Energetics and kinetics of monolayer formation in vapor-liquid-solid nanowire growth

The vapor-liquid-solid growth of semiconductor nanowires proceeds via the sequential addition of individual atomic or biatomic monolayers at the interface between the nanowire stem and a liquid catalyst droplet. Each monolayer growth cycle comprises a period of nucleation and extension of the monolayer followed by a waiting time before the next nucleation. The cyclic evolution of the liquid-solid system and the duration of these periods affect the properties of individual nanowires and the statistics of nanowire ensembles. Based on recent in situ growth experiments, we investigate theoretically the thermodynamics and kinetics of monolayer formation, with emphasis on nanowires of III-V materials. We first study the thermodynamics of the process in a quasistatic approximation. We calculate and minimize the work of formation of a partial monolayer, accounting for the depletion and refill of the liquid phase and all relevant interface energies. We focus on the regime where the liquid does not contain enough nanowire material at nucleation to form a full monolayer and where growth comprises a fast stage where a partial monolayer forms rapidly using the available material and a slower stage during droplet refill. We explore the effect of the nonmonotonous variation of the total monolayer edge energy on these two growth stages and determine the partition between liquid and monolayer of the atoms provided during refill. We then develop a kinetic model that accounts in simple terms for the competition between delivery and attachment of atoms to the monolayer and droplet refill. We perform fully analytical calculations that describe a continuum between the pulsed regime explored in our thermodynamic study and a smooth regime where the monolayer extends at the pace of refill. The various characteristic times of the monolayer cycle are calculated in presence of desorption of the volatile group V species from the liquid droplet. Simplified asymptotic formulas corresponding to fast attachment or low desorption are derived. Desorption is found to affect weakly the monolayer formation time but strongly the waiting time.