2. Random rough surfaces

The interface between two heterostructure layers is typically not pristine (or flat) at the atomic scale. Rather, it is typically rough, in the sense that the height of the interface exhibits fluctuations around a mean value. The roughness of such an interface depends on various factors, such as the relevant experimental fabrication processes (e.g. chemical-mechanical planarization), the heterostructure growth conditions (e.g. the temperature, pressure, and presence of impurities), the material stack of the heterostructure layers, strain, and so on.

Experimentally, these rough surfaces are fractal in nature, meaning that the fluctuations of the interface height around its mean exhibit self-similarity across lengthscales spanning several orders of magnitude [CTG+24, GFW+85]. In QTCAD, a rough surface height profile \(h\left(x,y\right)\) is thus characterized by five parameters:

the mean height \(\left< h \right>\);
the root-mean-square amplitude of height fluctuations \(\sqrt{\left< \left( h - \left< h \right> \right)^2 \right>}\);
the Hurst exponent \(H\), which is related to the fractal dimension \(D_f\) through \(D_f = 3 - H\) (note that \(2 \leq D_f \leq 3\));
the minimal lengthscale \(\lambda_{\mathrm{min}}\) over which self-similarity is observed, which is typically on the order the the interatomic distance;
the maximal lengthscale \(\lambda_{\mathrm{max}}\) over which self-similarity is observed, which is typically set by the relevant experimental fabrication processes, such as chemical-mechanical planarization.

These parameters are defined when creating a RoughSurface object in QTCAD, specifically as the arguments of its constructor method.

To generate a random rough surface from these five parameters, we use the following formula [IJLB+24]:

(2.13)\[h\left(x,y\right) = C \sum_{n=1}^{2^m} G_n \left\lVert \mathbf{k}_n \right\rVert^{-2\left(1+H\right)} \cos \left[ 2\pi \mathbf{k}_n \cdot \left(x,y\right) + U_i \right]\,.\]

In this formula:

\(C\) is a constant chosen so that the root-mean-square amplitude of height fluctuations is equal to the desired value;
\(m\) is the base-\(2\) logarithm of the number of wavevectors \(\mathbf{k}_n\) sampled to generate the rough surface;
\(G_n\) is randomly sampled from a Gaussian distribution with zero mean and unit variance;
\(\mathbf{k}_n\) is sampled within an annulus of inner radius \(\frac{2\pi}{\lambda_{\mathrm{max}}}\) and outer radius \(\frac{2\pi}{\lambda_{\mathrm{min}}}\);
\(U_i\) is randomly sampled from a uniform distribution between \(0\) and \(2\pi\).

For illustration purposes, we show below a random rough surface generated in QTCAD and use it to construct a Si–Ge heterostructure. The 2D contour and 3D surface plots of this rough surface are generated using the plot method with the argument type set to 2D and 3D, respectively.