Enhanced Morphological Stability in Sb-Doped Ge

Enhanced Morphological Stability in Sb-Doped Ge

ANDREW DEAL, ERCAN BALIKCI, and REZA ABBASCHIAN

The axial heat processing (AHP) crystal growth technique was used to investigate the mor-phological stability of faceted solid/liquid (s/l) interfaces. Six Sb-doped Ge single crystalscontaining 2.3 · 10)2 to 2.3 · 10)1 at. pct Sb were grown at pulling rates of 10 to 20 mm/h.These include two bicrystals specifically designed to investigate the effect of slight misorientationon stability. Faceted growth with a kinetic supercooling on the order of 0.15 K was achieved,and a characteristic two-dimensional W instability boundary, an inverted crater in threedimensions, was observed. The crystals exhibited enhanced morphological stability over thepredictions of the constitutional supercooling (CS) criterion and the Mullins and Sekerka (MS)stability criterion, with the highest stability in the center of the W. These results are examinedwith current analytical stability theories accounting for convection and kinetics. An alternatemodel is proposed based on anisotropic kinetics and the competition between lateral spreadingon a faceted interface and the amplification rate of an interfacial perturbation.

DOI: 10.1007/s11661-006-9013-5� The Minerals, Metals & Materials Society and ASM International 2007

I. INTRODUCTION

THE morphological stability of solid/liquid (s/l)interfaces is an extremely complex phenomenon. Fluidflow, heat and mass transfer, surface energy, s/l interfaceorientation, and kinetics all affect the stability of aninterface. Since the introduction of the constitutionalsupercooling (CS) criterion in 1953,[1] there have beenmany attempts to identify the conditions under whichinstability will occur. Mullins and Sekerka (MS) exam-ined the one-dimensional steady-state problem forcontinuous growth of a single-phase binary alloy, andthey proposed a criterion based on perturbation theorythat included a conductivity-weighted temperature gra-dient and accounted for capillarity.[2] Since then,researchers have extended the MS approach to examinethe time-dependent stability of interfaces, nonlineareffects, geometrical effects on stability, rapid solidifica-tion, convection and forced flow effects near theinterface, multicomponent systems, and kinetic effects,among other issues.[3] While experimental evidencesupporting many of these theories exists, additionalwork investigating kinetic effects is still needed. Thisparticular area is important to the semiconductorindustry, because semiconductors are facet-formingmaterials with large anisotropic kinetics.

The experimental investigation of interfacial morpho-logical stability is limited by the ability to accuratelydetermine the conditions near the s/l interface duringsolidification. Crystal growth techniques such as theCzochralski, vertical Bridgman, and float zone meth-ods[4] supply heat radially to the melt, leading to radial

temperature gradients across the s/l interface that aredifficult to measure experimentally without disturbingthe interface. Furthermore, natural convection cancause augmented segregation of the solute. Thus, it isdifficult to accurately determine the thermal and solutalfields at the onset of an instability during growth.Recently, a technique called axial heat processing

(AHP), also known as ‘‘axial heat flux close to the phaseinterface,’’ was developed to better control the thermaland fluid flow conditions near the interface. The AHP issimilar to the vertical Bridgman method, but it incor-porates a baffle immersed in the melt near the s/linterface.[5,6] Experiments using AHP have demon-strated that a baffle reduces the amount of naturalconvection by effectively reducing the melt height.[7, 8]

Additionally, a conductive or heated baffle reduces theradial temperature gradients inherent to other methodsof crystal growth, which further lowers the level ofconvection in the melt. Moreover, when thermocouplesare incorporated into the baffle, AHP provides a meansto directly control the thermal profile near the s/linterface without disturbing the growth.In the present study, AHP is used to investigate the

morphological stability of faceted interfaces in Sb-dopedGe. Section II provides relevant background informationon the morphological stability theory. Section III gives adetailed account of the experimental portion of thisinvestigation, with the results given in Section IV. Currentanalytical theories are used to examine these results inSection V, and an explanation based on interfacialkinetics and geometry is presented in Section VI. SectionVII summarizes the findings of this investigation.

II. THEORETICAL BACKGROUND

A. CS and Morphological Stability

Research on the morphological stability of interfacesspans more than half a century. Rutter and Chalmers[9]

ANDREW DEAL, Material Scientist, is with GE Global Research,Niskayuna, NY. ERCAN BALIKCI, Assistant Professor, is with theDepartment of Mechanical Engineering, Bogazici University, Istanbul,Turkey. REZA ABBASCHIAN, Dean and Professor, is with theBourns College of Engineering, University of California Riverside,Riverside, CA. Contact e-mail: [email protected] submitted February 2, 2006.

100—VOLUME 38A, JANUARY 2007 METALLURGICAL AND MATERIALS TRANSACTIONS A

were the first to develop a theory addressing theoccurrence of morphological instabilities in directionallysolidified alloys. They examined the effect of differentgrowth speeds, temperature gradients, orientations, andimpurities on the appearance of cellular structures indecanted tin s/l interfaces. Furthermore, they proposedthat an instability could appear if the liquid temperaturegradient imposed by the furnace was less than theequilibrium liquidus temperature gradient dictated bythe solute profile ahead of the s/l interface. Under thesecircumstances, constitutional supercooling would existahead of the s/l interface, meaning the actual liquidtemperature would be less than the equilibrium liquidustemperature based on the solute concentration. Thus, arandom protuberance of the s/l interface could grow intothe supercooled liquid, leading to interfacial instability.Together with Tiller and Jackson,[1] they developed aquantitative one-dimensional criterion for maintainingthe stability of a planar front in a binary alloy at steadystate, known as the CS criterion. The CS criterionmaintains that the ratio of the temperature gradient inthe liquid to the growth rate determines if morphologicalstability can occur at steady state. Thus, high furnacegradients and low pulling rates are needed for stability.

Mullins and Sekerka[2] later took a completely differ-ent approach to the subject. They used perturbationtheory to treat one-dimensional directional solidifica-tion, and they coined the term ‘‘morphological stability’’to describe the state of a stable plane front. They alsoaddressed a few of the assumptions in the CS criterion,making their own solution more robust. Specifically,they incorporated the wavelength and amplitude of theinstability perturbations, different temperature gradientsin the solid and liquid, and the effect of s/l surfaceenergy and latent heat of fusion on the liquidustemperature into their theory. The MS criterion predictsa planar, stable s/l interface when infinitesimal ampli-tudes of all possible perturbation wavelengths decay intime under an initial set of growth conditions. Formaterials having a thermal conductivity of the solid lessthan that of the liquid, the MS criterion predicts that as/l interface is more stable than what the CS criterionpredicts under the same conditions.

Several experimental efforts have supported the CSand MS criteria. Some of the early work included aqualitative analysis of morphological stability by Morrisand Winegard[10] with Pb )1.0 wt pct Sb. Their exper-

iments and analysis showed that the onset of instabilitieswas a linear function of velocity, as predicted by the CS

criterion. Coriell and co-workers[11–14] conducted morequantitative experiments on the stability of ice cylinders.They found a good agreement with the MS criterionadapted to a cylindrical geometry. Experiments on Al-Tialloys by Shibata et al.[15] quantified the differencebetween the MS and CS criteria for the first time. Sincethen, several directional solidification experiments havesupported the MS criterion, although, as De Cheveig-ne[16] pointed out, some of these results occurred undersubcritical conditions where linear theory does not hold.

More recently, Holmes and Gatos[17] conducted adetailed supercritical experiment on Ga-doped Ge that

provided quantitative verification of the MS criterion.To match the theoretical predictions with the experi-mentally observed wavelengths of the cells, however,they had to adjust the value of the diffusion coefficient.

B. Convective Treatments

The MS criterion assumes the liquid is free ofconvection and solute redistribution is controlled purelyby diffusion. However, many attempts have been madeto incorporate the convective effects into the MSperturbation theory approach. Delves[18] and Hurle[19]

were the first to develop analytical solutions. Hurle’scriterion, which Hurle later corrected with Corriel andSekerka,[20] used the BPS boundary layer model toestimate the level of convection and included this in theMS perturbation theory approach. Compared to the MScriterion, Coriell, Hurle, and Sekerka’s (CHS) criterionpredicts that convection can either decrease or increasethe stability of the s/l interface for alloys with adistribution coefficient less than one.More recently, Favier and Rouzaud (FR) developed

another analytical criterion.[21] However, unlike the BPSboundary layer assumption used in the CHS criterion,they used a ‘‘deformable’’ boundary layer, i.e., one thatbecame perturbed with the s/l interface. At low levels ofconvection, the CHS and FR criteria predict virtuallyidentical results. However, unlike the CHS criterion, theFR criterion predicts that convection will always stabi-lize an interface, and that the extent of stability increaseswith increasing convection. This concept is supported byground-and space-based experiments with Ga-doped Ge

by Carlberg.[22,23] Additionally, Tewari and Chopra[24]

examined Pb-Sn and succinonitrile-acetone, and theirresults were in good qualitative agreement with the FRcriterion.

C. Kinetic Treatments

The MS criterion also assumes that local equilibriumexists at the s/l interface. Many materials, such assemiconductors, grow in a faceted manner under non-equilibrium conditions. Accordingly, several authors haveincorporated growth kinetics into stability theory. Tarshisand Tiller[25] considered the possibility that only certainwavelengths could appear on faceted s/l interfaces,governed by the dislocation density for screw dislocationassisted growth (SD) and the nucleation rate in two-dimensional nucleation and growth. Their analysisshowed source-limited kinetics of this type could signif-icantly stabilize an s/l interface. Likewise, Chernovtheoretically examined the effects of layer-spreading

kinetics on the stability of faceted shapes[26] and foundthat kinetics could increase their stability. Coriell andSekerka[27] incorporated faceted kinetics into the MScriterion. However, the criterion for marginal stability of asupercooled interface, where an instability is just predictedto form, was unaffected by kinetics. The kinetic factorsappeared in the denominator of their criterion, and onlythe numerator determines the marginal stability limit.More recently, the effects of anisotropic kinetics on

the stability of vicinal s/l interfaces were investigated.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 38A, JANUARY 2007—101

Coriell and Sekerka[28] demonstrated that perturbationson interfaces with anisotropic kinetics could travelparallel to the normal direction of growth. Chernov[29]

analytically investigated the effects of parallel fluid flowon step bunching instabilities and concluded that flowopposite to the direction of step motion stabilized theinterface. Conversely, flow in the direction of stepmotion destabilized the interface. This was consistentwith previous experimental observations in solutiongrowth.[30] This kinetic stabilization complements thecapillarity effects considered in the MS criterion, whichdominate for smaller perturbation wavelengths.

Coriell, Murray, and Chernov (CMC)[31] explored theseconcepts and developed an analytical stability criterion fora binary alloy at steady state in stagnant melts. They alsoestablished criteria for solution growth[32] and purematerials.[33] In their criterion, the kinetic stability is verysensitive to the vicinal angle of the s/l interface. Very smallvicinal angles, typically less than 1 deg, are predicted tostabilize s/l interfaces beyond what the MS criterionpredicts. Abbaschian, Corriell, and Chernov[34] used theCMC criterion to explain the stability of an observedvicinal interface grown in microgravity.

III. AHP EXPERIMENTS

Turning now to the present investigation, a schematicrepresentation of the AHP method used is shown inFigure 1. Heat was supplied radially from a multizonetubular furnace (1), and it was conducted through agraphite baffle (2) to the growth domain (3). In oneexperiment, a heater enclosed in the baffle was used tosupply additional axial heat. Crystal growth was

achieved by lowering the crucible assembly (4) awayfrom the fixed baffle assembly (5) and furnace while heatextraction simultaneously occurred at the base of thegraphite crucible via a conductive, water-cooled rodassembly (6). Additionally, the thermal profile alongthe crystal/melt boundaries was monitored using K-type, INCONEL* -sheathed thermocouples (7) embed-

ded in the crucible and baffle. More detailed descriptionsof the experimental equipment are found elsewhere.[5,35]

To produce Sb-doped Ge single crystals using AHPfor the stability study, two different seeding approacheswere used. The growth direction was chosen as [111],perpendicular to the predominating (111) facet plane ofGe. (The convention here is that the [111] vector pointedtoward the liquid during growth.) In the first approach,a full-diameter single-crystal disk of Ge was used as aseed. Each disk was 10-to 15mm high and slightly n type(less than 2.3 · 10)5 at. pct Sb). In the second approach,bicrystal Ge substrates were specially prepared to studythe effect of slight misorientation on morphologicalstability. Each substrate was fabricated by longitudi-nally sectioning a disk seed to expose the ( 11�2) plane.The half opposite of the [ 11�2] direction was then tiltedapproximately 3 deg toward [ �1�12 ] by carefully handpolishing its base. The two halves were then bondedtogether with a graphite paste. Graphite paste was alsoapplied to the circumference of the bicrystal substrate.After curing the paste, the circumference was lightlysanded to fit the crucible so that the final diameter was38.05±0.05 mm. The result is schematically shown inFigure 2.Six single crystals of Sb-doped Ge were produced for

this investigation: four from disk seeds and two frombicrystal substrates. Their designations are DS-1, DS-2,DS-3, DS-4, BI-1, and BI-2, where DS stands for diskseed and BI is short for bicrystal. All were grown in an

Fig. 1—Schematic of the AHP technique. Heat supplied radially isconducted by the baffle to the center of the growth area, and thenevenly distributed to the interface. Refer to the text for a descrip-tion.

Fig. 2—Side view representation of a bicrystal substrate. The lefthalf is tilted 3 deg from the right half about the [ 1�10 ] axis. Graph-ite paste was used to fill the resulting gaps.

*INCONEL is a trademark of Inco Alloys International,Huntington, WV.


inert argon atmosphere, and the initial pulling rate, R,and initial concentration, Co, for each sample are listedin Table I. These ranged from 10 to 20 mm/h and2.3 · 10)2 to 2.3 · 10)1 at. pct Sb, respectively. Onecrystal, DS-1, was grown using a powered AHP baffleheater.

For each crystal, a Ge-Sb master charge, approxi-mately 10 g of the alloy containing all of the Sballocated for the experiment, was prepared via encapsu-lation in an evacuated quartz tube. The purpose of themaster charge was to prevent Sb evaporation when thefurnace was brought to the growth temperature. Addi-tional bulk Ge was required to obtain the target initialconcentration for the experiment, either 2.3 · 10)2 or2.3 · 10)1 at. pct Sb, as indicated in Table I. The bulkGe and the master charge were loaded on top of theseed, which was placed at the base of the crucible. Thebaffle was initially positioned on top of the charge. Asthe furnace was heated, the top of the charge becamemolten. Once this occurred, the baffle was positionedcloser to the seed, either by raising the crucible orlowering the baffle, until it was in contact with theunmelted material. This process was repeated as thefurnace temperature increased, until the baffle waspositioned 10 to 20 mm above the base of the crucible.Final adjustments were made to the furnace or baffleheater to melt a portion of the seed and establish thedesired thermal profile for growth. After the establishedthermal profile was allowed to stabilize for a minimumof 1 hour, growth was initiated by lowering the crucibleat the intended rate while the baffle remained stationary.When growth was completed, the furnace was allowedto slowly cool to room temperature and the crystal wasremoved from the crucible. Note that all of the crystalspresented in this study grew detached from the crucible,

as discussed elsewhere,[36] and were easily removable.To characterize the solute distribution after growth,

each sample was longitudinally sectioned, and thesurface of one-half was polished to a 0.3-lm mirrorfinish. Resistivity measurements using a four-point probewere made at 3.81-mm intervals along the longitudinaland radial directions of a grid spanning the samplesurface. These data were then converted to Sb concen-tration (atoms/cc) according to the empirical relation-

ships established by Cuttriss[37] and Sze and Irvin.[38]

Special etching was used to reveal detailed micro-structural features. It was found that conventionaletchants for Sb-doped Ge such as H2O:KOH:K3-

Fe(CN)6 (25:3:2 by mass) and variants based on HF,HCl, or H2O2 were not sensitive enough to lowconcentrations of Sb. Instead, the crystal halves were

electrolytically etched in a dilute Na2SO3 and H2Osolution (1:79.2 by mass) opposite a graphite cathode,with an applied current of approximately 2 amps. Theetching was performed in 2-min intervals between whichthe samples were rinsed and examined. The processcontinued until no new sample features appeared, andthe total etching time ranged from 6 to 16 minutes.For samples BI-1 and BI-2, electron backscatter

diffraction (EBSD) was used to assess the orientationof the tilted and untilted halves after a period of growth.The mating longitudinal sections of these crystals wereradially sectioned, polished to 0.3lm, and analyzedusing a scanning electron microscope with EBSDcapabilities.

IV. RESULTS

A. Morphology

The sectioned and etched cross sections of the six38.1-mm-diameter crystals grown using AHP are pre-sented in Figures 3(a) through (f). The top row ofcrystals (Figures (a) through (c)) was grown at 12, 10,and 20 mm/h, while the bottom row (Figures (d) through(f)) was grown at 15 mm/h. The initial concentration ofsample DS-1 in Figure 3(a) was 2.3 · 10)2 at. pct Sb,and the initial concentrations of the other five were2.3 · 10)1 at. pct Sb. Initial axial temperature gradientsin the liquid were 20 to 35 K/cm. The high Sb sensitivityof the electrolytic etchant brought out the horizontalgrowth striations, which have spacing on the order of0.1 mm. These were not visible with conventionaletchants. The striations, visible due to small differencesin Sb segregation attributed to ±0.5 K thermal fluctu-ations during growth, revealed the evolution of eachcrystal’s s/l interface morphology. Although the initialinterface shape varied from crystal to crystal, within afew millimeters of growth, an approximately planar s/linterface was established in each one. Fringe patterns,identified in the figures, outlined large macrofacetedregions in the centers of the crystals. Additionally, duringgrowth, each crystal’s s/l interface became increasinglyconcave. (Here, the convention is that concave featureshave axially higher sides and a lower center.)Two of the crystals, DS-1 (R = 12 mm/h) and DS-2

(R = 10 mm/h), were completely single under AHPconditions, i.e., before the baffle left the melt and theremaining liquid solidified in a Bridgman manner(approximately 40 and 45 mm above the initial interface,respectively). The other four exhibited evidence ofmorphological instability after a shorter period of stablegrowth, outlined in Figures 3(c) through (f) by dashedlines. A detailed example of the stable/unstable transi-tion is shown in Figure 4, which is a composite macro-graph of the left side of sample DS-4 (R = 15 mm/h)Cells initiated from the curved portions of the striations(s/l interface) as small perturbations and developed intosharp terraces, disappearing into a striationless regionthat preceded a blocky morphology. The initial spacingof the cells ranged from 0.17 to 0.30 mm; however, thecells at the center of the crystals had uneven spacings.

Table I. Initial Pulling Rate and Concentration

Sample R (mm/h) C0 (At. Pct Sb)

DS-1 12 2.3 · 10)2

DS-2 10 2.3 · 10)1

DS-3 20 2.3 · 10)1

DS-4 15 2.3 · 10)1

BI-1 15 2.3 · 10)1

BI-2 15 2.3 · 10)1


Interestingly, the instability boundary of each crystalformed a W shape, where cells initiated at midradialpositions and the crystal centers remained stable forlonger. Subsequent radial cross sections confirmed thatthe instability boundary outlined an inverted crater inthree dimensions. The bicrystals, BI-1 and BI-2 (Fig-ures 3(e) and (f)), also exhibited a W instability bound-ary, with the tilted and untilted halves virtually identicalto each other.

Some lineage features, sharp lines revealed by theetchant, were visible in the crystals, as indicated. Theseinitiated at the crystal edges, migrated toward the centerduring growth, and are presumably dislocation lines.However, these features had no observable effect on thestability of the interface.

B. Segregation

Results from the four-point probe analysis are pre-sented in Figures 5(a) through (f). Isoconcentrationcontours are superimposed on each etched crystal crosssection. The concentration values of the lowest contoursin each subfigure are 1.8 · 10)4, 6.8 · 10)4, 6.8 · 10)4,5.7 · 10)4, 5.7 · 10)4, and 9.1 · 10)4 at. pct Sb, respec-tively. The concentration intervals between successivecontour lines are, respectively, 1.4 · 10)4, 3.4 · 10)4,6.8 · 10)4, 5.7 · 10)4, 5.7 · 10)4, and 9.1 · 10)4 at. pctSb. The contours become increasingly concave along thegrowth direction for each crystal, similar to the s/linterface morphological trend. However, a comparisonof the interface striations with the isoconcentrationcontours indicates that segments of a s/l interfacesolidified at significantly different concentrations. Spe-cifically, more solute was incorporated into the center ofthe interface throughout growth, resulting in isoconcen-tration contours that were more concave than theinterface. Moreover, the radial segregation of the dopantincreased as a function of distance solidified. This wasquantified by defining the percent segregation as

pct radial segregation zð Þ

¼ max CS rð Þ½ � �min CS rð Þ½ �average CS rð Þ½ � � 100 pct

½1�

where CS(r) is the measured solid concentration as afunction of radial position. A plot of percent segregationagainst distance solidified for each sample is presented inFigure 6.

Fig. 4—Composite micrograph of the left instability region in sam-ple DS-4.

Fig. 3—Etched single-crystal halves: (a) DS-1, (b) DS-2, (c) DS-3, (d) DS-4, (e) BI-1, and (f) BI-2. Fringe patterns (F) and lineage features (L)are indicated. Dashed lines outline morphological instability patterns, which have a characteristic W shape in two dimensions.


C. Orientation

The EBSD analysis of samples BI-1 and BI-2 wasperformed on radial cross sections taken 1 mm belowcell initiation at z = 30 mm and z = 25 mm in Fig-ures 3(e) and (f), respectively. Pole figures were obtainedfrom both the tilted and untilted halves of each crystal,as shown in Figure 7. These confirmed that the orien-tations of each seed half were maintained during stablegrowth. One half of each crystal had a [111] normal, andthe other half was titled approximately 3 deg toward[ �1�12 ], as designed.

D. Growth Parameters

As subsequently explained in detail, the crystal height,H, the solid and liquid temperature gradients, GS andGL, and the melt height, h, at the center of the crystalwere calculated as a function of the growth time (t) fromthe following heat balance equation at the s/l interface:

KSTiðHÞ � TC tð Þ

H tð Þ

� �S�KL

TB tð Þ � TiðHÞZB tð Þ � H tð Þ

� �L¼ RLV ½2�

where, referring to Figure 1, TB, ZB, and TC are theexperimental baffle temperature, baffle position, andcrucible temperature, respectively. The terms Ks and KL

are the thermal conductivities of the solid (17 W/m K)and liquid (39 W/m K), LV is the latent heat per unit

volume (2.5 · 109 J/m3),[39] Ti is the interface tempera-

ture, and R is the experimental growth rate. Note that Ti

is a function of the crystal height. This accounts forsolute buildup at the interface, which increases as soluteis rejected from the freezing solid during growth. As thecrystal height increases, the interface temperature (melt-ing point) decreases according to

TiðHÞ ¼ T om þ mLkCSðHÞ � DTk ½3�

where Tmo is the melting temperature of pure Ge

(1211 K), mL is the slope of the phase diagram liquidusline ()4.0 K/at pct), k is the equilibrium distributioncoefficient (0.003), CS is the solid concentration at theinterface, and DTk is the kinetic supercooling of theinterface. The kinetic supercooling of these experimentswas determined to be less than 1 degree due to the slowgrowth rates imposed, as explained in Section F.Consequently, DTk has a negligible effect on the crystalheight and temperature gradients, and it is neglected forthe purpose of calculating H, h, GL, and GS.The specific procedure for calculatingH, h,GL, and GS

was as follows. The experimental four-point probemeasurements provided the solid concentration at agiven crystal height CS(H), and the corresponding Ti(H)value was found using Eq. [3]. To determine the time ofgrowth for a given crystal height, a Ti(H) value wassubstituted into Eq. [2], which was then solved for theremaining unknown, H(t). The experimental valuesTB(t), ZB(t), and TC(t) were then iterated in time untilH(t) matched the corresponding position, H, of theinitial solid concentration measurement used to calculate

Fig. 5—Isoconcentration curves superimposed on the etched sample halves. Each sample’s lowest curve value and the fixed interval between linesin atoms atomic pct Sb are, respectively, as follows: (a) 1.8 · 10)4, 1.4 · 10)4; (b) 6.8 · 10)4, 3.4 · 10)4; (c) 6.8 · 10)4, 6.8 · 10)4; (d) 5.7 · 10)4,5.7 · 10)4; (e) 5.7 · 10)4, 5.7 · 10)4; and (f) 9.1 · 10)4, 9.1 · 10)4.


Ti. Then, the growth parameters H, h, GL, and GS werecalculated from the relationships

h tð Þ ¼ ZB tð Þ � H tð Þ ½4�

GL tð Þ ¼ TB tð Þ � TiðHÞZB tð Þ � H tð Þ ½5�

GS tð Þ ¼ TiðHÞ � TC tð ÞH tð Þ ½6�

The results of these calculations are plotted inFigures 8(a) through (f) against growth time foreach crystal. The open symbols show the values ofthe parameters H (mm), h (mm), GL (K/cm), and GS

(K/cm) calculated using the preceding procedure.Additionally, the lines show the same growth param-eters calculated assuming a constant interface tem-perature of Tm

o, i.e., neglecting the effect of solutebuildup.It can be seen that as the crystals grew, the

temperature gradients, GL and GS, decreased and themelt height, h, increased. This was expected, as thesystem maintained a heat balance at the interface byretarding the solidification of the low-conductivitysolid. For sample DS-1, with the low initial concen-tration of 2.3 · 10)2 at. pct Sb, solute buildup had anegligible effect on H, h, GL, or GS. However, thesolidification of the other samples, with an initialconcentration 10 times greater, was retarded, as shownby the deviation of the open symbols from the lines.When considering the solute buildup in these samples,the temperature gradients decreased less and the meltheight increased more during growth. The growth rateof the s/l interface decreased as well. The solutebuildup decreased the interface temperature (meltingtemperature) by as much as 27 degrees for those pulledabove 12 mm/h. This retarded the solidification, help-ing to maintain the heat balance at the interface andkeep the temperature gradients closer to their initialvalues.The local interface velocity and temperature gradi-

ents normal to the s/l interface at different radialpositions were obtained from experimentally measuredquantities:

Rn r; tð Þ ¼ Rið0; tÞDH rð ÞDH 0ð Þ cos hð Þ ½7�

GL;n r; tð Þ ¼ TB r; tð Þ � Ti r; tð ÞZBðr; tÞ � H r; tð Þ

� �cos hð Þ ½8�

GS;n r; tð Þ ¼ KL

KSGL;n þ

LV Ri

KScos hð Þ ½9�

The termRi(0, t) is the velocity of the interface center,determined by the slope of the H curves in Figures 8(a)through (f). The terms DH(r) and DH(0) are themeasured difference in the height of adjacent striations,taken at a radial position and the center, respectively.The angle h was measured between the tangent to theinterface striation at a given radial position and thehorizontal. The temperature of the baffle at differentradial positions, TB(r, t), was determined using aparabolic interpolation between measurements fromcentral and edge thermocouples, TB and TBE shown inFigure 1. The normal interface velocity, Rn, temperaturegradient in the liquid, GL,n, and temperature gradient inthe solid, GS,n, were used to determine the stability of theinterface at different radial positions, as described inSection V.

Fig. 6—Percent segregation for each crystal.

Fig. 7—EBSD pole figures from bicrystal radial sections just belowinstability. All are [111] upper hemisphere stereographic projectionswith the growth direction coming out of the page: (a) tilted left halfof BI-1, (b) untilted right half of BI-1, (c) tilted left half of BI-2, and(d) untilted right half of BI-2.


E. Kinetic Supercooling

By examining the nature of the morphologicalinstabilities that formed in the crystals, it was possibleto estimate the kinetic supercooling of the interface.Referring to Figure 4, the evolution of the cells fromperturbations to developed terraces is clearly visible,similar to that of the Czochralski-grown Sb-dopedSi.[40] The terraces in this figure, of sample DS-4, hadapproximately 0.29 mm plateaus and 0.09 mm heights(hst). The gradient in the liquid when these formedwas approximately 17 K/cm. Assuming that thekinetic supercooling was a maximum at the base ofthese steps and negligible at their tips, DTk iscalculated via

DTk ¼ GLhst ½10�

This results in a value of 0.15 K, which is consistent withthe value of 1.8 K reported for Ge (111) crystals, grownmore than eight times faster at 97 mm/h.[41]

The supercooling is also in agreement with theories ongrowth kinetics. The expressions of Cahn et al.[42] forlateral growth (LG) and normal SD of an interface are

RLG ffiBDLDSSLDT

aRT2þ 1ffiffiffi

gp

� �¼ bstDT ½11�

RSD ffiBDL DSSLð Þ2 DTð Þ2

4pR�T cSL NA=XSð Þ1þ 2

ffiffiffigp

g

� �½12�

where DL is the diffusion coefficient (5.5 · 10)9 m2/s);R* is the gas constant; NA is Avogadro’s number; cSL isthe interfacial energy (0.181 J/m2); DSSL is the transfor-mation entropy ()30.52 J/mol K), W S is the atomicvolume of the solid (2.26 · 10)29 m3/atom); and b is thestep kinetic coefficient, which for Ge is approximately

0.25 m/Ks.[43,44] The term B is a correction factor and gis the diffuseness parameter, measuring the sharpness ofthe interface (0 is completely smooth and 1 is infinitelysharp.) Using the value of 0.15 K for DT in eqs. [11] and[12], it is possible to simultaneously solve for g and B:g = 0.25 and B = 1.35. The diffuseness parameter iswithin the range of 0.09 to 0.34 determined for facetedGa (111) interfaces,[45] and B is well above its theoreticallower limit in this case. Thus, the value of 0.15 Kis substantiated, and neglecting DTk in Eq. [3] isreasonable.

V. DISCUSSION

A. Stability Criterion Formulations

The CS criterion of Rutter, Chalmers, Tiller, andJackson[1] and MS’s[2] original morphological stabilitycriterion are now applied to assess the experimentalstability observed. To explore the possibility that con-vection or anisotropic kinetics may have had an effect,FR’s[21] criterion and CMC’s[31] criterion for vicinalinterfaces are considered as well.All four of these theories are applicable to quasi-

steady-state experiments in which one process variablechanges very gradually until an instability occurs. Thevalue of that parameter is then compared with a criticalvalue calculated theoretically to provide a quantitativeassessment of the system’s stability. In the experimentspresently under analysis, the concentration in the solidbuilt up slowly due to the low distribution coefficient ofSb in Ge, and the value at the onset of cell formationswas directly measurable. Therefore, the critical value ofsolid concentration at the onset of instability, CS

*, is thefocus of this analysis.

Fig. 8—Growth parameters for the six crystals: (a) DS1, (b) DS2, (c) DS3, (d) DS4, and (e) B1 (f) B2. Assuming a constant interface tempera-ture: H (mm) = solid lines, h (mm) = dotted lines, GL (K/cm) = dot-dash lines, and GS (K/cm) = dashed lines. Including the effect of solutebuildup: H = circles, h = triangles, GL = squares, and GS = diamonds.


The CS, MS, and FR criteria are directly solvablefor the critical parameter CS

* via algebraic manipula-tion. In this marginal stability formulation, the CScriterion[1] is

C�S ¼kDLGL

mLR k � 1ð Þ ½13�

where, as defined previously, k is the equilibriumdistribution coefficient, DL is the diffusion coefficient,GL is the temperature gradient in the liquid ahead of theinterface, mL is the slope of the phase diagram liquidusline, and R is the growth rate. Similarly, the MScriterion[2] is

C�S ¼DLk 2TmCSLx2 þ KSGSþKLGL

�K

� �� x� � R 1�kð Þ

DL

�

2mLR k � 1ð Þ x� � RDL

� ½14�

where

x� ¼ R2DLþ R

2DL

� �2

þx2

" #1=2½15�

where x is the wavenumber of the perturbations, KS andKL are the thermal conductivities of the solid and liquid,the average thermal conductivity is �K ¼ ðKS þ KLÞ=2 ,and the capillarity constant GSL = cSL/LV. The FRcriterion[21] reduces to

C�S ¼�k2 DL KLGLþKSGSð Þ

2V �K � 4TmCSLRp2

K2DL

h i

keff mL k � 1ð Þ 1� 2kf �

HþQf exp �D 1þQð Þ

2

h in o ½16�

where

keff ¼k

k þ ð1� kÞ expð�DÞ½ � ½17�

Q ¼ K2 þ 16p2

K2

� �1=2½18�

H ¼ Q coth QD=2ð Þ ½19�

f ¼ H þ 2k � 1 ½20�

where k is the wavelength of the perturbation (2p/x) andd is the extent of the solute boundary layer ahead of theinterface. Note that D = dR/DL and L = k R/DL = 2pR/x DL are simply the nondimensional solute boundarylayer and wavelength, respectively. These are includedhere for an easier comparison with FR¢s originalnotation.

All the parameters in the CS and MS criteria areknown material properties or are experimentallydetermined, as discussed in Section IV. Thus, CS

* isdirectly calculable from Eqs. [13] and [14], and inSection B, it is directly compared to the experimen-tally measured critical concentration. However, theextent of the solute boundary layer, d (contained inthe quantity D), in the FR criterion is an unknown

quantity due to the presence of convection. Toprovide insight into the effect of convection on theobserved experimental stability, d is adjusted so thatthe calculated CS

* values match the critical concen-trations measured from the crystals. In other words, avalue for d is found that predicts the stabilityexperimentally observed. Resultant values of d arereported in Section B.The CMC criterion has a very nonlinear depen-

dence on concentration. Additionally, interfacial vic-inal angles less than 1 deg, where kinetic effectsbecome important, are difficult to measure withreasonable accuracy. To overcome these difficultiesand employ the CMC criterion[31] to analyze theexperimental results, it is expressed in terms of thevicinal angle:

h ¼ arctan~x2R3

2DLSox mLGCbstð Þ2 1� SoR=2xDLð Þ2

" #1=48<:

9=;½21�

where

GC ¼ �RC�S 1� kð Þ

kDL½22�

So¼ � KSGSþKLGLð Þ=2�K½ ��TmCSLx2þmLGC

� =mLGC

½23�

~x ¼ x� R=2DLð Þ 1� 2kð Þ ½24�

and bst is the step kinetic coefficient, a materialproperty. The result amounts to the degree of tiltnecessary to achieve the stability experimentallyobserved, which is used to assess kinetic effects. InSection B, Eq. [21] is used to analyze the experi-mental results by substituting a measured value ofCS

* and reporting the resulting value of h. This isthen compared with the measured angle of theinterface.

B. Analysis of Experimental Results

First, a qualitative comparison of the W instabilityboundaries with the isoconcentration curves gives in-sight into the stability of the crystals. As seen inFigures 5(c) through (f), the highest isoconcentrationcurves in each figure follow the instability boundariesalong the peripheral regions of the W boundary.However, these same contours clearly lie below thecentral instabilities. Thus, the central portions of thecrystals are stable at higher concentrations than thesides. Quantitatively, the ratio of the concentrations atthe centers of the instability boundaries to the midradialconcentrations (the base points of the W) are 2.6, 2.6,1.7, and 1.6 for samples DS-3, DS-4, BI-1, and BI-2,respectively. For the bicrystals, the untilted and tilted


halves were virtually identical, indicating that the 3 degorientation difference had a negligible effect on stability.

Values of Rn, GL,n, and GS,n are calculated from Eqs.[7] through [9] at the base points and center of each Wpattern observed. Additionally, the cell spacing (orwavelength of the instability) is measured at thesepoints. These are listed in Table II, and the materialproperties for the Sb-doped Ge crystals are listed inTable III. These values are substituted into the CScriterion and into the MS criterion to assess the stabilityachieved before interfacial breakdown in samples DS-3,DS-4, BI-1, and BI-2.

The results of these two stability criteria are presentedin Table IV, together with the measured critical con-centrations. Left, right, and center refer to the basepoints and center of the W instability boundary,respectively. The predictions of the CS and MS criteriafall short of the measured values. The difference isquantified by the stability ratio, the measured Cs

*

divided by the predicted Cs*. This ranged from 1.9 to

4.8 for the MS criterion and 3.0 to 7.2 for the CScriterion, as shown in the table. The experimentallyobserved Cs

* values are significantly greater than thepredictions of either criterion; the stability is enhanced.For example, the ratio of 3.6 for the center region ofsample BI-1 corresponds to a 10 mm difference in thepredicted height for instability. Though the MS criterionpredicts a more stable interface with a higher criticalconcentration than the CS criterion, it still cannot matchthe experimental values. Furthermore, the center of theeach W pattern has a greater stability ratio than itscorresponding base points. Thus, the centers of thecrystals exhibit even greater enhanced stability.To assess whether convection in the melt or the

kinetics of the Ge-Sb system contributed to the observedenhanced stability, the FR and CMC stability criteriaare employed. The results of these criteria, namely, theboundary layer thickness, d, and vicinal angle, h,calculated to match the experimentally measured criticalconcentration, are presented in Table V along withexperimentally estimated values for the diffusion bound-ary layer, DL/R, and the vicinal angle.Examining the FR criterion first, the predicted

boundary layer thickness calculated to provide thestability observed ranged from 0.022 to 0.054 mm.These values are several orders of magnitude belowthe DL/R estimates for the diffusion boundary layer, 1.2to 3.3 mm, which would exist in the absence ofconvection. The relative effects of convection anddiffusion in the melt during the experiments is estimatedfrom the Grashof number, Gr:

Gr ¼ buoyancy force

viscous force¼ gHT GLh4

t2½25�

Here, g is the gravitational constant, QT is the thermalexpansion coefficient of the melt (1.1 · 10)4 K)1), m is

Table II. Parameters at the Onset of Morphological

Instability

Sample RegionRn

(mm/h)GL,n

(K/cm)GS,n

(K/cm)k

(mm)

DS-3 left 15 16 42 0.21DS-3 center 15 18 48 0.20DS-3 right 17 15 42 0.17DS-4 left 11 14 38 0.25DS-4 center 9 17 42 0.19DS-4 right 9 13 34 0.17BI-1 left 9 14 35 0.23BI-1 center 9 15 39 0.20BI-1 right 9 13 33 0.26BI-2 left 7 10 25 0.23BI-2 center 6 12 30 0.22BI-2 right 8 11 29 0.22

Table IV. CS and MS Stability Criteria Results

Sample Region Cs*—Measured

(Atomic pct Sb)Cs

*—CS(Atomic pct Sb)

Ratio(Cs

*/Cs*—CS)

Cs*—MS

(At. Pct Sb)Ratio

(Cs*/Cs—MS)

DS-3 left 5.4 · 10)3 1.5 · 10)3 3.6 2.5· 10)3 2.2DS-3 center 1.3 · 10)2 1.8 · 10)3 7.2 2.7· 10)3 4.8DS-3 right 4.8 · 10)3 1.3 · 10)3 3.7 2.1· 10)3 2.3DS-4 left 8.2 · 10)3 1.9 · 10)3 4.3 3.0· 10)3 2.7DS-4 center 1.9 · 10)2 2.7 · 10)3 7.0 4.3· 10)3 4.4DS-4 right 6.6 · 10)3 2.2 · 10)3 3.0 3.4· 10)3 1.9BI-1 left 8.6 · 10)3 2.3 · 10)3 3.7 3.4· 10)3 2.5BI-1 center 1.4 · 10)2 2.5 · 10)3 5.6 3.9· 10)3 3.6BI-1 right 6.3 · 10)3 2.1 · 10)3 3.0 3.2· 10)3 2.0BI-2 left 1.0 · 10)2 2.1 · 10)3 4.8 3.2· 10)3 3.1BI-2 center 1.5 · 10)2 3.0 · 10)3 5.0 4.5· 10)3 3.3BI-2 right 8.8 · 10)3 2.2 · 10)3 4.0 3.4· 10)3 2.6

Table III. Material Properties

Property Value

k 0.003DL 5.5 · 10)9 m2/smL )4.0 K/at. pctTm 1211 KG SL 7.24 · 10)11 mKL 39 W/m KKS 17 W/m KcSL 0.181 J/m2

LV 2.5·109 J/m3

bst 0.25 m/K s


the kinematic viscosity of the melt (1.3 · 10)7 m2/s), GL

is the temperature gradient in the melt, and h is the meltheight. For Gr values greater than 104, the diffusionboundary layer is expected to be truncated by convectiveflow.[46] Experimental minimum values of the meltheight (5 mm) and liquid temperature gradient (10 K/cm) give an experimental lower limit for Gr of 4.0 · 104.This indicates that convection was present during thegrowth, supporting the smaller boundary layers calcu-lated with the FR analysis. Note that if the AHP methodwere not employed in this investigation, the melt heightwould have been significantly larger, increasing the Grnumber by several orders of magnitude due to the strongdependence on h in Eq. [25]. Thus, although convectionwas not eliminated by AHP, it was significantly reduced.

While the boundary layer thicknesses calculated fromthe FR criterion are in agreement with the presence ofconvection during growth, they must be examined withrespect to the solute buildup in the crystals. As shown inTable V, the calculated boundary layer at the centerinstability of each crystal (center) is smaller than at thebase points of the W (left and right). A smallerboundary layer results from more convective mixing,which removes rejected solute from the interface. Thus,for the results of the FR criterion to fully explain theenhanced stability observed, the solute buildup at thecenters of the crystals should be less than at the sides.However, the exact opposite was observed experimen-tally, as shown by the isoconcentration contours inFigure 5. Solute buildup in the center was significantlygreater than at the sides, and this contradicts the FRanalysis.

Turning now to the CMC criterion, the vicinal anglescalculated to predict the measured critical concentra-tions range between 0.09 and 0.24 deg. Accordingly, forthe CMC criterion to explain the experimental results,the s/l interface must be less than a quarter of a degreefrom a singular orientation to provide the enhancedstability. The experimental s/l interfaces were concavewhen cells formed. Specifically, the angle between thetangent to the interface and the [111] orientation at thecell initiation site, measured from the growth striations,ranged from less than 1 to 21 deg, as shown in Table V.It is possible that the CMC mechanism accounts for thestability observed in the center regions of samples DS-3

and DS-4, because the calculated vicinal angles are inthe range of the experimental values. However, theCMC criterion cannot explain the enhanced stabilityobserved at the base points of the W pattern, where themeasured angle ranges from 6 to 18 deg. These anglesare roughly two orders of magnitude greater than whatthe CMC criterion predicts. According to the CMCcriterion, any interfacial angle greater than ~3 deg fromthe singular orientation should not enhance stability butmatch the predictions of the MS criterion,[31] and thiswas not experimentally observed.Samples BI-1 and BI-2 were designed to be a more

definitive test for the CMC criterion. These grew frombicrystal seeds having one-half with a [111] normal andthe other tilted approximately 3 deg from [111]. Pre-experimental predictions based on the CMC criterionwere that the tilted half of each sample would form cellsat the MS critical concentration, while the complemen-tary half would exhibit enhanced stability. However, theexperimental results did not follow these predictions.Both halves of each crystal exhibited approximatelyequal amounts of enhanced stability, with critical con-centrations higher than predicted by the MS criterion.

VI. KINETIC STABILIZATION

A. Criterion Development

An alternative mechanism for the observed enhancedstability is now proposed. It is reasoned that kinetics area dominate contributor to the enhanced stabilitybecause the crystals grew with faceted interfaces. Thefollowing arguments lead to a criterion based on theanisotropic kinetics of stepped interfaces.The case of a perfectly flat s/l interface is treated by

the MS criterion. The interface considered is rough andisotropic, with all atomic sites equally available foratomic attachment. Thus, it grows continuously. If thisinterface becomes perturbed due to local fluctuations inthe thermal and solutal fields, the MS criterion predictswhether the perturbation will grow or decay. Theamplification rate for a given wavelength, k, is definedby the quantity _e=e in the expanded form of the MScriterion:[2]

Table V. FR and CMC Stability Criterion Results

Sample Region Cs*—Measured(At. Pct Sb)

d—FR(mm)

DL/Rn

(mm)h—CMC(Deg)

h—Measured(Deg)

DS-3 left 5.4 · 10)3 0.039 1.3 0.21 13DS-3 center 1.3 · 10)2 0.022 1.3 0.13 <1DS-3 right 4.8 · 10)3 0.032 1.2 0.24 18DS-4 left 8.2 · 10)3 0.041 1.8 0.15 13DS-4 center 1.9 · 10)2 0.022 2.2 0.09 <1DS-4 right 6.6 · 10)3 0.036 2.2 0.18 11BI-1 left 8.6 · 10)3 0.040 2.2 0.14 13BI-1 center 1.4 · 10)2 0.028 2.2 0.11 3BI-1 right 6.3 · 10)3 0.054 2.2 0.17 12BI-2 left 1.0 · 10)2 0.033 2.8 0.12 16BI-2 center 1.5 · 10)2 0.030 3.3 0.10 8BI-2 right 8.8 · 10)3 0.034 2.5 0.14 13


where l=(1)k). The term _e=e physically representsthe inverse of the time required for the perturbationto increase by a factor of e (SI units are 1/s.)Negative values are interpreted to indicate a stableinterface. Note that the marginal stability criterion ofEq. [14], where perturbations are just predicted toappear, is derived from Eq. [26] by simply setting _e=eto zero.

If, however, the s/l interface is faceted, havinganisotropic surface energy, atoms will preferentiallyattach to step ledges and kink sites, and lateral spread-ing will dominate growth. Specifically, the growthvelocity parallel to the interface, RLG, will be severalorders of magnitude faster than the normal componentof velocity, Rn. This is significantly different fromcontinuous growth, where there is little differencebetween the two velocities, as shown schematically inFigure 9. Consequently, if perturbations form on afaceted interface, they will act as sources for lateralspreading. For an interface of perfectly singular orien-tation, the spreading rate for a given supercooling isobtained from the following relationship:

RLG ¼ bstDT ½27�

where bst is 0.25 m/s K for Ge. It follows that RLG forpure Ge with 0.15 K of supercooling, the value for theexperiments under consideration, is 1.35 · 105 mm/h.This is roughly four orders of magnitude larger than thepulling rates, which ranged from 15 to 20 mm/h.

For continuous growth, a perturbed and unstableinterface will have a positive amplification rate definedby _e=e . If the initial perturbation peaks are only oneatomic step high, then, in e=_e seconds, they will be twosteps high. However, if the interface is faceted, thenrapid lateral spreading will occur between the pertur-bations due to the anisotropic nature of the interface.

Steps from adjacent peaks may spread into each otherand decrease the overall height of the perturbations.The time it takes for steps from adjacent peaks,separated by a wavelength, k, to spread into eachother is simply

ts ¼k

2RLG½28�

In other words, in time ts, the amplitude of theperturbation is reduced by the height of one step. Itfollows that if ts is shorter than the time it takes for theperturbation amplitude to double to two steps, theperturbation will decay. Thus, rapid lateral spreadingcan increase the stability of singular, faceted interfacesover rough ones.For vicinal and tilted faceted interfaces, the lateral

spreading rate may be significantly slower than therelationship in Eq. [27]. These interfaces are constructedof a series of steps so that they form an angle, h, with thesingular orientation [111], as shown in Figure 10. Tomaintain this angle during growth, the ratio of the axialgrowth rate parallel to the singular direction, Rz, to thegrowth rate of the steps, RL, must follow Rz/RL = tan h,based on the interfacial geometry. The lateral growth rateparallel to the interface, RLG, is determined by multiply-ing RL by cos h. In the present experiments, Rz is on theorder of 10 mm/h. Accordingly, an interface tilted just1 deg would have a step velocity of 573 mm/h: RLG isreduced by three orders of magnitude. Note that spread-ing parallel to a vicinal interface will preferentially occurin one direction due to geometrical constraints. Conse-quently, the time it takes for interfacial spreading betweentwo adjacent perturbation peaks becomes

ts ¼k

RLG¼ k

RL cos hð Þ ¼kðsin hÞ

Rz½29�

_ee¼

Rx �2TmCSLx2 x� � RlDL

�� KSGSþKLGL

�K

� �x� � Rl

DL

�þ 2mLGC x� � R

DL

�h iKS GS�KLGL

�K

� �x� � Rl

DL

�þ 2xmLGC

½26�

Fig. 9—Continuous growth (top) has similar lateral and normalgrowth rates. Faceted growth (bottom) has rapid lateral spreading. Fig. 10—Geometry of a vicinal interface.


It follows that a faceted or tilted interface is stable whenthe lateral spreading between perturbation peaks isfaster than the perturbation amplification. Using Eq.[26] as a first approximation for the amplification rate,this concept can be written as a criterion for stability:

_ee

<Rz

k sin hð Þ ½30�

According to this kinetic stability (KS) criterion, stabil-ity increases with smaller vicinal angles, smaller ampli-fication rates, and larger supercoolings. A stabilityparameter based on Eq. [30] allows for a more quan-titative analysis:

KS parameter ¼

_ee� Rz

k sin hð ÞRz

k sin hð Þ

½31�

This KS parameter is plotted as a function of solidconcentration in Figure 11. The parameters Rz =10 mm/h, GL = 14 K/cm, GS = 36 K/cm, andk = 0.21 mm, which represent average experimentalvalues, are constant in this representation. The fourdifferent curves are calculated for different angles, whichcorrespond to different RLG/Rn ratios: h = 26.5 deggives RLG/Rn = 2.0, h = 11.3 deg gives RLG/Rn = 5.0,h = 5.7 deg gives RLG/Rn = 10.0, and h = 0.573 deggives RLG/Rn = 100.0 for the solid, dashed, dash-dot,and dotted curves, respectively. The curves intersect at avalue of )1.0 and the critical concentration predicted bythe MS criterion, 3.2 · 10)3 atomic pct Sb, which iseasily seen by setting _e=e to zero in Eq. [31]. If the KSparameter is negative, then a stable interface is pre-dicted, similar to the MS criterion. As seen in Figure 11,increasing the RLG/Rn ratio, or similarly decreasing thevicinal angle, results in a higher critical concentration atwhich the KS parameter becomes positive. Thus, inter-faces with low vicinal angles, where RLG is several ordersof magnitude greater than Rn, are kinetically stabilized.

B. Results and Analysis

The critical concentrations necessary for marginalkinetic stability (a KS parameter of zero) at the basepoint of the W boundary are determined from Eq. [30]using the experimental parameters in Table II. Thesepredicted concentrations ranged from 5 · 10)3 to

2.5 · 10)2 at. pct Sb and are reported in Table VI, alongwith the measured values. The measured and predictedcritical concentrations are again compared through thestability ratio, which is simply the ratio of the measuredcritical concentration to the predicted value. The stabil-ity ratio for KS criterion ranged from 0.3 to 1.4, giving abetter prediction of the observed stability than the MScriterion (refer to Table IV). The KS parameter, calcu-lated from Eq. [31] and listed in Table VI, ranged from)0.5 to 0.3. These KS parameter data are reasonablyclose to zero, the condition of marginal stability, inagreement with the experimental results.The KS parameter is also determined as a function of

radial interface position for both halves of each sample,using experimental data along the striations containingthe base points of theW instability boundary. The resultsare plotted in Figures 12(a) through (h), with the observedbase point positions indicated by vertical lines. The KSparameter values listed first in Table VI lie on these lines.Note that the shapes of the curves indicate a single pointwhere an instability is predicted to initiate in each half: themaximum KS parameter value. These maximum valuesand their radial difference from the observed positions areincluded in Table VI. As shown, the maxima of the curvesare no more than 4 mm from the observed positions ofinitiation and are generally closer to the marginal stabilitycondition of zero, correlating even better with the

Fig. 11—KS stability parameter as a function of solid concentrationfor different vicinal angles, giving different normal growth (Rn) tolateral growth (RLG) ratios: h = 26.5 deg gives RLG/Rn = 2.0 (solidline); h = 11.3 deg gives RLG/Rn = 5.0 (dashed line); h = 5.7 deggives RLG/Rn = 10.0 (dash-dot line); and h = 0.573 deg gives RLG/Rn = 100.0 (dotted line).

Table VI. KS Criterion Results

Sample Region Cs*—Measured(At. Pct Sb)

Cs*—KS

(At. Pct Sb)Ratio

(Cs*/Cs

*—KS)KS

ParameterMax. KSParameter

Radial Difference(mm)

DS-3 left 5.4 · 10)3 9.7 · 10)3 0.6 )0.3 )0.2 )1.5DS-3 right 4.8 · 10)3 5.0 · 10)3 1.0 0.0 0.0 0.0DS-4 left 8.2 · 10)3 1.2 · 10)2 0.7 )0.2 0.0 )3.5DS-4 right 6.6 · 10)3 2.5 · 10)2 0.3 )0.5 )0.4 3.5BI-1 left 8.6 · 10)3 1.3 · 10)2 0.6 )0.2 )0.2 )2.0BI-1 right 6.3 · 10)3 1.6 · 10)2 0.5 )0.4 )0.4 1.5BI-2 left 1.0 · 10)2 6.9 · 10)3 1.4 0.3 0.1 )2.5BI-2 right 8.8 · 10)3 1.3 · 10)2 0.7 0.1 0.4 4.0


experimental results. The curves for the bicrystals alsocorrectly predict stability at the center of both the tiltedand untilted halves. The 3 deg difference in orientation onthe left halves raises the KS parameter slightly, but notenough to cause a central instability.

Insight into the shape of the KS parameter curves isobtainable from Figure 13. Here, using the left side ofDS-4 as an example, the MS perturbation amplificationrate and lateral spreading rate (left and right sides of Eq.[30], respectively) are plotted as a function of radialinterface position. Again, the position of the singleexperimentally observed point of instability, the left basepoint of the W shape, is indicated by the vertical dottedline at r = )8.5 mm. The shaded region below the curverepresents where the MS criterion predicts instabilityalong the interface. It encompasses all but the peripheral

2 mm, which contradicts the experimental observation.On the other hand, the lateral spreading rate issignificantly greater than the MS-predicted amplifica-tion rate everywhere except at r = )12 mm, where thesetwo values are approximately equal. This point corre-sponds to the maximum in Figure 12(c), which is thesingle point where instability is predicted to initiate.Regarding the center points of the W instability

boundary, the central instabilities appear over basicallyflat interfaces. As a result, the KS parameter is close to–1.0 in every case, and extremely high stability ispredicted—even more than what is observed (refer toFigure 11, the h = 0.573 deg case). However, asFigure 4 shows, these central instabilities have more ofa blocky morphology than their base point counter-parts. Moreover, the base point instabilities grow

Fig. 12—KS parameter for each sample half along the interface where instability first occurs: (a) DS-3 left, (b) DS-3 right, (c) DS-4 left, (d) DS-4right, (e) BI-1 left, (f) BI-1 right, (g) BI-2 left, (h) BI-2 right. Observed positions of instability are indicated by vertical lines.


toward the central region and eventually transition tothis blocky structure. Taking the three-dimensionalnature of this growth into account, it is conceivablethat these central areas are simply instabilities that havegrown from peripheral portions of the crystal not visiblein the cross section taken. In other words, the centralregions have not become morphologically unstable, justovergrown. Considering this, the KS criterion correctlypredicts the W shape of the instabilities observed, whichthe other criteria fail to do.

In light of these results, the observedW pattern can beexplained by the competition between the perturbationamplification rate and lateral spreading. Toward thecenter of a given interface (r = 0), a critical concentra-tion was reached at which the MS criterion predictedinstability. However, the angle of the interface was toolow in this region and lateral spreading providedstability. Conversely, toward the edge of a giveninterface, the angle of tilt was significant, but a lowerconcentration did not cause perturbation amplificationaccording to the MS criterion. At a midradial position,as experimentally observed, the solute buildup resultedin a positive amplification rate and lateral spreading wasslow enough to allow perturbations to grow. Thus, byaccounting for lateral spreading, the KS criterioncorrectly predicts the W shape instability boundaryobserved.

VII. SUMMARY AND CONCLUSIONS

Six Sb-doped Ge single crystals were grown using theAHP technique to minimize convection in the melt.Their growth was faceted, with a kinetic supercooling onthe order of 0.15 K. Of the six crystals, morphologicalinstabilities were observed in four of them, with acharacteristic ‘‘inverted crater’’ shape in three dimen-sions or W shape in two dimensions. The central regionsof these crystals demonstrated enhanced morphologicalstability. For the bicrystals, the tilt introduced by theseed remained throughout the growth, but the orienta-

tion difference seemed to have no observable effect onthe stability in the central region.The enhanced morphological stability observed in the

Sb-doped Ge single crystals could not be explained bythe FR or CMC stability criterion. While the FRcriterion predicted boundary layers that might accountfor the convection present, they did not match the solutebuildup observed. Regions predicted to have a smallboundary layer, i.e., a large amount of convection, had ahigh solute buildup and vice versa. The CMC criterionpredicted vicinal angles that were less than a quarter of adegree from the singular [111] orientation. These pre-dictions were more than an order of magnitude lowerthan the experimental measurements at the onset ofinstability. Moreover, there was no observable differ-ence in the stability of bicrystal halves in the centralregion, where the orientation was 3 deg apart. Conse-quently, the convective and vicinal kinetic explanationsof the FR and CMC criteria for the enhanced stabilityare unsatisfactory.An alternative model based on the anisotropic kinet-

ics of stepped interfaces was proposed. This KS criterionpredicts the enhanced critical concentration more accu-rately than the MS criterion, and it correctly predicts theobserved W pattern of instability. It is also consistentwith the experimental stability of the bicrystals. Theenhanced stability observed in Sb-doped Ge is, there-fore, attributed to anisotropic kinetics. TheW instabilityboundary is attributed to the competition betweenlateral spreading and the perturbation amplificationrate of a vicinal, faceted interface.

ACKNOWLEDGMENTS

Support for this research was sponsored by NASAunder Grant No. 450975812, entitled ‘‘MorphologicalStability of Faceted Interfaces.’’ The authors also thankHamid Garmestani, Gilberto Branco, and Bob Goddard(National High Magnetic Field Laboratory, Tallahas-see, FL), for timely assistance with the EBSD mea-surements, and Sveta V. Bykova, CTR ‘‘Thermo’’(Alexandrov, Russia), for suggesting the sodium sulfite-based etchant.

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Fig. 13—Lateral spreading and perturbation amplification rates forthe left side of sample DS-4 along the interface where instability firstoccurs. The shaded region shows where the MS criterion predictsinstability.


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Enhanced Morphological Stability in Sb-Doped Ge

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