The mathematical model for one interface must be specified through one of
ba.SelfAffineFractalModel(...)
ba.LinearGrowthModel(...)
These models define the behavior of autocorrelation function of roughness or, better to say, its Fourier spectrum in the full range of spatial frequencies from 0 to infinity.
ba.SelfAffineFractalModel(sigma, hurst, lateral_corr_length, max_spatial_freq=0.5)
where
sigma, $\sigma$, is the root-mean-square amptitude of out-of-plane
(transversal) fluctuation.hurst, $H$, is a fractal exponent H with 0<H<=1.
The smaller $H$ is, the more serrate the surface profile looks.lateral_corr_length, $\xi$, is the lateral (in-plane) correlation length.max_spatial_freq, $\nu_{max}$, is a cut-off spatial frequency in nm⁻¹.See the specular roughness
example for
a complete simulation using SelfAffineFractalModel.
This is the K-correlation model of Palasantzas 1993. The autocorrelation spectrum is
$$ S(\nu)=\dfrac{4 \pi H \sigma^2\xi^2}{(1+(2\pi\xi\nu)^2)^{1+H}} ,\quad for \quad \nu<\nu_{max} $$
In case there is no cut-off, the real-space roughness correlation function at the interface is expressed as: $$ < U(x, y) U(x’, y’)> = \sigma^2 \frac{2^{1-H}}{\Gamma(H)} \left( \frac{\tau}{ξ} \right)^H K_H \left( \frac{\tau}{ξ} \right), \quad \tau=[(x-x’)^2+(y-y’)^2]^{\frac{1}{2}} $$
where $U(x, y)$ is the height deviation at position $(x, y)$.
The main property is that it remains nearly constant at low $\nu$ and transitions into a straight line on a log-log scale at high $\nu$, exhibiting an inflection point between these two regions.
The cut-off $\nu_{max}$ sets the spectrum to zero above that spatial frequency. It is a real frequency, not a circular one, in nm⁻¹; the default 0.5 nm⁻¹ corresponds to a lateral period of 2 nm, the scale of the atoms or molecules that build the interface.
The parameter $\sigma$ is the rms height of the full, uncut spectrum. Everything except the spectrum itself is computed with the rms height of the cut spectrum, $$ \sigma_{\nu_{max}}^2 = \int_0^{\nu_{max}} S(\nu), 2\pi\nu, d\nu = \sigma^2 \left[1 - \left(1 + (2\pi\xi\nu_{max})^2\right)^{-H}\right], $$ namely the specular reflection and transmission through the transient profile, and the damping of the diffuse scattering amplitudes with the vertical momentum transfer. For $\nu_{max} \gg 1/(2\pi\xi)$, $\sigma_{\nu_{max}}$ approaches $\sigma$; with the default cut-off and $\xi = 12$ nm it is $0.99,\sigma$. A cut-off below $1/(2\pi\xi)$ removes most of the roughness: for $\xi = 12$ nm and $\nu_{max} = 0.003$ nm⁻¹, $\sigma_{\nu_{max}} = 0.16,\sigma$. The interfaces then become correspondingly smoother for the specular reflectivity, and the diffuse scattering changes through the reflected and transmitted amplitudes also at spatial frequencies below the cut-off. Lowering the cut-off is therefore not a way to restrict the diffuse scattering to low spatial frequencies while keeping the specular reflectivity. For the linear growth model below, the same integral is evaluated numerically.
ba.LinearGrowthModel(particle_volume, damp1, damp2, damp3, damp4, max_spatial_freq=0.5)
where
particle_volume, $\Omega$, is the volume of particle (atom, molecule, cluster)
coming to the surface during film deposition,damp1-damp4, $a_1-a_4$, are damping coefficients, responsible for replication
of underlying roughness and the growth of the new independent one,max_spatial_freq, $\nu_{max}$, is a cut-off spatial frequency in nm⁻¹.This is the model described by Stearns 1993 and extended by Stearns and Gullikson 2000.
The model describes the evolution of the roughness spectrum on the top surface of a growing film. Its autocorrelation spectrum depends not only on the model parameters but also on the film thickness and the spectrum of the underlying interface. Therefore, the model cannot be applied directly to the substrate.
The autocorrelation spectrum is
$$ S_{above}(\nu)=S_{below}(\nu)e^{-b(\nu)t} + \Omega\dfrac{1-e^{-b(\nu)t}}{b(\nu)}, \quad for \quad \nu<\nu_{max} $$ where $t$ - film thickness, $b(\nu) = \sum{a_i\nu^i}$ - relaxation function
An essential property of the model is that it describes not only autocorrelation but also cross-correlation properties. Therefore, it does not require cross-correlation to be specified separately.
The multilayer linear-growth
example shows
LinearGrowthModel above a self-affine substrate.