- Email: [email protected]
Orientation dependence of the distribution coefficient obtained from a spin-1 Ising model
j. . . . . . . .
CRYSTAL G R O W T H
Journal of Crystal Growth 174 (1997) 28-34
ELSEVIER
Orientation dependence of the distribution coefficient obtained from a spin-1 Ising model K.M. Beatty*, K.A. Jackson Department of Materials Science and Engineering. University of Arizona, Tucson, Arizona 85721, USA
Abstract
The nonequilibrium distribution coefficient (kneq) as a function of solid-liquid interface velocity and orientation was investigated for a spin-1 Ising model for binary alloys using Monte Carlo computer simulations. The crystal structure and thermodynamic properties were chosen to correspond to bismuth doped silicon with keq = 7 × 10 -'~. Values for kne q w e r e obtained for several orientations of the solid/liquid interface, including (1 1 1) and (0 0 1). For the same growth velocity, kneq was found to be greatest for solid/liquid interfaces parallel to the (1 1 1) plane. The orientation dependence is related to variations in the kink site density at the interface. The simulation results are compared with experimental results reported by Aziz et al. PACS: 02.70.Lq; 05.70.Ln; 64.70.Dv; 64.75. + g; 81.10.Aj Keywords: Segregation; Distribution coefficient; K-valve; Orientation; Facet effect; Non-equilibrium; Ising model;
Monte Carlo; Solidification
I. Introduction
The composition of the solid formed during the solidification of a binary liquid depends on transport processes in the melt and on the rate at which the two species join and leave the solid. Nonequilibrium segregation effects related to solidliquid interfacial rate processes include the 'facet effect' in Czochralski growth of silicon and InSb [1,2], sectoring in solution growth, and near
* Corresponding author.
diffusionless transformations in rapid solidification. Several models have been developed for this phenomena [3-10]. Comprehensive experimental work on the orientation dependence of the segregation coefficient has been done by Aziz et al. [9-11]. They have studied the orientation dependence of the segregation coefficient for Si doped with Ga, In, Sn, As, Sb, and Ge [12-14]. They have used the aperiodic step growth model (ASGM) to fit their data. Although this model provides a satisfactory fit to most of their results, it involves two adjustable parameters that cannot be predicted a priori and that differ markedly for the various dopants.
0022-0248/97/$17.00 Copyright ((~) 1997 Elsevier Science B.V. All rights reserved PII S0022-0248(96)01 057-3
K.M. Beatty, K.A. Jackson/Journal of C~stal Growth 174 (1997)28 34
Jackson et al. [-15, 16] have studied the nonequilibrium segregation coefficient in a spin-1 Ising model (which will be referred to as the JGT model). The JGT model exhibits an orientation-dependent segregation coefficient for a 2-D square lattice with bond energies appropriate for Bi doped silicon [-17]. The segregation coefficient is higher on the smooth face (the (1 0) face for a square lattice) in accord with experimental results. In this paper, the segregation coefficient has been determined for the JGT model using thermodynamic properties of Bi doped silicon. The Monte Carlo simulations were done for the (1 1 1), (2 2 3), (1 1 2), (2 2 4), and (1 0 0) faces of the diamond cubic structure in order to compare directly with experiments of Aziz et al.
2. Description of the JGT model The JGT model is a spin-1 Ising model where there are four possible states for each lattice site (as opposed to two possible states for a spin-½ Ising model). The rate equations for the model give the frequency at which different types of events occur on the lattice. The frequency with which liquid atoms convert to solid atoms is given by v/Ls= v, e x p ( - ~ . ~ ) e x p ( - - -
4'ij\
(')
The frequency with which atoms change from solid to liquid is given by _ 4)ij \ US L : l'n exp -- ~ ]£~B~),
(2)
where 4)~j is the energy of the bond between atom i and atom j, AS the entropy of fusion, kB the Boltzmann's constant, and T the temperature. The energy of the bond between two atoms depends only on the types of atoms involved and the relevant phase diagram. (In the present paper the liquid is assumed to be an ideal solution, whereas the solid is assumed to be a regular solution.) The four atom states are: solid of species A; solid of species B; liquid of species A; and liquid of species B. v, is the normalizing frequency, which determines the time scale. Diffusion occurs as the exchange of a liquid atom with one of its liquid
29
neighbors. The frequency of a diffusive jump is given by F. This gives a diffusion coefficient [18] Fa 2 D-
2d'
(3)
where a is the nearest neighbor distance and d is the dimensionality of the system. We define a kink site as a liquid site at the solid--liquid interface with 1 liquid neighbors and ½ solid neighbors.
3. JGT model parameters for Bi doped silicon In order to compare the simulation results with experiment, the entropy of fusion, growth rates, and the diffusion coefficient must be correlated between the two. The roughening transition for melt growth is given by the Jackson ~-factor, surfaces for which > 2 are smooth, and for which ~ < 2 are rough. Using a spin ½Ising model, Van Enkenfort and Van Der Eerden found the roughening transition for the diamond cubic (1 l 1) face at L / k T ~ ~ 5.4 [-19]. Scaling from this result and the entropy of fusion for silicon [20J the appropriate L/kTM to model silicon in lsing model simulations is found to be approximately 7.2. This value gives reasonable results for the relative growth rates for Si normal to the (1 1 1) and normal to the (1 0 0) face. This comparison is a good test for the entropy of fusion value used in the simulations [21, 22]. These results will be reported elsewhere. Using the scaled entropy of fusion, the bond strengths are set to give the correct melting point for silicon, and also the correct equilibrium segregation coefficient (7 x 10 4). All liquid-solid bond energies are assumed to be zero. Experimentally the maximum growth rate of silicon on the (1 00) face is about 17m/s [23]. From molecular dynamics modeling Gilmer et al. found a corresponding liquid diffusion coefficient, D, as a function of temperature of the form Do e x p ( - 6 5 0 0 / T ) [24]. In order to scale the model growth rates to experimentally determined silicon growth rates, we assume the arrival frequency at a kink site has the temperature dependence given by Gilmer for diffusion, and a maximum velocity of 17 m/s.
30
K.~L Beatty, K.A. Jackson/Journal of Crystal Growth 174 (1997) 28 34
In the J G T model, the diffusion jump distance in the liquid is the nearest neighbor distance, a, whereas in a real liquid the diffusional mean free path, A, is much shorter. However at the interface, when an atom makes a short diffusional motion, A, to join the crystal, the interface advances a distance a. As a result, the free path in the liquid enters into the relationship between the diffusion coefficient and the growth rate as v, = aD/A 2. In the J G T model A = a. In order to account for this effect, the diffusional jump rate in the simulations has been decreased by a factor of 0.14 to fit the experimental nonequilibrium segregation data. This is the same as the value of A/a used by Grabow et al. [22] to fit both molecular dynamics and experimental data for the silicon (1 0 0) growth rates to the conventional equation for the growth rate of a crystal.
4. Determination of the segregation coefficient It might appear to be a simple task to calculate the nonequilibrium distribution coefficient, kneq, for the computer simulations since the position of all the atoms is known. However, Jackson et al. showed that for diffusionless growth the concentration of liquid at the interface greatly exceeds the bulk liquid concentration. Calculating the segregation coefficient by dividing the concentration of the solid grown by concentration of the liquid at the interface gives a segregation coefficient much less than 1 for diffusionless growth r15], whereas the segregation coefficient must be 1, since none of the atoms move. To circumvent this problem, the concentration of the liquid was extrapolated to the interface. The extrapolated interface concentration, C~(0), can be found from a least squares fit of the equation:
,4, where CL(x) is the average concentration of the liquid at a distance x from the first liquid layer, C~ is the far-field concentration of the liquid, and 2 is the boundary layer width. For steady-state growth, 2 should be equal to D/v; at short times a reasonable approximation is 2~x/-D~. This
method can be used to evaluate the liquid concentration even under transient conditions, but it is sensitive to small changes in concentration when 2 is small. However, when 2 is small, it is possible to achieve a steady state diffusion profile in the simulations. In this case concentrations can be modeled by matching the flux into and out of the first liquid layer at the interface:
C~(O) = CL -~ (CL -- Cs)6 D/v '
(5)
where C[ is the concentration of the second liquid layer, and Cs is the concentration of the solid grown. For 6 we use half the distance between the atoms in the second layer of liquid and their nearest solid atom. The segregation coefficient, kneq, w a s calculated from the extrapolated concentration of the liquid at the interface as C~(0) and Cs. The concentrations in the liquid were evaluated by averaging at least 50 snapshots of the distribution of atoms in the liquid. CLE(0) and Cs were evaluated at four separate intervals in each run. For the last two intervals the growth velocity, C[(0), and Cs were relatively constant. The values for these last two intervals were averaged over four separate runs for each orientation and temperature. The average of these eight measurements was used to determine the velocity and the segregation coefficient. Large lattices (approximately 400 000 lattice sites) were used to minimize inaccuracies due to fluctuations in Cs and CLE(0).
5. Results and discussion Fig. 1 is a plot of the segregation coefficient for bismuth doped silicon as a function of velocity for growth normal to the (1 1 1), (2 2 3), (1 1 2), (1 1 4) and (1 0 0) faces. The magnitude of the segregation coefficient increases with the smoothness of the face. The substantial variation of the segregation coefficient with orientation is in agreement with previous results of Jackson et al. [17]. Fig. 2 is a plot of the k-value versus orientation for growth rates of 5 and 1.7 m/s; results from the Ising model simulations as shown together with the experimental data of Aziz et al. The orientation
K.M. Beatty. K.A. Jackson/Journal qfl Co,stal Growth 174 (1997) 28--34 0.6
I
I (111)
0.5
I
I
31
l
•
(223)
o
(112)
i
(114)
n
(100)
•
/ / / / l
0.4
0.3
>
0.2
0.1
0 0
1
2
.3
I
I
4
5
Velocity ( m / s ) Fig. 1. k-value for several orientations as a function of velocity for the spin-1 Ising model. The lines serve to guide the eye. The lines are drawn t h r o u g h keq = 7 × 10 ,t at zero velocity.
0.6
I
I
I
I
I
JGT Model Simulation 1.7 m / s - 5 m/s Aziz 1.7 m / s • Aziz 5 m / s O
0.5
0.4
>
0.3
0.2 o
o
0.1 m
o
I o
1o
I
I
I
:~ 40 Degrees from ( l l 1) 20
I
so
Fig. 2. k-value for different orientations for growth rates of 1,7 and 5 m/s. The orientations included are (1 1 1), 12 2 3), [I 1 2), t 1 1 4) and (1 00), which are at inclinations of 0 , 11.4, 1 9 . 5 , 3 5 . 2 , and 54.7 to the (1 1 1), respectively.
dependencies obtained from the experimental and simulation data are consistent. The velocity dependence of the segregation coefficient for bismuth doped silicon, however, does not agree with the data of Aziz et al. [12, 25]. The presence of this
discrepancy, and the fact that it cannot be eliminated by changing the diffusion coefficient in the simulations, has been reported previously by Jackson et al. [17]. Although the reason for the discrepancy is not clear, it is known that segregation
K.M. Beatty, K.A. Jackson/Journal of Crystal Growth 174 (1997) 28-34
32
coefficients as determined from experimental results depends strongly on the diffusion coefficient which is assumed for the impurity in the liquid [26]. In the case of the diffusion of bismuth in a silicon melt, the diffusion coefficient is inferred from laser melting experiments [25].
I''""!
......... •
Fig. 3 is a plot of the kink site density, p, as a function of temperature and solid-liquid interface orientations. The kink site density, p, is defined here as the number of liquid atoms at the solid-liquid interface which are bonded to two solid atoms in an area ao2, where ao is the diamond cubic lattice
I
I
I
I
..........D....i....D ....................• ....................• ....................• ....................• ....................•
0.8
¢0
"Fn
0.6
0.4
EI"'""I2P'"
f"-""'O
0 ...........El......fiil......eI.....................0 .....................0 .....................{El-....................Q .....................[]
.......... • ........... @ ' " ' " O " " " g
.....................
U .....................
" 0 ..................... O .................... • ..................... @
0 " ' " ' ~ ..........~' ..........~'-'"0 ...................
.. &.....................i ....... ,...--~ .............
...........&.........
~......A- .........
0.2 .&.... & _ _ _ . . . ~ .......... - i ¢ "
J
I
t
1650
1600
1550
o 1700 ~ M.P.
{111)
•
{22a)
o
(112)
•
(114)
[]
(100)
•
I 1500
Temperature (K)
Fig. 3. K i n k site d e n s i t y (the n u m b e r o f k i n k sites in a n a r e a a02 o f t h e i n t e r f a c e ) at d i f f e r e n t t e m p e r a t u r e s for s e v e r a l o r i e n t a t i o n s . T h e a r r o w m a r k s t h e m e l t i n g p o i n t o f p u r e silicon, 1688 K . T h e k i n k site d e n s i t y is a m e a s u r e o f s u r f a c e r o u g h n e s s . T h e lines s e r v e to g u i d e t h e eye.
o
i
I
I
I
I
-0.2 -0.4 -0.6 -0.8 -1
/~
(223) 0 (n2) •
_z6
-1.2
(114)
El
(]~o)
-1.4 -1.6 -2.6
i
J
J
-2.2
-1.8
-1.4
|oglo
[~'
-1
Uk]
Fig. 4. T h e s e g r e g a t i o n coefficient, k, as a f u n c t i o n o f t h e d i m e n s i o n l e s s v e l o c i t y , u, a n d t h e d i m e n s i o n l e s s k i n k site v e l o c i t y , Uk. T h e line is a g u i d e to t h e eye.
K.M. Beatty, K.A. Jackson / Journal ol +Crystal Growth 174 (l 997) 2 8 - 3 4
parameter. It is happenstance that p ~ 1 for the (1 0 0) face. The data of Fig. 3 provide a measure of surface roughness for different solid-liquid interface orientation and temperatures. The difference in roughness of the various faces diminishes with increasing undercooling. From Fig. 1 it can be seen that at larger growth velocities (i.e., larger undert_(lll)/t.(lOO) coolings) there is a decrease in the ,_~+:_ a u v ~,eq /'~,eq • This suggest that the value of this ratio is related to the kink site density. If one assumes that growth is due to the movement of kink sites, the kink site velocity can be defined as the growth velocity divided by the kink site density. On smooth surfaces the kink sites must move much faster to maintain a given growth velocity. It has been proposed that this difference in kink site velocity may be responsible for the differing levels of impurity incorporation on different faces [5]. (Such proposals are often based on the motion of steps, but the arguments are similar.) If the growth is due only to the motion of kink sites, the maximum growth velocity normal to a surface is v+pfL where v + is the arrival rate at a kink site, p is the kink site density at the surface, and O is the volume per atom. The growth rate is greatest on the roughest face where p = Or (which we take to be l/ao from Fig. 3). From the growth rate on the roughest face, we can construct a dimensionless velocity u = v/v+prO, where t is the growth velocity. We can also construct a dimensionless kink site velocity Uk =t:/V+p[2. Fig. 4 shows that the segregation coefficients collapses to a master curve when plotted versus the product of u and Uk. There are large systematic deviations (related to p) from a single curve if the segregation coefficient is plotted against either u or Uk.
6. Conclusions The spin-1 Ising model used by Jackson et al. (JGT model) with a diamond cubic structure exhibits a segregation coefficient which increases with growth velocity and surface smoothness in agreement with prior simulations and with experiment. The segregation coefficient for growth normal to the (1 0 0), (1 1 4), (1 1 2), (2 2 3), and (1 1 1) faces (listed in order of increasing smoothness) increased
33
with the smoothness of the face. For the simulations completed in this study the segregation coefficient can be accurately determined from the product of the dimensionless velocity and the dimensionless kink site velocity. As found in former studies, the velocity dependence of the segregation coefficient is not in accord with the results of Aziz, but it is in accord with the data of Baeri et al. and White et al. [12, 25]. The reason for this discrepancy is unclear. The orientation dependence of the segregation coefficient obtained from the simulations agrees with the experimental results of Aziz et al. [14].
Acknowledgements This work was supported by NASA.
References [1] K.F. Hulme and J.B. Mullin, Philos. Mag. 4 (1959) 1286. [2] J.A. Burton. Prim and Slichter. J. Chem. Phys. 21 24/25 (1953} 1987. [3] A. Trainor and B.E. Bartlett, Solid State Electronics 2 (1961) 106. [4] P.J. Hohnes, J. Phys. Chem. Solids 24 (1963) 1239. [5] T. Abe, J. Crystal Growth 24/25 (1974) 463. [6] R.N. Hall, Phys. Rev. 88 {1952) 139. [7] K.A. Jackson, G.H. Gilmer and H.J. Leamy, Laser and Electron Beam Processing of Materials IAcademic Press, New York, 1980). [8] R.F. Wood, Appl. Phys. Lett. 37 (19801 302. [91 M.J. Aziz, J.Y. Tsao, M.O. Thompson, P.S. Peercy and C.W. White, Phys. Rev. Lett. 56 {19861 2489. [10] M.J. Aziz and T. Kaplan, Acta Met. 36 (1988) 2335. [11] L.M. Goldman and M.J. Aziz, J. Mater. Res. 2 (1987) 5241. [121 P. Baeri, G. Foti, S.U. Campisano, J.M. Poate and A.G. Cullis, Appl, Phys. Lett. 38 {1981} 800. [13] R. Reitano, P.M. Smith and M.J. Aziz, J. Appl, Phys. 76 (19941 1518. [14] M.J. Aziz and C.W. White, Phys. Rev. Left. 57 (1986) 2675,
[15] K.A. Jackson, G.H. Gilmer, D.E. Temkin, J.D. Weinberg and K. Beatty, J. Crystal Growth 128 (1993) 127. [16] K.A. Jackson, G.H. Gilmer and D.E. Temkin. Phys. Rev. Lett. 75 (1995) 2530. [17] K.A. Jackson. G.H. Gilmer, D.E. Temkin and K.M. Beany, J. Crystal Growth 163 (19961 461. [18] P. Shewmon, Diffusion in Solids 2nd ed. (Minerals Metals and Materials Society, Warrendale. PA, 1989).
34
ICM. Beatty, K.A. Jackson/Journal of Crystal Growth 174 (1997) 28-34
[19] W.J.P. Van Enckevort and J.P. Van De Eerden, J. Crystal Growth 47 (1979) 501. [20] R. Hultgren, P. Desai, D.T, Hawkins, M. Gleiser, K.K. Kelly and D.D. Wagman, Selected Values of Thermodynamic Properties of The Elements (American Society for Metals, Metals Park, OH, 1973). [21] G.H. Gilmer, MRS Symp. Proc. 13 (1983) 249. [22] M.H. Grabow, G.H. Gilmer and A.F. Bakker, MRS Symp. Proc. 141 (1989) 349. [23] J.A. Yater and M.O. Thompson, Phys. Rev. Lett. 49 (1982) 1496.
[24] M.H. Grabow, G.H. Gilmer and A. F. Bakker, Molecular dynamics simulations of rapid solidification of silicon, in: Int. Workshop on Computational Materials Science (1990) pp. 27-47. [25] L.D. Hess, J.F. Gibbons and T.W. Sigmon, Eds., Laser and Electron Beam Solid Interactions and Materials Processing, MRS Symp. Proc., Vol. 1 (North-Holland, Amsterdam, 1981). [26] D.P. Brunco, M.O. Thompson, D.E. Hoglund, M.J. Aziz and H.J. Gossman, J. Appl. Phys. 78 (1995) 1575.
CRYSTAL G R O W T H
Journal of Crystal Growth 174 (1997) 28-34
ELSEVIER
Orientation dependence of the distribution coefficient obtained from a spin-1 Ising model K.M. Beatty*, K.A. Jackson Department of Materials Science and Engineering. University of Arizona, Tucson, Arizona 85721, USA
Abstract
The nonequilibrium distribution coefficient (kneq) as a function of solid-liquid interface velocity and orientation was investigated for a spin-1 Ising model for binary alloys using Monte Carlo computer simulations. The crystal structure and thermodynamic properties were chosen to correspond to bismuth doped silicon with keq = 7 × 10 -'~. Values for kne q w e r e obtained for several orientations of the solid/liquid interface, including (1 1 1) and (0 0 1). For the same growth velocity, kneq was found to be greatest for solid/liquid interfaces parallel to the (1 1 1) plane. The orientation dependence is related to variations in the kink site density at the interface. The simulation results are compared with experimental results reported by Aziz et al. PACS: 02.70.Lq; 05.70.Ln; 64.70.Dv; 64.75. + g; 81.10.Aj Keywords: Segregation; Distribution coefficient; K-valve; Orientation; Facet effect; Non-equilibrium; Ising model;
Monte Carlo; Solidification
I. Introduction
The composition of the solid formed during the solidification of a binary liquid depends on transport processes in the melt and on the rate at which the two species join and leave the solid. Nonequilibrium segregation effects related to solidliquid interfacial rate processes include the 'facet effect' in Czochralski growth of silicon and InSb [1,2], sectoring in solution growth, and near
* Corresponding author.
diffusionless transformations in rapid solidification. Several models have been developed for this phenomena [3-10]. Comprehensive experimental work on the orientation dependence of the segregation coefficient has been done by Aziz et al. [9-11]. They have studied the orientation dependence of the segregation coefficient for Si doped with Ga, In, Sn, As, Sb, and Ge [12-14]. They have used the aperiodic step growth model (ASGM) to fit their data. Although this model provides a satisfactory fit to most of their results, it involves two adjustable parameters that cannot be predicted a priori and that differ markedly for the various dopants.
0022-0248/97/$17.00 Copyright ((~) 1997 Elsevier Science B.V. All rights reserved PII S0022-0248(96)01 057-3
K.M. Beatty, K.A. Jackson/Journal of C~stal Growth 174 (1997)28 34
Jackson et al. [-15, 16] have studied the nonequilibrium segregation coefficient in a spin-1 Ising model (which will be referred to as the JGT model). The JGT model exhibits an orientation-dependent segregation coefficient for a 2-D square lattice with bond energies appropriate for Bi doped silicon [-17]. The segregation coefficient is higher on the smooth face (the (1 0) face for a square lattice) in accord with experimental results. In this paper, the segregation coefficient has been determined for the JGT model using thermodynamic properties of Bi doped silicon. The Monte Carlo simulations were done for the (1 1 1), (2 2 3), (1 1 2), (2 2 4), and (1 0 0) faces of the diamond cubic structure in order to compare directly with experiments of Aziz et al.
2. Description of the JGT model The JGT model is a spin-1 Ising model where there are four possible states for each lattice site (as opposed to two possible states for a spin-½ Ising model). The rate equations for the model give the frequency at which different types of events occur on the lattice. The frequency with which liquid atoms convert to solid atoms is given by v/Ls= v, e x p ( - ~ . ~ ) e x p ( - - -
4'ij\
(')
The frequency with which atoms change from solid to liquid is given by _ 4)ij \ US L : l'n exp -- ~ ]£~B~),
(2)
where 4)~j is the energy of the bond between atom i and atom j, AS the entropy of fusion, kB the Boltzmann's constant, and T the temperature. The energy of the bond between two atoms depends only on the types of atoms involved and the relevant phase diagram. (In the present paper the liquid is assumed to be an ideal solution, whereas the solid is assumed to be a regular solution.) The four atom states are: solid of species A; solid of species B; liquid of species A; and liquid of species B. v, is the normalizing frequency, which determines the time scale. Diffusion occurs as the exchange of a liquid atom with one of its liquid
29
neighbors. The frequency of a diffusive jump is given by F. This gives a diffusion coefficient [18] Fa 2 D-
2d'
(3)
where a is the nearest neighbor distance and d is the dimensionality of the system. We define a kink site as a liquid site at the solid--liquid interface with 1 liquid neighbors and ½ solid neighbors.
3. JGT model parameters for Bi doped silicon In order to compare the simulation results with experiment, the entropy of fusion, growth rates, and the diffusion coefficient must be correlated between the two. The roughening transition for melt growth is given by the Jackson ~-factor, surfaces for which > 2 are smooth, and for which ~ < 2 are rough. Using a spin ½Ising model, Van Enkenfort and Van Der Eerden found the roughening transition for the diamond cubic (1 l 1) face at L / k T ~ ~ 5.4 [-19]. Scaling from this result and the entropy of fusion for silicon [20J the appropriate L/kTM to model silicon in lsing model simulations is found to be approximately 7.2. This value gives reasonable results for the relative growth rates for Si normal to the (1 1 1) and normal to the (1 0 0) face. This comparison is a good test for the entropy of fusion value used in the simulations [21, 22]. These results will be reported elsewhere. Using the scaled entropy of fusion, the bond strengths are set to give the correct melting point for silicon, and also the correct equilibrium segregation coefficient (7 x 10 4). All liquid-solid bond energies are assumed to be zero. Experimentally the maximum growth rate of silicon on the (1 00) face is about 17m/s [23]. From molecular dynamics modeling Gilmer et al. found a corresponding liquid diffusion coefficient, D, as a function of temperature of the form Do e x p ( - 6 5 0 0 / T ) [24]. In order to scale the model growth rates to experimentally determined silicon growth rates, we assume the arrival frequency at a kink site has the temperature dependence given by Gilmer for diffusion, and a maximum velocity of 17 m/s.
30
K.~L Beatty, K.A. Jackson/Journal of Crystal Growth 174 (1997) 28 34
In the J G T model, the diffusion jump distance in the liquid is the nearest neighbor distance, a, whereas in a real liquid the diffusional mean free path, A, is much shorter. However at the interface, when an atom makes a short diffusional motion, A, to join the crystal, the interface advances a distance a. As a result, the free path in the liquid enters into the relationship between the diffusion coefficient and the growth rate as v, = aD/A 2. In the J G T model A = a. In order to account for this effect, the diffusional jump rate in the simulations has been decreased by a factor of 0.14 to fit the experimental nonequilibrium segregation data. This is the same as the value of A/a used by Grabow et al. [22] to fit both molecular dynamics and experimental data for the silicon (1 0 0) growth rates to the conventional equation for the growth rate of a crystal.
4. Determination of the segregation coefficient It might appear to be a simple task to calculate the nonequilibrium distribution coefficient, kneq, for the computer simulations since the position of all the atoms is known. However, Jackson et al. showed that for diffusionless growth the concentration of liquid at the interface greatly exceeds the bulk liquid concentration. Calculating the segregation coefficient by dividing the concentration of the solid grown by concentration of the liquid at the interface gives a segregation coefficient much less than 1 for diffusionless growth r15], whereas the segregation coefficient must be 1, since none of the atoms move. To circumvent this problem, the concentration of the liquid was extrapolated to the interface. The extrapolated interface concentration, C~(0), can be found from a least squares fit of the equation:
,4, where CL(x) is the average concentration of the liquid at a distance x from the first liquid layer, C~ is the far-field concentration of the liquid, and 2 is the boundary layer width. For steady-state growth, 2 should be equal to D/v; at short times a reasonable approximation is 2~x/-D~. This
method can be used to evaluate the liquid concentration even under transient conditions, but it is sensitive to small changes in concentration when 2 is small. However, when 2 is small, it is possible to achieve a steady state diffusion profile in the simulations. In this case concentrations can be modeled by matching the flux into and out of the first liquid layer at the interface:
C~(O) = CL -~ (CL -- Cs)6 D/v '
(5)
where C[ is the concentration of the second liquid layer, and Cs is the concentration of the solid grown. For 6 we use half the distance between the atoms in the second layer of liquid and their nearest solid atom. The segregation coefficient, kneq, w a s calculated from the extrapolated concentration of the liquid at the interface as C~(0) and Cs. The concentrations in the liquid were evaluated by averaging at least 50 snapshots of the distribution of atoms in the liquid. CLE(0) and Cs were evaluated at four separate intervals in each run. For the last two intervals the growth velocity, C[(0), and Cs were relatively constant. The values for these last two intervals were averaged over four separate runs for each orientation and temperature. The average of these eight measurements was used to determine the velocity and the segregation coefficient. Large lattices (approximately 400 000 lattice sites) were used to minimize inaccuracies due to fluctuations in Cs and CLE(0).
5. Results and discussion Fig. 1 is a plot of the segregation coefficient for bismuth doped silicon as a function of velocity for growth normal to the (1 1 1), (2 2 3), (1 1 2), (1 1 4) and (1 0 0) faces. The magnitude of the segregation coefficient increases with the smoothness of the face. The substantial variation of the segregation coefficient with orientation is in agreement with previous results of Jackson et al. [17]. Fig. 2 is a plot of the k-value versus orientation for growth rates of 5 and 1.7 m/s; results from the Ising model simulations as shown together with the experimental data of Aziz et al. The orientation
K.M. Beatty. K.A. Jackson/Journal qfl Co,stal Growth 174 (1997) 28--34 0.6
I
I (111)
0.5
I
I
31
l
•
(223)
o
(112)
i
(114)
n
(100)
•
/ / / / l
0.4
0.3
>
0.2
0.1
0 0
1
2
.3
I
I
4
5
Velocity ( m / s ) Fig. 1. k-value for several orientations as a function of velocity for the spin-1 Ising model. The lines serve to guide the eye. The lines are drawn t h r o u g h keq = 7 × 10 ,t at zero velocity.
0.6
I
I
I
I
I
JGT Model Simulation 1.7 m / s - 5 m/s Aziz 1.7 m / s • Aziz 5 m / s O
0.5
0.4
>
0.3
0.2 o
o
0.1 m
o
I o
1o
I
I
I
:~ 40 Degrees from ( l l 1) 20
I
so
Fig. 2. k-value for different orientations for growth rates of 1,7 and 5 m/s. The orientations included are (1 1 1), 12 2 3), [I 1 2), t 1 1 4) and (1 00), which are at inclinations of 0 , 11.4, 1 9 . 5 , 3 5 . 2 , and 54.7 to the (1 1 1), respectively.
dependencies obtained from the experimental and simulation data are consistent. The velocity dependence of the segregation coefficient for bismuth doped silicon, however, does not agree with the data of Aziz et al. [12, 25]. The presence of this
discrepancy, and the fact that it cannot be eliminated by changing the diffusion coefficient in the simulations, has been reported previously by Jackson et al. [17]. Although the reason for the discrepancy is not clear, it is known that segregation
K.M. Beatty, K.A. Jackson/Journal of Crystal Growth 174 (1997) 28-34
32
coefficients as determined from experimental results depends strongly on the diffusion coefficient which is assumed for the impurity in the liquid [26]. In the case of the diffusion of bismuth in a silicon melt, the diffusion coefficient is inferred from laser melting experiments [25].
I''""!
......... •
Fig. 3 is a plot of the kink site density, p, as a function of temperature and solid-liquid interface orientations. The kink site density, p, is defined here as the number of liquid atoms at the solid-liquid interface which are bonded to two solid atoms in an area ao2, where ao is the diamond cubic lattice
I
I
I
I
..........D....i....D ....................• ....................• ....................• ....................• ....................•
0.8
¢0
"Fn
0.6
0.4
EI"'""I2P'"
f"-""'O
0 ...........El......fiil......eI.....................0 .....................0 .....................{El-....................Q .....................[]
.......... • ........... @ ' " ' " O " " " g
.....................
U .....................
" 0 ..................... O .................... • ..................... @
0 " ' " ' ~ ..........~' ..........~'-'"0 ...................
.. &.....................i ....... ,...--~ .............
...........&.........
~......A- .........
0.2 .&.... & _ _ _ . . . ~ .......... - i ¢ "
J
I
t
1650
1600
1550
o 1700 ~ M.P.
{111)
•
{22a)
o
(112)
•
(114)
[]
(100)
•
I 1500
Temperature (K)
Fig. 3. K i n k site d e n s i t y (the n u m b e r o f k i n k sites in a n a r e a a02 o f t h e i n t e r f a c e ) at d i f f e r e n t t e m p e r a t u r e s for s e v e r a l o r i e n t a t i o n s . T h e a r r o w m a r k s t h e m e l t i n g p o i n t o f p u r e silicon, 1688 K . T h e k i n k site d e n s i t y is a m e a s u r e o f s u r f a c e r o u g h n e s s . T h e lines s e r v e to g u i d e t h e eye.
o
i
I
I
I
I
-0.2 -0.4 -0.6 -0.8 -1
/~
(223) 0 (n2) •
_z6
-1.2
(114)
El
(]~o)
-1.4 -1.6 -2.6
i
J
J
-2.2
-1.8
-1.4
|oglo
[~'
-1
Uk]
Fig. 4. T h e s e g r e g a t i o n coefficient, k, as a f u n c t i o n o f t h e d i m e n s i o n l e s s v e l o c i t y , u, a n d t h e d i m e n s i o n l e s s k i n k site v e l o c i t y , Uk. T h e line is a g u i d e to t h e eye.
K.M. Beatty, K.A. Jackson / Journal ol +Crystal Growth 174 (l 997) 2 8 - 3 4
parameter. It is happenstance that p ~ 1 for the (1 0 0) face. The data of Fig. 3 provide a measure of surface roughness for different solid-liquid interface orientation and temperatures. The difference in roughness of the various faces diminishes with increasing undercooling. From Fig. 1 it can be seen that at larger growth velocities (i.e., larger undert_(lll)/t.(lOO) coolings) there is a decrease in the ,_~+:_ a u v ~,eq /'~,eq • This suggest that the value of this ratio is related to the kink site density. If one assumes that growth is due to the movement of kink sites, the kink site velocity can be defined as the growth velocity divided by the kink site density. On smooth surfaces the kink sites must move much faster to maintain a given growth velocity. It has been proposed that this difference in kink site velocity may be responsible for the differing levels of impurity incorporation on different faces [5]. (Such proposals are often based on the motion of steps, but the arguments are similar.) If the growth is due only to the motion of kink sites, the maximum growth velocity normal to a surface is v+pfL where v + is the arrival rate at a kink site, p is the kink site density at the surface, and O is the volume per atom. The growth rate is greatest on the roughest face where p = Or (which we take to be l/ao from Fig. 3). From the growth rate on the roughest face, we can construct a dimensionless velocity u = v/v+prO, where t is the growth velocity. We can also construct a dimensionless kink site velocity Uk =t:/V+p[2. Fig. 4 shows that the segregation coefficients collapses to a master curve when plotted versus the product of u and Uk. There are large systematic deviations (related to p) from a single curve if the segregation coefficient is plotted against either u or Uk.
6. Conclusions The spin-1 Ising model used by Jackson et al. (JGT model) with a diamond cubic structure exhibits a segregation coefficient which increases with growth velocity and surface smoothness in agreement with prior simulations and with experiment. The segregation coefficient for growth normal to the (1 0 0), (1 1 4), (1 1 2), (2 2 3), and (1 1 1) faces (listed in order of increasing smoothness) increased
33
with the smoothness of the face. For the simulations completed in this study the segregation coefficient can be accurately determined from the product of the dimensionless velocity and the dimensionless kink site velocity. As found in former studies, the velocity dependence of the segregation coefficient is not in accord with the results of Aziz, but it is in accord with the data of Baeri et al. and White et al. [12, 25]. The reason for this discrepancy is unclear. The orientation dependence of the segregation coefficient obtained from the simulations agrees with the experimental results of Aziz et al. [14].
Acknowledgements This work was supported by NASA.
References [1] K.F. Hulme and J.B. Mullin, Philos. Mag. 4 (1959) 1286. [2] J.A. Burton. Prim and Slichter. J. Chem. Phys. 21 24/25 (1953} 1987. [3] A. Trainor and B.E. Bartlett, Solid State Electronics 2 (1961) 106. [4] P.J. Hohnes, J. Phys. Chem. Solids 24 (1963) 1239. [5] T. Abe, J. Crystal Growth 24/25 (1974) 463. [6] R.N. Hall, Phys. Rev. 88 {1952) 139. [7] K.A. Jackson, G.H. Gilmer and H.J. Leamy, Laser and Electron Beam Processing of Materials IAcademic Press, New York, 1980). [8] R.F. Wood, Appl. Phys. Lett. 37 (19801 302. [91 M.J. Aziz, J.Y. Tsao, M.O. Thompson, P.S. Peercy and C.W. White, Phys. Rev. Lett. 56 {19861 2489. [10] M.J. Aziz and T. Kaplan, Acta Met. 36 (1988) 2335. [11] L.M. Goldman and M.J. Aziz, J. Mater. Res. 2 (1987) 5241. [121 P. Baeri, G. Foti, S.U. Campisano, J.M. Poate and A.G. Cullis, Appl, Phys. Lett. 38 {1981} 800. [13] R. Reitano, P.M. Smith and M.J. Aziz, J. Appl, Phys. 76 (19941 1518. [14] M.J. Aziz and C.W. White, Phys. Rev. Left. 57 (1986) 2675,
[15] K.A. Jackson, G.H. Gilmer, D.E. Temkin, J.D. Weinberg and K. Beatty, J. Crystal Growth 128 (1993) 127. [16] K.A. Jackson, G.H. Gilmer and D.E. Temkin. Phys. Rev. Lett. 75 (1995) 2530. [17] K.A. Jackson. G.H. Gilmer, D.E. Temkin and K.M. Beany, J. Crystal Growth 163 (19961 461. [18] P. Shewmon, Diffusion in Solids 2nd ed. (Minerals Metals and Materials Society, Warrendale. PA, 1989).
34
ICM. Beatty, K.A. Jackson/Journal of Crystal Growth 174 (1997) 28-34
[19] W.J.P. Van Enckevort and J.P. Van De Eerden, J. Crystal Growth 47 (1979) 501. [20] R. Hultgren, P. Desai, D.T, Hawkins, M. Gleiser, K.K. Kelly and D.D. Wagman, Selected Values of Thermodynamic Properties of The Elements (American Society for Metals, Metals Park, OH, 1973). [21] G.H. Gilmer, MRS Symp. Proc. 13 (1983) 249. [22] M.H. Grabow, G.H. Gilmer and A.F. Bakker, MRS Symp. Proc. 141 (1989) 349. [23] J.A. Yater and M.O. Thompson, Phys. Rev. Lett. 49 (1982) 1496.
[24] M.H. Grabow, G.H. Gilmer and A. F. Bakker, Molecular dynamics simulations of rapid solidification of silicon, in: Int. Workshop on Computational Materials Science (1990) pp. 27-47. [25] L.D. Hess, J.F. Gibbons and T.W. Sigmon, Eds., Laser and Electron Beam Solid Interactions and Materials Processing, MRS Symp. Proc., Vol. 1 (North-Holland, Amsterdam, 1981). [26] D.P. Brunco, M.O. Thompson, D.E. Hoglund, M.J. Aziz and H.J. Gossman, J. Appl. Phys. 78 (1995) 1575.