........
ELSEVIER
CRYSTAL
GROWTH
Journal of Crystal Growth 172 (1997) 313-322
Methods for liquid-solid interface shape and location
discrimination during eddy current sensing of Bridgman growth
Haydn N.G. Wadley *, Kumar P. Dharmasena
Intelligent Processing of Materials Laboratoo,, School of Engineering and Applied Science, University of Virginia, Charlottesville, Virginia
22903, USA
Received 18 July 1995; accepted 17 April 1996
Abstract
Axisymmetric finite element calculations of the multifrequency eddy current sensor response during vertical Bridgman
growth have been conducted for GaAs and CdTe. These are representative of materials that are either ideally (the GaAs
case) or marginally suited (CdTe) to eddy current sensing during semiconductor growth by a vertical Bridgman process. The
simulations reveal two potential strategies for separately discriminating the interface shape and location. One is based upon a
comparison of the sensor's high and low frequency imaginary impedance components. The former characterizes the interface
location and the latter its shape. The second approach exploits the existence of an inflection point (or a peak) in the
imaginary impedance response of an absolute (or a differential sensor) as an interface passes through it. This latter approach
is less affected by test circuit contributions to the sensor's high frequency response. Both strategies with either sensor type
lead to reasonably precise location/shape characterization for GaAs. The differential sensor coupled with the peak position
method offers the best precision for less favorable material systems like CdTe. Even for this worst case material, the
interface location can be determined to +1 mm and its curvature estimated with sufficient precision to be of use for
characterizing vertical Bridgman growth processes.
1. I n t r o d u c t i o n
The vertical Bridgman method is a widely used
technique for growing single crystal semiconductor
materials [1-5]. In this method, a vertical multizone
furnace is used to establish an optimized axial temperature gradient through which is translated an ampoule containing the liquid semiconductor. As the
lower tip of the ampoule enters the furnace cold
zone, single crystal solid is nucleated and a l i q u i d solid interface propagates upward through the ampoule. Today, the yield and quality o f material grown
* Corresponding author.
by this technique is maximized by empirically optimizing the temperature gradient and the axial translation rate for each material system. Essentially, this
involves repeated experimentation until a satisfactory
material can be grown. Once obtained, temperature
set points within the furnace, and furnace translation
rate schedules are rigidly controlled from run to run.
When applied to the CdTe system, this approach
to process optimization/control has failed to result
in high yields of acceptable quality single crystal
material. The reasons for this are still not fully
understood. Recent modelling of the growth process
indicates a strong l i q u i d - s o l i d interface curvature
sensitivity to the translation rate [6]. This arises from
0022-0248/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved
PII S0022-0248(96)00496-4
314
H.N.G. Wadley, K.P. Dharmasena /Journal q['Crvsml Growth 172 (1997) 313-322
the very low thermal conductivity of the CdTe system which retards the dissipation of the latent heat
released during solidification and promotes non-ideal
concave interface shapes. Reports of significant melt
undercooling in the high purity CdTe systems, coupled with the possibility of spontaneous solid nucleation and rapid transient growth, raise the possibility
that the liquid-solid interface location (its velocity)
and its curvature may be uncontrolled by current
growth strategies.
This problem has stimulated interest in the use of
eddy current methods for sensing the location and
shape of liquid-solid interfaces in CdTe [7]. The
approach exploits the large change in electrical conductivity that accompanies the melting of this and
many other semiconductors [8]. Electromagnetic finite element methods (FEM) have been applied to
the design of encircling sensors for locating and
characterizing the interface [7,9]. These calculations
have revealed that large changes in the sensor's
impedance accompany a change of interface location
or shape, and they have identified the best frequency
ranges for observing these effects for a variety of
semiconductor materials. CdTe has a low electrical
conductivity compared to many other semiconductors, and multifrequency measurements in the 100
kHz to 10 MHz range are needed to characterize the
interface [7]. Data collected at the high frequency
end of this range have been shown to be dependent
on the interface's location but relatively insensitive
to its shape. At lower frequencies ( ~ 500 kHz), the
sensor's response depends on both location and
shape. Subsequent experiments with an electromagnetically equivalent model system have verified that
these predicted trends can be observed experimentally, though at high frequencies (above 3 MHz), test
circuit impedances and sensor parasitics cause an
unmodelled shift in the sensor's measured output
[10].
These studies indicate that provided the contributions from interface location and shape can be separated, the signals from eddy current sensors appear
well suited for monitoring the growth of CdTe.
Several monitoring strategies could be pursued. For
example, a sensor could be positioned at a fixed
location with respect to the ampoule, and the passage
of the interlace along the ampoule monitored, or the
sensor could be continuously repositioned along the
ampoule to coincide with the interface, and the
sensor's position then continuously monitored to
track interface location (and measure its velocity). In
either case, an anomaly-free protocol is needed for
separately discriminating the location of the interface
within the sensor and determine its curvature.
Here, the finite element method is used to calculate the multifrequency output of eddy current sensors as a liquid-solid interface is propagated upwards through the sensor. This simulation of a stationary sensor's response during a simulated growth
run is repeated for three different interfaces (flat,
concave, and convex) and performed for two material systems (GaAs and CdTe). Earlier studies [7]
have shown GaAs to be an ideal material system for
eddy current sensing because its high conductivity
enables the use of low frequencies where test circuit
impedance/parasitics are absent during measurements. CdTe appeared to be a more marginal material system and represents a critical test of the eddy
current method. The sensor responses from these
simulated growth runs reveal two possible strategies
for separating the contributions of the interface's
location from its curvature. One uses the frequency
dependence of the complex impedance; the second
exploits the observation of a frequency dependent
inflection point/peak in the imaginary impedanceinterface location relationship for the absolute/differential sensor designs [7]. The location of these
points is found to provide good discrimination between interface location and curvature even for the
more problematic CdTe material system.
2. Analysis methodology
The problem analyzed is shown in Fig. 1. It
consists of a cylindrical sample (of diameter, D =
76.2 mm) contained within an axisymmetric eddy
current sensor. The sample has two regions with
electrical conductivities of either the solid or liquid
semiconductor at its melting point. These conductivity values were the same as those reported in an
earlier study [7]. The interface between the two
conductivity regions was allowed to be flat, concave,
or convex with a convexity parameter 0 = z/D of
_+0.333 and 0 (z is the maximum difference in axial
coordinates of the interface across the sample). The
H.N.G. Wadley, K.P. Dharmasena /Journal of Crystal Growth 172 (1997) 313-322
Z[
R 101.6
R 50.8
R 45.7
R 38.1
Air
g = 4nx10-7 H/m
c = 0 S/m
76.2
Differential
pick up coil
~r
~.3b
6.35
~Of~ f
pick up coil
76.2
~- Driver coil
Solid
Differential pick up coil
(~s
D i m e n s i o n s in ram.
315
liquid-solid interface was then introduced at a height
h=-38.1
mm below the sensor's midpoint (see
Fig. 1 for the definition of h) and the two sensors'
responses at 500 kHz again calculated. The interface
was systematically advanced in 3.2 mm increments
upwards through, and eventually beyond, the sensor
and the two 500 KHz responses obtained. Finally,
the response due to an entirely solid sample was
obtained. This procedure was repeated for ten other
excitation frequencies to cover the 10 kHz to 10
MHz range, and for interfaces with convexities of
_+0.333. The electrical conductivities of the test
system were then changed to the values of GaAs,
and the entire procedure was repeated for 12 frequencies ranging from 200 Hz and 1 MHz. Finally,
the sensors' responses with the samples removed
were obtained for all test frequencies used in the
calculations. In all, a total of 877 separate FEM
calculations were conducted. Using similar procedures to those described in Ref. [7], the absolute and
differential sensor impedances normalized by their
empty coil values were obtained from these FEM
results.
3. A b s o l u t e sensor
Fig. 1. Finite element model geometry.
sample's magnetic permeability ( ~ ) was taken to be
4 ~ X 10 7 H m ~ which is that of free space.
The multifrequency responses of two types of
sensor were analyzed. Both used a seven-turn driver
(or primary) coil with a 6.35 mm coil spacing through
which was passed a fluctuating unit current. The
resulting electromagnetic field was calculated using
the same FEM procedures described in Ref. [7]. An
absolute (single pickup coil) sensor response was
obtained from the electromagnetic field's vector potential at the pickup coil located midway along the
length of the primary coil, Fig. 1. The differential
(two axially displaced, opposingly wound pickup
coils) sensor response was obtained from the difference in the vector potentials at each pickup coil's
location (near either end of the primary coil), Fig. 1.
The finite element model was used to first calculate the 500 kHz responses of the absolute and
differential sensors for a homogeneous liquid sample
with the electrical conductivity of CdTe. A flat
Fig. 2a shows the normalized imaginary component of the absolute sensor's impedance for CdTe at
a test frequency of 500 kHz as a function of interface
location for each interface shape. Results for GaAs at
10 kHz are shown in Fig. 2b. At these two frequencies, both materials exhibit similar behaviors. Initially, when the interface is beyond the range of the
sensor's electromagnetic field (say h = - 8 0 mm),
the imaginary impedance has a value that depends
only upon the liquid's electrical conductivity and the
fill factor of the coil. If the coil's fill factor is
known, the value of the impedance at a fixed frequency moves downwards from its null (i.e. empty
coil) value of 0 + jl as the conductivity increases.
This data could be used to determine the liquid's
electrical conductivity (and to thus infer the factors
affecting it) during applications of the sensor to a
real growth process. As the interface propagates
upwards through the sensor, h goes from negative to
positive, the impedance increases, goes through an
inflection and asymptotically approaches a constant
316
H.N.G. Wadley, K.P. Dharmasena / Journal qf Co,stal Growth 172 (1997) 313-322
1.0
1.0
I
(a) CdTe, f = 500kHz
o
o
o
0.9
"O
I
(b) GaAs, /= lOkHz
0.9
"o
c~
E 0.8
"5
0.8
"5
E
¢~
i
I
O
0.6
° I
0=0,0
0.6
o~
.=_
E~
_~3-o33
.__
EI)
_.E 0.5
0.7
E
O
•
0.5
o !-'so,id
0.4 ~
-80
~
-40
0
40
Interface position, h
0.4
-80
80
-40
(rnm)
0
40
80
Interface position, h (mm)
Fig. 2. Calculated impedance variation of the absolute sensor with interface position for simulated growth runs of three interface shapes. (a)
C d T e (500 kHz), (b) G a A s (10 kHz).
value corresponding to that of the solid. The
impedance in this latter case again depends only
upon the test material's electrical conductivity and
the coil's fill factor. If the latter remains constant
(i.e. the test sample and secondary coil diameters do
not change), an impedance measurement in this region could be used to infer the solid's electrical
conductivity and gain insight into the factors that
control it (e.g. the average temperature in the sensed
volume).
1.0
t.0
I
(b)GaAs
0.9
f
o
oE 0.9
"O
fl)
Q.
__.E 0.8
o
E
_
I Inflecti°n
100kHz
J
=
0.8
"6
E
O
ro
Q.
0.7
ro
E
O
O
~,
O
O
J
~,
0.6
0.4
-30
5MHz
e=°'° e ~
~
I
i
I
I
-20
-10
10
20
Interface position, h (mm)
_E
e=o.o.
~I
t0kHz
.E
0.5
2kJ
J
E
500kHz
0.6
t~
.~_
E
0,7
500Hz
0.5
h
30
0.4
-30
I
-20
1MHz [
-10
I
I
t0
20
!
30
Interface position, h (mm)
Fig. 3. Variation o f the imaginary impedance component of the absolute sensor with interface position for a flat interface at four frequencies.
(a) C d T e , (b) G a A s .
317
H.N.G. Wadley, K.P. Dharmasena /Journal of Crystal Growth 172 (1997)313 322
The results presented in Fig. 2 indicate that when
the interface is located far from the sensor's center
(e.g. h = _+40 mm or more), the imaginary
impedance is approximately independent of both its
location and curvature. However, as the interface
approaches the location of the pickup coil, a strong
dependence upon both location and curvature is
seen. For both material systems, the imaginary
impedance associated with a sample containing a
convex interface first begins to increase (towards the
sensor's null value of unity on the imaginary axis) as
the interface approaches the sensor's midpoint from
below. If the location of the interface (i.e. h) were
known independently, the imaginary impedance
component at these test frequencies (10 kHz for
GaAs and 500 kHz for CdTe) is directly related to
the interface's convexity. For example, if the outside
edge of the interface coincide with the pickup coil
location (h = 0.0 ram), the imaginary impedance for
GaAs would increase from 0.615 for a concave
interface to 0.715 for a convex one. A smaller, but
still significant shift would occur for CdTe. It can
also be seen that the point of inflection shifts to the
left as the interface's curvature changes from concave to convex. Therefore, either the value of the
impedance or the position of the inflection point are
interfacial curvature dependent.
The response to interface location depends upon
1.0
I
test frequency for both materials. Fig. 3a shows the
imaginary impedance-position relationship for CdTe
with a flat interface at test frequencies of 50 kHz,
100 kHz, 500 kHz, and 5 MHz. Analogous results
for GaAs at 500 Hz, 2 kHz, 10 kHz, and 1 MHz are
shown in Fig. 3b. At very low frequencies (below 50
kHz for CdTe and 500 Hz for GaAs), the rate of
change of the electromagnetic flux within the test
material is sufficiently low that weak eddy current
induction occurs, and both samples are almost electromagnetically transparent. In these cases, the decrease in electrical conductivity associated with passage of the liquid-solid interface through the sensor
results in a very small change in the nearly null
response of the sensor. At high frequency (above 1
MHz for GaAs and 5 MHz for CdTe), the very high
rate of change of flux induces intense eddy currents
that are concentrated close to the sample surface
(due to the skin effect). In this frequency range, the
sensor's response depends strongly on the coil's fill
factor (i.e. the sample diameter) and progressively
less on conductivity as the frequency increases. Thus,
the sensor's response is only moderately affected by
the location. The largest changes in response are
seen at intermediate frequencies where both the eddy
current density and the volume within which it exists
are both large.
Fig. 4 shows high and intermediate frequency
I
(a) CdTe
I1)
o
¢-
0.9
0.8
1] ~
(b) GaAs
0.9 -
® h
>'1 e~..i_
8=0.0 •
,f = 500kHz
o
_-0.333
II
+0.333 []
e = 0.0 ~ I
~
1.0
+0.333 [
i 81:l::ecl!one
_E
f f , - , ~
0.8 -- 0 . 3 3 3
0 i Solid
Inflection
¢" 0.7
0.7
0
r~
EO
o~
E
O
o
0.6
._~
0.6
.E
.EE 0.5
0.5
I
0.4
-30
I
-20
r
I
i
l
-10
0
10
20
Interface
position, h
(mm)
o.4
30
-30
i
-20
-10
f = 1MHz
I
i
i
0
10
20
Interface position, h
30
(mm)
Fig. 4. Variation of the imaginary impedance component of the absolute sensor with interface position for three interface shapes at two
frequencies. (a) CdTe, (b) GaAs.
H,N.G. Wadlev K.P. Dharmasena / Journal of Co,stal Growth 172 (1997) 313-322
318
results for each interface shape. It can be seen that
for GaAs, Fig. 4b, a 1 MHz measurement is independent of interface shape but is a unique (though weak)
function of location, whereas the intermediate frequency (10 kHz) impedance monotonically increases
as either the interface changes from concave to
convex, or as its position moves upwards through the
sensor. Thus, measurements at the two frequencies
shown in Fig. 4b, in combination with the pre-calculated responses for different interface shapes, could
be used to separately determine the interface location
and shape. The frequencies required to do this for
GaAs are below those where test circuit
impedance/sensor parasitics are likely to perturb the
response [ 10].
CdTe exhibits a less ideal response, Fig. 4a. Even
at 5 MHz (where test circuit impedance/sensor parasitics begin to significantly contribute to experimental measurements), a small residual dependence upon
interface shape is observed. This results in greater
uncertainty in the interface's location determination,
and since this needs to be known before the interface
can be characterized, it results in a reduced ability to
characterize the interface's curvature. This is further
compounded by the smaller differences in intermediate frequency (500 kHz) sensor response to each
interface shape. Measurements at very high fre-
20.0
........
I
.......
i
........
I
........
i
quency (suitably corrected for test circuit/sensor
parasitic shifts) might overcome this difficulty, and
the precision of the interfacial curvature could be
improved by developing an analysis for a range of
test frequencies.
An alternative, potentially simpler approach is to
examine the inflection point in the imaginary componenrs position dependence. It is clear from Figs. 3
and 4 that this is a function of test frequency and
interface shape. To investigate it in more detail, the
imaginary impedance-position relationships for each
interface shape, frequency, and material were numerically differentiated with respect to interface position
and the resulting peak location (and thus the data's
inflection point) determined. This inflection point
position is plotted versus test frequency for CdTe in
Fig. 5a and for GaAs in Fig. 5b.
Fig. 5 shows that at high frequencies the inflection point position becomes a progressively weaker
function of the interface shape. The inflection point
for data collected at 1 or 2 MHz during the simulated
growth of GaAs corresponds to an interface that is
level with the pickup coil location. This result is
valid within _+0.5 mm for all three interface shapes.
At lower frequencies, the results of Fig. 5b indicate
that a large shift in the inflection point occurs as the
interface's curvature changes from convex to con-
20.0[
........
EE ts,o
o.o
E E
O Q
=o-o
0 = -0.333
"5 ~ o.o
;ensor
position
8~
"~ r~
~..E_
N.~_
-5,0
"~ ~
-=. _
........
10 2
0.0
>~'g
-,o.o
-20.0
I
........
t
........
I
........
I
.....
P
'o.oI
g~
~8
........
| (b) GaAs
(a) COTe
I
lO3
-
r
1001'
105
Frequency (Hz)
106
107
_~
*0.333
102
, , ,,,,,,I
103
El !
_~
0=o.oo,lliml~
......
I
~,7
-15.o I- 0=+o.333
2001
104
s.°,
j ,
......
I
10 `=
........
o333oL-'"
I
........
10 s
e h
~ ," /
/ ~
| o
I
10 s
........
-1
/
I
107
Frequency (Hz)
Fig. 5. Relative position of the inflection point on the imaginary impedance component response of the absolute sensor versus frequency for
three interface shapes. (a) CdTe, (b) GaAs.
H.N.G. Wadlev K.P. Dharmasena /Journal of Co'stal Growth 172 (1997) 313-322
cave. For this material system, measurement of the
high frequency inflection point (for example, by
axially translating the sensor along the ampoule)
could be used to position the sensor at the interface's
location, and lower frequency data (say 10 kHz)
could then be used in conjunction with Fig. 5b to
infer the curvature. If the axial position of the sensor
were simultaneously monitored, the strategy would
enable separate discrimination of interface location
and curvature.
The lower electrical conductivity CdTe system,
Fig. 5a, exhibits a similar behavior to GaAs but at
significantly higher frequency. For this system, even
inflection point data collected at 10 MHz exhibits a
_+ 1.5 mm position variability due to interfacial curvature. This degree of uncertainty may still be sufficiently precise for some applications (for example,
ensuring that solidification occurs at an optimal location in the furnace), but would enable only qualitative insights into the interface's curvature to be
obtained from lower frequency (say 500 kHz) data.
Efforts to make measurements at higher frequency to
reduce the variation in the inflection point data would
have to contend with test circuit contributions to the
overall response, These are likely to dominate measurements in the 10-25 MHz range for most experimental setups. Alternatively, extrapolating inflection
point data collected in the 1-10 MHz range out to
4.0
I
~ 50 MHz would reduce uncertainty in location and
might enable a more precise determination of curvature.
4. D i f f e r e n t i a l s e n s o r
The physical basis of the discrimination approaches proposed above lies in the expulsion of
electromagnetic flux by the skin effect. At high
frequencies, a uniformly distributed excitation field
of the primary coil exists near its center when no
sample is present. When a solidification interface
resides within the sensor, the skin depths at high
frequency are small in both the solid and the liquid,
and the electromagnetic field samples only the outer
surface of the test material. In this situation, the
interface's location deep within the sample has little
or no interaction with the excitation field, and the
sensed response depends only upon the fraction of
solid and liquid in a thin annular region at the
sample's surface (i.e. it depends only upon the location of the interface within the sensor and not the
radial dependence of the location). Unfortunately,
the low conductivity (large skin depth) of CdTe
requires the use of too high a frequency for this to be
easily accomplished.
If this limitation is to be overcome, some way is
4.0
I
I
(b) GaAs, f = lOkHz
(a) CdTe, .f = 5OOkHz
0
o
c:
"0
Q.
~
~e h
~
2.0-
E
0
0
r-
I
3.0-
E
"6
"E
0
to
0
0
¢-.
+0.333 n
1.0
[
j
-
/ ¢
¢~
o~
o
e
319
3.0
:~[]'
~i.;;~
-~o~+°'~Us~d
~
E
E
~
I
2.0
e~
E
0
.~_
1.0
t~
E
E
0.0
-80
f
-40
I
0
Interface position, h
40
(mm)
0.o
80
-80
I
I
-40
40
Interface position, h
80
(mm)
Fig. 6. Calculated impedance variation of the differential sensor with interface position for simulated growth runs of three interface shapes.
(a) CdTe (500 kHz), (b) GaAs (10 kHz).
H.N.G. Wadley, K.P. Dharmasena / Journal Of C<vstal Growth 172 (1997) 313-322
320
4.0
I
t03
"o
r'J
o=oo.!
°
S
OkHz-
o
r03
ea
1.0
3.0
;>;. . . .
OOe
'~, o e
o
I
h
)
10kHz
!Solid I o /
g
2.0
c~
2kH2
E
0kHz
O
o
~ 1 [ z ~ k H z
O3
I
Peak
"6
ss0,
~
O
I
E
s~t
E
O
._~
h
i°5: -
Peak
2.0
I
(b) GaAs
EL
el.
03
I..iqUid;I
l eo
i Solid
3.0
E
"6
E
o
4.0
I
(a) CdTe
0Hz
03
E
5MHz
03
E
1.0 I
MHz
E
0.0
-30
I
-20
I
-10
I
10
Interface position, h
I
20
I
-10
0,0
30
-30
-20
(mm)
I
0
Interface position, h
I
10
I
20
30
(mm)
Fig. 7. Variation of the imaginary impedance component of the differential sensor with interface position for a fiat interface at four
frequencies. (a) CdTe, (b) GaAs.
needed to increase the rate of decay of the excitation
field into the sample. In earlier work, the placement
of pickup coils at the ends of the excitation coil was
shown to result in a more rapid convergence of the
impedance with frequency for the different shaped
interfaces [7]. This arose because the "fringe" field
4.0
I
I
I
(a) CdTe
o
r03
"0
m
4.0
I
I ,
' Uquid
+0.333 O
0=0.0 •
-0.333 o
3.0
~
¢ ~
~
-
,
oo h
o
-
Q_
E
r-"
0
Q.
E
0
2.0
E
0
O3
03
E
3.0
"6
2.0
o
E"
0
03
f=l~
o
e~
Peak
.~
I
(b) GaAs
"6
"o
en
at the top and bottom of the sensor decays more
rapidly with radial distance in the sample than predicted by a skin effect alone. This additional mechanism of radial field decay can be exploited in the
differential sensor approach to better discriminate
between interracial location and curvature.
03
~
o3
03
E
1.0
o.o
-30
I
-20
I
-10
I
0
Interface position, h
I
10
(mm)
I
20
30
f
1.0
] = 1MHz
o.o
-30
I
-20
I
-10
0
Interface position, h
I
10
I
20
30
(mm)
Fig. 8. Variation of the imaginary impedance component of the differential sensor with interface position for three interface shapes at two
frequencies. (a) CdTe, (b) GaAs.
H.N. G. Wadley, K.P, Dharmasena / Journal oJ Crvsml Growth 172 (1997) 313-322
,
20.0
, ,
..... =
..... ~
........
~
........
i
20.0
.......
(a) CdTe
'*~
~
10.0
'*O 8
8 8
".,=
o
........
I
...... I
.......
, ~0.
• 5.0
o.o
E E
5.0b
~~ 8 t~
3ensorposition
o o
oo
o.o
C). O)
0=0.0
/
Sensor
~_~.
=-
.E
I
"x
~
;~
"~
I T ' C :t~
........
10,0
~o
E E
........ I
t (b) GaAs
I
321
-10.0
ri-"
t--'~ ~
t
-lS.O
._
+0.333
[]
-10.0
~o
-20
-15.0
........
102
I
103
...... f
104
........
I
10 s
................
10 s
h
i-""
iso,,,Ii o -
o = +0.333
-0.333 o L
o.
-20.0
position
®
-20.0
107
........
102
Frequency (Hz)
t
.t03
........
t
104
........
i
10 s
........
i
106
........
107
Frequency (Hz)
Fig. 9. Relative position of the peak in the imaginary impedance component response of the differential sensor versus frequency for three
interface shapes. (a) CdTe, (b) GaAs.
Fig. 6 shows the variation in the imaginary
impedance of a differential sensor when concave,
flat, or convex interfaces are propagated through the
sensor. The result for CdTe corresponds to an excitation of 500 kHz whilst that for GaAs was obtained at
l0 kHz. In both material systems, the passage of the
interface through the sensor results in a peak in the
impedance. When the interface lies outside the sensor (so that the sensed region is either solid or
liquid), a near null response is obtained. As the
interface passes close to the sensor's midpoint, a
peak in impedance is observed, and at the two
frequencies referred to above, the position of the
peak is a significant function of the interface's curvature. The position within the sensor where the peak
is seen also depends upon the test frequency, Fig. 7.
The position dependence of the impedance at high
and intermediate frequencies is shown for three interfaces in both CdTe and GaAs in Fig. 8. It can be
seen that for both materials the high frequency data
is nearly independent of interface curvature when the
interface is located near the center of the sensor. At
lower frequency, the position of the impedance peak
is a strong function of the interface's position and
curvature.
Fig. 9 shows the position of the impedance peak
as a function of frequency for each interface and
both materials. The fringe field effect combined with
the differential scheme results in a lowering of the
frequency at which the peak positions converge for
both materials. Thus, using the results shown in Fig.
9, data collected at 500 kHz could be used to locate
the interface's position to better than _+0.5 mm in
GaAs. For CdTe, data collected in the 5 - 1 0 MHz
range would enable the location to be deduced to
better than _+1 ram. For both material systems,
lower frequency data (say 500 kHz for CdTe and 10
kHz for GaAs) could be used in conjunction with the
calculations of Fig. 9 to deduce the interface's curvature. The precision of the curvature characterization
will depend upon the accuracy of the location determination. It is estimated to be on the order of 0.05 of
the convexity parameter (i.e. approximately 4 mm
for a 76 mm diameter sample) assuming the location
is determined to _+ 1 mm.
5.
Conclusions
An electromagnetic finite element method has
been used to calculate the response of absolute and
differential eddy current sensors during the simulated
vertical Bridgman growth of CdTe and GaAs. GaAs
was found to be an ideal material system for eddy
322
H.N.G. Wadley, K.P. Dharmasena /Journal of Co'sml Growth 172 (1997) 313-322
current sensing because of its high liquid electrical
conductivity and large liquid:solid conductivity ratio. For semiconductor systems of similar (or higher)
conductivity, eddy current data collected between
500 kHz and l MHz with either the absolute or the
differential sensor scheme can be used to locate the
position of solidification to better than + 0 . 5 ram.
Less ideal systems such as CdTe require the exploitation of fringe fields at the ends of the excitation
coil together with a differential sensing scheme to
achieve similar location precision. Once the interface's location is obtained, lower frequency data can
be used to deduce the interface's curvature with
acceptable precision for both material classes.
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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