j. . . . . . . .
C R Y S T A L
G R O W T H
ELSEVIER
Journal of Crystal Growth 172 (1997) 303-312
Eddy current sensor concepts for the Bridgman growth of
semiconductors
Kumar P. Dharmasena, Haydn N.G. Wadley *
Intelligent Processing of Materials lxlboratory. School of Engineering and Applied Science, Unit~ersity of Virginia, Charlottest ille, Virginia
22903, USA
Received 18 July 1995; accepted 17 April 1996
Abstract
Electromagnetic finite element methods have been used to identify eddy current sensor designs for monitoring CdTe
vertical Bridgman crystal growth. A model system consisting of pairs of silicon cylinders with electrical conductivities
similar to those of solid and liquid CdTe has been used to evaluate the multifrequency response of several sensors designed
for locating and characterizing the curvature of liquid-solid interfaces during vertical Bridgman growth. At intermediate
frequencies (100-800 kHz), the sensor's imaginary impedance monotonically increases as interracial curvature changes from
concave to convex or the interface location moves upwards through the sensor. The experimental data are in excellent
agreement with theoretical predictions. At higher test frequencies ( ~ 5 MHz), the test circuit's parasitics contribute to the
sensor's response. Even so, the predicted trends with interface location/curvature were found to be still preserved, and the
experiments confirm that the sensor's high frequency response depends more on interface location and has only a small
sensitivity to curvature. Multifrequency data obtained from these types of sensors have the potential to separately
discriminate the location and the shape of liquid-solid interfaces during the vertical Bridgman growth of CdTe and other
semiconductor materials of higher electrical conductivity.
1. Introduction
H g ] _ ~ C d , T e infrared (IR) focal plane arrays are
presently manufactured on IR transparent CdTe or
C d ~ _ y Z n , T e substrates obtained from large polycrystalline or single crystal boules grown either by a
vertical or horizontal Bridgman method [1-4]. In the
vertical Bridgman process, a multi-zone furnace is
used to create an axial temperature gradient and
crystal growth is then attempted by translating the
furnace (and its associated themral profile) relative
* Corresponding author.
tO a stationary ampoule containing the molten charge
(or sometimes vice versa). In general, the ability to
obtain a high yield of acceptable quality single crystal material has been linked to the velocity (i.e. rate
of change of the interface's axial location) and shape
of the l i q u i d - s o l i d interface during crystal growth
[3,5-9]. In the vertical Bridgman process, the liqu i d - s o l i d interface shape is determined by heat flow
near the interface, which is in part governed by the
position where solidification occurs within the furnace's temperature gradient and by the growth velocity; it may be convex, flat, or concave depending
upon the thermal conditions existing near the interface [9]. Non-planar interfaces are undesirable be-
0022-0248/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved
PII S0022-0248(96)00495-2
304
K.P. Dharmasena, H.N.G. Wadle)'/Journal of O3"sml Growth 172 (1997) 303-312
cause they can facilitate the nucleation of secondary
grains at the ampoule wall [8] and contribute to the
development of large radial thermal stresses and
thus, the generation of dislocations.
For materials which have a low thermal conductivity (e.g. such as CdTe), it can be difficult to
enforce the optimal thermal field during crystal
growth and to ensure that solidification initiates and
continues at the best location within the temperature
gradient of the crystal grower. Reports of low substrate yields from crystals grown by this process may
have arisen in part from a variability in the interface
position (with respect to the furnace's thermal gradient) and the development of non-ideal interracial
shapes during growth. These issues have stimulated
an interest in the development of in situ sensing
methods for continuously locating the interface position and characterizing its curvature throughout crystal growth processes [10,11].
The large difference between the electrical conductivities of most liquid and solid semiconductors
has motivated an exploration of eddy current sensors
for the non-invasive sensing of liquid-solid interfaces during crystal growth [12-24]. Recently, electromagnetic finite element modelling has been used
in a detailed theoretical study of eddy current sensor
concepts for the vertical Bridgman configuration [16].
The approach involved the building of an axisymmetric ampoule model containing liquid and solid
regions with uniform (but different) electrical conductivities and a variety of interface shapes. This test
"sample" was encircled with a seven-turn eddy
current exciting coil through which a variable frequency current was passed. The resulting electromagnetic interactions with the sample were calculated and the voltages induced in axisymmetric
pickup coils obtained over the 200 H z - 2 MHz frequency range. The study covered a variety of test
material systems and identified the frequency ranges
where interface shapes and locations could best be
observed. It revealed that a high frequency region
existed where the sensor's response was heavily
weighted by the interface's location while at lower
frequencies, the response was dependent upon both
interface location and shape. The study also investigated the use of one (absolute) and two (differential)
coil pickups to assess sensor design criteria that
might maximize the sensor's response to interface
effects and enhance its location/shape discrimination capability.
The modelling study revealed that the viability of
an eddy current approach depends on the liquid's
electrical conductivity and its liquid : solid conductivity ratio [16]. CdTe had the lowest liquid electrical
conductivity of the materials evaluated, and was
concluded to be the most difficult candidate for eddy
current sensing. However, the study suggested that
even for this material, the approach might be successful provided relatively high frequencies could be
used (e.g. 0.1-5 MHz). The principal concern was
the (unmodelled) significance of other test circuit
impedances, which can make important contributions
to a sensor's response at high test frequencies and
potentially mask the contributions of the interface.
Thus, an experimental study has been conducted to
evaluate one of the sensor concepts proposed by the
modelling approach. The study seeks to determine
the significance of the test circuit's impedance and
parasitic contributions to an absolute sensor's response obtained during the monitoring of a low
electrical conductivity system such as CdTe.
Since the growth of high purity CdTe crystals by
the vertical Bridgman method is costly, typically
takes several days, and the interface position and
shape are not independently known, it is expensive,
time consuming, and problematic to try and validate
a proposed sensor's response via crystal growth experiments. Instead, experiments have been conducted
with a model system consisting of pairs of silicon
samples that were carefully machined to create various interface geometries. By doping the silicon crystals it was possible to achieve electrical conductivities similar to those of either solid or liquid CdTe at
its melting point and thus create an electromagnetically equivalent problem to the one encountered in
the growth environment. This could be done while
still retaining a complete description of the test
system's geometry and all of its relevant physical
properties. The sensor's response to this model test
system was then calculated and compared to that of
the experimental setup.
2. The electromagnetic finite element model
Electromagnetic finite element modelling provides a convenient tool for evaluating eddy current
K.P. Dharmasena, H.N.G. Wadley /Journal of Co'stal Growth 172 (1997)303-312
sensor responses for a potential sensor and sample
geometry provided the electromagnetic properties
(electrical conductivity and the magnetic permeability) of the sample material and the sensor's geometry
and excitation frequency are known. It can be used
to rapidly evaluate the performance of different sensor designs and sample geometries. For example, the
sample's diameter, length and a variety of interface
shapes ranging from convex, flat and concave can all
be simply and precisely modelled [16].
The finite element analysis solves Maxwelrs
equations for the magnetic vector potential in a
prescribed domain where the system loads and
boundary conditions are specified. By applying Faraday's law, the induced voltage in a secondary coil
can be calculated from the magnetic vector potential,
and for a known current flowing through the primary
coil, the sensor's transfer impedance as a function of
excitation frequency can be determined. The details
of the method and the calculation procedures have
been reported elsewhere [16]. The results are most
simply displayed on an impedance plane diagram
showing the real and imaginary parts of the complex
transfer impedance for each calculated frequency.
The impedance components are usually normalized
by the imaginary part of the empty coil's impedance
to emphasize the contributions of the sample to the
results.
The sensors analyzed to date have been of a two
coil coaxial design. (The advantages of a two coil
system have been described in earlier work [16].)
One coil (the primary coil) consisted of 7 turns of a
relatively large diameter, low resistivity wire and
was used to generate a fluctuating electromagnetic
field. A second coaxial coil system detected the
perturbation of the primary coil's field resulting from
the presence of the conducting sample. The pickup
coil system consisted of either a single-turn coil
placed at the primary coil's midpoint (a so-called
"absolute" sensor, Fig. 1) or a "differential" sensor
with a pair of opposingly wound coils placed at
opposite ends of the primary coil.
Calculations of the multifrequency response of
both "absolute" and "differentiar' sensor arrangements have been performed [16]. The model treated
each loop in the multiple turn winding of the primary
coil as a circular current ring. The total number of
turns were thus modelled as a stack of "current
305
101.6
O 91.4
O76.2
:
Liquid
~t = 10,000S/m
Drivercoil:7tums[i ~
(18AWGcopper)~ ~
wound4tums/in. ~
-. ,% .
"Absolute"
pickup
coil/
(24AWGcopper)
"i!i '"I
+6.35
{
Dimensions in m m
Fig. 1. The geometry of the model problem. An axisymmetric
seven turn primary coil in conjunction with a central coaxial
pickup coil (absolute sensor) interrogates a cylinder with regions
of low (representative of solidified CdTe) and high (representing
liquid CdTe) conductivity.
rings" with a spacing equal to that of the primary
coil turns. Neither the interwinding (parasitic) capacitance, the frequency dependence of the coil's a.c.
resistance, nor the test circuit's resistance, inductance or capacitive contributions to the apparent
impedance were included in the electromagnetic field
computations.
3. Experimental procedures
3.1. Samples
The model test systems were built using four 76.2
mm diameter <111} oriented doped silicon samples
with resistivities representative of either the liquid or
solid phases of CdTe at its melting point [25]. The
supplier's (Lattice Materials Corporation) specifications for the two low resistivity samples (representative of the liquid) were 0.01 _+ 0.005 ~ . cm while
the two high resistivity (i.e. solid) samples had resistivities of 0.25 _+ 0.05 1) • cm. These are equivalent
to sample conductivities in the rather broad ranges of
6667-20000 S / m and 333-500 S / m , respectively.
K.P. Dharmasena, H.N.G. Wadley/Journal of Crystal Growth 172 (1997) 303-312
306
2
a s = 410 - 432 S/m
(a)
I
mm). This collection of test samples allowed assessment of interfaces with 0 = + 0.33 and 0 = 0.
3.2. Absolute sensor design
oz : 9 7 1 6 - 1 0 , 2 2 7 S / m
(b)
~s = 3 6 0 - 3 8 0 S / m
o"t : 9 2 4 5 - 9 7 3 2 S l m
(c)
(d)
D i m e n s i o n s in m m
Fig. 2. Cylindrical silicon samples used to construct test material
geometries with electrical conductivity discontinuities analogous
to those encountered in the vertical Bridgman growth of CdTe.
Subsequent four-point probe measurements indicated
that the two lower conductivity samples had measured conductivities of 360-432 S / m while the two
higher conductivity samples had conductivities of
9 2 4 6 - 1 0 227 S / r e .
The samples were machined so that the ends of
one pair of cylindrical samples (one of low conductivity and one of high conductivity) formed matching
convex and concave surfaces. The other pair of
samples had flat surfaces, Fig. 2. By pairing the
samples shown in Figs. 2a and 2b it was possible to
simulate a flat interface. By placing the sample in
Fig. 2d on top of that in Fig. 2c it was possible to
represent a convex solid-concave liquid interface
and by reversing the order of samples in Figs. 2c and
2d, a concave solid-convex liquid interface could be
obtained. An interface curvature parameter, 0, was
defined as the ratio of the interface height, z ( = 0
and _+25.4 mm), to the crystal diameter, D (76.2
The absolute eddy current sensing concept analyzed in earlier modelling work [16] consisted of a
multiple turn excitation (primary) coil and a single
turn pickup (secondary) coil. The primary coil had
seven turns of 18 gauge (1 mm diameter) copper
wire wound on a 101.6 mm outer diameter × 95.25
mm inner diameter × 50.8 mm long PMMA preform. A 1.04 mm wide, 0.508 mm deep, 7 turn
helical groove with a spacing of 4 turns per inch was
machined on the outer preform surface to secure the
primary coil windings and prevent movement of the
individual turns.
The secondary coil preform had a 95.25 mm outer
diameter, a 88.9 mm inner diameter and a 50.8 mm
length. The middle section of its outer surface was
reduced to 91.2 mm to provide an annular gap for
the placement of pickup coil wires. The absolute
sensor pickup coil consisted of a 0.5 mm diameter
(24 gauge) single loop of copper wire wound level
with the center turn of the primary coil.
3.3. Sensor instrumentation
The multifrequency transfer impedance of the two
coil eddy current sensor has been measured by assembling an eddy current instrument from off the
shelf circuit testing components, Fig. 3a. A model
HP4194A multifrequency impedance/gain-phase analyzer operating in the gain-phase mode was used to
make gain and phase measurements. The primary
solenoid was driven by the impedance analyzer's
oscillator to impress an alternating electromagnetic
field on the test material over a wide range of
frequencies (50 k H z - 1 0 MHz). A model 25A100
Amplifier Research Inc. RF power amplifier was
used to enhance the drive signal strength at low
frequencies. The current flowing through the primary
coil was obtained by recording the voltage drop
across a low inductance precision (1 1~) resistor
connected in series with the primary coil. This measurement was recorded on the reference channel of
the analyzer. The emf induced in the secondary coil
was measured on the analyzer's test channel. The
analyzer was programmed to record the amplitude
K.P. Dharmasena, H.N.G. Wadley /Journal of Crystal Growth 172 (1997) 303-312
ratios of the test and reference channel voltages, and
to then compute their gain, g, and their phase angle
difference, 05, for 101 logarithmically spaced frequencies starting at 50 kHz and ending at 10 MHz. A
total of 32 samples were averaged with an integration time of 5 ms for each sweep frequency.
A complete set of gain-phase measurements were
obtained first without a test sample (to give a reference empty coil reading for the gain, go, and phase,
050, at each frequency). The test sample was then
inserted and the gain and phase re-measured. For
each test frequency, the real (Re) and imaginary (Ira)
normalized impedance (Z) components were found
from
g
Re(Z) =
sin( 05 - 050),
( 1)
go
g
I m ( Z ) = - - c o s ( 05 050).
(2)
-
307
"~"
Sample
Electrical conductivity, c
Magnetic permeablltiy, g
-
-
leasure:
Gain (Tch/Rch) VS. frequency (f) wJw.o, sample (g/go)
Phase vs. frequency if) wJw.o, sample (¢/%)
g0
The real and imaginary components of the transfer
impedance (computed from Eqs. (1) and (2)) are
functions of four independent measurements; g, the
ratio (i.e. the gain), and 05, the phase difference of
the test and reference channel voltages measured
with a sample present, and go and 050, measured
with the empty sensor. Each of these measurements
has an associated measurement accuracy, which depends on the input signal levels of the reference and
test channels, the test frequency range, the number of
samples averaged, and the integration time of the
measurement [26]. The gain and phase measurement
accuracies (k g and A05) can be computed from the
reference and test channel voltage and phase accuracies specified in the HP4194A operation manual for
a 1 Mf~ input impedance measurement i [26]. Errors
in both the empty and sample filled sensor gain and
phase are incurred. If these errors are assumed to be
a According to the impedance/gain-phase analyzer manual [26],
the measurement error could be improved if a 50 ~ input
impedance measurement configuration was used. However, this is
only obtained at the expense of introducing a finite current in the
secondary coil circuit of the sensor. At high temperatures, this
results in a sensitivity to temperature induced resistance changes
in the secondary coil, and adversely affects the measured test
voltages. In addition, a lower input impedance increases the
"loading" effect of circuit components (resistance, capacitance,
and inductance of the cables) between the pickup (secondary)
coil(s) and the instrument.
~'h"lr ' ~ l~"lF2EIW/IY~I1¢g~I1~ ;I'~ Rr/~=
Real Z / imaginary Z o = (g/go)sin(~- %)
Imaginary Z / imaginary Z o = (g/go)COS(~-$o)
b)
1.0
~
~
l
j
--
Abso/ute
Sensor
n
I
creasing f,
0,0
I
1
I
1.0
0,0
Real Z / Imag Z o
Fig. 3. The eddy current test setup used a HP4194A impedance
analyzer to measure the multifrequency gain/phase response of
the sensor. Data were normalized by the empty coifs response
and the resulting complex transfer impedance plotted on an
impedance plane diagram.
random, the error in the imaginary impedance component is [27]
AIm(Z)= AgTglm(Z
0
+ Ag07g0Im(Z)
2
2]
-
308
K.P. Dharmasena, H.N.G. Wadley/Journal of Co'stal Growth 172 (1997) 303 312
The total error in the imaginary impedance component computed from these four measurements can be
calculated by substituting Eq. (2) into expression (3).
A similar expression can be used to compute the
measurement uncertainty for the real impedance
component by replacing the Ira(Z) terms in Eq. (3)
with Re(Z) and substituting Eq. (1) for Re(Z) in Eq.
(3).
The normalized impedance curve for an absolute
sensor encircling a cylindrical sample is shown in
Fig. 3b. The curve has a characteristic " c o m m a "
shaped trajectory. If no eddy currents are induced
when a sample is placed in the sensor (i.e. if it has
infinite resistivity or a d.c. primary coil excitation is
used), the normalized impedance is located at a point
with coordinates (0 + j l ) and is located on the imaginary axis. Either an increase in frequency or a
reduction in the test material's resistivity (i.e. replacement of the solid phase of a semiconductor by
its liquid) will shift the sensor's impedance clockwise around this characteristic curve. When a liquidto-solid transformation occurs within the interrogated
volume of the sensor, the resulting conductivity decrease has been predicted to result in a monotonic
increase (toward the null value) in the sensor's imaginary impedance component [16]. Interface shape and
position effects have been predicted to be most
simply observed from variations of the sensor's
imaginary impedance component between 0.1 and 5
MHz.
3.4. Experiments
An initial gain-phase measurement was made with
the empty sensor mounted with its axis vertical. The
effects of an interface's location were then investigated by placing one of the pairs of silicon samples
(e.g. Figs. 2a and 2b) co-axially within the sensor
and adjusting its height using a rack and pinion
mechanism. This enabled systematic variation of the
position of the interface relative to the center of the
primary coil without adjustment of the sensor's position. It was a preferred test method since changing
the sensor's position relative to a stationary sample
might have introduced varying electromagnetic coupling between the test circuit lead wires which might
have affected the self-inductance of the test circuit.
For the first measurements, the platform height
was adjusted so that the outer edge of the surface of
contact between the two cylinders was 12.7 _+ 0.025
mm below the center of the primary coil. Gain and
phase measurements were then made at this position.
Keeping the sensor undisturbed, the height of the
sample was subsequently increased in 6.35 _+ 0.025
mm increments to a series of new positions and the
gain-phase measurements repeated. In all, measurements were made for two positions of the interface
below the sensor, one at the same level as the
sensor's midpoint, and for two positions above the
sensor.
To obtain insight into the significance of test
circuit contributions to the sensor responses, the
initial series of experiments were performed using
0.61 m long connecting cables between the eddy
current sensor and the gain-phase analyzer. The effect of the test circuit's capacitance, inductance and
resistance was investigated by substituting longer
coaxial cables (1.83 m in length) and repeating the
measurements.
4. Results
4.1. Interface position effects
The effect of interface location upon an absolute
sensor's response to a flat interface ( 0 = 0 . 0 ) is
summarized in Fig. 4. It shows experimentally measured normalized imaginary impedance versus frequency data (shown at every 5th frequency measurement) from the absolute sensor design when a flat
interface was positioned at five different heights
within the sensor. FEM predicted responses using the
methodology described in Ref. [16] and the conductivities measured with the four-probe technique are
also shown for comparison (as solid lines). The
position of the interface can be characterized by a
distance, h, measured from the center of the (stationary) primary coil to the interface, Fig. 1.
The five curves span a total interface translation
of 25.4 mm. The curve corresponding to h = - 12.7
mm might represent a situation encountered early
during crystal growth before the interface had advanced significantly into the sensor. At this stage,
the electromagnetic field of the sensor samples more
of the higher conductivity liquid resulting in a large
K.P. Dharmasena, H.N.G. Wadley /Journal of Crystal Growth 172 (1997) 303 312
1.0
Experimental data
t-
0.8
h = +12.70mm
(D
E
~d o.6
E
E
O
t~
~
O
FEM
0.4
.__
~ 0.2
E
0.0
103
O=O.O~
h ~~e
iso01 o
104
10s
106
107
108
Frequency (Hz)
Fig. 4. A comparison of measured and FEM predicted imaginary
impedance component frequency response for a flat interface
located at various axial positions using an absolute sensor.
inductance change and the biggest shift (from unity)
in the imaginary impedance for all of the test frequencies. An upward shift in the imaginary
impedance component as the interface translated upwards through the sensor was seen for all test frequencies. This is consistent with a reduction in the
"effective" conductivity sampled by the sensor's
electromagnetic field as the lower conductivity material propagates through the sensor. A maximum sensor sensitivity to position was observed at a frequency around 600 kHz. Above this frequency, the
curves for each position converged slowly. Even at 5
MHz, a significant dependence upon location was
seen. Below 600 kHz, the curves for the different
locations converged rapidly and showed no dependence upon location below about 10 kHz.
Excellent agreement between the model predictions and the experimental data was observed in the
5 kHz-3 MHz frequency range. Above 3 MHz, the
measured data deviated slightly from the calculated
values and above 5.5 MHz, experimental impedance
values were observed to increase with frequency
whereas the predicted data continued to decrease.
Table 1 shows the estimated uncertainties in the
imaginary impedance component for the absolute
sensor in the 5 kHz-5 MHz frequency range for
309
0 = 0.0 and h = 0. The estimated error decreases
from 0.032 at 5 kHz to 0.018 at 5 MHz and is
approximately 3.25% of the nominal impedance at
each frequency. For an operating frequency of around
700 kHz, the error analysis indicates an ability to
sense a change in the position of a CdTe liquid-solid
interface to within 2-3 ram.
To understand the origin of the high frequency
behavior in the experimental impedance curve, the
gain and phase frequency responses for the empty
sensor are shown in Fig. 5. The gain is seen to
linearly vary with frequency up to about 3 MHz as
expected, but a non-linear behavior is observed at
higher frequencies. A maximum gain occurred at 5.5
MHz, after which a drop-off in the response was
observed. An analogous effect in the phase response
was also seen at high frequencies. These anomalous
responses are consistent with a circuit approaching
resonance. The resonant frequency will be governed
by the sensor's and circuit's lumped inductance,
capacitance and resistance parameters. For the sensor, these depend upon the diameter of the sensor
windings, the number of coil turns, the spacing
between the windings, and the capacitive coupling
between the two coils. Contributions from the lead
wires (resistance, capacitance and inductance) connecting the sensor to the analyzer are length dependent. Thus in addition to the inductive coupling of
the coils with the sample (which is calculated with
the finite element method), frequency dependent parasitic capacitive effects, stray inductive effects, and
lead wire impedance all contribute to the overall
response as the frequency is increased.
To confirm the origin of this anomalous behavior,
the coaxial cables were tripled in length (to 1.83 m)
and the empty sensor gain and phase measurements
Table 1
Uncertainties in the imaginary impedance of the absolute sensor
for 0 = 0 . 0 a n d h = 0
Frequency (kHz)
Nominal impedance
Probable error
5
22.9
104.5
478.2
2186.7
5045.5
1.0003
0.9836
0.8675
0.7203
0.6027
0.5435
0.0325
0.0321
0.0287
0.0238
0.0199
0.0177
K.P. Dharmasena, H.N.G. Wadley/Journal of Crystal Growth 172 (1997) 303-312
310
repeated. The transition to non-linear behavior was
seen to occur at a slightly lower frequency when the
length was increased. The magnitude of the deviation
from linearity also increased as the lead length was
increased. In the lower frequency region where the
gain remained linear, very good agreement between
the experimental results and the electromagnetic
(finite element) model calculations was again seen.
Fig. 6 shows a comparison of the electromagnetic
FEM calculated impedance with experimental data
obtained with both cable lengths (0.61 and 1.83 m).
It shows that significant discrepancies between model
predictions and experimental measurements are likely
to be encountered above 2 MHz. However, even
above 2 MHz the trends associated with interface
location are still evident, and so this effect does not
necessarily preclude the use of higher frequency
measurements, but rather implies the need for careful
calibration if the high frequency data are to be used
to infer information about interface location.
These discrepancies between FEM predictions and
the experiments are important only at higher frequencies ( > 2 - 3 MHz). Earlier work [16] has shown
that as the liquid conductivity decreases, it becomes
increasingly necessary to perform measurements at
higher frequencies to separate the contributions to
the sensor's response from the interface's location
from its shape (curvature). From the measurements
1021
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106
107
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Frequency
(Hz)
Fig. 5. T h e f r e q u e n c y r e s p o n s e o f the e m p t y a b s o l u t e s e n s o r ' s
g a i n a n d p h a s e f o r d i f f e r e n t l e n g t h s o f c a b l i n g c o n n e c t i n g the
s e n s o r to its m e a s u r e m e n t i n s t r u m e n t a t i o n .
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cables of length...
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Frequency (Hz)
Fig. 6. T h e f r e q u e n c y d e p e n d e n c e o f an a b s o l u t e s e n s o r ' s i m a g i n a r y i m p e d a n c e c o m p o n e n t for t w o test c i r c u i t c a b l e lengths.
reported above, the effective upper frequency that
can be used without resorting to calibration is seen to
be limited by circuit resonances. For high conductivity materials like GaAs, the test circuit effects occur
at frequencies well above those that would be used
for interface characterization. However, for low electrical conductivity materials like CdTe it is important
to design the sensor and its test circuit so that these
resonances are made to occur at as high a frequency
as possible to avoid adversely impacting the fidelity
of the sensor's response in the region where it is
dominated by interface location.
4.2. Interface shape effects
{~
E
~10-'
1.0
The effects of interface shape upon the measured
and predicted sensor response for the absolute sensor
are shown in Fig. 7. Very good agreement between
measurements and predictions is again seen at frequencies below about 3 MHz. In the frequency
region where the agreement between measurement
and prediction is good, changing the interface from
concave through flat to convex leads to an increase
in impedance component values. This is a direct
result of the decrease in the fraction of the high
conductivity liquid sampled by the sensor's electromagnetic field [16]. For samples of the conductivity
K.P. Dharmasena, H.N. G. Wadley / Journal of Co,stal Growth 172 (1997) 303-312
1.0
311
differing shape is still preserved in the experimental
data.
O
t~ 0.8
"1:3
(1)
5. S u m m a r y
E
"*O 0.6
E
E
O
Ca.
OE 0.4
I
FEM calculated curve,
Liquid
0
r-
'~:~ 0.2
O=0.0~"~e
E
i
h=0
s J
o
; ..... I
0.0
10a
i
,
L J JK~l I
104
........
I
10s
........
I
........
106
I
107
.....
108
Frequency (Hz)
Fig. 7. The effect of interface shape upon the measured and
calculated imaginary impedance component of the absolute sensor
for h = 0.
tested here, the strongest sensitivity to shape occurs
at intermediate frequencies ( ~ 200 kHz). In this
frequency region, the calculated and measured responses are in excellent agreement and test circuit
contributions can be safely ignored.
As the frequency was increased, both the calculated and measured impedances converged as a consequence of the skin effect which confines the electromagnetic field within the sample to a thin annular
shell near the outer surface of the sample. In this
interrogated volume, all the interfaces have similar
fractions of high conductivity material and so the
response to the detailed internal shape of the interface is lost.
At high frequency ( > 2 - 3 MHz), the experimental data again deviated from the predicted behavior
because of the approach of a coil and test circuit
resonance. Again, the relative trends predicted by the
finite element method are seen to be still preserved
even at the high frequencies where test circuit
impedances and parasitic effects are significant.
These unmodelled phenomena introduce discrepancies between the measurements and calculations.
However, the data still appears useful, even when the
theory is no longer strictly valid, because the predicted convergence of the responses for interfaces of
Benchtop eddy current measurements have been
conducted with 76 mm diameter doped silicon single
crystals to explore the responses of an absolute
sensor during vertical Bridgman growth of CdTe-like
semiconductors. They have been compared with finite element electromagnetic model calculations of
the experimental test setup and very good agreement
was obtained over a wide range of interface position
and interface shapes. The sensor was found to have a
maximum response to interface curvature and position in the range 200-600 kHz. At higher frequencies, the measurements were increasingly independent of interface shape and depended only upon
location. However, at the highest test frequencies,
unmodelled circuit effects significantly contribute to
the sensor's response and a departure between FEM
predicted and measured behavior exists. In this region, it is necessary to use either an experimental
calibration or include a complete circuit analysis in
an improved model to avoid incurring errors in eddy
current sensor deduced interface locations.
Acknowledgements
This work has been performed as a part of the
research of the Infrared Materials Producibility Program conducted by a consortium that includes Johnson Matthey Electronics, Texas Instruments, I I - V I
Inc., Loral, the University of Minnesota, and the
University of Virginia. We are grateful for the many
helpful discussions with our colleagues in these organizations. The consortium work has been supported by A R P A / C M O under contract MDA972-91C-0046 monitored by Raymond Balcerak.
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