Retrograde solubility in semiconductors
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INTRODUCTION
TRUMBORE found while determining the solubilities of many dopants in silicon and germaniumv~ that most of the impurities tested in his study displayed a phenomenon known as retrograde solubility. Retrograde solubility describes the change in solute concentration in a solid above the eutectic temperature. Initially, as a melt begins to solidify, the solute concentration in the solid increases. Usually, the maximum in solute concentration occurs at the eutectic temperature; however, for solutions that have retrograde solubility, as the material cools, the solute achieves a maximum concentration and then begins to decrease until the eutectic temperature is reached (Figure 1).v-rl Since both dopant and impurity concentrations can be affected by retrograde solubility, an understanding of this phenomenon is extremely important to semiconductor processing. As an example of the extreme effect retrograde solubility can have, it is known that the maximum concentration of Cu in Si occurs above the eutectic and is 30 times greater than the concentration of solute at the eutectic temperature. [7] Previous explanations for retrograde solubility are insufficient, as they are based on tautological arguments which only describe the effects of this phenomenon and never its origin. The first example of retrograde solubility was discovered in 1926 by Jenkins, where he observed that the Cd concentration in Zn increased above the eutectic temperature.tSI However, Jenkins attributed this new and unusual behavior to a peritectic reaction involving an aUotropic transformation of Zn, instead of a solubility change. I21 Thunnond and Struthers attributed retrograde solubility to "the fact that the chemical potential must go through a maximum. . . . ,,[3,7] This argument is circuitous in that the maximum in chemical potential is caused by the maximum in concentration. Raoult's law states that the activity, a, of the solvent is equal to the concentration in the limit of pure solvent. The chemical potential,/x, is a function of a, and therefore,/x is also a function of concentration in this limit. Thus, the maximum in/z is a result of the maximum in concentration. Alternatively, Meijering concluded that "retrograde solid solubility is expected in binary systems where
AINDREA L. McKELVEY, Research Assistant, is with the Department of Materials Science and Mineral Engineering, University of California at Berkeley, Berkeley, CA 94720-1760. Manuscript submitted December 4, 1995. 2704---VOLUME 27A, SEPTEMBER 1996
the solid solubility is small and the eutectic temperature considerably below the melting point of the pure solid solvent."I91 This is perhaps consistent with experimental observations; however, it does not explain the source of retrograde solubility. Weber claimed that "retrograde solubility is observed if the temperature determined by equation (1) is
T(x~ (max)) =
aHT / k / M-/,il \ AS: / k - ln~zM_/f + ) d / f )
(1)
higher than the eutectic temperature . . . . due to a large enthalpy of formation, M-/~,t, (energy r
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