Sticking coefficient

Sticking coefficient is the term used in surface physics to describe the ratio of the number of adsorbate atoms (or molecules) that adsorb, or "stick", to a surface to the total number of atoms that impinge upon that surface during the same period of time.^[1] Sometimes the symbol S_c is used to denote this coefficient, and its value is between 1 (all impinging atoms stick) and 0 (none of the atoms stick). The coefficient is a function of surface temperature, surface coverage (θ) and structural details as well as the kinetic energy of the impinging particles.

Derivation

When arriving at a site of a surface, an adatom has three options. There is a probability that it will adsorb to the surface ( $P_a$ ), a probability that it will migrate to another site on the surface ( $P_m$ ), and a probability that it will desorb from the surface and return to the bulk gas ( $P_d$ ). For an empty site (θ=0) the sum of these three options is unity.

P_a + P_m + P_d=1

For a site already occupied by an adatom (θ>0), there is no probability of adsorbing, and so the probabilities sum as:

P_d'+P_m'=1

For the first site visited, the P of migrating overall is the P of migrating if the site is filled plus the P of migrating if the site is empty. The same is true for the P of desorption. The P of adsorption, however, does not exist for an already filled site.

P_{m1}=P_m(1-\theta)+P_m'(\theta)

P_{d1}=P_d(1-\theta)+P_d'(\theta)

P_{a1}=P_m(1-\theta)

The P of migrating from the second site is the P of migrating from the first site and then migrating from the second site, and so we multiply the two values.

P_{m2}=P_{m1} \times P_{m1}=P_{m1}^2

Thus the sticking probability ( $s_c$ ) is the P of sticking of the first site, plus the P of migrating from the first site and then sticking to the second site, plus the P of migrating from the second site and then sticking at the third site etc.

s=P_a(1-\theta)+P_{m1}P_a(1-\theta)+P_{m1}^2P_a(1-\theta)...

s=P_a(1-\theta)\sum_{n=0}^{\infin} P_{m1}^n

There is an identity we can make use of.

\sum_{n=0}^{\infin} x^n =\frac{1}{1-x}\forall x<1

\therefore s=P_a(1-\theta)\frac{1}{1-P_{m1}}

The sticking coefficient when the coverage is zero $s_0$ can be obtained by simply setting $\theta=0$ . We also remember that

1-P_{m1}=P_a+P_d

s_0=\frac{P_a}{P_a+P_d}

\frac{s}{s_0}=\frac{P_a(1-\theta)}{1-P_{m1}}\frac{P_a+P_d}{P_a}

If we just look at the P of migration at the first site, we see that it is certainty minus all other possibilities.

P_m1=1-P_d(1-\theta)-P_d'(\theta)-P_a(1-\theta)

Using this result, and rearranging, we find:

\frac{s}{s_0}=[1+\frac{P_d'\theta}{(P_a+P_d)(1-\theta)}]^{-1}

\frac{s}{s_0}=[1+\frac{K\theta}{1-\theta}]^{-1}

K\overset{\underset{\mathrm{def}}{}}{=}\frac{P_d'}{P_a+P_d}

References

↑ sticking coefficient IUPAC Compendium of Chemical Terminology 2nd Edition (1997), Accessed 30 September 2008

King-Ning Tu, James W. Mayer, and Leonard C. Feldman, in Electronic Thin Film Science for Electrical Engineers and Materials Scientists, Macmillan, New York, 1992, pp. 101–102.