Theoretical Studies on the Quantum Capacitance of Two-Dimensional Electrode Materials for Supercapacitors

Abstract

In recent years, supercapacitors have been widely used in the fields of energy, transportation, and industry. Among them, electrical double-layer capacitors (EDLCs) have attracted attention because of their dramatically high power density. With the rapid development of computational methods, theoretical studies on the physical and chemical properties of electrode materials have provided important support for the preparation of EDLCs with higher performance. Besides the widely studied double-layer capacitance (C_D), quantum capacitance (C_Q), which has long been ignored, is another important factor to improve the total capacitance (C_T) of an electrode. In this paper, we survey the recent theoretical progress on the C_Q of two-dimensional (2D) electrode materials in EDLCs and classify the electrode materials mainly into graphene-like 2D main group elements and compounds, transition metal carbides/nitrides (MXenes), and transition metal dichalcogenides (TMDs). In addition, we summarize the influence of different modification routes (including doping, metal-adsorption, vacancy, and surface functionalization) on the C_Q characteristics in the voltage range of ±0.6 V. Finally, we discuss the current difficulties in the theoretical study of supercapacitor electrode materials and provide our outlook on the future development of EDLCs in the field of energy storage.

1. Introduction

In today’s world, non-renewable energy sources are decreasing. Energy supply is closely related to environmental issues and basic human needs. Improving the conversion efficiency of energy sources and developing new energy sources has become an urgent problem that needs to be solved in the midst of the energy crisis [1]. In this energy-dependent world, electrochemical devices for energy storage have played a crucial role in overcoming the depletion of fossil fuels [2]. Compared with conventional batteries, supercapacitors have the advantages of high power density, long cycle life, and fast charge and discharge rates. However, the energy density of supercapacitors is usually low, which is a major obstacle to their development [3]. Supercapacitors are usually classified into three types: (1) electrical double-layer capacitors (EDLCs) with ion adsorption through the electrode surface; (2) pseudocapacitors with surface Faraday redox reactions on the electrodes; (3) hybrid supercapacitors that are a mixture of the above two [4,5]. In this paper, we mainly focus on the electrode materials for EDLCs.

For the charging process on EDLCs, the anions and cations in the electrolyte are adsorbed to the positive and negative surfaces, respectively, forming a double-layer due to the external voltage difference. After charging, the anions on the double-layer produce a potential difference between the two plates to store energy. Discharging is the opposite process of charging. Because of the fast rate of this simple physical adsorption process, the EDLCs usually have high power densities. For the electrode materials, carbon materials with a large specific surface area and good electrical conductivity are generally the best choice for the fabrication of EDLCs [4,6,7,8].

EDLCs have high output power, fast charge and discharge rates, and long service lives but poor energy density [9,10,11]. Therefore, increasing the energy density of EDLCs has become a key research goal. The energy density of EDLCs is determined by the operating voltage and the specific capacitance of the electrode/electrolyte system [12]. The total interface capacitance (C_T) of EDLCs is related to the quantum capacitance (C_Q) and the double-layer capacitance (C_D), with the expression of 1/C_T = 1/C_Q + 1/C_D [13,14,15,16,17]. C_Q, also known as electrode capacitance, reflects the finite quantum state process of the electron-filled system [18,19]. The theoretical prediction of increasing the total EDLC capacitance by increasing the C_Q of the electrode material has been experimentally confirmed [20]. C_Q is proportional to the density of electronic states. A large number of quantum states near the Fermi level can lead to a high C_Q. The electronic structure of a material can be modified by changing the dopants, functional groups, defects, etc. of the structure, thus changing the specific surface area and surface morphology of the electrode material. The larger the specific surface area, the better the energy storage performance of the electrode material [21,22].

Usually, materials with a thickness of a few atomic layers are considered as two-dimensional (2D) materials [3]. Since the discovery of graphene in 2004, 2D materials have experienced rapid development. For example, 2D carbon materials demonstrate excellent properties, such as high specific surface area and high electrical conductivity [23]. Since the working mechanism of EDLCs is an electrostatic effect, the anions and cations on the electrode material surface move to the positive and negative electrodes during the charging and discharging process, forming electric double-layers at the interface. Thus, 2D electrode materials with larger specific surface areas are more suitable for EDLCs than their 3D counterparts [24,25,26,27]. However, 2D electrode materials are highly susceptible to stacking due to their high edge activity, resulting in a decreased specific surface area and capacity during reagent application. Numerous studies have been devoted to maintaining or increasing the specific surface area of the electrode materials and improving the circulation rate of anions and cations by changing the morphology of the electrode surface. In this paper, we focus on the aforementioned modification measures (defects, doping changes, adsorption of functional groups, etc.) and atomic exchange on the electrode materials. We summarize the classification of electrode materials and highlight the materials with better performance and greater potential for experimental application.

2. Theoretical Basis

Among many low-dimensional materials, the differential quantum capacitance (C_diff) can be defined by

$$C_{diff}=\frac{d\mathsf{\sigma }}{d{\Phi }_{\mathrm{G}}}={\mathrm{e}}^2\mathrm{D}\mathrm{O}\mathrm{S}\left(-{\mathrm{V}}_{\mathrm{e}}\right)$$

(1)

in which dσ and dΦ_G represent the differential charge density and differential local potential, respectively.

Thus the magnitude of C_diff is dependent on the density of states. The density of states is essentially the number of different states that an electron is allowed to occupy at a given energy level, i.e., the number of electron states per unit volume of energy. Due to quantum confinement effects and the limitation of the low density of states, the significant movement of Fermi levels in two-dimensional materials could accumulate a sufficient number of carriers to provide better energy density, thus improving the performance of supercapacitors.

The excess charge density can be expressed as:

$$∆Q={\int }_{-\infty }^{+\infty }D\left(E\right)\left[f\left(E\right)-f\left(E-e{\varphi }_G\right)\right]dE$$

(2)

where D(E) represents the density of states of the system, ƒ(E) is the Fermi-Dirac distribution function, E is the electronic energy with respect to the Fermi level, and e is the fundamental charge. For 2D materials, C_diff can be obtained by the following equation:

$$C_{diff}=e^2{\int }_{-\infty }^{+\infty }D\left(E\right)F_T\left(E-e{\varphi }_G\right)dE$$

(3)

where F_T(E) is the thermal spreading function, which is obtained from the following:

$$F_T\left(E\right)={\left(4{\kappa }_{{\rm B}}{\rm T}\right)}^{-1}{sech}^2\left(\frac{E}{2{\kappa }_{{\rm B}}{\rm T}}\right)$$

(4)

It is also common for researchers to analyze the energy storage capacity of supercapacitors by calculating C_int. The C_int is obtained by integrating the C_diff over the charge and discharge cycles [28,29].

$$C_{int}(V)=\frac{Q}{V}=\frac{1}{V_e}{\int }_0^V{C}_{diff}(V^{\prime })d{V}^{’}$$

(5)

In this paper, C_diff is equally defined as C_Q.

3. Research Progress of 2D Electrode Materials for EDLCs

3.1. Graphene-like 2D Main Group Elements and Compounds

Graphene with sp² hybridization is a typical representative of 2D materials. [30] In the past decade, graphene-based electrode materials have become a popular research direction for supercapacitor electrode materials. More recently, scientists have tried to discover various analogs with six-membered ring structures that are similar to graphene, such as silicene, germanene, phosphorene, etc. On the other hand, the 2D main group compounds with graphene-like structures, such as 2D carbon nitride (CN), have also been synthesized in recent years. There are also many studies on the performance of 2D CNs as electrode materials.

3.1.1. Graphene

Graphene is prone to stacking due to its high edge activity, resulting in a decrease in specific surface area and capacity. Since graphene oxides are usually used as precursors for graphene preparation, there are many vacancies in graphene-based materials. Many studies have focused on maintaining the dispersion of graphene [31,32].

In 2009, Xia et al. measured the C_Q of monolayer and bilayer graphene, and the curve of C_Q-potential was symmetrical and v-shaped (Figure 1a) [33]. Many scientists have experimentally demonstrated that the capacitance of carbon electrodes can be improved by doping nitrogen (N) atoms or the functionalization of N-containing groups [34,35,36,37,38,39,40,41]. Zhang et al. demonstrated that N-doping changes the electronic structure of graphene and increases the carrier density, which changes the C_Q and leads to an increase in the interfacial capacitance value (Figure 1b) [42]. Yang et al. theoretically investigated the effects of N-doping configuration, N-doping concentration, vacant concentration, and transition metal atoms (Cu, Ag, Au) adsorption on the electronic structure and C_Q of graphene [14]. Their results show that N-doping, vacancy defects, and transition metal atom adsorption can significantly enhance the C_Q of graphene. Among them, the maximum value of C_Q increases from 32.68 to 113.1 μF/cm² as the N-doping concentration increases from 1.4% to 12.5% (Figure 1c,d). Mousavi-Khoshdel et al. investigated the changes in C_Q of functionalized graphene with monovalent functional groups (-C₆H₅, -C₆H₄NH₂, -C₆H₄NO₂, -NH₂) and divalent functional groups (-C₆H₄, -C₆H₂F₂, -C₆H₂Cl₂, -C₆H₃CH₃) [43]. Their results show that the C_Q values of functionalized graphene are higher than that of pristine graphene in both cases. A schematic diagram of the structures of the three types of groups is shown in Figure 1e. Chen et al. investigated the interaction and C_Q of N and S co-doped graphene [44]. The maximum C_Q of pristine graphene is 14 μF/cm² and the minimum C_Q is 2.5 μF/cm². The C_Q values of pyridine-N-doped graphene and pyrrolic-N-doped graphene at the Fermi level are about 41.4 and 38.2 μF/cm², respectively. In a subsequent study, they found that the C_Q value of N/S co-doped graphene could be higher than that of single N-doped graphene, with the highest C_Q value being 95.8 μF/cm². However, the C_Q does not improve more when another N or S atom is added to the co-doped system (Figure 1f).

Hirunsit et al. studied the C_Q variation of Al-, B-, N-, and P-doped single-vacancy (V) and multilayer graphene. [28] They showed that Al₁, V, Al₃V, and N₃V modification can increase the C_Q by a large amount (>40 mF/cm²), and the N₃V structure showed the highest C_Q value, which was 82.18 mF/cm² (0.26 V). The construction of multilayer graphene also improves the C_Q (Figure 1g–i). Hu et al. investigated the effect of transition metal (Mn, Fe, Co, Ni) and N atom (TMN_x, x = 1–4) co-doping on the C_Q of graphene [45]. The co-doped systems showed an increase in C_Q, with a maximum value of 180.50 μF/cm² for CoN₂ g at −0.3 V (Figure 1j). A similar study was carried out by Wang et al., who explored the C_Q changes of transition metals after in-plane doping and out-of-plane doping on graphene [46]. Their conclusions show that the C_Q of in-plane doping is larger than that of out-of-plane doping, where the charge (Q) of Sc-doped graphene could reach 85 μC/cm² at negative bias (Figure 1k). Song et al. studied the variation of C_Q of epoxy (- O -)- and hydroxy (-OH)-modified graphene oxide [47]. The results show that the modified graphene oxide also has a higher C_Q than the original structure. There is a significant increase of C_Q with the increasing oxidation degree on both positive and negative bias (Figure 1l,m).

Sruthi et al. found that the C_Q of graphene can be significantly enhanced by doping on the pristine graphene surface with N, Cl, and P atoms [48]. Additionally, very large C_Q (>600 μF/cm²) can be achieved when doping N, Cl, and P atoms near room temperature. Xu et al. investigated the C_Q of graphene doped/co-doped with B, N, P, S atoms and vacancy [49]. They also obtained the C_D in a classical 1 M NaCl aqueous solution by using molecular dynamics simulations. Then, the C_T was calculated. Graphene that has been 3N-doped with a single vacancy is supposedly the best candidate as an EDLC electrode (Figure 1n). Zhou et al. investigated the effects of doping (B, N, Al, Si, P, S), vacancies, and Stone–Wales defects on the C_Q of graphene and found that Stone–Wales defects could also improve the C_Q of graphene, but not better than doping or vacancy. The maximum C_Q of Si-VG is 169.76 μF/cm² at −0.29 V. The maximum C_Q is 168.90 μF/cm² at −0.06 V, when the VG concentration is 5.9% (Figure 1o,p). Zhang et al. determined a variety of materials suitable for supercapacitor applications by systematic calculations and generalizations [50]. They explored the C_Q of 56 species of transition metal atoms and vacancy-doped/co-doped graphene, named TM@G and TM@VG, respectively (Figure 1q). Sruthi et al. explored the effect of different co-doping ratios on the C_Q of graphene [51]. When the dopant ratio C:O:N is 50:8:4, the C_Q of the system at the Fermi energy level can reach 423.73 μF/cm² (Figure 1r).

One of the inevitable problems in manufacturing and using graphene materials is the stacking of layers, which significantly affects the structure the electrochemical properties. Cui et al. explored the effect of stacking on multilayered graphene [52]. They assumed a two-layer ab-stacked graphene model, where the top layer is defective and the bottom layer is perfect. They showed that the C_Q of the pristine bilayer graphene increases linearly with voltage, reaching a maximum value of 37.7 μF/cm² at 1.0 V. The peak of D2_III has a maximum C_Q of 56.1 μF/cm² at a voltage of 1 V (Figure 1s,t). Zhou et al. explored co-doping with N, P, S and transition metals (Ti, V, Cr, Mn, Co, Ni) in monolayer and multilayer graphene [53]. Their study showed that doping with transition metals (TM) improves the C_Q more than co-doping with N, P, and S, and the Ti/Ni and N/P/S co-doped systems exhibit excellent C_Q. However, the C_Q of the multilayer system decreases due to the interactions between the adjacent layers of dopants. In a study by Zeng et al., it was found that the capacitance of B (N)-doped graphene as an anode (cathode) can reach a record C_Q of 4317 F/g (6150 F/g) [54].

3.1.2. Silicene

Inspired by graphene, silicene is made from 2D layered nanosheets. Silicene sheets with different structures have been successfully synthesized on various substrates. Silicene with a buckling layer structure has a high surface area [55]. It is considered as an excellent anode material for Li-ion batteries because it has enough space to adsorb Li-ions and prevents structural breakage induced by the insertion of Li-ions. Similar to graphene, it is also expected to be one of the ideal electrodes for EDLCs.

Yang et al. explored the effects of vacancy and dopants (N, P, B, and S) concentration on the C_Q of silicene [56]. Their results show that the maximum C_Q of silicene increases with the defect concentration from 1.91 μF/cm² at −0.38 V to 102.65 μF/cm² at −0.19 V. When the pyridine-N doping concentration is 5.6%, the maximum C_Q is 73.28 μF/cm² (−0.07 V). The C_Q is higher than that of the pristine silicene in all modified structures (Figure 2a). Momeni et al. explored the C_Q of pristine silicene, defective silicene, and XSi₃-like silicene (X = Al, B, C, N, P) structures [57]. Their results show that the alternative doped XSi₃-like silicene structures have higher C_Q compared to pristine silicene (C_Q = 1200 F/g) and graphene (C_Q = 500 F/g). The AlSi₃ system reaches a maximum C_Q of 2573 F/g under positive bias. They also showed that the large C_Q of XSi₃-like silicene originates from the high electronic states at the Fermi level of 2p and/or 3p orbitals of X and Si atoms, as evidenced by projected density of state analysis (Figure 2b,c). Xu et al. explored the C_Q of silicene with metal atom (Ti, Au, Ag, Cu, and Al atoms) adsorption and single-vacancy doping. [58] It was found that a single vacancy with metal adsorption can significantly increase C_Q. When the Ti concentration is increased from 2% to 12.5%, the maximum value of C_Q increases from 52.2 μF/cm² at −0.12 V to 132.2 μF/cm² at 0.12 V (Figure 2d).

3.1.3. Germanene

Silicene and Germanene are of great interest as 2D layered nanosheet materials inspired by graphene. Germanene is more prominent than silicene and graphene in terms of its spin–orbit interaction. The large spin–orbit gap (24 meV) of germanene makes it a typical alternative material with the quantum spin Hall effect [59,60,61,62,63]. Moreover, germanene is more easily to be functionalized and has been synthesized by different chemical methods [64,65,66,67]. In order to further investigate the electrochemical properties of germanene and probe for more superior performance electrode materials, numerous researchers have investigated the C_Q of germanene with doping, co-doping, and vacancy defects.

Si et al. explored the effect of single vacancy (SV), adsorption of Ti, Au, Ag, Cu, Al atoms, and different doping concentrations on the C_Q of germanene [61]. Similar to graphene and silicene, vacancies can increase the C_Q of germanene, especially in the positive bias range. The C_Q of Ti- and Cu-doped SV germanene is superior to that of Au-, Ag-, and Al-doped ones. Moreover, Ti-doping is more stable in graphene, silicene, and germanene (Figure 2e). Zhou et al. found that transition metal (Ti, Cr, Mn, and Co) doping enhanced the C_Q better than B/N/Al doping [68]. The maximum C_Q can reach 91.47 μF/cm² (0.2 V) for Ti-doping near the Fermi level. The co-doped system improves C_Q more than single-doping (Figure 2f). Si et al. further investigated the effects of doping/co-doping, vacant defects, and multilayer structure on the electronic structure and C_Q of germanene [69]. Their results show that N-doping can significantly improve the C_Q of germanene. In a study of single and multilayered germanene co-doped with NAl, NNAl, NPAl, and NSAl, it was found that the interlayered interactions contributed more to the increase in C_Q (Figure 2g).

3.1.4. Stanene

Stanene is a novel material that has received increasing attention in recent years. It has been successfully realized by epitaxial growth on Bi₂Te₃ (111) substrates [70,71,72]. Stanene exhibits several remarkable features, including large spin–orbit gaps, topological superconductivity, quantum anomalous Hall behavior, giant magnetoresistance, and efficient thermoelectricity [73]. Additionally, there have been numerous studies showing that measures such as doping with metal atoms can have a large effect on the structural and electrochemical properties of Stanene [74,75].

Zhou et al. verified the effect of vacancies and the single-doping and co-doping of light element atoms (B, N, Al, Si, P, S) and transition metals (Ti, V, Cr, Mn, Fe, Ni) on the geometry, electronic structure, and C_Q of stanine [76]. Their results show that vacancy, doping, and co-doping can improve the C_Q of stanene and that co-doped defective stanene exhibits better C_Q at negative potentials than at positive bias, indicating that it can be used as a good anode material. The maximum C_Q of BFeVSn is 76.52 μF/cm² (0.29 V) under positive bias conditions (Figure 2h,i). Zhou et al. also investigated the effect of N/P/S and line co-doping with heavy metals (Ti, V, Fe, Ni) on stanine [77]. The effect of line co-doping to improve the C_Q of the system is more obvious, where the maximum C_Q at a positive bias of SSTiSn is 77.18 μF/cm², which could be attributed to the increased electronic states of the Ti dopant and adjacent Sn atoms (Figure 2j–l).

3.1.5. Boronene

Due to the high carrier concentration, boronene has been used in plasma devices, extending the functionality to the visible region. Boronene is predicted to be an excellent candidate for Li-ion batteries due to its high Li capacity [78,79,80].

Kolavada et al. theoretically analyzed the C_Q of δ-6 boronene with different layer numbers in aqueous electrolytes (AEs) and ionic liquid electrolytes (ILEs) [81]. In both AE and ILE systems, C_Q enhances as the number of layers increases from 1 monolayer (ML) to 4 ML. When the number of layers is 4 ML, C_Q can reach more than 600 μF/cm² in both systems (Figure 2m).

3.1.6. Phosphorene

Phosphorene is a relatively new member to the group of 2D materials discussed in this study. Its strong in-plane anisotropy makes phosphorene a unique material for novel electronic devices [82,83,84,85,86]. Zu et al. fabricated supercapacitors by using phosphorene as electrodes and with the discharge capacity of 3181.5 F/g in a three-electrode configuration [87].

Ramesh et al. computationally examined the effect on phosphorene when half-metal (Si)-dopants, active-nonmetal (S)-dopants, and two transition metal (Ti, Ni) dopants replace the P atom [88]. The C_Q of pristine phosphorene is approximately symmetric, with a minimum value of 2.27 μF/cm² at the Fermi level. The C_Q of all substitution systems is higher than that of the pristine phosphorene, with the highest C_Q value of 92.1 μF/cm² for Ti-doping at 0.4 V (Figure 2n).

3.1.7. Main Group Compounds

In recent years, four 2D carbon nitride (CN) structures, hg-C₃N₄, tg-C₃N₄, C₂N, and C₃N have been experimentally synthesized to further enrich the 2D electrode materials for supercapacitors [89,90,91]. These CN structures have high specific surface areas and excellent electrochemical stability. Therefore, CNs are considered good electrode materials.

Chen et al. investigated the effects of B and O doping on the electronic properties and C_Q of 2D CNs. They found that doping with B or O could convert CNs from semiconductors to metals, thus improving the electrical conductivity [92]. The C_Q values of B-doped CNs are all higher than those of B-doped monolayer graphene. The increased C_Q can mainly be attributed to the strong hybridization between the dopant and the adjacent C and N atoms (Figure 2o,p).

Majdi et al. investigated the electrochemical properties of a new 2D Fe-doped boron carbide monolayer (FBC₃ML) [93]. The maximum C_Q of FBC₃ML increases to 150.09 μF/cm² compared to the original BC₃ML, and the C_Q-V curve becomes symmetric (Figure 2q).

3.2. Transition Metal Carbides or Nitrides (MXenes)

MXenes are 2D-layered materials derived from transition metal carbides, nitrides, or carbonitrides [94]. MXenes can be produced by selectively removing the A-layer from MAX phases, the 3D precursors of MXenes, noted as M_n+1AX_n phases (n = 1, 2 and 3). MAX phases are generally divided into three types: 211, 312, and 413 structures. M denotes early transition metal elements (such as Sc, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, etc.); A denotes elements of group 13 or 14, such as Al or Si; and X refers to C, N, or their mixtures [2]. MXenes are often used in the field of energy storage because of their special physical and chemical properties.

The first MXene, Ti₃C₂, was isolated from Ti₃AlC₂ powder by immersing it in a hydrofluoric acid solution [94]. Subsequently, many MXene family members have been synthesized using selective etching methods and many new MXene structures have been theoretically predicted [2]. When performing chemical etching methods, the metal atoms on the MXene’s surface can easily react with -H, -O, -F, and -OH groups in solution and terminate them on the MXene’s surface, thus giving rise to functionalized MXenes, M_n+1X_nT_x, where T is the surface termination group [95,96,97]. This surface functionalization usually has an impact on the energy storage capacity of MXenes. Since the MAX phases usually have carbon vacancies (V_C) [98,99], MXenes derived from the MAX phases are considered to have the same nature of carbon vacancies. The treatment of MXene materials via doping and vacancy also cause changes in the C_Q of the materials. The current research on the C_Q of MXene materials after modulation is more comprehensive. In this section, the effect of various modulation means on the C_Q performance of MXene electrode materials will be discussed separately according to the different M elements.

3.2.1. Ti_n+1C_nT_x

As the first successfully prepared MXene, Ti₃C₂ and its isomer Ti₂C have received increasing attention as electrode materials for supercapacitors [100,101].

Si et al. focused on the modulation of the two Ti-C MXenes materials using doping, vacancy, and adsorption methods [102]. Their calculations show the pristine structures have higher C_Q compared to the functionalized Ti₃C₂ and Ti₂C. The C_Q of the functionalized structures decrease in an order of OH > F > H > O. The maximum C_Q of OH groups adsorbed on Ti₃C₂ and Ti₂C near the Fermi level is 264.414 μF/cm² (0 V) and 276.960 μF/cm² (−0.12 V) (Figure 3a,b). On the other hand, the adsorption of metal atoms on the surface of Ti₃C₂ and Ti₂C can also change their C_Q considerably. It is shown that on Ti₃C₂, the adsorption of Al atoms significantly increases the C_Q (398.193 μF/cm² at −0.072 V) due to the increase of the low potential local electronic states. A similar trend also appears for Ti₂C, where Ti₂C-Al has a maximum C_Q of 444.192 μF/cm² (0.312 V) near the Fermi level (Figure 3c,d). Furthermore, they have found that the adsorption of Ca atoms on Ti₃C₂F₂ significantly improves the energy storage performance of the system with a C_Q value of 488.153 μF/cm². Nevertheless, the C_Q of the Ti₂C system shows no clear change (Figure 3e,f) [102].

Bafekry et al. investigated the oxygen vacancies in the Ti₂CO₂ monolayer and confirmed its semi-metallic properties by calculating the density of states of the system [103]. To further investigate the effect of O vacancy concentration on the properties of Ti₂CO₂, Su et al. systematically explored the C_Q of Ti₂CO₂ with different O vacancy concentrations [104]. Their results demonstrate that the O vacancy concentration has a strong effect on C_Q. The pristine Ti₂CO₂ has a low C_Q under negative bias, with a maximum value of 4204.4 μF/cm² (0.53 V). When one oxygen vacancy (5.56%) is introduced, the maximum C_Q values increase to 4263.85 μF/cm² (−0.1 V) and 4142.0 μF/cm² (0.43 V) for negative and positive potentials, respectively. However, the maximum C_Q decreases when two or three oxygen vacancies are introduced. The DOS of Ti₂CO₂ near the Fermi level mainly originates from O-p_z and Ti-d orbitals. They speculate that the introduction of oxygen vacancies increases the charge transfer between adjacent O and Ti atoms (Figure 3g,h) [104]. While Li et al. theoretically investigated the C_Q of Ti₂CO₂ monolayer with carbon vacancy line (CVL) [105]. The introduction of CVL improved the C_Q and Q of the system under negative and positive bias within ±0.6 V. The CVL4 system improves the maximum C_Q of 469.7 μF/cm² under negative bias compared to the pristine Ti₂CO₂ monolayer (Figure 3i). Li et al. calculated the C_Q of pristine Ti₂CO₂ (PT) and C-vacant Ti₂CO₂ (VT) monolayers adsorbed by Li atoms. [106] From their results, it can be seen that the maximum C_Q of PT-LT monolayer is 10,993 μF/cm² (0.48 V), the maximum C_Q of VT monolayer is 7592 μF/cm² (0.36 V), and the maximum C_Q of VT-LC1 is 6866 μF/cm² (0.54 V), within the positive bias voltage (Figure 3j,k).

3.2.2. Sc_n+1C_nT_x

Sc-based MXenes have the lightest M atom. Among them, Sc₂CF₂ is a semiconductor with strong anisotropic carrier mobility and thermal conductivity. The electron mobility of Sc₂CF₂ in the zigzag direction is almost four times higher than that of phosphorene in the armchair direction, and its thermal conductivity is higher than that of most low-dimensional metals and semiconductor materials. Due to its excellent properties in electronic devices, Sc₂CF₂ has received much attention in recent years [107,108].

Cui et al. studied the exchange defects of Sc/F, Sc/C, and C/F atoms (Figure 4a) [109]. Their study shows that the atomic exchange has little effect on the semiconducting properties. Additionally, the maximum C_Q of pristine Sc₂CF₂ under negative bias is 739.39 μF/cm² (−0.48 V). The atomic exchange of C/F atoms and C/Sc atoms reduces the maximum C_Q of Sc₂CF₂ monolayer under negative bias to 488.60 μF/cm² (−0.44 V) and 282.57 μF/cm2 (−0.48 V), respectively. The atomic exchange between F and Sc atoms increases the maximum C_Q of Sc₂CF₂ monolayer under negative bias to 1037.76 μF/cm² (−0.48 V). Cui et al. also studied the C_Q of Sc₂CT₂ (T = F, P, Cl, Se, Br, O, Si, S, OH) monolayers (Figure 4b) [110]. They found that the maximum C_Q of the Sc₂C monolayer in aqueous electrolyte (±0.6 V) is 1025.01 μF/cm² and 1297.03 μF/cm² under negative and positive bias, respectively. Compared with the original Sc₂C structure, the maximum C_Q under the negative bias of Sc₂CT₂ (T=P, Cl, Se, Si) increases, with the maximum C_Q of Sc₂CP₂ being 3800.34 μF/cm². The maximum C_Q of the Sc₂CSi₂ monolayer increases to 1708.82 μF/cm² under positive bias voltage, but the maximum C_Q of all other systems decreases under the positive bias voltage. However, all of them have a larger maximum C_Q than the system with Sc₂CF₂ for atomic exchange.

Subsequently, Cui et al. theoretically studied the C_Q of Sc₂CF₂ with different atomic vacancies (Figure 4c) [111]. They observe that, at positive bias, pristine Sc₂CF₂ has almost no C_Q, while the introduction of vacancies increases it. The maximum C_Q of Sc₂CF₂-VF at positive bias is 493 μF/cm² (0.40 V). It is noteworthy that the C_Q of all systems with vacancies at the Fermi level is larger than that of the original system. Rui et al. investigated the C_Q of doped Sc₂CF₂ with 13 transition metal atoms (Figure 4d) [112]. Fe-doped Sc₂CF₂ shows a symmetry C_Q-V curve, with a maximum C_Q of 5407.6 μF/cm² at 0 V. All the 4d transition metal atoms-doped Sc₂CF₂ structures show asymmetric C_Q-V curves. The maximum C_Q of the Mo-doped system at negative bias is 6917.88 μF/cm² at −0.2 V. For the Nb-doped system, the maximum C_Q is 2599.72 μF/cm² at 0.52 V under positive bias (Figure 4e). For 5d transition metal dopants, the maximum C_Q is 1833.15 μF/cm² (0.48 V) in the Re-doped system. Cui et al. also studied the geometry, electronic properties, and C_Q of Sc₂CF₂ with intrinsic defects [113]. They found that the C_Q fluctuations are more pronounced for systems with defects. Among them, the maximum C_Q of Sc₂CF₂-dF→C and Sc₂CF₂-dC→Sc monolayers are 924.69 μF/cm² (−0.48 V) and 1024.03 μF/cm² (0.56 V), respectively (Figure 4f). In their study, the charge (Q) of the Sc₂CF₂-dF→C monolayer was mainly stored in the negative potential (Figure 4g).

3.2.3. Hf_n+1C_nT_x

Most functionalized MXenes have metallic properties, while Hf₂CO₂ has a moderate band gap and good thermal conductivity [114,115].

Liu et al. studied the C_Q of Hf₂CT₂ (T = -O, -F, -S, -Cl, -OH, -Se) [116]. Their results show that the maximum C_Q value of Hf₂C is 549 μF/cm² (0.56 V). At positive bias, the C_Q of the other functionalized systems (T = -O, -S, -Se) are smaller than the original system. The C_Q of Hf₂CO₂ is almost zero at positive bias. The maximum C_Q of Hf₂CSe₂ at negative bias is 564 μF/cm² (−0.56 V). The maximum C_Q of Hf₂C(OH)₂ is 804 μF/cm² at 0.44 V (Figure 5a). In experiments, mixed terminations are often randomly attached to the surface of MXenes during etching; thus, there are various configurations of surface coverage and mixed terminations. The surfaces of MXenes are often bound by mixed terminations, mainly -O, -F, and -OH [117]. Therefore, Liu et al. considered MXene groups with a mixed termination of -O, -F, and -OH [116]. The results show the symmetrical characteristics of C_Q-V curves for all three groups, with the highest C_Q at zero potential of 778.82, 552.17, and 177.97 μF/cm², respectively (Figure 5b).

Liu et al. also computationally investigated the effect of N doping concentration on the electronic properties and C_Q of Hf₂CO₂ [118]. Based on the calculation results, the C_Q of Hf₂CO₂ systems with doping concentrations of 11%, 22%, 33%, and 44% are relatively low at negative bias. The maximum C_Q of the pristine Hf₂CO₂ system (PH) is 84.06 μF/cm². The maximum C_Q values of PH-33% and PH-100% at positive bias are 423.62 μF/cm² and 441.16 μF/cm², respectively. The maximum C_Q of PH-78% is 1208 μF/cm². Thus, it indicates that the N-doping concentration also changes the C_Q of Hf₂CO₂ monolayer. (Figure 5c) Subsequently, Liu et al. further investigated the maximum C_Q of PH and the doped systems at different temperatures [118]. They noted that the maximum C_Q decreased with increasing temperature for all systems except PH-22%. Among all systems, the maximum C_Q is 1535.2 μF/cm² at 233 K for the PH-78% system (Figure 5d). As with their previous studies, they also considered the case of mixed terminals. Their results show that the maximum C_Q is increased for all other systems after N-doping. As the doping concentration increases, the C_Q-V curve becomes more symmetrical (in the range of ±0.6 V).

More interestingly, Cui et al. explored the C_Q of Hf₂CO₂ monolayers under bi-axial strain [119]. Strain is a common strategy to modulate the properties of materials, which can tune the electronic structure of the material, thus affecting many physical properties of the material. For example, it has been experimentally demonstrated that the introduction of tensile strain can lead to a transition from direct to indirect bandgap in the MoS monolayer, which expands the light absorption range and reduces the complexation of photogenerated carriers. Cui et al. have demonstrated that strain can significantly modulate the electronic structure of Hf₂CO₂ monolayers, which has a very important impact on the material properties [119]. The results show that the maximum C_Q values of strain-free Hf₂CO₂ are 1.57 μF/cm² (0 V) and 78.99 μF/cm² (−0.6 V) under positive and negative bias, respectively. Under positive bias, the maximum C_Q of the Hf₂CO₂ monolayer increases at all strains except 3%. Under negative bias, the maximum C_Q of the Hf₂CO₂ monolayer increases at all strains except −6%, −4%, and −2% (Figure 5e). Li et al. explored the effect of adsorption of NH₃ on pristine Hf₂CO₂ and varying its biaxial stress [120]. In the range of ±0.6 V, the C_Q-V curve of Hf₂CO₂ under strain is asymmetric. The C_Q at negative potentials is significantly higher than that at positive potentials, and the maximum C_Q of Hf₂CO₂ under free strain is 38.75 μF/cm² at −0.57 V. The maximum C_Q increases gradually with increasing tensile strain and reaches a maximum of 244.27 μF/cm² at +5% strain (Figure 5f,g).

3.2.4. Zr_n+1C_nT_x

Zr₂CO₂ is an excellent functionalized MXene with many excellent properties. Due to its excellent photovoltaic properties and high hole mobility, it can be considered as a suitable photocatalyst [121,122].

Xu et al. investigated the electronic properties and C_Q of pristine, doped, and single C vacant (VC) Zr₂CO₂ [123]. The doped atoms were chosen as Y = Si, Ge, Sn, N, B, S, F (Figure 6a,b). The doped atoms had a significant effect on C_Q and Q. The maximum C_Q of pristine Zr₂CO₂ was 407 μF/cm² (−0.6 V) and 32.3 μF/cm² at the Fermi level. The introduction of C vacancies increased the C_Q at positive bias. The introduction of all the considered dopant atoms increased the maximum C_Q, and B doping at negative bias increased the maximum C_Q to 1993 μF/cm². The maximum C_Q of S-doped structure was 3293.7 μF/cm² (0.4 V). At 0 V, a significant increase in C_Q can be observed in the systems doped with VC, F, N and S atoms. Xu et al. also explored the maximum C_Q of pristine Zr₂CO₂, Zr₂CO₂-VC, and doped Zr₂CO₂ at different temperatures [123]. Similar to the trend of C_Q with temperature for N-doped Hf₂CO₂, they noted that the maximum C_Q of each of the studied Zr₂CO₂ systems decreased gradually with increasing temperature. (Figure 6c)

Li et al. investigated the effect of C, O, and Zr vacancies (V_C, V_O, V_Zr) on the C_Q of Zr₂CO₂ monolayers [124]. Their study shows that the introduction of atomic vacancies increases the maximum C_Q in the range of ±0.6 V. In particular, the maximum C_Q of PZ-VZr at positive bias is 586 μF/cm² (−0.30 V). The maximum C_Q of PZ-VC MXene under positive bias is 422 μF/cm² (0.53 V), while the maximum C_Q of PZ-VO MXene is 359 μF/cm² (0.57 V) (Figure 6d,e). Yin et al. investigated the C_Q of Zr₂CO₂ with an atomic exchange [125]. According to the plot of C_Q-V, they point out that, in the range of ±0.6 V, the C_Q of the pristine Zr₂CO₂ tends to be zero in positive bias, and in negative bias, the highest C_Q is 76.8 μF/cm². The highest C_Q is 737.1 μF/cm² in negative bias for the C-O1 exchanged system and 814.6 μF/cm² in negative bias for the Zr-O₂ exchanged system. The atomic exchange in Zr₂CO₂ greatly improves the top C_Q at negative bias in ±0.6 V. The Zr-O₂ exchanged system has the highest C_Q of 425.3 μF/cm² at 0 V (Figure 6f,g).

3.2.5. Nb_n+1A_nT_x

The first synthesized 2D niobium carbide was the thicker Nb₄C₃. The Nb₂C and Nb₃C₂ systems have an extremely high theoretical capacitance of Li atoms. Additionally, the surface termination has a considerable effect on the energy storage performance [126].

Xin et al. explored the C_Q properties of different thicknesses of Nb_n+1C_n (n = 2, 3, 4) [127]. They noted that the C_Q values of all intrinsic niobium carbides were higher than that of functionalized ones in the positive bias voltage range. Except for Nb₅C₄, the other three niobium carbides have higher C_Q values at positive bias than at negative bias. The functionalized Nb_n+1C_n shows a higher C_Q than the intrinsic system only at potentials below −0.4 V. In the positive bias range, functionalization causes a significant decrease in the C_Q of the system. To quantitatively compare the C_Q, they calculated the theoretical integrated C_Q of the positive and negative electrodes from 0 to 0.83 V and from −0.62 to 0 V, respectively. Their results illustrate that the C_Q of the intrinsic niobium carbide gradually decreases with the increasing number of layers in the positive potential region. For different thicknesses of functionalized Nb_n+1C_n MXenes, the C_Q is smaller than the intrinsic state. The C_Q of intrinsic Nb₂C is up to 1828.4 F/g at the positive electrode and 1091.1 F/g at the negative electrode (Figure 7a–c).

Transition metal nitrides, such as vanadium nitride, titanium nitride, and tungsten nitride, have been studied as electrode materials for EDLCs. It has been demonstrated that cobalt doping can increase the capacitance of niobium nitride. Bharti et al. calculated the C_Q of Nb₂N and Nb₄N₃ and investigated the effect of Co-doping on their C_Q [128]. Their calculations show that the C_Q value of Nb₂N is remarkably high (1196.28 μF/cm², −1 V) and the C_Q of Nb₄N₃ is much lower than that of Nb₂N. When they increased the number of layers of Nb₂N and Nb₄N₃, they found that the C_Q kept increasing with the more layer numbers. The C_Q of both Nb₄N₃ (Nb₄N₃-2Co) and Nb2N (Nb₂N-2Co) increase after Co-doping at the Fermi level, with the C_Q of Nb₂N-2Co reaching 1052.2 μF/cm² (Figure 7d–f).

3.2.6. Mo₂C and V₂C

Two-dimensional transition metal carbides (TMCs) have high melting points and good electrical conductivity and chemical stability [129,130,131]. With its excellent electrochemical properties, Mo₂C has been experimentally prepared as an electrode material for capacitors. In a experiment prepared by Lu et al., the capacitor with Mo₂C as the electrode material had a high specific capacitance and excellent cycling stability, and its performance was significantly better than most carbide-based asymmetric supercapacitors [132]. As early as 2015, it was shown that V₂CT_x could be used as the positive electrode of sodium ion capacitors [133]. The results of Ai et al.’s study show that the specific capacitance of V₂C in 1 M Na₂SO₄ is high (223.5 F/g) and the cycling stability is good (capacitance retention could be maintained at 94.7% after 5000 cycles) when the current density is 100 mA/g [134].

Bharti et al. discussed the C_Q of Mo₂C and V₂C [135]. They highlighted that the C_Q of intrinsic Mo₂C and V₂C reaches 3243.99 μF/cm² and 3465.51 μF/cm² at the Fermi level, respectively. In the positive potential range, the C_Q decreases rapidly and drops to 0. Similar to Nb₂N and Nb₄N₃, the C_Q of Mo₂C and V₂C enhances with the increasing number of layers. The C_Q values of both Mo₂C and V₂C at the Fermi energy decreased after O-functionalization. Similar to the pristine system, the O-functionalized V₂C and Mo₂C had higher C_Q at negative bias (Figure 7g,h).

3.3. Transition Metal Dichalcogenides (TMDs)

Transition metal-based materials are considered to have higher energy density than other materials [136,137,138,139]. Among them, transition metal dichalcogenides (TMDs) are a class of graphene-like structures that have been commonly used in terms of electrode materials for supercapacitors in recent years. Although TMDs usually store energy in the form of intercalation with alkali metals, they exhibit quantum effects that are reflected in their capacitive behavior.

MoS₂ is a typical representative of TMDs and exists in three main phases (2H, 3R, and 1T) with unique capacitive properties. Graphene-like MoS₂ materials have special structures, fast ionic conductivity, and high specific capacitance [140,141]. In addition, the electron correlation between Mo layers in the sandwich structure facilitates the carrier transport [142,143,144]. It is an excellent electrode material for supercapacitors and has attracted more and more attention.

MoS₂ was the subject of a comprehensive study by Xu et al. [145]. They first investigated the relationship between the C_Q and the potential of MoS₂ containing different dopants (where Ti, Au, Ag, Cu, and Al replace Mo atoms), single-vacancy V_Mo. Their results show that the C_Q of pristine MoS₂ is almost zero in the region near 0 V and increases at a higher voltage. For the C_Q of Al-doped MoS₂ and single-vacancy V_Mo, the local C_Q maxima near 0 V are 157.7 μF/cm² and 156.5 μF/cm², respectively. Subsequently, they observed the effect of Al doping on the C_Q in both pristine and single-vacant (VS) MoS₂ monolayers with different Al concentrations of 1.3%, 2.1%, 3.7%, and 8.3%. The C_Q value increases from 44.76 μF/cm² (0.17 V) to 227.85 μF/cm² (0.53 V) with the increase in doping concentration (Figure 8a–e). Secondly, they investigated the doped VS-MoS₂, where Ti, Au, Ag, Cu, and Al replace the S atoms. The C_Q increased in all systems except when doped with Ti, which is similar to that of S substitution in pristine MoS₂ (Figure 8f). Finally, they calculated the local C_Q maxima of 200.89 μF/cm², 132.77 μF/cm², and 254.29 μF/cm² in B-, N-, and P-substituted S atoms in the B-, N-, and P-doped MoS₂ monolayer (doping concentration kept at 3.7%), respectively. At the Fermi level, the B-doped system had higher C_Q and a clear advantage in terms of positive potential (Figure 8g). Therefore, they continued to investigate the effect of B-doping concentration on C_Q. The C_Q value increased gradually with the increase of B-doping concentration.

It is worth noting that MoS₂ should be considered as a van der Waals (vdW) 2D material. Biby et al. investigated the C_Q of multilayered MoS₂ with embedding and co-embedding in relation to vdW forces [146]. The C_Q of the three-layered 1T phases was as high as 2080 F/g, and the C_Q of 3R-MoS₂ was slightly higher than that of 2H-MoS₂ under negative bias. Subsequently, in their investigation on the effect of vdW forces on the C_Q, they pointed out that the absence of vdW forces increased the strength of the density of electric states. Thus, for 1T-MoS₂, the absence of vdW leads to a higher C_Q at positive bias, and in the case of 2H and 3R-MoS₂, the absence of vdW shifts the Fermi level, leading to a higher C_Q in the negative potential window. This finding also directly emphasizes the importance of vdW forces in the accurate calculation of 2D material properties (Figure 8h,i). Finally, they investigated the intercalation of cations (Li⁺, Na⁺, K⁺ and H⁺) in the three phases of MoS₂ and mixed Li⁺ and Na⁺ intercalation. They conclude that the C_Q of the 1T phase is increased near the Fermi level with Na⁺ intercalation. Additionally, the 2H and 3R phases have a larger improvement, mainly in the positive bias voltage. There are three different intercalation modes in the case of mixed doping, and the C_Q values of the co-intercalated system are higher than 2H Li-MoS₂ and close to 2H Na-MoS₂. The maximum C_Q of LiNa-MoS₂ (HASI) can reach 3163 F/g (Figure 8j) [146].

Irham et al. showed that the introduction of defects in h-FeS increased the C_Q at positive bias up to 2280 F/g, but the C_Q at negative bias decreased [147]. Cr-doped FeS has a maximum C_Q of 3076 F/g (0.6 V) at positive bias. P-type dopants (Co or Ni) do not significantly increase the C_Q in the positive voltage range. In their study of the integrated C_Q, they found that the C_int also changes nonlinearly with the doping concentration at positive bias, where the C_int can reach 1013 F/g with the Cr-doping concentration of 6.24%. They attribute the emergence of these nonlinear changes to the appearance of electronic off-domain states that hinder the increase of C_int (Figure 8k).

4. Conclusions

In this paper, we reviewed studies investigating the C_Q of 2D electrode materials through using theoretical calculations. In general, there are two solutions to enhance the performance of electrode materials for supercapacitors. One is to develop new electrode materials with higher performance, and the other is to modify the already found electrode materials, mainly by means of doping, adsorption, defects, atom exchange, etc. As far as the available studies are concerned, for graphene-like main group elements and compounds, all of the modification measures mentioned above can improve the C_Q of the electrode material. In contrast, in MXene materials, not all modification measures are able to improve the material performance. For example, the C_Q of functionalized Ti₃C₂ is not higher than that of the pristine structure. However, the introduction of functional groups during the preparation process is inevitable. Thus, it is necessary to investigate the C_Q of the functionalized MXenes to select the preparation precursors with the least impact on energy storage performance. There are not many studies on C_Q in transition metal-based supercapacitor electrode materials. Most TM-based materials are considered as pseudocapacitance supercapacitor materials. However, both pseudocapacitance and electric double-layer processes exist in such supercapacitors. So calculating the C_Q of TM-based electrode materials can lead to a more accurate prediction of the theoretical capacitance and provide more possibilities for the development of supercapacitors.

Nowadays, theoretical calculation plays an important role in scientific research. On the one hand, theoretical calculations can help interpret the results of existing experimental phenomena, and on the other hand, they facilitate the prediction and development of new materials. Most of the current studies on the theoretical calculation of electrode materials for supercapacitors focus on predicting new materials. Although the high performance of electrode materials has been calculated theoretically, there are still great difficulties in the preparation process. In addition, the current research only pertains to the electrode part, and the performance of the whole capacitor has not been completely considered. Further research should pay more attention to the feasibility and stability of material modification. Secondly, various variables that may cause influence should be fully considered so that the theoretical prediction is closer to the real situation. And thirdly, the amount of attention given to the overall performance and process of supercapacitors should be improved. With the advancement of science and technology, theoretical calculations have become more accurate and fast. This also provides us with a more thorough explanation of supercapacitor performance and greater feasibility for designing functional materials with various properties. We expect that theoretical calculations will provide a greater contribution to the development of supercapacitors.