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.

Figure 1

Schematics of the three stages of the formation of a ML. The thick red line marks the external droplet area, taken into account in the energy calculations.

Figure 2

Variation with ML coverage of the normalized ML edge energy. Composite full black curve: fit to the experiments obtained in Ref. [16]. Dashed-dotted blue curve: Eq. (6) with $n=1,A\simeq 3.3748,B\simeq 2.2993,a_0=-B,a_1=1-A+B$. Dashed red curve: the same with $n=2,A\simeq 3.0231,B\simeq 1.9475,a_0=-B,a_1=1-A+B-a_2,a_2\simeq -0.6746$. Short dashed green curve (log): fit by function $G_e^L$, with $A\simeq 2.4301,B\simeq 1.2193$. Dashed-dotted-dotted black curve (sqrt): function $G_e^u=A^{\prime }{\theta }^{1/2}$, with $A^{\prime }$ adjusted to reproduce experimental fit at low coverage. Schematics of the partial ML (shaded area) at low and high coverages, with thick red and orange lines marking ML edge portions at TPL (red hexagon) and away from TPL, respectively. Inset: variations of the derivatives of the normalized edge energy with respect to $\theta$ for fits $n=2$, “log” and “sqrt” (same line types).

Figure 3

Schematics of the variation of the work of formation $\mathrm{\Delta }{G}_{0i}$ of a ML with coverage (full curves, left scale) in three cases (numbers), depending on difference of chemical potential between liquid and solid $\mathrm{\Delta }{\mu }^{\left(i\right)}$ [dashed-dotted lines, capitals, right scale: (A) constant; (B) affected by liquid depletion] and on dependence of ML edge energy $G_e$ upon ML size $i$, as illustrated by $d{G}_e/di$ [dashed lines, small letters, right scale: (a) $G_e$ increasing as $i^{1/2}$; (b) nonmonotonic variation, as in Fig. 2]. The three cases are (1, black line) (A) and (a); (2, blue line) (B) and (a); (3, red line) (B) and (b). The extrema of $\mathrm{\Delta }{G}_{0i}$ correspond to the intersections of the $\mathrm{\Delta }{\mu }^{\left(i\right)}$ and $d{G}_e/di$ curves. The three maxima at left, all occurring within the orange rectangle, give critical size $i^{★}$ in cases (1) to (3); the blue arrow marks the minimum occurring at $i=i_s$ in case (2). Units are arbitrary but consistent between left and right axes.

Figure 4

Self-catalyzed WZ GaAs NWs of radius $R=20\text{nm}$ growing at $T=883\text{K}$. Variation with ML coverage of the difference of chemical potential per III-V pair between liquid and solid [$\mathrm{\Delta }{\mu }^{\left(i\right)}$, calculated with $b=0$ and ${\alpha }_b=3^{3/4}/{\left(2\pi \right)}^{1/2}$, full curves] and of the derivative of the total ML edge energy with respect to ML size $i$. $\mathrm{\Delta }{\mu }^{\left(i\right)}$ is shown for five different As contents of the liquid droplet at nucleation, expressed as equivalent coverage ${\theta }_0$ (indicated for each curve). The edge energy is taken as monotonic $G_e^u$ with ${\gamma }_0=0.2\text{J}{\text{m}}^{-2}$ (short orange dashes) or nonmonotonic $G_e^{\left(2\right)}$ with ${\gamma }_0=0.2\text{J}{\text{m}}^{-2}$ (long magenta dashes) or ${\gamma }_0=0.03\text{J}{\text{m}}^{-2}$ (black dashed-dotted curve).

Figure 5

Self-catalyzed WZ GaAs NWs of radius $R=20\text{nm}$. Variations of ML coverage (top panels) and group V liquid content (bottom panels) as a function of refill, for edge energies ${\gamma }_0=0.03\text{J}{\text{m}}^{-2}$ (left panels) and $0.2\text{J}{\text{m}}^{-2}$ (right panels), growth temperatures $T=713\text{K}$ (blue curves) and $883\text{K}$ (red curves), and for various group V liquid contents at nucleation, indicated in terms of equivalent coverage (${\theta }_0$) near each curve. The short bars at the right of the bottom panels give the equilibrium concentrations of the bulk liquid with solid GaAs, at the two temperatures. The insets show the variations with coverage of factor ${\chi }_j$ which determines the distribution between ML and liquid of the newly provided species (horizontal scales as in main panels). ${\alpha }_b$ as in Fig. 4.

Figure 6

(a) Evolution of $\theta$ and ${\theta }_L$ with normalized time $t/t_{\text{ML}}$ in absence of group V desorption and at $t_{\text{ML}}/\tau =50$, for three NW radii $R$ shown near each couple of curves. For $R=20\text{nm},{\theta }_{\text{eq}}=0.34$ and ${\theta }_0=0.7$, and we assume these coverages to scale with $R$, as discussed in the text. For $R=20$ and $40\text{nm}$, the ML grows faster at the beginning before reaching a stopping size, and then expands much slower at the rate of refill. The droplet content drops to near equilibrium and stays constant in the slow ML growth stage. After that, ${\theta }_L$ increases linearly at the rate of refill. At $R=60\text{nm},{\theta }_0-{\theta }_{\text{eq}}=1.08>1$ and the stopping effect disappears: the ML covers almost instantaneously the whole facet and the liquid content never drops to equilibrium. (b) Same as (a) at a fixed NW radius of 20 nm for different $t_{\text{ML}}/\tau$ shown near each curve. At a small $t_{\text{ML}}/\tau$ of 10, the stopping effect is not pronounced and ${\theta }_L$ drops to a minimum far above ${\theta }_{\text{eq}}$ (dashed line). At a large $t_{\text{ML}}/\tau$ of 200, there is a clear timescale hierarchy between the fast growth stage to a stopping size and slow growth at the rate of refill, while ${\theta }_L$ decreases almost instantaneously to equilibrium.

Figure 7

Evolution of $\theta$ and ${\theta }_L$ with normalized time $t/t_{\text{ML}}$ in presence of group V desorption at ${\theta }_{\text{eq}}=0.34,{\theta }_0=0.7,t_{\text{ML}}/\tau =100,R=20\text{nm}$, and two different ${\theta }_v=5$ (low desorption, dashed lines) and ${\theta }_v=0.71$ (high desorption, solid lines). The ML growth is almost identical in both cases. Refill takes slightly longer than in the absence of desorption at ${\theta }_v=5$ so that the initial liquid content at nucleation (indicated by horizontal dashed line) is recovered at $t\cong 1.1t_{\text{ML}}$. At ${\theta }_v=0.71$, refill is very long due to saturation of the ${\theta }_L$ curve, and the nucleation conditions are recovered only at $t\cong 2t_{\text{ML}}$. The line of circles shows the asymptote of $\theta$ given by Eq. (30) in the regime with high desorption.