Introduction

Since \(\hbox {H}_{{3}}\)S exhibits superconductivity with a high critical temperature (\(T_{\textrm{c}}\)) of 203 K at 155 GPa1, it has been established as a blueprint for high-temperature superconducting phases under high pressure. In this context, the predicted hot superconducting phase in metal hydrides has garnered significant attention in condensed matter physics. For example, \(\hbox {LaH}_{{10}}\) was discovered to exhibit superconductivity with a high \(T_{\textrm{c}}\) of 250 K under high pressure2. Additionally, recent studies on clathrate hydrogen in metal hydride display high \(T_{\textrm{c}}\) under compression as seen in \(\hbox {MgH}_{{6}}\)3, \(\hbox {CaH}_{{6}}\)4, \(\hbox {HfH}_{{6}}\)5, \(\hbox {YH}_{{6}}\)6 Currently, investigations into the role of H-rich, such as those in compressed \(\hbox {LaSc}_{{3}}\hbox {H}_{{24}}\), reveal superconductivity with high-\(T_{\textrm{c}}\) of 331 K at 250 GPa7. In this vein, several stable hydride compounds with evolving \(T_{\textrm{c}}\) values have been successfully predicted using first-principles calculations based on density functional theory (DFT). For instance, \(\hbox {LaBH}_{{8}}\) exhibits a \(T_{\textrm{c}}\) of 126 K at 50 GPa8, \(\hbox {YBeH}_{{8}}\) shows a \(T_{\textrm{c}}\) of 201 K at 200 GPa9, and \(\hbox {YCeH}_{{8}}\) demonstrates a \(T_{\textrm{c}}\) of approximately 24 K at 100 GPa10. Very recently, \(\hbox {LaBeH}_{{8}}\) has not only been synthesized under pressure11 but has also paved the way for advancements in DFT calculations12, demonstrating a potentially remarkable high \(T_{\textrm{c}}\) of 110 K at 80 GPa11.

Most hydrogen-based compounds have been extensively studied under high-pressure conditions. Recent investigations have predicted superconductivity in alloy hydrides such as \(\hbox {Mg}_{0.5}\hbox {Ca}_{0.5}\hbox {H}_{{6}}\), which exhibits a \(T_{\textrm{c}}\) of 288 K at 200 GPa13. Similarly, \(\hbox {Sc}_{0.5}\hbox {Y}_{0.5}\hbox {H}_{{6}}\) demonstrates a \(T_{\textrm{c}}\) of 127 K at 500 GPa14. Moreover, the substitutional alloy superhydride \(\hbox {YSrH}_{{22}}\) demonstrates superconductivity with a \(T_{\textrm{c}}\) of 240 K at 175 GPa15. Ternary hydrides, such as (La,Th)\(\hbox {H}_{{10}}\), are also predicted to be high-\(T_{\textrm{c}}\) superconductors, with a critical temperature of 242 K at 200 GPa16. Building on the (La,Th)\(\hbox {H}_{{10}}\) system, studies have shown that \(\hbox {LaThH}_{{20}}\) may achieve a high-\(T_{\textrm{c}}\) of 197 K at 200 GPa. Recently, (La,Ce)\(\hbox {H}_{{x}}\) has been synthesized at pressures below 130 GPa, showing structural similarities to (La,Ce)\(\hbox {H}_{{9}}\). Notably, superconductivity was observed with a \(T_{\textrm{c}}\) of 176 K at 100 GPa17. Additionally, Bi et al.18 reported that (La,Ce)\(\hbox {H}_{{9}}\) exhibits a \(T_{\textrm{c}}\) range of 148-178 K under pressures between 97-172 GPa. Further investigations identified (La,Ce)\(\hbox {H}_{{9}}\) as a hexagonal structure, with \(\hbox {CeLaH}_{{18}}\) (simplified as \(\hbox {Ce}_{0.5}\hbox {La}_{0.5}\hbox {H}_{{9}}\)) displaying a \(T_{\textrm{c}}\) of 127 K at 120 GPa19.

Very recently, Y–Th–H ternary compounds have been predicted to exhibit superconductivity with a \(T_{\textrm{c}}\) of approximately 222 K at a pressure of 200 GPa, as observed in \(\hbox {YThH}_{{18}}\)20. Additionally, \(\hbox {YHfH}_{{18}}\) shows superconductivity with a high-\(T{_{\textrm{c}}}\) of 130 K at a pressure of 400 GPa21. The majority of ternary superhydrides show that \(\hbox {H}_{{18}}\) may form under adequate pressure conditions. This raises questions about the structural stability and superconducting properties of \(\hbox {ThCeH}_{{18}}\). If found stable, \(\hbox {ThCeH}_{{18}}\) could exhibit behavior similar to that of \(\hbox {CeLaH}_{{18}}\) and \(\hbox {YThH}_{{18}}\). Accordingly, this study explores the potential stability of \(\hbox {ThCeH}_{{18}}\) from ambient pressure to high-pressure conditions.

In this study, we conducted a systematic investigation into the stability and superconducting properties of (Th,Ce)\(\hbox {H}_{{9}}\), utilizing the chemical template theory22. This theory highlights the role of chemical templates in designing two-metal superhydrides, suggesting that the combination of two template-active metals could enhance interactions within the interstitial regions of the metal lattice. We applied a simplified 1:1 ratio of Th to Ce, inspired by previous work on \(\hbox {Ce}_{0.5}\hbox {La}_{0.5}\hbox {H}_{{9}}\)19. Using this ratio, we initiated structural predictions to explore the potential for improved conductivity and stability in these systems. This emphasis is motivated by the the Allen–Dynes modified McMillan (ADM) and semi-empirical McMillan (McM) formalisms, which have demonstrated excellent performance for conventional metallic structures. Additionally, we theoretically examined the superconducting gap by applying the isotropic Migdal-Eliashberg (ME) formalism23,24, in conjunction with electron-phonon coupling (EPC) calculations performed using the stochastic self-consistent harmonic approximation (SSCHA)25,26. In the quest for superconductivity, we aim to identify energetically stable compositions within \(\hbox {ThCeH}_{{18}}\). By employing a combination of random structure search and evolutionary algorithm methods based on DFT, we establish a robust framework for this investigation. Through theoretical analysis, we predict a new ground-state structure based on formation energy and examine the potential impact of electronic properties on the superconducting phases derived from these calculations.

Results and discussion

To begin structural predictions, we utilized the Ab initio Random Structure Searching (AIRSS) method to theoretically explore the structural stability of \(\hbox {ThCeH}_{{18}}\). This approach identified two distinct structures for \(\hbox {ThCeH}_{{18}}\): a hexagonal structure with the space group \({P\bar{6}m2}\) and a tetragonal structure with the space group \({P\bar{4}m2}\), as illustrated in Fig. 1a, b. Using the AIRSS methodology, a prediction of \(\hbox {ThCeH}_{{18}}\) structures was conducted, followed by an independent search utilizing the Universal Structure Predictor: Evolutionary Xtallography (USPEX) algorithm. This search assumed no prior structural information other than the required number of atoms. The \({P\bar{6}m2}\) and \({P\bar{4}m2}\) structures were identified and confirmed as the most stable configurations. Additionally, the structural parameter is presented in Table 1. Notably, the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) and \({P\bar{4}m2}\)-\(\hbox {ThCeH}_{{18}}\) structures can form from the components Th + Ce + 18H under pressures exceeding 10 GPa, as shown in Fig. 1c. We further examined the stability of these structures against decomposition into \(\hbox {ThH}_{{9}}\) and \(\hbox {CeH}_{{9}}\), as a function of pressure. In this context, the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) is the most thermodynamically stable phase, favored over the other structures. Consequently, the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure could plausibly be synthesized under suitable physical conditions. The formation of high-pressure compounds can often be understood through a reaction pathway expressed as A + B + C \(\rightarrow\) D, where A, B, and C represent the reactants, and D denotes the resulting compound. In this study, the specific reactants A, B, and C correspond to Th, Ce, and H, respectively, while D represents \(\hbox {ThCeH}_{{18}}\). This high-pressure formation pathway can be simplified as Th + Ce + 18H \(\rightarrow\) \(\hbox {ThCeH}_{{18}}\), which ultimately reduces to \(\hbox {Th}_{0.5}\hbox {Ce}_{0.5}\hbox {H}_{{9}}\).

Fig. 1
figure 1

Predicted clathrate hydride structure: (a) the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) and (b) \({P\bar{4}m2}\)-\(\hbox {ThCeH}_{{18}}\) structures. (c) Phase transformation for the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) and \({P\bar{4}m2}\)-\(\hbox {ThCeH}_{{18}}\) structures and decomposed \(\hbox {ThH}_{{9}}\)+\(\hbox {CeH}_{{9}}\) with respect to Th+Ce+18H as a function of pressure. (d) The partial electronic band and partial density of state of \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) at the pressure of 100 GPa. (e) The partial density of states of \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) represents the d- and f-electrons of Th and Ce atoms at a pressure of 100 GPa.

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Table 1 Crystal structures of \(\hbox {ThCeH}_{{18}}\) predicted at a pressure (P) of 100 GPa.
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Fig. 2
figure 2

Electron localization function (ELF) of \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) at a pressure 100 GPa is plotted in (a) (101) plane and (b) (101) plane, respectively. Panels (c–e) display the Fermi surfaces with orbital weight of Th-5f, Ce-4f, and H-1s orbitals, respectively. Panels (f–h) show the harmonic phonon dispersion and phonon linewidths, with the electron-phonon coupling (EPC) strength indicated by blue circles. The corresponding partial phonon density of states (PHDOS) is also presented, where Th, Ce, and H contributions are shown in red, green, and blue, respectively. Additionally, the Eliashberg spectral function and EPC constant are plotted as a function of pressure, represented by steel blue and orange lines, respectively. Panels (i–k) display the anharmonic phonon dispersion and linewidths, with EPC strength indicated by orange circles. The PHDOS and Eliashberg spectral function are shown using the same color scheme as in the harmonic case.

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To screen the physical properties of \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\), we scrutinized the electronic structure through the band structure partial density of states. We first investigated the band structure. It is evident that the band dispersion shows crossings between the valence and conduction bands around the Fermi level at a pressure of 100 GPa. This observation corresponds to the partial density of states attributed to Th, Ce, and H atoms at the Fermi level. Notably, the H atom contribution is significant at the Fermi level, as shown in Fig. 1d. Regarding the partial density of states, it sheds light on the hybridization observed between Th and Ce atoms. In this context, we further presence of electrons in d-f hybridization under pressure, illustrating the localized \(\rightleftharpoons\) delocalized nature of the f electrons27. Having established that \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) exhibits Th–Ce hybridization, we now proceed with a more detailed analysis of its electronic characteristics. This behavior is identified as the transfer of d-electrons into f-electron states. Notably, both Th and Ce possess substantial f-state densities, attributed to the near-complete overlap of their d-states within the f-states, as can be seen in Fig. 1e.

In this stage, we theoretically examined the chemical bonding using the electron localization function (ELF) method. This method analyzes the Th–Ce bonding, Th–H bonding, Ce–H bonding, and H-bonding within these systems based on the framework introduced by Becke28. To gain deeper insights into the bonding characteristics of the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure, as evidenced by Fig. 2a and b, the ELF provides valuable information about the tendency of electrons to accumulate within these crystals compared to a uniform electron gas with the same density. As a result, \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) exhibits intermolecular of H. This evidence shows that it may support the formation of Cooper pairs, thereby significantly enhancing the \(T_{\textrm{c}}\). Keeping this in mind, one should deeply consider a cage-like structure of hydrogen, as reported in Ref14, which demonstrates that the bonding profile can achieve high \(T_{\textrm{c}}\). This highlights the importance of a profound understanding of superconducting behavior with localized hydrogen atoms. In this situation, \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) exhibits intermolecular H–H bonding with nearest neighbor (NN) distances of 1.007 \(\text{\AA }\) throughout its structure. In contrast, clathrate hydrides feature NN distances of 1.218 \(\text{\AA }\) and 1.521 \(\text{\AA }\) in sites \(\hbox {ThH}_{{9}}\) and \(\hbox {CeH}_{{9}}\), respectively. From this perspective, intermolecular H–H bonding in \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) and the bonding in clathrate hydrides can be categorized as strong and moderate, respectively. This observation suggests the absence of lone pairs, which could otherwise suppress Cooper pair formation, as seen in (Zr,Hf)\(\hbox {H}_{{3}}\)29. Consequently, \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) is expected to achieve a high \(T_{\textrm{c}}\), akin to \(\hbox {KH}_{{20}}\) and \(\hbox {KH}_{{30}}\)30. Previously, we discussed the design of two-metal superhydrides to explore the influence of mixed metals in these compounds22. This approach introduces the chemical template theory, which outlines the utilization of chemical templates for constructing two-metal superhydrides. According to this theoretical framework, the combination of two template-active metals can strengthen interactions within the interstitial regions of the metal lattice, as evidenced by the ELF. The bonding characteristics of Th, Ce, and H reveal a mix of ionic and covalent bonding types. This is evident from the H–H bonding, which demonstrates strong covalent character, while Th–H exhibits weak covalent bonding, and Th–H also displays typical weak ionic bonding19,31.

As discussed in the electronic structure, the partial density of states at the Fermi level highlights the interplay between the localized and delocalized nature of the f-electrons. To further explore this behavior, we conducted an in-depth analysis of the Fermi surface with orbital weight. Notably, the Fermi surface plays a crucial role in the evolution of the superconducting state, as it is closely linked to the band structure and the electron density of states at the Fermi level, as previously reported in Refs.32,33. Our calculations suggest that the Fermi surface of \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) features a prominent pancake-like pocket around the \(\Gamma\)-point, attributed to the presence of an electron pocket. Additionally, a hole-type pocket emerges, contributing to Fermi surface nesting, which is situated around the pancake-like pocket and exhibits distinct contributions from each orbital. It is important to note that the Fermi surface with orbital weight is a dimensionless quantity, with values ranging from 0 to 0.012 for \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\). Specifically, Th-5f makes a weak contribution, Ce-4f contributes moderately, while H-1s plays a dominant role. This observation is particularly significant, as it directly influences the phonon spectrum and impacts the EPC constant.

Regarding the superconducting state of \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\), the first step in this section involves providing the \(T{_{\textrm{c}}}\) profile across a range of pressures, from low to high. In this context, we assess the structural dynamic stability by analyzing the lattice dynamics. The \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure is dynamically stable, as it exhibits no imaginary frequencies at a pressure of 50 GPa. Continuing this result, we consider an impact of EPC on the phonon line width is noteworthy, as discussed in previous studies34,35. Examining the phonon linewidth reveals that the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure makes a significant contribution to the large EPC in the optical phonon mode, primarily originating from the H atom. Subsequently, calculations of the Eliashberg function were performed, including the evaluation of the integrated EPC as a function of frequency. The results show a dramatic increase in the optical phonon regime, as highlighted in Fig. 2f–h. Notably, the findings highlight a dual feature associated with the H atoms, with the phonon linewidth clearly reaching a maximum corresponding to the highest EPC constant. In this way, \(T_{\textrm{c}}\) is examined by directly solving the ADM equation. The majority of the EPC constants are found to have values not exceeding 1.5. Consequently, the ADM is described as neglecting the \(f_1\) and \(f_2\) parameters36.

To shed light on the anharmonic effects, we analyzed the phonon characteristics shown in Fig. 2i–k. The \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure was relaxed at finite temperature via free energy minimization using the SSCHA method. As a result, notable phonon softening is observed, particularly in the acoustic phonon regime near the K-point at 50 GPa-an effect similar to that reported in \(\hbox {ScH}_6\)37. This anharmonic softening indicates that \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) has strong potential to enhance the \(T{_{\textrm{c}}}\). This is further supported by the calculated EPC constant of 1.59, with most EPC values exceeding 1.5, highlighting strong coupling behavior. Furthermore, with increasing pressure, the softened phonon modes shift toward higher frequencies, which in turn leads to a reduction in the EPC strength. In addition, anharmonic effects reveal extended phonon characteristics-particularly for hydrogen atoms in the optical phonon regime-when compared to the harmonic approximation.

Under these circumstances, it worth to nothing that selecting the Coulomb pseudopotential parameter \(\mu ^{*}\) is crucial for achieving in \(T_{\textrm{c}}\). In this case, we theoretically scrutinized the use of the Coulomb pseudopotential parameter \(\mu ^{*}\), set to 0.1, which is analogous to the parameter \(\mu\) defined in \(\mu = N(0)\langle \langle V_{kk′}^c \rangle \rangle _{\text {FS}}\). Here, \(\mu\) is approximately 1, and \(V_{kk′}^c\) represents the effective screened Coulomb interaction. The Coulomb parameter \(\mu\) is incorporated into the pseudopotential parameter \(\mu ^{*}\), which accounts for corrections to the repulsive Coulomb interaction in the material38, as shown in \(\mu ^* = \frac{\mu }{1 + [1 + \log (\omega _{el}/\omega _D)]}\). Where \(\omega _{el}\) and \(\omega _D\) represent the Coulombic response frequency and the Debye frequency, respectively, with typical values around 10 eV. Consequently, \(\mu ^{*}\) is estimated to be approximately 0.1. The Allen–Dynes approach assumes a universal \(\mu ^{*}\) value of 0.10 for most metals36. However, it is important to note that adopting \(\mu ^{*}\) = 0.13 is generally sufficient to account for phonon-mediated superconductivity, particularly in hydrogen-rich compounds39. Within this framework, we explore the prediction of \(T_{\textrm{c}}\). Our analysis reveals that the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure achieves an EPC value of 0.95 and exhibits superconductivity with a \(T_{\textrm{c}}\) of approximately 70 K, using \(\mu ^{*}\) = 0.10, under a pressure of 50 GPa. With increasing pressure, the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure exhibits typical \(T_{\textrm{c}}\) values of approximately 60 K and 43 K at pressures of 100 GPa and 150 GPa, respectively. Additionally, we conducted a trial using a \(\mu ^{*}\) value of 0.13 to evaluate \(T_{\textrm{c}}\) at varying pressures. As a result, we observed that the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure exhibited superconductivity with \(T_{\textrm{c}}\) values of approximately 61 K, 48 K, and 32 K at pressures of 50 GPa, 100 GPa, and 150 GPa, respectively. These \(T_{\textrm{c}}\) values were lower compared to those obtained using \(\mu ^{*}\) = 0.10. This variation in \(T_{\textrm{c}}\) indicates that a \(\mu ^{*}\) value of 0.10 is adequate for our analysis. As the correction formula for the Coulomb pseudopotential \(\mu ^{*}\) was previously introduced, we evaluated \(\mu ^{*}\) using the computed values of \(\omega _{el} = 5285.25\) THz and \(\omega _D = 56.66\) THz. This yields a renormalized Coulomb pseudopotential of \(\mu ^{*} = 0.153\). We further identified that the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure exhibits representative \(T_{\textrm{c}}\) of approximately 51.8 K at a pressure of 50 GPa, as determined using the AMD method.

To further investigate the potential for superconductivity under pressure, it is noteworthy that the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure exhibits the highest \(T_{\textrm{c}}\) at 50 GPa. The main results obtained using the ADM method are summarized in Table 2. From this perspective, it is revealed that its \(T_{\textrm{c}}\) decreases with increasing pressure because \(\lambda\) decreases. Herein, it should be noted that as \(\lambda\) decreases, \(\omega _{log}\) increases. However, as the pressure increases to 200 GPa, \(T_{\textrm{c}}\) decreases to approximately 39 K. In contrast, \(\hbox {YThH}_{{18}}\) demonstrates superconductivity with a \(T{_{\textrm{c}}}\) of 222 K at 200 GPa. Additionally, attention should be given to the vibrational behavior of H atoms, as their intermediate optical phonon modes significantly contribute to the superconductivity observed in \(\hbox {YThH}_{{18}}\)20. Supporting this observation, the intermediate optical phonon modes of H atoms in \(\hbox {CaMgH}_{{12}}\) have been shown to contribute to superconductivity, resulting in a \(T_{\textrm{c}}\) of 288 K at 200 GPa13. Therefore, the intermediate optical phonon modes of H atoms may play a remarkably important role in the evolution of \(T_{\textrm{c}}\).

As discussed above, the ADM equation is well-suited for strong EPC when \(\lambda > 1.5\). However, for \(\hbox {ThCeH}_{{18}},\) where \(\lambda\) remains below 1.5, the semi-empirical McM approach40 is more appropriate. Our analysis reveals that \(\hbox {ThCeH}_{{18}}\) exhibits superconductivity with a \(T_{\textrm{c}}\) of 97 K at a pressure of 50 GPa for \(\mu ^{*} = 0.10\), with a corresponding Debye temperature (\(\Theta _D\)) of approximately 1827 K. When \(\mu ^{*} = 0.13\), \(T_{\textrm{c}}\) decreases to 82.3 K. Meanwhile, our theoretical analysis yields a \(\mu ^{*} = 0.153\). Based on this value, we find that the \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) structure exhibits a representative \(T_{\textrm{c}}\) of approximately 71.5 K at 50 GPa. Furthermore, our results indicate that \(T_{\textrm{c}}\) decreases with increasing pressure. This finding underscores the relevance of the McMillan equation, as it predicts a higher \(T_{\textrm{c}}\) than the ADM equation in this system. In principle, the McM approach provides a sufficiently accurate estimation of \(T_{\textrm{c}}\) when \(\lambda\) remains below 1.5, as observed in \(\hbox {ThCeH}_{{18}}\). The main results obtained using the McM method are summarized in Table 2. Furthermore, we investigated the impact of anharmonicity on the \(\hbox {ThCeH}_{{18}}\) structure, which yields \(T{_{\textrm{c}}}\) values of 87.4 K and 80.2 K for \(\mu ^{*} = 0.10\) and 0.13, respectively. For \(\mu ^{*} = 0.153\), a \(T_{\textrm{c}}\) of 74.6 K is obtained. It is also worth noting that the ADM formula incorporates the parameters \(f_1\) and \(f_2\), with values of 1.09 and 1.27, respectively. While these findings highlight the relevance of the McMillan equation, it also predicts a significantly higher \(T_{\textrm{c}}\) of 168.6 K. However, since the EPC constant \(\lambda\) exceeds 1.5, the McMillan-based estimate may overstate the actual \(T_{\textrm{c}}\), making its accuracy questionable in the strong-coupling regime. A summary of the anharmonic effects on \(T_{\textrm{c}}\) is provided in Table 3.

Fig. 3
figure 3

Superconducting energy gap as a function of temperature for \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) under the harmonic approximation at (a) 50 GPa and (b) 150 GPa, and under the anharmonic correction at (c) 50 GPa and (d) 150 GPa.

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Table 2 Electron-phonon interaction parameters, logarithmic average phonon frequencies, and calculated \(T_{\textrm{c}}\) values for \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) obtained using various theoretical methods within the harmonic approximation.
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Table 3 Electron-phonon interaction parameters, logarithmic average phonon frequencies, and calculated \(T_{\textrm{c}}\) for \({P\bar{6}m2}\)-\(\hbox {ThCeH}_{{18}}\) obtained using various theoretical methods including anharmonic corrections.
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Having identified the highest \(T_{\textrm{c}}\) in the crystal structure at a pressure of 50 GPa using the ADM method, it is noteworthy that \(T_{\textrm{c}}\) can be derived from either the ADM method or the isotropic ME formalism. In this context, we proceed to further investigate \(T_{\textrm{c}}\) using the isotropic ME formalism, which directly incorporates the electron-phonon matrix elements, phonon frequencies, and polarization vectors derived from the diagonalization of the Fourier-interpolated SSCHA force constants. The isotropic ME equations function for electron-phonon interaction can be calculated by including anharmonic effects in the evaluation of electron-phonon properties, yielding the anharmonic spectral function. This function is represented within the full spectral framework, as recently detailed in Ref.41, Furthermore, the SSCHA method allows for the determination of the system’s free energy while accounting for quantum zero-point motion and anharmonic effects, providing a more accurate description of lattice dynamics. The full spectral function is expressed as: \(\alpha ^2F(\omega ) = \frac{1}{N_q} \sum _{ab\mathbf{q}\mu } \frac{e^a_{\mu }(\mathbf{q}) \Delta _{ab}(\mathbf{q}) e^b_{\mu }(\mathbf{q})^*}{2\omega _\mu (\mathbf{q}) \sqrt{m_a m_b}} \delta (\omega - \omega _\mu (\mathbf{q})),\) where \({\Delta ^{ab}(\mathbf{q})}\) is the average of the deformation potential over the Fermi surface, the phonon frequencies, \({\omega _\mu (\mathbf{q})}\), and the polarization vectors, \({e^a_{\mu }(\mathbf{q})}\) and \({e^b_{\mu }(\mathbf{q})},\) are derived from the diagonalization of the Fourier-interpolated SSCHA force constants at the wavevector \({\mathbf{q}}\). The atomic masses are denoted by \({m_a}\) and \({m_b}\). In the equation, the Dirac delta function is approximated using a Gaussian broadening. Moreover, the McM and ADM methods may significantly underestimate \(T{_{\textrm{c}}}\) if \(\hbox {ThCe}_{{18}}\) is observed in experimental studies. For example, in the case of \(\hbox {LaCeH}_{{18}}\), the \(T_{\textrm{c}}\) calculated using the McM and ADM methods was found to be underestimated compared to experimental observations. Building on the ME calculation, the superconducting order parameter is defined by the gap equation: \(\Delta ^{n}(k) = \varphi (k, i\omega _n)/\mathbf{Z}(k, i\omega _n),\) where \(\mathbf{Z}(k, i\omega _n)\) represents the electron mass renormalization function and \(\varphi (k, i\omega _n)\) denotes the pairing function, as detailed in Refs.24,42,43. The corresponding imaginary-axis Eliashberg equations are given as follows:

$$\begin{aligned} Z(i\omega _n)&= 1 + \frac{\pi }{\beta i\omega _n} \sum _{n'=-n_c}^{n_c} \frac{i\omega _{n'} Z(i\omega _{n'}) \lambda (i\omega _n - i\omega _{n'})}{\sqrt{[\omega _{n'} Z(i\omega _{n'})]^2 + [\varphi (i\omega _{n'})]^2}} \end{aligned}$$
(1)
$$\begin{aligned} \varphi (i\omega _n)&= \frac{\pi }{\beta } \sum _{n'=-n_c}^{n_c} \frac{\varphi (i\omega _{n'}) \left[ \lambda (i\omega _n - i\omega _{n'}) - \mu ^*\right] }{\sqrt{[\omega _{n'} Z(i\omega _{n'})]^2 + [\varphi (i\omega _{n'})]^2}} \end{aligned}$$
(2)

It is worth noting that the Matsubara summation is divided into two parts: lower-frequency terms (up to \(n_{\textrm{c}}\)) are computed numerically, while the contributions from higher frequencies are treated analytically44. In this study, it is important to ensure that the Matsubara frequency cutoff is chosen to be roughly ten times the maximum phonon frequency, without exceeding this limit7. The \(T_{\textrm{c}}\) is estimated based on the superconducting gap at zero temperature, incorporating harmonic approximation, \(\Delta _0,_{har}\). The ME approach predicts a superconducting gap with a \(T_{\textrm{c}}\) of 112 K using \(\mu ^{*}\) value of 0.10 at a pressure of 50 GPa, corresponding to \(\Delta _0,_{har}\) of 18.92 meV, as demonstrated in Fig. 3a, b. Additionally, a ratio of \((2\Delta _0/{k_{\textrm{B}}}{T_{\textrm{c}}})_{har}\) = 3.92, surpassing the ideal Bardeen–Cooper–Schrieffer (BCS) value of 3.5345. This observation suggests that \(\hbox {ThCeH}_{{18}}\) showcases strong-coupling superconductivity. In the ME method, the EPC constant, commonly denoted as \(\lambda\), is directly obtained from the fully spectral function. In this context, we refer to it as \(\lambda ^{SSCHA}_{har}\), highlighting that the full spectral function is combined with SSCHA. This integration ensures a comprehensive treatment of \(\lambda _{har}\) across all phonon modes while also incorporating the highest phonon frequencies. As a result, \(\lambda ^{SSCHA}_{har}\) is higher than \(\lambda _{har}\), which is derived from the spectral function used in the ADM and McM calculations. Consequently, this leads to the highest \(T_{\textrm{c}}\) values among the calculations. The main results obtained using the ME method are summarized in Table 2. In this context, we can examine the hierarchy of factors influencing the isotropic ME approach, which demonstrates a high \(T_{\textrm{c}}\), making it an effective framework for evaluating the EPC. This effectiveness is partly attributed to the inclusion of phonon modes influenced by anharmonicity. This section provides the theoretical perspectives that align with experimental findings, offering an essential basis for future exploration of the \(\hbox {ThCeH}_{{18}}\) phase.

In the context of the superconducting gap, we denote the anharmonic EPC as \(\lambda ^{\text {SSCHA}}_{\text {anhar}}\), which reflects that the full spectral function is calculated within the SSCHA framework. This methodology allows for a comprehensive evaluation of \(\lambda_{\text {anhar}}\) across all phonon modes, including significant contributions from high-frequency phonons. Furthermore, the superconducting transition temperature was estimated from the zero-temperature superconducting gap that incorporates anharmonic effects, denoted as \(\Delta _{0,\text {anhar}}\). Using the ME formalism with a Coulomb pseudopotential \(\mu ^{*} = 0.10\), a \(T_{\textrm{c}}\) of 136 K is predicted at 50 GPa, corresponding to a superconducting gap \(\Delta _{0,\text {anhar}}\) of 25.15 meV, as illustrated in Fig. 3c, d. The impact of anharmonic effects on \(T_{\textrm{c}}\) is outlined in Table 3.

Finally, similar to other compounds such as (La,Ce)\(\hbox {H}_{{9}}\)17,18,19, our structural predictions and superconducting property analysis indicate that \(\hbox {ThCeH}_{{18}}\) could be synthesized at relatively lower pressures. While most hydride superconductors exhibit stability only under extremely high pressures, high-pressure synthesis techniques may provide a viable pathway for achieving stable phases under more accessible conditions. The dynamic and kinetic stability of these materials suggests that certain crystalline structures, initially stabilized at moderate to high pressures, may maintain their stability upon decompression, making their synthesis at lower pressures feasible.

Conclusion

In summary, this study employs the AIRSS and USPEX methods, combined with first-principles calculations, to predict the \(\hbox {ThCeH}_{{18}}\) phase. The exploration has revealed a hexagonal structure with a space group of \(P\bar{6}m2.\) Our analysis indicates that \(P\bar{6}m2\) \(\hbox {ThCeH}_{{18}}\) is theoretically stable above 10 GPa and dynamically stable at a pressure of 50 GPa. This study theoretically investigates the role of hydrogen atoms in optical phonon modes under anharmonic effects. These effects induce phonon softening, which in turn contributes to the enhancement of the \(T_{\textrm{c}}\). It highlights the significant impact of hydrogen atoms, showcasing their critical influence on the evolution of \(T_{\textrm{c}}\). A key finding is the dual-feature superconductivity, exhibiting a \({T_{\textrm{c}}}\) of 136 K. Our results demonstrate that applying pressure could facilitate the synthesis of \(\hbox {ThCeH}_{{18}}\) and holds significant potential for experimental realization across a broad range of superhydrides, enhancing superconductivity under low-pressure conditions.

Computational methods

To predict structural configurations, we employed the AIRSS formalism, which offers extensive coverage of the potential energy surface (PES) and ensures an unbiased sampling process46,47. This approach is based on density functional theory (DFT) and is implemented in the Vienna Ab Initio Simulation Package (VASP)48. We utilized the projector augmented wave (PAW) method49 along with the generalized gradient approximation (GGA) based on the Perdew, Burke, and Ernzerhof (PBE) exchange-correlation functional50. Our study focused on \(\hbox {ThCeH}_{{18}}\), examining structures with 1–2 formula units at pressures of 0, 50, 100, 200, and 300 GPa. In total, approximately 10,000 structures were generated across these pressure conditions. Notably, a fixed-composition search was conducted to identify stable phases, with the scope extended to higher formula units if no stable phases were detected for 1–2 formula units. The calculations employed a plane-wave basis set with a cutoff energy of 250 eV, and the initial Brillouin zone (BZ) sampling was performed with a k-point resolution of 2\(\pi\) \(\times\) 0.07 \(\text{\AA }^{-1}\), ensuring energy convergence within 0.001 eV per formula unit (f.u.). \(\hbox {ThCeH}_{{18}}\) were searched by adopting the evolutionary algorithm (AE), as implemented in the USPEX51 combined with the VASP code48. The favorable structures of \(\hbox {ThCeH}_{{18}}\) are searched at 50, 100, 200, 300 GPa, respectively. Generations were generated using a combination of 50\(\%\) heredity, 30\(\%\) randomization, and 10\(\%\) softmutation. Each structure underwent a three-step relaxation process in VASP, achieving energy convergence within 0.001 eV per formula unit (f.u.).

Screening on these stable phases, we investigated the potential for phase decomposition by calculating their relative enthalpy to pure elemental Th, Ce, H, as well as \(\hbox {ThH}_9\) and \(\hbox {CeH}_9\) at various pressures. The calculations were performed using a plane-wave basis set with a cutoff energy of 700 eV. A 12\(\times\)12\(\times\)12 k-point mesh was applied for the face-centered cubic (fcc) structures of Th and Ce, while a 12\(\times\)12\(\times\)8 k-point mesh was used for the hexagonal close-packed (hcp) structures of H, \(\hbox {ThH}_9\), and \(\hbox {CeH}_9\). The PAW method49 was utilized, along with the generalized gradient approximation within GGA-PBE50 for the exchange-correlation functional.

To investigate the electronic properties, the electronic band structures and density of states were analyzed for all calculated structures. The structures were optimized using the quasi-Newton method52 within first-principles calculations. A plane-wave basis set with a cutoff energy of 700 eV and a 12\(\times\)12\(\times\)8 k-point mesh was used. The GGA-PBE formalism and ultrasoft pseudopotentials53, including 12 valence electrons for Th, 12 valence electrons for Ce, and 1 valence electron for H, were employed. These calculations were implemented in the Cambridge Serial Total Energy Package (CASTEP)54.

To ascertain the critical temperature of superconductivity, density functional perturbation theory (DFTP) is conducted55. Implementing the GGA-PBE scheme and the ultrasoft pseudopotential with 12 valence electrons (6\(s^2\)6\(p^6\)7\(s^2\)5\(f^1\)6\(d^1\)) for Th, 12 valence electrons (5\(s^2\)5\(p^6\)4\(f^1\)5\(d^1\)6\(s^2\)) for Ce56, and 1 valence electron (1\(s^1\)) for H, as implemented in quantum espresso (QE)57. A plane-wave energy cutoff of 80 Ry was performed. A self-consistency threshold of 1\(\times 10^{-18}\) was applied. Ensuring a comprehensive k-points mesh encompassing all k and k+q grid points, essential for subsequent electron-phonon and spectral function calculations reliant on the k-point part covering the q-point grid, we conducted EPC matrices and phonon calculations with varying mesh resolutions for \(\hbox {ThCeH}_{{18}}\) using 24\(\times\)24\(\times\)16 and 12\(\times\)12\(\times\)8 k-point meshes, along with 4\(\times\)4\(\times\)2 q-meshes. This calculation, which has proven effective in accurately determining the EPC, employs the ADM equation36 and the semi-empirical McM approach40. Both methods have demonstrated excellent performance in describing superconductivity in conventional metallic structures.

To account for anharmonic effects, phonon dispersions were computed using the SSCHA method combined with ab initio calculations from QE. The SSCHA simulations were conducted at finite temperatures within the canonical NVT ensemble. The convergence of physical quantities was confirmed using a final ensemble consisting of 200 randomly generated structures. The optimization was deemed converged when the Kong–Liu ratio reached 0.6, and the normalized gradient of the free energy with respect to the auxiliary dynamical matrix dropped below \(10^{-5}\) relative to its stochastic uncertainty. These calculations were based on the ground-state structure predicted from crystal structure searches, incorporating non-perturbative second-order force constants as described in Ref.58. The free energy minimization in SSCHA was performed using self-consistent DFT evaluations in 80-atom supercells, which provided the atomic forces required for the stochastic optimization process.

A detailed discussion on the Eliashberg equation which characterizes the Eliashberg spectral function \(\alpha ^{2}F\)\((\omega),\) is thoroughly detailed in Ref.24. Additionally, the influence of EPC on phonon linewidth is significant and has been extensively discussed in previous studies34,35. We thoroughly analyzed the intrinsic superconducting gap using the isotropic ME formalism23,24, combined with EPC calculations based on SSCHA dynamical matrices and electron-phonon matrix elements41. We investigated phonon-mediated superconductivity and applied an effective Coulomb potential, \(\mu ^{*} = 0.10\) to 0.153, to solve the isotropic ME equations. The superconducting gap as a function of temperature was determined by numerically solving these equations.