Mechanism of high-temperature superconductivity in compressed H₂-molecular–type hydride

Abstract

The discovery of compressed atomic-type hydrides offers a promising avenue toward achieving room-temperature superconductivity, but it necessitates extremely high pressures to completely dissociate hydrogen molecules to release free electrons. Here, we report a remarkable finding of compressed H₂-molecular–type hydride CaH₁₄ exhibiting an unusual transition temperature (T_c) of 204.0 kelvin. The peculiarity of its electronic structure lies in the pronounced emergence of near-free electrons, which manifest metallic bonding, but molecular hydrogen fragments persist. This finding indicates that the necessary condition for superconducting transition is forming the Fermi sea with Cooper pairs rather than the monatomic hydrogen. Notably, the formation mechanism of free electrons can be effectively explained by the finite-depth potential wells model. Intriguingly, this H₂-molecular–type hydride can downgrade the required pressure to 80 gigapascal while maintaining a high T_c of 84 kelvin, well above the liquid-nitrogen temperature. Our study has established a high-temperature superconducting paradigm and opened the prospect for achieving high-T_c superconductors in H₂-molecular–type hydrides at low pressure.

The superconducting origin and associated formation mechanism of near-free electrons in molecular hydrides have been elucidated.

INTRODUCTION

The realization of room-temperature superconductivity stands as one of the most formidable challenges in the realm of condensed matter physics and materials science. According to the Bardeen-Cooper-Schrieffer (BCS) theory, hydrogen, with the highest Debye temperature plays a critical role in enabling the emergence of high superconducting transition temperature (T_c) (1, 2). Despite experimental exploration of solid hydrogen at high pressures exceeding 400 GPa (3), the metallization of solid hydrogen is still uncertain. The hydrogen-rich hydrides have been considered as an alternative owing to the proposed chemical pre-compression effect (2), which has been widely embraced as a promising route for achieving high-T_c superconductivity at lower pressures (4–7). At present, the emergence of atomic-type hydrides has opened a promising avenue for achieving room-temperature superconductivity (8, 9). Representative examples include the H₃S, a compressed covalent-type hydride with a T_c of 203 K at 155 GPa, and LaH₁₀, an ionic hydride characterized by H-clathrate structure with T_c ranging from 250 to 260 K at pressures between 170 and 180 GPa (8, 10–12). The compressed atomic-type hydrides have demonstrated the potential for achieving room-temperature superconductivity. However, the prerequisite for achieving high-temperature superconductivity in these atomic-type hydrides entails the dissociation of molecular hydrogen to release free electrons, necessitating extremely high pressures (13–15). This is particularly exemplified by MH₁₈ (M: rare-earth/actinide metals), which demonstrates room-temperature superconductivity but demands high-pressure exceeding 350 GPa (16).

It is generally considered that the presence of high T_c in atomic-type hydrides is in terms of the high H-derived electronic density of states (DOS) at the Fermi level (E_F), the large phonon energy scale of the vibration modes and the resulting enhanced electron-phonon coupling (EPC) (1, 17). Nowadays, the pursuit of high-T_c superconductors in atomic-type superhydrides devoid of molecular hydrogens (H₂ units) has emerged as a guiding principle. This is because H electrons in molecular hydrogens occupy low-lying σ-bonding orbitals (18–20), which are far from the E_F and thus considered unfavorable for achieving high T_c (21, 22). The abovementioned physical phenomenon elucidates the profound impacts of electronic structure on the superconductivity in H₂-molecular–type hydrides. Despite that the H-derived DOS at E_F is essential for high T_c in phonon-mediated BCS superconducting hydrides, more quantitative analysis is desired to provide a thorough explanation. The molecular hydrogen system has a possibly robust electron-phonon interaction associated with the strong covalent bonding within the molecules. Numerous studies have collected strong evidence pointing to high-T_c superconductivity, including the recent theoretical prediction of near room-temperature superconductivity of 270 K in molecular hydride NaH₁₀ at 400 GPa (23) and the previously proposed molecular hydrogen Cmca phase with 242 K at 450 GPa (24). Very recently, the T_c of 91 K at 170 GPa was observed in the molecular hydride C2/c-BiH₄, and it was experimentally characterized that the hydrogen lattice is entirely composed of H₂ molecules (25). However, the full scenario is still far from being clear and firmly established. In particular, the origins of superconductivity and the associated mechanisms governing the emergence of free electrons in the H₂-molecular–type compounds remain poorly understood and confused.

In this work, we predicted an unusual H₂-molecular–type C2/c phase in compressed hydride CaH₁₄ and verified its thermodynamic stability by referencing the other known Ca-H compounds (13, 26, 27). On the basis of EPC calculations, this unprecedented phase exhibits a remarkably high T_c of 204.0 K at 300 GPa and 192.6 K at 200 GPa. To address the abovementioned confusions, we propose a finite-depth potential wells model to explain the unusual emergence of near-free electrons with characteristic metallic bonding characteristics in the H₂-molecule hydride system. We elucidate the mechanism underlying the generation of near-free electrons: The pressure-induced decrease in interatomic potential energy and the strong Pauli repulsion leads to an increase in the kinetic energy of electrons, resulting in the emergence of nearly free electrons. Our first-principles calculations unveil that the near-free electrons scattered by molecular hydrogen–derived phonons to participate in EPC to induce the high T_c. Further analysis indicates that the phonon contribution on EPC stemmed from the large electron-phonon matrix element derived from intermediate-frequency molecular hydrogen–derived phonon vibrations and enhanced by phonon softening caused by Fermi surface (FS) nesting to scatter near-free electrons. Intriguingly, this H₂-molecular–type hydride can reduce the required pressure even down to attainable synthesis range of large cavity press and maintain high T_c of 60 K at 50 GPa and 84 K at 80 GPa above the liquid-nitrogen temperature. Our findings have clarified the formation mechanism of near-free electrons acting as the Fermi sea with Cooper pairs, thereby shedding light on the intricate causality governing high-temperature superconductivity in H₂-molecular–type hydrides. This discovery paves an avenue for realizing high-temperature superconductors at relatively low pressures.

RESULTS

Crystal structure and thermodynamic stability

Here, our extensive structure search identifies a superhydride CaH₁₄ that is stabilized by pressures above 100 GPa, which is confirmed by the constructed convex hull in Fig. 1A (13, 26–28). We have found three thermodynamically stable modifications for CaH₁₄ compound, P-1-i/ii and C2/c monoclinic lattices, as illustrated in Fig. 1B. These structures have more energy stability than the previously reported R-3 phase (26). The phase transition pressure of P-1-i is located at 112 GPa, and above that, P-1-ii phase is favorable in formation enthalpy, whereas C2/c is thermodynamically stable at pressures above 211 GPa. For C2/c phase at 300 GPa, the relaxed lattice parameters are as follows: a = 6.86570 Å, b = 5.63620 Å, c = 3.34160 Å, α = 90.0000°, β = 117.7662°, and γ = 90.0000°. The Wyckoff positions occupied by Ca and H atoms are shown in the Supplementary Materials (see table S1). In Fig. 1C, the Bravais lattice contains four formula units of CaH₁₄ including 60 atoms, and the Ca-H distances are greater than 1.746 Å. In particular, the intermolecular distances of H₂ units are significantly greater than 1.000 Å, while the intramolecular distances range from 0.774 to 0.855 Å at 300 GPa, indicating that the hydrogen sublattices are composed of molecular hydrogen fragments. We further calculated the variation of the nearest neighbor H-H distance with pressurization and compared this result with the H-H distance in I4₁/amd phase of atomic hydrogen and H-III C2/c phase of molecular hydrogen. As depicted in Fig. 1D, the result demonstrates that the C2/c-CaH₁₄ compound exhibits characteristics of a typical H₂-molecular–type hydride, as evidenced by the proximity between its intramolecular H-H bond length and the covalent bond length observed in H-III phase at 300 GPa. The intramolecular H-H distances gradually increase upon depressurization yet remain comparable to those of H-III phase.

Fig. 1. Thermodynamic stability and structural properties of Ca-H hydrides.

(A) Phase stabilities of various Ca-H hydrides at different pressures. (B) Calculated formation enthalpies of CaH₁₄ structures as a function of pressures relative to the CaH₂ + 6H₂. (C) The structure of C2/c-CaH₁₄ at 300 GPa corresponding to sublattices of H₂ units. (D) The H-H distances in CaH₁₄ compared to the values in atomic hydrogen H-IV (I4₁/amd) phase and molecular hydrogen H-III (C2/c) phase at different pressures.

Mechanism for the generation of near-free electrons as Fermi sea

To examine the bonding characteristic of this H₂-molecular–type hydride C2/c-CaH₁₄, we calculated the electron localization function (ELF), crystal orbital Hamilton population (COHP) and integral DOS analysis between Ca-H and H-H atoms (see table S2) (29–33). The result of integral DOS analysis confirms that the H₂ unit and Ca simultaneously experience distinct charge deficiencies, thus forming Ca^+0.82 and (H₂)^+0.60 cations, nurturing an unusual emergence of electrons within the integrated crystal cavity at high pressures. Figure S1 manifests the features of Ca─H metallic bonding, as demonstrated by the electrostatic attraction between free electrons and metal ions arranged in a lattice. On the contrary, strong covalent bonds are formed between H-H atoms based on highly localized charges in ELF and large occupancy of electrons on σ-bonding orbitals, resulting in a significantly strong bonding strength corresponding to an integral COHP value of −5.40 eV per pair at 300 GPa. This bonding strength is in close agreement with the molecular hydrogen bonding strength of 5.60 to 6.20 eV per pair calculated in the H-III phase (23), thereby further confirming that the hydrogen framework within the C2/c-CaH₁₄ phase predominantly consists of molecular H₂ units. Notably, a typical characteristic of metallic bonding, delocalized near-free electron gases, has been formed between adjacent H₂ units when considering the 0.5 standard of ELF, indicating that a large amount of free electron gases form conductive channels with metallic bond. The unique near-free electrons are completely different from the interstitial quasi-atoms in compressed lithium and sodium, where the valence electrons occupying the hybridized orbitals are repulsed by core electrons into the lattice interstices without itinerate characteristics, thus showing insulator properties (34, 35).

To elucidate the underling generation mechanism of near-free electrons, we calculated the effective local potential within the framework of Kohn-Sham density functional theory. The finite effective potential wells shaped by H₂ units and Ca atoms are depicted in Fig. 2A, where the interstitial region exhibits potential ridges connecting all wells throughout the crystal at approximately −10.0 eV, as illustrated in Fig. 2B. In addition, the E_F at 11.3 eV is much higher than the maximum effective potential at −3.4 eV (see Fig. 2C), indicating a pronounced near-free electron gas behavior across a wide range of occupied states spanning from −3.4 to 11.3 eV. The results are consistent with our expectations regarding the two mechanisms: (i) With the pressures increasing, the potential wells move closer to each other, resulting in the formation of lower ridge connections as illustrated in Fig. 2D. (ii) The strong Pauli repulsion among electrons leads to the occupation of higher energy levels characterized by a high E_F. As a result, the increased kinetic energy of electrons induced by pressure here exceeds the limit of effective potential wells shaped by cations of Ca^+0.82 and (H₂)^+0.60 and are expelled from the potential, representing a free electron gas on a Jellium-like background charges.

Fig. 2. The Kohn-Sham effective local potential, including ionic, Hartree, and exchange-correlation potentials.

(A) The distribution of effective potential in C2/c-CaH₁₄ system. (B) The equipotential surface at −10.0 eV. (C) The section of 3D (top) and 2D (bottom) potential surface for selected Ca atoms. (D) Schematic illustration of potential decrease at the ridge as two finite potential wells approach, where $d_0$ (Å) is a given initial distance between two potential wells (H₂ units or Ca cations).

The near-free electrons nature of the charge density distribution gives rise to an intriguing electronic band structure showing metallic characteristics, as evidenced by Fig. 3A and fig. S2, where the bands overlap at E_F. The projected DOS revealed that the electronic states primarily associated with near-free electrons are responsible for the large contribution to the total DOS near the E_F. In addition, H-s states are hybridized with Ca-d states forming sd-hybridized electric states to further contribute to the total electronic DOS. Two steep conduction bands intersecting the E_F shape the characteristic “hole conductivity” around Γ point and “electron pocket” in the intermediate position especially along Γ → Z, Γ → A → M and V → Γ directions. The multiple energy bands crossing the E_F suggest that a substantial number of electrons occupy the vicinity of the E_F. This result is further confirmed by the presence of a pronounced peak in the DOS near the E_F, predominantly originating from near-free electrons, which contribute to approximately 52.7% of the total DOS. The concurrent presence of steep bands along with a peak of DOS in close proximity to the E_F indicates a large EPC strength maybe bound up with phonon modes that favor the formation of a superconducting gap, suggesting the potential of CaH₁₄ to be a promising high-temperature superconductor (36, 37).

Fig. 3. Electronic structures, phonon dispersion, and superconducting gaps of C2/c-CaH₁₄ at 300 GPa.

(A) Electronic band structure and projected DOS of C2/c-CaH₁₄ phase and corresponding to the (B) 3D FSs of C2/c-CaH₁₄ phase. (C) Phonon dispersion, projected phonon densities of states (PDOS), and Eliashberg spectral function α²F(ω) at 300 GPa. The size of the red dot on the phonon spectra represents the contribution of the phonon linewidth in proportion to the spherical scale. (D) Calculated superconducting gaps Δ of CaH₁₄ as a function of temperature.

The origin of high-temperature superconductivity

As expected, the C2/c-CaH₁₄ exhibits an estimated T_c of 195 K at 300 GPa when using the McMillan-Allen-Dynes formula with a typical μ^* of 0.1 (38). Subsequently, we further test the superconducting gaps by numerically solving the isotropic Eliashberg equations, as displayed in Fig. 3D (39), in which the result T_c of 204.0 K is well comparable with that 195.0 K obtained via McMillan-Allen-Dynes formula. To ferret out the role of molecular hydrogen–derived phonon vibrations played on CaH₁₄, we calculate the lattice phonons in conjunction with projected phonon DOS, Eliashberg function spectral function α²F(ω), and integral EPC constant λ in Fig. 3C. The results demonstrate a conspicuously high EPC coefficient λ of 1.79 and a logarithmic frequency ω_log reaching up to 1279 K. To gain insight into the contribution of phonon vibrations in different frequency ranges to EPC, the phonon spectrum can be categorized into three regimes: the low-frequency region (0 to 16 THz) corresponding to translational vibration mode of Ca/H atoms, the medium-frequency H₂-derived diverse interatomic interactions (16 to 95 THz), and the high-frequency region (100 to 120 THz) characterized by stretching vibration mode of intramolecular H₂ units. At 300 GPa, we observed a monotonic increase in the integral λ(ω) within the frequency range of 10 to 90 THz, indicating that the medium-frequency H₂-derived phonon vibrations predominantly contribute to the EPC, accounting for 78% of the total λ(ω), but the stretching vibration mode above 100 THz makes a negligible contribution to λ(ω), amounting for only 5%.

To unveil the pressure effect on evolution of superconductivity in C2/c-CaH₁₄, we focus on discussing the changes of essential factors, including the strength of EPC λ, characteristic phonon frequency ω_log and N(E_F) at E_F. In Fig. 4A, as the pressure increased from 100 to 200 GPa, we found an increased number of N(E_F) at E_F while ω_log experiencing a prominent downward trend. In contrast, application of the pressure from 200 to 300 GPa, an obvious decline in N(E_F) corresponding to the descending λ while the ω_log increased with pressurization. As is apparent, a notable enhancement in the final T_c is observed upon compression despite the superconducting factors showing diversified evolution trends. These results indicate that both phonon contributions and electronic effects affect the superconductivity within different pressure ranges. Next, we investigate the phonon effect and electronic contribution to elucidate superconducting origin in the pressurization range of 100 to 300 GPa based on the Hopfield expression, $\mathrm{\lambda }=\frac{N({\mathrm{E}}_{\mathrm{F}})<I^2>}{M{\mathrm{\omega }}^2}$, where N(E_F) represents the purely electronic contribution derived from the DOS and the factor $\mathrm{\eta }=\frac{{\mathrm{{\rm I}}}^2}{{M\mathrm{\omega }}^2}$ related to the lattice (40).

Fig. 4. Superconducting parameters, nesting functions, and phonon linewidth of C2/c-CaH₁₄ at high pressures.

(A) The calculated T_c, logarithmic average frequency ω_log, N(E_F) at E_F and the EPC parameter λ of CaH₁₄ at different pressures. (B) The nesting function of C2/c-CaH₁₄ along high-symmetry q trajectories and (C) corresponding strength of linewidth ${\mathrm{\gamma }}_{\mathit{qv}}$ at 100, 200, and 300 GPa.

In the pressure range from 100 to 200 GPa, the η (3.615 eV⋅f.u.) at 200 GPa is 2.06 times as much as η (1.753 eV⋅f.u.) of that at 100 GPa, which agree well with the evolution in T_c that increased by a multiplicative factor of 1.95. This result confirmed the remarkable role of the lattice contribution to the EPC. The soft modes of phonon dispersion happened near 14 THz, extending from Q_Г → Q_z, Q_V → Q_Г → Q_C and Q_A → Q_M → Q_L directions. These soft modes were further enhanced at 200 GPa, contributing to the EPC and resulting in an increase in the strength of λ, which can be reflected through a larger α²F(ω) and linewidth ${\mathrm{\gamma }}_{\mathit{qv}}$ at 200 GPa compared with that at 100 GPa (see fig. S3). Despite the phonon softening also observed at approximately 70 THz, this part has a negligible impact on the EPC due to the nearly unchanged integral λ. To elucidate the underlying causation of phonon softening near the 14 THz, we calculated the FS nesting in Fig. 4B based on the nesting function ${\mathrm{\xi }}_{\mathbf{q}}=\frac{1}{N}{\sum }_{\mathbf{k},i,j} \mathrm{\delta } ({\mathrm{\epsilon }}_{\mathbf{k},i}-{\mathrm{\epsilon }}_F)\times \mathrm{\delta }({\mathrm{\epsilon }}_{\mathbf{k}+\mathbf{q},j}-{\mathrm{\epsilon }}_F)$. The nesting strengths at different pressures reflected the nesting degrees of FSs, where the nesting trajectories corresponding to the softening phonon modes are conspicuous at 200 GPa than that at 100 GPa. Meanwhile, the electronic bands and related FSs have been described in figs. S4 and S5, where two nearly parallel conduction bands cross the E_F along Q_Г → Q_z, Q_V → Q_Г → Q_C and Q_A → Q_M → Q_L and their related FSs are nearly parallel to each other at these positions, indicating that the soft phonon modes are a result of the FS nesting. Moreover, we notice that the strength of phonon linewidth ${\mathrm{\gamma }}_{\mathit{qv}}$ at 200 GPa is distinctly larger than that case at 100 GPa. According to the linewidth expression ${\mathrm{\gamma }}_{\mathbf{q}\mathrm{\upsilon }}=\mathrm{\pi }{\mathrm{\omega }}_{\mathbf{q}\mathrm{\upsilon }}{\sum }_{\mathit{mn}}{\sum }_{\mathbf{k}}{\mid g_{\mathit{mn}}^{\mathrm{\upsilon }}(\mathbf{k},\mathbf{q})\mid }^2\mathrm{\delta }({\mathrm{\epsilon }}_{\mathit{k},i}-{\mathrm{\epsilon }}_F)\times \mathrm{\delta }({\mathrm{\epsilon }}_{\mathit{k}+\mathit{q},j}-{\mathrm{\epsilon }}_F)$, apart from the contribution of the FS nesting effect to the strengthening of phonon linewidth ${\mathrm{\gamma }}_{\mathit{qv}}$, electron-phonon matrix element $g_{\mathit{mn}}^{\mathrm{\upsilon }}(\mathbf{k},\mathbf{q})$ as another critical factor plays an important role in EPC (41). In the pressure range from 200 to 300 GPa, because of the sharply decrease of N(E_F) at 300 GPa than that at 200 GPa, the FS nesting strength in Fig. 4B was weakened as shown in fig. S3. This observed result also aligns with a decline in the contribution of the peak reduction of the α²F(ω) and phonon linewidth ${\mathrm{\gamma }}_{\mathit{qv}}$ to EPC. Considering that the electron-phonon matrix element $g_{\mathit{mn}}^{\mathrm{\upsilon }}(\mathbf{k},\mathbf{q})$ and FS nesting collectively govern the phonon linewidth ${\mathrm{\gamma }}_{\mathit{qv}}$ to influence EPC, the weaker strength of FS nesting at 300 GPa than that at 200 GPa in Fig. 4B reflects the increased contributions of $g_{\mathit{mn}}^{\mathrm{\upsilon }}(\mathbf{k},\mathbf{q})$ to EPC within this pressure range.

DISCUSSION

In contrast to the H-derived electronic states that dominate superconductivity in atomic-type hydrides H₃S (8) and LaH₁₀ (11), our simulations demonstrate that near-free electrons predominantly offer Cooper pairs to induce high T_c in C2/c-CaH₁₄ phase. To further elaborate the role of near-free electrons in superconductivity, we further calculated and compared the superconductivity of different molecular-type hydrides at 200 GPa, as shown in table S3. The typical H₂-molecular–type TeH₄ (42) and SnH₄ (43) also exhibit near-free electrons behaviors. These findings suggest that the necessary condition for superconducting transition is the formations of free electrons as Fermi sea based on BCS theory (44) rather than the monatomic hydrogen in hydrogen-rich system. At 200 GPa, when the phonon effects make the same contribution to EPC in TeH₄ and SnH₄, it was observed that an increase in near-free electrons indeed resulted in a high T_c. This result may confirm that an increase in near-free electrons within E_F leads to a gradual enhancement of superconductivity in H₂-molecular–type hydrides.

In summary, in contrast to the established guideline for achieving high-temperature superconductivity in atomic hydrogen–type hydrides, we report a remarkable finding of an unusual class of H₂-molecular–type superhydride CaH₁₄, with a predicted high T_c of 204.0 K at 300 GPa, as identified through advanced crystal structure search methods. We identify unusual itinerant electrons that act as fully near-free electrons to mainly mediate metallic interactions. Our first-principles calculation unveils that the high T_c is mainly attributed to strong EPC stemming from the large electron-phonon matrix element driven by intermediate-frequency molecular hydrogen–derived phonon vibrations and phonon softening caused by FS nesting, thus scattering itinerant electrons to form Cooper pairs. The formation mechanism of near-free electrons originated from the pressure-induced sinking effect on interatomic potential wells, rendering it unfeasible to confine electrons due to their high kinetic energy endowed by strong Pauli repulsion. These results elucidate that the necessary condition for superconducting transition is forming the Fermi sea with Cooper pairs rather than the monatomic hydrogen. Intriguingly, this H₂-molecular–type hydride can reduce the required pressure even down to attainable synthesis range of large cavity press and maintain the high T_c of 60 K at 50 GPa and 84 K at 80 GPa above the temperature of liquid nitrogen. Our current findings provide powerful insights for exploration of superconductivity predicted in molecular hydrides and could also reignite and attach great importance to the understanding of the superconductivity of molecular hydrogen. The unique high-pressure behavior of electrons and EPC interaction in molecular hydrides adds unprecedented categories to the search for room-temperature superconducting materials.

METHODS AND MATERIALS

Ab initio calculations

The binary Ca-H system was investigated using an evolutionary structure prediction algorithm implemented in the USPEX code, in conjunction with the particle swarm optimization method executed in the CALYPSO code (45–47). The first-principles calculations were performed on VASP code using the projector augmented wave (PAW) method and Perdew-Burke-Ernzerhof functional within the Kohn-Sham density functional theory (48–51). The PAW potentials for Ca and H atoms considered the 3s²3p⁶4s² and 1s¹, respectively, as valence electrons. The kinetic cutoff energy of 1000 eV and Monkhorst-Pack k meshes with a grid spacing of 0.10 Å⁻¹ were then used to ensure the self-consistent field tolerance of 0.1 × 10⁻⁴ eV/atom. The LOBSTER code was used to perform COHP analyses, which provided insight into the bonding status of the system (31). In particular, COHP information was obtained using the plane-wave (PW) method, and atom information was extracted from delocalized PW basis sets. The lattice-dynamical and superconducting properties were estimated using density functional perturbation theory (DFPT), which was implemented in the Quantum-ESPRESSO code (52, 53). Ultrasoft pseudopotentials were used with a kinetic energy cutoff of 100 Ry, and the charge density was integrated on a centered 12 × 12 × 16 k-point mesh. The first-order potential perturbation and dynamical matrices were calculated using DFPT on an irreducible 3 × 3 × 4 centered q-point mesh. In particular, the model potential formed by superimposing two 1D-Gaussian potential wells: $V(x)=G(x;-d/2)+G(x;d/2)$, where $G(x;d)=-\frac{1}{\sqrt{2\mathrm{\pi }}\mathrm{\sigma }}{e}^{-\frac{{(x+d)}^2}{2{\mathrm{\sigma }}^2}}$, σ = 0.15 Å, d = d₀, 0.5d₀, 0.4d₀, d₀ = 1.5 Å. Using this model potential, we can easily see the formation of the lower connecting ridge.

Supplementary Materials

This PDF file includes:

Figs. S1 to S6

Tables S1 to S3