Kris's Research Notes

April 10, 2012

Nucleation Regime. Part 2

Filed under: GaAs Simulations — Kris Reyes @ 11:41 pm

This note is a follow-up to this post. In that note, we developed an expression for when nucleation is possible by expressing the rate of change in concentration c_r as the net effect of atoms impinging upon the nucleus (given by an As flux and Laplace’s equation) and atoms leaving the nucleus (given by the Gibbs-Thomson relation). In this note, we discuss the problems with the discussion in that post and describe alternative models for surface nucleation.

Recall in our previous note, we expressed the nucleation condition by a function \phi(F, T, r) where r was the radius of the droplet so that \phi > 0 implied growth of a nucleus while \phi = 0 meant decay. For a fixed flux and temperature, the radius r_c such that

\phi(F, T, r_c) = 0

was called the critical radius. We modeled the droplet size as a function of temperature

\displaystyle r(T) = \left(D_G/F_G\right)^\alpha,

where D_G was the diffusion coefficient for Ga-on-Ga diffusion and F_G was the deposition rate of Ga during droplet growth. The parameter value \alpha \approx 1/4 was found empirically from the simulations. From this, we obtained a necessary condition of F, T for growth: \phi(F, T, r(T)) > 0. This is the condition for the droplet to be sufficiently large in order to contain a nucleus of critical size (it is equivalent to r(T) > r_c since \phi_r > 0).

From this condition, we considered the critical curve \phi(F, T, r(T)) = 0 in the (\log(F), \frac{1}{T}) plane and argued (correctly) that it is essentially linear. The slope of the line was given by the expression

\displaystyle \frac{\alpha E_{Ga} - E_{As}}{k_B} \approx -3\times 10^3,

for choices of the energy barrier for Ga-Ga diffusion E_{Ga} = 0.9 eV and energy barrier for As diffusion in liquid Ga E_{As} = 0.7 eV. While the slope is steep, it is not sufficiently steep to match our data. Indeed, we had considered instead (without proper justification) an alternative barrier E_A = 1.7 in order to match the data more closely.

Moreover, the condition r(T) > r_c, while necessary for nucleation, is not very strong as it is not directly a statement about the onset of nucleation. That is, asking for a domain to be sufficiently large to host a nucleus of critical size is not the same as asking whether (through random events) a nucleus of critical size is favored to form. In addition, the critical radius for GaAs is suspected to be small, on the order of 1 to 2 atoms. Hence, this condition is not that useful to determine a nucleation regime.

A Simplified Domain

As a simplification, we consider instead a semi-infinite system consisting of a slab of liquid Ga on top of a GaAs substrate:

Let L be the thickness of the liquid Ga. We assume uniformity along the horizontal direction so that the concentration c(y) of As atoms depends only on the height y. Within the domain, we assume Laplace’s equation holds:

\displaystyle c_{yy} = 0,

and on the boundaries:

\displaystyle \ell D_{As} c_y(L) = F_{As},

\displaystyle c(0) = c_0,

where c_0 is the equilibrium concentration of As atoms above the solid/liquid interface

The solution

\displaystyle c(y) = \frac{F_{As}}{\ell D_{As}}y + c_0

means the concentration gradient is linear within the liquid.

Equilibrium Concentration

In equilibrium (i.e. when F_{As} = 0), the concentration c_0 is governed by the solubility of As from the substrate in liquid Ga. In equilibrium, the rate of detachment from the substrate equals that of attachment onto the substrate, and hence c_0 satisfies

\displaystyle R_{D} - c_0 R_A = 0,

where R_D, R_A are the rates of detachment, attachment respectively, which are in Arrhenius form

\displaystyle R_D = R_0 \exp\left[- \frac{E_D}{k_B T}\right], R_A = R_0 \exp\left[- \frac{E_A}{k_B T}\right],

where E_D, E_A are the energy barriers for detachment and attachment respectively and R_0 \approx e^{26} is a constant prefactor. Hence, the equilibrium concentration may be written as

\displaystyle c_0 = \exp\left[\frac{E_A - E_D}{k_B T}\right].

We may obtain the energy barriers empirically from simulation results. We consider the configuration depicted above, with L = 32 and set the flux F_{As} = 0. Allowing the simulation to run for 600 seconds, we measure the frequency and associated rates of attachment and detachment events. The (harmonic) average of these rates \bar{R}_D, \bar R_A are used as empirical estimates for the attachment, detachment rates above. From this, we may solve for the energy barriers, yielding empirical values:

E_A = 0.33 eV,

E_D = 1.29 eV.

These are very close to the computed energy barriers for the following transitions:

Indeed, most of the attachment/detachment events in the simulations are of this form (I say this because of the way attachment/detachment events are “measured” in the simulations, events other than attachment/detachment from a perfectly flat interface could be counted.)

These barriers yield an equilibrium concentration of c_0 = 5.5 \times 10^{-10} at 523K. We may alternatively measure the attachment/detachment rates in the presence of an As flux when nucleation is first present. These rates and corresponding barriers describe the equilibrium concentration in the liquid when surface nucleation occurs. In this case, the barriers are

\displaystyle E^{nuc}_A = 0.55 eV,

\displaystyle E^{nuc}_D = 1.15 eV.

These barriers were for T = 498K and F_{As} = 0.1 monolayers/second and were selected because they minimized the difference

\displaystyle E_{nuc} = (E^{nuc}_A - E_A) - (E^{nuc}_D - E_D),

among all conditions which facilitated surface nucleation, and hence described the minimal concentration observed to allow nucleation. Here, E_{nuc} = 0.36 eV. This critical concentration c_{nuc} present when nucleation occurred may be written as

\displaystyle c_{nuc} = c_0 \exp\left[ \frac{(E^{nuc}_A - E_A) - (E^{nuc}_D - E_D)}{k_BT}\right] = c_0 \exp\left[\frac{E_{nuc}}{k_B T}\right].

Nucleation Condition

From this, we get a condition for nucleation by comparing the concentration at the surface c(L) with that of the critical concentration c_{nuc}. If c(L) > c_{nuc}, then surface nucleation will occur. From above, we know

\displaystyle c(L) = \frac{F_{As} L}{\ell D_{As}} + c_0,


\displaystyle c_{nuc} = c_0 \exp\left[\frac{E_{nuc}}{k_B T}\right],

so that the necessary condition for surface nucleation is

\displaystyle \frac{F_{As} L}{\ell D_{As}} + c_0 >  c_0 \exp\left[\frac{E_{nuc}}{k_B T}\right]

We may write

\ell = 1,

c_0 = \exp\left[\frac{E_A - E_D}{k_B T}\right],

\displaystyle D_{As} = R_0 \exp\left[-\frac{E_{As}}{k_B T}\right],

where E_{As} = 0.7 is the energy barrier for As diffusion thorough liquid Ga.

For the growth conditions we consider, \frac{F_{As} L}{\ell D_{As}} \gg c_0, so that the conditions above is well approximated (after performing the proper substitutions):

\displaystyle F_{As} >  \frac{R_0}{L} \exp\left[\frac{E_A - E_D - E_{As} + E_{nuc}}{k_B T}\right],

or alternatively:

\displaystyle \log F_{As} >  \frac{E_A - E_D - E_{As} + E_{nuc}}{k_B }\frac{1}{T} + \log R_0 - \log L.

Plugging in all parameter values, we obtain:

\displaystyle \log F_{As} > -\frac{1.3}{k_B}\frac{1}{T} + 24.5.

Simulation Results

We simulated the system discussed at the beginning on a domain of width 128 atoms, a GaAs substrate 16 monolayers thick, and a liquid Ga slab L = 32 monolayers thick. We deposited As atoms with flux F_{As} varying between 0.1 and 4.0 monolayers/second at a temperature T varying between 423 and 623 K. We deposited material for 30 seconds, and for each choice of (T, F_{As}), we indicate whether surface nucleation takes place. This results in a phase diagram in the (1./T, \log F_{As}) plane which we display below, along with the critical curve derived above (shown in green).

We see that all nucleation occurs above and to the right of the critical line.

Variable Droplet Radius

Recall the radius R of a Ga droplet formed from the deposition of Ga at a rate of F_{G} = 0.1 monolayers/second at temperature T is given by

\displaystyle R = r_0 \left( \frac{D_G}{F_G} \right)^\alpha,

where r_0 is some constant of proportionality and D_G is the diffusion coefficient

D_{G} = R_0 \exp\left[ \frac{-E_G}{k_B T}\right],

for the energy barrier E_G = 0.9 eV of Ga-on-Ga diffusion. The power \alpha = 0.25. Substituting R = L into the above expression for the nucleation condition yields:

\displaystyle \log F_{As} >  \frac{E_A - E_D - E_{As} + E_{nuc} + \alpha E_{G}}{k_B }\frac{1}{T} + \log \frac{R_0^{1-\alpha}F_G^\alpha}{r_0}.

Plotting this line against our droplet crystallization simulation results (counting the number of anti-site defects to measure the presence of nucleation), we observe a good fit (fixing r_0 = 0.5)

Here, blue crosses correspond to growth conditions favorable to nucleation, while black dots indicate no nucleation present.


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