## 1 Introduction  
Laser-driven ion accelerators are characterized by their large accelerating field gradient and have attracted significant attention over the past two decades [1; 2; 3; 4]. Compared with conventional accelerators, laser-driven ion accelerators have a few unique features, including higher ion density and shorter bunch duration or more compact size [4; 5]. These features of such novel ion sources can be applied in various prospective applications, ranging from proton radiography for cancer treatment [6] to fast ignition of nuclear fusion [7; 8] and nuclear physics [9], etc. Aiming at these applications, quasi-monoenergetic ion beams with high energy are required. For instance, such ion beams can ignite compressed fuel to generate fusion energy [10]. Radiography for cancer treatment is a representative example, for which laser-driven ion beams should have an energy spread of only 1% [3; 4; 6]. Due to the advancements in both laser technology and targetty, a variety of schemes have been proposed and successfully achieved in recent laser-driven ion acceleration experiments [11; 12; 13; 14]. However, although the protons can be accelerated to a cut-off energy of near 100 MeV [12; 15], they are usually not monoenergetically distributed. To achieve quasi-monoenergetic ion beams, several mechanisms have been investigated, such as collisionless shock acceleration (CSA) [16; 17; 18; 19; 20], radiation pressure acceleration (RPA) [21; 22; 23; 24; 25], and RPA-wakefiled hybrid acceleration [26]. In one of the most-investigated mechanisms, light-sail mode of RPA, the acceleration of ions to ultrahigh energies is allowed, where a nanoscale solid foil could be continuously accelerated by an ultrashort, ultraintense laser pulse under the ideal conditions [21; 22; 27]. With the development of a multi-petawatt-class laser system, intensities of \(10^{22}\)\(\sim\)\(10^{23}\) W cm\({}^{-2}\) can be reached, for which the generation of monoenergetic GeV ion beams is one of the primary applications based on light-sail mode of RPA [28]. However, this model is usually plagued by strong electron heating due to the onset of transverse instabilities [29; 30; 31] and finite spot size [32]. These effects can lead to thermal expansion of the thin foil and relativistic transparency for the ultra-intense laser. As a consequence, the charge-separated field is destroyed and the obtained ion energy spectrum has a broad energy spread. Significant efforts [33; 34; 35; 36] have been devoted to overcoming the limitations of this mechanism to improve the ion beam qualities, such as transverse divergence, the maximum energy, and the energy spread. Most of these scenarios are still demanded to be verified in the future experiments with the developing laser technology. Therefore, how to produce high-energy monoenergetic ion beams is still an open question in both theory and experiment. In this paper, we propose a stable radiation pressure acceleration (SRPA) scheme by using a combination target of a \(CH_{2}\) thin foil and a large-scale near-critical-density (NCD) plasma. In the primary stage, an ultra-intense laser pulse with a Gaussian shape in transverse space illuminates the thin foil, and the protons of the thin foil are pre-accelerated to sub-relativistic velocities by the light-sail mode of RPA. With the help of the NCD plasma, the strong plasma heating can be obviously alleviated and thus the pre-accelerated sub-relativistic protons can stay in the cold background. In the NCD plasma, the laser pulse propagates relatively slowly due to the relativistically transparency and, hence, the speed of the longitudinal electric field of large amplitude is limited, which is the other key factor to further accelerate the sub-relativistic protons effectively. The accelerating field with limited speed, together with the alleviated plasma heating, guarantees efficient SRPA of the protons.