## 3 Particle-In-Cell Simulation Results  
To verify the efficiency of the stable-radiation-pressure ion acceleration by using a foil + NCD combination target, we carry out a series of 2D PIC simulations using the code EPOCH [42]. The circularly polarized (CP) laser pulse has a wavelength \(\lambda=1\)\(\upmu\)m. Its amplitude profile is \(a(t,y)=a_{0}exp(-t^{2}/\tau^{2})exp(-y^{2}/R^{2})\). In typical simulations, a 30 fs laser pulse has the amplitude \(a_{0}=100\) and the waist \(R=6\lambda\). The wavefront of the laser pulse arrives at the vacuum-foil interface \(x=0\) at \(t=0\). The thin foil is composed of carbon ions \(C^{6+}\) and protons \(H^{+}\) in a 1:2 number ratio with \(n_{e}=n_{H^{+}}+6n_{C^{6+}}=200n_{c}\). The semi-infinite NCD plasma only contains \(C^{6+}\) and \(e^{-}\) with a uniform density profile \(n_{e}=2n_{c}\). In each simulation, a 30 \(\upmu\)m \(\times\) 44 \(\upmu\)m simulation box moves along the \(x\)-axis at the speed of light and is divided into 3600 \(\times\) 3520 cells. The box parameter set as \(-29\)\(\upmu\)m \(\leq x\leq 1\)\(\upmu\)m in x-direction at \(t=0\). The thin solid foil in the front region (\(0\leq x\leq 0.085\)\(\upmu\)m) and a semi-infinite NCD plasma at \(x>0.085\)\(\upmu\)m. Each cell has 225 macro-particles in the foil region and 25 macro-particles in the semi-infinite plasma region. Figure 2a,b display the energetic ion density distribution for the cases without and with the NCD plasma, respectively. In the laser interaction with the thin foil, although the protons in the focal spot area can be accelerated by the light-sail mode of RPA, the efficient light-sail acceleration ceases as the thin foil deforms and becomes transparent. So, as shown in Figure 2a,c, the protons are expanded to have a low density of a few \(n_{c}\) even in the laser propagating axis and the achieved proton energy spectrum is broad. By contrast, when the NCD plasma is adopted, the proton density is as high as \(20n_{c}\) around the laser propagating axis at \(t=40T_{0}\), as shown in Figure 2b. At this time, the protons are still accelerated and they have quasi-monoenergetic spectrum with the peak of \(\sim\)1.2 GeV (see Figure 2d), suggesting that the NCD plasma is essential to sustain a stable radiation pressure acceleration (SRPA). Meanwhile, the energy conversion efficiency from the laser pulse to the proton beam can reach up to 16% in this SRPA scheme. Figure 1: (**a**) The accelerating region of the ions in the \(\xi-p_{i}^{\prime}\) phase space with the plasma density \(n_{e}=2n_{c}\). The boundary of the accelerating region is defined by \(H(\xi,p_{i}^{\prime})=1\) (black solid lines) and an ion will be accelerated (red dashed lines) by the electric field if \(H(\xi,p_{i}^{\prime})<1\). (**b**) The required minimum momentum \(p_{i}/m_{i}c\) as functions of the background density \(n_{e}\), obtained by Equation (2). The laser normalized amplitude is taken as \(a_{0}=100\). The replenishment of cold electrons in the NCD plasma plays a crucial role for suppressing the plasma heating. In Figure 3a without the NCD plasma, the plasma temperature is as high as tens of \(m_{e}c^{2}\), where the plasma temperature \(T_{e}\) in PIC simulations is defined by the averaged kinetic energy of electrons in the center-of-momentum frame [46]. This considerable plasma heating causes the thermal expansion of the plasma and the decrease in the plasma density, which is indicated by the black and blue lines in Figure 3a. Then, the light-sail RPA process is severely disturbed and the quality of accelerated protons will also be destroyed in the later interaction period. While the NCD plasma is adopted, the plasma heating is significantly alleviated, as observed in Figure 3b. The plasma temperature is reduced by a factor of four inside the plasma peak and about halved at the laser-plasma interface.