The modeling study of Czochralski (Cz) crystal growth is reported. The approach was to relate in a quantitative manner, using models based on first priniciples, crystal quality to operating conditions and geometric variables. The finite element method is used for all calculations

Dorsey, D.

Dudukovic, M. P.

Ramachandran, P. A.

Srivastava, R. K.

English

NASA Technical Reports Server

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ADVANCEDSILICONSHEET N8 ? " 1 6 4 1 2
"i CZOCHRALSKI CRYSTAL GROWTH: MC)DELING s:'rUDY
' CHEMICAL REACTION ENGINEERING LABORATORY
, DEPARTMENT OF CHEMICAL ENGINEERING
WASHINGTON UNIVERSITY
ST. LOUIS, MO
M. P. Dudukovic, P. A. Ramachandran, R. K. Srivastava,
and D. Dorsey
Summary of Accomplishments: Modeling the Cz Growth of Silicon
Modeling Approach
During the past eighteen months the CREL group at Washington University
- has established itself as a leader i_. mathematical modeling of CZ growth
of silicon. The unique approach of this modeling group was to relate
in a quantitative manner, using models based on first principles,
crystal quality to operating conditions and geometric variables, Superior
quality Sl crystals need to be of uniform diameter, of high, axially
and radially uniform, resistivity and defect free. The resistivity
profiles can readily be related to imbedded dopant and impurity concen- i
i
trations, while the dislocations are a strong function of the interface !
shape and residual stresses in the crystal. In turn the dopant dis-
; tributions, interface shape and diameter are governed by the heat balance
at the interface. Thus, it can be . lcluded that crystal quality can
directly be related, in a quantitative way, to the hydrodynamics, mass and
heat transfer in the melt and to heat transfer in and from the crystal.
_;=ese phenomena are dependent on: geometric variables which are fixed
for a g£ven puller (e.g., crucibl_ and enclosure shape, heater shape,
/
etc.), process variables which are governed by the selected operating
'_ schedule (e.g., melt depth, height of exposed crucible vail, etc.),
manipulated variables which could _e used for control (e.g., heater power,
) ,
]987006957-3]8
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crystal and crucible rotation speed, crystal pulling rate, heater position,
gas flow rate, etc.). The variables to be controlled are crystal diameter
and interface shape since they directly affect dopant distribution and
I
residual stresses.
The systematic modular approach to this complex problem developed
at CKEL is shown schematically in Figure i. The sketch of the puller is
showing Figure 2. The finite element method is used for all calculations.
The heat balance at the interface, whicb is the key element that governs
crystal quality can be stated as:
(Heat of Solidification) x (Rate of Solidification) -
(Heat Loss into - (Heat Transfer from
Crystal from the Melt to the Inter-
Interface) face)
aT_ a T
oV G AHcos _ - (- kc _n - (- km_ n) (i)
I - II - III
Here term II is given by the heat transfer model for the crystal while
term III is determined by the hydrodynamic and heat transfer model for
the melt. Clearly, terms II and III dictate the interface shape through
angle _ and achievable pulling rate VG at the fixed diameter. Equation
(i) allows the general problem to be approached in a modular, sequential
fashion. The rigorous mo.-.l for the heat transfer in the crystal can be
solved first and coupled with an approximate model for the melt. Melt
models can be gradually improved and incorporated into the overall
model as shown in Figure I.
The above approach has been used successfully by the CREL group in
modeling CZ of silicon and it should be directly extrapolable to modeling
of GaAs.
320
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1987006957-319
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Major Aq¢ompltshme',ts
i. A rigorous model based on first principles was established to predict
_ the heat transfer in the crystal and the interface shape [Ramachandran ,_
_. and Dudukovlc, J. Crystal Growth 71 (1985): 399-408]. The model is the
first to have considered properly both direct and reflected radiation in
the crystal enclosure. The model established two dimensionless parameters:
/"
_., a dimensionless flux at the interface and a Blot number as the key factors
in determining the interface shape. This is confirmed by experimental
data. The effect of Including reflected radiation on axial temperature
profiles is shown :inFigure 3. The effect is even more dramatic for inter-
: face shape.
2. A coupled problem for the puller was solved based on an approximate
conductor dominated heat transfer in the melt [Srlvastava, Ramachandran
and Dudukovic, J. Crystal Growth 73 (1985); 487-504]. Both, the Laplace-
Young equa_l_c _or the melt meniscus shape and Gebhart's enclosure theory
for radiation ,Jere used. This paper provides a detailed account for the
effect of all variables an interface shape. It is shown that reflected
>
radiation is Important ezpeclally in the vicinity of the interface.
3. Radiation view factors for short crystals during necking and shoulder
growth ar_ reported for the first time [Srivastava, Ramachandran and
Dudukovic, J. Crystal Growth (i_86) in print]. These are essential in
simulating properly the critical initial stages of growth.
4. The detailed model developed and described under item 2 was simplified.
Useful simple relations were developed for growth rate and interface
shape as shown in Figure 4. The various parameters A, B, b, CI, C2, C3
i.._ can be correlated with operating conditions. The simple models can then
,,r
o me"
be run on line. [Srivastava, Ramachandran and Dudukovic, J. Electrochem.
I'
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_I 321
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1987006957-320
} ADVANCED SILICON SHEET _
i Soc. Solid State Sci. Techn. 133 (1985): 1007-1015]. T_e success of thei
simple models in matching the predictions of _he detail models for growth
_J
rate versus radius and for interface shape versus radius are shown in
Fl_ure 5 and 6, respectively.
5. A novel concept of improving the controllability of the diameter and
/i achieving decoupled diameter and interface shape control is introduced
and numerically demonstrated [$ri_astava, Ramachandran and Dudukovlc,
._ J. Crystal Growth (1986) in print]. The concept relies on regulating
the convecting cooling component of the crystal by usin_ gas Jets.
Varla¢ions of gas flow rate or gas pressure can also be used. Predicted i
: increases in productivity with increased Jet coollng are illustrated in
Figure 7.
6. A preliminary steady state model fcr simulation of the hydrodynamics
i' of the melt has been established [Dorsey, Ramachandran and Dudukovic, AIChE
One Day Syrup., St. Louis, April, 1985].
!
k
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1987006957-321
, ADVANCED SILICON SHEET ORIG_JAL ;_ ,_+
OF POORQuALrr_
List of Publications Resulting from the Cz Modeling Work
at CREL at Washin_ton University
, I. Ramachandran , P. A. and M. P. Dudukovi6, "Simulation of Temperature
, Distribution in Crystals Grown by Czochralskl Method", J. Crystal
Growdh 71 (!985): 399-408.
2. Srlvastava, R. K., P. A. Ramachandran and M. P. Dudukovi_, "Interface
Shape in Czochralski Grown Crystals: Effect of Conduction and
•_,_'i_ Radla_ion".J.C_ys_alGrowth73(1985):4S7-S04.
3. Srivastava, R. K., P. A. Ramachandran and H. P. Dudukovi_, "Radiation
View Factors in Czochralski Cry_tal Growth Apparatus for Short
Crystals", J. Crystal Growth (1986). In print.
4. Srlvastava, R. K., P. A. Ramachandran and M. P. Dudukovi_, "Czochralskl
Growth of Crystals: Simple Models for Growth Rate and Interface Shape",
" J. Electrochem. Soc.: Solid-State and Technology 133 (1985):
1009-1015.
5. Srlvastava, R. K., P. A. Ramachandran and M. P. DudukovIE, "Simulation
of Jet Cooling Effects on Czochralskl Crystal Growth", J. Crystal
Growth (1986). Accepted for publlcatlon.
6. Dorsey, D., P. A. Ramachandran and M. P. Dudukovi_, "Numerical
S_mulatlon of the Hydrodynamics and Heat Transfer in Czochralski-
Melts: Preliminary Results", AIChE One Day Symp., St. Louis,
April 17, 1985.
+
- Overall Schematic of Cz Modeling
Geo_crtc I Hlnzpulated Input Varlables: Selected Operatznq
Vlriab},lll I -Cructble temp., T c $chllule lind Process
- Pill. l-rate, Vp V4rL4b|es
- Heat transfer Ln qas, ha
et¢.
Y
- Hydrodynam.cs of the Melt
- Heat and MasS Tranl{er Ln MII
- Hilt TransEct in and {_Om CL'_ItI].
/ to 1_ Con_.roltech "1 Induced Stresses, rl ,i
;_ - Crystal dLamter
.... Inter t_¢l shape
• " * Dopant "
diltribution Outpdt Variables to
..'-r be control led=
- dLllOCltionl
. - tvlnn_n 9
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1987006957-322
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Schematic of Cz Single Crystal Puller
.- (---Indicates Radiation Interaction Between Various Surfaces)
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d_,
Enclosing
/ Walls
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Grids
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• d 4
CRYSTAL
' Exposed Portion '
o_ Crucible _'
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Si =elt
T /f
Crucible Be=tter
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1987006957-323
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PredictedCrystal Surface TemperaturesBasedon Gebhartand
Stefan Modelsand ExperimentalResldtsof Will,ares and RPuser
[from Ramachandranand Dudukovic,
J. Crystal Growth (1985)" 399-408]
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NOD[L PIIZDICTIONS (GElltART)
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AXIAL DISTMICZ FROH IELT SURFACE, c,*
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Simple Models for Growth Rate and Interface Shane
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Comparison of Pulling Rate Versus Crystal Radius
(V Versus R)
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# Reynolds A(cuZ/hr) S(cm/hr) era s
Humber
1 - 125.5 18.6 C.78
it 70 2 100 128.9 18.0 O. 79
• 3 500 132.1 15.5 0.83
"l " 4 1000 137.9 15.5 0.88
; \ 5 1500 142.7 14.5 0.91
it i \" 5 3000 159.4 13.1 O. 89
80! \ 7 6000 182.3 10.1 O. 98
\ 8 9 98.6 7.4 1.07
_J _ \ 9 12000 217.5 5 44 1.14
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_, Variation of Interface Shape with Crystal Radius
, and Jet-Cooling
, 1..5 4.
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1987006957-326
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Variation of Growth Rate with Crystal Radius
.,- _ at Different Levelsof Jet-Cooling
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2 100 8.567 x 10 -4
L 3 500 1.456 x 10-3
4 1000 2.0t, O x 10"3
•_' _ 1500 2.._$3 x 10-3
_,! eC 3000 3.896 x 10 -3
• 7 6000 6.178 x 10-3
_ : 9000 8.199 x 10 -312000 1.007 x 10-3
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HIGH-EFFICIENCY SOLAR CELLS ....
Paul Alexander, Chairman
r
Six presentations were made at this session on progress in high-efficiency
solar cell research.
'_ The presentation by the University of California, Los Angeles (UCLA), was
on optimization methods for solar cell numerical models. A description was
_i given on optimizing algorithms of solar cell models to simultaneously vary
design variables such as impurity concentrations, front junction depth, back
junction depth, and cell thickness to maximize predicted cell efficiency. The
optimization methodology was demonstrated on Solar-Cell Analysis Program in One
Dimension (SCAPID), which is a computer program written by Purdue University,
that relates a speci[ic solar cell efficiency to a given set of input
conditions such as cell thickness, impurity concentration, junction depth, etc.
The UCLA optimization method, by iterating a set of input conditions from
SCAPID, provided a maximum solar cell efficiency related to ranges of input
conditions that were derived from the SCAPID inputs. UCLA presented curves of
computer runs that demonstrated their optimization techniques.
The University of Pennsylvanla described their work in identifying,
developing, and analyzing techniques for measuring bulk recombination rates,
and surface recombination velocities and rates in all regions of high-
efficiency silicon solar cells. Professor M. Wolfe described their recent
work of improving the accuracy of their previously developed DC measurement
: system by adding blocked interference filters. The system has been further
automated by writing software that completely samples the unknown solar cell
regions with data of numerous recombination velocity and lifetime pairs. The
results can be displayed in three dimensions and the best fit can be found
: numerically using the simplex minimization algorithm. They also described a
/ theoretical methodology to analyze and compare existing dynamic measurement
techniques.
Stanford University presented their work on carrier transport and
recombination parameters in heavily doped silicon. They presented data of
minority carrier diffusivity in both p- and n-type heavily doped silicon
covering a broad range of doping concentrations from 1015 to 1020 atoms
per cm 3. One of the highlights of their results showed that minority
carrier diffusivities are higher by a factor of 2 in silicon compared to
majority carrier diffusivities.
Pzofessor R. Swanson, of Stanford University, described the construction
of a _2.2_ efficient single-crystal silicon solar cell fabricated at Stanford.
The cell dimensions were 3 x 5 mm and I00 _m thick with a base lifetime of
500 Bs. The cell featured light trapping between a texturized top surface and
_/ a reflective bottom surface, small point contact diffusions, alternating between
,; n-type ,_d p-type in a polka-dot pattern on the bottom surface, and a surface
.... passivation on all surfaces between contact regions. A Vo& of 0.681 V, a
_'_ Jsc of hi,5 mA/cm 2, and a fill factor of 0.786 was reported for the 22.2Z
i_ efficient cell.
I
--_ 329
l
]987006957-328
j
_L HIGH-EFFICIENCY SOLAR CELLS
The University of Washington presented their work on investigating tt,e
passivation of silicon surfaces using SiN x. They used high-frequency
capacitance voltage (CV) measurements to determine the interface state density
at midgap (Vss). They indicated that the SiNx/Si interface ey-hibits values
t of Dss larger for n-type substrates than for p-type substrates and that,
further, after heat treatment, this diffez=,,ce becomes even more pronounced.
Also, it was found that the interface state density for SiNx/P-type silicon is
essentially independent of dopant concentration, and the Dss for SiNx/n-type
silicon interface increases from 1012 to 1013 states cm-2 eV-I as the
phosphorus concentration increascs from 1016 to 1018 cm-3. These results
suggest that SiNx can passivate a P+/n cell more effectively than an n+/p
cell. The University of Washlngcon also indicateJ that they are working on
_ special structures that will allow measurement of both surface _ecombination
velocity and interface state density.
Westinghouse described their work on achieving higher efficiency cells by
directin_ efforts toward identifying carrier loss mechanisms, designing new cell
structures, and developing new processing techniques. Their work, using
techniques such as deep-level transient spectroscopy (DLTS), laser-beam-induced
current (LBIC), and transmission electron microscopy (TEM), indicated that
' dislocation_ in web material rather than twin planes were primarily responsible _ i
for limitin 8 diffusion lengths in web. They described t_eir work in hydrogen i
implantation which indicated that they can improve lifetimes and cell
efficiencies from 19 to 120 tan,and 8 to I0.3_ (no AR), respectively, by
implanting hydrogen at 1500 eV and a beam current density of 2.0 mA/cm 2. Some
of their processing improvements included use of a double-layer AR coating
' (ZnS and MgF 2) and an addition of an aluminum back surface reflector. Cells
of more than 16% efficiency have been achieved.
l"
i
/
,P
e
330
9..... ' .. • ®I_ .,
1987006957-329


Czochralski crystal growth: Modeling study

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