Dynamics of partially faceted melt–crystal interfaces III: Three-dimensional computational approach and calculations

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doi:10.1016/j.jcr

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(S. Brandon).

Journal of Crystal Growth 284 (2005) 235–253

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Dynamics of partially faceted melt–crystal interfaces III:Three-dimensional computational approach and calculations

Oleg Weinstein, Simon Brandon�

Department of Chemical Engineering, Technion– Israel Institute of Technology, 32000 Haifa, Israel

Received 20 December 2004; received in revised form 5 June 2005; accepted 17 June 2005

Available online 10 August 2005

Communicated by G.B. McFadden

Abstract

The modeling of partially faceted melt–crystal interfaces in bulk melt growth systems has been addressed in a number

of recent publications. In particular, in Weinstein and Brandon [J. Crystal Growth 268(1–2) (2004) 299], a method for

self-consistent two-dimensional dynamic analysis of such systems while accounting for both macro- and nano-scale

phenomena, which result from the coupling between competing kinetic mechanisms and associated thermal fields, was

presented. In this manuscript, we report on an extension of this approach to three-dimensional systems. The method is

first described in detail after which it is applied to model processes involving the vertical gradient freeze growth both of

silicon and of yttrium aluminum garnet. In axisymmetric situations, results are shown to successfully reproduce

calculations obtained using the previous two-dimensional modeling approach. Additional results demonstrate a number

of important three-dimensional nano- and macro-scale features of the melt–crystal interface. These include

observations of the dominant role of the coldest dislocation step source in the case where more than one such

dislocation line intersects an advancing facet, a demonstration of the effect of growth rate on the morphology of a

multi-faceted interface, and a simple explicit analysis of step flow on an evolving facet.

r 2005 Elsevier B.V. All rights reserved.

PACS: 02.70.Dh; 81.10.Aj; 81.10.Fq

Keywords: A1. Facets; A1. Finite element method; A1. Kinetics; A1. Modeling; A2. Growth from the melt

e front matter r 2005 Elsevier B.V. All rights reserve

ysgro.2005.06.031

ng author. Tel./fax: +972 4 8292822.

ss: [email protected]

1. Introduction

The importance of modeling facet formationduring bulk melt growth has been recognized inrecent years. The possibility for non-equilibriumsolute segregation on facets [1], the appearance of

d.

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O. Weinstein, S. Brandon / Journal of Crystal Growth 284 (2005) 235–253236

twinning associated with facets in a number ofsemiconductor growth systems (e.g. [2–4]), andlarge-scale facet-related changes in system geome-try and associated transport phenomena (e.g. [5]),all provide good reasons for modeling facetingduring melt growth.

Modeling facet formation while accounting forthermal fields evolving during directional meltgrowth dates back to the early 1970s [6–8].Generally speaking, these first analyses were basedon the estimation of a facet size based on a pre-defined idealized two-dimensional temperature fieldimposed as a constraint, and were therefore not selfconsistent in nature. In recent years, a number ofgroups have reported on advances made in the self-consistent, coupled analysis of heat transfer andfaceting in melt-growth systems. Their studies wereall limited to two-dimensional geometries (e.g.[9–13]) except for those of Lan and co-workers[14,15] who reported on the quasi-steady-state three-dimensional analysis of faceting of a melt–crystalinterface in a confined crystal growth geometry.These last two investigations dealt with the behaviorof a number of transport mechanisms, using ascheme according to which facets are perfectly flatsurfaces whose size and position are determined by aprescribed level of maximum undercooling.

Our recently developed algorithm [13,16], for two-dimensional modeling of faceting, takes into accountcoupling between different kinetic mechanismsinvolved in facet formation and evolution. However,as it is evident in the experimental literature (see e.g.[5,17,18]), partially faceted melt–crystal interfacesoften promote three-dimensional geometries even ifthe crystal growth set-up is two-dimensional (e.g.axisymmetric) in nature. Therefore, realistic model-ing of crystal growth systems involving partialfaceting will usually require three-dimensional ana-lysis techniques. In this manuscript, we describe anew method, based in part on the two-dimensionaltechnique described in Ref. [16], which allows for theanalysis of three-dimensional systems.

Fig. 1. Mathematical representation of the system (left) and

corresponding plot of the time-dependent furnace temperature

profile (right).

2. Model system and related governing equations

The model processes analyzed here involve asimplified vertical gradient freeze growth system,

which includes a vanishingly thin-walled andcylindrically shaped ampoule as well as an idea-lized axisymmetric furnace temperature profile. Amathematical description of the relevant system,including the idealized furnace temperature pro-file, is shown in Fig. 1.

The temperature distribution T in the melt (Dm)and solid (Ds) is determined by

ðrCpÞiqT

qt¼ r � ki rT ; i ¼ m; s, (1)

where r is the density, Cp is the heat capacity, ki isthe thermal conductivity, r is the gradientoperator and the subscripts m and s denote meltand solid (crystal) phase, respectively.

The system’s outer boundaries (qDs and qDm)communicate with the idealized furnace tempera-ture profile via radiative and convective heattransfer mechanisms according to

� ki rT � n ¼ hc½T � T f ðtÞ� þ �s½T4 � T f ðtÞ4�,

i ¼ m; s, ð2Þ

where n; hc; � and s are the unit normal vectoralong the boundaries (pointing away from thesystem), the convective heat transfer coefficient,the ampoule wall’s emissivity and the Stefan–Boltzmann coefficient, respectively. The idealizedfurnace temperature, T f (see Fig. 1), is given by the

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O. Weinstein, S. Brandon / Journal of Crystal Growth 284 (2005) 235–253 237

same relation as in Ref. [13]:

T f ðz; tÞ

¼

Tmin for Tc0 þ Gz � Gt

pTmin;

T c0 þ Gz � Gt for TminoT c0 þ Gz

�GtoTmax;

Tmax for Tc0 þ Gz

�GtXTmax;

8>>>>>>>>>>><>>>>>>>>>>>:

ð3Þ

where G ¼ ½ThðtÞ � T cðtÞ�=L is the time-indepen-dent thermal gradient, ThðtÞ ¼ Th0 � Gt andT cðtÞ¼T c0 � Gt denote the time-dependent hotand cold zone temperatures, respectively, Th0 ¼

Th ðt ¼ 0Þ ¼ Tc0 þ GL, T c0 ¼ T cðt ¼ 0Þ, Tmax andTmin are constants which determine a range overwhich the furnace temperature is allowed to vary,G is the furnace cooling rate, and L is the lengthof the ampoule.

Along the melt–crystal interface (qDsm) twoconditions must be met, the first involving abalance of heat flow across the interface,

ð�km rTm þ ks rT sÞ � nsm ¼ DHvVn, (4)

and the second, relating the undercooling alongthe interface to the normal growth rate,

Vn ¼ bðy;DTÞDT , (5)

where DHv, Vn and b are the volumetric latentheat of fusion, local normal growth rate, and localkinetic coefficient, respectively. y is the misorienta-tion angle between the direction normal to theinterface and the relevant low-index face orienta-tion, and DT ¼ Tmp � T I is the difference betweenthe melting point temperature (Tmp) and the localinterfacial temperature (T I).

An initial condition for all transient calculationsshown here was obtained (as in Refs. [16,13]) bycalculating a steady-state thermal field and (facetfree) interface shape stemming from the solution ofthe relevant quasi-steady-state problem with G ¼ 0.

3. Molecular attachment kinetics

In our recent two publications [16,13], we reviewmolecular attachment kinetic mechanisms active in

the growth of both rough and atomically flatmelt–crystal interfaces in the context of directionalmelt-growth systems. Further information onfundamentals of melt-growth kinetics is availablein the classical crystal growth literature (e.g. inRef. [19]). In this manuscript, we briefly reviewimportant details of melt-growth kinetics with anemphasis on specific issues related to three-dimensional systems.

In melt growth, the normal growth velocity canbe related to the driving force for growth (theinterfacial undercooling) according to Eq. (5). Thisformulation is restricted to the (not uncommon)case according to which growth kinetics areaxisymmetric with respect to the singular orienta-tion. Using a local cartesian coordinate systemðx; y; zÞ, with the singular orientation parallel tothe z coordinate, and Hðx; y; tÞ denoting the localheight of the interface as measured along thez coordinate, it is trivial to show that the mis-orientation angle can be determined according to

j tanðyÞj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqH

qx

� �2

þqH

qy

� �2s

¼ jrHj, (6)

where r � exq=qx þ eyq=qy, ex is the unit vector inthe x direction and ey is the unit vector in the y

direction. Note that in the case where the singularorientation is aligned with the growth direction,the local coordinate system and height of theinterface in Eq. (6) become the ðx; y; zÞ coordinatesystem and position of the interface (Hðx; y; tÞ, seeFig. 1) used to describe the system in Section 2.

Once the local misorientation angle and under-cooling values are known, it is possible to calculateVn via Eq. (5), though the local kinetic coefficientbðy;DTÞ must first be determined. We consider bto mirror a combination of four possible kineticmechanisms. These include two mechanisms oper-ating along the singular orientation: growth bytwo-dimensional nucleation (2DN) and disloca-tion-driven growth (DG), the step flow mechanism(SM) characteristic of vicinal surfaces, as well asthe rapid growth mechanism associated withatomically rough interfaces (rough). The localcombination of these four mechanisms is achievedexactly in the same fashion as described in ouranalysis of two-dimensional systems [16,13]. For

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example, using a piecewise differentiable ap-proach, the local kinetic coefficient (bðy;DTÞÞ isgiven by

bðy;DTÞ ¼ min½brough;max½bSMðyÞ,

b2DNðDTÞ;bDGðDTÞ��, ð7Þ

where brough;bSM;b2DN;bDG are the local kineticcoefficients calculated according to each of thefour mechanisms described above. The coefficientfor growth of a rough interface (brough), which isartificially reduced in magnitude (see Ref. [16]), isassumed to be independent of misorientation andundercooling. As in the case of Refs. [16,13], thesimplifying assumption that brough is isotropic, aswell as its reduced value do not affect our results,since, due to the (still) relatively large values ofbrough that we use, rough parts of the interfacesimply coincide with the melting point isotherm.The dependence of the other three coefficients on yand DT is given by

b2DN ¼ B expð�A=DTÞ, (8)

bDG ¼ C DT , (9)

and

bSMðyÞ ¼ bstj sinðyÞj, (10)

where A;B;C can be treated as constants and, asin the case of the two-dimensional modelingapproach [16,13], step flow is assumed to involveonly elementary, non-interacting steps therebyrendering the step kinetic coefficient bst constant.

There are at least two approximations, relatedto the physics of melt–crystal interfacial motion,which are made here and should be mentioned.First, it can be shown that surface energy effectsare often unimportant in situations similar tothose discussed here (see Refs. [20,21]); we there-fore neglect these effects in our analysis. Inaddition, in this manuscript, we assume there areno heterogenous nucleation effects at the ampou-le–crystal–melt three-phase contact line. Thisassumption, which is reasonable in certain casesfor confined growth systems (e.g. when anencapsulant is used [3]) makes possible theintersection of facets with the contact line.

Investigating the general situation, in whichheterogenous nucleation at the three-phase contactline is possible, is beyond the scope of thismanuscript (see Ref. [20] for a partial treatmentof this issue).

4. Numerical approach

The Galerkin finite element method is applied inthe solution of Eqs. (1)–(5). Unstructured meshes(generated using GAMBIT [22]) involving tetra-hedral elements are defined to span both regions ofmelt and solid; sample meshes are shown in Fig. 2.In the calculations presented in this manuscript,9376 3D elements were used with a corresponding770 2D elements applied along the melt–crystalinterface. The temperature field is representedusing three-dimensional (Lagrange–quadratic) ba-sis functions:

Tðx; y; z; tÞ ¼XNT

i¼1

T ðiÞðtÞFðiÞðx; y; z; tÞ, (11)

with NT, T ðiÞðtÞ, and fFðiÞðx; y; z; tÞg denoting thetotal number of nodes in the entire mesh, thetemperature at each node and the set of globalbasis functions, respectively.

The vertical position of the melt–crystal inter-face, Hðx; y; tÞ is represented using a set ofLagrange global basis functions, fGðiÞðtÞg,

Hðx; y; tÞ ¼XNH

i¼1

H ðiÞðtÞGðiÞðx; y; tÞ, (12)

where NH is the number of interface nodes andH ðiÞ is the value of the function Hðx; y; tÞ at node i.

The partial derivative of temperature withrespect to time is given by [23]

qT

qt¼ _T � _z

qT

qz, (13)

where _T and _z are time derivatives of thetemperature and axial coordinate position on themoving mesh, respectively; for the sake ofsimplicity, we move mesh nodes in the axial (z)direction only. _z, _T are determined using a simple

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Fig. 2. A sample three-dimensional unstructured mesh used for

YAG growth simulations for the case of growth in a direction

coinciding with a single singular orientation (a) External three-

dimensional view of the entire mesh. (b) Top view of

melt–crystal interface and related interfacial mesh (thick line:

quadratic elements, thin line: additional linear line segments).

(c) Three-dimensional view of melt–crystal interface and related

interfacial mesh (only linear line segments are shown).


backward finite difference formulation:

_z ¼zm � zm�1

Dt, (14)

_T ¼Tm � Tm�1

Dt, (15)

where m is an index referring to the time stepnumber and Dt is the time step. A conservativeapproach, involving a fixed time step size of 0.2 sfor yttrium aluminum garnet (YAG) growth and0.1 s for silicon growth, was applied in all thecomputations presented here.

After discretization, Eq. (1) is weighed by thebasis functions and integrated over both the melt

and crystalline domains yielding a set of algebraicresidual equations:

RTi

¼

ZDm

ðrCpÞm_T � _z

qT

qz

� �FðiÞ þ km rT � rFðiÞ

� �dV

þ

ZDs

ðrCpÞs_T � _z

qT

qz

� �FðiÞ þ ks rT � rFðiÞ

� �dV

þ

ZqDm

½hcðT � T f Þ þ �sðT4 � T4f Þ�F

ðiÞ dA

þ

ZqDs

½hcðT � T f Þ þ �sðT4 � T4f Þ�F

ðiÞdA

�

ZqDms

DHv VnFðiÞ dA ¼ 0, ð16Þ

which is evaluated at each time step using theNewton–Raphson technique together with PETSc[24] for solution of the resultant linear equations.Boundary conditions involving temperature gra-dients (Eqs. (2,4)) are naturally implementedwithin the formulation of Eq. (16).

4.1. Calculation of interface motion

Interface motion is calculated using an ap-proach based in part on the method discussed inRef. [16]. The vector of vertical (z) positions ofnodes along the melt–crystal interface at timestep m, is related to the same vector evaluated attime step m � 1 as well as to the vector ofinterfacial temperature values (T I) at these nodesat time step m:

zm ¼ Qðzm�1;TmI Þ, (17)

where Q is an appropriate operator. This equation,together with Eq. (14), makes it possible tocalculate _z, generate a new mesh associated withtime step m, determine Vn, and solve Eq. (16) alsoassociated with time step m.

As in Ref. [16], although Eq. (17) refers to nodaldisplacements within the time frame ofDt ¼ tm � tm�1, these movements are composedof several displacements, each performed over asmall (internal) time step, usually much smallerthan Dt. The upper limit of this internal time stepsize is determined using the approach described inRef. [16].

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Although the interface is represented by quad-ratic basis functions, for the purpose of calculatinginterface motion, it is approximated using a set oflinear line segments constructed by joining eachinterface node to adjacent nodes using straightlines (see Figs. 2b, c, 3a). Looking at a specificinterfacial node, a different interface slope (i.e.misorientation) value can be calculated, for eachadjacent node, using the line segment joining the

Fig. 3. Determination of the misorientation angle (y): (a) Two-

dimensional projection of linear line segments joining adjacent

interfacial nodes (n0–n6) where n0 is a node of interest

surrounded by the other nodes (n1–n6). (b) Three-dimensional

view of nodes n0–n6, linear line segments, singular surface (with

associated normal unit vector nf ) and misorientation angles

associated with line segments joining nodes n1–n6 to node n0. (c)

Schematic of quadratic interpolation (y as a function of j) used

for determination of misorientation angle at the position of

node n0.

two nodes (Fig. 3b). As in Ref. [16], the algorithmfor interfacial motion is based in part on anupwinding-like approach, according to which onlyadjacent nodes that are above the given node(relative to the relevant facet surface) can affect itsmotion. This statement simply reflects the ideathat the motion of a point on a vicinal surface willbe affected by steps flowing towards it rather thanaway from it. A crude estimate for y can thereforebe obtained by choosing the largest misorientationvalue (y2 in Fig. 3) from those calculated withadjacent nodes above the node in question. Thisestimated y value can then be used with Eq. (5) tocalculate nodal displacement (of node n0 in Fig. 3).Note, however, that it is possible to obtain a moreprecise estimate for y (ymax in Fig. 3c) by using athree-point approximation based on the adjacentnode with the maximal misorientation value,together with the two closest adjacent nodes(nodes n1; n2; n3 in Fig. 3).

The operator in Eq. (17) is semi-implicit innature since interface displacement depends oninterfacial undercooling at the end of the current(m) time step. Therefore, in practice, interfacemotion is first calculated using Tm�1

I as an estimatefor Tm

I in Eq. (17). After calculating the newinterface position and associated thermal field(labeled by the index m), an iterative procedure isimplemented, according to which the abovesequence of local and global interface displace-ment actions is repeated several times for the samelarge-scale time step (tm � tm�1). At each iteration,the nodal temperatures (Tm

I ) are updated until thecalculation becomes self consistent within themargin of a pre-determined error. However, as inthe two-dimensional computations [16], in certaincases (e.g. convex interface motion) it is found thatan explicit interface motion operator (according towhich zm ¼ Qðzm�1;Tm�1

I Þ) is sufficient. In thiscase, an iterative procedure is unnecessary, andcalculating a new interface position and associatedthermal field becomes a non-iterative process.

5. Results and discussion

Two very different material systems are exam-ined. As in Refs. [16,13], we investigate the growth

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of both silicon and YAG. Unless stated otherwise,system parameters and physical properties corre-spond to those listed in Ref. [16]; note that theradii of the (cylindrical) YAG and silicon growthsystems studied here are 1.0 and 1.2 cm, respec-tively. In this section, we present results aimed atemphasizing differences between the two systems,verifying the validity of the three-dimensionalalgorithm and testing three-dimensional effectsassociated with competition between growth me-chanisms.

5.1. On certain differences between YAG and

silicon growth systems

One of the striking differences between the twosystems considered here, from the point of view offaceting phenomena, results from the shape of themelting point isotherm which is often concave inthe case of silicon and convex in the case of YAG.It is this difference that promotes, in the event ofgrowth in the singular orientation, the appearanceof a center facet in YAG versus a ring-shaped facetin silicon. These features can be clearly observed inthe results of the simulated growth runs depictedin Fig. 4, where the center facet (YAG) and ringfacet (silicon) are visible in the macro-scale inter-face profiles shown in Figs. 4a and b, respectively.Here, as in most other calculations discussed inthis manuscript, growth is assumed to start froman initial condition involving an immobile am-poule, in a steady-state thermal field, in which thefacet-free stationary melt–crystal interface followsthe melting-point isotherm. Growth is initiated byincreasing, at t ¼ 0, the furnace cooling rate (G)from zero to a finite constant value. The resultsshown in Fig. 4 were all obtained after steady-stategrowth conditions were achieved.

It is interesting to investigate mechanisticfeatures of crystal growth phenomena exhibitedin Figs. 4a, b. In these calculations, the crystallinematerial is assumed to be free of dislocations.Therefore, step generation on the facets is gov-erned by 2DN. Looking at a nano-scale (in thevertical direction) picture of the facets (Figs. 4c, d),it is apparent that both the ring-shaped silicon-surface facet and the circular YAG-surface facetare composed of a single growth hill whose apex,

the point of step generation, is located at thecoldest point of the surface. This is further clarifiedwhen observing the distribution of temperaturealong the interfaces (Figs. 4e, f). The coldest point(and apex), in the case of YAG, is in the center ofthe interface, while in the case of silicon this pointis on the periphery of the interface. Note that inthe case of silicon, the coldest position is notunique, and is selected during the onset of growthas a result of small numerical fluctuations in thetemperature field. Steps generated at the apex ofthe growth hill flow towards warmer parts of theinterface where they stop flowing either due to astep sink or due to the dominating effect of therough-growth mechanism at large enough yvalues. In the case of YAG growth (Fig. 4c), allsteps stop flowing when they reach the rough partof the interface on the periphery of the (center)facet. According to Fig. 4d, in the silicon growthsystem steps are annihilated when they encounterthe rough (central) part of the interface as well asthe step sink that is positioned directly across theinterface from the step source. Finally, note thecurved nature of the facet nano-profiles, which is aresult of the non-uniform undercooling along thevicinal (facet) surfaces, and is similar to curvatureof growth hills observed in two-dimensionalsimulations [13].

5.2. Testing the three-dimensional algorithm

against two-dimensional calculations

In Ref. [13], we presented a number of calcula-tions showing the effect of sudden switching, on anadvancing facet, between 2DN-driven step genera-tion and dislocation-driven step generation.Here, we select this non-trivial (to compute)scenario to test our three-dimensional algorithm’sability to reproduce results obtained by two-dimensional calculations (via the method de-scribed in Ref. [16]). The two case studiesconsidered here involve crystals growing in adirection coinciding with a single singular orienta-tion, thereby promoting the appearance of facetssimilar in nature to those shown in Fig. 4.After increasing G (from zero), an initiallystationary and facet-free melt–crystal interfaceadvances while exhibiting a dislocation-free facet.

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Fig. 4. Melt–crystal interface and temperature profiles for YAG (a,c,e) and silicon (b,d,f) after steady-state dislocation-free growth

conditions are achieved (t ¼ 2500 s for YAG and t ¼ 1200 s for silicon) for the case of growth in a direction coinciding with a single

singular orientation. (a,b) Macro-scale melt–crystal interfacial profiles. (c,d) Nano-scale profiles of the facet area (i.e. growth hills). (e,f)

Temperature profiles along the melt–crystal interface.


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The facet, which at first advances via 2DN-drivenstep generation, is at some point in time suddenlyexposed to a dislocation step source.

Results from the first of these cases, involvingthe simulated growth of a YAG crystal (e.g. in the[1 1 2] direction), are exhibited in Fig. 5. The timehistory of the axial velocity, shown in Fig. 5a,nicely exhibits important features already dis-cussed in Ref. [13]. These include (in chronologicalorder):

1.
The initial time lag during which the facet doesnot advance due to insufficient undercooling.
2.
The steady-state growth achieved after the facetstarts advancing. Note that the same, time-independent, velocity is exhibited by both roughand faceted parts of the interface during thistime-period.
3.
The sudden increase in growth velocity on thefacet when a dislocation step source is intro-duced at its center. Recall [13] that thisphenomenon is due to the fact that thedislocation step source is initialy exposed tothe large undercooling necessary for sustainingthe induced growth rate via 2DN-driven stepgeneration. Achieving the same growth ratewith dislocation-driven growth requires a muchlower undercooling value.
4.
The melt-back of rough parts of the interfaceassociated with the sudden increase in thevelocity of the facet. This was attributed inRef. [13] to significant levels of latent heatreleased at the facet.
5.

1Note that the two-dimensional projections of the three-

dimensional growth hills in Figs. 6e, f are performed on the

plane, parallel to the z axis, running both through the system’s

centerline and through the apex of the facet (the top of the 2DN

growth hill in Fig. 6e and the top of the dislocation-driven

growth hill in Fig. 6f).

The long-time decay of axial velocities of allparts of the interface to the same steady-statevalue observed earlier in the growth.

From the point of view of this manuscript, themost important feature observed in Fig. 5a is thefact that the three-dimensional algorithm does anexcellent job of reproducing results calculated,during all stages of the growth scenario, with thetwo-dimensional algorithm described in Ref. [16].

Further confirmation of the robustness andaccuracy of the three-dimensional algorithm isprovided in Figs. 5b–e. Perfect matching betweenthree- and two-dimensional calculations is ob-served when looking at the time history of the

undercooling at the facet center (Fig. 5b). Note thetransients associated with the achievement ofundercooling necessary for sustaining the inducedgrowth rate, either (initially) via 2DN-drivenstep generation or (finally) via dislocation-drivengrowth. A sequence of macroscopic melt–crystalinterface shapes (Fig. 5c) as well as the distributionof interfacial temperatures during the two steady-state growth periods (Fig. 5d) also exhibit perfectmatching between two- and three-dimensionalcalculations. Finally, in Fig. 5e it is apparent thatnano-scale features of the growth hill on the facetare extremely well captured by the three-dimen-sional algorithm (as compared with the two-dimensional method) even during the non-trivialtime-dependent event of suddenly induced disloca-tion-driven step generation.

The second case study used to test the three-dimensional algorithm against two-dimensionalcalculations involves the simulated growth ofsilicon (e.g. in the [1 1 1] direction) which, asalready shown in Fig. 4, tends to exhibit a ring-shaped facet. After an initial period during which a2DN-driven facet develops, a dislocation stepsource suddenly appears at some point on thefacet, away from its periphery (at r ¼ 0:8 cm) andat a distance rotated (in the x–y plane) both fromthe 2DN-driven step source and from the resultantstep sink (which is positioned directly across theinterface from the step source). In Fig. 6, wepresent information similar in nature to thatshown in Fig. 5. The time history of axial velocities(Fig. 6a), time history of the undercooling at thefacet’s outer edge (Fig. 6b), snap-shots of steady-state interface shapes (Fig. 6c) and temperaturedistributions (Fig. 6d), and nano-scale snap-shotsof growth hills1 (Figs. 6e, f), all show very goodagreement between three- and two-dimensionalcalculations. The main features observed in thecase of silicon growth in Ref. [13] are also observedhere. The slight differences between the twocalculations of facet-edge undercooling, shown in

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Fig. 6b, are most probably due to the fact that, ona nano-scale, the problem is not truly axisym-metric. The step source, both in the case of 2DN-driven growth and in the event where a screwdislocation is responsible for step generation,consists of a single point on the ring-shaped facet.This results, as is evident in Figs. 6e, f or moreclearly in Fig. 4d, in a non-axisymmetric growthhill. Two-dimensional axisymmetric calculations(exhibited in Fig. 6) are, on the other hand, basedon the assumption that, with the exception of theposition r ¼ 0, all positions at which a step sourceappears represent a constant r-value ring (circle)along which several step sources operate. The verygood matching between two- and three-dimen-sional calculations in Fig. 6 may therefore seemeven rather surprising. This is, in our opinion, anindication that the number of step sources is notnecessarily an important parameter. It is their‘‘strength’’ that matters. In other words, featuresof growth are most affected by the kineticparameters associated with the ‘‘strongest’’ stepsource, as well as the level of undercooling towhich it is exposed. This point is further exploredin the next section.

5.3. Three-dimensional nano-scale phenomena

The situation where a single dislocation stepsource drives the growth of a facet is interestingthough hardly the common case. Other than in theevent of dislocation-free crystal growth, mostsituations involve several dislocation lines inter-secting an advancing melt–crystal interface. In thenext set of results we examine the situation wherethe growth of a center facet on a YAG melt–crys-

Fig. 5. Comparison of three-dimensional calculations with those ach

Results correspond to an axisymmetric scenario in which YAG is grow

the ½1 1 2� direction) where, after an initial period of dislocation-free gr

the melt–crystal interface. (a) Evolution of axial velocities. (b) E

Macroscopic melt–crystal interface profiles. Shaded surfaces: two-dim

shapes. Curves: melt–crystal interface shapes calculated using two-di

200 s. (d) Melt–crystal interfacial temperature profiles. Shaded sur

dimensional profiles. Curves (dashed): profiles calculated using two

profiles calculated immediately after the appearance of the dislocation

side views) of three-dimensional crystal shapes. Curves (dashed): m

algorithm.

tal interface, initialy governed by 2DN-driven stepgeneration, is suddenly altered by the appearance(on the facet) of three different dislocation stepsources. A sequence of nano-scale interface pro-files and the corresponding distribution (over thefacet) of axial velocities are depicted in Fig. 7. Thefacet surface, which is initialy composed of a singleand dislocation-free axisymmetric growth hill,suddenly sprouts three new, rapidly growing,growth hills whose slopes are significantly steeperthan that of the underlying original hill. Here,as in the cases discussed with respect to Figs. 5and 6, suddenly appearing dislocation step sourcesare initialy exposed to relatively large under-cooling values (originally sustained for 2DN-driven growth at the induced rate). It is theselarge undercooling values that drive the rapidgrowth rate and associated steep slopes of the newhills.

Probably the most striking and importantfeature revealed in Fig. 7 is the fact that, of thethree emerging growth hills, only the fastestgrowing one survives. The two dislocation stepsources, which are exposed to relatively lowerlevels of undercooling, generate growth hills thatare rapidly overtaken by the advancing slopes ofthe fastest rising growth hill. After an initialtransient only the coldest dislocation step sourceremains effectively active. Note that we assumethat one step source does not affect the funda-mental kinetic relation associated with another,adjacent, step source. In addition, recall that stepflow is assumed to involve only elementary, non-interacting steps. Within the restrictions of theseassumptions, it appears that the number of dis-location step sources on a facet is not important.

ieved with the two-dimensional method described in Ref. [16].

n in a direction coinciding with a single singular orientation (e.g.

owth, a dislocation step source abruptly appears at the center of

volution of centerline undercooling values (DTðr ¼ 0; tÞ). (c)

ensional projections (i.e. side-views) of three-dimensional crystal

mensional algorithm, where solid curves were calculated every

faces: two-dimensional projections (i.e. side views) of three-

-dimensional algorithm. (e) Nano-scale melt–crystal interface

step source. Shaded surfaces: two-dimensional projections (i.e.

elt–crystal interface shapes calculated using two-dimensional

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Fig. 6. Comparison of three-dimensional calculations with those achieved with the two-dimensional method described in Ref. [16].

Results correspond to a scenario in which silicon is grown in a direction coinciding with a single singular orientation (e.g. the ½1 1 1�

direction) where, after an initial period of dislocation-free growth, a dislocation step source abruptly appears on the facet surface (at

r ¼ 0:8 cm). (a) Evolution of axial velocities. (b) Evolution of edge undercooling values (DTðr ¼ 1:2 cm; tÞ). (c) Macroscopic

melt–crystal interface profiles. Shaded surface: two-dimensional projection (i.e. side-view) of three-dimensional crystal shape. Curve

(dashed): melt–crystal interface shape calculated using two-dimensional algorithm. (d) Melt–crystal interfacial temperature profiles.

Shaded surfaces: two-dimensional projections (i.e. side views) of three-dimensional profiles. Curves (dashed): profiles calculated using

two-dimensional algorithm. (e) Nano-scale melt–crystal interface profiles calculated prior to the appearance of the dislocation step

source. Shaded surface: two-dimensional projection (i.e. side-view) of three-dimensional crystal shape. Curve (dashed): melt–crystal

interface shape calculated using two-dimensional algorithm. (f) Nano-scale melt–crystal interface profiles calculated immediately after

the appearance of the dislocation step source. Shaded surface: two-dimensional projection (i.e. side view) of three-dimensional crystal

shape. Curve (dashed): melt–crystal interface shape calculated using two-dimensional algorithm.

Fig. 7. Short time-scale dynamics of interfacial nano-profiles and associated thermal fields for the case of YAG growing in a direction

coinciding with a single singular orientation (e.g. the ½1 1 2� direction) where, after an initial period of dislocation-free growth, at

t ¼ t0 ¼ 2500 s, three dislocation step sources simultaneously appear on the facet surface (at r ¼ 0:2; 0:3; 0:4 cm). Nano-scale profiles of

facet/growth hills (top sequence) and associated axial velocity fields (bottom sequence) 0:001; 0:005; 0:013; 0:019 s after appearance of

dislocations.


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It is their position, or more precisely the positionof the coldest one, that matters.

The nano-scale consequences of a single dis-location suddenly appearing on an otherwisedislocation-free silicon ring-shaped facet are care-fully revealed in Fig. 8. Here, the dislocation stepsource appears at some point on the facet, awayfrom its periphery and at a distance both from the2DN-driven step source and from the resultantstep sink (in the same position discussed in Section5.2 and Fig. 6). As can be seen in Fig. 8, thedislocation step source generates a new growth hillwhich quickly dominates the growth process. The

Fig. 8. Short time-scale dynamics of interfacial nano-profiles and asso

coinciding with a single singular orientation (e.g. the ½111� direction

t ¼ t0 ¼ 1200 s, a single dislocation step source appears on the facet

(top sequence) and associated axial velocity fields (bottom sequence)

positions at which the slopes of the new hill meetthe old, underlying, growth hill rapidly movearound the facet until they collide with each otherthereby forming a new step sink. The end result isa fast-growing hill whose apex and lowest pointare rotated with respect to those on the originalgrowth hill. Both in the case of YAG (Fig. 7) andin the case of silicon, the spatial separationbetween the different, competing, growth hills isclarified when looking at the distribution of axialvelocity over the facet. In both cases, a sharpdifference between velocities associated with thedifferent growth hills marks a clear boundary

ciated thermal fields for the case of silicon growing in a direction

) where, after an initial period of dislocation-free growth, at

surface (at r ¼ 0:8 cm). Nano-scale profiles of facet/growth hill

0:004; 0:02; 0:052; 0:076 s after appearance of dislocation.

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between them. Finally, it is interesting to note, inboth figures, that the new fastest moving growthhill dominates the growth process at time scaleswhich are somewhat shorter than the time scale forrelaxation of the system back to the inducedsteady-state growth velocity; this is the basevalue surrounding the spike-shaped surges asso-ciated with the new growth hill(s) on the firstframe (on the lower left) in the series of picturesdepicting the dynamics of the axial velocity inFigs. 7 and 8.

5.4. Response of melt– crystal interface to changes

in growth rate

As is well known, h1 1 1i growth of YAG is oftencharacterized by a melt–crystal interface exhibitingthree f2 1 1g facets arranged in a three-foldsymmetric pattern around the growth axis. Thispattern of facets is associated with the appearanceof an unwanted strained core in Czochralskigrown YAG crystals. It is therefore important todeepen our understanding of this phenomenonand possibly learn methods for its control.Although experimental analyses of this problemwere reported more than three decades ago (e.g.[25,26]), it is only recently that numerical analysistechniques capable of addressing three-dimen-sional faceting phenomena have been successfullyapplied in the analysis of this phenomenon [14,15].However, these studies are restricted to quasi-steady-state growth and, moreover, do not ac-count for details of growth kinetics on the inter-face. Here, we use this phenomenon as a useful testfor the ability of our scheme to tackle realisticgrowth scenarios involving three-dimensional fa-ceting patterns associated with multiple singularorientations. In addition, we use the advantages ofour method to investigate dynamics of this systemas it responds to changes in the induced crystalgrowth rate. Results from the analysis, shown inFig. 9, clearly demonstrate the ability of themethod to capture three-dimensional patternsinvolving multiple facets even when the regionsof influence of different singular orientationsoverlap; this situation, which yields intersectionbetween facets, is clearly seen in Fig. 9e, f. It isinteresting to look at the dynamics of the growth

rate on the facet and on rough parts of theinterface (Fig. 9g). Here we clearly see the fastresponse of rough growth kinetics to changes inthe induced growth rate. In contrast, singular-direction growth (in this case via dislocation stepsources) shows a more sluggish response to thesechanges. In retrospect this result is not surprisingsince, as seen in previous figures, singular growthmechanisms require substantial undercooling va-lues to achieve a given induced growth rate. Theresponse of the interface rate of motion is there-fore limited by the speed at which the interfaceachieves the necessary degree of undercooling.This is clearly understandable when viewing thedynamics of singular orientation undercoolingshown in Fig. 9h. Interestingly, the last change ininduced growth rate is followed by a non-mono-tonic response in the growth rate of the roughpart of the interface (see Fig. 9g). This can beunderstood to be a result of the fact that as theinduced growth rate is increased, the rough partof the interface shrinks in magnitude (eventuallydisappearing), and this non-monotonic responseoccurs when the point of measurement (point 2),previously reflecting rough-growth kinetics, be-comes a point where two facets intersect (i.e. a stepsink).

5.5. Vicinal surface growth: comparison with an

explicit step-flow model

In the method described in this manuscript, aswell as in the two-dimensional approach discussedin Ref. [16], step-source and step-flow kinetics aredescribed without actually following individualsteps. Interface motion is calculated based on localinterface orientation values which reflect, in thecase of a vicinal surface, the local step density. Inthe next set of results, we closely examine thisapproach as compared with an analysis explicitlyinvolving individual steps.

The local velocity of a single step in the plane ofa relevant facet (i.e. perpendicular to the singulardirection) is given by

V st ¼ bst Tmp �gstTmp

rhDHv� T

� �¼ bst DT 1 �

r

r

� �,

(18)

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Fig. 9. Dynamics of YAG grown in the h1 1 1i direction with a time-dependent furnace cooling rate. Singular-direction growth is

dominated by dislocation-driven step sources. In this case G ¼ 20K=cm and, for 5000Xt40 s, G ¼ 4:0K=h; for 8000Xt45000 s,

G ¼ 8:0K=h; for t48000 s, G ¼ 24K=h. (a,c,e) Macroscopic melt–crystal interface profile. (b,d,f) Temperature profile along

interface. (a,b) t ¼ 5000 s. (c,d) t ¼ 8000 s. (e,f) t ¼ 12000 s. (g) Time history of axial velocity on faceted ‘‘1’’ and rough ‘‘2’’ parts of

interface. (h) Time history of undercooling at the center of the facets.


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Fig. 10. Comparison of calculated growth hills with topography simulated via explicit step-flow simulation. Top half corresponds to

step edges calculated via Eq. (18) and marked at constant Dt intervals. The bottom half corresponds to z-value contours, marked at

constant Dz intervals and obtained from the three-dimensional calculation, together with corresponding mesh of linear line segments.

(a) YAG, Dt ¼ 0:02 s, Dz ¼ Vz Dt ¼ 10 nm and (b) silicon, Dt ¼ 0:01 s, Dz ¼ Vz Dt ¼ 20 nm.

2Since the temperature fields are ‘‘frozen’’ at the values

obtained at t ¼ 1200 s (silicon) and t ¼ 2500:025 s (YAG),

silicon is truly growing under steady-state dislocation-free

conditions while the YAG growth hill (facet) is approximately

advancing via steady-state dislocation-driven growth.


where gst is the step edge energy (per unit length),r is the step radius of curvature, r ¼ gstTmp=hDHv DT and h is the step height. Using datafor silicon cited in Refs. [16,7], r ¼ ð3:58 �

10�6 K cmÞ=DT and it is therefore safe to assumethat the effect of curvature will be important onlyvery close to the step source (small r), or possiblyat the junction between the facet and rough partsof the interface (small DT), or at the intersection ofa step with either another step or an externalboundary (small r). Note that, since there is aproblem with availability of data for YAG, we usethe same value of r for both YAG and siliconsystems.

In the results shown in Fig. 10, we selectcalculations based on growth scenarios alreadydiscussed in Section 5.3 and Figs. 7, 8, andobtained using the algorithm described in Sections2–4. Specifically, for YAG we choose the interfacetemperature distribution at t ¼ 2500:025 s (i.e.0.006 s after the last frame in Fig. 7) and for

silicon we choose the interface temperature dis-tribution at t ¼ 1200 s (i.e. 0.004 s before the firstframe in Fig. 8). Using these temperature distribu-tions, we simulate the generation of a single,circular step at the step source (i.e. coldest)location and follow its lateral motion as dictatedusing Eq. (18).

Considering the points in time at which thecalculations were made (and looking at Figs. 7 and8), it is reasonable to assume that on the time scaleassociated with a single step sweeping across thefacet, the axial velocity of the facet is both uniformand constant2, i.e.

Vz ¼ V stj tanðyÞj ¼ const. (19)

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Consider two consecutive step edges at time t. Thedistance between them is l and, in this casej tanðyÞj ¼ h=l. The step velocity is given byV st ¼ l=Dt, where Dt is the time it takes the stepto advance the distance between its edge and that ofthe following step (i.e. the distance l). Inserting thelast two expressions into Eq. (19) yields the resultV z ¼ const ¼ h=Dt, i.e. Dt ¼ const. This suggeststhat by marking the position of the simulatedmoving step, at constant time intervals, we shouldbe able to reproduce a topographic map of thegrowth hill. It is interesting to compare this map tothe topography obtained directly from the calcula-tion based only on the algorithm described inSections 2–4. In Fig. 10 we show, both for a YAGcenter facet and a silicon ring-shaped facet, howsurface height contours obtained from the originalcalculation agree fairly well with snapshots of theadvancing step, taken at corresponding constanttime intervals.3 This gives additional credibility tothe interface-advancement algorithm described inSection 4 and confirms the above estimate that, forcases discussed here, step-edge curvature, which isignored in Sections 2–4, indeed has only a smallimpact on the kinetics of vicinal surface advance-ment. The small discrepancies in step shapes andpositions in Fig. 10b, near the outer facet edge andat the step sink, are possibly a result of thecurvature term in Eq. (18) due to the small r valueassociated with the intersection of steps with theexternal boundary of the facet. Finally, we shouldmention that the wiggles at the junction betweenthe facet and the rough part of the interface (outeredge-YAG, inner edge-silicon) are due to meshresolution issues, in the three-dimensional calcula-tions, at these interfacial positions.

6. Conclusions

In Ref. [16] we described, in detail, a newmethod for two-dimensional modeling of partiallyfaceted melt–crystal interfaces, advancing within

3Note that Dt values used in Fig. 10 are larger than the time it

takes a step to advance the distance l, and therefore Dz4h. In

addition, the V z values used in this figure to compute Dz are the

relevant growth hill axial velocities measured from the three-

dimensional calculations.

the framework of large-scale confined crystalgrowth systems. In this manuscript, we extendthis method, rendering it applicable to three-dimensional models of crystal growth. Resultsobtained with this method, and presented here,clearly show the following main characteristics:

�
A number of non-trivial benchmarks demon-strate an excellent agreement between calcula-tions obtained using the two-dimensionalalgorithm described in Ref. [16] and the newthree-dimensional approach presented here.
�
Analysis of nano-scale dynamics reveals that inthe event that multiple dislocation step sourcesexist on an advancing facet, it is only the coldestone which is of consequence once short time-scale dynamics have ended. This observationmay obviously change if certain underlyingrestrictions of our approach are relaxed. Theseinclude the assumption that steps do notinteract one another and, more importantly,that step sources are far apart so that theunderlying fundamental kinetic relations of agiven source is unaffected by the proximity ofanother step source.
�
Investigation of ½1 1 1� growth of a YAG crystalnicely demonstrates the ability of our method tocapture the correct three-dimensional geometryassociated with three identical facets advancingon the melt–crystal interface. The added valueof our approach, as compared with that of Lanand co-workers [14], is twofold. First, ouranalysis is time dependent and is thereforecapable of capturing dynamic occurences suchas the changes in induced growth velocityconsidered in Fig. 9. Moreover, the fact thatour technique hinges on fundamental kineticmechanisms is reflected in the observation of adifferent dynamic response of facets as com-pared with rough portions of the interface.
�
An explicit step-flow model demonstrates therobustness and accuracy of our three-dimen-sional algorithm, which does not explicitlyaccount for individual steps. In addition, thistest confirms a simple estimate according towhich step-curvature effects are mostly (albeitnot entirely) unimportant in the problemsstudied here.

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Acknowledgements

This Research was supported by The IsraelScience Foundation founded by The Academy ofSciences and Humanities.

References

[1] D.T.J. Hurle, B. Cockayne, in: D.T.J. Hurle (Ed.),

Handbook of Crystal Growth vol. 2a: Bulk Crystal

Growth, Basic Techniques, North-Holland, Amsterdam,

1994.

[2] D.T.J. Hurle, J. Crystal Growth 147 (3/4) (1995) 239.

[3] J. Amon, F. Dumke, G. Muller, J. Crystal Growth 187

(1998) 1.

[4] H. Chung, M. Dudley, D.J. Larson Jr., D.T.J. Hurle, D.F.

Bliss, V. Prassad, J. Crystal Growth 187 (1998) 9.

[5] MT. Santos, C. Marin, E. Dieguez, J. Crystal Growth 160

(3/4) (1996) 283.

[6] J.C. Brice, J. Crystal Growth 6 (1970) 205.

[7] V.V. Voronkov, Krystallografiya 17 (5) (1972) 909.

[8] A.A. Chernov, Ann. Rev. Mater. Sci. 3 (1973) 397.

[9] Y. Liu, A. Virozub, S. Brandon, J. Crystal Growth 205

(1999) 333.

[10] A. Virozub, S. Brandon, Model. Sim. Mater. Sci. Eng. 10

(2002) 57.

[11] O.N. Budenkova, M.G. Vasilyev, V.S. Yuferev, E.N.

Bystrova, V.V. Kalaev, in: V.P. Ginkin (Ed.), Proceedings

of the Fifth International Conference on Single Crystal

Growth and Heat & Mass Transfer, Obninsk, Russian

Federation, 2003.

[12] Y. Ma, L.L. Zheng, D.J. Larson, J. Crystal Growth 266

(1–3) (2004) 257.

[13] O. Weinstein, S. Brandon, J. Crystal Growth 270 (1–2)

(2004) 232.

[14] C.W. Lan, C.Y. Tu, J. Crystal Growth 233 (3) (2001) 523.

[15] C.W. Lan, C.Y. Tu, Y.F. Lee, J. Heat Mass Transf. 46 (9)

(2003) 1629.

[16] O. Weinstein, S. Brandon, J. Crystal Growth 268 (1–2)

(2004) 299.

[17] A.G. Petrosyan, G.O. Shirinyan, K.L. Ovanesyan, A.A.

Avetisyan, Kristall und Technik 13 (1) (1978) 43.

[18] B. Cockayne, M. Chesswas, D.B. Gasson, J. Mater. Sci. 4

(1969) 450.

[19] M.C. Flemings, Solidification Processing, McGraw-Hill,

New York, 1974.

[20] O. Weinstein, Ph.D. Thesis, Technion-Israel Institute of

Technology, accepted for publication.

[21] O. Weinstein, A. Virozub, S. Brandon, J. Crystal Growth,

submitted.

[22] http://www.fluent.com

[23] D.R. Lynch, K. O’neill, Int. J. Num. Meth. Eng. 17 (1981)

81.

[24] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G.

Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc,

2001, web page: http://www.mcs.anl.gov.petsc

[25] R.F. Belt, R.C. Puttbach, D.A. Lepore, J. Crystal Growth

13–14 (1972) 268.

[26] J. Basterfield, M.J. Prescott, B. Cockayne, J. Mater. Sci. 3

(1) (1968) 33.

http://www.fluent.com

http://www.mcs.anl.gov.petsc

Dynamics of partially faceted melt–crystal interfaces III: Three-dimensional computational approach and calculations

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