Journal of Crystal Growth 130 (1993) 553—566
North-Holland
jo~*~ o~
CRYSTAL
GROWTH
Simulation of the eddy current sensing of gallium arsenide Czochralski
crystal growth
Haydn N.G. Wadley and Kumar P. Dharmasena
School of Engineering and Applied Science, University of Virginia, Charlottesville, Virginia 22903, USA
Received 6 July 1992; manuscript received in final form 7 December 1992
The feasibility of eddy current sensing of (1) the melt surface position and (2) the liquid—solid interface shape of 3-inch gallium
arsenide crystals being grown by the Czochralski technique has been investigated using an axisymmetric finite element method. The
results show clearly that differential sensor designs operating at high frequency (— 1 MHz) are very sensitive to the distance
between the sensor and the surface of the melt providing the opportunity to precisely monitor and control this important variable
of the growth process. The calculations also show a weaker effect of interface shape upon the imaginary impedance component at
lower frequencies (1—10 kHz). Its physical basis is due to the different skin depths of solid and liquid GaAs. We show that a
sensor’s response to this interface effect can be enhanced by appropriate design of the differential sensor’s pick-up coils.
1. Introduction
Emerging monolithic microwave integrated circuits and low power, high speed computation is
creating a demand for semi-insulating GaAs
wafers. Presently, these materials are 40—50 times
the cost of silicon, and suffer wide variations in
their electrical and physical properties. While
there is likely to always be some additional costs
associated with the processing of compound
semiconductors, it can be significantly lessened by
increasing yield, eliminating wafer to wafer vanability and reducing the degree of human supervision of the slow crystal growth process.
Yield losses for crystals grown by the high
pressure, liquid encapsulated Czochralski
(HPLEC) process are principally due to the loss
of single crystal orientation during growth, uncontrolled variation of crystal diameter, and handling damage during subsequent post solification
processing. For instance it is not unusual to observe a transition to polycrystallinity half-way (or
even less) through the growth of a boule with a
subsequent halving of yield. This loss of single
crystal growth has been linked to a change of
0022-0248/93/$06.00 © 1993
—
liquid—solid interface shape. Provided the growing crystal is convex, single crystal growth occurs.
However, because of changes in the heat transport of the liquid during growth, it is not unusual
for the crystal to develop a concave curvature at
the edge of the crystal. This results in the nudeation of new orientation grains and the loss of
useful material.
Researchers have been pursuing intelligent
processing of materials (1PM) approaches to address these issues [1]. In the 1PM approach, a
suite of advanced sensors are employed to directly measure critical parameters of the growth
process. This new information is combined with
process understanding (in the form of predictive
mathematical models and process heuristics) in a
supervisory controller. The controller combines
multivariable real-time control algorithms with an
artificial intelligence-based supervisor to facilitate
growth of material with a goal state combination
of electrical properties and ideally near theoretical yield. Ultimately, such a supervisor promises
to lessen or eliminate the need for human participation in the growth process provided suitable
sensors are available. In particular, a need exists
Elsevier Science Publishers B.V. All rights reserved
554
H.N.G. Wadley, K.P. Dharmasena
/ Simulation
for sensing the melt height with respect to the
furnace and the liquid—solid interface shape.
There are many physical differences between
solid and liquid gallium arsenide near its melting
temperature that could be considered a basis for
sensing melt position and interface shape. They
include (together with their measurement methodologies):
• refractive index (optical scattering);
density (X- or y-ray tomography);
elastic moduli (ultrasonic imaging);
emissivity (infra-red imaging);
electrical conductivity (eddy current measurements).
The optical and infrared techniques suffer from
a high optical attenuation in crystalline material
near the melt temperature and the difficulty of
seeing through the boric oxide melt overlayer of
the LEC process. Of the remainder, the use of
uncooled encircling eddy current sensors appears
to be the most promising because of the large
difference in electrical properties of solid and
liquid gallium arsenide. It can also be implemented noninvasively.
We report on feasibility studies of eddy current methods for sensing interface shape and the
melt heigh during GaAs crystal growth. It extends
the previously reported experimental eddy current responses of an encircling sensor during
Czochralski silicon crystal growth [2—61.In this
earlier work, the sensor data were primarily used
to calculate the axial and radial thermal profiles
within the growing crystal. Here, we seek to evaluate other potential uses of such sensor approaches, and in particular, the determination of
the interface shape. Similar concepts have been
proposed for controlling the interface of HgCdTe
during solidification, though in that case the geometry was different to the one studied here [7].
The eddy current method relies upon the large
difference in electrical conductivity of solid and
liquid gallium arsenide at the melting point [8,91.
Fig. 1 shows that at the melting transition, galhum arsenide undergoes a semiconductor—metal
transition with a large (15-fold) change of electrical conductivity. At the melting point, solid galhum arsenide has a reported electrical conductivity(cr) of
0.55 x i03 Q~ cm~.Upon melting,
—~
of eddy current sensing of GaAs CZ crystal growth
a semiconductor—metal transition occurs and
there is a 15-fold jump in conductivity to a value
of 8.3 x iO~Q’ cm~. Therefore, the induced
currents of a fluctuating electromagnetic field
within a gallium arsenide volume element that
encloses a liquid—solid interface will depend
strongly upon the liquid fraction. One hopes that
a means might then be found to determine the
melt level and to infer the interface shape by
monitoring the eddy current interaction with the
growing crystal.
We have performed calculations of the response of encircling eddy current sensor designs
to changes in the shape of the liquid—solid interface and melt level. We find for (what we call)
absolute sensors utilizing a single secondary coil
that only small effects due to interface shape are
seen, but there is a great sensitivity to the separation distance between the sensor and the surface
of the melt. This sensitivity to melt level is of
interest for better methods of melt height control
are urgently needed to control crystal growth.
Differential (two opposingly wound secondary
coils) sensor designs are found to have enhanced
sensitivity to interface shape though the results
are also dependent upon melt surface-sensor distance. Our results indicate that provided the crystal diameter and meniscus profile do not vary, the
melt height can be measured with very good
(potentially 10—20 ~.tm) precision from high frequency (100 kHz to 1 MHz) data. Once the melt
I
8
-
6
-
‘~J
—
Graphite
-
Metalhc
-
2
Mercu~
-
—Semiconciucting
-
—
-
-
0
1100
1175
I
1250
1325
1400
Temperature (°C)
Fig. 1. Temperature dependence of electrical conductivity for
gallium arsenide [8,91.
H.N.G. Wadley, KP. Dharmasena
/ Simulation of eddy current sensing of GaAs CZ crystal growth
height is known, the results show that the inter-
dation [101.We note in fig. 1 that the conductivity
of solid gallium arsenide is close to that of
graphite while the conductivity of the liquid is
similar to mercury. This suggests a laboratory
bench system for experimental study that consists
of a mercury bath (represening the melt) and a
graphite cylinder (representing the crystal). By
machining the graphite, it is possible to reproduce the interface shapes of interest during the
Czochralski growth of GaAs (fig. 2), and by vertically translating the sensor, the effect on melt
height. Normally, one seeks interfaces such as
face shape can then be inferred from differential
sensor measurements at lower frequency (in the
2—10 kHz region) provided one’s instrumentation
can measure the complex impedance to order 1
part in 102.
2. The model system
We have conducted our calculations on a
model system that is suited for experimental vahi-
76mm
76mm
Solid
T
122mm
122mm
121mm
A 121mm
Liquid
51mm
-
-
-
____________
-
Liquid
(a)
(b)
76mm
76mm
Solid
I,
127mm
Solid
134mm
127mm
L~,,~T36mm
~A16mm
Liquid
Liquid
-.--.-/
-..--.-.~-..-,.--
R2Omm
(c)
555
(d)
Fig. 2. Interface shapes chosen for study of sensing concepts.
556
H.N. G. Wadley, K.P. Dharmasena
/ Simulation of eddy current sensing of GaAs CZ crystal growth
that shown in fig. 2b. As the radius of curvature
increases (fig. 2a), better crystal quality is achieved
but at the expense of a loss of diameter control.
If heat flow in the melt is not properly controlled,
the interface, while remaining convex at its center, turns over near the crystal surface becoming
locally concave, and depending upon its severity,
results in polycrystal nucleation. Interfaces shown
in figs. 2c and 2d simulate to a greater or lesser
extent this eventuality. The “crystal” studied here
is assumed to be nominally 76 mm (3 inches) in
diameter corresponding to today’s state of the
art. The “melt” is assumed to be of 51 mm (2
inch) depth. The boric oxide layer over the melt
was assumed to have perfect insulating properties,
The ideal sensor should be uncooled to avoid
perturbing growth conditions. Therefore, its response ought not to be sensitive to fluctuations in
its temperature. A two-coil (primary/secondary)
sensor design has been found useful for this
[11,12]. One can measure the current that flows
in a primary coil (for example, by measuring the
voltage across a precision resistor to ground).
0 108mm
0 91mm
~
(a)
0 108mm
0 91mm
22~m
0125mm
(b)
38Lm]J~
(C)
Connecting the secondary coil to a high
impedance (say 1 Mfl) voltage measurement instrument allows the induced voltage to be measured independent of the temperature induced
changes of the secondary coil resistance. The
ratio of these quantities gives the transfer
impedance of the sensor, which, to first order, is
independent of the sensor’s temperature. Second
order changes to the sensor’s impedance can occur because of thermal expansion effects. These
can be minimized by design of the sensor, or
calculated and compensated for during use.
The sensor designs for study are shown in fig.
3. They are all of an encircling type and can be
positioned at selected heights above the liquid
surface. For some calculations, only a single
(lower) secondary was considered (fig. 3a). For
others, a second opposingly wound secondary,
either 22 mm or 38 mm above the lower one, was
placed in series (figs. 3b and 3c). Lastly, a secondary arrangement with the two coils spread 17
mm radially was considered (fig. 3d). The best
preform material for the sensor is boron nitride;
a very good electrical insulator that is electromag-
(d)
ii
Fig. 3. Encircling sensor designs used for the study: (a) absolute sensor; (b) differential sensor; (c) differential sensor; (d)
differential sensor.
H.N.G. Wadley, KP. Dharmasena
/ Simulation
netically transparent over the frequencies and
temperatures of interest. Like the boric oxide, we
have taken its conductivity to be zero.
of eddy current sensing of GaAs CZ crystal growth
557
by that of the empty sensor, Z0 (i.e., when the
sensor is far removed from the sample):
2ii’Nrw
3. Calculation method
For two-coil (primary and secondary) eddy current sensors, the quantity measured is a transfer
impedance, Z defined as:
Z
=
EMF induced in secondary coil
current flow in primary coil
V~
=
Z0 —ItO+Jw
D
S 7 0~
[Im(A0) —J Re(A0)]
where A0 is the vector potential for the empty
coil, R0 is the empty coil resistance, and L0 the
empty coil inductance. In general wL0 >> R0 and
the normalized impedance:
z
R +jwL
+
—
=
R0 + jwL0
“
From Faraday’s law:
=
Z
N~04~
=
-y--
—
(2)
—~--~
p
—-~
[—Im(Aave)
wL0
+j
wL0
Re(Aave)]/Re(Ao).
(6)
The response of a differential sensor is found
by summing the impedance of the individual sec-
where N~is the number of turns in the secondary
coil and 4~is the magnetic flux (Wb). The flux
linked by the secondary coil can be calculated
from the magnetic vector potential A (Wb m i)
of the primary coil’s field and the secondary coil
geometry:
ondaries. For two secondary coils, of the same
radius and number of turns wound in opposition:
Z z, + z2
~=fAd1,
—[Im(A~e)—jRe(A~e)J},
(7)
where A~eand A~eare the vector potentials at
the two secondary coil locations. This can be
normalized with respect to the empty coil
impedance in the usual way.
=
2
—
(3)
where the integral is taken around the path of the
secondary coil. For sinusoidal currents flowing in
an axisymmetric sensor:
z
2 N
=
~
5~5~
[Irn(Aave)
—
j Re(A~,~)]
‘p
=
R
+
jwL
(4)
where r5 is the secondary coil radius, w the
angular frequency (radians/s), Aave is the average vector potential over the cross section of the
secondary coil wire, R is the coil resistance and
L the coil inductance. The real part of the
impedance (R) corresponds to eddy current losses
in the sample (heating), while the imaginary cornponent (wL) corresponds to the change of phase
between voltage in the secondary and current in
the primary,
When one conducts measurements on a test
sample, it is useful to normalize the impedance
=
~
N
~ {[Im(AW
I~
ave
) —j Re(A”)ave )}
We have used an axisymmetric finite element
code (MAGGIE developed by the MacNealSchwendler Corp.) to compute the magnetic vector potential needed by eq. (6) or (7) to compute
impedance.
Roughly 1000 grid points were used with care
being taken to ensure that in regions of significant current induction the grid points were closely
spaced compared with the skin depth in that
region for the highest frequencies considered. In
setting up the finite element model, we were
careful not to allow the grid to change between
different melt height and interface shape models.
Fig. 4 shows the finite element grid used in the
interface region. By varying the electrical conductivities of the elements, we could investigate the
range of interfaces of interest without incurring a
558
H.N. G. Wadley, K.P. Dharmasena
/ Simulation
problem
a superpositionsurface
of theseseparation
two subproblems.
As theas sensor—liquid
increases (curves B, C and D), several things happen to the curves. First, at high frequencies (above
kHz) the data approach the imaginary axis and
extrapolated intercept moves up (towards the
inductance of an empty coil). At high frequencies,
the sensor’s response is dominated more by geometry than sample conductivity. When the sensor is near the melt, the strongest induced cur-
10
the
~IdAIr
Liquid
____________________
______________________________
/~if~H~
frequency increases, the current density here increases, but this is offset by the skin effect which
expels the flux towards the surface and reduces
fromresistive
the in
sample
and
the
impedance
approaches
the
rents
are
the
losses
(more
within
conductive)
the liquid.
liquid.
In the
As the
absence of a crystal, the flux is eventually driven
__________________________________
_____
~__________________________________
___________
____________
of eddy current sensing of GaAs CZ crystal growth
I
_
I
I
the imaginary axis at infinite frequency making an
intercept determined only by the sensor—melt
-j——
Fig. 4. Finite element model used in the interface region.
50 Hz
1 0
100 Hz
________________________________
I
200 Hz
change of mesh and potential errors attendant
with this. For the calculations, crystal conductivity was taken as 1.123 x iO~fl-’ cm~ and that
of the melt as 1.062 x iO’~fi
cm’.
09
-
0.8
-
—
—
500 Hz
0.7
C
D
-
-
C
a)
1kHz
C
4. Absolute sensor
006-
-
E
0
4.1. Melt level
0.5
-
Using the methods described, we have calculated the impedance for an absolute sensor with a
single secondary coil in the bottom location
(nearest the liquid). The normalized impedance is
plotted on an impedance plane diagram for excitation frequencies between 50Hz and 2 MHz for
a range of lift-off (i.e. melt height) values in fig. ~
Curve A corresponds to a sensor located 4.8 mm
(3/16 inch) above the liquid. It looks superficially
like the curves one calculates for either a sensor
located above a conducting plane or encircling a
conducting cylinder [13], and indeed, to first order, one can think of the gallium arsenide sensing
~
a
0.4
2kHz
A
B
a
5 kHz
-
-
-
E
0.3
10kHz
-
20kHz
0.2
01
-
-
50kHz
100 kHz
200kHz
\\~5OOkHz
1 MHz
2 MHz
A
Lift Off
B
C
0
LifiOff=11.lmm
LiftOff- 17.5mm
=
4.8mm
-
-
Lift Off -302mm
_____________________________________
I
I
0.00.0
0.1
02
03
Real Z Component
Fig. 5. Normalized impedance curves for an absolute sensor
and a flat interface.
H.N. G. Wadley, KP. Dharmasena
/ Simulation
surface separation. As the coil is raised, the flux
that links the sample is reduced and the intercept
moves up towards that of an empty coil. This is a
well-known result that is frequently used to measure the separation between a coil and a conducting surface [12]. In the presence of the crystal, we
find that the intercept stops its approach to the
empty coil inductance. If, for the moment we
ignore the liquid, then we have a (weakly) conducting cylindrical crystal in a solenoid. Now as
the frequency is increased, the flux within the
crystal is increasingly expelled by the skin effect
(it requires higher frequencies (than the melt) to
do this because the skin depth, ~
(2/wu~,.L)”2
where o-~ is the crystal conductivity and ~i the
magnetic permeability, is larger than for the liquid at the same frequency due to the crystal’s
lower conductivity). The impedance, at very high
frequency again approaches the ordinate and intercepts the imaginary axis at the value (1
r~/r~)
where r~is the crystal radius and r~the
sensing coil radius [131. This also is a well-known
result that has been frequently exploited to measure the radius of conducting cylinders [11]. The
combination of these two flux exclusion effects
results in the observed variation of the high fre=
—
of eddy current sensing of GaAs CZ crystal growth
quency intercept with the imaginary axis. We see
that for sensors located closer to the melt, there
is good sensitivity to small changes in melt level
(if the sensor position is fixed) whilst for large
melt—sensor distances one could measure the
crystal diameter (though this would not be very
useful for process control since it corresponds to
crystal grown several hours in the past).
At lower frequency, the sensor’s response is
affected by both sample geometry and conductivity. This is because the real part (resistive) of the
impedance due to dissipitation of the eddy currents in the sample becomes more significant. We
can intuitively understand the overall shape of
the curves in fig. 5 by again considering the
behavior, at intermediate frequencies, of separate
conducting cylinders/ planes [13]. For both, the
real part of the impedance goes through a maximum and a knee is formed on the impedance
curve. The frequency at which the maximum is
reached depends upon the sample’s conductivity.
It occurs at lower frequency for more conductive
cylinders and planes. When the sensor is near the
higher conductivity melt, the knee in the curve is
controlled by a combination of the melt and the
crystal. At higher sensor locations, the knee fre-
D Lift Off 30.2mm
1o~
559
io4
io~
io6
Frequency (Hz)
Fig. 6. Imaginary component of normalized impedance versus frequency for the absolute sensor and four interface shapes.
560
H.N. G. Wadley, KP. Dharmasena
/
Simulation of eddy current sensing of GaAs CZ crystal growth
quency becomes increasingly dominated by the
crystal’s lower conductivity, and the impedance
curves cross. In practice, this intermediate frequency eddy current data, while strongly affected
by the melt and crystal conductivities, are difficult
to analyze because of the sensitivity to the (usu
ally unknown) geometry (melt surface—sensor
separation and crystal diameter).
-_________
_________
____________
4.2. Interface shape
_________
7/7//f//f
________
0.2065+jO.5287
0.2078+jO.5345
O.2O7O+JO.5232
0.2058+jO.5169
(b)
~,.
(c)
Fig. 7 Lines of constant magnetic flux resulting from the
interation of the coil, crystal and melt: (a) low frequency; (b)
intermediate frequency; (c) high frequency.
10 kHz
0.1343+jO.3430
0.1366+jO.3443
O.l3l8+JOLI400
0.1295+iO.3379
face that is seen by the electromagnetic field
becomes
increasingly
concentrated
at the
crystal
outer surface,
and in the
limit, becomes
infinitesimal. In this high frequency limit only the crystal
_______________________________________________
Interface Normalized impedance
No.
f = 500 Hz
f 2 kHz
I
—~.
towards the periphery of the crystal and more so
the surface of the melt (fig. 7b). The level of the
conductive liquid within a skin depth of the edge
of the crystal significantly perturbs the fields and
the closer this region is to the sensor, the higher
is the imaginary Z-component. As the frequency
increases (fig. 7c), the annular region of the inter-
Table 1
Normalized impedance values for the absolute sensor
0.2229+jO.7505
0.2204+jO.7548
0.2254+J0.7477
0.2238+jO.7431
__________
(a)
In fig. 6, we show the frequency dependence of
the imaginary impedance component for four
sensor—liquid surface separations and the four
interface shapes. Table 1 lists the actual normalized impedance values at a lift-off of 4.8 mm.
There is a small (approximately 3%) variation in
impedance due to the interface shape changes at
intermediate frequency (2—5 kHz), but at high
andlowfrequencies,thecurvesforthefourinterfaces overlap at the resolution of the graph (and
eddy current instrumentation). The differences at
intermediate frequencies also decrease with increasing sensor—melt separation. We note that in
the intermediate frequency range the value of the
imaginary Z-component varies inversely with the
liquid level (just inside the crystal) at the solid—
liquid interface (fig. 4).
These differences in impedance for different
interface shapes again arise from a perturbation
to the electromagnetic field by the crystal and the
underlying melt (where there is a 3- to 4-fold
difference in skin depth). At low frequency, the
eddy current density is low everywhere in the
sample and the electromagnetic fields penetrate
easily both the low conductivity crystal and the
liquid as shown in fig. 7a. As the frequency increases, the skin effect begins to concentrate flux
2
3
4
_____
f
=
H.N.G. Wadley, KP. Dharmasena
/ Simulation
of eddy current sensing of GaAs CZ crystal growth
diameter, melt level and meniscus shape effect
the impedance. But at the lower frequencies, the
topology of the interface is revealed, and the
possibility of its measurements exists provided
other factors (e.g. crystal diameter, meniscus and
melt level) affecting the eddy current response
are sufficiently controlled or known,
These results lead us to the conclusion that an
absolute sensor exhibits significant enough sensitivity to melt level to make it a useful in-situ
monitoring tool. The sensitivity to interface
shapes is rather small, however, and resolving the
interface shape would be difficult to accomplish
with this sensor design. The sensitivity to interface shape changes could also be masked by
other factors that influence the sensor’s response
(e.g. melt level, crystal diameter or menisus geometry), and unless these are controlled or independently measured, anomalous results would be
expected. Currently, the liquid level is usually not
known or controlled to better than 2 mm (1/16
inch) during growth, and observed changes of
impedance at intermediate frequency due to these
melt level fluctuations could easily be mistaken
for changes of interface shape. If one uses high
frequency data to infer melt level to significantly
better than 2 mm (1/16 inch), the prospects for
I
I
I
interface sensing would be greatly improved.
However, sensor designs that enhance the sensitivity to interface shape at intermediate frequency and to melt height at high frequency (to
increase the precision of its measurement), and
that reject the effects of other factors influencing
the perturbation to the electromagnetic fields
would be desirable. We consider several of these
in the section that follows.
5. Differential sensor
Differentail sensors are well known to provide
a discrimination against some eddy current signal
components and to have enhanced sensitivity to
lift-off. We have explored the feasibility of using
this approach to better resolve the interface
shape. We have investigated cases in which the
two secondaries are either axially or radially displaced.
5.1. Axial separation
The calculated response of a differential sensor with a 22 mm axial secondary spacing is
shown in fig. 8. The curves no longer lie between
~
-
1.4
-
I
I
______________
~
~
16
I
I
1=10kHz
f=1~
f-2OkHz
,/(“
1=500Hz
B
-
f=100k~\
L~1O$
1
N
12~
-
\
i~ /~
~kHz~
y’500kHz
0.8-
-.04
I
-0.3
I
-0.2
I
-0 1
-
~( tJ
1-50Hz
0
-
~
f=200H~~~
E
561
J
0.0
Y
I
0 1
A
Lift Off - 4.8mm
B LiftOff=11.lmrn
-
C Lift Off- 17.5mm
0 Lift Off = 30.2mm
I
0.2
I
0.3
I
0.4
0.5
Real Z Component
Fig. 8. Normalized impedance curves for a differential sensor with a 22 mm axial separation and a flat interface.
562
H.N.G. Wadley, K.P. Dharmasena
/ Simulation
of eddy current sensing of GaAs CZ crystal growth
1 and 0 on the imaginary and real axes reflecting
the measurement of a difference in induced voltages which may be negative in some instances
and much greater than the empty coil in others
(i.e. when the empty coil impedance is very much
smaller than with the sample present). We note
the strong effect of melt—sensor separation which
is clearly seen over the entire frequency range.
We also note that the imaginary impedance component now increases with decrease in melt surface—sensor distance. This occurs because the
secondary coil nearest to the melt is much more
affected by the sample than the one further from
the melt. The calculated response is due to this
difference in individual coil response.
In fig. 9, we show the impedance curves for
each of the four interfaces and in table 2 list the
normalized impedance values at a lift-off of 4.8
mm (3/16 inch) for test frequencies of 0.5, 2 and
10 kHz.
We again see that at intermediate frequencies
(2—10 kHz) there is a separation in the curves for
the different interfaces, but that at both low and
high frequencies the curves again converge. For
this sensor design, we find that the most desirable
18~
1.6
‘‘‘I
interface (fig. 2b) has the lowest impedance and
as the interface liquid level near the outer crystal
edge rises, the imaginary impedance component
progressively increases for the reasons discussed
in section 4.2. We have found that this interface
shift can be increased by locating the second
secondary further from the first (fig. 3c). For
example in fig. 10 we show the calculated response for a secondary separation of 38 mm (1~
inch). Now, at 2 kHz the shift in normalized
imaginary impedance component component is
0.3, provided the melt surface—sensor distance is
small. We see again that the curves for the different interfaces converge at high frequency and in
the high frequency limit, the impedance then
depends only on melt level.
5.2. Radial separation
The sensitivity to melt height fluctuations could
be reduced by using differential coils positioned
so that they are similarly affected by the melt,
i.e., a differential secondary coil arrangement with
a radial separation (fig. 3d). Figs. 11 and 12 show
the impedance curves for a flat interface calcu-
I
‘I
~
A
B
Lift Off - 4.8mm
o
Lift Off — 30.2mm
Lift Off — 11.1mm
C Lift Off 17.5mm
—
~14-
—
0
-
E
0
0 12
N
>~
—
A
~-—---~.
—
—
a
C
~?=L°
-
~=
B
-
C
—
+
08—
D
06’’~
102
I
1o~
I
I
io4
io~
1o6
Frequency (Hz)
Fig. 9. Imaginary component of normalized impedance versus frequency for the differential secondary with a 22 mm axial coil
separation and four interface shapes.
/ Simulation of eddy current sensing of GaAs CZ crystal growth
H.N. G. Wadley, KP. Dharmasena
‘‘‘‘‘‘I
70
®J_~_ ~
-
‘‘‘‘‘‘I
i,,~o,
+
~A
563
‘‘‘‘‘‘I
e~
A LiftOff =4.8mm
B LiftOff=11.lmm
~
C LiftOff=17.5nim
-
D Lift Off 30.2mm
-
60
Ii
~‘~-
~=~=
3
1o
102
~::
I
io~
106
Frequency (Hz)
Fig. 10. Imaginary component of normalized impedance versus frequency for the differential secondary with a 38 mm axial coil
separation and four interface shapes.
lated for such a sensor. We observe that the
imaginary component of impedance still however
depends upon the melt level (it decreases with a
decrease in melt surface—sensor distance). A
comparison of the sensor’s response to the four
interface shapes is shown in fig. 13 for two melt—
1~~~
~
~
-0.3
~
100H~
L~ItOIt
00
-0.4
I
~
-0.2
20kHz—’
10 kHz —
-0.1
~5~Hz
5kHz
0.0
2kHz
I
0.1
I
0.2
I
0.3
0.4
Real Z Component
Fig. 11. Normalized impedance curves for the differential sensor with a 17mm radial separation and a flat interface.
564
H.N.G. Wadley, KP. Dharmasena
0C
2
‘
of eddy current sensing of GaAs CZ crystal growth
~
1O~
io
‘
/ Simulation
i0~
io6
Frequency (Hz)
Fig. 12. Imaginary component of normalized impedance versus frequency for the differential secondary with a 17 mm radial coil
separation and a flat interface.
sensor separation distances. Table 3 lists the normalized impedance for the four interfaces at a
lift-off of 4.8 mm (3/16 inch).
It can be seen that the variation in impedance
due to changes of interface shape at intermediate
frequencies (2—5 kHz) is
45% at a lift-off of
4.8 mm (3/16 inch) (fig. 13). This interface resolution becomes progressively smaller as the melt
—~
‘‘‘‘‘‘I
2.0.1
1 .8
surface—sensor distance increases. In the frequency range of greatest interface sensitivity, we
see from fig. 13 that there is a relatively small
dependence upon melt level. Increasing the separation of the secondaries enhances the sensitivity
to the interface until the outer secondary begins
to be influenced by other factors (e.g., furnace
hardware). We believe that a 17 mm separation
‘‘‘‘‘‘I
‘‘I
Lift Off = 30.2mm
—
16-
00
—
-
‘
‘
102
‘‘‘‘‘‘‘1’I~~
~
io~
io6
Frequency (Hz)
Fig. 13. Imaginary component of normalized impedance versus frequency for the differential secondary with a 17 mm radial coil
separation and four interface shapes.
H.N.G. Wadley, K.P. Dharmasena
/ Simulation of eddy current sensing of GaAs CZ crystal growth
Table 2
‘
Normalized imped~t~values for the differential sensor (22
mm axial coil separation)
Inter- Normalized impedance
face f = 500 Hz
f = 2 kHz
f
=
10 kHz
No.
1
2
—0.3181+j1.3957 —0.1683+jl.7202 0.1518+il.6694
0.3I54+jl.3897 —0.1706+jl.7147 0.1504+jl.6658
3
4
—0.3212+jl.3997 —0.1681 +jl.7269 0.1531 +jl.6737
—0.3332+jl.4051 —0.1652+jl.7292 0.1516+jb.6773
sensor design is close to the optimum for interface discrimination for current crystal puller technology.
These simulations have allowed an understanding to be developed of the interaction between the electromagnetic fields of various eddy
current sensor concepts and the features of importance to crystal growth in the gallium arsenide
system. All of the sensor concepts have a high
frequency response that is sensitive (particularly
so for axially displace differential secondaries) to
the melt level and could provide a convenient
means of determining this. The determination of
the interface shape by eddy current methods looks
more promising with the radially displaced differential secondary concepts. Provided the melt level
is already known (or held constant), the 2 kHz
imaginary impedance of a 17 mm radially separated secondary sensor increases from 0.197 to
0.286 in going from a good to poor interface
shape.
The simulations performed above were based
on several simplifying assumptions including:
• Circular cylindrical crystal geometry (in pratice, slightly triangular or square shapes are
frequently found).
Table 3
Normalized impedance values for the differential sensor (17
mm radial coil saparation)
Interface
No.
Normalized impedance
f = 500 Hz
f = 2 kHz
f
1
2
3
4
0.3186+jO.4118
0.3334+jO.3971
0.3058+jO.4207
O.2899+jO.436l
0.0044+jO.2083
0.0079+jO.1992
0.0172+jO.2263
0.0281+jO.2394
0.1233+jO.2272
0.1213+jO.1969
0.1162+jO.2549
0.1175+jO.2862
=
10 kHz
_____________________________________________
.
565
Coaxial alignment of crystal and sensor.
Absence of temperature gradients in the melt
and crystal.
No crystal rotation.
To some extent, each assumption is only approximated, and some comment is needed about
their significance to the conclusions of the research
.
.
Since the enc,rcl,ng des,gn of the eddy current
sensor measures c,rcumferentially averaged sample parameters, small deviations from a circular
cross section leading to a polygonal cross section
of the same area cause very weak perturbations
to the sensor’s response. Provided the cross-sec-
tional area of the crystal does not change, the
sensed response will be very close to that calculated. This circumferential averaging also compensates for misalignment of the growth and sensor axes. The shortening of the crystal—sensor
distance at one side of the sensor is simultaneously accompanied by a lengthening on the other.
Because the response of the sensor is dominated
by the conductivity of the melt, changes in the
crystal thermal gradient are likely to have a negligible effect on the sensor’s response. More serious would be changes in the “average” temperature of the melt sampled by the sensor since the
melt conductivity is temperature dependent.
These changes are likely to be small and we do
not anticipate that under normal conditions of
growth they would pose a serious problem to the
interpretation of the sensor’s response. The last
potential error arises from crystal rotation within
the sensor. Viewed from the sensor, this appears
as an increase or decrease in sample current
density (depending upon direction of rotation). It
will be a weak effect because of the small ratio of
rotational current to that induced by the sensor
and at this preliminary stage of the study seems
reasonable to ignore [14].
These calculations and conclusions have been
developed for the gallium arsenide system. However, they apply in principle, to other materials
provided there is an increase of conductivity during melting (e.g., InP). The detection of the liquid—solid interface shape depends quite sensitively upon the o’ /u ratio. Ratios of 5—100 are
best sutted to thts Capproach. We caution that
.
566
H.N.G. Wadley, KY. Dharmasena
/ Simulation
further calculations are needed for the systems
where o-,/oi~< I (i.e., most metals) and designs
proposed here may be suboptimal.
of eddy current sensing of GaAs CZ crystal growth
berg and M. Mester during this research. This
program of research was supported under US
Army Research Office/DARPA contract number L03/ 86/ C/ 0022.
6. Summary
References
The response of an absolute (single secondary
coil) eddy current sensor encircling a gallium
arsenide crystal being grown from the melt has
been calculated. A strong effect of melt height
upon the sensor’s response has been predicted,
indicating a possible method for the measurement of this important quantity. Calculations of
the sensor’s response to changes of liquid—solid
interface shape reveal a small effect upon the
imaginary impedance component. It has been
found that the magnitude of the sensor’s response to both melt height and interface shape
can he enhanced by the use of differential sensors
and several designs have been examined. The
optimal for interface shape determination is a
radially displaced differential sensor. The degree
to which the interface can be characterized in
practice will depend upon the accuracy of the
liquid level determination and control of other
factors affecting eddy current response. Using the
high frequency data of an axially separated differential sensor provides a possible solution to this.
Although the calculations have been conducted
for gallium arsenide, they apply to any system for
which there is a significant increase in conductivity upon melting. They should not be used to
asses systems where the conductivity decreases
upon melting.
Acknowledgements
We are very grateful for the helpful comments
of Tex McCary, Arnold H. Kahn, Howard Gold-
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bulk crystal growth, in: Proc. Symp. on Intelligent Processing of Materials, Eds. H.N.G. Wadley and W.E.
Eckhart (TMS. Warrendale, 1990) p. 83.
[2] K.S. Choe, J.A. Stefani, T.B. Dettling, J.K. Tien and J.P.
Wallace, J. Crystal Growth 108 (1991) 262.
[3] J.A. Stefani, J.K. Tien, K.S. Choe and J.P. Wallace, J.
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[4] K.S. Choe, J.A. Stefani, J.K. Tien and J.P. Wallace. J.
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Crystal Growth 88 (1988) 30.
[6] K.S. Choe, J.A. Stefani, J.K. Tien and J.P. Wallace, J.
Crystal Growth 88 (1988) 39.
[71D.E. Wilier, Intelligent growth of HgCdTe by the travelling heater method, in: Proc. Symp. on Intelligent Processing of Materials, Eds. H.N.G. Wadley and W.E.
(TMS, Warrendale,
1990)
p. 91.Rzaev
[8] Eckhart
V.M. Glazov,
V.G. Pavlov,
F.R.
and
I.R.
Suleimanov, Soviet Phys.-Semicond. 14 (1980) 162.
[9] V.M. Glazov, S.N. Chizhevskaya and N.N. Glagoleva.
Liquid Semiconductors (Plenum, New York, 1969).
[10] H.N.G. Wadley, K.P. Dharmasena and H.S Goldberg.
The eddy current sensing of gallium arsenide crystal
growth, in: Proc. Symp. on Knowledge Based Control of
Solidification, Eds. B.G. Kushner and C. Levi (TMS,
Warrendale, 1992).
[11] H.N.G. Wadley, A.H. Kahn, Y. Gefen and M.L. Mester.
Eddy current measurement of density during hot isostatic
pressing, in: Review of Progress in Quantitative Nondestructive Evaluation, Vol. 7B, Eds. DO. Thompson and
D.E. Chimenti (Plenum, New York, 1988) p. 1589.
[12] K.P. Dharmasena and H.N.G. Wadley. Eddy current
sensing of intermetallic composite consolidation, in: Review of Progress in Quantitative Nondestructive Evaluation. Vol. bA, Eds. DO. Thompson and D.E. Chimenti
(Plenum, New York, 1991) p. 1111.
[13] C.V. Dodd and W.E. Deeds, J. AppI. Phys. 39 (1968)
2829.
[141A.H. Kahn. private communication.