This book, together with its companion volume The science of crystallization: microscopic interfacial phenomena, make up a complete course that will teach an advanced student how to understand and analyze scientifically any of the phenomena that are observed during natural or technological crystallization from any medium and via any technique. It is an advanced text that goes into considerable detail concerning the many elements of knowledge needed to understand both quantitatively and qualitatively a crystallization event. This particular volume, having briefly reviewed the important findings of the companion volume, then deals specifically with convection, heat transport and solute transport to describe both steady state and transient solute distributions in bulk crystals, small crystallites of various shapes and thin films. The author then integrates all these factors to describe interface stability for interfaces of different shapes plus the dominant morphological characteristics found in crystals during either single phase or polyphase crystallization. The concepts are extended to embrace biological crystallization and the connecting links between the morphological features of polymer crystallization and simple organic molecule crystallization. The generation of physical and chemical defects is treated for both bulk crystals and thin films, to show both the origins of these faults and some procedures for eliminating them. A variety of mathematical examples are utilised to help the student gain a quantitative understanding of this topic area. Both the present book and its companion volume are much more broadly based and science oriented than other available books in this field, and are therefore more able to address any area of application, ranging from the production of dislocation-free single crystals in bulk or film form, at one extreme, to structurally sound large metal ingots, at the other. This book and its companion can be used independently of each other, and together they provide the basis for advanced courses on crystallization in departments of materials science, metallurgy, electrical engineering, geology, chemistry, chemical engineering and physics. In addition the books will be invaluable to scientists and engineers in the solid state electronics, optoelectronics, metallurgical and chemical industries involved in any form of crystallization and thin film formation.
The science of crystallization: macroscopic phenomena and defect generation
The science of crystallization: macroscopic phenomena and defect generation William A. Tiller Department of Materials Science and Engineering Stanford University
CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521381390 © Cambridge University Press 1991 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1991 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Tiller, William A. The science of crystallization: macroscopic phenomena and defect generation / William A. Tiller. p. cm. ISBN 0-521-38139-8. - ISBN 0-521-38828-7 (pbk.) 1. Crystallization. I. Title. QD921.T519 1992 548'.5-dc20 91-26320 CIP ISBN-13 978-0-521-38139-0 hardback ISBN-10 0-521-38139-8 hardback ISBN-13 978-0-521-38828-3 paperback ISBN-10 0-521-38828-7 paperback Transferred to digital printing 2006
To My Loving Wife Jean
Contents
1. 1.1 1.2 1.3 1.4 1.5 1.6 2. 2.1
2.1.1 2.1.2
2.1.3
Preface Symbols Introduction Coupling equations Interface attachment kinetics Surface energetics Surface creation Surface adsorption layer Solute partition coefficients Convection and heat transfer Convection Boundary layer relationships Natural convection Case 1 (Isothermal vertical plate) Case 2 (Forced and buoyancy-driven flows combined) Case 3 (Recirculating flow effects) Case 4 (Benard convection and the Rayleigh instability condition) Case 5 (Marangoni convection) Case 6 (Temperature oscillations due to natural convection) Case 7 (Vertical Bridgeman configurations) Forced convection Case 1 (Laminar motion produced by a rotating disc)
XV
xvii 1 1 5 10 11 13 14 19 19 24 26 27 29 30
32 34 37 39 42 43
x
2.1.4 2.1.5 2.1.6 2.2 2.2.1 2.2.2
2.2.3 2.2.4 2.2.5 3. 3.1 3.1.1
Contents Case 2 (Influence of a flow perpendicular to the surface of a rotating disc) Case 3 (Ekman layer spiral shear flow) Extension to film formation Combined forced and natural convection flows Interaction of flows with the crystal/melt interface Heat transport Basic equations One-dimensional heat transport approximation Case 1 (Linear heat dissipation) Case 2 (Small Peclet number, steady state) Case 3 (Finite Peclet number) Case 4 (Steady state rotating disc flow effects) Three-dimensional heat transport approximation Interface shape considerations Microscopic meniscus-lift picture Problems Steady state solute partitioning Normal freezing The steady state distributions (planar front, V = constant) Case 1 (Dsc — ^oo = SAGo = 0; conservative) Case 2 (DSc = ^oo = 0, conservative, SAG0 ^ 0) Case 3 (Dsc = SAGo = 0, conservative, Uoo ^ 0)
3.2 3.2.1 3.2.2 3.2.3 3.3 4. 4.1
51 53 59 59 69 73 74 75 76 77 79 79 81 83 88 92 95 95 97 98 100 106
Case 4 (Dsc = ^oo = SAGo = 0, chemical reaction in liquid) Non-planar interfaces (V — constant) Idealized eutectic Idealized macroscopic interface Idealized interface layer motion Film formation Problems Macroscopic and microscopic solute redistribution Initial transient (V = constant, planar front, no mixing)
115 118 120 122 122 123 135 139 141
Case 1 (Dsc = SAGo — 0, conservative)
142
Case 2 (Dsc = DLC^AGO = 0, conservative) Case 3 (-D.se = 0, conservative, E = constant)
145 146
Case 4 (DSc = 0, conservative, 6AG0 ^ 0)
148
Contents
4.2
4.3 4.4 4.4.1 4.5 4.5.1 4.6
4.6.1 4.6.2 4.6.3 4.7 4.7.1 4.7.2 4.7.3 5. 5.1 5.1.1 5.1.2 5.1.3 5.2 5.2.1 5.2.2 5.2.3 5.2.4
xi
Case 5 (Dsc — &i» ^2 KB
-
A
- grid spacing
Boltzmann's constant a drag coefficient potential function components inverse Debye length (A^1)
^
~ 'Jmax I imin
XD XeD A* A A* AA/A
-
/i fiQ /JLQ , //* A/if, /x
- chemical potential - standard state chemical potential - magnetic permeabilities - excess chemical potential per bond - viscosity
v v v v vr
-
£ £
- distance coordinate in a moving frame of reference - a surface roughening parameter
p pc Pm
- density, radius of curvature of a filament - radius of curvature of the tip of a filament - mass density
a a a* (d^f/dz)rnax), kinks can also form spontaneously on the terrace so that phase change can occur anywhere on the interface. Qualitatively, the smaller is SI/KT, the greater is the kink density, the smaller are 7^ and 7/, the smaller is the anisotropy in 7^ or 7/ and the greater is the crystallization rate for a given value of AGKFor those cases where a continuous supply of ledges occurs easily, the
8
Introduction
velocity of crystallization is proportional to both the density, pe, and height, /i£, of these ledges on the interface and the rate of movement of these ledges; i.e., V = hiptVt = PKAGK
(1.3a)
where and Afc - a'/X1*
(1.3c) 1
at small driving forces. Here, a and a represent atomic spacings while A^ and A& are the distances between ledges and between kinks, respectively. This formula applies in the domain where AGK/HT 1, we must use (2h£aa'B\ (-AGA\ \ f-AGK\] From Eq. (1.3d), one can readily see how the kink density (A^T1) enters the expression for V. For the case where the velocity of crystallization is limited by the supply of ledges, via either two-dimensional nucleation or screw-dislocations, we find that (a) two-dimensional nucleation: V = V0 exp[-(7ra" 7f/ftTAG*)]
(1.4a)
Vo = a AI0
(1.46)
where
(b) screw dislocations: V = f3'K\AGK\AGK
(1.4c)
where • ko because solute transport across the distance AZs + AZL is rapid compared to the rate of movement of the interface and essentially thermodynamic equilibrium prevails over this dimension. For interface velocities that are large relative to effective transport velocities but small relative to the maximum transition velocity for solute attachment and detachment at the interface, V*,fc$—> ki, the equilibrium interface partition coefficient defined as the concentration ratio immediately at the interface and illustrated in Fig. 1.8 for the four general combinations of interface field. The qualitative variations of ki with V for these four types of interface field are shown in Fig. 1.9.^ The transition of ki to ko + Aki occurs when transport communication is lost across the field zone in the solid/1) The further transition of ki to ki occurs when transport communication is also lost across the field zone in the liquid. Finally, the
16
Introduction
I/® C
V* because ki —> 1 and the solute becomes trapped. When one applies these considerations to film formation, interface fields exist both normal to the ledges in the plane of the adlayer and normal to the bulk film/adlayer interface. This is one of the reasons why it is so difficult to dope crystals during film growth. It is in such cases that Fig. 1.8 (substituting the adlayer for the nutrient phase) and Fig. 1.9 are expected to apply most strongly, although there are also many geological and solution growth as well as melt growth crystal examples/1) This chapter has been essentially a review of the most important findings discussed in the companion book/1) There, one will also find the background science on both the thermodynamics of bulk media as well as surfaces and the kinetics of chemical reactions as well as nucleation. Thus, we are in position to integrate these microscopic interfacial phenomena with the macroscopic aspects of crystal growth via the coupling equations (see Eqs. (1.1) and (1.2)). Chapters 2, 3 and 4 allow us to appreciate quantitatively AGSV and the surrogates ATT, AXc, A C T and ACc while Chapters 5 and 6 allow us to investigate the morphology of crystals via use of the coupling equations. Finally, Chapter 7 shows us how these various solute, temperature and stress profiles interact to generate defects in our growing and grown bulk crystals. Chapter 8 extends the exposition of Chapter 7 to thin films.
Solute partition coefficients
(1,4)
17
(1,3)
1 -
y*
(2,4)
y
CS
y CL
(2,3)
y*
Fig. 1.9. Schematic variation of the net solute partition coefficient, hi as a function of interface velocity, V, for the four different kinds of interface field given in Fig. 1.8.
Convection and heat transfer
In the following sections on convection, the intent is to apply all of this understanding and analysis to the growth of crystals from a fluid or gaseous nutrient. Most of this convection information comes from studies on fluids in contact with rigid and non-reacting boundaries. However, a crystal/nutrient interface is a reacting interface which can store heat and matter and thus can act as a type of "storage ballast" for the energy contributions imbedded in theflowpatterns of the adjacent fluid. Thus, the numerical details of the fluid flow in a box with reacting walls may be somewhat different from those in a box with non-reacting walls. However, for pedagogical simplicity, most of the chapter will deal with flows adjacent to non-reacting walls so that the author can simply "lift" work from many other studies and then weave it into a fabric that has relevance to crystal growth. At the end of the convection section, we shall return to the "reacting wall" consideration and present some data illustrating the importance of this factor.
2.1
Convection
The state of a moving incompressible fluid is fully described if, for each point in space and each instant of time, the following four quantities can be defined: the three components of the fluid velocity u and the pressure p. Conservation of matter is expressed for an incompressible fluid by divu = 0
(2.1)
20
Convection and heat transfer
and the equations of motion for an element of fluid are given by the Navier-Stokes equation
(
du
->
-A
-•
->
— + u • Vu \ = pf - Vp + V(f}Vu) (2.2) Here, the left side comprises the product of the mass of the unit and its acceleration while the right side represents the sum of the external forces acting on the unit. Following Levich/1) the term V(//Vw), where fj is the viscosity of the fluid, accounts for the effect of viscous forces. The internal friction due to the viscous nature of the fluid is manifested only when one region of fluid moves relative to another. Faster moving layers entrain slower moving ones so that momentum is transferred from the faster to the slower layers. This unique value of volume force arises in those fluids where the transfer obeys Newton's laws of motion. Such fluids are called Newtonian fluids which include water, aqueous solutions of inorganic and many organic substances, a number of organic liquids, alcohols, hydrocarbons, liquid metals, glycerine, glasses, gases and certain resins. In most examples, we shall use the parameter v = fj/p which is the kinematic viscosity of the fluid (p = density). The negative of the pressure gradient is that volume force which acts on the fluid element when the pressure changes from point to point. It is not the pressure itself but only its gradient which is required in the equation of motion (since we want the net force on an element of volume) while the local pressure may be reckoned from an arbitrary datum. The vector / represents the volume force exerted on the element of fluid. Gravity is one example of a volume force while the electromagnetic Lorentz force due to induced currents is another. In general, any gradient of extended chemical potential can be considered as such a body force. The gravity effect is of great importance in natural convection which is driven by a buoyancy density difference and impeded by the viscous inertia of the fluid. The viscosity dissipates kinetic energy while the buoyancy force releases internal energy. One example of / due to spatial temperature and concentration differences is J=-gp[l3*T(T-f)+(3*c(C-C)} (2.3a) fc = -(d£np/dT)c (3*c = -(d£np/dC)T
{2.3b) (2.3c)
where T and C refer to the average values and g is the gravitational con-
Convection
21
stant. We shall see later that this gives rise to Benard convection. While this is a volume drive, we can also think in terms of a surface or interface drive due to the variation of 7 with temperature and concentration along the surface, i.e.,
We shall see later that this gives rise to Marangoni convection. Numerous experimental studies of the flow of Newtonian liquids past the surface of a solid body wetted by them have established that the layer of fluid immediately adjacent to the surface remains motionless. Velocity measurements have shown that the thickness of this stationary layer is quite small, ~ several molecular layers. The absence of slip past the surface is highly important to fluid flow, in general, and the boundary condition that is generally assumed at solid/liquid interfaces is (2.5) uR = 0 where i£# is the relative velocity since the solid may be moving. Thus, the fluid exerts on each unit area of the solid a force which is numerically equal to the rate of momentum transfer across the surface. For example, defining pxz as the x-component of momentum transfer across a unit surface which is perpendicular to the z-axis, (dux duz Pxz = (pux)uz + V -^ \~ ~^\ oz ox The first term on the right side of Eq. (2.6) is the x-component momentum transfer accompanying physical transfer (convection) of fluid volume across a surface perpendicular to the y-axis. The second term represents the momentum transfer caused by fluid viscosity. The viscous properties of the fluid assure the transfer of a portion of the momentum from regions of greater velocity to regions of lesser velocity. In this book we wish to make use of the similarity of different flows of a viscous fluid. For this, the equations for flow must be expressed in dimensionless form so that all dimensional variables that appear in the hydrodynamic equations must be expressed in terms of factors characteristic of these variables. Defining £ as our characteristic dimension (size of crystal around which flow occurs or tube in which flow occurs) and /Too as our characteristic velocity (far-field stream velocity, for example), all linear dimensions and velocities take the form Zi = Zi/£ and Ui = Ui/Uoc, respectively. Expressed in dimensionless form, the steady flow of an incompressible fluid without body force yields equations of
22
Convection and heat transfer
the
aut
dp
i a^
where P = p/pU^ is the dimensionless pressure and ite is the Reynolds number R e = ^
(2.7c)
In this example of the steady flow of an incompressible fluid there is only one controlling number, the Reynolds number. Thus, the dimensionless shear force, r, acting on one square centimeter of surface past which the fluid streams is equal to r = f(Re)
(2.8)
and is completely determined by the magnitude of Re. In more complex cases such as for unsteady flow or for flow in the presence of an external field of volume forces, etc., other controlling parameters enter along with the Reynolds number. In these cases the flows are similar if the geometrical conditions are similar, the initial and boundary conditions are identical and all the controlling dimensionless numbers have the same respective numerical values. As an additional example to illustrate the point, if we consider the free fall of a sphere of density p and diameter a in a fluid of density difference Ap and viscosity 77, the terminal velocity, [/, of the sphere is completely determined by two dimensionless numbers, Re = Ua/v and Gr = g(Ap/p)a3/v2 where Gr is called the Grashof number. This terminal velocity is governed by the dimensionless relation Re = f(Gr)
(2.9)
At high Reynolds numbers, Re ^> 1, viscosity effects play a secondary role and, if they are eliminated, the Navier-Stokes equation transforms to the Euler equation (first order differential equation) for an ideal fluid with dimensional velocity potential
having a solution v , U dd u2 h^r-H h - = constant (2.10) p ot 2 p where / = —grad U and yields Bernoulli's theorem if the flow is steady (d(f)/dt = 0). This theorem is analogous, in some degree, to the energy principle in ordinary mechanics showing that, as we go from regions of higher flow velocity to regions of lower velocity, the pressure of the fluid changes in the opposite direction.
Convection
23
To show that the ideal fluid approximation is inadequate for some problems even at very high Reynolds numbers, we need only consider the example of a sphere moving steadily through an ideal fluid. Reasoning based on the Bernoulli equation shows that the force acting on the front hemisphere is exactly matched by the force acting on the rear hemisphere so that the sphere shouldn't move. Clearly, then, the viscosity exerts a very significant influence in the region immediately adjacent to the surface of a solid body. Combining these two features, one comes to the conclusion that, near the surface of a solid body, there must be a thin zone in which the tangential velocity component undergoes a very abrupt change from a high value at the outer border of the zone to zero at the solid surface. The retardation of the fluid in this boundary layer is caused by viscous forces alone and, although the boundary layer occupies an extremely small volume, it exerts a significant influence on the motion of the fluid. The phenomena that take place in the boundary layer are the sources of hydrodynamic resistance to the motion of solids through fluids. In a thin boundary layer, all quantities change rapidly in the direction perpendicular to the wall while their tangential rate of change is comparatively small. Moreover, provided the dimensions of the body are large compared to the thickness of the boundary layer, the flow in the boundary layer may be regarded as laminar. Simplistically, the entire zone of motion may be roughly subdivided into two regions: a region of inviscid motion, where Euler's equation applies, and a boundary layer region of thickness Ma in the meniscus region. Case 6 (Temperature oscillations due to natural convection): As already mentioned, the rolling fluid due to natural convection loops leads to thermal fluctuations in the liquid. The magnitude of the temperature fluctuations may be small in metals, ST — c for £ —•> oc,F + ,G*, and iJ* could be expanded in powers of exp(—c^), to satisfy the £ —> oo condition, as well as in powers of £ to satisfy the boundary conditions at £ —> 0.
Convection
45
In this way, he was able to determine the functional forms for F*, G* and H* (see Fig. 2.15(b)) which are given in detail in Table 2.2. Near C = 0 we find that F* « 0.51C, G* « 1 - 0.62C and H* = -0.51C 2 so that uz « 0.51a; 3/2 z/- 1/2 z 2 , u r « 0.51rcc;3/2z/~1/2^ and u = ruo(l 0.62(JU1/2IS~1/2Z). The graphs in Fig. 2.15(b) show that the distance from the disc at which the circumferential component of the flow velocity drops to half its value at the disc is OO
Q = 2TTR
/
urdz =
0.886TT#VU;)1/2
(2.26d)
Jo
which is just the axial flow from £ —> oo over this disc area. The average flow velocity of this fluid is 0.886i/o;1//2 cm s" 1 so that, in a crucible of radius R* with a crystal of radius Rc, the average return velocity of the flow, UR that would be competing with the natural convection flow is (2.27)
In addition, the shearing stress on the disc due to the fluid is given by r^ ~ pruj26rn ~ prLoiyuu)1/2 = prz/ 1 / 2 o; 3 / 2 G / (0) and the resistive torque, M, of a disc of radius Rc, which is the product of the shearing stress, area and moment arm, will be M - rwR\ ~ pRAcu(vu)1/2
= 0.3087rpi£(i/a;3 ) 1/2
(2.28)
Finally, the transition to turbulence for this type of flow occurs for a Reynolds number, Re = R2C(JO/V ~ 3 X 105, so that, even for Rc « 12.5 cm, UJ would need to be extremely high for CZ Si crystal growth to generate turbulent flow due to forced convection. To consider the effect of crystal growth on these flows, people have considered the effect of a uniform suction condition over the surface of
46
Convection and heat transfer Table 2.2. Values of the functions determining the distribution of velocities and pressure near a disc rotating in a fluid at rest. ^ 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 oo
F.
G*
-#*
_p
0 0.046 0.084 0.114 0.136 0.154 0.166 0.174 0.179 0.181 0.180 0.177 0.173 0.168 0.162 0.156 0.148 0.141 0.133 0.126 0.118 0.111 0.104 0.097 0.091 0.084 0.078 0.068 0.058 0.050 0.042 0.036 0.031 0.026 0.022 0.018 0
1.000 0.939 0.878 0.819 0.762 0.708 0.656 0.607 0.561 0.517 0.468 0.439 0.404 0.371 0.341 0.313 0.288 0.264 0.242 0.222 0.203 0.186 0.171 0.156 0.143 0.131 0.120 0.101 0.083 0.071 0.059 0.050 0.042 0.035 0.029 0.024 0
0 0.005 0.018 0.038 0.063 0.092 0.124 0.158 0.193 0.230 0.266 0.301 0.336 0.371 0.404 0.435 0.466 0.495 0.522 0.548 0.572 0.596 0.617 0.637 0.656 0.674 0.690 0.721 0.746 0.768 0.786 0.802 0.815 0.826 0.836 0.844 0.886
0 0.092 0.167 0.228 0.275 0.312 0.340 0.361 0.377 0.388 0.395 0.400 0.403 0.405 0.406 0.406 0.405 0.404 0.403 0.402 0.401 0.399 0.398 0.397 0.396 0.395 0.395 0.395 0.395 0.395 0.394 0.394 0.393 0.393 0.393 0.393 0.393
n 0.510 0.416 0.334 0.262 0.200 0.147 0.102 0.063 0.032 0.006 -0.016 -0.033 -0.046 -0.057 -0.064 -0.070 -0.073 -0.075 -0.076 -0.075 -0.074 -0.072 -0.070 -0.067 -0.065 -0.061 -0.058 -0.052 -0.046 -0.040 -0.035 -0.030 -0.025 -0.022 -0.019 -0.016 0
0.616 0.611 0.599 0.580 0.558 0.532 0.505 0.476 0.448 0.419 0.391 0.364 0.338 0.313 0.290 0.268 0.247 0.228 0.210 0.193 0.177 0.163 0.150 0.137 0.126 0.116 0.106 0.089 0.075 0.063 0.053 0.044 0.037 0.031 0.026 0.022 0
Convection
47
^ Fig. 2.16. (a) Radial velocity distribution function, F*, near a rotating disc with uniform suction parameter, k, as a function of reduced distance f, and (b) circumferential velocity distribution function, G*S^
the rotating disc. They solved the hydrodynamic equations with F* = 0, G* = 1, H* = —k on £ = 0 and F* = G* = 0 on £ = oo. For small values of /c, around £ = 0 they found (2.29) For fc = 1, they found ai = 0.389, h = -1.175 and c = 1.295 instead of 0.886 as in the Cochran analysis. More complete data for F* and G* are given in Fig. 2.16. From this figure, we see that the effect of suction (freezing for metals) at the disc or crystal surface for materials that contract on freezing is not large. In fact, it is quite small for most metals. We can see this by letting uz = (VUJY^H* = (Av/v)V « 10~3 cms" 1 « lO" 1 ^ for UJ « 60 rpm. Thus, A;4 dissolved, it was homogeneously distributed after 1-2 ACRT cycles and, after 15 minutes, all the KMnO4 had dissolved to yield a homogeneous solution. In contrast, the uniformly rotated container showed unmixed regions with no coloration at all while the stationary container exhibited even less mixing. Clearly, the ACRT is an excellent technique for initial mixing of an alloy melt. An even more effective fluid mixing technique has recently been utilized by Feigelson et alS8^ They clamped a vertical cylindrical container
58
Convection and heat transfer
1.5 min
'
4 4min
15 min
66 min
Continuous rotation
i
ACRT
Stationary
Fig. 2.25. Successive mixing stages for continuous rotation (at left), ACRT (middle) and stationary (right) containers heated from below. (Courtesy of H. Scheel.)
Convection
59
containing water to a low frequency vibration table and, by injecting drops of dye, were able to show that complete mixing in the fluid required an order of magnitude less time than was found by using the ACRT technique. This technique has been applied successfully to the control of solute partitioning during alloy crystal growth from the melt. 2.1.4
Extension to film formation It should be clear that a flow velocity will exist in a vapor from which a crystal is growing whenever the vapor/crystal transition involves a change in the number of vapor phase molecules, i.e., either a change in volume or pressure. This flow will set in motion convection patterns in the gas phase independent of Benard or Marangoni flows. For the simplest case of solid —• vapor —> solid film formation, there is a flow velocity away from the source material towards the growing crystal. This is called Stefan flow and has been described in useful detail by Faktor and Garrett/ 9 ) For the simplest case of a long silica capsule containing a solid Ag source at one end and an Ag seed crystal at the other plus sufficient inert gas to yield a total pressure P, let us consider the gaseous transport. The Ag vapor is carried down the capsule by both flow of the whole volume of gas at a velocity U and also by diffusion. The inert gas is also carried down the capsule at flow velocity U and diffuses back at such a rate as to give no net flow. It can be shown that U = (lZT/P)JAg = (P/P — PAg)dPAg/dz where z refers to distance along the length of the capsule. Thus, since V oc JA9 for the film, one can see that the importance of the Stefan velocity factor to a particular film growth situation will depend upon the ratio PA9/P. For PA9/P • 1, the effect becomes very large. The important point, for our purposes here, is that gas phase convection can also be an extremely important consideration during film growth. 2.1.5
Combined forced and natural convection flows In the CZ crystal growth technique, the buoyancy-driven flow (combined with some Marangoni flow) yields an upwardly rising natural convection velocity near the crucible wall of UN = CgPrG£3/v where C is a geometry-dependent numerical factor, typically ~ 102, Q is the temperature gradient normal to the wall and £ is a typical crucible dimension (£ « crucible radius). Thus, for Q « 1 °C cm" 1 and £ « 10 cm, UN ~ 10 cm s~1 which is a very strong flow velocity near the wall. This
60
Convection and heat transfer
flow velocity decays with distance from the wall in a manner analogous to Fig. 2.4(b). The competing flow from the Cochran drive has a velocity UR given by Eq. (2.27) and this may be much smaller than J7jv» especially at radii R* » Rc. If we define UN(?) and UR(T) as UN(r) « UN(max) exp[-b(RM - r)]
(2.35a)
and UR(r) ~ 0.886(i/w)1/2 where RM is the radius at which UN = t/jv(max) and b is a constant, then the value of r, r#, at which the natural convection velocity and the forced flow velocity are equal can be determined; i.e.,
This will determine the relative sizes of the zones of natural convection and forced flow in the CZ technique. Figs. 2.26(a)-(c) illustrate, from Carruther's work/10 ) the decrease in natural convection dominance as the crystal rotation rate, a;, is increased. When one adds the additional factor of crucible rotation, as in Figs. 2.26(d) and (e), then the spiral Cochran-type flow at the bottom of the crucible is in the same upward/downward pattern as the natural convection flow but modifies the natural flow into a spiral pattern as long as the spiral shear does not have too large a pitch. In such a case (high crucible rotation rates), the flow pattern splits into a central forced flowdominated pattern and an outer natural convection pattern perturbed into an almost turbulent state by the spiral forced flow. The degree of complexity of these Taylor-Proudman-type flow cells for different cases of crystal/crucible isorotation and counterrotation is illustrated by Fig. 2.27.(10) In the floating zone technique, rotation of the rods is often applied to induce forced convection in the floating zone so as to (i) manipulate the steady Marangoni convection to a more advantageous pattern and (ii) suppress the oscillatory Marangoni convection so as to avoid crystal striations. Iso, single and counter-rotations of the two coaxial rods have been studied. Single rotation of the lower rod induces a radial outward moving secondary flow in the vicinity of the rotating lower rod and a radial inward moving flow in the vicinity of the upper stationary rod. The whole secondary flow is a toroidal vortex whose rotation is just counterdirected to the interacting Marangoni convection due to the
Convection
61
(a)
10 rpm
(b)
(d) 25 rpm
(c)
(e) 100 rpm
Fig. 2.26. Schematic CZ flow patterns from a transparent fluid model (a) thermal convection only, (b),(c) crystal rotation and (d),(e) crucible rotation. (Courtesy of J. Carruthers.)
zone heating. These flow situations are indicated schematically in Fig. 2.28(a)/11) In the large figure to the left, on the left side is given the secondary flow induced by single rotation alone while, on the right side, the Marangoni convective flow alone is given. The combined secondary flows are shown at the right for different values of 0J2 and different ratios of the rotational and Marangoni Reynolds numbers given by (ujr2/u) Rep = { ReM p\d-y/dT\AT(2r/fj*) ' } Owing to the competition between the two counterdirected flows, it
Convection and heat transfer
62
- Qx small
Q1 small . or zero
Q2 large (ii)
^
Q2 increasing Q1 increasing
Qx large
large
Q2 increasing
Qx increasing
Q1 large
^ large
small (iv)
=0 (v)
sma11
(viii)
Fig. 2.27. Taylor-Proudman convection cell formation in a melt as a function of crystal or crucible rotation (iso or counter). (Courtesy of J. Carruthers.)
is possible to reduce the toroidal Marangoni convection by increasing the rotation rate of the lower rod. At Reo/ReM = 0.75, the ratio of the axial forced flow velocity, VD, due to rotation to the velocity UM of the Marangoni convection, is VD/UM = 0.21. At higher rotation
Convection
63
rates, VD/UM increases and the rotating flow becomes more dominant. If the upper rod is rotated while the lower one remains at rest, the Marangoni convection and the rotation-induced secondary flow will be isodirectional, resulting in an intensification of the coupled flow. Most important, the oscillatory Marangoni convection can be suppressed by applying rotation to only the lower rod. Fig. 2.28(b) clearly demonstrates the capability. The upper diagram shows the temperature oscillation measured at Reu — 87.0. Its first harmonic frequency is 2.6 Hz according to Fourier analysis. After superimposing the single rotation of the lower rod at 1000 rpm (Ren/' R&M — 1-5), the temperature oscillation is completely suppressed. The amplitude is reduced to a negligible value compared to a low rpm case and Fourier analysis shows only the frequency of the rotation speed (1000 rpm = 16.66 Hz). This fact confirms successful suppression of the flow oscillation. Before closing this section, it is beneficial to consider the results by Kuroda, Kozuka and Takano^12) re the effect of temperature oscillations at the growth interface on crystal perfection. Dislocation-free crystals were grown via the CZ technique under various growth conditions, including changed heater position, pull rate plus crystal and crucible rotation rates. The density of microdefects was found to decrease as the amplitude of the temperature oscillation,
