One of the most used parameters, describing the properties of a surface, is the root mean square (RMS) roughness, R q , defined as22,23:

$${R}_{{\rm{q}}}=\frac{\sqrt{{{\sum }_{i=1}^{N}[{h}_{i}(x,y)-\overline{h}]}^{2}}}{N}$$ (1)

where h i (x, y) is the profile function defined at the point of coordinates (x, y) on the surface, N is the number of points, and \(\overline{h}\) is the average height value. Note that the roughness contains information about the height distribution, but it does not provide any information on the distance between the features on the surface. The scaling method, based on fractal theory, involves measuring the surface’s roughness at various times and at various radial lengths. This allows replacing the roughness with scale independent parameters, which show how the roughness changes as a function of space and time. This replacement eliminates instrumental dependences such as sampling interval, resolution and so on; indeed, the scaling exponents are determined only by the processes happening during nucleation of the film24.

In the framework of the scaling theory, the R q of a sample of size L × L is expected to depend on the measurements window size, l × l (assuming l << L) as2:

$${R}_{{\rm{q}}}(l) \sim {l}^{\alpha },\,for\,l < {l}_{sat}$$ (2)

where l sat is the window size at which the surface roughness does not change anymore because all surface features are correlated. The roughness exponent, α, is a number between 0 and 12, and can be calculated graphically from the slope of the logarithmic plot of the RMS roughness as a function of window size.

The time-dependent dynamics of the roughening process is described by2:

$${R}_{{\rm{q}}}(L,\,t) \sim {t}^{\beta },\,{\rm{for}}\,t < {t}_{{\rm{sat}}}$$ (3)

where t sat is the time required to reach full surface coverage, hence no roughness variation is expected after this time. Based on Eq. 3, the RMS roughness is expected to increase gradually as t increases, until it reaches saturation. The growth exponent, β, can be calculated graphically from the slope of the logarithmic plot of RMS roughness as a function of time for very small times (i.e. at the nucleation).

The scaling exponents α and β allow the definition of universal growing models. Four kinetic growth models have been theoretically studied for thin film growth, which depends on how and where particles approach, rest and stick to the surface and (or) to existing particle deposits, and how smoothing and fluctuation effects compete with each other in a particular surface1. Experimentally, the scaling parameters have been investigated in the case of few carbon-based materials, such as diamond-like carbons4,6 and clustered carbon25.

Frequency analysis

The surface properties can be analysed in the frequency space, e.g. by measuring the power spectra density, PSD(w), of the surface by Fourier transform, \(PSD(w)={\mathscr{F}}(C(r,\,t))\), where C(r, t) is the autocorrelation function, given by an average value of the product of two height measurements at a distance r apart and at fixed time. Herring26 described four distinct surface transport mechanisms that reduce surface roughness by using the frequency analysis. These surface kinetic models come from an analysis of the time and amount of material needed to produce a geometrically similar change in two different clusters on the surface. The word “cluster” in Herring’s work refers to any particle or to a number of particles, which have started to grow together26 and we used the same definition in our work. Every growth mechanism leaves distinct fingerprints on the topography. We consider two spherical clusters of radius R 1 and R 2 , respectively, where R 2 = λR 1 and λ is a scaling factor. For each growth mechanism there will be a relation between the time δt 1 , required to produce a certain change in the cluster 1, and the time δt 2 , required to produce a similar change in the cluster 2 of the form: δt 2 = λiδt 1 , where i is an integer, called Fourier index. The dominant growth model depends on the Fourier index value. Four cases have been identified: i = 1 is associated to viscous flux of an amorphous material, i.e. it has been assumed that temperature is high enough so that the atoms are reasonably free to rearrange themselves. In the case i = 2, there is evaporation-condensation, i.e. the vapour pressure of the two clusters are different and the amount of material evaporated from a cluster and condensed to the other will be different from the amount passing in the reverse direction. When i = 3, there is bulk diffusion, where the equilibrium between the chemical potentials of the two clusters are considered. Finally, i = 4 is associated to surface diffusion, where the rate of migration over a certain type of surface is proportional to the gradient of the chemical potential.

In order to apply this analysis model, the surface profiles are Fourier analysed and the coefficients for the individual profiles are averaged. If a log-log plot of the integrated power spectrum is a straight line, then the modulus of the slope gives i. Note that a correlation between the frequency analysis and the scaling approach is expected, as: i = 2(α + 1)27.

CVD Growth model

The following kinetic model has been proposed to explain the possible mechanisms of graphene growth as a function of the nucleation density28. This model assumes that graphene growth takes place on a hexagonal grid of active sites for graphene formation that represents the active sites for graphene formation on a flat surface28. The binding of carbon atoms at the active sites occurs from reactive carbon intermediates C x H y (\(x\ge 1\), \(y\ge 0\)) which are caused by the decomposition of CH 4 molecules on the Cu surface29,30,31. The variation of the graphene coverage θ over time is given by28:

$$d\theta /dt=sJ\theta [1-\theta (t)]$$ (4)

where J is the impingement rate at a single site and s is the sticking coefficient of reactive carbon species (i.e. the probability of binding at the grid sites). Experimentally, the variation in the graphene growth rate observed shows that s is not constant32. For this reason, two sticking coefficients need to be introduced: the first sticking coefficient, s 0 , describes the mechanism for capturing the active species at random vacant sites, and it is proportional to the fraction of empty sites given by s 0 (1 − θ(t))28. The second sticking coefficient, s 1 , describes the physisorption of active species on top of an already formed graphene island, and it is described by a coverage-dependent sticking coefficient as s 1 θ(t)[1 − θ(t)]28. By introducing these coefficients into Eq. (4), the graphene coverage variation with time is now given by:

$$d\theta /dt=J\{{s}_{0}[1-\theta (t)]+{s}_{1}\theta (t)[1-\theta (t)]\}$$ (5)

Equation 5 can be re-arranged as28:

$$d\theta /dt=[1-\theta (t)][{\theta }_{0}+\theta (t)]/[\tau (1+{\theta }_{0})]$$ (6)

where θ 0 = S 0 /S 1 defines the ratio of the sticking coefficients and describes the scaled binding time at a single adsorption site that depends on both sticking coefficients and the impingement rate, and τ = 1/J(S 0 + S 1 ) describes the scaled binding time at a single adsorption site that depends on both sticking coefficients and the impingement rate that is proportional to the concentration of active carbon species determined by the CH 4 flow rate28. The solution of Eq. 6 is28:

$$\theta (t)=1-(1+{\theta }_{0})/(1+\exp [(t-{t}_{0})/\tau ])$$ (7)