SINGLE CRYSTAL BY ZONE LEVELING 647 



The true growth rate may be affected by factors that cause variations 

 from steady state growth such as temperature and gas flow fluctuations. 

 The need to control these variables has already been mentioned because 

 of their effect on zone volume; their effect on growth rate is thus a second 

 reason for their control. 



Cracks or similar discontinuities in the unmelted charge act as barriers 

 to heat flow. Thus they cause a local rise in temperature and lengthening 

 of the liquid zone as the crack approaches the zone, until it is closed by 

 melting. The resulting transient increase in liquid volume (and in p of 

 the product) may be of the order of 10 per cent. 



(b) Cross-Sectional Com'position Uniformity 



Difficulty may be expected in controlling the cross-sectional uniform- 

 ity of the zone leveled ingot chiefly when the third assumption is invalid, 

 i.e., when Cl throughout the liquid is non-uniform. As shown in the next 

 paragraph, the true Cl must always rise locally near the solidifying inter- 

 face due to the solute diffusion which is necessary when k < \. However, 

 it is possible to improve the validity of assumption 3 both by slowing 

 Ihe groAvth rate and by stirring the liquid zone. 



One can form an estimate of a theoretically reasonable growth rate 

 in terms of the rate of diffusion of impurities in liquid germanium. It 

 should be noted that movement of a liquid zone containing a solute 

 whose segregation coefficient is small implies a general movement by 

 diffusion of essentially all the solute atoms away from the solidifying 

 interface at a speed ecjual to the rate of motion of the zone. Even slow 

 zone motion corresponds to a high diffusion flux of the solute through 

 the Uquid. As a consequence, the solute concentration must rise in front 

 of the advancing solidification interface to a concentration Cl' (see Fig. 5) 

 until a concentration gradient is reached sufficient to provide a diffusion 

 flux equal to the growth rate. Fick's Law of diffusion is useful here to 

 calculate the extent of the rise in C/,/ at the growth interface, assuming 

 the liquid to be at rest. The ratio of the maximum concentration to the 

 bulk concentration may be taken from Fig. 5. If the maximum is to 

 l)c no greater than 10 per cent above the mean, a maximum growth 

 rate of 2 X 10~^ mils per second or 7 X 10"^ inches/hour would be 

 r(3(juired. Clearly, this rate is far too slow to provide an economical 

 means of growing single crystals. For a practical process, it will be neces- 

 sary to use non-equilibrium conditions at growth rates that must result 

 HI appreciable concentration differences within the liquid zone. Of course, 

 the slower the growth rate the smaller will be the diffusion gradient and 

 the higher will be the expected cross-sectional uniformity. 



