High-Tc superconductivity by metalizing σ-electrons

Raising superconducting transition temperature ( T _c ) is an important task of fundamental research on superconductivity. It is also a prerequisite for the large-scale application of superconductors. Since the microscopic mechanism of high- T _c superconductivity is unknown, the conventional approach for increasing T _c is either to apply high pressure to a material that has the potential to become superconducting or to push it close to an antiferromagnetic or some other quantum instability point by chemical doping. In this article, the authors point out that another general approach for raising T _c is to lift the σ -bonding bands to the Fermi level or to me talize the  σ -bonding electrons. This approach can increase the probability of finding a novel high- T _c  superconductor because the coupling of σ -bonding electrons with phonons is generally strong and the superconducting transition induced by this interaction can occur at relatively high temperatures. After elucidating the underlying mechanism, the authors discuss a number of schemes to metalize σ -bonding electrons and present their recent prediction for the crystalline and electronic structures of two potential high- T _c superconductors ,  Li ₂ B ₃ C  and  Li ₃ B ₄ C ₂ , with T _c higher than 50 K.

Keywords superconductivity, electron-phonon coupling, first-principles calculations

Exploring and discovering new superconductors, especially high-temperature superconductors, is a goal continuously pursued in superconductivity research. This endeavor serves as a driving force behind the advancement of superconducting physics. In 1986, Swiss physicists Bednorz and Müller made a groundbreaking discovery of high-temperature superconductors based on copper oxides ^[1] . Their finding ignited a research boom in high-temperature superconductivity and deepened the study of strongly correlated quantum physics. In 2008, Japanese physicist Hosono and his colleagues discovered iron-based superconductors ^[2] , further expanding the scope of research in high-temperature superconductivity. High-temperature superconductivity has garnered widespread attention for two primary reasons. Firstly, the discovery and development of economically viable high-temperature superconductors are crucial prerequisites for expanding the range of applications for superconductors, thus bearing significant practical implications. Secondly, uncovering novel high-temperature superconductors is typically accompanied by the revelation of new phenomena and effects. Studying these phenomena and effects facilitates a deeper understanding of the microscopic mechanisms underlying high-temperature superconductivity and guides the exploration and discovery of new microscopic quantum effects and methodologies. 

Superconductivity is a macroscopic quantum phenomenon characterized by zero direct current resistance and perfect diamagnetism, known as the Meissner effect. The study of superconductivity traces back to 1911 when the Dutch physicist Onnes observed the phenomenon in mercury at temperatures near 4.2 K, marking the inception of superconductivity research. High-temperature superconductors, which have relatively high critical temperatures ( T _c ), lack a strict temperature range definition, but generally refer to materials with a  T _c  close to or above 40 K. The cuprate superconductor HgBa ₂ Ca ₂ Cu ₃ O _{8+ δ}  holds the record for the highest  T _c  , reaching 133 K under ambient pressure ^[3] . Its critical temperature rises to 164 K under pressure ^[4] . Iron-based superconductors represent another class of high-temperature superconductors, exhibiting a maximum T _c of 55 K ^[5] , approximately one-third of the highest transition temperature observed in copper oxide superconductors.

Revealing and elucidating the microscopic mechanism underlying superconductivity is a fundamental topic in condensed matter physics. In 1957, Bardeen, Cooper, and Schrieffer (BCS) proposed that the origin of superconductivity is the formation of bound electron pairs (Cooper pairs) due to an attractive interaction between two electrons with opposite spins and momenta near the Fermi level generated through the exchange of virtual phonons ^[6] . Their theory explains the experimental phenomena observed in element and alloy superconductors. However, electron-phonon coupling is not the only interaction capable of inducing electron pairing. For example, electron pairing in copper oxide and iron-based superconductors may result predominantly from antiferromagnetic fluctuations, although a microscopic picture still needs to be established. 

The superconducting phase transition temperature is primarily determined by two key parameters: the effective coupling constant responsible for superconducting pairing and the characteristic energy scale near the Fermi surface that facilitates the formation of Cooper pairs. In the context of electron-phonon interaction systems, these parameters correspond to the electron-phonon coupling constant and the characteristic coupling frequency of phonons. The strong-coupling theory of superconductivity allows for an approximate calculation of these parameters through first-principles density functional theory calculations. However, determining these parameters in copper oxides, iron-based materials, and other superconductors, where electron-phonon coupling may not be the dominant mechanism, poses a challenge that requires further research and exploration.

There are at least two approaches that can be used to enhance the critical temperature of superconductivity: 

The first approach is to apply high pressure to materials that have the potential to form superconducting pairs, especially in pure elements or compound materials composed of light elements, to increase the superconducting phase transition temperature. High pressure is an effective means of manipulating material properties. It can simultaneously enhance the interaction between superconducting electron pairs and the characteristic energy scale. It can also stabilize crystal structures that are unstable or metastable under normal pressure, revealing novel physical phenomena. For example, semiconductor materials like germanium and silicon can become superconductors under high pressure, with superconducting transition temperatures of 6.7 K and 5.3 K, respectively. In 1968, Ashcroft predicted solid hydrogen could become metallic under high pressure and exhibit room-temperature superconductivity ^[7] . However, achieving the necessary pressure for the metallic solid hydrogen is extraordinarily challenging and has yet to be achieved experimentally. One approach to overcome this diffculty is to select hydrogen-rich materials that utilize attractive interactions generated by chemical bonding between hydrogen and other elements to lower the pressure required for metalizing hydrogen. A German research group recently discovered superconductivity in hydrogen sulfide at a high pressure of 200 GPa, with a transition temperature of 190 K ^[8] . If their experimental results are confirmed, it would be the highest superconducting phase transition temperature discovered.

The second approach involves elevating  T _c  by introducing doping or other means to bring the material close to antiferromagnetic or other quantum critical points. This proximity amplifies the pairing energy of superconducting electrons through antiferromagnetic or other quantum fluctuations. Antiferromagnetic interaction serves as a potent force between two electrons. In the case of copper oxide superconductors, the characteristic energy scale of antiferromagnetic interaction is approximately 130 meV ^[9] , which is roughly an order of magnitude higher than the electron-phonon interaction observed in metals. Consequently, compared to systems reliant on electron-phonon interactions, the pairing energy of superconducting electrons resulting from antiferromagnetic interaction is substantially greater. However, a strong antiferromagnetic interaction may lead to the formation of antiferromagnetic spin ordering, which proves detrimental to the emergence of superconductivity. Thus, to effectively harness the influence of antiferromagnetic interaction on superconducting electron pairing while avoiding the formation of long-range antiferromagnetic order, it is critical to approach, but not surpass, the critical point of the antiferromagnetic phase transition. 

However, both of these approaches encounter certain limitations. First, achieving and controlling pressures above 100 GPa is extremely challenging experimentally. The critical pressure for hydrogen transition into solid metal has yet to be determined. Theoretical calculations predict that the critical pressure is above 500 GPa ^[10] , but the actual value may be higher than the theoretical estimate. Such high pressures pose significant challenges for high-pressure experiments. Currently, the highest reliably attainable pressure using diamond anvil cell techniques is approximately 350 GPa ^[11] . In 2012, Dubrovinsky and colleagues improved the hardness anisotropy of single crystal diamonds by using nanocrystalline diamond particles as anvils. They designed a two-stage high-pressure device to generate static pressures up to 640 GPa ^[11] . Despite this, there have been no experimental reports on solid metallic hydrogen. Second, controlling the growth of materials to approach antiferromagnetic or other quantum critical points remains challenging. As reliable rules or guidelines for achieving this outcome are still unavailable, discovering new high-temperature superconductors through this approach is serendipitous. The findings of copper oxide and iron-based superconductors exemplify this fortuitous nature of discovery.

Is there a more feasible experimental approach that offers a higher probability of successfully synthesizing or discovering new high-temperature superconductors? The answer is yes: to metalize the  σ -bonding electrons in a material, driving them into conducting electrons and ultimately leading to supeconductivity. 

The σ -bond is a spin singlet formed by two electrons with opposite spins, bound together by the head-to-head overlapping of two orbitals along the axis linking the two orbitals, which may have the same or different orbital symmetries. It has a considerable overlap and corresponds to a strong covalent bond. For example, the bond energy of σ -bonds formed by the  sp ³  hybridized orbitals in the diamond is approximately 3.9 eV. In a crystal, the σ -bonding electrons further hybridize and form energy bands called the σ -bonding bands or simply the σ -bands. The σ -bonding electrons are sensitive to lattice vibrations of the ions, hence coupling strongly with phonons. This coupling is generally the primary interaction that stabilizes the crystal structure. However,  σ -bonding electrons are typically below the Fermi level and do not contribute to conductivity. Therefore, to achieve superconductivity through metalized  σ -bonding electrons, we must use various methods, such as chemical doping, to move the σ -bonding bands to the Fermi level. While the metalization of  σ -electrons will inevitably reduce the electron-phonon interactions and weakens the pairing energy of electrons, assuming the band structure of the σ -electrons does not change significantly, the remaining electron-phonon interactions can still be strong. Once forming superconducting pairs, the corresponding superconducting energy gap can be potentially high.

MgB ₂  is a superconductor that arises from the metallization of the σ -electrons within the boron layers. It exhibits a remarkably high superconducting transition temperature of 39 K, which stands as the highest among all phonon-mediated superconductors discovered under ambient pressure ^[12] . In cuprate superconductors, the 3 d _{x 2－ y 2}  -orbitals of copper atoms hybridize with the 2 p  oxygen orbitals to form a  σ -bonding band. Upon hole doping, this σ -bonding band is elevated to the Fermi level and becomes conducting. In particular, a doped hole on the in-plane oxygen 2 p -orbital couples with a copper 3 d _{x 2－ y 2}  electron to form a Zhang-Rice singlet ^[13] , whose propagating and pairing leads to high- T _c  superconductivity. Therefore, there are at least two examples that support the notion of achieving high-temperature superconductivity through the metallization of σ -electrons. However, it has not been widely recognized that this scenario applies  universally  and can be effectively utilized in the quest for new high-temperature superconductors.

Metallization by lifting the σ -electron band to the Fermi level depends on the chemical compositions and crystal structures of the materials studied and does not follow a fixed model. The following discussion will explore several specific examples to illustrate potential methods toward metallization.

(1) Crystal field effect: The interaction between electrons and the crystal field can vary across different bands. Manipulating this interaction can lower the energy of valence bands or raise the energy of conduction bands, leading to the metallization of σ -bonding electrons. A prominent example demonstrating high- T _c  superconductivity through this effect is MgB ₂ . In MgB ₂ , boron atoms form a two-dimensional honeycomb structure, while magnesium atoms are positioned above the hexagonal centers of the boron atoms between adjacent boron layers. The Mg ²⁺  ions strongly attract the π-electrons but exert a weaker attraction on the σ -electrons within the boron layers. Consequently, this interaction significantly lowers the Fermi energy, resulting in an energy overlap between the σ and π bands and leading to the metallization of the σ -bands [see Fig. 1(a)] ^[14] . These metalized σ -electrons couple with the  E _2g  phonon mode on the boron layers [see Fig. 1(b)], ultimately enabling MgB ₂  to exhibit superconductivity at 39 K ^[15] .

Figure 1 (a) Band structure of MgB ₂ . Red and blue lines represent the bands mainly contributed by the σ and π electrons of boron. The linewidth is proportional to the weight of σ or π electrons. The Fermi level is set to zero. (b) Phonon spectrum of MgB ₂ . The red linewidth represents the coupling strength between  E _2g  phonon modes and electrons.

(2) High pressure: High pressure can induce structural changes in crystals, stabilize metastable structures, and modify the band structure by increasing the electron bandwidth. Metallization of the σ  band occurs when the increase in bandwidth is significant enough for the conduction and valence bands (often containing the σ bands) to start overlapping. An illustrative example is the hydrogen sulfide under high pressures, which exhibits superconductivity with a remarkable transition temperature of 190 K. Both experimental and theoretical investigations indicate that H ₂ S decomposes under high pressure and the primary superconducting compound is H ₃ S ^{[8, 16 —18]} . Notably, the σ band, formed by hybridizing hydrogen  s  electrons with sulfur  p  electrons, was found to undergo metallization under high pressure ^[18] . If the superconducting properties of H ₃ S at 190 K and 200 GPa are experimentally confirmed, it will be an example of σ -electron metallization through high pressure.

(3) Charge doping: Charge doping plays a crucial role in the investigation of cuprate and iron-based superconductors. It is an effective method for adjusting the position of the Fermi level and metalizing the σ  band. In 2004, Ekimov and his colleagues discovered superconductivity in boron-doped diamonds (with a boron doping level of 2.8%) with a transition temperature of 4 K ^[19] . This superconductivity results from the raising of the valence band, specifically the σ band formed by the hybridized  sp ³  orbitals, above the Fermi level due to the boron doping, rather than the contribution of impurity bands ^[20—22] . Theoretical calculations suggest that increasing the concentration of boron doping in diamonds to 20%—30% can result in a higher transition temperature ^[23] . However, achieving such high boron doping concentrations in diamonds poses significant challenges. In the case of cuprate superconductors, hole doping lift the σ band formed by the hybridization of copper’s 3 d _{x 2－ y 2}  electrons and oxygen’s 2 p  electrons to the Fermi level. Nevertheless, the mechanism of electron pairing for superconductivity based on this foundation remains unclear. 

The preceding discussion underscores the universal nature of metalizing σ -electrons in the search for new high-temperature superconductors. In line with this, Pickett and his colleagues predicted that Li _0.5 BC, achieved by introducing 50% lithium vacancies to LiBC, represents a strongly coupled electron-phonon superconductor with a superconducting temperature potentially reaching 100 K ^[24] . In its undoped state, LiBC is a semiconductor with a crystal structure resembling that of MgB ₂ , featuring intercalated lithium layers positioned above the boron-carbon honeycomb layers [see Fig. 2(a)]. The top valence band is formed by the hybridized  sp ² σ -bonding electrons [see Fig. 2(d)]. The introduction of lithium vacancies causes the σ -bonding states to intersect the Fermi level. However, the experimental observation of superconductivity in Li _0.5 BC has not yet been accomplished due to the significant structural alterations in the boron-carbon layers caused by the partial absence of lithium atoms, resulting in a complete modification of the band structure and the prevention of σ band metallization  ^[25] . This example underscores the significance of maintaining structural stability while achieving σ band metalization through hole doping. 

Figure 2 (a-c) Crystal structures of LiBC, Li ₂ B ₃ C, and Li ₃ B ₄ C ₂ . (d-f) Band structures of LiBC, Li ₂ B ₃ C, and Li ₃ B ₄ C ₂ . The Fermi level is set to zero. The width of the red line denotes the weight of  sp ² -hybridized σ orbitals in the corresponding Bloch state.

An approach proposed to hole dope LiBC without introducing crystal instability is substituting part of carbon atoms with boron atoms instead of creating lithium vacancies  ^[26] . Theoretical calculations have identified two structure-stable materials, Li ₂ B ₃ C and Li ₃ B ₄ C ₂ , as depicted in Figure 2(b) and 2(c), respectively, which show promising potential for achieving high superconducting transition temperatures. In Li ₂ B ₃ C, one boron-carbon layer is replaced by a boron layer for every two boron-carbon layers in LiBC. In Li ₃ B ₄ C ₂ , the boron-carbon layers maintain a 2:1 ratio of boron to carbon. The band structures of these materials are illustrated in Figure 2(e) and 2(f), respectively. Encouragingly, first-principles calculations of the electron-phonon interaction reveal a strong coupling between the  σ -bonding electrons and phonons in both materials. Further calculations, employing the Eliashberg theory for strong coupling superconductivity, suggest that both materials exhibit superconducting transition temperatures above 50 K ^[26] . Notably, the phonon spectra of these materials display no imaginary frequencies, indicating their dynamical stability. Should experimental investigations confirm the theoretical predictions, these materials would represent a significant breakthrough in the discovery of phonon-mediated high- T _c  superconductors under ambient pressure.

Ensuring the stability of the crystal structure is critical during the metalization of the σ bands and demands careful attention. Several crucial considerations arise in this context: Firstly, the strong coupling between σ electrons and phonons can induce structural transitions, potentially compromising the stability of the crystal. It is preferred to mitigate this instability by prioritizing optical phonon modes that exhibit a strong coupling with electrons while minimizing the influence of acoustic phonon modes. Acoustic phonons have lower frequencies and are more prone to further softening under electron-phonon coupling, potentially triggering structural transitions. Secondly, the σ bonds play a vital role in maintaining the stability of the crystal structure. Attempting to metalize all σ bands is neither practical nor desirable. Instead, a selective approach is necessary, focusing on metalizing one or a few σ bands closest to the Fermi level. This selective strategy preserves the structural integrity of the crystal while facilitating the desired metalization process. Thirdly, a σ band exhibits minimal dispersion in two-dimensional systems along the direction perpendicular to the two-dimensional plane. Consequently, upon metallization, the σ electrons in two-dimensional systems tend to possess a higher density of states at the Fermi level than in three-dimensional systems. Therefore, reducing the dimensionality can effectively increase the density of states on the Fermi surface and enhance the superconducting transition temperature.