are compared to
that from the reference Zr sample. The excellent agreement between theory and
experimental
results is confirmed: By fitting the data to Eq. (16), with the d value
determined for the Zr
sample obtained from the last row of Table I, we were able to precisely
determine the Q
24
CA 02305477 2000-04-17
component zero-crossing positions 0°,Q. The tan~(((dog +d)~rITJ values
shown in Fig. 15, were
compared to the theoretical interpolations given by Eq. (18), in order to
calculate the
Im~S(~JIRe~S(~J values which are reported in Table II. The Im~S(~JIRe(S(~J
values obtained for
the CS sample reveal a trend as a function of the modulation frequency, which
is quite consistent
with the hardness profile. A shift from the homogenous sample response is
expected when the
thermal-wave diffusion length is on the order of the depth where the hardness
profile shows
significant variations. Considering that the nominal thermal diffusivity value
of the Zr-2.SNb
alloy is 0.093 cm2/sec [ASM Handbook, IOTA; Edition, (Joseph E. Davis et al.,
Ed., ASM
International, Materials Park, OH, 1992) Vol. II, p. 666.], at 0.5 kHz the
thermal diffusion length
is on the order of 75 Vim, i.e. commensurate with the hardness depth.
Therefore, the
Im~S(~JIRe~S(~J value at f = 0.5 kHz is higher than -1 expected from a semi-
infinite
homogeneous solid, or, equivalently, the phase lag is less than -45° (-
42.8° ). This is consistent
with the fact that the hardened layer has an effective thermal diffusivity
lower than that of the
bulk [See; for example, G. Busse and H. G. Walther in Pro~C;ress in
Photothermal and
Photoaco2rstic Science and TechnoloQV Vol. I: Principles and Perspectives of
Photothermal and
Photoaco~istic Phenomena, (A. Mandelis, Ed., Elsevier, New York,1992), Chap.
5, pp. 207 -
298]. At higher modulation frequencies, 5 kHz and 10 kHz, the thermal
diffusion length becomes
20 Vim. The corresponding hardness profile does not show large variations over
this distance.
This means that the sample can be assumed homogeneous over this depth and,
accordingly, the
Im~S(~J)JIRe~S(~J ratio assumes values roughly corresponding to -45° .
On the contrary, the
Im~S(~JIRe~S(~J data obtained for the N7 sample do not reveal any measurable
trend with
modulation frequency, reflecting the corresponding flat hardness profile
behavior.
For the purposes of this application of the present invention, conventional
PTR frequency
scans were further performed for comparison by using the same setup and the
same 250-mW
optical power for all the measurements. The only change was in the excitation-
laser-beam
modulation waveform, a 50% duty-cycle square wave. The experimental data,
normalized by the
data obtained from the Zr reference, are reported in Fig. 16. The systematic
high-frequency-
amplitude differences of the two curves in Fig. 16(a) are not meaningful, as
extensive satellite
PTR experiments with these and different Almen-intensity shot peened Zr-2.SNb
samples have
shown that there exists no consistent trend of the signal with degree of
hardening. Furthermore,
unlike the Im~S(~J)JIRe~S(f~J ratios of Table II, the presence of a hardness
depth profile in the
CS sample, Fig. 12, cannot be measured from the similar amplitude trends of
both samples
throughout the entire frequency range in Fig. 15(a). The N7 > CS amplitude
ratio in Fig. 16(a) is,
however, consistent with the lower slope of the CS curve in Fig. 14, which
indicates a lower-
amplitude thermal-wave signal at 500 Hz for the CS shot peened Zr-2.SNb alloy.
The phase
channel, Fig. 16(b), also clearly shows insensitivity to the differences in
hardness between the
two shot-peened samples. Doubtless, any such differences are masked by the
large data scatter in
this figure.
In conclusion, the PTR experimental calibration of the novel common-mode-
rejection
demodulation technique was shown to be a very promising high-detectivity
measurement method
for low-dynamic-range and poor-SNR signals, such as those obtained with
thermal-wave
diagnostics. Results with two shot-peened Zr-2.SNb samples have shown that
this technique is
sensitive enough to resolve minute differences in thermophysical properties
resulting from
mechanical structure changes of these materials after shot peening and to
monitor hardness depth
profiles by means of the value of the Im(S(~J)JIRe~S(~J ratio at several
frequencies.
CA 02305477 2000-04-17
Conventional single-ended frequency-scanned PTR detection proved unable to
resolve these
differences.
ii) Frequency-swept ("Chirp")/PTR combination method
A combination of chopped illumination and frequency swept ("chirped")
detection has
been used for a quantitative kinetic PTR study based on the real-time
monitoring of the temporal
evolution of the low-injection minority-carrier transport properties of two
silicon wafers which
exhibited PTR transients. Depending on crystal growth and wafer manufacturing
conditions,
some lower quality Si wafers exhibit mild or strong temporal transients under
the PTR probe.
PTR frequency scans were performed in the steady state following the complete
saturation of the
PTR transient. The two 6" Si wafers used in this study were provided by Mitel
SCC (Bromont,
Quebec, Canada). One wafer was unprocessed 10-15 S2-cm n-type (100) wafer with
oxygen
content between 30-to-38 ppma. The wafers were polished using a colloidal
suspension of Si02
as the polishing slurry. After polishing the wafers were cleaned, first in a
wet bench and then in a
scrubber. In the wet bench the wafers go through three solutions with water
rinses in between.
The first one removes particles and organics and leaves a thin oxide layer.
The second solution is
used to etch this oxide and dissolve metals like Cu that tend to diffuse into
the oxide. The third
solution removes metals and leaves the wafers hydrophobic. The other wafer was
24-40 S2-cm p-
type silicon, with a thermally grown oxide. This wafer was ramped from 800 to
1175 °C under
low concentration OZ and N2 for 40 min. This was followed by oxidation at 1175
°C in 02 for 6
hours and 50 minutes and a ramp-down from 1175 to 800 °C in N 2 for 1
hour and 20 minutes.
The experimental setup for the PTR method used to obtain conventional
frequency scans
has been described previously [S. J. Sheard and M. G. Somekh, Infrared Phys.
28, 287 (1988)].
The instrumental technique of photothermal chirped frequency sweep has also
been described in
detail by A. Mandelis, IEEE Trans. Ultrasonics, Ferroelectrics, Freq. Control,
UFFC-33, 590
(1986)]. The new combined apparatus for the simultaneous monitoring of
transient evolution and
chirped-PTR correlation and spectral analysis is shown in Fig. l7. An Ar+-ion
laser emitting at
514 nm was used as the excitation source. The beam was chopped at fo=83 Hz
with a mechanical
chopper. Then it was expanded, collimated and focused by a lens on the sample
surface. The
incident power was 40mW. A square-waveform chirp from a dual-channel FFT
analyzer was
used to drive the acousto-optic modulator, which produced periodic frequency
sweeps of the
probe beam in the range 100 Hz to 100 kHz. The low-power laser beam was used
to induce
temporal effects on the sample. The amplitude, A(fo,t), and phase, ~(fo,t), of
the PTR signal in
the lock-in amplifier were recorded as functions of time. For each
intermittent (quasi-steady)
measurement 1000 frequency chirps were co-added and averaged, and the spectral
transfer
function, H(f), was generated and stored in the FFT analyzer. Quasi-steady
chirps were obtained
intermittently at various time-windows during the transient from the
unoxidized n-type wafer as
illustrated in Fig. 18. For the n-Si wafer, five consecutive chirps during a
single continuous
transient from start to saturation were obtained on a previously unirradiated
spot. Each chirp
measurement lasted 5 minutes. After the 5~' chirp, the optical source was
blocked for 1 hour.
Then the transient and five more chirps (not shown in Fig. 18) were measured
again on the same
spot of the wafer. When steady state was reached (>4000 s) a lock-in frequency
scan was
obtained as shown in Fig. 19. For the p-Si wafer only the lock-in temporal
behavior of a
previously unirradiated spot was monitored. No chirps were introduced because
the response was
flat , Fig. 18.
26
CA 02305477 2000-04-17
The amplitude and the phase of the PTR frequency response of the unprocessed n-
Si at
steady-state, Fig.l9, were fitted simultaneously, by using the computational
mufti-parameter
fitting procedure (described in the next section below) to a three-dimensional
theoretical model,
taking into account the laser beam spotsize, the thickness of the wafer, the
photoexcited
minority-carrier plasma-wave generation, and the optical-to-thermal energy
conversion
following lattice absorption. The following values were used to obtain the
best fit for both signal
channels: lifetime i=110 p,s; photoexcited minority carrier diffusion
coefficient Dp 10 cm2/s;
front-surface recombination velocity S1=320 cm/s; and sample thickness L=570
pm. Figure 18
shows that the transient behavior of the n-Si sample is semi-reversible
following the cut-off of
the radiation and the one-hour recovery time. Spots with fully reversible
transient behaviour have
also been observed in the same wafer.
The phase and amplitude of the spectral transfer function H(f) of the PTR
signal from the
frequency-sweep measurements at the onset of the laser (curve 2) and at steady
state (curve 1)
are shown in Fig. 19. A point-by-point lock-in frequency scan was also
performed in the steady
state and at the same spot on the n-Si wafer. The steady chirped amplitude and
phase were
normalized to those of the higher-quality lock-in signal to eliminate the
instrumental transfer-
function effects of the dual-gate FFT analyzer. The steady state chirp
measurements were
smoothed with double adjacent 5-points filter and then fitted with a Sth order
polynomial
function. This polynomial function was the one used for the normalization
procedure. These
corrections were subsequently applied to all other quasi-steady PTR chirps.
The 3-D PTR model
was used for theoretical best fits to the data to obtain the quasi-steady and
steady-state carrier
transport parameters. Dramatic quantitative changes in the front-surface
recombination velocity
were found. On the contrary, the minority carrier lifetime (T) and the and
diffusion coefficient
(Dp) did not exhibit any measurable change. The temporal evolution of the
front-surface
recombination velocity for the n-Si sample is shown in the inset of Fig.19.
When the laser beam
is turned on, the surface recombination velocity, S, starts decreasing steeply
from 830 cm/s, until
it reaches the saturation value (steady state) at 320 cm/s, a sign of
electronic-quality
improvement of the laser irradiated wafer surface.
In conclusion, we have presented an example of the Si-wafer diagnostic use of
frequency-
swept PTR in the form of combined frequency-swept and single-frequency-
modulated technique
suitable for the simultaneous kinetic measurement of surface-state annealing
temporal evolution
and minority-carrier transport properties at several time windows along the
transient generated
by low-power-laser-irradiation on n- and p-type silicon wafers subjected to
optical annealing. A
quantitative dependence of the front-surface recombination velocity decrease
on the total
annealing time in laser-irradiated unoxidized n-Si was extracted. The use of
frequency-swept
PTR to obtain fast frequency scans, time-averaged on the order of one minute,
at pre-determined
sites on a Si wafer and extract the local electronic and thermal transport
properties is a
straightforward extension of this example.
iii) Mufti-parameter computational method for thermo-electronic parameters
determination of semicondutor wafers
a) Description of the Method.
In order to obtain a particular set of parameters from PTR measurements of a
Si wafer, a
mufti-parameter fitting procedure based on the simulation trends explored by
was developed.
27
CA 02305477 2000-04-17
The total blackbody (Plank) radiation emitted from a silicon sample
illuminated with a
modulated laser beam arises from two sources: emission of IR radiation from
the photo-excited
carrier plasma-wave (injected excess carrier density) and from direct lattice
photon absorption
and optical-to-thermal (nonradiative) power conversion leading to temperature
rise (a thermal
wave)[ S. J. Sheard, M. G. Somekh, and T. Hitler, Mat. Sci. and Eng, B 5, 101;
(1990) A.
Mandelis, Solid State Electron. 42, 1 (1998); M. Hitler, M. G. Somekh, S. J.
Sheard, and D. R.
Newcombe, Mat. Sci. and Eng. B 5, 107 (1990)] Sheard and co-workers observed
experimentally
that under infrared photothermal radiometric (PTR) detection, carrier emission
dominates and the
thermal-wave contribution can be neglected for some Si samples. This
observation was
addressed theoretically recently [see, A. Salnick, A. Mandelis, H. Ruda, and
C. Jean, J. Appl.
Phys. 82, 1853 (1997); A. Salnick, A. Mandelis, and C. Jean, Appl. Phys. Lett.
69, 17 (1996)].
These authors generated a composite plasma- and thermal-wave PTR model of
semiconductors
and showed that the plasma-wave signal component can dominate in high-quality
materials
virtually at all modulation frequencies. However, in this model the radial
spatial variation of
laser-generated excess carriers and of the temperature rise was not considered
[T. Ikari, A.
Salnik, and A. Mandelis, J. Appl. Phys. 85, 7392 (1999)] have presented a
general theoretical
model for the laser-induced PTR signal from a semiconductor wafer of finite
thickness using a
three-dimensional geometry. In this model, carrier diffusion and
recombination, as well as heat
conduction, along the radial and axial directions in the sample were taken
into account using
cylindrical coordinates. A pair of conventional coupled plasma- and heat
diffusion-wave
equations were written and solved in Hankel space. In this theoretical
framework, the plasma and
thermal components can be written as follows:
Spry (~) = A' ~F{~~ ~)J~ (~') d~, + ~Z T(~., ~~i ~~~~ {24)
0 0
In Eq. (24), A is the effective detector area: A=Tray, where a is the detector
radius; Jl(x) is the
Bessel function of the first kind and order one; and F(~,, ~) and T(A, c~) are
the radial Hankel
transforms of corresponding frequency-dependent solutions to the plasma-wave
and thermal-
wave boundary-value problems under optical excitation by a Gaussian laser
beam. These
functions have been reported previously [ A. Salnick, A. Mandelis, and C.
Jean, Appl. Phys. Lett.
69, 17 (1996); T. Ikari, A. Salnick, and A. Mandelis, J. Appl. Phys. 85, 7392
(1999)]. They
contain the thermal and electronic transport parameters of the electronic
solid: recombination
lifetime (i), carrier diffusion coefficient (D",), front surface recombination
velocity (S1), back
surface recombination velocity {S2) and thermal diffusivity (a). In practice,
for accurate
measurements, both thermal-wave and plasma-wave contributions must be
considered in the
interpretation of PTR data (amplitude and phase) from semiconductor samples,
such as Si
wafers. The electronic quality of industrial semiconductor wafers varies
widely and they
frequently exhibit strong thermal behavior, especially at low modulation
frequencies.
Semiconductor PTR is an excellent candidate technique for non-destructive
mufti-parameter
measurements in electronic materials because it can offer measurements of
several important
properties, which are crucial for device fabrication control. The major
problem, facing the
implementation of the three-dimensional model by Ikari et. al. for use with
experimental
radiometric data, is the reliability of the measured values of any and all of
the foregoing
electronic transport parameters, in view of the intrinsic non-uniqueness of
the theoretical fit
28
CA 02305477 2000-04-17
(maximum of five unknown material parameters plus the constants C1 and C2 in
Eq. (24)) to only
two data channels (amplitude and phase) available to the experimenter.
The present invention presents a computational methodology developed to
address precisely
the uniqueness problem of the PTR signal interpretation. The effects of the
various transport
parameters on the shape of the frequency response curves (amplitude and phase)
are studied
theoretically. Then a robust computational best-fit algorithm is described,
based on the specifics
of signal sensitivity dependence on a given transport parameter across
particular regions of the
modulation frequency spectrum. As a result, the conditions for unique fits and
reliable parameter
measurements are deduced and examples of such measurements are given.
Theoretical Simulations. A pair of conventional coupled plasma and heat
diffusion equations
based on Eq. (24) can be written and solved in Hankel space. The three
dimensional PTR signal
is finally obtained by taking a weighted superposition of the plasma and
thermal contributions
[A. Mandelis, Solid State Electron. 42, 1 (1998)]
SPTR (~) - C~SPlarma (r°) + CtS~ermal (~)
where parameters CP and Ct represent the weight of each component (plasma and
thermal)
contributing to the PTR signal and SPTR represents a vector in the sense of a
function with
complex argument. In Eq. (25), the plasma-wave component (SP,~",a) is obtained
from the
Hankel transform N (z,~,;w) of the 3-dimensional N( r ,t~) by integrating over
the thickness of
the wafer, which takes into account deep-lying Plank radiation emission from
photogenerated
and diffused carriers, according to Kirchhoff's law of Detailed Balance. The
result for the plasma
contribution in the Hankel space is
A + a b"L
F(~~ w) = N~Z~ ~; w~Z = ( ) 2 (26)
hv'n(D~b~ +Sl~n A -A e-2b~L '
0 2 1
the parameters in equation (26) are defined as follows.
Dn bn + S, 27
nb" S~ ( )
- Dn bn + SZ
2 Dn bn _ S2
where ~" _ (1 + ic~zn)l D"z" , D" is the minority electron carrier
diffusivity, zn is the carrier
lifetime, S1 and S2 are the front and back surface recombination velocities.
The thermal
component is calculated in the same manner:
1-8 6~t BbaG -1 1-B 6"G 1-8 6"L
T /~,, Cl) = B~ + Bz + B3 + B4 a b"L (2g)
~t ~t ~n ~n
The parameters in the above equation are defined as follows.
29
CA 02305477 2000-04-17
h~ _ hze_b'L
Bi - 1 _ e_z~L
-b,L
B = ht a _ hz a _bt~
z 1 _ e-zb'L
-~.ZdZ l8
B = Ege Az
3 ~ci~vkr" b" -ba (17"b" -Sy Az -Ale-zb"L
B B3
a -- A2 (29)
hl - - ~ z" (b,2, - b~ ~B3 + B4e_zb~L )- ~" (B3 - B4e_ZbnL
t
[S2T~(b~ -bt ~B3 +Ba)-b~(B3 -Ba)]G'-b"~
t
b2 = ~2 +a2.
t t.
This 3-D PTR model takes into account the finite size of the exciting laser
beam, the effective
detector size, and the sample thickness [T. Ikari, A. Salnik, and A. Mandelis,
J. Appl. Phys.
( 1999)]. The parameters involved in a typical mufti-parameter fitting
procedure are:
recombination lifetime (i), ambipolar carrier diffusion coefficient in n- orp-
type material (D~,p),
front surface recombination velocity (St), thermal (Ct) and plasma (CP)
contribution coefficient,
back surface recombination velocity (S2) and thermal diffusivity (a).
b) Application of the mufti-parameter best-fit PTR metrology to uniquely
determine
thermo-electronic parameters of high and low resistivity silicon wafers
By way of example for the purposes of the present invention, the theoretical,
experimental and computational PTR methodology of the present invention is
applied to two
samples, a high resistivity (25-44 ohm-cm) and a low resistivity (14-24 ohm-
cm) wafer. Both
wafers were thermally annealed and had a polished front surface and a rough
(matte) back
surface. These wafers contained centerpoint oxygen concentration between 24 to
32 ppma and
carbon concentration of 0.5 x 1017 atoms/cm3. Thermal dry isochronal annealing
was carried out
using a horizontal furnace BDF-200 (see Fig. 20). The gas inlet was located at
the back of the
furnace ("source") and the gas exit was located at the opposite end of the
furnace ("door").
Inside the tube there was a negative static pressure of 0.3 psi to evacuate
the gas. The system
was, nevertheless, considered to be under STP conditions, without vacuum. The
polished side of
the wafers was facing the door. Thus, the gas was flowing toward the back of
the wafers. Each
tube could accommodate four quartz boats and each boat could hold 25 wafers.
The spacing
between two adjacent wafers was 3 mm. The tube was surrounded by heating
elements, which
provided uniform temperature across its length. All wafers were subjected to
the following
thermal cycle: they were first exposed to 800 °C for 10 minutes. After
reaching thermal
equilibrium, the wafer temperature was ramped-up from 800 to 1175 °C
under low flows of OZ
and N2 gases for 40 minutes, at a rate of 5 °C/mim. The temperature was
stabilized at 1175 °C
and dry oxidation was induced in pure 02 for 6h:50 minutes. At the end of the
oxidation
process, the furnace temperature was ramped-down to 800 °C in pure N2
at a rate of 5 °C/min.
CA 02305477 2000-04-17
Both surfaces of the wafers used in this work were exposed to the same gas
flow and temperature
conditions, under the isochronal dry oxidation process. Therefore, they were
expected to exhibit
similar surface recombination velocities, even if the front surface had been
polished chemically
and mechanically. PTR diagnostics of both surfaces of the aforementioned
wafers were
performed and the results are discussed below.
The computational best-fit procedure includes the following steps: a)
selection of initial
values within the adequate range of the physical; b) variation of the thermal
and plasma
coefficients (Ct and CP), until a good fit is obtained at the low-frequency
range of both
experimental phase and amplitude; c) variation of the recombination lifetime
value, until the
characteristic knee of the experimental curve is best-fitted and is consistent
with the
2~f~i ~ 1 criterion. These procedures are then followed by re-adjusting the
thermal and plasma
coefficients to match phase and amplitude at the low frequency range; d)
variation of D" until the
intermediate region of the phase and the amplitude signal converge to the
experimental data.
Then phase and amplitude are adjusted while C, and CP are also varied, as
required; e) variation
of S1, until the signal phase spread between two typical frequency range
extremes {10 Hz - 100
kHz) is matched to the experimental spread. Since the amplitude is also
affected by this
parameter, this procedure is followed by a re-adjustment of the amplitude at
low frequencies by
varying Ct and Cp, as required; f) variation of the back-surface recombination
velocity, S2, for
fine-tuning at low frequencies, only when the data are sensitive to S2. This
is followed by a re-
adjustment of both phase and amplitude at low frequencies by varying Ct and CP
, if necessary;
and g) fine-tuning of the fit by repeating steps c-to-f.
It is well-known that lifetime values vary across a silicon wafer [A. Salnick,
A. Mandelis,
F. Funak, and C. Jean, Appl. Phys. Lett. 71, 1531 (1997)]. This has also been
observed during
the development of the present PTR metrologic technology. However, this
variation has a special
significance in the case of PTR amplitude measurements. When mufti-point
measurements
across the surface of a single sample are performed, an extra channel of
information is available,
that is the relative positions of the flat (low frequency) region of the
amplitude curves scale
linearly with lifetime at a given point [A. Mandelis, Solid-State Electron.
42, 1 (1998)], see also
Fig. 20 and Fig 21. The relative values of the amplitude with respect to other
locations further
reinforce the consistency of the foregoing computational procedure by Gross-
correlation of the
measured lifetimes. Based on Fig. 21, amplitude scans can immediately yield
lifetime maps upon
calibration, as in the case of Fig. 22.
For a reliable and unique mufti-parameter fit it was found very helpful to
establish
realistic initial "seed" values for the various electronic parameters. The
thermal diffusivity is a
bulk property and simulations indicate that it has a weak influence in the PTR
si~nal (both
amplitude and phase). The values chosen for these simulations were 0.75 and
0.96 cm /s for low
and high resistivity wafers, respectively. These values are in close agreement
with those reported
in the literature [W. M. Bullis and H. R. Huff, J. Electrochem. Soc. 143, 1399
(1996); G. Zoth
and W. Bergholz, J. Appl. Phys. 67, 6764(1990); E. Yablonovich, D. L. Allada,
C.C. Chang, T.
Gmitter, and T. B. Bright, Phys. Rev. Lett. 57, 249 (1986)]. The range of
values reported in the
literature for the front surface recombination velocity of n- and p-silicon is
between 0.25 cm/s,
for a very passivated surface, and 107 cm/s for highly doped p-silicon. A low
value of the front
surface recombination velocity (i.e. 100 cm/s) for a p-silicon, which
represents a moderately
passivated surface, was chosen as our initial value. For the Si wafers of this
study, the back
surface was subjected to the same thermal process as the front surface.
Therefore, the S2 value of
100 cm/s, same as S,, was chosen to initialize the fitting procedure. Initial
relative values for Ct
31
CA 02305477 2000-04-17
and CP were 1 and 3 x 10'2° a.u, respectively. The reported values for
D~,P are in the range
between 8 cm2/s and 36.4 cm2/s. We chose 7.5 cm2/s (lower limit) as the
initial value for our
fitting procedure. Finally, to start the fitting procedure, values of 1500 ~s
and 100 p,s were
chosen for minority carrier lifetimes of high and low resistivity wafers,
respectively. Once the
initial values were chosen, the cyclic fitting procedure with the feedback
described above was
followed, steps a-to-f.
Results of the simulations using the aforementioned methodology has been
reported in
detail in. This procedure was performed manually in an electronic sheet
program. However, an
automated computer program for the sequential cyclic mufti-parameter fitting
procedure can be
implemented. An embodiment of the computational algorithm is appended to this
invention. It
will be appreciated by those skilled in the art that numerous other
configurations for this
algorithm may be used. The above example is meant to be non limiting and
illustrative only.
c) An application of the mufti-parameter best-fit PTR metrology to intact and
damaged wafers.
Front- and back-surface PTR measurements and recombination lifetime scanning
imaging . Frequency and imaging scans were performed at the center-point of a
test wafer (high
resistivity wafer), with the laser beam impinging successively on the front
surface and on the
corresponding spot of the back surface. The wafer used for these measurements
and their
preparation were described in the previous section (iv.b).
A small area on the back surface of the wafer was intentionally scratched very
lightly and
frequency scans were repeated. Silicon carbide paper with an average particle
size of 22 pm was
used to scratch the surface. The experimental PTR signal for this Si sample
obtained from these
frequency scans are shown in Fig. 26. The solid squares and inverted triangles
represent
frequency scans for both surfaces prior to damaging the back surface. The
solid circle and
upright triangles represent frequency scans for both surfaces after damaging
the back surface. In
the former case the experimental amplitude and phase curves are almost
identical indicating
similar electronic transport parameters in both directions. The solid lines
represent the best fits to
the experimental data following the aforementioned procedure. The measured
values of the
various parameters for the front and back surface before (front 1 and back 1),
and after (front 2
and back 2), scratching the back surface are shown in Table III.
Table III. Thermal and electronic transport parameters for long lifetimep-Si
(high resistivity
wafer), determined by 3-D PTR model and the mufti-parameter best fit; intact
sample (front 1,
back 11. and scratched hack c»rfar.P ~frnnt ~ h~r~lr 71
Amplitudea .~ D" S 1 S2 CP C
(m~ (cm2/s)( s) (cmz/s)' (cm/s)(a.u) (a.u)
~ (cm/s)
Front 56.652 0.96 950 3.2 90 ----- 4.5x10' 3.1
1
Back 66.505 0.96 950 3.2 110 ----- 4.5x10' 3.0
1
Front 37.388 0.96 80 3.2 90 ----- 0.1 SxlO'2.0
2
Back 12.776 0.12 12 3.2 3x10 ----- 3.0x10- 1x10
2
The fitting values obtained for S2 (1000 cm/s) were in the non-sensitive
region according
to simulations of the theory (see Fig. 25). Therefore no conclusion regarding
this parameter
could be made. The electronic parameters obtained for the front and back
surface (before
scratching) were similar except that S1 was slightly higher. These results
indicate that the dry
32
CA 02305477 2000-04-17
oxidation process (described at the onset of section iii.b) did, indeed, have
the same effect on
both surfaces. The solid circles and solid upright triangles represent the
frequency scans after the
damage on the back surface. The PTR amplitude, in the high frequency range, of
the front-
surface signal (front 2) in the case with the damaged region is similar to
that obtained from the
case with the intact-wafer-surface (front 1). However, there are significant
changes regarding the
PTR signal obtained from the back surface (back 1) and the damaged region case
(back 2). In
Table III, the value of i for the scratched back surface represents an average
recombination
lifetime and could not be measured with accuracy. The low value found for a is
related to the
state of the surface, and represents the effective thermal diffusivity of the
rough surface and the
layer immediately below. This result shows that the PTR method is very
sensitive to near-surface
thermophysical properties. The value found for S1 (back 2) is as expected for
damaged surfaces;
it is also in agreement with the theoretical predictions for bare surfaces
(Figure 24, and Table
III). After the mechanical damage on the back surface of the wafer, the
recombination lifetime
decreased dramatically. A scanned image of the amplitude and phase over the
defect region is
shown in Fig. 22. This figure shows the sensitivity of the phase and amplitude
to the extended
defect area. The linearity of the PTR amplitude on the minority carrier
lifetime is shown in Fig.
21.
These results suggest that the carrier recombination lifetime is not only
affected by the
bulk lifetime, but also by the recombination on the front surface of the
sample exposed to the
laser beam. This result is consistent with the very shallow optical absorption
depth, (3-1 = 10-4 cm
at 514 nm. Under front-surface detection, the very-near-surface photo-injected
carrier lifetime
may be affected by the electronic state of the surface itself. The effective
lifetime, ie~ , is known
to be given by [H. Daio, A. Buckowski, and Shimura, J. Electrochem. Soc.,141,
1590 (1994)].
1 1 1
- -+-
Teff zb zs
[30]
where ib is the bulk lifetime, and is represents the surface lifetime. It is
possible that the PTR-
measured lifetime under 514-nm laser irradiation is more closely related to
is, rather than t0 Tb .
Measurements at longer wavelengths and greater optical absorption depths are
likely to be more
representative of bulk lifetime, as shown in our earlier work [A. Mandelis, R.
Bleiss, and F.
Shimura,J. Appl. Phys. 74, 3431 (1991)]. Such measurements are part of the
spirit of the present
invention and are not meant to be excluded from the applications discussed in
this section.
In conclusion, according to the results from the high resistivity sample
presented here,
Fig. 26 and Table III, the state of the back surface plays an important role
in determining the
thickness-averaged PTR carrier-recombination lifetime. Also the ability of the
present invention
methodology for lifetime mapping of damaged silicon wafers. This imaging
capability can be
easily extended to other applications as will be shown in the sections below.
d) Surface recombination velocity and minority carrier lifetime anti-
correlation
Diagnostics of oxidized silicon wafers. In order to demonstrate the capability
of
measuring surface recombination velocity and minority carrier lifetime using
the computational
methodology presented in this invention, results from high and low resistivity
wafers positioned
33
CA 02305477 2000-04-17
differently in wafer tubes inside a horizontal furnace are presented. These
results are part of a
more extensive study carried out for a major semiconductor manufacturer.
The oxidation and preparation process for the wafers utilized in this study is
similar to the
one reported previously in section iii.b. A schematic of the wafer tube and
the boat arrangement
inside the furnace is shown in Fig. 23 (Horizontal Furnace BDF-200). The test
batch of wafers
included groupings oxidized inside different tubes: Two wafers, one of low
resistivity (front/door
position, Fig. 23) and one of high resistivity (back/source position, Fig.
23), were placed near
the door location. Two similar wafers were placed in the same order near the
rear end of the tube
(source). The spacing between two adjacent wafers was 3 mm, and the distance
between the
wafer pair at the front and the pair at the rear was about 16.5 cm.
The results of lifetime and front surface recombination velocity are
discussed. The rest of
the parameters (a, ~, D" and S2) showed no significant variations with respect
to position in the
furnace. The range of values found for Dn for the various examined wafers was
between 3-12
cm2/s. Our simulations have shown that for this type of variations in D~ the
signal amplitude and
phase do not show a significant change. Similar results for Sz were also
found.
Four wafers (1, 2, 25 and 26) were analyzed with the 3D-PTR
model/computational
methodology. The lifetime values associated with the low resistivity wafers
(LR) 1 and 2 are
substantially lower that those with higher resistivity (HR) 25 and 26 (see
histogram, Fig. 27(b)).
The results indicated a monotonically decreasing lifetime from the center to
the edge of the HR
samples. We found that, in general, there is an inverse correlation between
lifetime and surface
recombination velocity. In Fig. 27, wafers 1 and 2 with lower lifetimes have
somewhat higher
front surface recombination velocities than wafers 25 and 26, histogram Fig.
27(a).
In conclusion, the longest lifetimes and lowest surface recombination
velocities were
measured for samples with high resistivity located near the source in a given
tube, compared to
those located near the door. We can speculate that this may be attributed to
significant turbulence
phenomena near the door location due to possible currents, and/or to heavy
metal contamination
of the door. There is a strong correlation between nominal wafer resistivity
and transport
properties: the longest lifetime values and lowest front surface recombination
velocities were
found in high-resistivity samples.
e) An application of the multi-parameter best-tit PTR method for Iron
Concentration
(imaging) Measurements on p-Si wafers.
A comparative study of electronic transport properties of p-Si wafers
intentionally
contaminated with Fe was performed using infrared photothermal radiometry
(PTR) and
micrometer photoconductance decay (p-PCD). Strong correlations were found
between PTR and
~-PCD lifetimes in a lightly contaminated wafer with no significant PTR
transient behavior. The
absolute PTR lifetime values were larger than the local averaged p.-PCD
values, due to the
different excitation wavelengths and probe depths. In a heavily contaminated
wafer the p-PCD
and PTR lifetime correlation was poorer. PTR measurements were highly
sensitive to Iron
concentration, most likely due to the dependence of the bulk recombination
lifetime on it. Rapid-
scanned (non-steady-state) PTR images of the wafer surface exhibited strong
correlations with
both p.-PCD lifetime and [Fe] concentration images in both heavily and lightly
contaminated
wafers. For the lightly and uniformly contaminated wafer, PTR scanning imaging
was found to
be more sensitive to the Iron concentration and lifetime variations than ~-PCD
34
CA 02305477 2000-04-17
Two p-type (boron-doped) Si wafers grown from magnetic Czochralski ingots, 5
and 6
inches in diameter (labeled 1 and 2, respectively), with resistivities between
10-20 S2-cm and
(100) crystallographic orientation were investigated. The wafers were oxidized
under standard
oxygen flow (500 cm3/min) in a mini furnace at 1000 °C for 70 minutes.
They were inspected
with a ~-PCD probe after processing. Sample 1 was placed in a quartz boat
vertically, while
sample 2 was placed between two Silicon Carbide (SiC) boats horizontally. As a
result, sample
1 received a relatively homogeneous Fe contamination of lower concentration
than sample 2.
This wafer was in contact with the SiC boats and thus received very
inhomogeneous and heavier
Fe contamination from both the solution and by contacting the boats.
The influence of Fe concentration on the thermoelectronic properties was
studied in the
low-injection regime (typically about 30 mW of optical power). Figure 28 shows
the p-PCD Fe
concentration and lifetime maps of the entire wafers surface (samples 1 and
2). In sample 2 six
radial points 1-6 and two regions were studied (1 cm x 2 cm, region A; and 1
cm x 1 cm, region
B) (see Fig 28a). p-PCD scans showed that sample 1 was much more uniform than
sample 2 and
was thus examined with PTR at eight locations along the radial direction as
well as inside a small
area (1 cm x 1 cm, Region A, Fig. 28b). Unfortunately, p-PCD could not yield
information about
Fe concentration values inside the light regions across sample 2, Fig. 28a;
and, to a much lesser
extent, along the rims of sample 1, Fig 28c. The II-like shape in wafer 2 is
the trace of a contact
between the SiC boat pedestal and the Si wafer during the oxidation process
and is a seat of
heavy Fe contamination. The existence of PTR signal transients in these
samples, a phenomenon
exhibited by some wafers with electronically poor surfaces [A. Cuevas, P. A.
Basore, G. Giroult-
Matlakowski, C. Dubois, J. App. Phys. 80, 1996] was observed, especially with
sample 2.
Therefore, frequency scans at each point used to calculate thermoelectronic
properties, were
carried out only after steady state signal conditions were established. Figure
29 shows the PTR
signal for the six radial positions in sample 2. The theoretical fits are
shown as continuous lines
in that figure. Table IV shows the thermal and electronic values associated
with each spot as
were obtained through the computational multi-parameter data fit explained
above. The average
local p-PCD carrier recombination lifetime values, extrapolated from the
images in Fig. 28 with
the help of calibration histograms (not shown) and the average [Fe]
concentration, calculated for
each spot in the same manner, are also given in Table IV. It is seen that the
lifetime trends
between PTR and p-PCD measurements are well correlated, even though the PTR
values are
consistently higher. Recall that the PTR laser source wavelength was 514 nm,
whereas p-PCD
data were obtained with optical excitation at 904 nm. At this excitation
wavelength, the optical
absorption coefficient of Si is [3 = 1.1 x 10z cm 1, and the p.-PCD skin depth
is ~.= 100 p.m [A.
Dargys, J. Kundrotas in Handbook on Physical Properties of Ge, Si, GaAs and
InP, Vilnius,
Science and Encyclopedia Publishers, p. 100 (1994)]. Both microwave and PTR
detection for
this wafer were efFected from the front (polished) surface. Given the long
excitation wavelength,
the p-PCD optical probe "sees" deeper into the substrate and can be reasonably
expected to yield
shorter lifetime values from thoroughly Fe-contaminated wafers than the very-
near-surface PTR
probe.
In conclusion, PTR scanning imaging at 515-nm optical excitation produces
amplitude
and phase images which may be directly related to the near-surface [Fe]
concentration
distributions and are in good-to-excellent agreement with ~-PCD-derived
recombination lifetime
and [Fe] images. Quantitative PTR measurements of the thermal and electronic
transport
parameters from steady-state frequency scans are well-correlated with local
averaged p.-PCD
lifetime values and ~-PCD-derived [Fe] concentrations for lightly and
uniformly contaminated p-
CA 02305477 2000-04-17
Si; they are not as well correlated with heavily and non-uniformly
contaminated samples. For the
lightly and uniformly contaminated wafer, PTR scanning imaging was found to be
more
sensitive to [Fe] concentration and lifetime variations than p-PCD-derived
images.
Table IV. Thermal and electronic transport parameters for wafer sample 2 (6-
inch diameter),
determined from the 3-D PTR model, for six radial positions 1-6, and region A
and B, Fig. 28a.
-PCD lifetime and [Fe] concentration values are also shown.
Location Amplitude a, ~P.,.RD" S1 Average 'c,,~Average [Fej(cni'
from center(m~ (cm2/s)(ws) (cm2/s)(cm/s)(~sl
0.00 (1) 32.745 0.75 71 3.10 300 30 <10
1.27 (2) 28.333 0.70 38 3.20 210 11 >1012
2.54 (3) 29.674 0.73 63 3.10 370 25-30 No-data
3.81 (4) 14.275 0.65 26 5.00 850 19 5x101'
5.08 (5) 19.707 0.70 45 4.60 750 25-30 No-data
5.58 (6) 19.120 0.70 46 3.40 560 25-30 No-data
Region A
a 26.632 0.70 70 3.40 300 19 SxlOlo
b 30.110 0.70 75 3.40 240 24 2xlOlo
c 26.818 0.70 70 3.40 330 12 5x1011
d 17.8681 0.50 35 3.40 280 18 6x1010
Region
B
a 32.850 0.80 52 3.0 300 13 3x1011
b 30.287 0.80 45 3.0 430 9 1x1012
c 11.261 0.20 6 8.0 360 6 >1x1012
d 9.830 0.80 4 10.0 100 4 >3x1012
f) Application of the mufti-parameter best-fit PTR method for
Thermoelectronic characterization of p-Si wafers annealed in the presence of
an electric field.
The exact nature of the Si-Si02 interface is not yet fully understood. A
simple picture of
thermal oxidation is single crystalline silicon followed by a monolayer of
SiOX, that is
incompletely oxidized silicon. This, in turn, is followed by a strained region
roughly 10-40 ~
thick, and the remainder is an overlayer of stoichiometric, strain-free,
amorphous Si02. For a
typical oxide semiconductor (OS) device, interface traps and oxide charges
exist that will, in one
way or another affect the operating characteristics of devices, such as metal
oxide
36
CA 02305477 2000-04-17
semiconductors (MOS). The basic classification of these traps and charges
involves [D. K.
Schroder, Semiconductor Material and Device Characterization (Wiley, New York,
1998);
Chap. 6, pp. 337-419.]: a) Interface trapped charges, Q;t, located at the Si-
Si02 interface with
energy states within the silicon bandgap. This type of charge exists in
interface (or surface)
electronic states, which can exchange charges with silicon in a short time.
Q;t can possibly be
produced by excess silicon (trivalent silicon), excess oxygen, and impurities.
The interface trap
states occupied by Q;t can be affected by the surface potential and can become
charged or
discharged. b) Fixed oxide charges, Qf, located at, or near, the interface.
These charges are
immobile under an applied electric field and do not interact through exchanges
with the
underlying Si lattice. c) Oxide trapped charge, Q°t, due to holes or
electrons trapped in the bulk
of the oxide. This charge does not normally interact with the underlying
silicon, under room-
temperature conditions. It can be annealed by allow-temperature
(<500°C) treatments. Finally, d)
mobile ionic charges, Qm, such as Na+, K+, Li+ , as well as negative ions and
heavy metals. These
charges are, nevertheless, typically immobile below 500°C. Early
measurements on clean Si
wafer surfaces in ultra high vacuum have confirmed that the interface trap
density can be very
high -of the order of the density of surface atoms (1015 atoms/cm2) [F. G.
Allen and G. W.
Gobeli, Phys. Rev. 127, 50 (1962).]. The foregoing classification of the
various types of interface
charges is helpful in understanding the mechanism of the laser radiometric
results obtained under
the influence of an external electric field in the present work.
A typical configuration of thermal annealing of industrial Si wafers under an
applied
electric field is shown in Gross-section in Fig. 30. The electric field was
created by a voltage
difference between the external grounded metallic electrode and the SiC boat
at V= 0, +1000 V,
or -500 V. The vertically positioned Si wafers inside a quartz boat prevented
direct contact of
the cool OZ gas flow with the heated sample wafers inside the SiC boat and
were only used as 02
gas flow dispersion devices or "Si shields". The samples used in this work
were usually
positioned vertically in the standard silicon carbide boat as shown in Fig.
30. Samples were
heated using quartz lamps as radiation heaters. Three p-Si wafers, 6" in
diameter, were thus
studied, as follows.
1. Sample p-Si #1 was annealed without an electric field.
2. Sample p-Si #2 was annealed under an applied electric field of +1000V.
3. Sample p-Si #3 was annealed under an applied electric field of -SOOV.
Thermal annealing was performed in standard pressure and Oz flow rate of 500
cm3/min, at a
temperature of 1050 °C for 70 min. The furnace was equipped with a tube-
reactor of 10"
diameter and a pedestal; both made of fused quartz. The three foregoing wafers
were examined
under the PTR probe at four different positions across the front surface, and
at one position
located near the center of the back surface. The insert in Fig. 31 shows the
locations of these
points. Positions A - D were scanned along the radial direction from center to
edge at 1.8-cm
intervals. Regions I (grey) and II (white) refer to broadly observed
uniformities in the
distributions of both the carrier recombination lifetime and the surface
recombination velocity of
the p-Si#I wafer, which was annealed without the application of a static
electric field.
Figure 31 shows the PTR signal amplitude (a) and phase (b) of Sample p-Si#3
(annealed
with a positive electric field), for the four positions located as in insert.
The continuous lines
represent the best-fit results using the 3D-PTR-model/computational mufti-fit
parameter
methodology of the present invention. The thermal and electronic parameters
obtained at these
positions, as well as at the one position in the back surface for all the
samples studied, (carrier
37
CA 02305477 2000-04-17
de-excitation or recombination lifetime, i, minority carrier (electrons)
diffusion length, Dn, front
surface recombination velocity, Si, and thermal diffusivity, a) are presented
in Table V.
Table V. Thermoelectronic parameters of various wafers under various electric
field conditions
Position T ~,s S I (cm/s)D" (cmZ/s)a (cmz/s
Wafer p-Si #1
No Electric Field
A 28 2900 3.6 .55
B 28 2500 3.6 .55
C 33.5 3500 7.0 .57
D 33.5 3500 7.0 .SS
Back 32 3000 5.0 .SS
Wafer pSi #2
With Electric Field
(+1000 V)
A 53 4300 10 .70
B 54 4200 10 .70
C 52 4200 10.4 .75
D 51 4200 10.4 .70
Back 38 4500 6 .40
Wafer p-Si #3
With Electric Field
(-500 V)
A 45 2200 8.3 .65
B 46 1800 7.5 .65
C 45 1700 7. S .65
D 44 1900 7.5 .65
Back A 44 3200 6.0 .50
According with Fig. 31 (a) and (b) there exists good electronic uniformity
across this
wafer and the measured localized lifetimes lie between 28 and 33 p.s. The
results indicated that
there exist roughly two regions in this sample: Region I (central part of the
wafer) with relatively
short lifetime and short front surface recombination velocity, located at r <
3 cm from the center
(see inset in Fig. 31); and Region II located at r>3 cm, in which both the
lifetime and the surface
recombination velocity values are longer.
The same procedures were applied to other wafers. The lifetime improvement for
both
wafers thermally annealed in the presence of an electric field of either sign
is remarkable.
Between the two polarizations of the electric field, the positive bias was
most effective when
applied to the SiC boat. The back surface response was somewhat different.
Although the initial
value of the lifetime at the single measured point was not different from
those measured along
one radius of the front surface of the p-Si#1 wafer, the improvement upon the
application of the
electric field was not as strong as that on the front surface, and its
effectiveness with respect to
polarity was reversed.
The most striking trend with the surface recombination velocity results is the
significant
increase of S1 for the positive polarity of the field, and the also
significant decrease for the
negative polarity, both with respect to the zero-field case. The values of S
on the back surface of
the biased wafers p-Si#2 and p-Si#3 show the same relationship as the front-
surface values,
however, they both are higher than that in the zero-field case.
38
CA 02305477 2000-04-17
In conclusion, the methodology presented in this invention is able to measure
the
thermoelectronic parameter in samples annealed under the present of an
electric field. Using this
methodology it is also possible to detect differences between electronic
transport parameters due
the surface conditions.
g) Application of the multi-parameter best-fit PTR method for monitoring of
ion
implantation in Si with carrier plasma waves.
Ion implantation is a very important technological process in the modern
microelectronics industry. It is widely recognized that integrated circuit
performance and yield
are strongly dependent on the accuracy and uniformity of the implanted ion
dose. This is
specially true for some critical implantation steps such as the low-dose
implant adjustment of the
treshhold voltages of the integrated circuits.
For the purpose of this invention, we present quantitative experimental
results on the
sensitivity of the PTR set up and computational methodology (described
previously, see section
iv.c) to the implantation dose and energy.
The PTR measurements were obtained from the near-center region of some Si
wafers (B-
doped, p~ 14-24 Ohm-cm, thickness S 10-520 pm) implanted with phosphorus to
various doses
from 5x10'° ions/cm2 to 1x10'6 ions/cm2 at three different implantation
energies: 50, 100, and
150 keV. The phosphorus implantation was performed through a thin oxide layer
at room
temperature. PTR amplitude (a) and phase (b) experimental results for one non-
implanted silicon
wafer and silicon wafers implanted with P+ ions of SO keV energy to various
doses are shown in
Fig. 32. For the non-implanted wafer the PTR-amplitude exhibits carrier plasma
behaviour [ A.
Salnick, A. Mandelis, and C. Jean, Appl. Phys. Lett. 69, 2522 (1996); A.
Salnick, C. Jean and A.
Mandelis, Solid-State Electron. 41, 591 (1997); A. Mandelis, A. Othonos, C.
Christofides, and J.
Boussey-Said, J. Appl. Phys. 80, 5332 (1996)] with the PTR amplitude saturated
at low
modulation frequencies and the PTR phase tending to saturate at -90°
along the high frequency
edge. For the implanted wafers a smooth transition from the plasma-dominated
PTR-signal
behavior at low doses to the nearly pure thermal signal at high implantation
doses is observed
(see Fig. 32). The solid lines in Fig. 32 correspond to the best fit obtained
from the
computational multi-parameter fitting methodology discussed previously in this
invention. In the
carrier plasma-dominated frequency region, the PTR signal has been found to be
extremely
sensitive to the damage introduced by ion implantation even at low doses and
energies. At 10
kHz modulation frequency the difference between the PTR amplitudes from the
non-implanted
wafers and the wafer implanted with the lowest dose/energy (5x10'° cm2,
50 keV) is more than
one order of magnitude, thus allowing for the monitoring of ion implantation
with dose and/or
energy much lower than these values.
The variations of carrier lifetimes with implantation dose and energy are
shown in Fig.
33(a). These results were obtained from the quantitative analysis performed by
using the
computational methodology presented in this invention. As the implantation
dose/energy
increases, the carrier lifetime remains unchanged and equal to that of a
reference wafer (~ 10 its)
up to a threshold value of the dose 010'2 cm 2) and then starts to decrease
with a rate which is
implantation rate dependent. This effect is related to the fact that the PTR
technique is measuring
the photoexcited carrier lifetime in layers lying deeper than the thickness of
the implanted layer
(<1 p,m). Thus the value of i in implanted Si wafers is unaffected by damage
introduced by ion
39
CA 02305477 2000-04-17
implantation to the uppermost layer until the effective depth of the
electronically sensitive
defects significantly exceeds the thickness of the implanted layer at high
doses and/or energies.
The PTR amplitude in the plasma dominated region (10 kHz) as a function of the
implantation dose for various implantation energies is presented in Fig.33
(b). The lattice
damage induced by the ion beam causes the plasma wave signal to decrease below
that of the
non-implanted reference wafer in a pronounced manner. The functional
dependencies exhibited
by the data in Fig. 33 (b) are smooth, monotonically decreasing and gradual,
such that a very
good sensitivity of the PTR amplitude to the implantation dose is maintained
over more than five
orders of magnitude dose.
h) Application of the PTR/computational methodology to Scribeline
Characterization
of Integrated Circuits.
There are four basic operations performed on a wafer during the fabrication
process:
layering, patterning, doping and heat treatments. Layering is the operation
used to add thin layers
to the wafer surface. These layers are insulators, semiconductors or
conductors including
interconnects; they are made of different materials and are grown or deposited
by a variety of
techniques [P. Van Zant in: Microchip Fabrication , McGraw-Hill, New York,
1997, p. 102).
Patterning is the series of steps that result in the removal of selected
portions of the added layers.
This process is also known as photomasking, masking, photolithography, and
microlithography.
It is the patterning operation that creates the surface parts of the device
that make up a circuit.
This operation sets the critical dimensions of devices. Errors in the
patterning process can cause
changes in the electrical functionality of the devices and of the circuit.
Contamination in any and
all of the process steps can introduce defects. Contamination problems tend to
be .magnified by
the fact that patterning operations are performed on the wafer several times
in the course of the
wafer fabrication process.
Recombination lifetimes were monitored within the scribelines of various
processed wafers,
for reliable diagnostics of the onset of furnace (and/or other process)
contamination, with the
PTR computational methodology discussed in this invention. The samples used in
this work were
four 4" wafers of p-type Si, with patterned device structures. The wafers had
been oxidized with
a 1000-~r gate oxide. Polycrystalline Si (polysilicon) was deposited and
patterned to form pads
of different sizes and shapes. Figure 34(a) shows an optical microscope
photograph of two
different sizes of scribelines. One type of scribeline was 120 ~m in width;
the other type was
about 68 p.m wide. Throughout this work, bright structures in photographs are
(light scattering)
poly-Si pads or devices, whereas darker regions are the more highly reflecting
Si02 - Si
interfaces. Figure 34(b) is a schematic of the Gross-sectional geometry of the
wafer. The poly-Si
layer thickness was about 4500 t~.
Radiometric images were generated using a manual scanning system. These images
are
PTR amplitude and phase scans at a fixed laser-beam-intensity modulation
frequency. We have
shown that the amplitude scales linearly with the recombination lifetime in
some ranges of
parameters, Fig.2l. Therefore, an x-y amplitude scan of Si substrate (with or
without the
presence of oxide) when it is properly calibrated in units of ~s, yields, in
principle, a
recombination lifetime image of the scanned region. Such a radiometric image
has been called a
"thermoelectronic image ", or "thermoelectronic scan ". The resolution of each
spot was 20 pm.
Beam size was estimated to be 48 p.m using a CCD camera and optical scan
measurements
through a 5-p.m pinhole. The laser power on the wafer was about 40 mW, which
corresponds to a
CA 02305477 2000-04-17
low injection level. On scanning across typical wafer structures shown in Fig.
35, the CCD
camera was used to determine the desired location and to guide the laser beam
inside or around
the neighborhood of a given scribeline.
The sample wafers were scanned along and across a scribeline, through poly-Si
and
oxide-covered regions. Figure 35 shows the topology of a typical small area
near the crossing of
two scribelines, one of which contains test inserts. Frequency scans were
carried out at six
locations (a - f): three (a, b, c) across the insert-free scribeline (120-~m-
wide) very near the
crossing point and within the silicon oxide region; and three (d,e,f) at
various poly-Si locations.
The purpose of these scans was to explore the capabilities of PTR for
measuring recombination
lifetimes in and around scribeline locations with the goal of using these
values as very
convenient benchmarks for wafer contamination monitoring during (or after)
processing.
Figure 36 shows the PTR frequency amplitude and phase obtained at the six
locations of
Fig. 35: three scans (open symbols) were performed through the oxide layer
outside the
scribeline {points a, c, f in Fig. 35); and three more scans (solid symbols)
on three poly-Si pads
of different sizes and 4500-A thickness scribeline (points d, e, and f in Fig.
35). One scan was
performed inside the scribeline (point b on the straight line A in Fig. 35).
Continuous lines {over
the open symbols) represent the multi-parameter best fits of the experimental
data for Si02
locations using the 3-D PTR/computational method described in this invention.
Point (d) lies
inside a small poly-Si pad (150 ~m x 150 pm) close to the scribe line. Point
(e) lies inside a
larger poly-Si pad (300 p,m x 300 p.m). Finally, point (f) lies inside a wide
poly-Si strip
perpendicular to the investigated scribeline. It is interesting to see that
the PTR signals from
these poly-Si regions exhibit different behavior in Fig. 36. The smallest pad
(d) and the
intermediate size pad (e) exhibit both thermal and carrier plasma behavior,
with the free-carrier
contributions clearly appearing at frequencies > 500 Hz. On the other hand,
the poly-Si strip (fj
exhibits purely thermal contribution throughout the entire frequency range 10
Hz - 100 kHz.
These systems consist of three layers (Si substrate + gate Si02 + poly-Si).
For this reason, it was
not possible to fit the experimental data using our 3D-PTR model, which
predicts the response of
a single-layer Si wafer. An extension of the single-layer PTR model to
electronically active
multi-layers along the lines of our earlier theoretical treatment of modulated
thermoreflectance
signals from similar geometries is currently under development. The fact that
the responses of
these three poly-Si layers were different may be likely due to the
increasingly greater depletion
of the photoexcited free carrier contribution with increasing lateral
dimensions of the layer.
Earlier work [A. Salnick, C. Jean and A. Mandelis, Solid-State Electron., 41,
591, 1997] showed
that lifetime measurements through poly-Si layers in MOS capacitor structures
are possible,
since any photoexcited carrier within the Si substrate can emit infrared
radiation through the pad,
which can be captured by the PTR detection electronics. Under this hypothesis,
scattered light
propagating across a small-size poly-Si pad can eventually penetrate the oxide-
Si underlayer and
generate carriers, thus creating the partly plasma-wave scan of Fig. 36,
curves (d, e). As the
lateral dimensions of the poly-Si layer increase, the probability of the
essentially spherically
propagating scattered photons reaching the substrate diminishes due to the
increased lateral
(radial) scattering in the polysilicon. As a result, more of the incident
optical power is converted
readily into heat within the poly-Si layer, creating the purely thermal scan
of Fig. 36, curve (f).
This purely thermal behavior of the strip (fj is consistent with the expected
very high optical-to-
thermal (nonradiative) conversion efficiency of the laser radiation in the
presence of a very high
density of free thermal carriers within the poly-Si layer (nearly metallic
behavior).
41
CA 02305477 2000-04-17
Electronic parameters for the probed Si02 locations (a-c) are shown in Table
VI.
According to these results the Si02 regions inside and around the scribeline
exhibit the same
lifetime (~35 p,s), without any great variations in the values of the Garner
dii~usion coefficient.
Surface recombination velocity S1, however, does vary substantially as a
function of location. No
transient behavior was found in any of the examined locations, thus indicating
good quality
surfaces with low surface defect state densities, as expected from oxidized
Si02 - Si interfaces.
Table VI. Electronic and thermal parameters for three points located across
the scribeline
see Fi . 3 5 .
Location Amplitudea i D" S1
(cmz/s)(~.s)(cmz/s)(cm/s)
a 10.613 0.55 36 2.0 440_
b 10.475 0.55 35 2.4 630
c 9.730 0.55 35 2.0 600
Thermoelectronic Images. One region close to the central part of a wafer was
scanned
with a step of 20 p.m, 300 p.m x 340 ~m in area (Region A in Fig 34(a),
located near the center of
the wafer. This region was chosen for PTR scanning imaging because it
encompasses a square
poly-Si test pad, a poly-Si scribeline rim, and half the width of a 120-p,m
wide scribeline. The
PTR signal amplitude and phase images of this region obtained at the
modulation frequency of 1
kHz are shown in Figs. 37(a) and 37(b), respectively. The low amplitude and
large phase lag
values correspond to the poly-Si regions as expected from the efficient
nonradiative conversion
at this location leading to the absence of measurable photoexcited free-
carrier densities and the
domination of the signal by the thermal-wave component within the detection
area of the infrared
detector. On the other hand, high amplitude and small phase-lag values are
associated with direct
probing of the Si - Si02 interfaces, due to the domination of the PTR signal
by the free-carrier
plasma component. These reszrlts demonstrate that PTR imaging can be used to
ident j~ areas of
high and low electronic activity across a particz~lar patterned region of a
processed wafer. It is
important to note that neither the amplitude nor the phase levels inside the
half scribeline image
are equal to those from the oxide patches outside the scribeline.
iii) PTR Depth Profilometry and PTR Multi-Layer metrology by heuristically
eliminating roughness
In the reconstruction of depth profiles of thermophysical properties in
solids, there are
two aspects to consider: (a) a forward problem (theoretical model) must be
formulated; and (b)
an inverse method (numerical model) must be applied to retrieve the inverse
variable (thermal
diffusivity).
(a) Theoretical model for discrete homogeneous layer on a semi-infinite
inhomogeneous layer
42
CA 02305477 2000-04-17
The regions surrounding the investigated layer are an air-solid interface and
a solid-
backing interface as shown in Figure 38.The a.c. temperature fields in each
region air (a), rough
layer (1) and investigated inhomogeneous layer (2) are:
TQ(x, cc~) = D a°°(x+,~
x<-d~ x+d_<0(31a)
Tl (X, co) = B a°~x + Ce °~x ; - d < x < 0 (31b)
T2 (x ~) = 1 (C' 1 eHz(x) - C'2 a Hz (x) ] ~ 0 <- x < 00
' 2 e2 (x - (31 c)
Equation (31 a) is the bounded (finite as x -' ~°) solution to the
thermal-wave equation for
homogeneous semi-infinite medium [A. Mandelis, J. Math. Phys. 26, 2676 (1985)]
and equation
(31b) is the solution for a finite homogeneous region. In equation (31a) and
(31b) a~ is the
complex wave number defined in equation (4) with a.~ the thermal diffusivity
of the j-th medium
(j:0,1). Equation (31c) is the result of a treatment of the inhomogeneous
layer thermal wave field
in terms of the Hamilton-Jacobi formulation as shown by A. Mandelis, J. Math.
Phys. 26, 2676
(1985) and applying the appropriate subscript (2)(see Fig. 38) to the
expressions for identifying
the investigated layer. Constants D, B and C depend on the boundary and
limiting conditions of
the system and Cl and C2 are as defined by A. Mandelis, J. Math. Phys. 26,
2676 (1985).
The boundary conditions for the investigated region at x=-d, 0 are from
continuity of
temperature and heat flux:
Tl (x - d, a~) = T°(x = -d, co), (32a)
-ki dT I (x d' ~) + ko dT°(x = -d, co) i e~~r
dx dx = 2 Qo
(32b)
T' (x = 0' ~) = T 2 (x = 0' ~)~ (32c)
kl dTl (x = 0, co) - k dT2(x = 0, co)
2
(32d)
where Qo represents the thermal source fluence at the material surface [W/m2]
assuming 100%
laser power absorption. In the limit x -' °° the ac temperature,
T2(x) generated should be zero.
Applying this condition to equation (31c) yields,
p~ e-r~r4 -z° ~ 1 + e-~2(°°~
- 1 - e-~z(~)
. (33)
43
CA 02305477 2000-04-17
Substituting equation (33) to (31c) gives,
_ Zp g-Nz(x) - g-l~z(°~~H2(x)~
T2 (X) - ~ 1- e-~z(~)
e2(X)
(34)
To be used in the boundary conditions the first derivative of T2(x) with
respect to x is taken and
results in
C~ _ ZO Cl~ e-HZ(x) - e-12H2(ao)-Hz(x)1
T 2 (X) e2 X C~ ~ 1 - g-2H2(oo)
g-Hz(x) - e-~2H2(ao)-H2(x)l
+ T p ~ 1 - e-2H2 (~) ~ ~ ~e21 /2 (X) ] .
(35)
An approximation is now made in neglecting the second part of equation (34) by
setting the
thermal effusivity derivative equal to zero:
[~,21l2 (X) ] N O
(3 6)
This assumption amounts to a requirement for nonsteep local variations of the
effusivity. This is
easily satisfied when the thermophysical field is evaluated at small
incremental depth slices
where it is not expected that local steep diffusivity gradients may exist [A.
Mandelis, S.B. Peralta
and J. Thoen, J. Appl. Phys. 70 , 1761 (1991)]. Solving for the constants by
using the boundary
conditions and substituting in equation (34), the temperature distribution at
layer (2) becomes:
R2 (X) e-H2(x) - e-[2H2(oo)-H2(x)1
T '2 ~X~ - k2 (~)Q2 (~) [ 1 . a _2H2 (oo)
b21 (0)e-aid
X[ 1+b2 OF 1-b -2Q~d
1 ~ ) 2) - ( 21 (0)F2)e
(3 7)
where,
__ 1 + e-~z~°°>
F2 1- a 2Hz(~~ (38a)
b21(0 = k2(0)Q2(0) = b
kl ~ l 201
e2(~) (38b)
R2 (X) e2 (X) .
(38c)
44
CA 02305477 2000-04-17
In deriving equation (37) the air-solid interface was assumed negligible. This
is a valid
assumption since in most cases the thermal coupling coefficient bol '~ 1 (near
adiabatic
conditions). Similarly, by substitution, the temperature distribution in the
homogeneous layer (1)
from equation (31b) becomes:
~~~x+~ -E- r'21 (~)g~l~d x)
( ) _ QO
Tl x - 2k1 ~1 [ 1- T'21 {~)e-2ald (39)
where,
1 - b 21 ~~)F2
r21 (~) 1 + b21 (0)F2 , (40)
Although it will be seen that the results are valid for arbitrary thermal
diffusivity depth
profiles, for this analysis the following simple simulated functional
dependence of the solid
inhomogeneous region thermal diffusivity is assumed [see, A. Mandelis, F.
Funak and M.
Munidasa, J. Appl. Phys. 80, 5570 (1996)]:
a2(x) = as(x) = aoCl 1 ~0 x~ (41)
such that as(~) = a~, as(0) = ao
and
O = Q ~ - 1 _ (42)
The parameter q is a constant that determines the rate of thermophysical decay
if ao ~ a~ or
growth if ao ~ a~.
By defining a form for the inhomogeneous thermal diffusivity the integral for
H(x) [A.
Mandelis, J. Math. Phys. 26, 2676 (1985)] gives H2(°°) ''
G° which is also valid for a constant
homogeneous thermal diffusivity in layer (2). Thus from equation (38a) F2=1
and equation (40)
r21 (0) = y2i (0) = yzoi. The resulting temperature, for the inhomogeneous
layer (2) from equation
(37) simplifies to,
Qo R2{x) b21 (4) e~ld HZ(x)
T2(x) - k2(0)~2(0) (1 + b21(0)) - (1- b21{4))e-2~1a
Qo R2(x) e-a~a HZtx)
kl~l (1 +b21(0))(1-Y21(4)e-ZQ,d)~ (43)
CA 02305477 2000-04-17
The superposition principle is implemented in solving the complete expression
for the thermal
wave field in an inhomogeneous solid bounded by regions shown in Fig. 38.
According to this
principle, any complicate linear boundary-value problem can have a solution
written as a linear
combination of solutions to a number of simpler boundary value problems. The
general solution
of the thermal wave field for the regions shown in Fig. 38 is then,
T(x) = aT2(x, co) + bT°(x, co) + cT~(x, to), (44)
where To and T~ are the temperature distributions with constant thermal
diffusivities ao and a~
in layer (2) respectively, and the expressions are
_ Qo e-oid_ozox'
T°(x'~) klQ1(1 +b2o1)(1 -Y201e-2a~d)~ (45a)
_ ~0 2_a1d_az~ox
T~(x'~) - k10'1(1 +b2oo1)(1 -Y2cole-2ald)~ (45b)
where bzo, and yzo~ are as defined in equation (38b) and (40) (F2=1)
respectively. b2ol and y2ol
are defined similarly by replacing 0 with °° in equations (38b)
and (40) respectively.
Determination of the constants (a, b, c). Constants a, b and c are determined
by the various
limiting case requirements of the problem. In the limit of very large
distances from the surface,
x -' °°, equation (41 ) gives a constant diffusivity profile of
a~ and equation (44) leads to
lim{a T~~X' ~~ + b T fix' ~~ + c } = 1. 46a
°° ~ ( )
By substituting equations (43), (45a) and (45b) and by setting b=0 to satisfy
boundness results in
a = ( 1 _ c) Z a _QZ~~
R2 (x)
(46b)
where
Jco - 2q IIl(I Q2oo I), (46c)
- (1 + b201)(1 - y201e-2a~d)
(1 + b2~1)(1 - y2oole 2Q'd) (46d)
In the very high frequency limit ~ -' °°, the penetration depth
of the thermal wave is zero which
results in
46
CA 02305477 2000-04-17
T(0~ ~ -~ oo) = T°(0, c~). (47)
Substituting (46b) in equation (44) and since ~'2~ 1 °° as ~ '-'
°° it can be shown that
To (0~ ~)
c = T~(0, cv) (48)
In the very low frequency limit m -' 0, the penetration depth is infinite
resulting in
T(0, co -> 0) - T~(0, co). (49)
Substituting (46b) and (48) in equation (44) and since ~z~ -' 0 as w' 0 it can
be shown that
1 T°(0' cv) Z - T°°(0' cc~) = Z
T~ (0 ~ ~) ~ oo To (0~ ~)
R2( ) (so)
which results in
R2(°°) = 1. (51)
Finally, substituting all the determined constants from equations (48), (51)
in equation
(44) and calculating the field at the front surface x = -d
T(-d~ ~) - Qo 1 + y2oi a Z~,a
_2~1~ (1 + (Z- 1)e~z~'°° )
- 2kW 1- y2oie (52)
where d cannot -' d'
(b) Numerical method
Experimentally the amplitude and phase which correspond to the surface
temperature
distribution, T(o,~) are obtained. The theoretical values of the data pair are
calculated by
Z.(0~~) _ ~M(~)lera~(~)~ (53)
where M(co) is the amplitude and ~~(~) is the phase at an angular frequency
co. At each
frequency the amplitude, phase and the derivative of phase are use to
calculate ao, ar. and q.
Although a profile of the form in equation (41) is assumed, the actual profile
is updated at each
frequency by recalculating the parameters ao, aL and q. The reconstruction
method used to solve
for the parameters oco~~, ai,t;~ and q~ is a multidimensional secant method,
known as Broyden's
method [A. Mandelis, F. Funak and M. Munidasa, J. Appl. Phys. 80, 5570
(1996)], and is based
on minimizing the difference between the experimental and theoretical data for
amplitude, phase
as follows,
47
CA 02305477 2000-04-17
~M~(~i)~ - ~Mrh(~i)~ = 0, (54a)
~~~a(~i)~ - ~ ~~th(~i)~ = 0, (54b)
The calculation of the depth parameter x~ is performed based on the fact that
as modulation
frequency decreases the thermal wave probing depth increases. Starting at the
highest frequency,
w° the shortest depth is the shortest thermal diffusion length, i.e.
_ 2a°
x° - ~° (55)
The next (lower) frequency, w~+~ corresponds to an increased thermal wave
depth
tai - 2ai
xi+i = xi + ~y y
which is used to calculated a~+1 in Equation (41). Once the a~+i is calculated
the method returns
to calculate in recursive iteration the increased thermal wave depth as,
2aJ+1 2al
xi+i = xi + ~i+1 - ~i . (57)
In reconstructing depth profiles from data it is important to first find a
reliable set of initial
values for a°, aL, and q. This could be achieved by finding the best
theoretical fit (Eq. (52)-
forward problem) to the first few end points (high frequency) using a single
profile of the form
ofEquation (41) [M. Munidasa, F. Funak and A. Mandelis, J. Appl. Phys. 83,
3495(1998)].
(c) Instrumental System
The instrumental setup for this application is of low spatial resolution since
this is a one-
dimensional problem. The pump beam spot size is made much larger than the
maximum profile
depth to maintain the one-dimensional heat diffusion formalism assumed in the
theory. The
instrumental apparatus is shown in Fig. 39. An Ar+ laser, modulated by an
acousto-optic
modulator (AOM-Isomet 1201E-1), is directed onto the sample surface. The
emitted infrared
radiation from the sample is collected and focused by two Ag-coated, off axis
paraboloidal
mirrors onto a liquid nitrogen HgCdTe (Mercury-Cadmium-Telluride) infrared
detector (EG&G
Judson J15D12-M204-SOSOU) with an active area of lmmxlmm. The detector signal
is
preamplified before being sent to a lock-in amplifier (Stanford Research
System SR850), and the
outputs, amplitude and phase, are recorded at a range of laser frequencies.
The experimental
surface temperature response on the sample is normalized by the surface
temperature response of
a reference sample giving for each frequency an amplitude ratio and phase
difference. This
normalizing procedure is necessary for the correcting of all frequency
dependencies other that
due to the sample [A. Mandelis, F. Funak and M. Munidasa, J. Appl. Phys. 80,
5570 (1996)].
48
CA 02305477 2000-04-17
With this experimental arrangement, a dynamic experiment can be performed at
one
location on the sample. The experiment generates depth-dependent information
by scanning
acousto-optic modulator frequency ("a frequency scan"). Two channels of
information
(amplitude and phase) are then obtained.
(d) Experimental Results
The case hardening process of carburizing, which is the absorption and
diffusion of
carbon into solid ferrous alloys by heating, is examined. The microstructure
of the surface is
changed by the process, producing carbon gradients and therefore changing the
thermal
diffusivity of the surface layer. A preliminary study shows that there is an
anticorrelation
between the thermal diffusivity and the hardness of the treated layer [M.
Munidasa, F. Funak and
A. Mandelis, J. Appl. Phys. 83, 3495(1998)]. Another important factor to
examined is the effect
of roughness on the depth profiles since thermal profiles are influenced by
surface roughness.
Roughness is monitored at the high frequency and Figure 40 shows the different
responses for
two different roughness levels (200 grit and 600 grit). The roughest surface
shows a peak in the
phase data which affects the signal beyond the roughness depth and deviates
from the theory of a
homogeneous sample (constant phase). Therefore, it is necessary to use a
finite thickness layer
theory for depth reconstruction in order to obtain a reliable profile beyond
the depth of the
roughness. The effects of roughness are investigated and incorporated to the
experimental data.
At high frequencies the penetration depth is close to the surface so lateral
heat diffusion
is negligible but at low frequencies the penetration depth is deep into the
material and lateral heat
diffusion is pronounced. To ensure one-dimensionality the size of the beam
must be larger than
the deepest penetration. Not only is the beam the important consideration
here, but also the beam
shape. The laser source has a Gaussian profile so what is needed
experimentally is a top hat
distribution of the beam. To alleviate these problems a thick diffuser with a
lens is placed at the
path of the beam to broaden the beam and reduce its Gaussian profile. As the
beam is diffused
more both the amplitude and phase graphs approach one-dimensional theory. The
three-
dimensionality effects are, as expected, more pronounced at the low
frequencies.
Depth profiles of rough untreated AISI 8620 steels. With knowledge of the bulk
thermal diffusivity and the thickness of the surface roughness (600 and 200
grit) of an untreated
AISI 8620 a reconstruction is performed. The bulk thermal diffusivity was
measured
independently and was found to be a;nf-12.5x10-6m2/s. The reconstruction is
based on
reconstructing from the high frequency end by fitting the ao at the interface
with the geometry
(d= roughness thickness) shown in figure 36. With this method the effect of
surface roughness is
greatly reduced from the system. For the samples in question, the input
parameters of the 600 grit
roughness layer were thermal diffusivity, aa=2.15x10-6m2/s, thermal
conductivity kd=6.95W/mK
and independently measured thickness d=l.6pm and for the 200 grit the
parameters were
ad=2.15x10-6m2/s, thermal conductivity kd=4.6W/mK and independently measured
thickness
d=S~m. It is observed that as roughness increases the thermal conductivity of
the sample
decreases. The experimental data with the theoretical fit which assumes two
homogeneous layers
(roughness and bulk) is shown in figure 38. The forward theoretical fit is in
excellent agreement
with the experimental data. Small discrepancies exist at the high frequency
end where the
roughness is more difficult to model. A reconstruction of the untreated AISI
8620 sample was
49
CA 02305477 2000-04-17
then performed (figure 39). In a reconstruction the experimental data are
numerically inverted to
obtain the corresponding thermal diffusivity profile.
The simulation theoretically eliminates the roughness layer, which is assumed
to be
homogeneous with low thermal parameters, thus the reconstruction shown above
commences
after the roughness layer. It is seen from the reconstruction that the
thermophysical properties are
disturbed up to about SOp.m and that the bulk material is undisturbed
approaching the
experimentally independent measured value of a=12.5 x10-6m2/s. The near
surface fluctuation
can be attributed to the insufficient modeling of the roughness. This is not a
relevant issue for
hardness measurements since the interest is usually above SOpm. Such a
reconstruction can serve
as a guide to determine the extent roughness influences a specific profile. As
roughness increases
the reliability of the reconstruction is less. The reason for this is dual:
(1) the forward model is
not represented well in the higher frequency spectrum and the randomness of
roughness is more
evident, (2) the ill-posedness of the inverse problem increases since more
degrees of freedom are
introduced.
(e) Heuristic approach to eliminate roughness from experimental data
The method outlined above although effective for small roughness scale can be
erroneous
for the larger roughness as it appears in the signal response. With the idea
that roughness is a
random system, the effect of inhomogeneity and roughness is investigated. In a
frequency
domain method both the roughness and the inhomogeneity is felt throughout the
spectrum. A
simplistic approach of deconvolving the roughness from the inhomogeneity would
not be valid
since this is a non-linear system. The theoretical model represents roughness
as a constant layer
on top of an inhomogeneity and with a low-level roughness the results are
satisfactory as is seen
in Fig. 40. As the level (thickness) of roughness increases, the thermal-wave
spectrum becomes
more involved, especially at high frequencies resulting in an erroneous
thermal diffusivity
profile. In this application of PTR diagnostics a heuristic approach is taken
and tested for various
levels of roughness and inhomogeneity. The theoretical results show great
promise and as a
result the method is implemented to reconstruct experimental data.
The roughness method is based on recognizing distinct features (phase maxima)
from the
frequency spectrum. Since roughness is associated with the surface of a sample
the effects are
seen the strongest at high frequencies whereas the low frequency is mostly
related to substrate
inhogeneities. The objective of the method is to deconvolve the roughness
spectrum from the
underliying profile (homogeneous or inhomogeneous). To demonstrate the method
simulations
of an inhomogeneous profile using a single profile of the form of equation
(41) as derived in
equation (52), with three roughness cases were made. Figure 42 shows the
amplitude and phase
of case 1. Curve 1 shows the response of an inhomogeneous sample with
roughness, d=1.6p,m.
Curve 2 is the inhomogeneous field with no roughness. This would represent the
ideal
experimental situation where no roughness effects are present. Curve 3 is the
homogeneous field
with only roughness. In order to retrieve and eliminate roughness from curve 1
a theoretical
fitting to the higher frequency end which is associated with the roughness is
made. An effective
thermal conductivity for roughness is found to model the high frequency phase
response. This is
curve 4 and it requires that an effective thermal conductivity is modified
from curve 3. Curve 4 is
an effective homogeneous field similar to 3 but with higher thermal
conductivity. Table VI
identifies the thermal properties for all three cases under conditions l, 2, 3
and 4. Finally, the
effective curve 4 is normalized with the total field 1. The response is as
follows:
CA 02305477 2000-04-17
IMrorat(~)I e'°~~o~d(~) _ ~ Mrorat(~) ~e~(e~r~d(~r~~~)1
Tfinat~~(0~ ~) _ ~MeH(~)l e~e~ytm) Mefl(W) , (58)
where the each temperature distribution is as defined in equation (52). Curve
5 is the result of the
operation of equation (58) with dividing for the amplitude and subtracting for
the phase. In this
way the desired result of the inhomogeneity with no roughness is obtained.
Comparing with
curve B these two results are in excellent agreement with each other. The
method is then tested
for a higher level roughness. Figure 43, shows case 2 where the roughness is
d=7~,m. Apart from
the roughness thickness the thermal properties are identical to case 1 as seen
in table VII. The
final result (curve 5) is in good agreement with the expected theoretical
value (curve 2) with
small deviations in the low frequency end. The interesting observation is that
the effective
thermal properties of curve 4 of case 2 are identical to case 1. A more
complicated situation is
then examined where the amplitude and phase does not show any characteristic
maximum to
distinguish from the inhomogeneities (Fig. 44). The knowledge that the same
inhomogeneities
affect the roughness spectrum in a similar manner is used in this case. Curve
4 is constructed
with the same effective properties are in case 1 and 2. The result is that for
this case the
deviations from the theoretical value are satisfactory with small variations.
C..:..~.,.~nf,nne eftnWn FltlC 4~-42
'fable v 11: 1 ~ v m m~ ~l J==.u.~.~~~.... ...- - -
nermar -a_. - Case 3
Case 2
Thermal Pro ertiesCase 1 _ aL=6.15x10-6m2/s
2/ 15x10-6m2/s
~ =6
m . 0x10-6m2/s
s aL =4
aL=6.15x10 2/ a
2 ~
-6
m s .
/s m o
ao=4.0x10 ao=4.Ox10 3
A: Total field q=2x103 q=2x103 q=2x10
ad=2.1x10~m2/s ad=2.1x10~m2/s aa=2.1x10~m2/s
kd=4 . 8 W/mk kd=4. 8 W/mk kd=4. 8 W/mk
d=1.6 m d=7 m d=13 m
15x10-6m2/s aL=6.15x10-6m2/saL=6.15x10'~m2/s
=6
B:Inhomogeneity aL ao=4.0x10-6m2/s ao=4.0x10-6m2/s
. 3
ao=4.0x10-6m2/s
q=2x103 q=2x103 q=2x10
d=O~.m d=0 m d=0 m
15x10-6m2/s aL=6.15x10-6m2/saL=6.15x10~m2/s
a
=6
L ao=6.15x10~m2/s ao=6.15x10~m2/s
.
15x10~m2/s
=6
C: Roughness ao aa=2.1x10-6m2/s ad=2.1x10'~m2/s
.
ad=2.1x10-6m2/s
kd=4. 8 W/mk kd=4. 8 W/mk kd=4.8 W/mk
d=1.6 m d=7 m d=13 m
D: Effective ker,=5.96W/mK ke,~=5.96W/mK ke,~=5.96W/mK
Although the above method proves to be very effective in theoretical
application of
inhomogeneous substrate with a rough layer, a more general expression for
modeling the
roughness can be obtained. Since roughness can be viewed as a random effect a
Gaussian noise
is fitted to the effective frequency-domain roughness profile (curve 4). The
field created has to
be a non-symmetrical field and thus the expression for amplitude and phase
respectively, is as
follows:
st
CA 02305477 2000-04-17
(~0 - ~ l ci) z
Me~(~o) = Mo + ~ ~ W~ri a ~~i (59)
(~o - cv2ci)z
~c~e~(u~o) = O~o + ~ ~ W2 a wz 60
( )
where Mo and O~o are the amplitude and phase offsets, W is the width, A is the
area and c~~ is
the center of the Gaussian. The summation of Gaussians is greater than one so
that the
asymmetry of the field is obtained. For more accurate results, the offset
values can be derived
from the first point of theoretical fit of the effective roughness. This can
only be an
approximation. The method is then applied to experimental data. Figure 45
shows the
elimination of roughness on experimental data. The sample has roughness of
1.6~m on an
inhomogeneous substrate. A Gaussian fit is performed on the roughness part of
the data. The
gaussian needed to perform such on operation on these data is a double
Gaussian whose values
can be found in table VIII. Figure 46 shows the method of elimination on
rougher data (d=Sp,m)
with an inhomogeneous substrate. Although the phase roughness is fitted to a
double Gaussian as
above, the amplitude for this data requires a summation of three Gaussians to
satisfactory
perform the operation. A double Gaussian would have been able to fit the data
but the higher
frequencies would have suffered from deviation from the experimental data. The
values for the
Gaussians can be found in table VIII.
Table VIII Gaussian fit parameters of experimental data shown in Figs. 46 and
47.
Gaussian sample 22 sample 12
Fit
_ __
Am litude Phase Am litude Phase
0 0.98 0.08 0.98 0.11
xcl 4.89 4.61 4.81 4.96
w 1 3 .69 2.03 0.81 2.28
A1 0.29 21.29 0.12 41.86
xc2 5.06 8.47 4.34 5. 81
w2 1.51 3.09 1.32 2.03
A2 0. 54 -429.12 0.29 -82.02
xc3 - - 3.3 8 -
w3 - - 2.47 -
A3 - - 0.13 -
Depth profiles of carburized AISI 8620 steels through a roughness layer. A
sample
matrix is constructed as a function of roughness and case hardness depth.
Samples with two
levels of roughness (200 grit and 600 grit) were carburized at three different
depths (0.02", 0.04"
and 0.06"). The sample matrix is shown in Table VIII. The samples are AISI
8620 steel alloy
from the same slab. Experimental frequency scans on the samples were taken on
the rough
surfaces before (Fig. 40) and after the case hardening process (Fig. 47).
Above 1000Hz strong
52
CA 02305477 2000-04-17
effects of roughness are seen. Comparing these data with the untreated ones
(Fig. 40) it is seen
that the phase shift has decreased. This can be attributed to the fact that
the thermal properties of
this layer have changed after carburizing. Roughness elimination is performed
on all the data as
seen in Fig. 48. The success of the method is clearly seen here where two
different levels of
roughness result in the same inhomogeneous experimental response. The result
is consistent for
all the inhomogeneous depths as seen in Fig. 48.
The reconstructions at the three depths are shown in Fig 49. This figure also
includes the
conventional microhardness test. The depth profiles of the hardened samples
exhibit an
anticorrelation between thermal diffusivity and hardness. It is seen that a
good one-to-one
correspondence between hardness and thermal diffusivity is present although
the curves are not
each other's mirror images. The anticorrelation relationship is consistent
with earlier results
produced [T.T.N. Lan, U. Seidel and H.G. Walther, J. Appl. Phys. 77, 4739
(1995); M.
Munidasa, F. Funak and A. Mandelis, J. Appl. Phys. 83 (5) 3495(1998)].
Thermal wave depth profilometry can be an invaluable application to surface
treatment
processes such as case hardening. In this process, important AISI steel types
underwent
industrially commonly used case hardening process and then a complete
experimental and
theoretical/computational analysis generated thermal diffusivity profiles. The
elimination of
roughness has been shown to be an important method of improving the
experimental data and
thus the reconstruction. The current methods used to characterize case
hardening are destructive
and therefore success in developing a correlation (anti-correlation) between
hardness and thermal
diffusivity profiles would mean a significant contribution to the steel
industry. An anticorrelation
between the thermal diffusivity profile of a hardened surface and its
microhardness is found.
Many approaches of the thermal diffusivity depth profiling have been
introduced over the years
with all the methods suffering from non-uniqueness, a distinct characteristic
of ill-posed
problems. By eliminating roughness the ill-posedness of the problem is
reduced.
Table IX: AISI 8620 steel sample matrix.
Case de thlRou ~ 0.02" 0.04" 0.06"
h
600 grit sample 22 sample 26 sample 28
200 grit sample 12 sample 16 sample 18
Test sam le 34 sam le 35 sam le 36
Thermal Spray coating roughness application. Thermal sprayed coatings of 316
stainless steel was applied to 9.5 mm thick, 1018 steel rectangular bars. The
stainless steel were
applied using the high velocity oxy-fuel (HVOF) process with the JP-5000 spray
system. In
order to account for the instrumental frequency dependence, the PTR signal of
a Zr alloy
reference sample was measured. For the low frequency range ( 1 to 1000 Hz) a
defocused beam
(~6 mm diameter after the diffuser) was used to minimize three- dimensional
effects of the heat
diffusion. A bare laser beam (~l mm diameter) was used for the higher
frequency range (1 to
100 kHz). All measured PTR signals from the thermal sprayed coatings were
normalized to the
Zr alloy reference sample.
53
CA 02305477 2000-04-17
The amplitude (a) and phase (b) of the normalized PTR signal of a stainless
steel sample
are shown in Fig. 50. The frequency structure for both signals in amplitude
and phase is
dependent on the thermophysical and geometrical properties of the sample. This
signal
frequency-structure is due to thermal-wave interference resulting from
coherent energy
confinement within the spray coating layer. At higher frequencies the surface
effects become
more dominant and the observed structure is more likely due to roughness
effects [J. A. Garcia,
A. Mandelis, B. Farahbaksh and C. Lewitz, Int. J. Thermophysics, 20, 5, 1999].
The roughness
elimination method was applied to this sample and the resultant corrected
experimental data (see
Fig. 51) was then fitted with a one-dimensional two-layer model. In this
instance one has two
channels of information, amplitude and phase and two unknown parameters, the
thermal
conductivity (k2) and diffusivity (a2) of the upper layer. These properties
can then be determined
uniquely and then correlated to the condition of the coating.
v) PTR application to micro-welds
The apparatus described in this invention (see Fig. 1) was used for frequency-
domain
PTR of gold/aluminum microjoints. Experimental PTR frequency scans as well as
PTR imaging
have been obtained for two sets of samples (8f2 and 4fZ). Samples 8f2, labeled
"good bonds",
had gold wire (25 microns diameter) bonded to aluminum foil of 0.1 mm
thickness. The force
and frequency used in the 8f1 bonds were 90 gf and 60kHz respectively. Samples
4f2, "bad
bonds", had the same material combination and same bonding parameters except
for the bonding
force which was 90 gf. Fig. 52 shows CCD camera images of four pins (gold
wire/aluminum
substrate) from samples 8fz. Pins 1-to-3 from sample 8fZ and pins 3-to-5 from
sample 4f2 (CCD
picture not shown) were examined using PTR.
A typical PTR image (amplitude and phase) of pin 1 of sample 8f2 is shown in
Fig. 53.
Then PTR frequency scans were performed on the aluminum foil at approximately
100-to-200
microns away from the edge of the splat for each micro- weld as well as 50
microns inside the
splat (see Gross-hair in Fig. 53).
The frequency scans performed at 50 microns inside the splat showed
significant
differences between the poor (60 gf)(frequency scan not shown) and good (90
gf) bondings, Fig.
54. The minimum in the phase-frequency scan was more pronounced for the poor
bonds when
compared to the minimum in the phase-frequency scan of the good bonds. This
result indicated
differences in interfacial conditions that may be related to thermal
resistances between the two
types of samples. The maximum change in phase (obtained from the PTR images)
of various
pins in sample 4f2 and 8fZ were 41-to-62 degrees and 55-to-85 degrees
respectively. These
results indicated that the information given by the PTR phase image can be
used to correlate to
the quality of the bond (bonding force).
In summary there is provided a metrologic methodology comprising of novel
combinations of new signal generation and analysis techniques, computational
software, and
photothermal radiometric instrumental configurations for measuring thermal and
electronic
properties of industrial semiconductor wafers and engineering materials.
The combination of frequency sweep ('Chirp ") and frequency scan methodology
for
rapid measurement of electronic and thermal transport properties of
semiconductor and
engineering materials presented in this invention involves providing a sample
such as a
54
CA 02305477 2000-04-17
semiconductor wafer or other engineering material, irradiating the sample with
an excitation
source (laser or other sources), generating a square-wave chirp from a dual-
channel fast Fourier
transform (FFT) analyzer to drive an acousto-optic modulator and produce
periodic frequency
sweeps (Chirps) of the laser beam in the range including (but not confined to)
do to 100 kHz,
generating a transfer function, H(f), by fitting the frequency-scan data from
a silicon wafer with
well known electronic and thermal parameters to a theoretical model, computing
the necessary
corrections to the amplitude and phase signal, and storing them in the FFT
analyzer and in a
personal computer, fitting the obtained signal from arbitrary semiconductor
samples to the same
theoretical model of the photothermal response, corrected for the instrumental
transfer function
to obtain the thermal and electronic parameters of these samples. This same
methodology can be
used for generating fast photothermal frequency scans from multilayered
inhomogeneous
materials, such as thermal barrier coatings and hardened steels.
The common rejection mode (dual pulse) methodology for detection of very weak
inhomogeneities among materials involves: providing a sample of the material,
irradiating the
sample with an optical or otherwise excitation source of thermal waves,
generating a real time
periodic waveform consisting of two incident pulses, detecting the signal
(photothermal) and
feeding it to a lock-in amplifier. This methodology is not confined to thermal-
wave signal
generation, but encompasses all manner of modulated signals, such as acoustic,
optical,
ultrasonic, X-rays and other signal generation method accessible to those
skilled in the art.
The multi-parameter computational methodology for determining a unique set of
thermal
and electronic parameters of industrial semiconductor (i.e. Si) wafers, from
frequency domain
measurements, involves: providing a semiconductor wafer (or sample),
irradiating the sample
with a periodic optical (laser) or other free-carrier raising energy source
generating a
photothermal signal, detecting said photothermal signal, inputting said signal
to a lock-in
amplifier, storing the frequency scans in a personal computer, and applying
the multiparameter
fitting procedure (by means of an electronic sheet or any other code program,
i.e C, Fortran).
The depth profilometry and roughness elimination method for determining
thermal
diffusivity profiles of rough samples involves: (a) providing a sample of
process-related
inhomogeneous material or multi-layer structures; (b) irradiating the sample
with a periodically
excited source (laser); (c) detecting the photothermal frequency sweep signal
with a lock-in
amplifier and storing the experimental data in a personal computer; (d)
processing the
experimental data with a heuristic approach to roughness so as to eliminate
the effects of
roughness; (e) applying to the processed data the theoretical/computational
model to reconstruct
the thermal diffusivity profile.
While these methodologies and some of their applications have been described
and
illustrated with respect to an embodiment of some radiometric instrumental
arrangements, it will
be appreciated that numerous variations of the instrument/methods may be made
without
departing from the scope of this invention.
