JOURNAL OF APPLIED PHYSICS
VOLUME 83, NUMBER 8
15 APRIL 1998
Temperature dependence of the elastic constants of solid and liquid
Cd0.96Zn0.04Te obtained by laser ultrasound
Douglas T. Queheillalta) and Haydn N. G. Wadley
Intelligent Processing of Materials Laboratory, School of Engineering and Applied Science,
University of Virginia, Charlottesville, Virginia 22903
~Received 30 October 1997; accepted for publication 9 January 1998!
Laser based ultrasonic techniques provide a potential noninvasive sensing methodology to monitor
the solid–liquid interface shape and position during crystal growth provided, the temperature
dependence of the ultrasonic velocity for both solid and liquid phases are known a priori. A laser
ultrasonic approach has been used to measure the ultrasonic velocity of single crystal solid and
liquid Cd0.96Zn0.04Te as a function of temperature. The longitudinal wave velocity was found to be
a strong monotonically decreasing function of temperature in the solid and liquid phases and
exhibited an abrupt almost 50% decrease upon melting. Using both longitudinal and shear wave
velocity values together with data for the temperature dependent density allowed a complete
evaluation of the temperature dependent single crystal elastic stiffness constants ~C 11 , C 12 , C 44! for
the solid and the adiabatic bulk modulus (K S ) for the liquid. In addition, evaluation of the
thermoelastic modulus (M T ) has given insight on the impact of dislocation generation from
excessive thermal gradient induced stresses during growth. The simplicity of making high
temperature laser ultrasonic measurements together with the large longitudinal wave velocity
difference between the solid and liquid phases suggests a laser based ultrasonic sensor has
significant potential for sensing the solid–liquid interface during the growth of CdTe alloys.
© 1998 American Institute of Physics. @S0021-8979~98!04208-X#
I. INTRODUCTION
accompanying change in density associated with thermal expansion.
One ultrasonic approach to solidification sensing has
sought to exploit the acoustic impedance difference across a
solid–liquid interface by detecting ultrasonic reflections
from the interface. A pulse-echo approach has been investigated by Parker et al.9 for Si, by Jen et al.10 and Carter
et al.11 for Ge. For both of these semiconductors, good solid
and liquid elastic constant and density ~hence ultrasonic velocity! data existed in the literature4–8 and the measured
time-of-flight ~TOF! of signals reflected from the solid–
liquid interface could be used to estimate the interface position. However, the large, usually poorly characterized axial
temperature gradient present during vertical Bridgman
growth introduced uncertainties in mapping TOF data into
the interface’s axial position, and little information could be
inferred about interfacial curvature by this method. An alternative approach is to exploit the ultrasonic delay when ultrasonic rays are partially transmitted through the liquid phase.
Mauer et al.12 and Norton et al.13 have reported measurement of the TOF for rays transmitted through a solid–liquid
interface during the controlled solidification of metals. Using
the predictions of a heat flow model, they were able to construct an interface geometry model and used a simple ray
tracing code, temperature dependent elastic constants, and
temperature dependent density data to compute the TOF values for a set of rays propagating between the known source
and receiver points on the boundary of the body. A nonlinear
least squares reconstruction technique was then able to fit for
the unknown coefficients of the interface geometry model
and thus robustly reconstruct the interface position and shape
Single crystal CdTe and substitutional solid solution
Cd12y Zny Te alloys ~with 0.03,y,0.05! are used for solid
state g-ray detectors and as substrate materials for the epitaxial growth of Hg12x Cdx Te ~with x50.2! thin film infrared
focal plane array ~IRFPA! sensors.1–3 As the size of imaging
detectors continues to increase, a need is emerging for large
area Cd1-y Zny Te wafers with low defect ~i.e., dislocations
and precipitate! densities, uniform Zn distribution, and high
infrared transmission coefficients. Although the bulk crystal
growth of CdTe and its alloys has been practiced for many
years, none of the various growth techniques which include
zone refining, Bridgman, and liquid encapsulated Czochralski methods, have yet demonstrated the reliable growth of
material with the quality needed for today’s IR detector
applications.3
The difficulties associated with the growth of premium
quality Cd12y Zny Te ~with y50.04! alloys has stimulated an
interest in the development of ultrasonic sensor technologies
to noninvasively monitor the solid–liquid interface location
and shape throughout the vertical Bridgman ~VB! growth
process. The principle underlying these ultrasonic sensing
methodologies is the often strong temperature dependence of
the ultrasonic velocity in the solid and liquid phases, and the
frequently large increase in the longitudinal wave velocity
upon solidification.4–8 These effects arise from temperature
dependent softening of the solid and liquid phases and the
a!
Electronic mail: [email protected]
0021-8979/98/83(8)/4124/10/$15.00
4124
© 1998 American Institute of Physics
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
from a small ~10! transmission ray path data set.
The emergence of optical ~laser! methods for the remote
excitation and detection of ultrasound now enables these
transmission approaches to be extended to vertical Bridgman
configurations. Recent ultrasonic ray path modeling and
bench top testing on simulated interfaces has identified several ultrasonic transmission sensor concepts capable of recovering the interface position and shape encountered during
vertical Bridgman growth.14,15 However reliable elastic constant and density data as a function of temperature are
needed in order to realize the potential of these laser ultrasonic sensing approaches. The temperature dependence of
the linear thermal expansion for both solid and liquid CdTe
has been previously determined by Williams et al.16 and Glazov et al.17 While some data relating the elastic constants of
CdTe to temperature exists for low temperatures ~,300
K!,18,19 none has been published for the elevated temperatures encountered during crystal growth and the role of Zn in
CdZnTe alloys has also not yet been established.
Elastic constants at ambient temperature or below are
often determined by ultrasonic methods. Both resonance
methods and pulse-echo methods have been used.20 However
both methods usually use piezoelectric transducers to excite/
detect ultrasound and are therefore experimentally difficult to
use at high temperatures since they require buffer rods, cooling systems, special bonds or momentary contact to keep the
piezoelectric transducers below their Curie point. The problem is further compounded by the high vapor pressure of Cd
in CdTe alloys as they approach their melting point.21 This
requires encapsulation in thick ampoules and precludes direct contact with the transducers.
The advent of Q-switched lasers allows the rapid deposition of optical energy onto a surface and the resultant production of energetic elastic waves ~ultrasound!. The two
dominant mechanisms of laser ultrasonic generation are thermoelastic expansion and ablation ~vaporization in liquids!.22
At low power densities (,106 W/cm2), localized heating in
the region of irradiation produces a thermal expansion. This
is constrained by surrounding material at ambient temperature creating thermoelastic stresses and the generation of
both longitudinal and shear waves. At higher power densities
(.108 W/cm2), the incident laser pulse increases the surface
temperature above the vaporization threshold and a small
region of material is ablated. During ablation, atoms are
ejected from the surface at high velocities creating a force
normal to the surface. Conservation of momentum induces a
large amplitude impulse into the sample and a strong longitudinal wave is generated.
Interferometric detection of ultrasound is generally accomplished with surface displacement interferometers.23
These interferometers either measure the phase modulation
directly or the Doppler shift of the beam reflected off the
surface which is displaced by the ultrasonic wave. By using
a combination of laser excitation and interferometric detection, laser based ultrasonics overcomes many of the physical
limitations of conventional piezoelectric tranducers and are
suitable for high temperature applications and hostile
environments.
D. T. Queheillalt and H. N. G. Wadley
4125
Here, we show that measurement at high temperatures
can be accomplished quite readily on quartz encapsulated
samples by the use of laser ultrasonics using lasers to noninvasively generate and detect ultrasound. In this work, the
temperature dependence of the ultrasonic velocity for solid
and liquid Cd0.96Zn0.04Te contained in quartz ampoules with
very small free volumes has been deduced from TOF data
collected in primary low index crystal orientations. Using
these velocity values together with data for the temperature
dependent density allowed a complete evaluation of the
single crystal elastic stiffness constants and the adiabatic
bulk modulus for liquid Cd0.96Zn0.04Te.
II. BACKGROUND
A. Wave propagation
In ideal ~nonviscous, nonabsorbing! liquids the pressure/
density cycle due to propagation of an ultrasonic wave takes
place adiabatically or at constant entropy.24 These longitudinal waves propagate with a constant velocity which is characteristic of the liquid. In the case of nonideal liquids, the
liquid may exhibit either viscous or absorption effects, i.e.,
some of the ultrasonic energy is dissipated ~local heating
effects! and the wave attenuates in amplitude and the ultrasonic velocity becomes frequency dependent ~dispersive
propagation!. Assuming nonideal wave propagation the
Navier–Stokes equation of hydrodynamics ~in the acoustic
approximation! is represented by
r0
S D
S D S
] 2 u~ x,t !
d P ] 2 u~ x,t !
5r0
2
]t
dr
]x2
1 b1
D
] ] 2 u~ x,t !
4
h
,
3
]t
]x2
~1!
where u(x,t) is the particle velocity, r 0 is the density of the
liquid in the absence of the sound wave, d P and dr are the
changes in the pressure and density due to u(x,t), b is the
bulk or volume viscosity, and h is the dynamic or shear
viscosity.24 Assuming one-dimensional plane harmonic wave
propagation ~u 1 is a function of x 1 only, u 2 5u 3 50!,
d P/ d r 5K S / r 0 , where K S is the adiabatic bulk modulus and
setting a characteristic frequency v v 5K S @ b1(4/3) h # 21 a
dispersion equation may be expressed as
k 25 v 2
S D
r0
1
.
K S 11i v / v v
~2!
Since v must assume real values, k is allowed to take on
complex values, k5k 1 2ik 2 , where k 1 and k 2 are both real
and positive, and solving for a plane wave propagating in the
x 1 direction, the longitudinal wave phase velocity can be
expressed in the form
v5
v
5
k1
A
2K S
@ 11 ~ v / v v ! 2 #
.
r 0 11 @ 11 ~ v / v v ! 2 # 1/2
~3!
As b→0 and h→0 and/or the condition v ! v v is satisfied
~low MHz region for most liquids! the longitudinal wave
phase velocity approaches that of an ideal liquid and is given
by
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
4126
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
D. T. Queheillalt and H. N. G. Wadley
TABLE I. Relationships between elastic constants and ultrasonic velocities for cubic materials.
v liq5
A
Direction of
wave propagation
Direction of
particle motion
Type of wave
^100&
^100&
Longitudinal
^100&
In plane ~100!
Shear
^110&
^110&
Longitudinal
^110&
^001&
Shear ~fast shear!
^110&
^ 1̄1̄0 &
Shear ~slow shear!
^111&
^111&
Longitudinal
^111&
In plane ~111!
Shear
KS
.
r
V l5
i51,2,3,
~5!
where r is the mass density of the solid, k5 v / v is the wave
number, v is the angular frequency, c i jkl are the components
of the elastic stiffness tensor, d j ( j51,2,3) are the components of the unit particle displacement vector d, and d i j is the
Kronecker delta.26 For a given propagation direction vector
n, the squares of the phase velocities of the bulk modes are
obtained as the eigenvalues of Eq. ~5!. The particle displacement vector d of a bulk mode is obtained as a unit eigenvector for the phase velocity ~eigenvalue! of the bulk mode.
Once the phase velocity and the displacement vector for a
specific mode are know, the Cartesian components of the
group velocity, vg , are then obtained from25,26
v gi5
c i jkl d j d l n k
,
rv
i51,2,3.
~6!
Equation ~6! may be greatly simplified for propagation in
high symmetry directions, where the phase and group velocities are equal.
The most straightforward and precise method for evaluating the elastic stiffness constants of anisotropic solids is
through the measurement of the ultrasonic velocity of pure
longitudinal and pure shear modes propagating in principal
symmetry directions, i.e., ^100&, ^110&, and ^111& for cubic
C 11
r
V s5
C 44
r
A
C 111C 1212C 44
2r
Vss5
A
A
A
A
V s5
The particle motions in elastic solids can be resolved
into three perpendicular components, one longitudinal and
two transverse.25,26 Each of the three modes has its own characteristic velocity, although in isotropic solids the two shear
velocities are equal. For anisotropic solids the phase velocity
( v ) of the three possible bulk wave modes ~quasilongitudinal, fast, and slow quasishear! in a direction defined by the
unit propagation vector n, with components n j ( j51,2,3)
can be found from the eigenvalues of the Christoffel equation:
A
A
V l5
V fs5
V l5
~4!
~ k 2 c i jkl n k n l 2 rv 2 d i j ! d j 50,
Velocity
C 44
r
C 112C 12
2r
C 1112C 1214C 44
3r
C 112C 121C 44
3r
materials. Cadmium telluride and its substitutional solid solution CdZnTe alloys have a zinc-blende ~cubic! structure. It
can be shown from Eqs. ~5! and ~6! that the three bulk wave
velocities ~one longitudinal and two shear! can be expressed
as a function of the elastic stiffness constants C 11 , C 12 , C 44
and the density r. Table I gives the expressions for the pure
longitudinal and pure shear velocities in the primary ^100&,
^110&, and ^111& directions for a cubic crystal structure. Note,
that the two shear wave velocities are equivalent in both
^100& and ^111& directions. Moreover, it can be shown that in
all cases in Table I, except for the shear wave in the ^111&
direction, the energy flux vector is in the same direction as
the wave normal.26 Thus, the measurement of three velocities
that are independently related to C 11 , C 12 , and C 44 allows
the determination of the three single crystal elastic stiffness
constants.
B. Linear thermal expansion and density
In the solid state, the sample’s thickness and thus density
are governed by the linear thermal expansion. Williams
et al.16 and Glazov et al.17 have reported data for the temperature dependence of the linear thermal expansion and
density for CdTe. Glazov et al.17 reported density data calculated from
r t5
r rt
,
~ 11Dl t ! 3
~7!
where r t is the density at a given temperature, r rt is the
density at room temperature, and Dl t is the measured elongation per 1 cm length of the sample due to heating.27 Glazov et al.17 used a reported room temperature density of
5.80 g/cm3 for CdTe, which is slightly lower than the now
accepted value of 5.854 g/cm3. 28 Therefore, the temperature
dependent density data was corrected. Figure 1 shows the
data for the linear thermal expansion for solid CdTe of both
Williams and Glazov.16,17 The linear thermal expansion data
~in %! in Fig. 1 is well fitted by a quadratic relationship:
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
D. T. Queheillalt and H. N. G. Wadley
4127
in the solid and liquid phases. Also shown in Fig. 2, is a
quadratic fit of the corrected data for solid CdTe represented
by
r ~ g/cm3 ! 55.857429.782631025 T26.300531029 T 2 ,
~10!
and a linear fit for liquid CdTe represented by
r ~ g/cm3 ! 55.828721.433731024 T.
FIG. 1. Temperature dependence of thermal linear expansion (Dl t /l 0 ) for
CdTe ~Refs. 16 and 17!.
Dl t /l 0 ~ % ! 520.0191815.573531024 T14.1866
31028 T 2 ,
~8!
where T is the temperature ~in °C!. Subsequently, the thermal expansion coefficient, a, of CdTe can be evaluated by
the differentiation of Eq. ~8! with respect to temperature
~note a change from °C to K! and is obtained in the linear
form
a ~ K21 ! 55.34531026 18.373310210T,
~9!
where T is the absolute temperature. At room temperature a,
calculated from Eq. ~9!, is 5.5931026 K21 which is comparable to 4.9631026 K21 reported by Williams et al.16
By numerically substituting the data of Eq. ~8! and a
room temperature density of 5.854 g/cm3 into Eq. ~7! the
corrected values of the temperature dependent density were
evaluated. Figure 2 shows both the corrected and original
data for the temperature dependence of the density for CdTe
~11!
For the work here, it is assumed that the linear thermal expansion and density for CdTe and Cd0.96Zn0.04Te are the
same.
Evaluating Eqs. ~10! and ~11! with a melting point of
1098 °C ~Ref. 29! for Cd0.96Zn0.04Te results in a solid density
of 5.742 g/cm3 and a liquid density of 5.671 g/cm3. Thus, the
density of the liquid phase is ;1.24% lower than the density
of the solid phase, consistent with a semiconductor
→semiconductor type of melting, i.e., one in which covalent
characteristics of the bond are partially preserved during
melting.8 It has been speculated that the covalent nature in
the molten state is likely to be retained as one-dimensional
chainlike or molecular structures or in some temperature regions as two-dimensional units.8 As a result, semiconductors
such as CdTe which retain their chemical bonding in the
liquid state exhibit an expansion during melting, which is
due to increases in the average interatomic distances and in
the number of defects ~primarily vacancies!.17 This is further
supported by the fact that the electrical conductivity increases exponentially with temperature in the molten state.30
In the liquid state, the sample’s thickness is defined by
the internal diameter of the quartz ampoule that contains the
charge. The thermal linear expansion ~in %! of quartz is
available in the literature and is represented by27
Dl t /l 0 ~ % ! 520.0189716.496631025 T18.0884
31029 T 2 ,
~12!
where T is the temperature ~in °C!.
III. EXPERIMENTAL PROCEDURES
A. Laser ultrasonic measurement technique
FIG. 2. Temperature dependence of density ~r! for CdTe ~Refs. 16 and 17!.
Ultrasonic time-of-flights ~TOFs! between precisely positioned source and receiver points were measured using the
laser ultrasonic system shown in Fig. 3 A ;10 ns duration
Q-switched Nd:YAG laser pulse of 1.064 mm wavelength
was used as the ultrasonic source. The energy per pulse
was ;10 mJ and the roughly Gaussian beam of the multimode pulse was focused to an approximate circular spot of 2
mm diameter. Thus, the source power density was
;100 MW/cm2. Although this power density was slightly in
the ablative regime, minimal surface damage incurred during
repeated pulsing. In addition, the quartz ampoule provided a
constrained surface promoting generation of large amplitude
longitudinal and shear waves.
The ultrasonic receiver was a Mach–Zehnder heterodyne
laser interferometer. It responded to the sample’s out-ofplane ~normal! surface displacement associated with wavefront arrivals at the receiver point.23 It was powered by a 5
mW HeNe laser which produced a continuous Gaussian
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
4128
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
D. T. Queheillalt and H. N. G. Wadley
FIG. 3. A schematic diagram of the high temperature laser ultrasonic test facility.
beam of 632.8 nm wavelength focused to a circular spot
;100 mm in diameter. The interferometer had a displacement sensitivity of 0.4 Å/mV and exhibited linear output for
displacements up to about 300 Å. For the experiments reported here, maximum surface displacements were on the
order of 140 Å. The signal from the interferometer was bandpass filtered between 10 kHz and 10 MHz and recorded with
a precision digital oscilloscope at a 2 ns sampling interval
using 8-bit analog-to-digital conversion. To improve the
signal-to-noise ratio, each waveform used for a TOF measurement was the average of ;50 pulses collected at a pulse
repetition rate of 20 Hz. A fast photodiode identified the
origination time for the ultrasonic signals.
B. Sample preparation and test methodology
Cd0.96Zn0.04Te samples were supplied by Johnson Matthey Electronics ~Spokane, WA!. The material consisted of
precompounded IRFPA substrate purity polycrystalline equatomic CdTe to which Zn and Te were added to reach the
target zinc concentration of 4 at. % 61 at. %. The subsequent precompounded ingot was then grown from the melt
using a commercial 17-zone VB growth furnace and
‘‘mined’’ into samples that measured 1031035 mm3 in
size. The samples were designed so that the thickness ~5 mm
was oriented in either the ^100&, ^110&, or ^111& directions.
One surface was lapped and the other chemical-mechanical
polished to provide a reflective surface for the laser interferometer. They were inserted in 10 mm i.d. square quartz ampoules with quartz plugs ~to fill in the free volume! and
sealed under 1026 Torr with a very small free volume to
inhibit the vaporization of Cd during subsequent heating
@Fig. 4~a!#. Ingots for liquid state measurements were prepared in the same manner, except that the polycrystalline
ingot ~;350 g! was contained in a 30 mm i.d. quartz ampoule @Fig. 4~b!#.
FIG. 4. A schematic diagram of the sample configurations for ~a! solid and
~b! liquid Cd0.96Zn0.04Te ~all dimensions in mm!.
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
D. T. Queheillalt and H. N. G. Wadley
4129
FIG. 6. Typical ultrasonic waveform denoting the arrival TOFs of the encircling and longitudinal waves in liquid Cd0.96Zn0.04Te.
heat source. Variations of the temperature in the region of
the sample was better than 61 °C. The solid samples were
heated at a rate of 2 °C/min, and allowed to equilibrate ;30
min at each test temperature ~every 25 °C!. The liquid
sample was heated at a rate of 10 °C/min up to 1070 °C and
allowed to equilibrate at this temperature for several hours.
Subsequent cyclic heating and cooling near the melting temperature was conducted at a rate of 1 °C/min, in 5 °C increments and allowed to equilibrate for ;1 h at each test temperature.
IV. RESULTS AND ANALYSIS
A. Wavefront identification
Figure 5 shows the normal displacement waveforms
measured at room temperature ~;21 °C! for samples oriented in the ^100&, ^110&, and ^111& directions. Figure 5~a!
shows the normal surface displacement waveform in the
^100& direction. The interferometer output shows two distinct
wavefront arrivals. The first arrival ~1L! corresponds to that
of the longitudinal wave and the second ~1S! to that of the
shear wave. In Fig. 5~b!, the ^110& direction, the first ~1L!
and third ~3L! reflections of the longitudinal wave along
FIG. 5. Typical ultrasonic waveforms denoting the arrival TOFs of the
longitudinal and shear waves in ~a! ^100&, ~b! ^110&, and ~c! ^111& crystal
orientations for solid Cd0.96Zn0.04Te.
Heating was accomplished in a programmable tubular
furnace. The furnace was equipped with a temperature controller that was able to maintain the temperature to within 1
°C. The Cd0.96Zn0.04Te samples were supported by a Al2O3
support assembly. The laser source and receiver gained access into the furnace through small ~;4 mm! slits at each
end of the furnace. The sample temperature was measured
with two type R thermocouples; one located on the generation side of the sample and the other on the receiver side. In
both cases, the thermocouples were in physical contact with
the outer surface of the ampoule and shielded from the direct
FIG. 7. Temperature dependence of the longitudinal wave velocity in ^100&,
^110&, and ^111& orientations for solid Cd0.96Zn0.04Te.
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
4130
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
FIG. 8. Temperature dependence of the shear wave velocity in the ^100&
orientation for solid Cd0.96Zn0.04Te.
with the fast ~FS! and slow ~SS! shear wave arrivals were
observed. In the ^111& direction, Fig. 5~c!, only the first ~1L!,
third ~3L!, and fifth ~5L! reflections of the longitudinal wave
were observed. No shear wave arrival was observed because
the energy flux vector is not in the same direction as the
wave normal.26
The normal displacement waveform measured in the liquid state ~1140 °C! is shown in Fig. 6. The waveform shows
two distinct arrival events. During laser ultrasonic generation
the thermal expansion of the liquid induces a force at the
constrained quartz-liquid interface generating an energetic
elastic wave in the quartz ampoule. This wave propagates
cicumferentially in the quartz ampoule with a velocity corresponding to that of the zeroth-order symmetric Lamb
wave.14,15 The arrival of this encircling wave is indicated by
the minor surface displacement at ;8 ms. The second event
at ;23 ms is due to the arrival of the longitudinal wave
propagating through the liquid along a straight path from the
source to the receiver. In both Figs. 5 and 6 the photodiode
pulse denotes the start time for the ultrasonic signals.
B. Ultrasonic time-of-flight and velocity
The temperature dependence of the ultrasonic velocity
was evaluated from the measured TOF data of longitudinal
waves in ^100&, ^110&, and ^111& orientations and the shear
wave in the ^100& orientation. The ingots’ instantaneous
length was calculated from Eq. ~8! at each target tempera-
D. T. Queheillalt and H. N. G. Wadley
FIG. 9. Temperature dependence of the longitudinal wave velocity in liquid
Cd0.96Zn0.04Te.
ture. The measured temperature dependence of the longitudinal wave velocity in ^100&, ^110&, and ^111& orientations is
shown in Fig. 7. Errors in the measured longitudinal wave
velocities were within 60.02 mm/ms. The measured temperature dependence of the shear wave velocity in the ^100&
orientation is shown in Fig. 8. Errors in the measured shear
wave velocities were within 60.01 mm/ms. Both the longitudinal and shear waves showed a monotonically decreasing
velocity with temperature. The existence of CdTe crystals of
a hexagonal structure at high temperatures has been
reported.31–33 It has been speculated that CdTe can easily
oscillate between hexagonal and cubic structures near its
liquid/solid transition temperature during growth processes
from congruent melts.34 However, no evidence of
solid→solid phase changes or other anomalous effects were
observed in the ultrasonic velocity data collected near the
melting point.
In the liquid state, the TOF was measured during both
heating and cooling. The longitudinal wave velocity was
then calculated assuming that the ingot diameter is governed
by the thermal expansion of the quartz ampoule, Eq. ~12!.
The temperature dependence of the longitudinal wave in the
liquid state is shown in Fig. 9. Errors in the measured longitudinal wave velocities were within 60.002 mm/ms. As the
temperature increased there was a loosening of the covalent
~semiconducting! liquid structure due to thermal motion with
a concomitant linear drop in the longitudinal wave velocity
in the temperature region monitored. No evidence of liquid-
TABLE II. Temperature dependent polynomial fits of the longitudinal and shear wave velocity data of
Cd0.96Zn0.04Te.
Type of wave
Solid
Longitudinal
Longitudinal
Longitudinal
Shear
Liquid
Longitudinal
Orientation
Temperature dependent velocity ~mm/ms!, T in °C
^100&
^110&
^111&
^100&
3.06721.91731024 T21.93331027 T 2
3.35622.52831024 T21.17231027 T 2
3.45322.44531024 T21.34431027 T 2
1.85921.27831024 T23.65131028 T 2
1.89825.08031024 T
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
FIG. 10. The effect of melting on the quasilongitudinal wave velocity for
solid and liquid Cd0.96Zn0.04Te. The solid quasilongitudinal velocity data
was taken from a polycrystalline ingot.
liquid polymorphisms or other anomalies were observed in
the liquid velocity data.
The ultrasonic velocity data for the solid and liquid
phases was well fitted by quadratic relationships. Table II
shows the polynomial fits for the temperature dependence for
the ultrasonic velocities in both the solid and the liquid
phases from Figs. 7, 8, and 9. Both the longitudinal and shear
wave velocities exhibited a strong monotonically decreasing
function of temperature in the solid and liquid phases. The
longitudinal wave velocity was found to be ;2 times faster
in the solid phase ~depending on the crystallographic orientation! at the melting temperature, Fig. 10. This ultrasonic
velocity ratio ( v s / v l ) between the solid and liquid phases
appears to be sufficient for the implementation of laser ultrasonic sensors to monitor the liquid–solid interface during
vertical Bridgman growth.
C. Elastic stiffness constants
The temperature dependent single crystal elastic stiffness
constants for the solid phase ~C 11 , C 12 , and C 44! can be
deduced from ultrasonic velocities measured in the three pri-
D. T. Queheillalt and H. N. G. Wadley
4131
FIG. 12. Temperature dependence of the adiabatic bulk modulus (K S ) for
liquid Cd0.96Zn0.04Te.
mary crystal directions and published temperature dependent
density using the simple inversion formula summarized in
Table I. The temperature dependence of the solid phase elastic stiffness constants ~C 11 , C 12 , and C 44! are shown in Fig.
11. Like the ultrasonic velocity data, the elastic stiffness constants are a monotonically decreasing function of temperature. The adiabatic bulk modulus (K S ) for the liquid state
can be evaluated from liquid velocity data using Eq. ~4! and
is shown in Fig. 12. Table III shows polynomial fits for the
temperature dependence of the elastic constants in both the
solid and the liquid phases.
The single crystal elastic stiffness constants ~C 11 , C 12 ,
and C 44! for solid and the adiabatic bulk modulus (K S ) for
liquid of Cd0.96Zn0.04Te have been fully evaluated from 20–
1140 °C covering the broad spectrum of temperatures encountered during vertical Bridgman growth. Also the phase
and group velocities of the three bulk wave modes can be
evaluated from Eqs. ~5! and ~6! for any crystallographic orientation and temperature. This data is a necessary prerequisite in accurate model development of laser ultrasonic sensing methodologies to monitor the solid–liquid interface
position and shape during VB solidification.
D. Thermoelastic modulus
It has long been recognized that excessive thermal gradient induced stresses may initiate dislocation generation
during the growth of semiconducting materials from the
melt.35 The criterion for dislocation generation is that the
resolved shear stress that arises from temperature gradients
TABLE III. Temperature dependent polynomial fits of the elastic stiffness
constant data of Cd0.96 Zn0.04 Te.
Elastic constant
Solid
FIG. 11. Temperature dependence of the single crystal elastic stiffness constants ~C 11 , C 12 , and C 44! for solid Cd0.96Zn0.04Te.
C 11
C 12
C 44
Liquid
KS
Temperature dependent elastic property ~GPa!, T in °C
55.16628.497310 23 T25.292310 26 T 2
36.78526.715310 23 T22.229310 26 T 2
20.24223.182310 23 T25.429310 27 T 2
18.87927.911310 23 T
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
4132
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
D. T. Queheillalt and H. N. G. Wadley
dislocation generation in CdTe and its alloys during growth
from congruent melts.35 Efforts to alleviate these dislocations
include doping to raise s CRSS or by reducing the thermal
gradients by thermal design of growth furnaces.
V. CONCLUSIONS
FIG. 13. Temperature dependence of the critical resolved shear stress
( s CRSS) and the thermoelastic modulus (M T $ 111% ) for Cd0.96Zn0.04Te.
in the solidified boule exceed the critical resolved shear
stress ( s CRSS) for slip on $111%, ^ 11̄0 & slip system of these
cubic materials. The thermal stress calculations entail the
single crystal elastic stiffness constants in the form of a factor designated as the thermoelastic modulus, M T . 36
Brantley37 has shown that the thermoelastic modulus is invariant within the $111% plane and can be expressed in terms
of the single crystal elastic stiffness constants in the form
M T $ 111% 5
S D
F
aE
12 v
$ 111%
5 a C 111C 121
C ~ C 1114C 12! 23C 212
2C11.5C 11
G
,
~13!
where E is Young’s modulus, v is Poisson’s ratio, a is the
thermal expansion coefficient, and C5C 4420.5(C 112C 12)
reflects the departure from isotropy.37,38 Now, the stress induced in the solidified crystal can be found by multiplying
Eq. ~13! by the temperature difference in the crystal.
Kuppurao et al.39,40 conducted a quasi-steady-state
analysis of the vertical Bridgman growth of 75 mm-diameter,
cadmium zinc telluride using finite element methods. This
analysis reveals that significant radial temperature differences ~3–15 K! exist in the radial direction of a solidifying
boule. Figure 13 shows a plot of M T $ 111% ~with a 3 K radial
temperature difference! as a function of temperature evaluated from the single crystal elastic stiffness constants, Table
III, and the thermal expansion coefficient, Eq. ~9!.
The single-most significant parameter governing dislocation generation due to thermal stresses is the critical resolved
shear stress, s CRSS . The critical resolved shear stress for
Cd0.96Zn0.04Te has been measured by Parfeniuk et al.41 and
Imhoff et al.42 between 350–1150 K. Assuming the s CRSS is
an exponential function of temperature,43 the data of Parfeniuk and Imhoff can be extrapolated up to the melting point
and is fitted by
log s CRSS ~MPa)55.174824.765331023 T,
~14!
where T is the absolute temperature. It is clear from Fig. 13
that s CRSS is significantly less than M T $ 111% above ;1100 K.
Thus this strongly suggests that large thermal gradients cause
A laser ultrasonic sensor has been used to determine the
ultrasonic TOF and hence velocity of Cd12y Zny Te with y
50.04 from 20 to 1140 °C. Both the longitudinal and shear
waves exhibited a monotonically decreasing velocity with
temperature. The solid and liquid exhibited normal semiconducting behavior and did not exhibit any structural changes.
When the ingots were heated through the melting point under
quasiequilibrium conditions an approximately twofold decrease was observed in the longitudinal wave velocity, due to
the absence of the shear modulus in the liquid phase. These
results suggest that the ultrasonic velocity ratio ( v s / v l ) between the solid and liquid phases are sufficient for the implementation of laser ultrasonic sensors to monitor the liquid–
solid interface during vertical Bridgman growth.
The single crystal elastic stiffness constants ~C 11 , C 12 ,
and C 44! for solid and the adiabatic bulk modulus (K S ) for
liquid Cd0.96Zn0.04Te have been fully evaluated from simple
inversion techniques of the temperature dependent ultrasonic
velocity in primary ^100&, ^110&, and ^111& orientations and
the temperature dependence of the density. The elastic stiffness constants showed a monotonically decreasing function
with increasing temperature in both the solid and liquid
phases. In addition, an analysis of the thermoelastic modulus
and the critical resolved shear stress indicate that severe thermal stresses are responsible for slip-induced dislocation generation in CdTe and its alloys during growth from congruent
melts.
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 MD A972-91-C-0046 monitored by
Raymond Balcerak.
1
S. Sen, W. H. Konkel, S. J. Tighe, L. G. Bland, S. R. Sharma, and R. E.
Taylor, J. Cryst. Growth 86, 111 ~1988!.
2
W. E. Tennant, C. A. Cockrum, J. B. Gilpin, M. A. Kinch, M. B. Reine,
and R. P. Ruth, J. Vac. Sci. Technol. B 10, 1359 ~1992!.
3
R. Triboulet, Mater. Forum 15, 30 ~1991!.
4
Yu. A. Burenkov, S. P. Nikanorov, and A. V. Stepanov, Sov. Phys. Solid
State 12, 1940 ~1971!.
5
V. M. Glazov, A. A. Aivazov, and V. I. Timoshenko, Sov. Phys. Solid
State 18, 684 ~1976!.
6
Yu. A. Burenkov and S. P. Nikanorov, Sov. Phys. Solid State 16, 963
~1974!.
7
N. M. Keita and S. Steinemann, Phys. Lett. A 72, 153 ~1979!.
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
J. Appl. Phys., Vol. 83, No. 8, 15 April 1998
8
V. M. Glazov, S. N. Chizhevskaya, and N. N. Glagoleva, Liquid Semiconductors ~Plenum, New York, 1969!.
9
R. L. Parker and J. R. Manning, J. Cryst. Growth 79, 341 ~1986!.
10
C. K. Jen, Ph. de Heering, P. Sutcliffe, and J. F. Bussiere, Mater. Eval. 49,
701 ~1991!.
11
J. N. Carter, A. Lam, and D. M. Schleich, Rev. Sci. Instrum. 63, 3472
~1992!.
12
F. A. Mauer, S. J. Norton, Y. Grinberg, D. Pitchure, and H. N. G. Wadley,
Metall. Trans. B 22B, 467 ~1991!.
13
S. J. Norton, A. H. Kahn, F. A. Mauer, and H. N. G. Wadley Proceedings
of the Symposium on Intelligent Processing of Materials, edited by H. N.
G. Wadley and W. E. Eckart ~TMS, Warrendale, 1990!, pp. 275–291.
14
D. T. Queheillalt, Y. Lu, and H. N. G. Wadley, J. Acoust. Soc. Am. 101,
843 ~1997!.
15
D. T. Queheillalt and H. N. G. Wadley, J. Acoust. Soc. Am. 102, 2146
~1997!.
16
M. G. Williams, R. D. Tomlinson, and M. J. Hampshire, Solid State
Commun. 7, 1831 ~1969!.
17
V. M. Glazov, S. N. Chizhevskaya, and S. B. Evgen’ev, Russ. J. Phys.
Chem. 43, 201 ~1969!.
18
H. J. McSkinnen and D. G. Thomas, J. Appl. Phys. 33, 56 ~1962!.
19
Yu. Kh. Vekilov and A. P. Rusakov, Sov. Phys. Solid State 13, 956
~1971!.
20
D. E. Bray and R. K. Stanley, Nondestructive Evaluation: A Tool in Design, Manufacturing, and Service ~CRC, Boco Ratton, FL, 1997!.
21
D. de Nobel, Philips Res. Rep. 14, 361 ~1959!.
22
C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications ~Adam Hilger, New York, 1990!.
23
J. P. Monchalin, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 33, 485
~1986!.
24
A. B. Bhatia, Ultrasonic Absorption: An Introduction to the Theory of
Sound Absorption and Dispersion in Gases, Liquids and Solids ~Dover,
New York, 1967!.
D. T. Queheillalt and H. N. G. Wadley
4133
F. I. Federov, Theory of Elastic Waves in Crystals ~Plenum, New York,
1968!.
26
B. A. Auld, Acoustic Fields and Waves in Solids ~Krieger, Malabar, Fl,
1990!.
27
Y. S. Touloukian, R. K. Kirby, R. E. Taylor, and T. Y. R. Lee, in Thermophysical Properties of Matter: Thermal Expansion of Non Metallic
Solids ~IPI/Plenum, New York, 1975!, Vol. 13.
28
R. L. Lide Editor, CRC Handbook of Chemistry and Physics, 71st ed.
~CRC, Boston, 1991!.
29
P. Capper and A. W. Brinkman, Properties of Narrow Gap Cadmiumbased Compounds, edited by P. Capper, EMIS Data Review Series, No.
10 ~INSPEC, IEE, London, UK, 1994!.
30
H. N. G. Wadley and B. W. Choi, J. Cryst. Growth 172, 323 ~1996!.
31
E. Kendall and J. Hendom, Phys. Lett. 8, 237 ~1964!.
32
J. Appel, Z. Naturforsch. 9, 265 ~1953!.
33
P. Hoschle and C. Konak, Czech. J. Phys., Sect. B B13, 850 ~1963!.
34
R. Triboulet, Mater. Forum 15, 30 ~1991!.
35
A. S. Jordan, R. Caruso, and A. R. Von Neida, Bell Syst. Tech. J. 59, 593
~1980!.
36
S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed.
~McGraw-Hill, New York, 1970!.
37
W. A. Brantley, J. Appl. Phys. 44, 534 ~1973!.
38
J. F. Nye, Physical Properties of Crystal ~Oxford Univ. Press, Oxford,
1957!.
39
S. Kuppurao, S. Brandon, and J. J. Derby, J. Cryst. Growth 155, 93
~1995!.
40
S. Kuppurao, S. Brandon, and J. J. Derby, J. Cryst. Growth 155, 103
~1995!.
41
C. Parfeniuk, F. Weinberg, I. V. Samarasekera, C. Schvezov, and L. Li, J.
Cryst. Growth 119, 261 ~1991!.
42
D. Imhoff, A. Zozime, and R. Triboulet, J. Phys. III 1, 1841 ~1991!.
43
H. G. van Beuren, Imperfections in Crystals, 2nd ed. ~North-Holland,
Amsterdam, 1961!.
25
Downloaded 25 Apr 2013 to 128.143.68.13. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions