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2 Theory of Transport Processes

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in Semiconductor Crystal Growth from the Melt Robert A. Brown Department of Chemical Engineering and Materials Processing Center, Massachusetts Institute of Technology, Cambridge, MA 02139

The quality of large semiconductor crystals grown from the melt for use in electronic and optoelectronic devices is strongly influenced by the intricate coupling of heat and mass transfer and melt flow in growth systems. This chapter reviews the present state of understanding of these processes, starting from the simplest descriptions of solidification processes to the detailed numerical calculations needed for quantitative modeling of processing with solidification. Descriptions of models for the vertical Bridgman-Stockbarger and Czochralski crystal growth techniques are included as examples of the level of understanding of industrially important techniques.

IVÎELT C R Y S T A L

G R O W T H O F S E M I C O N D U C T O R MATERIALS

is a mainstay of

the m i c r o e l e c t r o n i c i n d u s t r y . B o u l e s of nearly perfect crystals are g r o w n b y a v a r i e t y of techniques for c o n t r o l l e d solidification a n d are u s e d as substrates i n almost a l l d e v i c e fabrication technologies. T h e variety of crystalline m a terials p r o d u c e d i n this m a n n e r ranges from the u b i q u i t o u s s i l i c o n to m o r e exotic materials l i k e G a A s , I n P , and C d T e . A l t h o u g h the processing c o n ditions for each of these materials differ i n some details, the solidification systems are similar i n that the system dynamics a n d the q u a l i t y of the crystal p r o d u c e d are g o v e r n e d b y the same set of concepts d e s c r i b i n g the transport processes,

t h e r m o d y n a m i c s , a n d materials science. T h e s e u n d e r p i n n i n g s

make m e l t crystal g r o w t h one of the " u n i t operations" of electronic materials processing. Revised and reprinted with permission from AlChE J. 1988, 34, 881-911 0065-2393/89/0221-0035$16.35/0 Published 1989 American Chemical Society

Hess and Jensen; Microelectronics Processing Advances in Chemistry; American Chemical Society: Washington, DC, 1989.

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MICROELECTRONICS PROCESSING: C H E M I C A L ENGINEERING ASPECTS


M a n y r e v i e w papers a n d several books (1-3) have focused o n the science a n d technology of crystal g r o w t h . T h e purpose of this chapter is not to d u p l i c a t e these w o r k s b u t to focus o n the f u n d a m e n t a l transport processes that occur i n m e l t crystal g r o w t h systems, especially advances i n u n d e r standing that have o c c u r r e d d u r i n g the last decade of vigorous research. T h e chapter also accentuates the features a n d research issues that are c o m m o n to m a n y of the techniques u s e d today i n laboratories a n d i n d u s t r i a l p r o d u c tion. A t h o r o u g h u n d e r s t a n d i n g o f heat a n d mass transport i n m e l t g r o w t h processes is a p r e r e q u i s i t e to o p t i m i z a t i o n of these systems for c o n t r o l of crystal q u a l i t y , as m e a s u r e d b y the degree of crystallographic perfection o f the lattice a n d the spatial u n i f o r m i t y of e l e c t r i c a l l y active solutes i n i t . T h i s discussion concentrates o n the role o f transport processes i n c o n t r o l l i n g the stability of m e l t g r o w t h techniques a n d i n setting solute segregation i n the crystal. S e v e r a l r e v i e w papers have also discussed these aspects (4, 5). T h e c o n n e c t i o n b e t w e e n processing conditions a n d crystalline perfection is i n c o m p l e t e , because the l i n k is m i s s i n g b e t w e e n m i c r o s c o p i c variations i n the structure of the crystal a n d macroscopic processing variables. F o r e x a m p l e , studies that attempt to l i n k the t e m p e r a t u r e field w i t h dislocation generation i n the crystal assume that defects are created w h e n the stresses due to l i n e a r thermoelastic expansion exceed the critically resolved shear stress for a perfect crystal. T h e status of these analyses a n d the u n a n s w e r e d questions that m u s t b e r e s o l v e d for the precise c o u p l i n g of processing a n d crystal properties are d e s c r i b e d i n a later subsection o n the c o n n e c t i o n b e t w e e n transport processes a n d defect formation i n the c r y s t a l . . T h e s i m p l e models of transport processes i n c o n t r o l l e d solidification, solute r e d i s t r i b u t i o n , a n d process stability r e v i e w e d i n this chapter are based o n results of p i o n e e r i n g studies d u r i n g the last three decades. T h e analyses are rich i n p h y s i c a l insight b u t are semiquantitative because of the l i m i t i n g assumptions n e e d e d to p e r m i t closed-form analyses. T h e n e w possibilities for the large-scale n u m e r i c a l analysis o f these transport processes are b e g i n n i n g to make practical the d e t a i l e d s i m u l a t i o n of m e l t g r o w t h . E x a m p l e s of these results are i n c l u d e d to give perspectives of the c o m p l e x i t y i n v o l v e d i n quantitative m o d e l i n g o f these systems. Several examples of m e l t g r o w t h systems are d e s c r i b e d i n the next section to facilitate the d e s c r i p t i o n of the transport processes i n s u c c e e d i n g sections. T h e crystal g r o w t h systems are d i s t i n g u i s h e d according to w h e t h e r a s u r r o u n d i n g s o l i d a m p o u l e or a m e l t - f l u i d meniscus shapes the crystal near the solidification interface. M e t h o d s w i t h these features are classified as confined a n d m e n i s c u s - d e f i n e d crystal g r o w t h t e c h n i q u e s , respectively. T h e n , the f u n d a m e n t a l transport processes associated w i t h solidification of a d i l u t e b i n a r y alloy i n a t e m p e r a t u r e gradient are r e v i e w e d . T h e classical o n e - d i m e n s i o n a l analysis of d i r e c t i o n a l solidification is discussed for diffusionc o n t r o l l e d g r o w t h . M e c h a n i s m s for convection i n the m e l t are r e v i e w e d , a n d models that account for convection i n solute transport are p r e s e n t e d .


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I m p o r t a n t d e v e l o p m e n t efforts have gone into any c o m m e r c i a l crystal g r o w t h system to o p t i m i z e the i m p o r t a n t aspects of each process w i t h respect to the p r o d u c t i o n of compositionally u n i f o r m c y l i n d r i c a l crystals w i t h l o w densities of crystalline defects. Progress i n analysis a n d o p t i m i z a t i o n of real solidification systems is e x e m p l i f i e d i n later sections t h r o u g h descriptions of the v e r t i c a l B r i d g m a n - S t o c k b a r g e r and C z o c h r a l s k i g r o w t h methods. T h e meniscus-defined g r o w t h of the C z o c h r a l s k i m e t h o d leads to the issues of process stability a n d control for batchwise g r o w t h ; these points are b r o u g h t


out i n a subsection d e v o t e d to process stability a n d c o n t r o l .

Classification of Melt Crystal Growth Systems T h e m a n y technological innovations i n m e l t crystal g r o w t h of s e m i c o n d u c t o r materials a l l b u i l d on the two basic concepts of confined a n d m e n i s c u s d e f i n e d crystal g r o w t h . E x a m p l e s of these two systems are s h o w n schematically i n F i g u r e 1. T y p i c a l semiconductor materials g r o w n b y these a n d other methods are l i s t e d i n Table I. T h e discussion i n this section focuses o n some of the design variables for each of these methods that affect the q u a l i t y of the p r o d u c t crystal. T h e r e m a i n d e r of the chapter addresses the relationship b e t w e e n these issues a n d the transport processes i n crystal g r o w t h systems. S e v e r a l issues must be addressed. F i r s t , the heat-transfer e n v i r o n m e n t must y i e l d a w e l l - c o n t r o l l e d t e m p e r a t u r e field i n the crystal a n d m e l t near the m e l t - c r y s t a l interface so that the crystallization rate, the shape of the solidification interface, a n d the thermoelastic stresses i n the crystal can be c o n t r o l l e d . L o w dislocation a n d defect densities occur w h e n the t e m p e r a t u r e gradients i n the crystal are low. T h i s p o i n t w i l l b e c o m e an u n d e r l y i n g t h e m e of this chapter a n d has manifestations i n the analysis of m a n y of the transport processes d e s c r i b e d here. S e c o n d , the stoichiometry of the m e l t a n d of i m p u r i t i e s i n t r o d u c e d d u r i n g processing m u s t be c o n t r o l l e d to the l e v e l d e m a n d e d b y application. A l t h o u g h these constraints vary w i t h a p p l i c a t i o n , m o r e c o n t r o l is clearly better i n that the demands o n p u r i t y a n d spatial u n i f o r m i t y of the m a t e r i a l are b e c o m i n g m o r e stringent w i t h the increasing m i n i a t u r i z a t i o n of electronic devices.

Confined Crystal Growth Systems.

I n confined g r o w t h g e o m e -

tries, such as the variations of the directional-solidification m e t h o d (I), the material is l o a d e d into an a m p o u l e , m e l t e d , a n d resolidified b y v a r y i n g the t e m p e r a t u r e field e i t h e r b y translation of the a m p o u l e t h r o u g h the furnace (the B r i d g m a n - S t o c k b a r g e r m e t h o d ; F i g u r e la) or b y t i m e - d e p e n d e n t v a r i ation of the heater p o w e r (the gradient freeze method). A f t e r solidification, the m a t e r i a l is r e m o v e d from the a m p o u l e . C o n f i n e d m e l t g r o w t h systems have b e e n u s e d p r i m a r i l y for laboratory preparation of exotic materials a n d for alloys w i t h h i g h vapor pressures. F o r


38


.AMPOULE

GRAPHITE HEAT SHIELD

Φ

\ \ \ \

c ο Ν Ο Χ

GRAPHITE CRUCIBLE

Φ

HEATER


c •3

Ε D "•5

< Φ C

PEDESTAL

ο Ν ο ο

t

ι

t

FEED ROD MELTING INTERFACE

HEATER MENISCUS

SOLIDIFICATION INTERFACE CRYSTAL

Figure 1. Schematic diagrams of several commonly used systems for melt crystal growth of electronic materials: (a) vertical Bridgman, (h) Czochralski, and (c) small-scale floating-zone systems.


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Table I. Common Semiconductor Materials and Methods for Growing Them from the Melt Application


Material

Technique

Single-crystal silicon High-resistivity, single-crystal silicon Polycrystalline silicon

Integrated circuits Power transistors

Czochralski Floating zone

Photovoltaic devices

Group I I I - V materials (e.g., GaAs, InP)

Optoelectronic devices, integrated circuits

Group I I - V I materials (e.g., CdTe, HgCdTe)

Detectors

Variety of capillary growth methods Horizontal boat, horizontal Bridgman, liquidencapsulated Czochralski, vertical gradient freeze Vertical Bridgman, horizontal Bridgman

such materials, the c o n t r o l of the stoiehiometry i n the m e l t is difficult w i t h o u t c o n f i n e m e n t (Table I). D i r e c t i o n a l solidification a n d the gradient freeze t e c h niques are b e c o m i n g p o p u l a r for the g r o w t h of G a A s , I n P , a n d other materials for w h i c h l o w axial t e m p e r a t u r e gradients are n e e d e d to p r o d u c e crystals w i t h l o w dislocation densities. S o m e of the r e n e w e d interest i n these m e t h ods is a result of i m p r o v e m e n t s i n the c o n t r o l of the t e m p e r a t u r e gradients. C a r e f u l l y d e s i g n e d h e a t - p i p e furnaces have b e e n successfully u s e d i n B r i d g m a n - S t o c k b a r g e r systems (6) b u t have b e e n l i m i t e d to temperatures b e l o w 1100 °C. M u l t i z o n e resistance furnaces have f o u n d a p p l i c a t i o n i n g r a d i e n t freeze systems o p e r a t i n g at h i g h e r temperatures (7, 8) for the g r o w t h of I n P , GaP, and GaAs. T h e h o r i z o n t a l B r i d g m a n t e c h n i q u e , or boat g r o w t h t e c h n i q u e , is a variation i n w h i c h the a m p o u l e is l a i d h o r i z o n t a l l y w i t h respect to gravity a n d the t e m p e r a t u r e gradient i n the m e l t is changed b y v a r y i n g the t e m perature profile i n the s u r r o u n d i n g heater. T h e m e l t is contained i n an o p e n a m p o u l e , or boat, so that the c o m p o s i t i o n of the m e l t can b e e q u i l i b r a t e d w i t h the s u r r o u n d i n g a m b i e n t . F o r G a A s g r o w t h , a source of arsenic is p l a c e d i n the system a n d the vapor pressure of arsenic is c o n t r o l l e d i n d e p e n d e n t l y to m a i n t a i n the stoiehiometry of the m e l t . T h i s t e c h n i q u e is a h y b r i d of the classical gradient-freezing t e c h n i q u e , because of the m e t h o d for v a r y i n g the t e m p e r a t u r e profile, a n d of a m e n i s c u s - d e f i n e d g r o w t h m e t h o d , because o f the presence of the m e l t - a m b i e n t surface.

Meniscus-Defined Crystal Growth Systems. I n most c o n v e n t i o n a l m e n i s c u s - d e f i n e d g r o w t h systems, a seed crystal is d i p p e d into a p o o l of m e l t , a n d the t h e r m a l e n v i r o n m e n t is v a r i e d so that a crystal grows from the seed as it is p u l l e d s l o w l y out o f the p o o l . T w o examples o f m e n i s c u s d e f i n e d g r o w t h are s h o w n i n F i g u r e 1. T h e C z o c h r a l s k i (CZ) m e t h o d ( F i g u r e l b ) a n d the closely related l i q u i d - e n c a p s u l a t e d C z o c h r a l s k i ( L E C ) m e t h o d are batchwise processes i n w h i c h the crystal is p u l l e d from a c r u c i b l e w i t h Hess and Jensen; Microelectronics Processing Advances in Chemistry; American Chemical Society: Washington, DC, 1989.

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MICROELECTRONICS PROCESSING: C H E M I C A L E N G I N E E R I N G ASPECTS

decreasing m e l t v o l u m e . I n the L E C system, an oxide m a t e r i a l (usually B 0 ) is l a y e r e d over the m e l t to p r e v e n t the loss of volatile components i n a high-vapor-pressure system, such as G a A s a n d InP. I n these systems, the gas pressure is usually m a i n t a i n e d significantly above the a m b i e n t pressure.


2

3

T h e details for i m p l e m e n t a t i o n of crystal g r o w t h b y the C Z m e t h o d (9-11), the floating-zone m e t h o d (12, 13), a n d m e n i s c u s - d e f i n e d g r o w t h methods (14, 15) d e v e l o p e d for p r o d u c i n g t h i n sheets have b e e n r e v i e w e d . T h e advantages a n d disadvantages of various confined a n d m e n i s c u s - d e f i n e d g r o w t h methods d e t e r m i n e the types of materials that are p r o d u c e d b y each t e c h n i q u e . F o r e x a m p l e , m e n i s c u s - d e f i n e d methods have the advantage that the c o o l i n g crystal is free to e x p a n d a n d so is less l i k e l y to generate large thermoelastic stresses that l e a d to defect a n d dislocation generation. H o w ever, active c o n t r o l is n e e d e d to p r o d u c e crystals of u n i f o r m cross section, because the shape o f the crystal is constrained o n l y b y capillary action. I n the floating-zone ( F Z ) system, a m o l t e n p o o l is f o r m e d b y a c i r c u m ferential heat source that separates a m e l t i n g p o l y c r y s t a l l i n e feed r o d a n d a solidifying c y l i n d r i c a l crystal. I n small-scale, resistively heated zones, the p o o l o f m e l t is h e l d i n shape b y capillary forces a n d gravity a n d b y h y d r o d y n a m i c stresses caused b y flow i n the m e l t . T h e e x p e r i m e n t a l t e c h n i q u e s related to floating-zone systems are discussed i n the p i o n e e r i n g book b y Pfann (12). F l o a t i n g - z o n e systems w i t h c o n v e n t i o n a l resistance heaters are l i m i t e d o n e a r t h to the g r o w t h of crystals w i t h diameters less than ~ 1 c m , because deformation of the meniscus b y gravity causes loss of w e t t i n g of the crystal b y the s u r r o u n d i n g m e l t (16); floating-zone g r o w t h i n o u t e r space removes this r e s t r i c t i o n a n d is an area of active research (17). T h e shape of a small-scale floating zone is s h o w n i n F i g u r e l c . L a r g e - d i a m e t e r i n d u s t r i a l floating-zone systems have b e e n d e v e l o p e d w i t h radio frequency (rf) i n d u c t i o n h e a t i n g elements shaped so that the i n d u c t i o n c o i l has a smaller d i a m e t e r than the g r o w i n g crystal. T h e s e systems are u s e d for the g r o w t h of h i g h - p u r i t y s e m i c o n d u c t o r materials, such as h i g h resistivity s i l i c o n a n d g e r m a n i u m , a n d are d e s c r i b e d i n the book b y M u h l bauer a n d K e l l e r (13). T h e explanation for the success of these systems has c e n t e r e d o n the lévitation of the m e l t caused b y the M a x w e l l stresses i n d u c e d b y the c o i l ; h o w e v e r , appeal to meniscus stabilization because of this a d d i t i o n a l lévitation force m a y not be necessary because of the shape of the zone. T h e large m e l t i n g interface of the p o l y c r y s t a l l i n e feed r o d is t y p i c a l l y v e r y concave, so that the distance b e t w e e n the two m e l t - s o l i d interfaces is s m a l l , perhaps no m o r e than a c e n t i m e t e r . T h e m e l t i n g interface is c o v e r e d w i t h a t h i n film of m e l t , so that the meniscus s u r r o u n d i n g the b u l k fluid exists o n l y adjacent to the crystal a n d the m o l t e n zone m a y b e no larger i n the system w i t h r f h e a t i n g than i n the system w i t h resistive h e a t i n g , w h i c h is u s e d for s m a l l e r d i a m e t e r crystals. F r o m this p o i n t of v i e w , the r f c o i l is a c l e v e r m e t h o d of locally h e a t i n g a s m a l l m e l t p o o l that is b o u n d b y a meniscus w i t h a shape that balances capillary force w i t h gravity.


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Modeling of Melt Crystal Growth Systems. T w o k e y issues m u s t be addressed before a d e t a i l e d survey of m o d e l i n g of m e l t crystal g r o w t h is p r e s e n t e d . F i r s t , it m u s t be clear w h y a n d o n what levels m o d e l i n g can p l a y a role i n o p t i m i z a t i o n a n d c o n t r o l of systems for the g r o w t h of single crystals from the m e l t . T h e p o t e n t i a l for m o d e l i n g i n the d e v e l o p m e n t of these processes w i l l b e obvious f r o m a survey of the c u r r e n t state of the art i n g r o w t h of different s e m i c o n d u c t o r materials b y similar processes. Variants of the C Z process make a good case study. D u r i n g the last 25 years, the C z o c h r a l s k i process has b e e n o p t i m i z e d for the g r o w t h of s i n g l e crystal silicon to the p o i n t that today dislocation-free crystals w i t h 4 - 6 - i n . (10-15-cm) diameters are g r o w n f r o m 5 - 2 5 - k g batches of m e l t . T h e d i a m e t e r c o n t r o l is 0 . 1 % , a n d i m p u r i t y levels are c o n t r o l l e d to b e less than 5 Χ 1 0 carbon atoms p e r c m a n d 1 X 1 0 oxygen atoms p e r c m . B y contrast, G a A s crystals g r o w n b y the l i q u i d - e n c a p s u l a t e d C z o c h r a l s k i process are l i m i t e d to less than a 3 - i n . (7.6-cm) d i a m e t e r w i t h p o o r d i a m e t e r c o n t r o l a n d have 3 0 0 - 5 0 0 0 dislocations p e r square c e n t i m e t e r , d e p e n d i n g o n the l e v e l of dopants a d d e d to the m e l t . M o r e o v e r , other s e m i c o n d u c t o r materials, such as C d T e , have not b e e n g r o w n successfully i n C z o c h r a l s k i configura tions. T h e difference i n the difficulty of p r o d u c i n g silicon a n d G a A s wafers is reflected i n the difference b y a factor of 100 i n t h e i r cost p e r square centimeter! 1 5

3

1 8

3

T h e role of transport processes i n setting the l e v e l of difficulty for the g r o w t h of each of these materials was d e s c r i b e d qualitatively b y S. Motakeff at the Massachusetts Institute of Technology ( M I T ) i n terms of a plot s i m i l a r to F i g u r e 2, i n w h i c h the estimated t h e r m a l conductivities of several s e m i conductor materials are p l o t t e d against the approximate value of the critical resolved shear stress ( C R S S ) . M a t e r i a l s w i t h l o w conductivities a n d l o w values of C R S S are m o r e difficult to g r o w because of the larger t e m p e r a t u r e gradients (proportional to the conductivity) that w i l l occur d u r i n g processing and the l o w e r resistance of the crystal to the formation of dislocations (pro p o r t i o n a l to C R S S ) . A l t h o u g h this argument is e x t r e m e l y qualitative, the t r e n d is clear. A n a l y s i s of l i n e a r thermoelastic stress i n the crystal w i l l b e d e v e l o p e d further i n a later subsection o n transport processes a n d defect formation. T h e n e w e r materials u s e d i n the m i c r o e l e c t r o n i c i n d u s t r y can be p r o d u c e d o n l y w i t h better quantitative o p t i m i z a t i o n a n d c o n t r o l of the g r o w t h systems. M o d e l i n g w i l l play a substantial role i n the d e v e l o p m e n t of the next generation of crystal g r o w t h systems. Analysis of crystal g r o w t h systems transcends levels of d e t a i l r a n g i n g from t h e r m a l analysis of an e n t i r e crystal g r o w t h system to analysis of the dynamics of defects i n the crystal lattice. T h e s e models are r e p r e s e n t e d schematically i n F i g u r e 3 for a v e r t i c a l B r i d g m a n growth system. S e v e r a l i n t e r m e d i a t e - l e n g t h scales for m o d e l i n g this system have b e e n i n c l u d e d that are not obvious w i t h o u t f u r t h e r discussion of the transport processes. T h e s e scales i n c l u d e an i n t e r m e d i a t e scale for analysis of m e l t flow a n d solute


42

MICROELECTRONICS PROCESSING; C H E M I C A L ENGINEERING ASPECTS

'

1


Si

·

-

InP

• •

• • 0

ο (In-Doped

GaAs

GaAs)

CdTe ι

I

1

2

CRSS x l O , 7

.

I

. 3

dynes/cm

4 2

Figure 2. Thermal conductivity of several common semiconductor materials plotted against the best estimates for the critical resolved shear stress (CRSS) for the crystal. As explained in the text, matenah with low thermal conductivity and low CRSS are hardest to grow.

transport i n the m e l t w i t h b o u n d a r y conditions i m p o s e d b y c o u p l i n g b e t w e e n this l e v e l of d e t a i l a n d an e n t i r e system m o d e l a n d microscale m o d e l s of m e l t - c r y s t a l interface m o r p h o l o g y that account for surface forces a n d c r y s talline anisotropy. Analyses o n these t h r e e l e n g t h scales are based o n c o n t i n u u m conservation equations and are d e s c r i b e d i n this chapter. D e f e c t formation a n d dynamics i n the crystal a n d at the m e l t - c r y s t a l interface are molecular-scale events that are only adequately s i m u l a t e d b y lattice-scale models. A discussion of lattice-scale e q u i l i b r i u m a n d calculations of m o l e c u l a r d y n a m i c s is b e y o n d the scope of this chapter. A n i m p o r t a n t q u e s t i o n for any m o d e l i n g effort, especially one a i m e d at a quantitative d e s c r i p t i o n of c o m p l e x transport processes, is the l e v e l o f accuracy of the m o d e l . A s w i l l b e c o m e e v i d e n t i n the discussion of transport models a n d specific calculations, the values for t h e r m o p h y s i c a l p r o p e r t i e s a n d transport coefficients m u s t be k n o w n , as w e l l as the d e p e n d e n c e of these coefficients o n t e m p e r a t u r e a n d pressure. Information is l a c k i n g for this data base. C r i t i c a l m a t e r i a l properties for s e m i c o n d u c t o r materials are not k n o w n



Detailed Study of Convection Ô Dopant Segregation in the Melt

Melt/Solid Interface Region

system.

• Microscopic Analysis of Melt/Solid Interface Morphology During Solidification Studies of Cells and Dendrites

Microscopic Morphology

Figure 3. Three spatial scales for modeling melt crystal growth, as exemplified by the vertical Bridgman

• Thermal Analysis of Entire Growth Systems

Entire Furnace


44


w i t h any accuracy, a n d little data are available on the sensitivity of the parameters to t e m p e r a t u r e a n d concentration. F o r example, the value of the coefficient of t h e r m a l expansion for m o l t e n s i l i c o n is o n l y available f r o m two isolated measurements, w h i c h disagree b y a factor of three (18). T h e difference i n this n u m b e r is c r i t i c a l i n p r e d i c t i n g w h e t h e r steady or t i m e d e p e n d e n t flow occurs i n simulations of b u o y a n c y - d r i v e n c o n v e c t i o n i n C Z crystal g r o w t h . T h e r m o p h y s i c a l properties for i m p o r t a n t group I I I - V s e m i conductors are almost totally u n k n o w n ; J o r d a n (19) has c o m p i l e d the a v a i l able data a n d has estimated properties for G a A s a n d I n P . T h e value of s u c h estimates is n o w b e i n g q u e s t i o n e d . F o r e x a m p l e , recent measurements o f Downloaded by UNIV OF PITTSBURGH on May 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1989-0221.ch002

the viscosity of m o l t e n G a A s o v e r a range of temperatures show an o r d e r of-magnitude increase as the m e l t i n g p o i n t is approached (20), a n d the a u thors suggest that m o l e c u l a r association i n the m e l t is responsible for the increase i n viscosity.

Basic Transport Processes in Directional Solidification T h e simplest p i c t u r e of a d i r e c t i o n a l solidification process is shown schematically i n F i g u r e 4 a n d corresponds to m e l t translating t h r o u g h a t e m perature field established b y a s u r r o u n d i n g furnace that brackets the m e l t i n g t e m p e r a t u r e . W h e n convective heat transport is u n i m p o r t a n t , the t e m p e r ature i n the m e l t a n d crystal is i d e a l i z e d as a n e a r l y constant axial gradient for the purpose of d e s c r i b i n g the basic transport mechanisms. T h e q u a n t i tative details of the t e m p e r a t u r e field d e p e n d o n the t h e r m o p h y s i c a l p r o p erties of the m e l t , crystal, a n d the s u r r o u n d i n g a m p o u l e a n d o n the g r o w t h rate, t h r o u g h the release of latent heat at the solidification interface. I n a

Figure 4. Profiles of temperature, solute concentration, and density in onedimensional directional solidification of a binary alloy.


2.

BROWN

45


confined g r o w t h system, c o n v e c t i o n i n the m e l t is caused b y the translation of the a m p o u l e a n d b y b u o y a n c y - d r i v e n c o n v e c t i o n d u e to t e m p e r a t u r e a n d concentration gradients. T h e details of the convection p a t t e r n w i l l d e p e n d on heat transfer i n the system a n d o n the orientation of the m e l t a n d crystal w i t h respect to gravity. H e a t transfer i n m e l t crystal g r o w t h systems is b y c o n d u c t i o n i n a l l phases; convection i n the m e l t ; a n d convective, c o n d u c t i v e , a n d radiative exchanges b e t w e e n the various parts of the g r o w t h system. T h e goals of heat-transfer systems d e s i g n e d for any crystal g r o w t h configuration are to


establish a nearly constant temperature gradient laterally along the m e l t - c r y s t a l interface a n d to control the c o o l i n g rate of the crystal. T h e details of i m p l e m e n t i n g these conditions can be e x t r e m e l y c o m p l e x because of the c o m p l i c a t e d geometries i n systems l i k e the C Z a n d L E C techniques a n d because of the influence of radiative heat exchange. Progress i n u n d e r standing heat transport is discussed for specific systems i n the later sections on vertical B r i d g m a n - S t o c k b a r g e r a n d C z o c h r a l s k i crystal g r o w t h . T h e d i s cussion i n this section focuses o n species transport for solutes that form i d e a l solutions i n the m e l t a n d single-phase crystalline solids. T h e shape of the phase diagram b e t w e e n m e l t a n d solid is d e s c r i b e d b y the d e p e n d e n c e of the l i q u i d u s T (c) a n d solidus Tj(c) temperatures o n the solute concentration s

c. T h e l i q u i d u s a n d solidus curves are taken to be straight lines w i t h slopes m a n d mlfc, respectively, i n w h i c h k is the e q u i l i b r i u m p a r t i t i o n coefficient. T h e n the composition of m e l t ( c j a n d solid (c ) i n e q u i l i b r i u m are related s

by c

m

= cjk

(1)

(The s y m b o l ~ is u s e d throughout this chapter to denote d i m e n s i o n a l v a r i ables; its absence corresponds to a dimensionless formulation.) T h e phase diagram c o r r e s p o n d i n g to constant values of m a n d k is s h o w n i n F i g u r e 5. T a k i n g the segregation coefficient to be i n d e p e n d e n t of the local g r o w t h rate of the crystal is to assume that the solute concentration is i n local e q u i l i b r i u m at the interface, that is, that interface kinetics plays no role i n setting the c o m p o s i t i o n of the crystal. A l t h o u g h this assumption is p r o b a b l y accurate for the slow g r o w t h rates ( 1 - 1 0 μπι/s) o n a microscopically r o u g h crystal surface t y p i c a l for the g r o w t h of m a n y semiconductors, it is u n d o u b t e d l y p o o r at h i g h e r solidification velocities or w h e n the crystal grows along a facet so that the g r o w t h rate is g o v e r n e d b y the m o v e m e n t of ledges across the surface.

One-Dimensional, Diffusion-Controlled Crystal Growth.

Ne

glecting b u l k c o n v e c t i o n leads to an i d e a l i z e d p i c t u r e of diffusion-con t r o l l e d solute transport of a d i l u t e b i n a r y alloy w i t h the solute c o m p o s i t i o n c far f r o m an almost flat m e l t - c r y s t a l interface located at ζ = 0(1, 21). 0



46


SOLUTE CONCENTRATION C Figure 5. Idealized phase diagram for a binary alloy. The liquidus slope m and equilibrium segregation coefficient k are defined in the text. W h e n v i e w e d from a reference frame that is stationary w i t h respect to the solidification interface, the m e l t moves uniaxially t o w a r d the interface a n d the crystal moves away. T h e n solute transport i n the m e l t is g o v e r n e d b y the o n e - d i m e n s i o n a l balance equation dC

dc

2

D —- + V — = 0 dz dz

(2)

g

2

g

i n w h i c h D is the solute diffusivity i n the m e l t , c is the d i m e n s i o n a l c o n centration, ζ is the d i m e n s i o n a l axial coordinate, a n d V is the g r o w t h rate m e a s u r e d i n terms of the m e l t velocity. E q u a t i o n 2 is solved w i t h the f o l l o w i n g c o n d i t i o n of solute conservation at the m e l t - c r y s t a l interface g

•D — dz

(3)

i n w h i c h k is the e q u i l i b r i u m p a r t i t i o n coefficient defined b y e q u a t i o n 1. Because the g r o w t h rate V is taken to be i n d e p e n d e n t of the c o m p o s i t i o n at the interface, the results are l i m i t e d i n practice to d i l u t e alloy systems. g


2.

BROWN


47

T h e solute field that decays t o c far from the interface (z —> °°) is g i v e n b y : 0

c(z) = c „ [ l + ^ e x p ( ^ ) ]

(4)

E q u a t i o n 4 shows the existence of an exponential diffusion l a y e r adjacent to the interface so that the variation i n concentration caused b y solute rejection (k < 1) or i n c o r p o r a t i o n (k > 1) at the interface o n l y persists for an e-folding distance i n the o r d e r o f D/V

r

Analysis o f transients i n d i r e c t i o n a l solidification is c o m p l i c a t e d b y the Downloaded by UNIV OF PITTSBURGH on May 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1989-0221.ch002

c o u p l i n g b e t w e e n heat transfer, the shape of the m e l t - c r y s t a l interface, a n d solute transport. S m i t h et a l . (22) analyzed the i d e a l i z e d situation o f o n e d i m e n s i o n a l , d i f i i s i o n - c o n t r o l l e d transport i n w h i c h the diffusion o f heat is m u c h m o r e r a p i d than that o f solute, so that changes i n the heat-transfer e n v i r o n m e n t o r the a m p o u l e p u l l rate appear as a step change i n the m a c roscopic solidification rate at t h e interface. U n d e r these conditions, t h e transient solute profile is g o v e r n e d b y equations 2 a n d 3, w i t h an a c c u m u lation t e r m (del dt) a p p e a r i n g o n the right side of equation 2. S o l u t i o n of this equation b y separation of variables and the b o u n d a r y c o n d i t i o n (equation 4) gives an e-folding t i m e of DlkV

2

for the solute profile i n response to a step

change i n the p u l l rate. W h e n the alloy is n o n d i l u t e , the m e l t i n g temperature d e p e n d s o n the solute concentration adjacent to the interface t h r o u g h the shape o f the l i q uidus c u r v e . T h e n the transport o f heat a n d solute and the location o f the solidification interface do not d e c o u p l e (23); D e r b y a n d B r o w n (24) p r e s e n t e d a n u m e r i c a l a l g o r i t h m for the analysis o f this p r o b l e m . Convection in Melt Growth. C o n v e c t i o n i n the m e l t is pervasive i n a l l t e r r e s t r i a l m e l t g r o w t h systems. Sources for flows i n c l u d e b u o y a n c y d r i v e n c o n v e c t i o n caused b y the solute a n d t e m p e r a t u r e d e p e n d e n c e of the density; surface tension gradients along m e l t - f l u i d m e n i s c i ; forced c o n v e c tion i n t r o d u c e d b y the m o t i o n o f s o l i d surfaces, such as c r u c i b l e a n d crystal rotation i n the C Z a n d F Z systems; a n d the m o t i o n o f the m e l t i n d u c e d b y the solidification o f m a t e r i a l . T h e s e flows are i m p o r t a n t causes o f the c o n vection o f heat a n d species a n d can have a d o m i n a n t influence o n the t e m perature field i n the system a n d o n solute incorporation into the crystal. M o r e o v e r , flow transitions from steady l a m i n a r , to t i m e - p e r i o d i c , chaotic, and t u r b u l e n t motions cause t e m p o r a l nonuniformities at the g r o w t h i n t e r face. T h e s e fluctuations i n t e m p e r a t u r e a n d concentration can cause t h e m e l t - c r y s t a l interface to m e l t a n d resolidify a n d can l e a d to solute striations (25) a n d to the formation o f microdefects, w h i c h w i l l b e d e s c r i b e d later. Equations of Motion and Driving Forces for Convection. The presentation i n this section is l i m i t e d to the case w h e n the density variations i n

American Chemical Society, Library

1155 15th St, N.W. Processing Hess and Jensen; Microelectronics Washington. D.C. 20038 Advances in Chemistry; American Chemical Society: Washington, DC, 1989.

48


the melt caused by temperature and concentration gradients can be modeled by the Boussinesq approximation (26) in terms of the thermal and solutal expansion coefficients, β = -(l/p)(dp/dT) and β = (l/p)(dp/dc) , re spectively, in which ρ is pressure and Τ is temperature. Positive values of Pt correspond to a material with a density that decreases with increasing temperature; a positive value of β is appropriate for a solute with density that increases with increasing concentration. The governing equations and boundary conditions for modeling melt crystal growth are described for the CZ growth geometry shown in Figure 6. The equations of motion, continuity, and transport of heat and of a dilute solute are as follows: ί

pc

ε

pT


c

Po

(S)

+ ν · Vv = - V p +

W μ

(5) ν ν + p g[l - β

ζ ο

2

w

ο

Η

ο > r

m

χ

Ο

Ο

m

Ω

?

2 Β

1

M Ω

r

M

ο

ο

to


—•

intense

TIME-PERIODIC CONVECTION

Dimensionless Temperature Gradient (Ra)

weak

CELLULAR LAMINAR CONVECTION WEAKLY CHAOTIC

LARGE SCALE : INTERACTING ' VORTICES

j j

e

*

FINE SCALE TURBULENCE

— T U R B U L E N T

Figure 10. Diagrammatic representation of the regimes of solute transport in melt crystal growth in terms of the effects on radial segregation Ac and the effective segregation coefficient k f.

UNIDIRECTIONAL CONVECTION CAUSED B Y CRYSTAL GROWTH



64

field v(r, z)], w h i c h is g i v e n i n e q u a t i o n 17 i n the dimensionless form of e q u a t i o n 8:

(17)

I n e q u a t i o n 17, P e ( P e = V L / D ) is the P e c l e t n u m b e r for mass transfer, s

s

0

as d e f i n e d i n T a b l e I I , a n d scales the i m p o r t a n c e of convective solute trans port b y the v e l o c i t y i n the b u l k (scaled b y V ) to diffusion. T h e dimensionless 0


solute balance at the m e l t - c r y s t a l interface is the generalization of e q u a t i o n 3 for a steadily s o l i d i f y i n g interface

dD

η · Vc = P e c ( l g

v

k)n · e at 8 D g

(18)

b I

I n e q u a t i o n 18, η is the u n i t n o r m a l to the m e l t - c r y s t a l interface, e is the g

u n i t vector i n the d i r e c t i o n of crystal g r o w t h , a n d P e ( P e = V L/D) g

g

g

is the

dimensionless crystal g r o w t h rate or, a l t e r n a t i v e l y , the P e c l e t n u m b e r based o n the solidification velocity. T h e appropriate b o u n d a r y conditions along the o t h e r surfaces d e p e n d o n the geometry of the system a n d o n the c h e m i c a l interactions of these boundaries w i t h the m e l t . F o r m a n y melts, a m p o u l e a n d c r u c i b l e materials are i n e r t , a n d these surfaces c o r r e s p o n d to no-flux boundaries for solutes. C h e m i c a l interactions are i m p o r t a n t i n some systems. F o r example, the q u a r t z c r u c i b l e s u s e d i n silicon g r o w t h dissolve w h e n contacted w i t h the m e l t , a n d m o d e l i n g of oxygen i n c o r p o r a t i o n i n the crystal m u s t account for this flux, i n a d d i t i o n to the losses of oxygen to the a m b i e n t that o c c u r i n m e n i s c u s - d e f i n e d g r o w t h (69, 70). T h e c o n c e p t of a stagnant-film thickness, as p r o p o s e d

Stagnant Films.

b y N e r n s t (71), is the most w i d e l y u s e d characterization of the role of c o n v e c t i o n i n solute segregation d u r i n g crystal g r o w t h . T h i s a p p l i c a t i o n was r e v i e w e d b y W i l c o x (72). C o n v e c t i v e m i x i n g is assumed to be totally effective outside of a t h i n l a y e r adjacent to the m e l t - c r y s t a l interface i n w h i c h species transport is b y diffusion o n l y a n d the m e l t m o t i o n is caused b y solidification ( F i g u r e 11). I f the c o m p o s i t i o n of solute i n the b u l k (ζ > δ) is c , the steady0

state c o m p o s i t i o n profile i n the l a y e r is d e r i v e d b y solving equations 2 a n d 3 w i t h the f o l l o w i n g b o u n d a r y c o n d i t i o n : (19) T h i s b o u n d a r y c o n d i t i o n y i e l d s e q u a t i o n 20, as o r i g i n a l l y g i v e n b y B u r t o n et a l . (73): c(z) c

0

k 4- (1 -

k) e x p ( - V z / D )

~ k + (1 -

fc)exp(-V 8/D)

g

g


(20)

2.

BROWN

65

Theory of Transport Processes MELT

CRYSTAL STAGNANT FILM C ' j

C /k s

C

WELL-MIXED BULK

s

1

—

*. ζ +


Z=0 Figure 11. Representation of a stagnant film in solute transport in melt growth. The variables are defined in the text. T h e effective segregation coefficient k

eS

is d e f i n e d i n terms o f the stag

n a n t - f i l m thickness as

k + (1 -

(21)

fc)exp(-V 6/D) g

E q u a t i o n 2 1 , t h e most often u s e d r e l a t i o n for fitting axial segregation data from e x p e r i m e n t , is a m a z i n g l y successful for this purpose. F o r a

finite-length

a m p o u l e , equation 21 is r e w r i t t e n i n terms o f the fraction (f) o f the sample solidified to give t h e n o r m a l freezing expression for t h e c o m p o s i t i o n o f the crystal (c = kc ) g r o w n f r o m a m e l t w i t h i n i t i a l c o m p o s i t i o n c : s

0

m

c

= M

i -

(22)

f) ' ~ (k s

i]

0

F i g u r e 12 shows a sample profile from t h e g r o w t h o f g a l l i u m - d o p e d ger m a n i u m b y W a n g (6). T h e profile fits e q u a t i o n 2 1 , w i t h a n e q u i l i b r i u m segregation coefficient k o f 0.087 a n d k o f 0 . 0 9 - 0 . 1 1 . eS

A l t h o u g h t h e n o t i o n o f a stagnant film is a useful way o f t h i n k i n g about the role o f convection i n axial solute segregation, it has w e l l - k n o w n d e f i ciencies (72) that p r e v e n t it from b e i n g p r e d i c t i v e . T h e stagnant-film t h e o r y assumes that m i x i n g is perfect outside o f the stagnant film a n d that b u l k convection does not modify t h e v e l o c i t y field inside t h e layer. T h e s e as sumptions p r e v e n t the p r e d i c t i o n o f changes i n axial segregation (fe ) caused b y changes i n t h e i n t e n s i t y o f the flow w i t h o u t an a d d i t i o n a l e m p i r i c a l cor relation for δ as a f u n c t i o n o f convection from e i t h e r e x p e r i m e n t a l (73-75) or c o m p u t a t i o n a l (76, 77) data for t h e effective segregation coefficient a n d equation 21. A l s o , because the v e l o c i t y field i n the stagnant film is a s s u m e d to c o r r e s p o n d o n l y to t h e g r o w t h rate, t h e stagnant-film m o d e l gives n o eff




66

p r e d i c t i o n o f t h e concentration variations a l o n g the crystal surface that are caused b y lateral nonuniforrnities i n the

flow. A closed-form solution of the

Analysis of Solute Boundary Layer.

species balance (equation 7) w i t h the interfacial balance (equation 8) a n d the c o n d i t i o n that c

c

0

as £ —> oo is o n l y feasible w h e n e i t h e r c o n v e c t i o n

contributes o n l y w e a k l y to transport, as was the case i n the analysis of H a r r i o t t a n d B r o w n (66), o r the variation o f the concentration from u n i t y is confined to a b o u n d a r y l a y e r adjacent to the g r o w i n g crystal. B u r t o n et a l . (74) analyzed the specific case of solute segregation near a rotating crystal, as occurs i n C Z g r o w t h . T h e velocity field near the crystal was taken to be the s u m o f a u n i f o r m crystal g r o w t h rate V a n d the self-similar a x i s y m m e t r i c g

velocity field d u e to rotation of an infinitely large crystal surface, as d e s c r i b e d b y the asymptotic expansion of C o c h r a n (78). Because the axial v e l o c i t y is i n d e p e n d e n t o f radial distance f r o m the center of the c r y s t a l , the concentration field varies o n l y axially a n d is g i v e n b y the solution of the solute balance equation w i t h the approximate v e l o c i t y field of the form β, =

-V

g

-

(23)

V (zW) 0

i n w h i c h L is a characteristic l e n g t h scale for the b u l k flow ( L = ν / Ω , i n w h i c h Ω is the angular v e l o c i t y of the disk i n the similarity solution of C o c h r a n [78]) a n d V = ΩΚ, i n w h i c h R is the radius of the crystal. T h e closed-form solution of this p r o b l e m is w r i t t e n i n terms of exponential integrals, as 2

0


2.

BROWN

67


d e s c r i b e d b y B u r t o n et al. (74). T h e effective segregation coefficient is d e f i n e d b y an equation analogous to e q u a t i o n 21:

fceff

* + (l-Jfc)exp(-A)

=

( 2 4 )

T h e constant Δ i n equation 24 is defined as


Δ =

- to ( l - £

(25)

exp [ - (ξ + Η ))άξ) 3

i n w h i c h λ is the ratio o f the c o m p o n e n t of the velocity field d u e to the b u l k flow to that d u e to solidification, λ SE ( V / 3 V ) , a n d ξ is the variable of 0

g

integration. W i l s o n (79, 80) p o i n t e d out that Δ is not the dimensionless thickness of the diffusion b o u n d a r y layer scaled w i t h D / V , as o r i g i n a l l y suggested g

by

B u r t o n et a l . (74), except i n the l i m i t at w h i c h the velocity field i n the l a y e r is d o m i n a t e d b y the b u l k flow, that is, λ > >

1. I n this case, the analysis

reduces to the one first p r e s e n t e d b y L e v i c h (81), a n d the i n t e g r a l i n e q u a t i o n 25 is a p p r o x i m a t e d as follows:

Δ ~ Γ(1/3)

X »

1

(26)

I n e q u a t i o n 26, Γ( · ) is the g a m m a function. W h e n the ratio λ is r e w r i t t e n i n terms of the P e c l e t n u m b e r s ( P e = V L/D s

0

and P e

g

= V L/D) g

as λ

=

( P e / 3 P e ) , the scaling i n e q u a t i o n 26 is r e c o g n i z e d as the scaling for a s

g

b o u n d a r y layer adjacent to a n o - s l i p surface. I n the l i m i t at w h i c h the s o l i d ification rate dominates (λ <
0 for a n interface that is convex w i t h respect to the crystal. E q u a t i o n 34 i m p l i e s that the m e l t i n g p o i n t is increased b y c a p i l l a r i t y for short-wavelength disturbances (H » 1). T h i s increase i n m e l t i n g t e m p e r a t u r e compensates for the constitutional s u p e r c o o l i n g m e c h a n i s m a n d i m p l i e s a short-wavelength cutoff to the i n s t a bility. f

M u l l i n s a n d S e k e r k a (88, 89) analyzed the stability of a planar s o l i d i f i cation interface to s m a l l disturbances b y a rigorous solution of the equations for species a n d heat transport i n m e l t a n d crystal a n d the constraint of e q u i l i b r i u m t h e r m o d y n a m i c s at the interface. F o r t w o - d i m e n s i o n a l s o l i d i fication samples i n a constant-temperature gradient, the results p r e d i c t the onset of a sinusoidal interfacial instability w i t h a w a v e l e n g t h (λ) c o r r e s p o n d i n g to the disturbance that is j u s t m a r g i n a l l y stable as e i t h e r G is decreased


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or V is increased. T h e f o l l o w i n g discussion focuses o n t h e g r o w t h rate as the c o n t r o l p a r a m e t e r for transitions i n t h e interface m o r p h o l o g y . T h e curves o f n e u t r a l stability [V = V (X)] c o r r e s p o n d to disturbances that b e c o m e unstable for any f u r t h e r increase i n the g r o w t h rate. F i g u r e 13 shows a sample set o f n e u t r a l stability curves for the s u c c i n o n i t r i l e - a c e t o n e alloy for a range o f concentrations o f acetone. T h e c o m b i n a t i o n of succinon i t r i l e a n d acetone is a w e l l - c h a r a c t e r i z e d organic alloy u s e d extensively i n e x p e r i m e n t s that simulate m e t a l solidification (90). S e v e r a l features o f these curves are u n i v e r s a l . F i r s t , t h e curves are closed; a disturbance w i t h a particular w a v e l e n g t h λ becomes unstable at a critical g r o w t h rate, V ( \ ) , b u t is r e s t a b i l i z e d at the h i g h e r g r o w t h rate c o r r e s p o n d i n g to the top section of the c u r v e . S e c o n d , decreasing t h e concentration o f the solute decreases the size o f the r e g i o n o f unstable wavelengths, as expected from the c r i t e r i o n of constitutional s u p e r c o o l i n g . g

g

C


c

A l t h o u g h t h e balance equations are l i n e a r , i n t h e absence o f b u l k c o n v e c t i o n , t h e u n k n o w n shape o f the m e l t - c r y s t a l interface a n d the d e p e n d e n c e o f the m e l t i n g t e m p e r a t u r e o n the energy a n d c u r v a t u r e o f the surface make the m o d e l for m i c r o s c o p i c interface shape r i c h i n n o n l i n e a r structure. F o r a particular v a l u e of the spatial w a v e l e n g t h , a family o f cellular interfaces evolves f r o m t h e critical g r o w t h rate V (X) w h e n the v e l o c i t y is increased. C

T h e e v o l u t i o n o f these c e l l u l a r forms to d e e p cells a n d d e n d r i t e s w i t h increasing V is g o v e r n e d b y the full n o n l i n e a r equations. F i g u r e 14 shows g

G = 67 °K/cm C = 0.0005 Mole% Ε ο

= 0.005 Mole%-

>° ICT

0.05 Mole%

>Ο Ο

ω 10" > -0.1 10-4 ï-6

10 -4

10 -2

10^

WAVELENGTH λ (cm) Figure 13. Neutral stability curves computed by linear analysis for the succinonitrile-acetone system as a function of acetone concentration for fixed temperature gradient of G = 67°/ cm. Hess and Jensen; Microelectronics Processing Advances in Chemistry; American Chemical Society: Washington, DC, 1989.



72

Figure 14. Photographs of cellular and dendritic structures in a thin-film solidification experiment using succinonitrile-acetone reported by Trivedi and Somboonsuk (91).


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pictures of the c e l l - t o - d e n d r i t e transition w i t h increasing g r o w t h rate i n a t h i n solidification sample for s u c c i n o n i t r i l e - a c e t o n e (91). F i g u r e 14 shows shallow cells ( F i g u r e 14a) that d e v e l o p d e e p grooves ( F i g u r e 14b) a n d , finally, d e n d r i t i c side arms ( F i g u r e 14c) as the g r o w t h rate is increased. E v e n i n the t h i n sample, the d e n d r i t i c structures are not two d i m e n s i o n a l , as i n d i cated b y the side arms that p r o t r u d e out of the plane of the photograph. P r e d i c t i o n of these n o n l i n e a r transitions, especially the spatial w a v e l e n g t h of the m i c r o s t r u c t u r e s , is an area of active research. F u l l n u m e r i c a l solutions of n o n l i n e a r models for microscopic solidification (92-94) have s h o w n the d e v e l o p m e n t of d e e p cells for ranges of spatial wavelengths w i t h i n the unstable r e g i o n of the n e u t r a l stability c u r v e a n d have d e m o n s t r a t e d mechanisms for s p l i t t i n g o n i n d i v i d u a l cells (cell birth) a n d for m e r g i n g of c e l l pairs (cell death). D y n a m i c a l calculations indicate that there m a y b e no m e c h a n i s m for w a v e l e n g t h selection i n collections of shallow cells (95). D e e p cells, l i k e those s h o w n i n F i g u r e 14b, have b e e n c o m p u t e d as the g r o w t h rate is increased (93). T h e s e cells exist for ranges of w a v e l e n g t h a n d have r o u n d e d tips c o n n e c t e d to slender side walls l e a d i n g to a c e l l root w i t h a smooth b o t t o m . I n c l u s i o n of b u l k c o n v e c t i o n i n the models d e s c r i b i n g i n terface m i c r o s t r u c t u r e severely complicates the analysis. T h e c u r r e n t state of research i n this area is r e v i e w e d b y G l i c k s m a n et al. (96). S e v e r a l analyses demonstrate that n o v e l interactions b e t w e e n the local flow a n d the interface m o r p h o l o g y are possible w h e n the two p h e n o m e n a are c o u p l e d (97).

Connection between Transport Processes and Defect Formation in Crystals. A n a l y s i s of the c o n n e c t i o n b e t w e e n macroscopic transport processes i n the m e l t and c r y s t a l , such as the t e m p e r a t u r e field i n b o t h phases a n d solute transport near the m e l t - c r y s t a l interface, a n d defects i n the crystal r e q u i r e s an u n d e r s t a n d i n g of the mechanisms for the generation (in the b u l k crystal a n d at the interface), m o t i o n , c o m b i n a t i o n , m u l t i p l i c a t i o n , and a n n i h i l a t i o n of crystallographic defects a n d dislocations. T h e d e s c r i p t i o n of this c o u p l i n g for crystal g r o w t h has b e e n confined to the s i m p l e p i c t u r e from c o n t i n u u m t h e r m o e l a s t i c i t y for a perfect crystal i n w h i c h t e m p e r a t u r e gradients i n the crystal d u r i n g processing create stress i n the solid that causes dislocations w h e n its m a g n i t u d e exceeds the critical r e s o l v e d shear stress ( C R S S ) , w h i c h is evaluated i n the slip directions for the crystal. B i l l i g (98) i n t r o d u c e d this n o t i o n to e x p l a i n the generation of dislocations i n g e r m a n i u m crystals g r o w n b y the C z o c h r a l s k i m e t h o d . Several authors have u s e d this i d e a to d e v e l o p e d qualitative estimates for the bounds o n t e m p e r a t u r e gradients i n C Z g r o w t h . T s i v i n s k y (99) estimated the m a x i m u m d i a m e t e r crystal that can be g r o w n before the C R S S is reached. B r i c e (2) f o u n d a r e l a t i o n s h i p b e t w e e n the m a g n i t u d e of the shear stress i n the solid a n d crack formation. J o r d a n et a l . (100) r e f i n e d the thermoelastic stress calculation for the analysis o f the spatial d i s t r i b u t i o n o f dislocations i n G a A s g r o w n w i t h the L E C m e t h o d . T h e i r analysis was based o n a t w o - d i m e n s i o n a l m o d e l for the


74



t e m p e r a t u r e field throughout the g r o w n crystal that i n c l u d e d a l l of the c r i t i c a l parameters: p u l l rate, heat transfer b e t w e e n t h e crystal a n d t h e a m b i e n t , a n d t h e r m o p h y s i e a l properties. I n this a n d i n other related works (101-103), the crystal was assumed to b e a n isotropic m a t e r i a l w i t h a constant coefficient of expansion, so that t h e strains i n t h e m a t e r i a l are o n l y functions o f t h e local axial a n d radial t e m p e r a t u r e gradients. U n d e r these conditions, n o n constant displacements o f the crystal, w h i c h l e a d to nonisotropic stresses i n the s o l i d , are caused b y nonconstant t e m p e r a t u r e gradients. Jordan calculated these strains i n closed form b y a s s u m i n g that o n l y deformations i n t h e a z i m u t h a l plane o f the crystal are i m p o r t a n t . D u s e a u x (102) a n d K o b a y a s h i a n d I w a k i (103) c o n s i d e r e d b o t h radial a n d axial d i s placements b y n u m e r i c a l l y c a l c u l a t i n g the strains. F i g u r e 15 shows a d i r e c t c o m p a r i s o n b e t w e e n t h e p a t t e r n for t h e m a g n i t u d e o f the shear stress p r e d i c t e d b y t h e analysis o f J o r d a n et a l . (100) a n d t h e p a t t e r n of dislocations i n a G a A s crystal g r o w n b y t h e L E C t e c h n i q u e . A l t h o u g h a quantitative c o m p a r i s o n is n o t m e a n i n g f u l , t h e similar patterns for regions o f h i g h stress a n d h i g h dislocation density near t h e edge o f the crystal strongly suggest that thermoelastic stress plays a n i m p o r t a n t role i n dislocation formation. T h e i m p o r t a n c e o f the t h e r m a l c o n d u c t i v i t y o f the crystal a n d the C R S S i n d e t e r m i n i n g t h e degree o f difficulty o f g r o w i n g a specific m a t e r i a l from the m e l t is u n d e r s t o o d i n terms o f the r e l a t i o n s h i p b e t w e e n these parameters a n d t h e formation o f dislocations i n t h e crystal because o f excess stress. C l e a r l y , materials w i t h l o w e r values o f the C R S S must b e g r o w n i n systems w i t h l o w e r t e m p e r a t u r e gradients to p r e v e n t crystallographic slip. L o w v a l ues o f the c o n d u c t i v i t y make this difficult to achieve. T h e isoelectronic d o p i n g o f G a A s crystals w i t h i n d i u m i n t r o d u c e d b y M i l v i d s k i i et a l . (104) (see also references 105 a n d 106) was based o n t h e i d e a that i n d i u m causes solid-solution h a r d e n i n g o f t h e G a A s lattice a n d raises t h e C R S S . C r y s t a l s w i t h l o w dislocation densities have b e e n g r o w n b y m a n y groups u s i n g this m e t h o d . R e c e n t measurements o f the value o f the C R S S as a f u n c t i o n of t e m p e r a t u r e a n d i n d i u m concentration (107, 108) r e v e a l a twofold increase i n C R S S i n t h e t e m p e r a t u r e range just b e l o w t h e m e l t i n g point. T h i s difference is adequate to e l i m i n a t e profuse dislocation m u l t i p l i c a t i o n i n m o d e r a t e - d i a m e t e r crystals, b u t dislocation formation i n crystals g r o w n i n a low-stress e n v i r o n m e n t r e m a i n u n e x p l a i n e d . A l t h o u g h a p p e a l i n g from an e n g i n e e r i n g perspective, the analyses based o n l i n e a r thermoelasticity d o not address the action of defects a n d dislocations created b y m i c r o s c o p i c y i e l d p h e n o m e n a b e l o w the C R S S a n d o f those that are i n c o r p o r a t e d i n t h e crystal at t h e solidification front. I n t h e p r e v i o u s works c i t e d (104-108), t h e authors assume that n o defects exist at t h e m e l t - c r y s t a l interface a n d that t h e stresses o n this surface are zero. C o n stitutive equations i n c o r p o r a t i n g models for plastic deformation i n the crystal d u e to dislocation m o t i o n have b e e n p r o p o s e d b y several authors (109-111) a n d have b e e n used to describe dislocation m o t i o n i n t h e i n i t i a l stages o f



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TOP. I t e m LONG 3600 MC I U 600 tec I U

TOP. 3 6 cm LONG 7200 MC III 1700 M C > 1) for the m e l t . T h e results showed the transition from diffusion-controlled g r o w t h to weak convection, w i t h the associated m a x i m u m i n radial segregation, a n d finally the formation of a b u l k - c o n v e c t i o n c o n t r o l l e d b o u n d a r y layer along the solidification interface, as d e s c r i b e d qualitatively b y F i g u r e 10. T h e i m p r e s s i o n of convection adjacent to the m e l t - c r y s t a l interface o n radial segregation A c was accentuated b y the changes caused b y the formation of the secondary flow c e l l adjacent to it w i t h increasing t h e r m a l R a y l e i g h n u m b e r R a .


g

t

A quantitative p r e d i c t i o n of convection a n d solute segregation i n any m e l t g r o w t h system r e q u i r e s careful c o n t r o l of the t h e r m a l b o u n d a r y c o n ditions that define the d r i v i n g forces for the flow. A d o r n a t o a n d B r o w n (76, 77) r e f i n e d the finite-element-Newton m e t h o d (77) a n d e x t e n d e d it to i n c l u d e calculation of heat transfer i n the a m p o u l e (76) a n d t h e r m o s o l u t a l convection d r i v e n b y b o t h t e m p e r a t u r e a n d concentration gradients. Results are r e p o r t e d (76, 77) for the g r o w t h of g a l l i u m - d o p e d g e r m a n i u m i n the B r i d g m a n - S t o c k b a r g e r furnace of W a n g (6) a n d the g r o w t h of s i l i c o n g e r m a n i u m alloys i n the constant-gradient furnace designed b y R o u s a u d et al. (129). Samples of the isotherms a n d convection patterns c o m p u t e d for the B r i d g m a n - S t o c k b a r g e r system o f W a n g are s h o w n i n F i g u r e 16. T h e results are s h o w n w i t h the frame of reference that the m e l t - c r y s t a l interface is stationary a n d that m e l t a n d crystal are translated d o w n w a r d at the growth rate V . T h e t e m p e r a t u r e field shows two distinct regions for the radial t e m p e r a t u r e gradients. N e a r the interface, the t e m p e r a t u r e is highest at the center of the a m p o u l e because of the h i g h e r t h e r m a l c o n d u c t i v i t y of the m e l t a n d the i n t e r m e d i a t e c o n d u c t i v i t y of the b o r o n n i t r i d e a m p o u l e ; the sign of the radial t e m p e r a t u r e gradient is r e v e r s e d near the j u n c t i o n of the hot a n d adiabatic zones for the reason just g i v e n . O n l y the uniaxial convection caused b y solidification is present for R a < 1 Χ 1 0 ; h o w e v e r , two toroidal flow cells d r i v e n i n opposite directions b y the t e m p e r a t u r e field are seen for h i g h e r values of R a . T h e cells intensify w i t h increasing R a to the p o i n t that the streamlines c o r r e s p o n d i n g to the g r o w t h velocity ( V = 4 μπι/s i n this case) are confined to t h i n layers along the p e r i p h e r y of the cells. K

t

5

t

t

g

Solute concentration fields are s h o w n i n F i g u r e 17 for the flows i n F i g u r e 16. T h e diffusion-controlled profile for u n i d i r e c t i o n a l solidification is u n -




80

Figure 17. Solute concentration fields for the growth of GaGe in the system described in Figure 16. The growth rate (V ) is 4 \imls. G


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mistakable w h e n R a

t

=

1 Χ 10

5

( F i g u r e 17, left). R a d i a l segregation is

present e v e n w i t h o u t extensive b u l k c o n v e c t i o n (the isoconcentration curves intersect w i t h the solidification interface) because of the c u r v a t u r e o f the m e l t - c r y s t a l interface. A t h i g h e r R a y l e i g h n u m b e r s , the c e l l u l a r flows cause m i x i n g a n d increase the radial segregation ( F i g u r e 17, m i d d l e a n d right). A t Ra

t

= 1 Χ 1 0 , cores of constant solute c o m p o s i t i o n f o r m w i t h i n the 7

flow

cells separated b y the b o u n d a r y a n d i n t e r n a l layers. C a l c u l a t i o n s of the solute field for c o n v e c t i o n levels c o r r e s p o n d i n g to R a values u p to 2 Χ 1 0 , w h i c h 8

t

are values appropriate for Wang's furnace, are i m p o s s i b l e w i t h the

finite-


e l e m e n t m e s h u s e d i n this analysis because of the u n d e r r e s o l u t i o n of these layers. T h e radial segregation p r e d i c t e d b y these calculations is s h o w n i n F i g u r e 18 for three g r o w t h rates a n d the range of R a for w h i c h computations w e r e t

possible. T h e transitions d e s c r i b e d b y F i g u r e 10 are apparent. T h e meas u r e m e n t s o f W a n g (6) for radial segregation of g a l l i u m are s h o w n also i n this figure,

a n d c o m p a r i s o n of the radial segregations demonstrates that dopant

r e d i s t r i b u t i o n i n this system is c o n t r o l l e d b y intense l a m i n a r c o n v e c t i o n . A d o r n a t o a n d B r o w n (76) a t t e m p t e d to extrapolate the c o m p u t a t i o n a l results for radial segregation to the R a y l e i g h n u m b e r appropriate for the real furnace b y u s i n g the p o w e r - l a w scaling suggested b y the b o u n d a r y layer analysis d e s c r i b e d p r e v i o u s l y . T h e results for ac agree to w i t h i n 4 - 3 0 % for the t h r e e g r o w t h rates, w i t h o u t any attempt to adjust the t h e r m o p h y s i c a l properties of the system b e y o n d an i n i t i a l calibration of the furnace t e m p e r a t u r e profile. M o r e i m p o r t a n t l y , the exponent for the p r o p o r t i o n a l i t y to P e ranged from s

100,—

0t~ I0

4

,

*-

I0

5

1

1

6

I0 Rayleigh Number

I0

7

J I0

1

Figure 18. Radial segregation (Ac) as a function ofRa for the growth of GaGe in the furnace described by Figures 15 and 16. Solid dots correspond to the results of experiments by Wang (6). t


82


0.28 to 0.30 w h e n the velocity scale was taken as that for a v e r t i c a l buoyant b o u n d a r y layer (130), so that V ~ ^ β Κ ( Δ Γ ) ) , i n w h i c h β is the coefficient of t h e r m a l expansion a n d ( Δ Γ ) is the characteristic radial t e m p e r a t u r e dif ference d r i v i n g the flow. T h e s e values are i n reasonable agreement w i t h the s i m p l e asymptotic theory p r e s e n t e d earlier. 0

2

Γ

1 / 2


Γ

Transitions to t h r e e - d i m e n s i o n a l , t i m e - p e r i o d i c , a n d chaotic flows do occur i n v e r t i c a l B r i d g m a n g r o w t h systems for h i g h e r convection levels w i t h the m e l t above the solid or w h e n the m e l t is b e l o w the crystal, so that the axial t e m p e r a t u r e gradient leads to an unstably stratified m e l t (46) (see the discussion i n reference 5). T h e i m p l e m e n t a t i o n of an i m p o s e d axial magnetic field removes these oscillations (62) a n d reduces the m o t i o n to l a m i n a r flow. S e v e r a l investigators have a n a l y z e d the i m p a c t o f the magnetic field o n solute segregation i n B r i d g m a n g r o w t h n u m e r i c a l l y (131-133) a n d asymptotically (133). T h e structure of the asymptotic analysis follows the w o r k of H j e l l m i n g a n d W a l k e r (134), as discussed earlier. T h e flows a n d the consequent solute segregation caused b y t h e r m o solutal c o n v e c t i o n i n n o n d i l u t e alloys is o n l y b e g i n n i n g to be e x p l o r e d b y e x p e r i m e n t a l a n d c o m p u t a t i o n a l analyses. Recent results are discussed b y B r o w n (5).

Czochralski Crystal Growth: Case Study of Meniscus-Defined Growth System T h e presence of the meniscus i n C z o c h r a l s k i (CZ) g r o w t h brings n e w d i mensions to the p r o b l e m s of o p t i m i z a t i o n a n d c o n t r o l , as w e l l as the analysis of transport processes. Because of its r e l a t i v e l y enormous i n d u s t r i a l signif icance, C Z g r o w t h has r e c e i v e d the most attention of any m e l t g r o w t h m e t h o d . A l s o , the distinctions made b y u s i n g F i g u r e 3 b e t w e e n different levels of m o d e l i n g are clearest i n the discussion of C z o c h r a l s k i crystal g r o w t h . T h e discussion i n this section focuses o n the analysis of macroscopic transport processes a n d t h e i r influence o n the design a n d c o n t r o l of a C Z g r o w t h system. U n d e r s t a n d i n g the onset of m i c r o s t r u c t u r e d crystal g r o w t h is v e r y r e l evant also to C Z systems, especially for the g r o w t h of h e a v i l y d o p e d crystals, as i n the case of a d d i n g i n d i u m to G a A s to l o w e r the dislocation d e n s i t y , w h i c h was discussed e a r l i e r i n regard to defect formation i n the crystal. I n this case, segregation of i n d i u m i n t o the m e l t d u r i n g g r o w t h i n a L E C system eventually leads to m o r p h o l o g i c a l instability, as d o c u m e n t e d b y O n o et a l . (135). Meniscus Shape. F i g u r e 6 shows a schematic representation of the i n n e r p o r t i o n of a C Z furnace f o r m e d b y the m e l t , crystal, and c r u c i b l e . T h e crystal is attached to the m e l t b y the m e l t - a m b i e n t meniscus d e n o t e d b y dD . If the traction caused b y h y d r o d y n a m i c forces is neglected (a good m


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assumption for s e m i c o n d u c t o r melts w i t h h i g h surface tensions a n d l o w viscosities), the interface shape, shown i n F i g u r e 6 as ζ = fir, Θ, t), is set b y a balance of capillary force w i t h gravity, b y the radius a n d height of the crystal, b y the w e t t i n g of m e l t on the c r u c i b l e , a n d b y the w e t t i n g c o n d i t i o n at the m e l t - c r y s t a l - a m b i e n t t r i j u n c t i o n . T h e hypothesis for the w e t t i n g c o n d i t i o n at this i n j u n c t i o n is that a constant angle φ

0

( F i g u r e 6) is f o r m e d

b e t w e e n the tangents to the local m e l t - a m b i e n t a n d c r y s t a l - a m b i e n t s u r faces, w h i c h is i n d e p e n d e n t of g r o w t h rate a n d other macroscopic p a r a m e ters. T h e concept of a w e t t i n g angle was first p r o p o s e d b y H e r r i n g (136) to


describe crystal s i n t e r i n g a n d t h e n adapted b y B a r d s l e y et a l . (137)

for

meniscus-defined crystal g r o w t h . T h e theoretical justification for such an angle hinges o n the same arguments used to describe e q u i l i b r i u m contact angles f o r m e d at the t r i j u n c t i o n of a l i q u i d o n an i n e r t solid (138). T h e use of local e q u i l i b r i u m i n the d e s c r i p t i o n of this p o i n t is more restrictive i n the solidification process because of the d y n a m i c phase transformation that m u s t be p r o c e e d i n g e v e n o n a m o l e c u l a r l e v e l near the t r i j u n c t i o n . T h e i m p l i cations of n o n e q u i l i b r i u m m i c r o s c o p i c processes a n d b u l k h y d r o d y n a m i c s on this angle are not u n d e r s t o o d . E x p e r i m e n t a l justification for specification of the angle at the p o i n t of three-phase contact comes from the results of S u r e k a n d C h a l m e r s (139), w h i c h verify that a particular value of φ

0

m e a s u r e d macroscopically can be

associated w i t h the crystal g r o w t h of a m a t e r i a l i n a specific crystallographic orientation a n d that φ

0

is r o u g h l y i n d e p e n d e n t of g r o w t h rate.

T h e second justification for the angular c o n d i t i o n is that this c o n d i t i o n is necessary for the d e t e r m i n a t i o n of the radius of the crystal at the t r i j u n c t i o n as a function of heat-transfer conditions a n d p u l l rate. T h i s a r g u m e n t is s i m p l e . T h e dimensionless Y o u n g - L a p l a c e equation of capillary statics gives the shape of an a x i s y m m e t r i c m e l t - a m b i e n t meniscus as d f/dr 2

df/dr

2

[1 + (df/dr) i]l * 2

3/2

τ +

-r η[ l + + MizrW* (df/dr) ] 2

=

Β θ [ / ( Γ

'

T)

"

λ ο ]

( 3 5 )

i n w h i c h the radius of the c r u c i b l e R has b e e n taken as the l e n g t h scale and B o is the B o n d n u m b e r (Bo = ApgR /v) or the square of the ratio of this scale to the capillary l e n g t h l [l = ( σ / p g ) ] . T h e left side of equation 35 is the local m e a n curvature of the meniscus. T h e constant λ is a d i m e n sionless reference value of the pressure difference across the meniscus i f the curvature is zero a n d is d e t e r m i n e d b y the constraint that the m e l t encloses a fixed v o l u m e . E q u a t i o n 35 is a n o n l i n e a r second-order b o u n d a r y - v a l u e p r o b l e m , w h i c h r e q u i r e s two b o u n d a r y conditions for solution. F o r a n o n w e t t i n g c r u c i b l e m a t e r i a l these conditions are as follows: c

2

c

c

c

1 7 2

0

h[R(f, t),t]

=f(R,

τ),^(Ι,ί) = 0 or


(36)

84


T h e first c o n d i t i o n specifies the j o i n i n g of the meniscus a n d the crystal, a n d the second c o n d i t i o n specifies the w e t t i n g at the c r u c i b l e w a l l . T h e radius of the crystal at the i n j u n c t i o n fl(f, τ) is left u n d e t e r m i n e d a n d m u s t be set b y an auxiliary constraint, such as the w e t t i n g angle there. F o r g r o w t h o f a crystal w i t h shape R(z, τ) e v o l v i n g i n t i m e , the w e t t i n g angle condition* is w r i t t e n i n dimensionless form as

dR

dh

dt

dt


z = h(R_t)

j tan [φ(τ)

-

φ ]

(37)

0

r=R(h,ty

E q u a t i o n 37 describes the relationship b e t w e e n t h e rate of change of the crystal radius at the i n j u n c t i o n a n d the d e v i a t i o n of the local angle from the e q u i l i b r i u m v a l u e φ . I n this expression, φ(τ) is the d y n a m i c angle f o r m e d b e t w e e n the local tangents to the m e l t - a m b i e n t a n d c r y s t a l - a m b i e n t s u r faces, a n d V (T) is the dimensionless p u l l rate of the crystal. F o r steady-state g r o w t h , e q u a t i o n 37 s i m p l y sets the angle w i t h w h a t m u s t b e a s o l i d c y l i n d e r of constant radius. T h e i m p o r t a n c e of the d y n a m i c a l f o r m e q u a t i o n 37 is b r o u g h t out i n the next section. 0

8

W h e n the c r u c i b l e is large e n o u g h that its w e t t i n g p r o p e r t i e s have no influence o n meniscus shape, that is, the meniscus becomes flat across the m e l t surface, the possible shapes c o m e from the c o l l e c t i o n for u n b o u n d e d a s y m m e t r i c interfaces r e p o r t e d b y H u h a n d S c r i v e n (140) (see also reference 141). T h i s case is a g o o d a p p r o x i m a t i o n for C z o c h r a l s k i g r o w t h of large crystals i n w h i c h the r e g i o n o f influence of surface tension, σ , m e a s u r e d b y the capillary l e n g t h , l [l = ( σ / p g ) ] , is a p p r o x i m a t e l y 1 c m . H u r l e (142) d e v e l o p e d a closed-form a p p r o x i m a t i o n to these meniscus shapes, w h i c h are d e s c r i b e d b y the shape f u n c t i o n f(r) that is v a l i d i n this l i m i t . c

c

1 7 2

Heat-Transfer Analysis: Thermal-Capillary Models.

Numerous

analyses of various aspects of heat transfer i n the C Z system have b e e n r e p o r t e d ; m a n y of these are c i t e d b y e i t h e r K o b a y a s h i (143) or D e r b y a n d B r o w n (144). T h e analyses v a r y i n c o m p l e x i t y a n d p u r p o s e , f r o m the s i m p l e o n e - d i m e n s i o n a l or " f i n " approximations d e s i g n e d to give o r d e r - o f - m a g n i t u d e estimates for the axial t e m p e r a t u r e gradient i n the crystal (98) to c o m plex s y s t e m - o r i e n t e d calculations d e s i g n e d to o p t i m i z e heater design a n d p o w e r r e q u i r e m e n t s (145,146). T h e s y s t e m - o r i e n t e d , large-scale calculations i n c l u d e radiation b e t w e e n c o m p o n e n t s of the heater a n d the c r u c i b l e as semblies, as w e l l as c o n d u c t i o n a n d convection. T h e d y n a m i c s of the C z o c h r a l s k i system can be d e s c r i b e d o n l y b y heattransfer models that i n c l u d e the interaction of the shape of the m e n i s c u s , w h i c h are r e f e r r e d to as t h e r m a l - c a p i l l a r y models ( T C M ) , because o n l y these models give self-consistent d e t e r m i n a t i o n of the meniscus shape, crystal radius, a n d heat transfer i n each phase.


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B i l l i g (98) r e a l i z e d this p o i n t a n d used a o n e - d i m e n s i o n a l heat-transfer analysis for the crystal w i t h assumptions about the t e m p e r a t u r e field i n t h e m e l t to d e r i v e t h e t h e f o l l o w i n g relationship that has b e e n u s e d h e a v i l y i n qualitative discussions o f crystal g r o w t h d y n a m i c s : R « V

g

2

(38)

K i m a n d co-workers (147) p r e s e n t e d t h e f o l l o w i n g e m p i r i c a l c o r r e l a t i o n for


silicon g r o w t h

R =

A

(39)

V. + B i n w h i c h A a n d Β are constants that d e p e n d o n the details o f the system a n d are d e t e r m i n e d from e x p e r i m e n t s . K o b a y a s h i (J 43) p r e s e n t e d the first c o m p u t e r simulations that c o n s i d e r e d the d e t e r m i n a t i o n o f the crystal radius as part o f the analysis b u t a v o i d e d the c a p i l l a r y p r o b l e m b y c o n s i d e r i n g a flat m e l t - a m b i e n t surface, w h i c h is consistent w i t h φ = 90°. C a l c u l a t i o n s w e r e p e r f o r m e d for a fixed crystal radius, a n d t h e n t h e g r o w t h rate was adjusted to balance t h e heat flux into the crystal. C r o w l e y (148) was t h e first to present n u m e r i c a l calculations o f a c o n d u c t i o n - d o m i n a t e d heat-transfer m o d e l for t h e simultaneous d e t e r m i nation o f the t e m p e r a t u r e fields i n crystal a n d m e l t a n d o f the shapes o f the m e l t - c r y s t a l a n d m e l t - a m b i e n t surfaces for an i d e a l i z e d system w i t h a m e l t p o o l so large that n o interactions w i t h the c r u c i b l e are c o n s i d e r e d . She u s e d a t i m e - d e p e n d e n t f o r m u l a t i o n o f the t h e r m a l - c a p i l l a r y m o d e l a n d c o m p u t e d the shape o f an e v o l v i n g crystal from a short i n i t i a l configuration. 0

I n a series o f papers, D e r b y a n d B r o w n (144, 149-152) d e v e l o p e d a d e t a i l e d T C M that i n c l u d e d t h e calculation o f the t e m p e r a t u r e field i n t h e m e l t , crystal, a n d c r u c i b l e ; the location of the m e l t - c r y s t a l a n d m e l t - a m b i e n t surfaces; a n d t h e crystal shape. T h e analysis is based o n a finite-ele m e n t - N e w t o n m e t h o d , w h i c h has b e e n d e s c r i b e d i n d e t a i l (152). T h e heattransfer m o d e l i n c l u d e d c o n d u c t i o n i n each o f the phases a n d a n i d e a l i z e d m o d e l for radiation from t h e c r y s t a l , m e l t , a n d c r u c i b l e surfaces w i t h o u t a systematic calculation o f v i e w factors a n d difluse-gray radiative exchange (153). Results from a quasi steady-state m o d e l ( Q S S M ) v a l i d for l o n g crystals a n d a constant m e l t l e v e l (if some f o r m o f automatic r e p l e n i s h m e n t o f m e l t to t h e c r u c i b l e exists) v e r i f i e d t h e correlation (equation 39) for t h e d e p e n d e n c e o f the radius o n t h e g r o w t h rate (144) a n d p r e d i c t e d changes i n t h e radius, t h e shape o f the m e l t - c r y s t a l interface ( w h i c h is a measure o f radial t e m p e r a t u r e gradients i n t h e crystal), a n d t h e axial t e m p e r a t u r e field w i t h i m p o r t a n t c o n t r o l parameters l i k e t h e heater t e m p e r a t u r e a n d t h e l e v e l o f m e l t i n t h e c r u c i b l e . P r o c e s s i n g strategies for h o l d i n g the radius a n d s o l i d -


86


ification interface shape constant as the m e l t v o l u m e decreases t h r o u g h a t y p i c a l batchwise g r o w t h cycle are p r e s e n t e d b y D e r b y a n d B r o w n (149), as c o m p u t e d b y a strategy of a u g m e n t i n g the N e w t o n m e t h o d w i t h the a d d i tional constraints for each c o n t r o l parameter.


D u d o k o v i c a n d co-workers (154, 155) e x t e n d e d the analysis of a Q S S M to i n c l u d e difiuse-gray radiation. T h e y c o m p u t e d v i e w factors b y a p p r o x i m a t i n g the shapes o f the crystal a n d m e l t b y a few standard g e o m e t r i c a l elements a n d i n c o r p o r a t i n g analytical approximations to the v i e w factors. A t h e r t o n et a l . (153) d e v e l o p e d a scheme for a self-consistent calculation of v i e w factors a n d radiative fluxes w i t h i n the finite-element framework a n d i m p l e m e n t e d this s c h e m e i n the Q S S M . F i g u r e 19 shows sample isotherms a n d interface shapes p r e d i c t e d b y the Q S S M for calculations w i t h decreasing m e l t v o l u m e i n the c r u c i b l e , as occurs i n the batchwise process. Because the crystal p u l l rate and the heater t e m p e r a t u r e are m a i n t a i n e d at constant values for this sequence, the crystal radius varies w i t h the v a r y i n g heat transfer i n the system. T w o effects are noticeable. F i r s t , decreasing the v o l u m e exposes the hot c r u c i b l e w a l l to the crystal. T h e c r u c i b l e w a l l heats the crystal a n d causes the decrease i n

Figure 19. Sample isotherms and interface shapes computed for the QSSM for CZ crystal growth by Atherton et al. (153). The model includes detailed radiation between the surfaces of the melt, crystal, and crucible. Isotherms are spaced at 10 Κ increments in the melt and 30 Κ increments in the other phases.


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radius seen for dimensionless m e l t volumes ( V ) of 3.1 a n d 2.3. m

Second,

heat transfer f r o m the c r u c i b l e to the m e l t becomes ineffective for l o w v o l umes a n d causes a decrease i n the t e m p e r a t u r e of the m e l t a n d an increase i n the crystal radius. F o r the lowest v o l u m e , the m e l t t e m p e r a t u r e at the b o t t o m of the c r u c i b l e drops b e l o w the m e l t i n g point, a n d a l u m p of s o l i d forms at the center. T h e g r o w t h is t e r m i n a t e d i n the s i m u l a t i o n b y m e r g i n g this mass of s o l i d w i t h the m e l t - c r y s t a l interface. Insulation or the i n t r o d u c t i o n of heat along the b o t t o m leads to a monotonie decrease i n the radius to m u c h l o w e r m e l t v o l u m e s , w i t h an almost l i n e a r relationship (R ~

V ) m


w h e n the b o t t o m is heated. T h e results of a l l the t h e r m a l - c a p i l l a r y models discussed so far have neglected the influence of convection i n the m e l t i n t r a n s p o r t i n g heat to the solidification interface. T h e status of convection calculations that neglect the c o u p l i n g to global heat transfer a n d capillary consideration is discussed later. T h e u n i o n of t h e r m a l - c a p i l l a r y analysis w i t h d e t a i l e d convection calculations is discussed i n the subsection o n m e l t flow. Process Stability and Control.

O p e r a t i o n a l l y , automatic c o n t r o l of

the crystal radius b y v a r y i n g e i t h e r the i n p u t p o w e r to the heater or the crystal p u l l rate has b e e n necessary for the r e p r o d u c i b l e g r o w t h of crystals w i t h constant radius. T e c h n i q u e s for automatic d i a m e t e r c o n t r o l have b e e n used since the establishment of C z o c h r a l s k i g r o w t h . O p t i c a l i m a g i n g of the crystal or direct m e a s u r e m e n t of the crystal w e i g h t has b e e n u s e d to d e t e r m i n e the instantaneous radius. H u r l e (156) r e v i e w e d the techniques c u r r e n t l y u s e d for sensing the radius. B a r d s l e y et al. (J57,158) d e s c r i b e d c o n t r o l based o n the m e a s u r e m e n t of the crystal weight. T h e details of the c o n t r o l strategy have r e c e i v e d m u c h less attention. T h e theoretical (159) a n d e x p e r i m e n t a l (160) analyses of the transfer f u n c t i o n for C Z g r o w t h are notable exceptions. A l g o r i t h m s for m o d e l - b a s e d c o n t r o l are just b e i n g d e v e l o p e d . T h e i m p l e m e n t a t i o n of a c o n t r o l a l g o r i t h m u s i n g any sensing technology d e p e n d s o n the d o m i n a n t dynamics of the C z o c h r a l s k i process, especially the stability of the process to perturbations i n the t h e r m a l field a n d the interface shapes a n d the batchwise transient i n t r o d u c e d b y the decreasing m e l t v o l u m e . T h e c o n t r o l strategies differ substantially according to w h e t h e r the process is stable or i n h e r e n t l y unstable. T h i s issue is discussed h e r e i n terms of the several notions for stability i n t r o d u c e d b y H u r l e (10) that are relevant to m e n i s c u s - d e f i n e d crystal g r o w t h . T h r e e types of instabilities are d i s t i n g u i s h e d a c c o r d i n g to the t i m e scales for the disturbances: c a p i l l a r y instabilities of the meniscus, d y n a m i c i n s t a b i l i t y of the e n t i r e t h e r m a l - c a p illary system, a n d convective instabilities of the m e l t . Capillary Instability. T h e meniscus shape m a y b e c o m e unstable i n the sense that s m a l l perturbations to it cause the melt to separate f r o m the


88


crystal i n the t i m e scale of a c a p i l l a r y - i n d u c e d m o t i o n of the surface. T h i s t i m e scale is p r o p o r t i o n a l to the capillary w a v e speed, w h i c h is a few seconds for the t h e r m o p h y s i c a l properties of m o l t e n silicon. T h e capillary stability of meniscus shapes that are i m p o r t a n t i n C Z g r o w t h have b e e n a n a l y z e d b y several authors (161,162) u s i n g arguments f r o m the theory of energy stability for e q u i l i b r i u m configurations. M e n i s c u s shapes that are stable to c a p i l l a r y instabilities are feasible for a range of contact angles a n d interface heights. Dynamic Stability of Thermal Capillary System.

S u r e k a n d co-workers


(163, 164) i n t r o d u c e d the n o t i o n of dynamic stability for m e n i s c u s - d e f i n e d crystal g r o w t h systems. T h e y c o n s i d e r e d the question of w h e t h e r a m e n i s c u s d e f i n e d g r o w t h system subjected to a p e r t u r b a t i o n to the crystal shape at fixed p u l l rate w i l l r e t u r n to the initial shape or diverge unstably away f r o m it. T h e c o n c e p t of d y n a m i c stability does not i n c l u d e the effect of the b a t c h wise d r o p i n the m e l t v o l u m e o n the e v o l u t i o n of the p e r t u r b a t i o n , e v e n t h o u g h the t i m e scales for the decreasing m e l t v o l u m e a n d for the g r o w t h of the crystal are essentially the same, except w h e n the c r u c i b l e is m u c h larger i n d i a m e t e r than the crystal b e i n g g r o w n . T h e i n i t i a l analysis o f S u r e k (163) c o n s i d e r e d the influence of the w e t t i n g c o n d i t i o n at the t r i j u n c t i o n (equation 39) a n d the Y o u n g - L a p l a c e e q u a t i o n for meniscus shape b u t n e g l e c t e d the influence of heat transfer o n the d y namics of the system. W i t h o u t heat transport, the location of the m e l t i n g point i s o t h e r m is unspecified, a n d so the dynamics of the height of the I n j u n c t i o n h(f, τ) is u n k n o w n . O n the basis of an i s o t h e r m a l analysis, S u r e k s h o w e d that the floating-zone techniques a n d the d i e - d e f i n e d methods for g r o w i n g t h i n s o l i d sheets are stable to perturbations i n the meniscus h e i g h t but that the C Z m e t h o d is i n h e r e n t l y unstable. Refinements to the analysis that i n c l u d e d s i m p l e descriptions of the interactions b e t w e e n heat transfer a n d the meniscus height s h o w e d that conditions exist for stable operation (164 165) w i t h o u t batchwise transients b e i n g considered. y

Integration of a t i m e - d e p e n d e n t t h e r m a l - c a p i l l a r y m o d e l for C Z g r o w t h (150, 152) also has i l l u m i n a t e d the i d e a of d y n a m i c stability. D e r b y a n d B r o w n (150) first c o n s t r u c t e d a t i m e - d e p e n d e n t T C M that i n c l u d e d the transients associated w i t h c o n d u c t i o n i n each phase, the e v o l u t i o n of the crystal shape i n t i m e , a n d the decrease i n the m e l t l e v e l caused b y the conservation o f v o l u m e . H o w e v e r , the m o d e l i d e a l i z e d radiation to be to a u n i f o r m a m b i e n t . T h e t e c h n i q u e for i m p l i c i t n u m e r i c a l integration of the transient m o d e l was b u i l t a r o u n d the finite-element-Newton method used for the Q S S M . L i n e a r a n d n o n l i n e a r stability calculations for the solutions of the Q S S M (if the batchwise transient is neglected) s h o w e d that the C Z m e t h o d is d y n a m i c a l l y stable; s m a l l perturbations i n the system at fixed o p e r a t i n g parameters d e c a y e d w i t h t i m e , a n d changes i n the parameters caused the process to e v o l v e to the e x p e c t e d n e w solutions of the Q S S M . T h e stability of the C Z process has b e e n v e r i f i e d e x p e r i m e n t a l l y , at least


2.

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partially, b y the u n c o n t r o l l e d , stable g r o w t h of s m a l l - d i a m e t e r g e r m a n i u m (166) a n d silicon crystals ( M . W a r g o , u n p u b l i s h e d , Massachusetts Institute of Technology). A t h e r t o n et a l . (153) have e x t e n d e d the calculations to i n c l u d e diffusegray radiation b e t w e e n components of the enclosure a n d r e a c h e d essentially the same conclusions r e g a r d i n g the stability of the process. H o w e v e r , t h e y d i s c o v e r e d a n e w m e c h a n i s m for the d a m p e d oscillation o f the crystal radius caused b y the radiative i n t e r a c t i o n b e t w e e n the crystal surface just above the m e l t l e v e l a n d the hot c r u c i b l e w a l l . T h e s e oscillations are especially


apparent w h e n the v e r t i c a l t e m p e r a t u r e gradient i n the crystal is l o w , so that radiative heat transport has a d o m i n a t i n g influence. T h e d y n a m i c stability of the quasi steady-state process suggests that active c o n t r o l o f the C Z system has to account o n l y for r a n d o m disturbances to the system about its set points a n d for the batchwise transient caused b y the decreasing m e l t v o l u m e . D e r b y a n d B r o w n (150) i m p l e m e n t e d a s i m p l e p r o p o r t i o n a l - i n t e g r a l (PI) c o n t r o l l e r that c o u p l e d the crystal radius to a set p o i n t t e m p e r a t u r e for the heater i n an effort to c o n t r o l the d y n a m i c C Z m o d e l w i t h i d e a l i z e d radiation. F i g u r e 20 shows the shapes of the crystal and melt predicted without control, with purely integral control, and w i t h

Figure 20. Isotherms and interface shapes for the time τ = 1.0 for batchwise simuhtions of CZ growth. Results are shown for (a) uncontrolled, (b-d) integral control, and (e) proportional-integral control simulations. Isotherms are spaced as described for Figure 19. The figure is taken from Atherton et al. (153).



90


P I c o n t r o l . T h e oscillations i n t h e radius about t h e set p o i n t p r e d i c t e d w i t h integral c o n t r o l are not unexpected. B o t h t h e oscillations i n t h e c o n t r o l of systems g o v e r n e d b y l i n e a r models (167) a n d t h e o c c u r r e n c e of H o p f b i f u r cations i n n o n l i n e a r models (168) have b e e n p r e d i c t e d w i t h i n t e g r a l c o n t r o l . I n d e e d , t h e oscillations s h o w n for P I c o n t r o l are anticipated f r o m t h e a p pearance of a H o p f bifurcation i n t h e Q S S M w i t h increasing gain of t h e integral part o f t h e c o n t r o l l e r (150). S i m p l e P I c o n t r o l r e m o v e s t h e o s c i l l a tions a n d leads to a constant radius throughout a large p o r t i o n of the g r o w t h r u n . A t h e r t o n et al. (153) have extended these calculations to i n c l u d e diffusegray radiation a n d have s h o w n that P I c o n t r o l o f t h e crystal radius is m u c h m o r e difficult t h e r e , because t h e increasing influence of radiation at t h e m e l t l e v e l drops makes a single set o f gain values inappropriate for t h e w h o l e growth run. It remains to b e s h o w n that t h e d y n a m i c s p r e d i c t e d b y t h e r m a l - c a p i l l a r y models adequately represents a r e a l C Z system. A first step i n this d i r e c t i o n has b e e n taken b y T h o m a s et al. (169) b y m a k i n g a direct c o m p a r i s o n b e t w e e n the predictions o f a d y n a m i c T C M for the l i q u i d - e n c a p s u l a t e d C Z g r o w t h of G a A s a n d t h e response o f an e x p e r i m e n t a l system u s i n g a large (2-kilogauss) axial magnetic field. T h e p u l l rate a n d heater t e m p e r a t u r e i n t h e calculation w e r e v a r i e d w i t h t i m e a c c o r d i n g to p r e s c r i b e d histories taken f r o m e x p e r i m e n t a l data. T h e shape of the crystal a n d the t e m p e r a t u r e field p r e d i c t e d b y t h e s i m u l a t i o n are s h o w n i n F i g u r e 21 as a function o f t h e t i m e d u r i n g the batch process. T h e p r e d i c t e d shape o f t h e crystal agrees q u a l i t a t i v e l y w i t h that p r o d u c e d i n t h e e x p e r i m e n t . T h e quantitative value o f the radius at any t i m e was w i t h i n 1 5 % of the e x p e r i m e n t a l value; this difference is w e l l w i t h i n the e r r o r caused b y uncertainties i n t h e e x p e r i m e n t a n d b y p o o r k n o w l e d g e o f t h e r m o p h y s i c a l p r o p e r t y data u s e d i n t h e m o d e l . Convection and Segregation. Melt Flow. A l t h o u g h analysis of convection a n d segregation i n C z o c h r a l s k i g r o w t h has r e c e i v e d t h e most attention o f a n y m e l t g r o w t h system, t h e large size o f a t y p i c a l p u l l e r a n d the presence o f a meniscus m a k e t h e simulations t h e most difficult o f any system. T h e r e a l flows i n large-scale systems are most p r o b a b l y three d i m e n s i o n a l a n d t e m p o r a l l y chaotic; a d e t a i l e d analysis of such motions stretches t h e l i m i t s o f e v e n t h e largest calculations p e r f o r m e d today. C a l culations of a x i s y m m e t r i c flow that neglect t h e d e t e r m i n a t i o n o f t h e shapes of t h e m e l t - c r y s t a l a n d m e l t - a m b i e n t phase boundaries have b e e n c a r r i e d out b y a n u m b e r of investigators for some t i m e . T h e results before 1985 are r e v i e w e d b y L a n g l o i s (170) a n d focus o n t h e interactions o f flows d r i v e n b y c r u c i b l e a n d crystal rotation, buoyancy, a n d gradients i n surface tensions. Starting w i t h t h e i n i t i a l calculations of K o b a y a s h i (171 ), most of these analyses have focused o n p r e d i c t i n g t h e m u l t i c e l l u l a r structure o f t h e flows caused b y combinations o f various d r i v i n g forces.



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4 Hrs

0 Hrs

β Hrs

Figure 21. Isotherms and interface shapes for selected times during the hatchwise simulations of GaAs crystal in LEC growth. The figure is taken from Thomas et al. (169).

M o r e - r e c e n t simulations concentrate o n the onset of t i m e - p e r i o d i c c o n v e c t i o n , w h i c h marks the onset of a d o m i n a n t m e c h a n i s m for dopant striations i n the crystal, a n d o n the effect of an i m p o s e d magnetic field o n this transition. C r o c h e t et a l . (56) first c o m p u t e d the onset of an a x i s y m m e t r i c t i m e - p e r i o d i c m o t i o n i n C Z flow d r i v e n b y heating of the side w a l l . T h e flow m o t i o n is caused b y the c o m p e t i t i o n b e t w e e n flow cells created b y an instability that leads to the separation of the h y d r o d y n a m i c b o u n d a r y layer along the c r u c i b l e side w a l l . A c c u r a t e calculations must resolve this b o u n d a r y layer. A s i m u lation r e p o r t e d b y C r o c h e t et a l . (56) is s h o w n i n F i g u r e 22, w h i c h is a p l o t of the k i n e t i c energy i n the flow as a function of t i m e for G r = g β ( Γ T )R /v = 1 Χ 10 , in which T is the constant t e m p e r a t u r e of the c r u c i b l e w a l l . T h e oscillations d e v e l o p from some i n i t i a l state a n d t e n d t o w a r d constant values of the a m p l i t u d e a n d frequency. T h e flow patterns a n d t e m perature fields r e p o r t e d b y C r o c h e t et a l . (56) for several different times i n this s i m u l a t i o n are shown i n F i g u r e 23. T h e interaction b e t w e e n the two largest toroidal vortices is apparent. w a U

m

3

c

2

7

w a l l

T a n g b o r n (172) a n d Patera (173) have r e p r o d u c e d the oscillations o b served b y C r o c h e t at a l . b y u s i n g a spectral-element s i m u l a t i o n w i t h flat phase boundaries. T h e y d e m o n s t r a t e d that u n d e r r e s o l u t i o n of the b o u n d a r y layer leads to spurious oscillations at l o w e r G r a s h o f n u m b e r s . O t h e r s have



92


30.0

Dimensionless Time Figure 22. Kinetic energy as a function of time in a simulation of Czochralski bulk flow by Crochet et al. (55) for Gr = 1 Χ 10 . Points A and Β correspond to flows shown in Figure 23. 7

c o m p u t e d unsteady c o n v e c t i o n i n a s i m i l a r m o d e l for a x i s y m m e t r i c c o n v e c t i o n i n C Z b u l k flow. L a n g l o i s (44, 63) first d e m o n s t r a t e d n u m e r i c a l l y the e x p e r i m e n t a l l y o b served d a m p i n g of convection caused b y an i m p o s e d axial magnetic field o n unsteady c o n v e c t i o n a n d the possibility o f a c h i e v i n g steady, a x i s y m m e t r i c flows. M i h e l c i c a n d W i n e g r a t h (174) a n d P a t e r a (173) have p r e s e n t e d s i m i l a r results w i t h m o r e emphasis o n the transition b e t w e e n steady a n d oscillatory flows caused b y increasing the magnetic field strength. Increasing the field b e y o n d the l e v e l necessary to stabilize the flow decreases the m i x i n g i n the m e l t a n d leads to radial n o n u n i f o r m i t y of dopants i n the crystal, as w o u l d be the case for weak convective m i x i n g s h o w n i n F i g u r e 10. O r e p e r a n d S z e k e l y (131) d o c u m e n t e d this effect, a n d H j e l l m i n g a n d W a l k e r (44) p r e sented an asymptotic analysis for the flow a n d t e m p e r a t u r e fields i n this limit. N u m e r i c a l simulations that c o m b i n e the details of the t h e r m a l - c a p i l l a r y models d e s c r i b e d p r e v i o u s l y w i t h the calculation of convection i n the m e l t s h o u l d be able to p r e d i c t heat transfer i n the C Z system. Sackinger et a l . (J 75) have a d d e d the calculation of steady-state, a x i s y m m e t r i c convection i n the m e l t to the t h e r m a l - c a p i l l a r y m o d e l for quasi steady-state g r o w t h of a l o n g c y l i n d r i c a l crystal. T h e calculations i n c l u d e m e l t m o t i o n d r i v e n b y buoyancy, surface t e n s i o n , a n d c r u c i b l e a n d crystal rotation. F i g u r e 24 shows sample calculations for g r o w t h of a 3 - i n . (7.6-cm)-diameter silicon crystal as a function of the d e p t h of the m e l t i n the c r u c i b l e .


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A

Theory of Transport Processes Β

C

D

Figure 23. Streamlines and isotherms for selected times during the simulation shown in Figure 22. Flows correspond to points shown in Figure 22. T h e radial t e m p e r a t u r e difference i m p o s e d b y the heater arrangement drives a large t o r o i d a l flow c e l l that moves u p the w a l l a n d d o w n t o w a r d the center o f the c r u c i b l e . F o r the deepest m e l t v o l u m e s , the flow is separated along the w a l l . T h i s steady a x i s y m m e t r i c m o t i o n is p r o b a b l y not stable. T i m e p e r i o d i c oscillations caused b y the i n s t a b i l i t y of this separation have b e e n r e p o r t e d at values of the G r a s h o f n u m b e r b e l o w the values that c o r r e s p o n d to the calculations i n F i g u r e 24. A d d i n g an axial magnetic field leads to the reattachment of the flow c e l l to the w a l l of the c r u c i b l e a n d to the flow structure p r e d i c t e d b y H j e l l m i n g a n d W a l k e r (43) at h i g h field levels. T h e flow separation along the c r u c i b l e side w a l l disappears, a n d a sep aration along the b o t t o m appears as the m e l t l e v e l is l o w e r e d ( F i g u r e 24). T h e m e l t - c r y s t a l interface deforms to b e convex at the c e n t e r of the crystal as it senses the c o l d c r u c i b l e b o t t o m w i t h the decreasing m e l t d e p t h . T h i s interface flipping is w e l l d o c u m e n t e d i n e x p e r i m e n t a l systems a n d is also seen i n calculations based solely o n c o n d u c t i o n i n the m e l t .



94


Figure 24. Streamlines and isotherms for the growth of silicon in a prototype Czochralski system with self-consistent calculation of interface and crystal shapes by using the quasi steady-state thermal-capillary model and the con dition that the crystal radius remains constant. Calculations are for decreasing melt volume. The Grashof number (scaled with the maximum temperature difference in the melt) varies between 1.0 Χ 10 and 2.0 Χ 10 with decreasing melt volume. 7

7

F e w calculations of t h r e e - d i m e n s i o n a l c o n v e c t i o n i n C Z melts (or other systems) have b e e n p r e s e n t e d because of the p r o h i b i t i v e expense of such simulations. M i h e l c i c et a l . (176) have c o m p u t e d the effect of asymmetries i n the heater t e m p e r a t u r e o n the flow pattern a n d s h o w e d that crystal rotation w i l l e l i m i n a t e t h r e e - d i m e n s i o n a l convection d r i v e n b y this m e c h a n i s m . T a n g b o r n (172) a n d P a t e r a (J 73) have u s e d a spectral-element m e t h o d c o m b i n e d w i t h l i n e a r stability analysis to c o m p u t e the stability of a x i s y m m e t r i c flows to t h r e e - d i m e n s i o n a l instabilities. S u c h a stability calculation is the most essential part of a t h r e e - d i m e n s i o n a l analysis, because n o n a x i s y m m e t r i c flows are undesirable. Solute Transport. B u r t o n et al. (72) o r i g i n a t e d the b o u n d a r y layer analyses for solute transport i n C Z crystal g r o w t h d e s c r i b e d earlier. T h e analysis of the b o u n d a r y layer has b e e n the starting p o i n t for s t u d y i n g axial solute segregation i n C Z g r o w t h . W i l s o n (80) has r e f i n e d this analysis b y i n c l u d i n g a full n u m e r i c a l calculation of the self-similar velocity field due to crystal rotation instead of u s i n g the asymptotic approximation of C o c h r a n


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(78). W i l s o n (177, 178) also u s e d the transient, self-similar v e l o c i t y

95 field

caused b y crystal rotation to analyze the t i m e - d e p e n d e n t axial dopant striations caused b y a fluctuating local g r o w t h rate. B y simultaneous calculation of the flow a n d the solute field w i t h o u t (177) and w i t h (178) accounting for the possible back m e l t i n g of the s o l i d , she d e m o n s t r a t e d p e r i o d i c axial striations i n the concentration field that w e r e e x t r e m e l y similar to ones r e p o r t e d i n C Z g r o w t h for systems w i t h p o o r t h e r m a l s y m m e t r y that r e s u l t e d i n g r o w t h rate fluctuations p r o p o r t i o n a l to the crystal rotation rate (25). E a r l i e r , H u r l e et a l . (179) made the connection b e t w e e n dopant striations a n d t e m p e r a t u r e


oscillations b y u s i n g a s i m p l e r m o d e l . Analysis of solute transport i n the presence of a magnetic field has r e c e i v e d considerable attention. H u r l e a n d Series (85) and C a r t w r i g h t a n d H u r l e (180) used the self-similar form of the v e l o c i t y field near a rotating crystal as a framework for e x a m i n i n g the role of an axial magnetic field i n m o d i f y i n g axial segregation i n the crystal. L e e et a l . (181) r e p o r t e d the o n l y d e t a i l e d calculations of solute transport i n C z o c h r a l s k i g r o w t h for oxygen transport i n the p r e s e n c e of an axially a l i g n e d magnetic field. T h e i r results i n d i c a t e d that oxygen transport is r e d u c e d b y the d a m p i n g of c o n v e c t i o n a n d that the structure of the flow cells has a significant effect o n the transfer rate, i f the flow is steady. H o s h i k a w a et a l . (182) first d e m o n s t r a t e d the c o n t r o l of oxygen concentration i n s i l i c o n g r o w n b y the C Z m e t h o d b y u s i n g an a p p l i e d magnetic field. M o r e recent analysis b y K i m a n d L a n g l o i s (183) of b o r o n transport i n C Z g r o w t h w i t h an axial magnetic field shows large radial variations i n b o r o n concentration w h e n b u l k convection is weak. S u c h large variations have b e e n r e p o r t e d i n several as yet u n p u b l i s h e d e x p e r i m e n t a l studies.

Summary and Outlook P i o n e e r i n g efforts i n the u n d e r s t a n d i n g of transport processes i n m e l t crystal g r o w t h have e l u c i d a t e d the i m p o r t a n t f u n d a m e n t a l mechanisms i n these systems a n d are the b e g i n n i n g of c o m p r e h e n s i v e u n d e r s t a n d i n g of p r o p e r t y - p r o c e s s i n g relationships for electronic materials. T h e segregation analysis of B u r t o n et a l . (74), the analysis of the onset of m o r p h o l o g i c a l instabilities b y M u l l i n s and S e k e r k a (88, 89), a n d the d o c u m e n t a t i o n of the transitions to chaotic convection i n m e l t crystal g r o w t h b y W i t t a n d his collaborators (46-50) are examples of such s e m i n a l investigations. E a c h of these efforts has s p a w n e d nearly a generation of scientific exploration a n d e n g i n e e r i n g application of these concepts. A s this chapter indicates, the l e v e l of generic u n d e r s t a n d i n g of transport processes a n d macroscopic properties of the crystal is good. E v e n so, quantitative p r e d i c tions of the performance of specific crystal g r o w t h systems a n d the d e s i g n a n d o p t i m i z a t i o n of these systems based solely o n theoretical u n d e r s t a n d i n g


96


are not yet feasible. T h e design of systems for specific materials still involves a great d e a l o f e x p e r i m e n t a l i n p u t and e m p i r i c i s m s . T h e reasons for this l i m i t a t i o n are m a n y .


F i r s t , the role of system d e s i g n o n the details o f c o n v e c t i o n a n d solute segregation i n industrial-scale crystal g r o w t h systems has not b e e n ade quately s t u d i e d . T h i s deficiency is mostly because n u m e r i c a l simulations of the t h r e e - d i m e n s i o n a l , w e a k l y t u r b u l e n t c o n v e c t i o n present i n these systems are at the v e r y l i m i t o f w h a t is c o m p u t a t i o n a l l y feasible today. N e w d e v e l o p m e n t s i n c o m p u t a t i o n a l p o w e r may lift this l i m i t a t i o n . A l s o , the extensive use of a p p l i e d magnetic fields to c o n t r o l the i n t e n s i t y o f the c o n v e c t i o n actually makes the calculations m u c h m o r e feasible. E v e n i f the extensive calculations n e e d e d to define the d e p e n d e n c e of segregation a n d heat transfer o n o p e r a t i n g conditions can b e p e r f o r m e d , t h e y m a y not b e j u s t i f i e d i n the sense that the results cannot b e s u p p o r t e d e i t h e r b y the l e v e l of quantitative characterization of a t y p i c a l crystal g r o w t h system o r b y the accuracy of the data base for the t h e r m o p h y s i c a l p r o p e r t i e s of exotic s e m i c o n d u c t o r alloys, especially u n d e r h i g h - t e m p e r a t u r e processing conditions. T h e first deficiency requires extensive d e v e l o p m e n t s i n the t e c h nology of sensors for m o n i t o r i n g variables d u r i n g the crystal g r o w t h process, s u c h as the t e m p e r a t u r e gradients i n the crystal that are d i r e c t l y responsible for defect generation i n the s o l i d . C h a r a c t e r i z a t i o n o f the t h e r m o p h y s i c a l properties of these materials is clearly a p r e r e q u i s i t e for the accurate u n d e r s t a n d i n g of the processing conditions for any m e l t crystal g r o w t h system. T h e most conspicuous shortcomings of m o d e r n theory of crystal g r o w t h appear w h e n the links b e t w e e n macroscopic processing conditions a n d the formation of crystalline defects are c o n s i d e r e d . O t h e r than the i n t u i t i o n d e r i v e d b y a p p l y i n g the t h e o r y o f l i n e a r thermoelasticity to the calculation of stress fields i n the c o o l i n g solids, no guidelines exist for d e s i g n i n g g r o w t h systems. E l u c i d a t i n g the m e c h a n i s m s of defect formation a n d the i m p o r t a n c e of the c h e m i s t r y of the m e l t a n d h e n c e transport processes i n these m e c h anisms are i m p o r t a n t frontiers i n the processing of electronic materials.

Abbreviations and Symbols Β

magnetic

Βο c*

m a g n i t u d e of magnetic field characteristic concentration scale

field

d i m e n s i o n a l concentration

c c c

0

Q

i c(x, t) Ac* D

field

reference value of c (equation 4) far-field l e v e l o f concentration interfacial average concentration (equation 16) dimensionless concentration field characteristic scale for concentration difference m o l e c u l a r difiusivity of solute i n m e l t


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97

Theory of Transport Processes b o u n d a r y of c o m p u t a t i o n a l d o m a i n fraction of b o u n d a r y adjoining m e l t - s o l i d interface fraction

e. fir) F (v,

dimensionless a p p l i e d b o d y force field

dimensionless meniscus shape function i)

b


of b o u n d a r y a d j o i n i n g solid surfaces

u n i t vector i n d i r e c t i o n of gravity u n i t vector i n ζ d i r e c t i o n (equation 5)

F.(i)

dimensionless surface force

g G

acceleration of gravity dimensionless axial t e m p e r a t u r e gradient dimensionless m e l t - c r y s t a l interface shape f u n c t i o n

Mr) fi H AH

dimensionless m e a n c u r v a t u r e of interface d i m e n s i o n a l m e a n c u r v a t u r e of interface latent heat of fusion refers to the m e l t - c r y s t a l interface w h e n u s e d as a subscript

{

I I

i d e n t i t y tensor e q u i l i b r i u m partition coefficient (equation 1)

h L*, m

effective segregation coefficient capillary l e n g t h , ( t f / p g ) m

L

1/2

characteristic l e n g t h scale refers to the m e l t phase w h e n u s e d as a subscript

m η

slope of l i q u i d u s c u r v e u n i t n o r m a l vector to m e l t - a m b i e n t meniscus

Ν Ο

u n i t n o r m a l vector to m e l t - c r y s t a l interface

Ρ Ρο

dimensionless pressure field reference pressure d i m e n s i o n a l pressure field

Ρ r

o r d e r of m a g n i t u d e

dimensionless radial coordinate

R s t

dimensionless crystal radius refers to the solid of crystal phase w h e n used as a subscript

i

dimensional time u n i t tangent vector to m e l t - a m b i e n t meniscus

dimensionless t i m e

t Τ t y* r

τ T

0

m

m T(x, t) ΔΓ ΔΓ* 1

V

u n i t tangent vector to m e l t - c r y s t a l interface d i m e n s i o n a l value of Τ (equation 5) characteristic t e m p e r a t u r e scale reference value of Γ melting temperature m e l t i n g t e m p e r a t u r e of p u r e m a t e r i a l dimensionless t e m p e r a t u r e

field

m a x i m u m t e m p e r a t u r e difference characteristic t e m p e r a t u r e difference d i m e n s i o n a l v e l o c i t y vector


MICROELECTRONICS PROCESSING; C H E M I C A L ENGINEERING ASPECTS

98

v(x, t) y*

field

ν, v V ( x , t)

characteristic v e l o c i t y scale g r o w t h rate v e l o c i t y scale for b u l k m o t i o n dimensionless v e l o c i t y of solid surfaces

X

vector of dimensionless coordinates

*•

t h e r m a l c o n d u c t i v i t y of phase i

X

d i m e n s i o n a l vector of coordinates

ζ

d i m e n s i o n a l axial distance t h e r m a l difiusivity of phase i coefficient of solutal expansion

0

s


dimensionless velocity

α* βο β. Ί Γ

coefficient of t h e r m a l expansion

δ

m e l t - c r y s t a l interfacial energy m e l t - c r y s t a l c a p i l l a r y l e n g t h , y/AH dimensionless stagnant-film thickness

Δ

dimensionless b o u n d a r y l a y e r thickness

η λ

b o u n d a r y layer coordinate

λ

(

P e / 3 P e (equation 26) reference value of λ s

0

g

Ρο

variable of integration m o m e n t u m diffusavity, μ / ρ N e w t o n i a n viscosity of m e l t characteristic rotation rate reference density o f m e l t

Pi σ

mass d e n s i t y o f phase i interfacial tension of m e l t - a m b i e n t meniscus

σ*

characteristic value of σ reference value of σ

ξ V μ Ω

σο τ Φ Φο 0

ω

deviatoric stress tensor angle b e t w e e n m e l t - a m b i e n t a n d c r y s t a l - m e l t surfaces reference value of φ refers to reference values w h e n u s e d as a subscript refers to d i m e n s i o n a l values of parameter w h e n u s e d o v e r a variable

Acknowledgments T h e w r i t i n g of this p a p e r a n d the research contributions of the crystal g r o w t h group at the Massachusetts Institute of T e c h n o l o g y d e s c r i b e d w i t h i n i t w e r e made possible b y grants f r o m the M i c r o g r a v i t y Sciences P r o g r a m of the N a t i o n a l A e r o n a u t i c s a n d Space A d m i n i s t r a t i o n , the Defense A d v a n c e d R e search Project A g e n c y , a n d the M o b i l F o u n d a t i o n a n d b y a T e a c h e r - S c h o l a r A w a r d from the C a m i l l e a n d H e n r y D r e y f u s F o u n d a t i o n to m e . I am d e e p l y i n d e b t e d to these sponsors. I a m also grateful to the m a n y colleagues w h o


2.

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99

have c o n t r i b u t e d to this paper t h r o u g h t h e i r research a n d discussions w i t h m e ; these i n c l u d e L . J . A t h e r t o n , S. R. C o r i e l l , J . J . D e r b y , D . - H . K i m , G . B . M c F a d d e n , P. H . Sackinger, P. D . T h o m a s , C . A . W a n g , M . J . W a r g o , a n d A . F . W i t t . I a m is especially grateful to E . D . B o u r r e t for h e r critical r e a d i n g o f the m a n u s c r i p t a n d to D . M a r o u d a s w h o h e l p e d w i t h t h e refer ences.


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