Survey of the Thermodynamics and Kinetics of Crystallization of Si and Ge
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D. TURNBULL Division of Applied Sciences,
Harvard University,
Cambridge,
MA
02138,
USA
ABSTRACT The thermodynamic interrelations of the crystalline (c-sc), amorphous semiconducting (a-sc) and liquid metal (tm) states of Si and Ge, based on the existing thermal data, are reviewed. Then the kinetics of the interfacial processes in the growth of c-sc and a-sc into undercooled km and the conditions for transition from c-sc to a-sc growth are discussed.
The main content of my oral presentation was set forth in papers which have Here appeared or will appear elsewhere, see especially in references 1 and 2. I will give only a summary of the points developed in those papers which seem most pertinent to this symposium. We are concerned with the thermodynamic and kinetic interrelations of the crystalline semiconducting, c-sc, amorphous semiconducting, a-sc, and liquid metal, tm, phases of Ge and Si at virtually zero pressure. The present inform0 ation indicates that a-sc is less stable than c-sc from 0 K to temperatures well in excess of, Tck, the thermodynamic melting point of c-sc, but a-sc is more stable than the amorphous metallic phase at 0*K. It follows that, if crystallization were bypassed, there would be some temperature, Tat, well under Tc; below which transition from undercooled tm to a-sc would be thermodynamically favored. Also it is known that the transitions a-sc-'c-sc and tm+ c-sc take place by nucleation and growth and that the melting of c-sc occurs heterogeneously, i.e., by movement of a c-sc-tm interface. Perhaps the major thermodynamic issues are whether or not the a-sc, which, as formed, is generally in a configurationally frozen state, can relax to and exist in a metastable state and the nature of the transition from such a metastable state to tm--e.g. would the transition be thermodynamically continuous or discontinuous? If a-sc has a structure, e.g. as modelled by a continuous random network (CRN), such that its crystallization must occur reconstructively, it should be, just as are amorphous Si0 2 and GeO 2 , capable of reaching a fully relaxed metastable state. It was suggested (1) that the time constant for configurational relaxation of the a-sc state is not likely to exceed that for a diffusive jump in the c-sc phase. On this basis and the assumption that the activation energy, Q, for a diffusive jump far exceeds the thermal energy the following expression was obtained for the upper limiting temperature, Txi at which relaxation should be virtually complete:
Tx Z [L (Tx)] 112 x
LR
x
where T is the heating rate-assumed constant, R is the gas constant and T(Tx) is the time constant for relaxation at Tx. At the usual heating rates Tx should be of the order of the glass temperature, Tg, of a-sc which had been estimated (2) by an analogous procedure. When a-sc specimens are heated, rapid crystallization usually intervenes in some temperature range centering at a temperature Tkc somewhat below Tx. It is instructive to scale these temperatures with Tct: thus Trx =Tx/Tct and Trkc=Tkc/Tct. On this
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