## 1 Introduction  
Diamonds are third-generation semiconductor materials that are often called ultimate semiconductors. Due to their excellent mechanical, chemical, and electronic properties [1], such as high fracture strength, corrosion resistance, high thermal stability, high breakdown field, and high carrier mobility [2] diamonds have attracted considerable attention in many fields. Doping different impurities in diamond to get the required performance is always a research hotspot, for example the Group-4 vacancy color centers in diamond can be used for quantum communication and sensor application [3]. Wide band gap (5.5 eV) makes diamond an ideal electronic material [1]. By doping donor and acceptor atoms in diamonds, the band gap can be adjusted to obtain p-type or n-type diamonds. P-type diamonds have already been successfully produced by doping with boron atoms and is considered as a mature topic [4]. However, the fabrication of high-quality n-type diamonds remains challenging [5]. Doped impurities in n-type diamonds primarily include lithium (Li), sodium (Na), oxygen (O), nitrogen (N), sulfur (S), and phosphorus (P). They are present in diamonds either as interstitial atoms (such as Li and Na) or substitutional atoms (such as O, N, S, and P) [6]. N, S, and P attracted considerable attention in previous studies [7]. Unlike N, which is proven to be a deep donor with an activation energy of 1.7 eV [8] and S with an activation energy of more than 1 eV and low doping efficiency as demonstrated by theoretical research [9], P with a relatively lower donor level (0.6 eV) is currently considered the best substitutional donor [4]. Theoretical and experimental studies prove that the diamond growth rate appreciably increases as a function of doping phosphorus and thereby reduce the production costs [10]. [PERSON] et al. [11] show that n-type diamonds can be successfully produced by doping phosphorus using microwave-enhanced plasma chemical vapor deposition (MPCVD). However, it is difficult to obtain practical n-type diamonds by doping P atoms. [PERSON] et al. [12] proved that in P-doped (111) diamond surfaces, the carrier concentration and Hall mobility at 500 K are only 3.8 \(\times\) 10\({}^{8}\) cm\({}^{3}\) and 38 cm\({}^{2}\)/V, respectively, and the Hall mobility in (001) surfaces is even lower. [PERSON] et al. report that the Hall mobility of activated carriers is very low (28 cm\({}^{2}\)/V at 500 K) and decrease as the temperature decrease (vanishing below 300 K). They suggest that unsatisfactory electrical properties may be due to crystalline imperfections caused by the incorporation of a large amount of P atoms, because crystalline perfection has a significant influence on the electrical properties [11; 13]. The Hall mobility vary with the temperature, also suggesting that crystal defects are dominant factors in electron scattering [14]. Through positron lifetime measurement, [PERSON] et al. [15] show that CVD diamond films usually contain a high concentration of vacancies (>50 ppm). The vacancy-related defects in diamonds are commonly considered as deep acceptors that act as strong donor compensation centers. Ideal P-doped diamonds should contain a large amount of P atoms and also be free of vacancies [16]. Therefore, a thorough understanding of the formation and structure of vacancy-related complexes in P-doped diamonds are important and worth studying. H atoms are also the reason for carrier compensation [17]. A large amount of H exists during CVD diamond growth [11; 13], and more than 10\({}^{19}\) cm\({}^{-3}\) of hydrogen is detected on (001) surfaces [18]. It is also necessary to study H-related complexes in detail. Theoretically, the doping mechanisms can be summarized by two methods, doping on the surface and in the crystal [19]. In previous studies, PV\({}_{x}\)H\({}_{y}\) complexes in bulk diamonds were carefully investigated [16; 20], but PV\({}_{x}\)H\({}_{y}\) complexes on growing surfaces still lack detailed research.