The Crystal-Melt Interface in Si or Ge
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Mat. Res. Soc. Symp. Proc. Vol. 398 © 1996 Materials Research Society
microstructure of thin Si films resolidified after melting by pulsed laser irradiation. Heat flow analysis of the experiment showed that the undercooling was AT=505±35K, corresponding to Tr=0.30K. Devaud and Turnbull [6] undercooled bulk liquid Ge droplets coated with a B2 0 3 flux by AT=415K below the melting point TM=1210K, which corresponds to Tr=0.34. Similar results have been obtained by Lau et al. [7]. DATA ANALYSIS From a knowledge of the cooling rate, T, (measured, imposed or calculated) and of the volume, V, in which the nucleation even occurred it is possible to estimate the nucleation rate [3]: events
m(3.s
1
T (K/s)
(1)
3 (K) V (m 3)
Using the entire specimen volume and the measured or imposed cooling rates, the estimated nucleation rate for uncoated bulk Si is 108 m3 s-1 , for fluxed bulk Si, 4x10 9 m- 3 s-1, and for fluxed bulk Ge, 1011 m- 3s- 1. Using the volume of the individual grains in the solidified Si thin films and the modeled cooling rate, the estimated nucleation rate is 1028 m- 3 s-1 . The results for Si are summarized in Fig. 1.
7j
1030
E CD
1025
.2
a1) 1020 C 0
1015 1
0 10
10s5 9CO00 950
1000
1050
1100
1150
1200
1250
1300
1350
1400
1450
Temperature (K) Figure 1. Nucleation rates of silicon crystals from the melt derived from undercooling experiments. Open circle: laser-irradiated thin film [ref. 4]; triangle: uncoated bulk sample [ref. 2]; filled circle: flux-coated bulk sample [ref. 3]. The lines are fits from classical nucleation theory. Solid curve: a=0.38 J m- 2 ; dot-dashed curve: CV=0.34 J m- 2 ; dashed curve: a(T), according to eq. (3), using the parameters of Table 1 [from ref. 3].
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The rate of homogeneous crystal nucleation is given by the classical theory as:
]
I=1o exp[-• 3
(2)
where Io=1039 m- 3 s-1 is a prefactor determined by the jump frequency across the interface [8,9], a is the crystal-melt interfacial tension, and AGv is the difference in bulk free energy, per unit volume of the nucleating phase, between the two phases. Application of this expression to the estimated nucleation rates gives lower limits for a: 0.34 J m-2 for the uncoated bulk Si and the thin film Si; 0.38 J m- 2 for fluxed bulk Si; and 0.32 J m-2 for fluxed bulk Ge. The agreement between the values for the uncoated and thin film Si is illustrated in Fig. I (dotdashed line). The greater undercooling found in the fluxed Si, however, indicates that the earlier data did not correspond to homogeneous nucleation. The solid line in Fig. 1 indicates that the bulk and thin film data for Si cannot be both accounted for by a constant value of C. Only by allowing a to be temperature-dependent can this be achieved (dashed line). TEMPERATURE-DEPENDENCE OF THE INTERFACIAL TENSION The temperature-dependence of the interfacial tension has its origin in the variation of the structure and the corresponding thermodynamic parameters as a function of distance normal to the interface, as illustrated by Fig. 2. The main feat
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