Full text of DOI:10.1557/JMR.2002.0305

Step bunching in potassium dihydrogen phosphate crystal
growth: Phenomenology
a)
N.A. Booth
Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4004
A.A. Chernov
Universities Space Research Association, 4950 Corporate Drive, Suite 100,
Huntsville, Alabama 35806
P.G. Vekilov
Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4004
(Received 4 December 2001; accepted 28 May 2002)
We have developed a real-time phase-shifting interferometer capable of imaging
interfacial morphology with a depth resolution of approximately 25 Å, with a lateral
2
resolution of approximately 0.5 ␮m across a field of view of 2 × 2 mm , and with
time resolution of 0.1 s. The method is applied in situ to the (101) face of potassium
dihydrogen phosphate crystals growing from an aqueous solution. We image the
formation and evolution of solution-flow-induced step bunches and determine their
characteristic wavelength to be ␭ ס 45 ␮m. This wavelength is within the
c
range predicted by a stability theory on the basis of the balance between the diffu-
sion interaction between steps and capillarity. The value of ␭ suggests
c
that step–step interactions are the likely major factor for instability.
I. INTRODUCTION
This spectrum includes as its major component the typi-
cal wavelength at which loss of morphological stability
occurs. As a model system, we chose potassium dihy-
drogen phosphate (KDP). This is a well-studied salt–
Morphological stability of crystal interfaces during
growth largely determines the structure and perfection
1
–5
of the resulting crystals.
Smooth surfaces that grow
1
3
water system, and the solubility, viscosity, refractive
index, and other relevant properties are known in a broad
range of parameters. Furthermore, KDP crystals as large
as 65 cm are needed as frequency multipliers for the
controlled nuclear fusion projects carried out at Lawrence
by the generation of new layers and their lateral
spreading (step flow growth mode) lose stability due
to bunching of the growth steps. The formation of step
bunches, or macrosteps, may cause striations via the dif-
ferential trapping of impurities and defects, as well as the
formation of inclusions. Hence, understanding and con-
trol of step bunching is important for fields ranging from
corrosion, through electrochemistry, and to protein and
1
4,15
Livermore National Laboratory.
The efforts to grow
such crystals revealed that the main difficulty stems from
the formation of step bunches along the pathway of
1
4–16
6
–12
growth steps.
In this first study, we chose the (101)
semiconductor crystal growth.
faces of these crystals because their growth is less sen-
sitive to impurity action and intrinsic effects can be
Despite the extensive discussions of the step bunches
and their consequences, in the case of growth of crystals
from solution neither the onset nor the evolution of step
bunching has thus far been addressed in quantitative ex-
periments. Here we report features of our experimental
setup and first results of an ongoing investigation aimed
at quantitative insight into the step bunching during
growth of crystals from solution. In this first study, we
determine the spectrum of the wavelengths present on the
growing interface on which bunching is developed.
1
7,18
observed.
Typically, the height scales encountered on a smooth
crystal surface during growth range from approximately
1
0 Å, the elementary step height, to microns, typical
macrostep heights. The technique that would appear most
appealing for the study of such features is atomic force
microscopy (AFM), which allows probing with molecu-
1
7,19–22
lar resolution.
However AFM has two main dis-
advantages for in situ studies of the evolution of surface
instability: (i) The width of the view field is at most
approximately 100 ␮m, while step bunch evolution oc-
curs over a length scale of millimeters and centimeters.
(ii) The typical AFM scanning time is often minutes/
a)
Address all correspondence to this author.
e-mail: nbooth@mail.uh.edu
J. Mater. Res., Vol. 17, No. 8, Aug 2002
© 2002 Materials Research Society
2059

N.A. Booth et al.: Step bunching in potassium dihydrogen phosphate crystal growth: Phenomenology
image, whereas typical tangential step velocities are of
the order of 1 ␮m/s. Therefore, in situ AFM studies
would be confined to growth systems with very slow
kinetics.
Another method used in crystal growth studies is two-
beam Michelson interferometry, in which one of the mir-
2
3,24
rors is replaced by the studied crystal face.
An
advantage of this method is the capability to observe
growth in situ over a relatively large area. As typically
applied, the raw interferometry data is the spacing be-
tween the interference fringes. This limits the depth reso-
1
lution of the method to about ⁄
2
of the wavelength of the
illuminating light in the growth medium, i.e., approxi-
mately 0.2 ␮m.
2
5–27
Phase-shifting interferometry (PSI)
has a number
of advantages over conventional interferometry: high
measurement accuracy, rapid data capture, high contrast
2
7
fringes, etc. In early variations of this technique the
phase of the interference pattern was shifted by mechani-
cally varying the difference in optical path length be-
tween the two shoulders. This precluded studies of
growth systems with fast kinetics. To accelerate data col-
lection, in place of mechanically altering the path length,
the upgraded system separates the interference pattern
into three components that each pass through polarizers
FIG. 1. Schematic of the experimental setup: phase-shifting interfer-
ometer; growth cell with solution circulator; data collection system.
2
8,29
whose phases are shifted by ␲/2.
The system was
that from the reference mirror; hence, a filter is inserted
in the reference arm to attenuate the reference beam to an
intensity roughly equal to that from the crystal.
recently refined by the use of a specially made phase-
dividing 3-charge-coupled device (CCD) camera in place
3
0
of three individual units. Using custom optical and
electronic components, in situ monitoring of the crystal-
lization of several inorganic salts was accomplished.
In view of its advantages, we chose phase-shifting
interferometry as the experimental technique for
the planned studies of the step bunching during KDP
growth. We designed a new optical subsystem so that
the apparatus could be built with off-the-shelf compo-
nents and used five, instead of three, phase-shifted im-
ages to improve the fidelity of the surface relief
reconstruction.
To observe the genesis of step bunches, a vertical reso-
lution around 50 Å is required. For this, the reference
mirror must be flat to a similar value. We used dielectric
coated fused silica mirrors certified P–V flat to 1/50 ␭ (␭
being the He/Ne wavelength 632.8 nm) with a root mean
square (rms) surface roughness of less than 2 Å.
We use a linearly polarized 20-mW He/Ne laser,
whose beam is expanded via a spatial filter and colli-
mating lens to a diameter of approximately 1.5 cm.
The beam is then split into two equal components by a
cube beamsplitter, half being subsequently reflected by
the reference mirror and the other by the reference mirror
via the variable wave plate. The two components then
recombine in the beamsplitter and interfere; this interfer-
ence pattern is then expanded via a variable focus zoom
lens and captured on a CCD array. We place a variable
radius aperture at the focal point between the zoom lens
and the CCD; this stops reflections from the other sur-
faces in the optical pathway from entering the recorded
signal. We use a 1024 × 1024 pixel CCD array capable
of capturing 12 bit images at a rate of 60 frames/s. The
whole optical system is mounted on an air-supported
table to minimize mechanical vibrations.
II. EXPERIMENTAL TECHNIQUE
A. Optical system
Our optical setup (see Fig. 1 for a schematic diagram)
is based on a Michelson interferometer with two main
modifications. First, the mirror in one of the two refer-
ence arms is replaced by a growing crystal surface. Sec-
ond, a liquid-crystal variable wave plate, or phase shifter,
is placed in the reference arm. This wave plate is con-
structed using nematic liquid crystals whose birefrin-
gence can be changed by applied voltage. Hence, we are
able to modulate the phase of the reference beam relative
to that reflected from the crystal surface. The reflection
from crystal surfaces is typically much weaker than
Using off the shelf components including a variable
wave plate and a single CCD, we are able to create a
simpler device than those reported in Refs. 28 and 30
without compromising efficiency.
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J. Mater. Res., Vol. 17, No. 8, Aug 2002

N.A. Booth et al.: Step bunching in potassium dihydrogen phosphate crystal growth: Phenomenology
B. Solution preparation
is repeated three more times until the total phase shift is
␲; hence, we obtain a sequence of five images as shown
2
We use KDP as supplied by Prochem, High Point, NC.
A typical solution “charge” consists of 400 g of salt in
in Figs. 2(a)–2(e). The camera and waveplate are syn-
chronized via custom-made software optimized for speed
of capture. The two constraining factors are the speed of
the camera’s electronic shutter and the time taken by the
nematic crystals to stabilize after a given voltage is ap-
plied. With the camera set to operate at 60 frames/s, the
average time taken to capture a sequence of five images
is 0.10 s.
1.5 l of deionized water (resistance 18 M⍀). The solution
is mixed in a dedicated preparation chamber at 38.00 °C
for a number of days to ensure total saturation. The tem-
perature within the chamber is thermostated and stable to
within 0.04 °C. The solution is siphoned from the prepa-
ration chamber via a filter (MSI 0.22 ␮m) to remove any
particulate contamination capable of acting as a source of
heterogeneous nucleation.
The solution is then transferred to the solution reser-
voir of the growth system (the origin of the basic design
of the whole crystallization subsystem is described in
Ref. 31). We then overheat by 3–4 °C for more than 24 h
to ensure that any nucleation that has occurred during
the transportation period is reversed. After this “purifi-
cation” period the solution is cooled to 0.04 °C above its
saturation point and the crystal growth cell attached via
Tygon tubes (R3603) with an inner diameter of 6 mm.
The pump is turned on, and the solution washes the
crystals.
Typically our optical system is set up so that 1 mm
of the crystal surface is expanded to cover the CCD
detector (a magnification factor of approximately 10×) so
2
that approximately 1 pixel corresponds 1 × 1 ␮m area
on the crystal surface. Tangential step velocities for el-
−1
ementary KDP steps are of the order of 1 ␮m s ; hence,
steps will not have moved from one pixel to the next
during the time taken to capture each five-image
sequence.
Sequences of five images are combined to give one
“phase wrapped” image [illustrated in Fig. 2(f)] using,
for each pixel,
The crystal is then faceted by waiting for the surface to
soften” as the edges begin to dissolve and then reducing
“
␭
2͑I
2
− I
͒
4
tan−
1
.
(1)
h =
ͫ
ͬ
the temperature to 0.2 °C below the saturation point to
begin slow regrowth. The solution temperature inside the
system is thermostated and stable to 0.04 °C. The seed
crystals are mounted inside the growth cell with a (101)
plane facing the incident beam. The solution is pumped
4
␲n
2I − I − I
3
5
1
In Eq. (1), I . . . , I are the interferometric intensities of
1
5
a pixel in images 1, . . . , 5, h is the calculated height on
the respective point on the crystal surface, ␭ ס 632.8 nm
is the wavelength of the incident light, and n is the re-
fractive index of the solution. Note that the interferomet-
ric intensities enter Eq. (1) as differences between two
−1
around the system at a rate of approximately 20 cm s .
The flow rate is variable, and its affects on growth are the
subject of a subsequent paper. In all experiments reported
here the flow direction was from the apex of the top
pyramid formed by (101) faces down toward the (100)
faces. The pump, solution reservoir, and growth cell are
all fabricated from Teflon to ensure that the solution is
not contaminated in any way by substances leaching
from the walls. The laser beam is incident on the crystal
surface via an optically flat (1/10 ␭) quartz window. The
channel within the crystal cell through which the solution
images that are in counterphase. The value of I is dif-
ferent from I , and I is different from I and I only if the
2
4
3
1
5
respective intensity values are due to interference be-
tween the object and the reference beams. Hence, all
noninterference noise, and all artifacts coming from in-
terference between other parallel surfaces in the optical
pathway, are canceled. As a result, the signal-to-noise
ratio of the surface characterization is significantly
increased.
flows has a square cross-profile of approximately
2
1
.69 cm .
To reduce the noise further, we employ a simple
low-pass filter that removes any singular spikes from the
image. The routine is a simple “salt and pepper” type
filter that compares a pixel’s value to the average of its
eight nearest neighbors. If the pixel’s value differs from
this average by the height difference of one fringe, i.e.,
␭/2n ס 0.115 ␮m, it is replaced by the average; else,
it is left intact. This procedure reduces the image by
2 pixels in each dimension. For further details on the data
processing, such as calibration of the phase shifter and
evaluation of the depth resolution of the technique of
2.5 nm, limited by the roughness of the reference mirror,
see Ref. 32.
C. Data collection and processing
For this investigation, we collected sets of five phase-
shifted interference images that are subsequently com-
bined using a phase wrapping algorithm. Further
processing yields images of the morphology of the sur-
face. At the start of data collection, the liquid-crystal
wave plate (see Fig. 1) is set to give zero phase-shift and
an image of the interference between the reference mirror
and the crystal is captured. The voltage across the wave
plate is then altered to introduce a ␲/2 phase shift to the
reference beam and another image captured. This process
J. Mater. Res., Vol. 17, No. 8, Aug 2002
2061

N.A. Booth et al.: Step bunching in potassium dihydrogen phosphate crystal growth: Phenomenology
FIG. 2. (a–e) A series of five interferometric images from a (101) face of a KDP crystal taken at 1/60-s time intervals. Between each two images,
the optical density of the liquid-crystal wave plate in the reference arm of the interferometer was changed so that the phase of the interferogram
was shifted by ␲/2 from the previous interferogram. (f ) The “phase-wrapped” image, produced by applying Eq. (1) to all pixels in (a)–(e). In (f )
the gray scale codes for height and the discontinuities reflect the arctan behavior. The difference in height between the brightest and the darkest
pixels is ␭/2␲n ס 0.115 ␮m.
III. RESULTS AND DISCUSSION
Figures 3(c) and 3(d) highlight the sections of the crys-
tal taken from the steepest sector of the (101) surface.
The average slope of the vicinal face in this sector is
A. In situ imaging of the surface of growing
KDP crystals
−3
6
.3 × 10 , corresponding to an average spacing between
Figure 3 shows a phase-wrapped image of a (101)
face of a KDP crystal growing at 0.2 °C undercooling.
Using the solubility dependence on temperature (cour-
tesy of J. DeYoreo), we evaluate the supersaturation
elementary steps of height 5.1 Å of 833 Å. In this sector,
the step trains move in the same direction as the solution
flow; this is predicted to be an unstable regime for step
11
bunches. We see ripples at the lower half of these im-
−
1
(
1 − CCe ) ס 0.0025. The image shows the presence of
ages—these are step bunches, approximately 50-nm high.
The step bunches that are further down appear higher;
i.e., the step bunch height appears to increase as the bunches
move. This observation agrees with the general picture of a
a hillock that is most likely generated by one or more
screw dislocations outcropping in the upper right part
of the studied face and covers the whole facet. Due to
the anisotropy of the surface kinetics the hillock has the
asymmetric trigonal shape typical of (101) faces of KDP–
ammonium di-hydrogen phospate (ADP)-class crystals.
Figure 3(b) highlights a section of the surface, located
on the shallowest sector of the (101) face, where the steps
move fastest. On this sector the steps’ and solution flow
9
convective step bunching instability and with previous re-
8,33,34
sults from solution growth systems.
C. Quantification of the step bunching
For a quantitative characterization of the step bunch
height, shape, and evolution, we took linear profiles of
the intensity distribution from images similar to the one
in Fig. 3(a). Figure 4(a) shows such a profile along a
vertical line on the right of the rectangular frame in
Fig. 3, the line not shown in Fig. 3(a). The heights at the
points of discontinuity of the arctan function from Eq. (1)
are stitched to obtain the interface profile in Fig. 4(a).
The profile is plotted relative to the averaged vicinal face.
1
1
directions are opposite. Theory and previous observa-
23
tions suggest that under such a condition the surface is
in a stabilizing regime and step bunches do not form.
Correspondingly, the roughness in Fig. 3(b) is defined by
the limitations of the reference mirror, and we conclude
that any step bunches on that vicinal sector should be
lower than 5 nm.
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J. Mater. Res., Vol. 17, No. 8, Aug 2002

N.A. Booth et al.: Step bunching in potassium dihydrogen phosphate crystal growth: Phenomenology
FIG. 3. Illustration of the morphology difference between facets: (a) phase-wrapped image of a KDP crystal growing at 38.40 °C from a solution
with saturation temperature of 38.75 °C, i.e., at a supersaturation of 0.044; (b–d) images produced from respective rectangular areas by stitching
the height and subtracting the average slope. This imaging mode highlights deviations form the average slope, i.e., the step bunching. For details
and tests of data-processing algorithms, see Ref. 32. Note the absence of step bunches in (b), where solution flows in the direction opposite to that
of step motion.
Figure 4 shows variations of the profile between 10
and −10 nm in the left part of the trace and between
20 and −20 nm in the right part. These values are sig-
nificantly higher than the roughness of the mirror. Fur-
thermore, the maxima in Fig. 4(a) are in perfect
correlation with locations of step bunches in an image
similar to Fig. 3(d). We conclude that these maxima cor-
respond to step bunches.
The profile in Fig. 4(a) breaks in to two parts: an area
of low step bunch height at 0 < x < 580 ␮m and an
area of high step bunches at longer step pathways. Within
each range, the step bunch characteristic height is pre-
served. This is a deviation from the expected steady in-
crease in step bunch height. We carried out several
determinations of profiles similar to those in Fig. 4(a) at
another location of the facet shown in Fig. 3(a) and at the
steep vicinal sectors on the (101) faces of other crystals.
We found that if the z(x) profile is taken from a location
far from any of the crystal’s edges, the average step
bunch height is uniform and comparable to that in the
first part of Fig. 4(a). Sudden increases in step bunch
FIG. 4. (a) Typical surface profile along a vertical line close to the
area in Fig. 3(d). In this presentation, the steps move from left to right.
(b) Fourier transform of the profile in (a).
J. Mater. Res., Vol. 17, No. 8, Aug 2002
2063

N.A. Booth et al.: Step bunching in potassium dihydrogen phosphate crystal growth: Phenomenology
height are always related to the proximity of the crystal’s
just the step height so that ␤ p L is the overall kinetic
st 1
edge. Hence, we tentatively attribute this increase to hy-
drodynamic instabilities occurring at the edges of a crys-
tal. Detailed investigation of this and other effects is
now in progress.
coefficient of the bunch. From our data p is up to 3 times
1
−2
larger than the average slope; i.e., we take p Ӎ 2 × 10 .
1
13
−3
Thus, with L Ӎ 2 × 10 cm, one has ␤ p L/D Ӎ 0.6.
st
1
Being proportional to the bunch width and slope, this
figure suggests noticeable dependence of the bunch rate
on its parameters. Therefore, since the bunches vary, at
least slightly, in height and length, we may expect varia-
tions in the bunch rate. As a result, coalescence of
bunches could occur. The approximately constant bunch
heights observed so far, at least within their limited paths
on the face, suggest either existing a compensating split-
ting of bunches or slow rate of coalescence. These are the
subject of further studies.
The step-bunching wavelength of about 45 ␮m is
preserved in both parts of the profile in Fig. 4(a). This
conclusion is supported by the Fourier spectrum of
the profile in Fig. 4(b), which shows a single peak at
−
1
3
−1
a wave number of 0.022 ␮m ס 2.2 × 10 cm .
We assume that this wave number corresponds to
the profile perturbation harmonic that ultimately in-
duces the bunching instability and compare it to the
wave number characteristic of instability induced by in-
3
5
terfacial stiffness only. According to theory, this wave
number is
IV. CONCLUSIONS
1
/2
We have developed a differential phase-shifting inter-
ferometry technique capable of capturing in situ the to-
pography of the crystal surface during growth. The depth
resolution of each image is 2.5 nm, the lateral resolution
is 0.5 ␮m, and the capture frequency can be as high at 10
k ס [R(C − C)/C ⌫D]
,
(2)
x
s
eo
where R is the growth rate of the face and C , C, and C
s
eo
are respectively the densities of the crystal, of the solu-
tion above the growing unperturbed interface, and of the
equilibrium solution. The stiffness constant
−1
images s .
We applied this technique to the study the bunching
instabilities that occur during the layer growth of KDP
crystals from solution. In agreement with previous ob-
servations, we found that step bunching only occurs
when the directions of solution flow and step motion
coincide. The characteristic wavelength of the flow-
induced step bunches on the steep sector of the (101) face
of KDP crystals is 45 ␮m. Although step bunch height,
or the amplitude of the deviations from the average step
density, might vary depending on supersaturation, step age,
and hydrodynamic conditions, the wavelength found in
our experiments seems to be remarkably well preserved.
This wavelength is about twice as large as the one that
supposedly should induce instability controlled by cap-
illarity suggesting a kinetic nature of the instability ob-
served in this study.
⌫
ס ⍀␣/kT
,
(3)
−23
3
where ⍀ ס 9.68 × 10 cm is specific molecular vol-
ume of KDP and ␣ approximately 29 erg/cm is surface
stiffness assumed to be equal to surface energy extracted
from the activity of dislocation sources. Diffusivity
may be estimated from its relationship to the measured
viscosity at the experimental condition used: D ס 7 ×
cm /s. Taking C approximately C ס 21 wt%, one
obtains (C − C)/C ס 3.76. At T ס 38 °C ס 311 K,
ס 6.5 × 10 cm, and with R ס 3 × 10 cm/s ob-
served in our experiments in which k was measured, we
obtain k ס 5 × 10 cm . This is more than double the
typical experimental value of the bunching wave number,
2
3
6
1
3
−
6
2
1
0
eo
s
eo
−
8
−6
⌫
3
−1
x
3
−1
2
.2 × 10 cm . Note also that the bunching occurs when
the solution and step flow directions coincide, indicating
again that bunching is related to the interface kinetics
rather than to the capillarity-controlled instability. In-
ACKNOWLEDGMENTS
deed, according to the more general analysis of the onset
We thank Boris Stanojev for data processing software,
William Witherow for help with the phase-wrapping al-
gorithm, and James DeYoreo, Terry Land, and Nataliya
Zaitseva for helpful discussions and for providing ultra-
pure KDP. Support by the Biological and Physical Sci-
ences Division of NASA (Grant HAG8-1454) is
gratefully acknowledged.
3
of instability controlled by both factors, k ס 2.2 × 10
x
is located in the region of both capillarity and kinetic
instability, well below the line discriminating the capil-
larity-controlled stability and instability regions on the k
x
3
7
versus R plane at high shear flow.
−3
The bunch wavelength, approximately 4.5 × 10 cm,
is large enough to allow noticeable bulk diffusion inter-
action between steps and dependence of bunch velocity
on its local slope and the bunch width. Indeed, the step
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2064
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