Within the continuum model of FEBID/FIBID growth, the time and spatial dependence of the precursor coverage is governed by the following reaction-diffusion equation [

15]

where

$\theta $ represents the precursor density in units

${n}_{0}$; the maximum precursor surface density,

D, is the temperature-dependent diffusion coefficient;

$\tau $ is the temperature-dependent precursor residence time;

${\mathsf{\Phi}}_{e}$ is the electron flux density;

$\sigma $ is the averaged dissociation cross section;

s is the precursor sticking coefficient; and

${\mathsf{\Phi}}_{g}$ is the precursor flux density. We assume that only a maximum of one monolayer can adsorb, which is generally a valid assumption for FEBID precursors [

15]. In the present case of flat deposits, the spatial derivatives can be taken for a flat surface that does not evolve in time, which facilitates the numerical solution of this partial differential equation.

We aim to extract the parameters

${E}_{A}$ and

${\kappa}_{0}$ governing the temperature-dependent residence time

$\tau $ via a thermally activated behavior according to

This is most easily accomplished in the case of a stationary equilibrium coverage

${\theta}_{0}$ obtained from Equation (

1) under beam-off condition without the diffusion term, which is not appreciably reduced over one dwell event of duration

${t}_{D}$. Solving for

${\theta}_{0}$ from Equation (

1) in this stationary state, we obtain

For

$\theta \approx {\theta}_{0}$, corresponding to the reaction-rate limited growth regime, the local height

h of the deposit is given by

where

N denotes the number of repetitions of the dwell event at the given position and

V is the volume of deposit per dissociated precursor molecule. Using Equation (

3) and solving for

$1/\tau $, one obtains

which is the basis for our analysis. We now use the measured deposit heights (center region) at the different substrate temperatures and take the precursor-specific parameters listed in

Table 1. For the beam, we assume a Gaussian shape with full-width at half maximum of 7 nm corresponding to a well-focused beam at 5 kV and 30 pA beam current (experiment

b), taking the secondary electron (SE-I) exit region of the primary beam with 2 nm beam diameter into account. This leads to an electron flux area density of

$3.4\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}$ nm

${}^{-1}$s

${}^{-1}$. The precursor flux we calculate from the chamber pressure increases (see Methods and Materials section) following Reference [

19]. For the sticking coefficient, which can be assumed to be temperature independent [

20], we take

$s=1$, but refer to the Discussion section regarding this assumption. The dwell time is

${t}_{D}=$ 1

$\mathsf{\mu}$$\mathrm{s}$, and the loop number

N is

$6\times {10}^{4}$ for the CoFe-and Nb-precursor and

$3\times {10}^{4}$ for the Pt-precursor, respectively.

In

Figure 4, the quantity

$F\left(h\right)$, as defined in Equation (

5), is plotted vs. the substrate temperature in Arrhenius representation for the three precursors. Evidently, for the CoFe-and Nb-precursor, a linear dependence is quite apparent if the data point taken at the lowest substrate temperature is excluded, as the CoFe- and Nb-precursors show deviating growth behavior due to beginning condensation at the lowest substrate temperature. For the Pt-precursor, we only use the three data points taken at the highest substrate temperature and note that the focus of this work is not on the Pt-precursor for which a careful analysis for the activation energy and pre-exponential factor has been previously performed [

20].