## 3 PIC Simulation Results  
Simulations were performed using the fully relativistic 2D EPOCH PIC code [29] to investigate how the PCDS dynamics differ over a range of target thicknesses. In the case of the thinnest target considered (\(l\) = 10 nm), the simulation box was 5 \(\upmu\)m \(\times\) 7.68 \(\upmu\)m, with mesh cell size equal to 1 nm, and for all other targets (with thickness up to \(l\) = 500 nm), the box had dimensions of 2.2 \(\upmu\)m \(\times\) 7 \(\upmu\)m, with a mesh cell size of 4 nm. The smaller mesh size was necessary for \(l\) = 10 nm to have enough cells across the target thickness. There were 840 particles per cell per species for \(l\) = 10 nm, and a range of 440-17.6 particles per cell per species for \(l\) = 20-500 nm. The lower particles per cell numbers may induce additional noise into the system but the effect on the overall behaviour will be negligible. The pulse temporal and spatial profiles are Gaussian, with FWHM equal to 40 fs and 3 \(\upmu\)m, respectively and the peak of the pulse arrives at the target at \(t\) = 40 fs. The energy of the pulse was defined to produce a peak laser intensity of 6 \(\times\) 10\({}^{20}\) Wcm\({}^{-2}\), to match the experiment. Simulations were performed for circularly polarized light of 800 nm wavelength. The representative targets were initially defined as solid Al\({}^{13+}\) neutralized with a corresponding electron density, justified because the sharp rising edge intensity profile will result in full ionization of the target before significant expansion can occur. The initial electron temperature was 1 keV. Figure 3a,c,e,g show 2D spatial maps of the electron density and laser intensity in the 2D PIC simulations at a point in time (\(t\) = 65 fs) just after the \(l\) = 10 nm target undergoes RSIT, for stated target thicknesses. It can be seen that the laser radiation pressure compresses and drives forward the plasma electrons. As the laser pulse has a Gaussian intensity profile (both temporal and spatially), the electron velocity is peaked on axis, where the laser intensity is highest. In order to quantify changes to the velocity of the PCDS, the electron density along the laser axis is plotted as a function of time in Figure 4a,c,e,g. The relativistically corrected critical plasma density (\(n_{c}^{\prime}=\gamma n_{c}\), where \(\gamma\) is the electron Lorentz factor) was obtained and plotted as the black-dashed line. These figures thus show a time-space distribution and the velocity of the relativistically corrected PCDS is obtained directly from the gradient of the black-dashed line. The velocity acquired from this method is plotted as a function of time as the blue dot-dash line in Figure 4b,d,f,h, for the four example target thicknesses considered. The velocity of the PCDS calculated using the time-dependent HB model (Equation (1)) is shown as the solid green line. Further discussion on the time-dependency of the HB model can be found in references [22; 24]. To determine the PCDS velocity in the case of the LS mechanism, a Gaussian-like temporal profile was assumed. Substituting this profile into Equation (3), the pulse fluence, \(F\), is described as:  
\[F(t)=\int_{0}^{t}I(t^{\prime})dt^{\prime}=\int_{0}^{t}I_{0}e^{-\left(\frac{t^ {\prime}-t_{0}}{w}\right)^{2}}dt^{\prime} \tag{7}\]  
Carrying out the integral, the pulse fluence for a Gaussian pulse is equal to:  
\[F(t)=\frac{I_{0}}{2}\sqrt{\pi}w\left[\text{erf}\left(\frac{t-t_{0}}{w}\right) +\text{erf}\left(\frac{t_{0}}{w}\right)\right] \tag{8}\]  
where \(t_{0}\) is set to be equal to the FWHM laser pulse duration (40 fs) and \(w=\frac{FWHM}{2\sqrt{ln(2)}}\). Replacing this new expression for the pulse fluence in Equation (5), the LS velocity model using a Gaussian-like (time) pulse is obtained from Equation (4). The solution for each target thickness investigated is shown as the solid-red lines in Figure 4b,d,f,h. For the case of \(l\) = 200 nm, it can be seen in Figure 4h that the PCDS velocity in the 2D simulation is in very good agreement with that calculated using the HB model.