Indian Journal of Pure & Applied Physics
Vol. 43, August 2005, pp. 596-601
Simple diffusion wave analyzer
K Sreekumar
Laboratory of Photothermal Sciences, Department of Physics, University of Kerala, Thiruvananthapuram 695 581
Received 17 December 2004; accepted 7 June 2005
A simple, cost-effective, compact and efficient photothermal diffusion wave analyzer for 1 Hz-150 kHz frequency band
has been developed and analyzed. The sensitivity is 12 nV and the total error is about 1.5% at 60 dB dynamic reserve,
excluding the laser intensity variation that may be corrected. The input voltage and current noises are 15nV/ Hz and
2 fA / Hz , respectively. The low power consumption of about 4 W is advantageous for long time off-line applications. The
instrument has been used for the metrology of silicon wafers by radiometric simulations. Results regarding the studies on
transport parameters and contamination control are presented.
Keywords: Diffusion waves, Photothermal radiometry, Laser diode, Wafer metrology
IPC code: H01L29/861
1 Introduction
Diffusion waves arise when the classical diffusion
equation is coupled to an oscillatory force function
such that the time derivative is only first order. They
lack wave fronts, cannot be beamed and do not travel
very far, yet form the basis of several revolutionary
measurement technologies1-3. The most common
oscillatory driving function is a modulated light
source and the method branches to photothermal (PT)
diffusion wave sciences. Spectroscopy of thin film
semiconductors, opaque samples and biological
specimens, surface and subsurface imaging and
tomography, quantum yield studies of photonic
materials, characterization and quality control of
semiconductors for micro and nano electronics
applications are a few key areas illustrating how this
unique technique leads to remarkable advances in the
characterization and metrology of materials1.
In this paper, we present the design, performance
analysis and some applications of a laser
photothermal diffusion wave analyzer (DWA). The
instrument is simple, cost effective, compact and
efficient with performance figures comparable with
those of an average commercial assembly. This
analog design needs only locally available
components and very little skill for implementation
compared to its digital counterpart, which is more
stable and drift free.
2 Design of the DWA
The essential sections of the analyzer are a
modulated laser source, source of modulation with
sinusoidal and orthogonal outputs, low noise ac
amplifier and a lock-in or phase sensitive detector
(PSD) for signal processing (Fig. 1).
2.1 Modulated laser source
The source of excitation (pump) is a laser diode
(LD) that can be modulated over a wide bandwidth.
Circuits around U6, U7 and Q2 (Fig. 1) form a
grounded-load constant current source whose output
current is
I L = Rf V1 / Ri Rs [A]
…(1)
that drives the laser diode4. Shunting MOSFET Q3
causes sinusoidal negative modulation of the laser
intensity without exceeding the preset peak power of
the laser diode. The photodiode (built inside LD
module) current is sampled using U8 and U9 for
correcting against any possible intensity fluctuations.
Conventional automatic power control (APC) scheme
is not employed, as at lower frequencies (say, a few
Hertz) the feedback integration time needed is too
long to cause overshoot in the laser current that may
permanently damage the diode. The load current is
adjusted using V1. The soft-start circuit (Q4 and Q5)
offers protection from ON/OFF transients.
The maximum diode current is about 200 mA with
the present circuit and can be delivered up to a load of
about 60 Ω. We have demonstrated the driver with an
LNCQ-05 PS (Panasonic) laser diode (660 nm, 50
mW) whose operating current is 75 mA and threshold
current is 35 mA. At 1 kHz, about 95% sinusoidal
597
Fig. 1—Circuit schematic of the DWA
SREEKUMAR: SIMPLE DIFFUSION WAVE ANALYZER
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INDIAN J PURE & APPL PHYS, VOL 43, AUGUST 2005
modulation is achieved with 0.2V (peak to peak)
signal when V1 is 2V. Over a frequency band of 1
Hz-50 kHz, the total harmonic distortion is below 2%,
at 100 kHz it is about 3.5% and is about 6% at 150
kHz. A multi-element glass lens with numerical
aperture of 0.476 is used for collimation5.
Operational trans-conductance amplifier U5
generates triangular and quadrature square wave
references. The triangular signal is sine converted for
laser modulation17. The integrating capacitor C is 100
nF for 1-100 Hz, 10 nF for 100-1000 Hz, 1 nF for 110 kHz and 0.1 nF up to 150 kHz. VR1 tunes the
frequency.
2.2 Signal processing: Phase sensitive or lock-in detection
The principle of PSD is well discussed in literature
and it has been shown that for the detection to be free
from harmonic errors (fundamental only response)
with a square wave reference (as in the present case),
the signal should be sinusoidal6. The sinusoidal laser
modulation is advantageous in this regard.
The front end is a high performance
instrumentation amplifier U1 with gain 10. The
overall stable gain of the cascaded amplifier can be
varied from 0.01 to 104 so that the full-scale (FS)
input sensitivity is adjustable from 1 V to 1 μV. The
50 and 100 Hz notch filters (optional) are included for
suppressing line frequency components17. Spice
simulation shows that at 1 kHz, the input voltage
noise is about 15nV/ Hz and current noise is about
2 fA / Hz for a gain of 10. The achieved common
mode rejection ratio (CMRR) is about 75 dB and it
decreases by 20 dB/decade (approximately) above 1
kHz. Ground loop minimizing techniques like floating
guard, double-shielded cabling etc. may be used for
improving CMRR (Ref. 7). The gain linearity error is
less than 1% over the entire bandwidth.
The suitably amplified signal is subjected to phase
sensitive detection in U2 for in-phase detection. The
second order low pass filter around U3 has variable
time constant determined by R with equivalent noise
bandwidth 1/ 8 RCT . The slew rate and offset current
considerations of U3 limit the time constant between
1 ms and 50 s and U4 provides sufficient output
expansion. A similar section is used for quadrature
detection.
The operating frequency is limited at the lower end
by the stability of the signal generator and the upper
end pump signal distortion and amplifier bandwidth
are the major factors.
3 Performance Analysis
3.1 Dynamic reserve and output stability
The transfer function of the detection system,
assuming gain 2 of U2, is
VO (rms ) = 2Gac GdcVS cos φ [V]
…(2)
where Gac and Gdc are the ac and dc gains,
respectively, φ is the phase of the signal with respect
to the reference and VS is the signal amplitude. For
calculating the dynamic reserve (DR), we have
considered the value of an asynchronous voltage that
causes 5% deviation in the FS output. The ratio of this
asynchronous amplitude to the signal amplitude for
FS reading is the DR. For an FS input of 10 mV
(amplitude) the said asynchronous input (from a
signal generator) is 10 V (amplitude) and the
DR is 60 dB. With 10 mV (rms) input, the DC gain
required for an output FS display of 10.00 (rms) is
500. DR can be increased by decreasing ac gain
while increasing dc gain, keeping their product
constant.
Output stability is principally determined by the
drift in the output-offset voltage of U2, which is about
5 μV/°C (Ref. 8). For a change of 10°C in the
operating temperature, this causes about 0.35% error
for 60 dB DR.
3.2 Total error, sensitivity and power consumption of the
DWA
The total error is evaluated with time constant
10 s and 1μV FS range for 60 dB DR over 10°C
change in operating temperature. 15nV/ Hz input
noise contributes about 2 nV (rms) output noise after
low pass filtering (equivalent noise bandwidth is
1/80Hz ) and the error is about 0.2%. After 60 s, the
exponential convergence of the filter adds about 0.3%
error. The gain error is 1% and the drift is
0.35%. Orthogonality and DC gain errors are
negligible. The rms of these uncorrelated error is
about 1.2% and the sensitivity is 12 nV. The drift in
the laser intensity for 5 hours continuous operation is
about 2% (may be compensated using the photodiode)
and the total error of the analyzer amounts to be
about 2.5%.
The power consumption of the DWA is about 4 W
enabling battery based operation. This ensures more
safety to the LD, improved DR, portability and the
feasibility of long time (off-line) operation.
SREEKUMAR: SIMPLE DIFFUSION WAVE ANALYZER
4 Some Applications of the DWA: Photothermal
Radiometry (PTR) of Si Wafers
4.1 Theory of PTR of semiconductors
When a semiconducting material is excited with a
harmonically modulated beam of super band-gap
light, thermal and plasma waves are generated, the
former arises from direct lattice heating and the latter
from the periodically generated excess carriers
diffusing away from the source (charge plasma
diffusion wave), until they recombine with opposite
carriers or defects/impurity centers after an average
life-time (τ)9-11. Each recombining photo-excited
carrier is equivalent to a blackbody radiator and the
amplitude and phase of the corresponding IR emission
is a complex function of the carrier density (Δn)
which depends on the diffusion coefficient (D),
surface recombination velocity (s) and life-time (τ) of
the concerned carrier type. The quantitative noncontacting evaluation of these parameters is essential
at different stages of the production line of micro and
nano electronic devices. The technique has been
emerging as an innovative metrology tool in the micro
and nano electronic industrial scenario12,13.
It has been established that for a good quality
(defect free) silicon wafer, above a few hundreds of
Hertz, the plasma wave content dominates the slow
thermal wave contribution9-11. For a wafer of
thickness L with unpolished back surface having
strong optical absorption at the front (illuminated)
surface, the PTR signal can be simplified to yield
S=
CN
(1 − e − σL ) 2
[W]
Dσ( Dσ + s ) (1 + Re −2σL )
…(3)
where CN is an instrumental constant,
σ = (1 + j ω τ) / Dτ [m−1]
…(4)
599
4.2 Experimental Details
The infrared emission is collected using two offaxis parabolic mirrors (Fig. 2) and detected with an
HgCdTe element (2-12 μm) of 1 mm2 active area. The
laser spot size is made 3.5×5 mm (approximately) by
adjusting the collimating lens so that the onedimensionality criterion is ensured to satisfy the
model (spot diameter > detector size). The germanium
window blocks the pump laser component. The
frequency scan width is 200 Hz to 100 kHz, which is
the plasma-dominating region.
The PTR amplitude has been recorded at 10 kHz
for different pump powers and between 10 and 40
mW, a linear response has been observed. This
ensures a linear carrier generation, which is essential
for the successful application of the above theory14.
Figure 3(a & b) shows the PTR amplitude (⎜S⎪) and
phase [tan−1(ImS/ReS)] data against the modulation
frequency sweep and the best fit curves from the
center of a boron doped (10-15 Ωcm), p-Si wafer of
thickness 500±20 μm at 300 K. Simultaneous fitting
of the amplitude and phase eliminates multi-pair
solutions in the determination of τ and s. The best-fit
values are D=17 (±2.2%) cm2/s, s=105 (±2.4%) cm/s,
and τ=240 (±1.6%) μs.
Wafer annealing is an important process at different
device fabrication stages and heavy metal ions
(contaminants) diffusing into the wafer during
annealing act as recombination centers influencing τ
that in turn deteriorates the device performance. We
have intentionally annealed the above wafer at 800°C
for 1 hour. The SiO2 layer thus grown has an
approximate thickness of 0.3 μm. This has been
etched using an HF:H2O (6:1) solution (etch rate is
1200 Å/minute) and then rinsed in distilled water15.
The complete removal of SiO2 is verified by
resistance measurement.
Figure 4 shows the quadrature frequency response
of the PTR signal from unannealed and annealed
is the complex plasma wave vector associated with
the excess carrier concerned and
R=
Dσ − s
Dσ + s
…(5)
The real and imaginary components of S are
available at the in-phase and quadrature output of the
DWA, from which the amplitude and phase can be
calculated.
Fig. 2—Experimental arrangement for photothermal radiometry
600
INDIAN J PURE & APPL PHYS, VOL 43, AUGUST 2005
(a)
(b)
wafers whose peak (at fC) roughly indicates the
corresponding carrier life-time according to the
relation 2πf C τ = 1 , which may be used as a guessparameter for simulation10. Simultaneous amplitudephase simulation shows that the once annealed wafer
has τ=105 (±1.5%) μs, s=132 (±2%) cm/s and D=17.3
(±2.2%) cm2/s. After two cycles of annealing, τ=34
(±1.5%) μs and s=180 (±2.2%) cm/s without
considerable change in D. The result is in close
agreement with the fact that annealing is a source of
Si wafer contamination due to heavy metals and ions
(life-time killers) from the furnace atmosphere. At
660 nm, the optical absorption length is about 1.5 μm
and the dynamics of near surface excess carriers are
reflected in the PTR response. In all the above
measurements, data have been collected from the
center of the wafer with peak pump power of about 35
mW.
For typical microelectronic grade Si wafers, the
reported values of τ, D and s are, respectively, in the
range 1-1000 μs, 10-40 cm2/s and 1-500 cm/s16. At
low modulation frequencies less than fC (say, 500 Hz),
well in the plasma-dominating region, these values
correspond to a real valued σ and R becomes 1. Now
if L > diffusion length Dτ , Eq. (3) simplifies to
S ∝ τ/ D
Fig. 3—(a) Experimental and simulated PTR amplitude against
modulation frequency. The crowded portion is due to the
logarithmic scale; (b) experimental and simulated PTR phase
against modulation frequency
…(6)
It has been reported16 and verified in our
experiment that D is the least sensitive parameter to
processing and surface conditioning. Thus, S can be a
measure of τ if properly calibrated by actual
simulation. We have applied this technique to three
different points radially outwards for a wafer of
D=7.6 (±2.2%) cm/s2 and τ=110 (±2%) μs at the
center and the τ values are compared (Table 1) with
those obtained from simulation at these points. The
deviation may be due to the approximations made.
The fall in τ outwards may be related to the crystal
growth process and is consistent with the literature16.
Table 1⎯ Minority carrier lifetime estimated at radial points by
simulation (error <±2.4%) and amplitude measurement at 500 Hz
Fig. 4—PTR Quadrature response of the annealed and
unannealed wafers
Distance between the centers
of wafer and laser spot
cm
Simulated
lifetime
μs
Measured
lifetime
μs
1
2
3
88
70
47
67
62
35
SREEKUMAR: SIMPLE DIFFUSION WAVE ANALYZER
5 Results and Discussion
A cost-effective, simple, compact and efficient
diffusion wave analyzer has been designed, developed
and analyzed. The rough estimate is about 150
Dollars. For further cost reduction, an active LD,
typically of 630-700 nm at 30 mW, may be collected
from a damaged laser printer or CD writer. The
instrument is a general-purpose photothermal/
photoacoustic analyzer. The major source of error is
the drift in the demodulator chip. The analyzer has
been used for the metrology Si wafers and estimated
parameters are in good agreement with the literature
values. The quadrature channel has been used for the
fast estimation of minority carrier life-time, which is a
crucial parameter in device modeling. At 660 nm, the
absorption length in Si is about 1.5 μm and the
measured parameters are those of the near surface
excess carriers. The instrument has been used for
contamination analysis too. For probing deep lying
carriers, higher wavelengths may be used.
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