Solid–liquid interface reconstructions from ultrasonic
time-of-flight projection data
Douglas T. Queheillalt and Haydn N. G. Wadley
Intelligent Processing of Materials Laboratory, School of Engineering and Applied Science,
University of Virginia, Charlottesville, Virginia 22903
~Received 2 April 1997; accepted for publication 23 June 1997!
The large difference in the ultrasonic velocity of solid and liquid semiconducting materials results
in significant ray refraction and pulse time delays during propagation through solid–liquid
interfaces. This has led to an interest in the use of ultrasonic time-of-flight ~TOF! measurements for
deducing the interfacial geometry. A ray-tracing analysis has been used to analyze two-dimensional
wave propagation in the diametral plane of model cylindrical solid–liquid interfaces. Ray paths,
wavefronts, and TOFs for rays that travel from a source to an arbitrarily positioned receiver on the
diametral plane have been calculated and compared to experimentally measured TOF data obtained
using a laser generated/optically detected ultrasonic system. Examinations of both the simulations
and the experimental results reveals that the interfacial region can be identified from transmission
TOF data. When the TOF data collected in the diametral plane were used in conjunction with a
nonlinear least-squares reconstruction algorithm, the interface geometry ~i.e., axial location and
shape! were precisely recovered and the ultrasonic velocities of both solid and liquid phases were
obtained with error of less than ;3%. © 1997 Acoustical Society of America.
@S0001-4966~97!00710-8#
PACS numbers: 43.35.Yb @HEB#
INTRODUCTION
Many single-crystal semiconductors are grown by variants of a Bridgman technique in which a cylindrical ampoule
containing a molten semiconductor is translated through a
thermal gradient, resulting in the directional solidification
and growth of a single crystal.1 During this form of crystal
growth, the solid–liquid interface shape and velocity ~i.e.,
rate of change of its location! together with the local temperature gradient control the mechanism of solidification
~i.e., planar, cellular, or dendritic!, the likelihood of secondary grain nucleation/twin formation ~i.e., loss of single crystallinity!, solute ~dopant! segregation, dislocation generation,
etc., and therefore determines the resultant crystals’
quality.1–4 For crystals grown by the vertical variant of the
Bridgman ~VB! technique, optimum properties are obtained
with a low (;1 – 5 mm/h) constant solidification velocity
and a planar or near planar ~slightly convex toward liquid!
interface shape maintained throughout growth.5,6 The solidification rate and the interface shape are sensitive functions of
the internal temperature gradients ~both axial and radial! during solidification which are governed by the heat flux distribution incident upon the ampoule, the latent heat release at
the interface, and heat transport ~by a combination of conduction, buoyancy surface tension driven convection, and radiation! within the ampoule.7–9 The solid–liquid interface’s
instantaneous location, growth velocity, and interfacial shape
during growth are therefore difficult to predict and optimize,
especially for those semiconductor materials with low thermal conductivity ~i.e., CdTe alloys!.10 Thus the development
of technologies to noninvasively sense the solid–liquid interfacial shape and location ~its change with time gives the
solidification rate! during VB crystal growth have become
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recognized as a key step for developing a better understanding of the growth process and implementing sensor-based
process control.
Ultrasonic techniques are of potential interest for this
sensing application because of the significant difference in
the longitudinal wave velocity between the solid and the liquid phases of many semiconducting materials ~Fig. 1!.11–13
Thus severe ray bending and significant time delays occur
when ultrasound is propagated through a semiconductor
solid–liquid interface. In other material systems, this approach has already been successfully used to recover the interface geometry ~location and shape! during, for example,
the controlled solidification of polycrystalline aluminum.14,15
While single crystal semiconductors are elastically anisotropic, extensive modeling suggests that similar strategies are
likely to work in VB geometry solid–liquid bodies, provided
point sources and receivers are used, and an anisotropic generalization of Snell’s law is incorporated in ray tracing.16 The
advent of laser ultrasonics17 ~i.e., laser generation and interferometric detection18! offers both the needed ultrasonic
point sources/receivers and a noninvasive means for their
introduction into the high-temperature environment of crystal growth furnaces.
In previous work19 we have demonstrated the feasibility
of using laser ultrasonics to characterize solid–liquid bodies,
similar to those found during VB crystal growth. We have
also shown that because of the severe ray bending at the
solid–liquid interface, a detailed ray-tracing analysis is necessary in order to reconstruct the interface from ultrasonic
time-of-flight ~TOF! projection data. A detailed study of ray
paths and TOF analysis of ultrasound propagating in the diametral plane of solid–liquid bodies for a coincident source/
receiver indicated that data obtained from paths on the dia-
0001-4966/97/102(4)/2146/12/$10.00
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2146
FIG. 2. A schematic of the H2O/PMMA bench-top model showing the
solid–liquid interface position (z i ), interfacial curvature (h), and the velocity fields of the liquid and solid phases ( n l , n s ) ~all dimensions in mm!.
FIG. 1. Temperature and orientation dependencies of the quasi-longitudinal
wave velocities for solid and liquid silicon ~Si! and germanium ~Ge!.
metral plane with identical axial positions was insufficient to
precisely characterize all interface shapes. Interfaces that
curved into the solid ~concave interfaces! were shown to be
particularly difficult to characterize with this type of data.
Here, we extend the previous work on wave propagation
in model solid–liquid bodies to a situation where the receiver
is positioned at arbitrary locations relative to the source in
the diametral plane. The diametral plane is defined as the
sagittal plane through the diameter of the ampoule. A detailed study of ray paths, wavefronts, and the TOF of rays
that travel from a source to an arbitrary positioned receiver is
reported. It is shown that the interfacial region can be easily
identified from transmission TOF measurements because the
velocity of the liquid is much smaller than that of the solid.
Also since convex and concave interfaces result in distinctly
different TOF data profiles when source and receiver points
are axially displaced, the interface location and shape ~convex or concave! can be easily determined.
I. EXPERIMENTAL PROCEDURE
A. Model interfaces
A bench-top model with known solid–liquid velocities
was constructed from water and solid polymethylmethacrylate ~PMMA! contained in a 37.5-mm-radius cylindrical aluminum ampoule; see Fig. 1. Ultrasound was optically generated and laser interferometrically detected on the outer
surface of this aluminum ampoule. The interface convexities
h @defined such that h.0 corresponds to a convex interface
~Fig. 2!# examined here were 62, 65, 610 mm and planar
(h50 mm). The water and PMMA had a measured longitudinal wave velocity of 1.49760.01 mm/ms and 2.670
60.01 mm/ms at 21 °C, respectively, while the 2024-T6 Al
alloy had a measured longitudinal wave velocity of 6.35
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60.01 mm/ms and a shear wave velocity of 3.01
60.01 mm/ms at 21 °C. All experiments were conducted at
2162 °C.
B. Laser ultrasonic measurements
A laser ultrasonic sensing system ~Fig. 3! was used to
acquire TOF data between precisely positioned source and
receiver points. The ultrasonic source was a ;10-ns duration
Q-switched Nd:YAG laser pulse of 1.064-mm wavelength.
The energy per pulse was 15 mJ and the roughly Gaussian
beam of the multimode pulse was focused to an approximate
circular spot 0.5 mm in diameter. Thus the resultant source
power density was ;1500 MW/cm2. A glass slide and propylene glycol couplant were used to enhance the acoustic
signal strength.17
The ultrasonic receiver was a heterodyne laser interferometer. It responded to the sample’s out-of-plane ~normal!
surface displacement associated with wavefront arrivals at
the receiver point.18 It was powered by a 1-W single mode
argon ion laser ~operated at 0.25 W!, which produced a continuous Gaussian beam of 514-nm wavelength focused to a
circular spot ;100 m m in diameter. The signal from the interferometer was bandpass filtered between 10 kHz and 10
MHz and recorded with a ~LeCroy 7200! precision digital
oscilloscope at a 2-ns sampling interval using 8-bit analogto-digital conversion. To improve the signal to noise ratio,
each waveform used for TOF measurement was the average
of ;100 pulses collected at a pulse repetition rate of 20 Hz.
A fast photodiode identified the origination time for the ultrasonic signals.
Two sets of computer controlled translation stages were
positioned such that the ampoule could be translated along
the axial z axis allowing the fixed receiver’s position to be
located at any point on the diametral plane. Thus the source
could be independently translated to acquire a ‘‘fan-beam’’
ultrasonic TOF projection set. Multiple ultrasonic TOF projection scans were acquired for each interface in two sensing
configurations. The first configuration positioned the receiver
at four positions in the liquid ~z r 55, 10, 15, and 20 mm
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2147
FIG. 3. A schematic of the laser ultrasonic test facility used to evaluate the sensor concepts.
above the outer edge of the interface, z i !. The second positioned the receiver at four positions in the solid phase ~z r
525, 210, 215, and 220 mm!. For each receiver positions, the ultrasonic source was scanned from the liquid
phase (z s 540 mm) to the solid phase (z s 5220 mm) at
1-mm increments to obtain a TOF projection set.
II. RAY PATH AND TIME-OF-FLIGHT ANALYSIS
It is assumed that infinitesimal omnidirectional ultrasonic point sources and receivers were positioned on opposite sides of the diametral plane. When an ultrasonic pulse is
generated by the source, it was assumed to propagate in all
directions in the diametral plane. One of the directions will
usually result in a ray ~which may or may not be refracted,
depending on the source and receiver’s position! whose path
reaches the receiver point. Knowledge of this ray path is a
necessary first step for understanding ultrasonic propagation
through a solid–liquid interface, for predicting a ray’s TOF,
and, ultimately, for reconstructing the interface location and
curvature from TOF projection data.
If the coordinates of the source (x s ,z s ) and receiver
(x r ,z r ) points on the diametral plane are prescribed, then
determining the ray path between these two points constitutes a boundary-value problem. The solutions of boundaryvalue problems like this are usually preceded by solutions of
initial value problems20 in which initial ray angles at the
source point are prescribed and the ray paths are obtained by
solving for the refraction angles at the interface. After obtaining the ray paths for an arbitrary set of initial ray angles
emanating from the source point, the ray path between the
prescribed source/receiver points can be obtained using the
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shooting method.21 In this approach, an initial ray direction
is first arbitrarily chosen, and the distance between the receiver point and the intersection of the ray path with the
outer boundary calculated. This procedure is then repeated
using a slightly different initial ray angle until the distance is
smaller than a prescribed tolerance d, ;50 m m. The ray path
and wavefront analysis solved the initial value problem,
while the boundary-value problems solution was used later
for image reconstruction. A further complication arises because acoustic energy can only be transmitted across an interface when the refraction angle is less than p /2. This could
sometimes lead to an incomplete projection data set ~i.e.,
dark zone! with potentially deleterious consequences for
some interface reconstruction techniques. It will be shown
that this problem can be overcome using this fan beam approach.
The measured time-of-flight, t m , for a ray that propagates along a path of length, L m , is defined by
t m5
E
Lm
dl
,
n
m51,2,...,M ,
~1!
where dl is an infinitesimal element of the path, n is the local
ultrasonic velocity ~1/n is the local slowness! within the object, and M is the number of different rays. When an ultrasonic ray is incident upon a solid–liquid interface, both reflected and refracted rays propagate on the diametral plane
~defined by the incident propagation vector and the normal to
the interface at the intersection of the incident ray with the
interface!.22,23 We arrange for the diametral cross-sectional
plane on which ultrasonic rays propagate to coincide with the
x-z plane in a Cartesian coordinate system ~Fig. 2!. It is
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until refracted by the interface during propagation into the
solid. Note that some of the rays are refracted at the first
liquid–solid interface encountered and are then refracted
again at the second solid–liquid interface ~doubly transmitted rays!. Others are only refracted at the first liquid–solid
interface and are termed singly transmitted rays. Both doubly
transmitted rays and rays with direct paths may reach the
same boundary point. However, these different kinds of rays
are experimentally distinguishable because the doubly transmitted rays have different travel times and are weaker because they suffered extra energy losses due to refraction and
mode conversion at each interface. When the source was
located below the interface @Fig. 4~b!#, allowable ray paths
are either straight ~in the solid! or are refracted only once by
the interface.
Wavefronts separate ultrasonically disturbed regions
from those that are undisturbed. Thus the TOF of an ultrasonic signal can, in principle, be measured at any receiver
point on the sample’s periphery that is intersected by a wavefront. The wavefronts at any time after source excitation can
be obtained by connecting points along the ray paths with the
same travel time. The calculated quasi-longitudinal wavefronts at 10-ms intervals are shown in Fig. 4~a! and ~b!. Ray
paths and wavefronts for h52 mm and h510 mm interfaces
are similar in nature and are not shown.
2. Time-of-flight projections
FIG. 4. Ray paths and wavefronts on the diametral plane of a convex interface (h55 mm) for a source located in ~a! the liquid (z s 515 mm) and ~b!
the solid (z s 5215 mm).
assumed that during crystal growth, the thermal gradients are
small and axially symmetric resulting in relatively uniform
velocities on both sides of the spherically shaped interface,
simplifying the computation of t m . The detailed procedure
for calculating ray paths is similar to that of Ref. 19 and is
therefore omitted here.
A. Convex interfaces „ h >0…
1. Ray paths and wavefronts
An example of the numerically simulated ray paths and
wavefronts on the diametral plane for a convex interface (h
55 mm) are shown in Fig. 4~a! and ~b! for source points
located in either the liquid (z s 515 mm) or the solid (z s
5215 mm), respectively. For a source point located above
the interface @Fig. 4~a!# ray paths in the liquid are straight
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The most convenient manner in which to present the
TOF data is in the form of ultrasonic TOF projections. The
TOF projection data for convex interfaces ~h52, 5, and 10
mm! are shown in Fig. 5~a!–~d! for a receiver located above
the interface at locations z r 55, 10, 15, and 20 mm, respectively. In Fig. 5 the TOF of rays generated below the interface (z s ,0) corresponds to singly transmitted rays. When
the source is located above the interface (z s .0), two TOF
arrivals corresponding to doubly transmitted and direct rays
are shown. The experimental TOF projection data of Fig. 5
~the points! are seen to be in good agreement with the numerically simulated data predicted by the ray path model ~the
curves!.
From direct observation of the TOF projection data ~Fig.
5! the interface location (z i ) is identified by the transition of
direct and doubly transmitted rays to singly transmitted rays
at the interface edge (z i 50). Also, the TOF of doubly transmitted rays coincides with that of the singly transmitted rays
at z i 50 for each interface convexity, and corresponds to the
minimum time-of-flight in the projection data. The TOF projections of direct rays in the liquid are constant for all interface convexities at each receiver location. However, the TOF
projections of singly and doubly transmitted rays are shifted
to the left ~i.e., to shorter TOFs! as the interface convexity h
was increased. This can be intuitively explained. As the convexity is increased, the fraction of the total ray path length in
the lower velocity liquid decreases, while the fraction in the
higher velocity solid increases, resulting in an overall decrease in TOF. Also, it is interesting to note that as the receiver’s position was translated further from the interface
edge ~i.e., as z r goes from 5 to 10 to 15 to 20 mm! the overall
TOF projection for each interface convexity shifts to in-
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FIG. 5. Ultrasonic time-of-flight projection data of convex interfaces, h52, 5, and 10 mm, for a receiver located in the liquid at axial positions ~a! z r
55 mm, ~b! z r 510 mm, ~c! z r 515 mm, and ~d! z r 520 mm.
creased TOF values and the relative time spacing between
each interface ~h52, 5, and 10 mm! also increases.
The TOF projection data for convex interfaces ~h52, 5,
and 10 mm! are shown in Fig. 6~a!–~d! for a receiver located
below the interface at locations z r 525, 210, 215, and
220 mm, respectively. In Fig. 6 the TOF of rays generated
below the interface (z s ,0) correspond to direct rays. When
the source is located above the interface (z s .0) only singly
transmitted rays were observed. Again, the TOF projection
data show good agreement between the experimentally determined TOFs and the numerically simulated data predicted
by the model.
When the receiver is positioned below the interface ~Fig.
6! the interface location is again identified by the transition
of singly transmitted rays to direct rays at the interface
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edge (z i 50). The TOF of singly transmitted ray paths coincides with that of the direct ray paths at z i 50 for all interface convexities. The TOF projections of direct rays are constant for all interface convexities at each receiver location.
However, the TOF of direct rays has a minimum at z r because the ray path length is the smallest. Similar to before,
the TOF data projections of singly transmitted rays are
shifted to shorter TOFs as the interface convexity was increased. Again, as the convexity increases the ray travels
more in the solid and less in the liquid resulting in shorter
TOFs. It is interesting to note that as the receiver’s position
was axially translated further from the interface ~i.e., as z r
goes from 25 to 210 to 215 to 220 mm! the overall TOF
increases for each interface convexity. However, this affect
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2150
FIG. 6. Ultrasonic time-of-flight projection data of convex interfaces, h52, 5, and 10 mm, for a receiver located in the solid at axial positions ~a! z r
525 mm, ~b! z r 5210 mm, ~c! z r 5215 mm, and ~d! z r 5220 mm.
is not as pronounced when the receiver is located in the
solid.
B. Concave interfaces „ h <0…
1. Ray paths and wavefronts
Numerically simulated ray paths and wavefronts on the
diametral plane for a concave interface (h525 mm) are
shown in Fig. 7~a! and ~b! for a source located above the
interface (z s 520 mm) and below the interface (z s
5220 mm), respectively. In Fig. 7~a! ray paths in the liquid
are straight, whereas they are bent by the interface during
propagation into the solid. Notice that none of the rays are
doubly transmitted; they can only be refracted once at the
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liquid–solid interface ~singly transmitted rays!. However,
due to the divergent nature of the concave interface, there
may be a region of nonpropagation ~i.e., dark zone! in which
the wavefront does not intersect the outer boundary. The first
dark zone on the left (x,0) is due to ray bending caused by
the interface, while no rays travel through the second dark
zone (x.0) because the refraction angle in the solid is
greater than p /2. It is interesting to note that the second dark
zone increases in size as the convexity of the interface becomes increasingly negative ~from h525 mm to h
5210 mm! and/or when the source point was translated toward the interface ~i.e., as z s goes from 20 to 15 to 10 to 5
mm!. In Fig. 7~b! ray paths are straight ~in the solid! or are
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FIG. 7. Ray paths and wavefronts on the diametral plane of a concave
interface (h525 mm) for a source located in ~a! the liquid (z s 520 mm)
and ~b! the solid (z s 5220 mm).
refracted once by the interface. Again, the first dark zone on
the left (x,0) is due to ray bending caused by the interface.
No second dark zone (x.0) was observed for the h
525 mm interface. As before, when the convexity was increased negatively ~from h525 mm to h5210 mm! and/or
as the source point approached the solid–liquid interface,
another dark zone in the liquid (x.0) near the interface
edge (z i 50) was observed. The calculated quasilongitudinal wavefronts at 10-ms intervals are also shown in
Fig. 7~a! and ~b!. Ray paths and wavefronts for the h
522 mm and h5210 mm interfaces were similar in nature
~except for the magnitude of the dark zones! and are not
shown here.
2. Time-of-flight projections
The ultrasonic TOF projections for concave ~h522 and
25 mm! interfaces are shown in Fig. 8~a!–~d! for a receiver
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located above the interface ~z r 55, 10, 15, and 20 mm!, respectively. In Fig. 8 the TOF of rays generated below the
interface (z s ,0) corresponds to singly transmitted rays.
When the source was located above the interface (z s .0)
only direct rays were observed. The TOF projections for h
5210 mm are not shown because the magnitude of the dark
zone extended beyond that of the scanned region ~i.e., the
singly transmitted wavefronts did not intersect the outer
boundary in the scanned region, 40 mm.z s .220 mm!.
Both TOF projection sets show good agreement between the
experimentally determined TOFs and the numerically simulated data predicted by the ray path models.
With the receiver located above the interface ~Fig. 8! the
interface location may be deduced by the presence of a dark
zone just below the interface or by the absence of direct rays
below the interface. The TOF projects of direct rays in the
liquid are constant for both interface convexities and vary
accordingly with the receiver’s position. However, the TOF
projection of singly transmitted rays is shifted to the right
~i.e., longer TOFs! as the interface convexity h was increased. As the negative convexity was increased, the fraction of the total path length of the ray in the lower velocity
liquid increased, and the fraction in the higher velocity solid
decreased resulting in an overall increase in TOF. Again, it is
interesting to note that as the receiver’s position was translated further from the interface edge ~as z r goes from 5 to 10
to 15 to 20 mm!, the overall TOFs of singly transmitted rays
for each interface convexity shifts to increased TOF values.
The magnitude of the dark zone increased with increasing
convexity ~as h goes from 22 to 25 to 210 mm! and the
magnitude for each individual interface decreased as the receiver’s position was translated further from the interface
edge. It also appears that the rate of change in dark zone
magnitude with respect to the receiver’s position is dependent on the interface convexity.
The ultrasonic TOF projections for concave ~h522 and
25 mm! interfaces are shown in Fig. 9~a!–~d! for a receiver
located below the interface ~z r 525, 210, 215, and
220 mm!, respectively. In Fig. 9 the TOF of rays generated
below the interface (z s ,0) corresponds to direct rays. When
the source was located above the interface (z s .0), only
singly transmitted rays were observed. The TOF projection
data for the h5210 mm interface convexity are not shown
because the dark zone extended beyond that of the scanned
region. Both TOF projection sets show good agreement between the experimentally determined TOFs and the numerically simulated data predicted by the ray path models.
When the receiver was located below the interface ~Fig.
9! the interface location is identified by the presence of a
dark zone just above the interface or by the absence of direct
rays above the interface edge (z i 50). The TOFs for direct
rays in the solid are constant for both interface convexities
and shift accordingly with the receiver’s position. When no
dark zone was observed, the TOF of singly transmitted rays
coincided with that of direct ray paths at z i 50 for the interfaces. Again, the TOF data projections of singly transmitted
rays are shifted to longer TOFs as the interface convexity h
was increased negatively and/or as the receiver’s position
was translated further from the interface edge. In addition,
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FIG. 8. Ultrasonic time-of-flight projection data of concave interfaces, h522 and 25 mm, for a receiver located in the liquid at axial positions ~a! z r
55 mm, ~b! z r 510 mm, ~c! z r 515 mm, and ~d! z r 520 mm.
the overall magnitude of the dark zone increased with increasing convexity ~as h goes from 22 to 25 to 210 mm!
and the magnitude of each individual interface decreased as
the receiver’s position was translated further from the interface edge.
C. Discussion
An inspection of the ultrasonic TOF projection data reveals that both the magnitude and form of TOF curves are
unique for different sensing configurations and interface curvatures. The TOF projection data reveal when a solid–liquid
interface exists either by a transition in propagation modes
~i.e., doubly transmitted rays to singly transmitted rays or
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singly transmitted rays to direct rays! or the observance of
dark zones at the interface edge (z i 50). Determining
whether the interface was above or below the receiver point
can be done by inspecting the TOF curves. For either convex
or concave interfaces, the TOF curves for direct rays will be
symmetric about the receiver point and much larger when the
receiver is positioned in the slower velocity liquid. Also, if
we compare convex and concave interfaces, we find that for
a convex interface the TOF of transmitted rays ~both singly
and doubly! increases with decreasing interface convexity,
while for a concave interface the TOF of singly transmitted
rays decreases with decreasing interface convexity.
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FIG. 9. Ultrasonic time-of-flight projection data of concave interfaces, h522 and 25 mm, for a receiver located in the solid at axial positions ~a! z r
525 mm, ~b! z r 5210 mm, ~c! z r 5215 mm, and ~d! z r 5220 mm.
III. INTERFACE CURVATURE RECONSTRUCTIONS
A reconstruction method is needed to quantitatively determine the interface location, the interfacial curvature, and
velocity fields. There are a variety of techniques available for
reconstructing an object image from ultrasonic TOF projection data, including nonlinear least-squares fitting, algebraic
reconstruction techniques ~ART!, and convolution backprojection ~CB! methods.24,25 Convolution backprojection
methods have not been applied to situations where ray bending is significant, and although algebraic reconstruction techniques are beginning to incorporate ray bending,26 both of
these procedures require large data sets which are difficult to
obtain in the crystal growth environment. Reconstruction ap2154
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proaches that can be used with limited data and exploit the
often significant a priori information available are preferable. The crystal grower requires the axial location (z i ) of
the interface ~its change with time gives the solidification
velocity! and its approximate curvature (h). The liquid and
solid velocities ( n l , n s ) are also useful since they are related
to the local temperature and the crystals crystallographic orientation. In previous work, we used a nonlinear least-squares
reconstruction approach and determined that it often provides sufficiently accurate results from relatively limited data
sets in minimal computational time.
We assume an interface model of the type shown in Fig.
2, but where h, z i , n l , and n s are all unknown. For the ray
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2154
FIG. 10. Nonlinear least-squares interface reconstructions of convex interfaces for all receiver locations above the interface with interface convexities
of ~a! h52 mm, ~b! h55 mm, and ~c! h510 mm.
path models, the TOF depends nonlinearly on the interface
convexity (h), interface location (z i ), liquid ( n l ), and solid
( n s ) velocities, and the mean-square error given by
M
x 5
2
( @ t i 2 t̂ i~ x i ;h,z i , n l , n s !# 2 ,
i51
~2!
where t i are the measured TOFs and t̂ i are the numerically
simulated TOFs for an initial estimate of the interface parameters. To reconstruct the model unknowns from the ultrasonic TOF projection data, a Levenberg–Marquardt nonlinear least-squares reconstruction method was used.27 The
nonlinear least-squares method determines the best-fit parameters (h,z i , n l , n s ) by minimizing x 2 .
Using data (;30– 50 rays) for convex interfaces, the
reconstructed interface locations and curvatures when the receiver was above the interface are shown in Fig. 10~a!–~c!
for h52, 5, and 10 mm, respectively. The reconstructed parameters obtained when the receiver was below the interface
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FIG. 11. Nonlinear least-squares interface reconstructions of convex interfaces for all receiver locations below the interface with interface convexities
of ~a! h52 mm, ~b! h55 mm, and ~c! h510 mm.
are shown in Fig. 11~a!–~c!. Both Figs. 10 and 11 represent
eight individual reconstructions each, where some of the reconstructions overlap each other. It is clear that the reconstructed interface locations and curvatures coincide well with
those of the actual interfaces. For the reconstructed velocities, the mean liquid velocity was n l 51.487 ( s 50.040) and
the mean solid velocity was n s 52.677 ( s 50.037).
Using concave interface data (;10– 55 rays) the reconstructed interface locations and curvatures when the receiver
was above the interface are shown in Fig. 12~a! and ~b! for
h522 and 25 mm, respectively. The reconstructed parameters obtained when the receiver was below the interface are
shown in Fig. 13~a! and ~b!. Again, Figs. 12 and 13 represent
multiple reconstructions with some overlap in the reconstructed surfaces. The reconstructed interface locations and
curvatures coincide well with those of the actual interfaces.
For the reconstructed velocities, the mean liquid velocity was
n l 51.493 ( s 50.006) and the mean solid velocity was n s
52.713 ( s 50.037).
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FIG. 12. Nonlinear least-squares interface reconstructions of concave interfaces for all receiver locations above the interface with interface convexities
of ~a! h522 mm and ~b! h525 mm.
The reconstructed interface parameters and velocities for
all the models ~convex and concave! are precisely recovered
and the nonlinear least-squares reconstruction algorithm represents a robust approach for interface reconstructions using
ultrasonic TOF projection data. Using the boundary-value
solutions with free parameters (h, z i , n l , n s ) the reconstruction algorithm converged upon the correct interface
model ~convex or concave!, and thus recovered the interface
geometry ~i.e., solid–liquid interface location, curvature! and
velocity fields for all interface convexities from laser ultrasonic TOF projection data collected in the diametral plane.
IV. CONCLUSIONS
An extension of a previously developed laser ultrasonic
sensor methodology for sensing the solid–liquid interface
location and curvature similar to those encountered during
vertical Bridgman growth of CdTe alloys and other semiconducting materials has been conducted. A combination of ray
tracing, wavefront, and TOF analysis and experimental testing on model cylindrical solid–liquid interfaces with a laser
ultrasonic system was used to improve sensing concepts to
determine a solid–liquid interface’s location, curvature, and
velocity fields from ultrasonic TOF projection data collected
in the diametral plane. Because convex and concave solid–
liquid interfaces resulted in uniquely different TOF data profiles, the interface shape ~convex or concave! was readily
determined from the TOF data. When TOF data ~;10–60
rays! collected in the diametral plane was used in conjunction with a nonlinear least-squares reconstruction algorithm,
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FIG. 13. Nonlinear least-squares interface reconstructions of concave interfaces for all receiver locations below the interface with interface convexities
of ~a! h522 mm and ~b! h525 mm.
the interface geometry ~i.e., location and curvature! has been
successfully reconstructed and the ultrasonic velocities of
both solid and liquid obtained. The development and integration of this sensing methodology into semiconducting material growth promises to significantly advance process understanding and design.
ACKNOWLEDGMENTS
We are grateful to Brent Bollong and Art Sochia of
Johnson Matthey Electronics for helpful discussions concerning crystal growth. This work has been performed as part
of the Infrared Materials Producibility Program ~IRMP!
managed by Duane Fletcher ~JME! that includes Johnson
Matthey Electronics, Texas Instruments, II-VI 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 IRMP consortium
work, and our work within it, has been supported by ARPA/
CMO under Contract No. MD A972-91-C-0046 monitored
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