Superconductivity in Compression-Shear Deformed Diamond Superconductivity in Compression-Shear Deformed Diamond

Diamond is a prototypical ultrawide band gap semiconductor, but turns into a superconductor with a critical temperature T c ≈ 4 K near 3% boron doping [E. A. Ekimov et al. , Nature (London) 428 , 542 (2004)]. Here we unveil a surprising new route to superconductivity in undoped diamond by compression-shear deformation that induces increasing metallization and lattice softening with rising strain, producing phonon mediated T c up to 2.4 – 12.4 K for a wide range of Coulomb pseudopotential μ (cid:1) ¼ 0 . 15 – 0 . 05 . This finding raises intriguing prospects of generating robust superconductivity in strained diamond crystal, showcasing a distinct and hitherto little explored approach to driving materials into superconducting states via strain engineering. These results hold promise for discovering superconductivity in normally nonsuperconductive materials, thereby expanding the landscape of viable nontraditional superconductors and offering actionable insights for experimental exploration.

Diamond is a prototypical ultrawide band gap semiconductor, but turns into a superconductor with a critical temperature T c ≈ 4 K near 3% boron doping [E. A. Ekimov et al., Nature (London) 428, 542 (2004)]. Here we unveil a surprising new route to superconductivity in undoped diamond by compressionshear deformation that induces increasing metallization and lattice softening with rising strain, producing phonon mediated T c up to 2.4-12.4 K for a wide range of Coulomb pseudopotential μ Ã ¼ 0.15-0.05. This finding raises intriguing prospects of generating robust superconductivity in strained diamond crystal, showcasing a distinct and hitherto little explored approach to driving materials into superconducting states via strain engineering. These results hold promise for discovering superconductivity in normally nonsuperconductive materials, thereby expanding the landscape of viable nontraditional superconductors and offering actionable insights for experimental exploration. DOI: 10.1103/PhysRevLett.124.147001 Diamond crystal comprises a carbon bonding network that produces superior mechanical strength and a very large electronic band gap at ambient conditions, making it a superstrong ultrawide band gap semiconductor. Theoretical studies predicted superconductivity in select semiconductors in the early 1960s [1,2], but superconducting states in technologically important group-IV semiconductors long eluded experimental detection. Ekimov et al. reported in 2004 synthesis of boron doped diamond with near 3% hole concentration, hosting a superconducting state with a critical temperature T c ≈ 4 K [3], and theoretical works [4][5][6][7][8][9] suggest an electron-phonon coupling mechanism. Similar phonon mediated superconductivity was later observed in silicon at 5.7-8.4 at.% boron doping, with a much lower T c of 0.35 K due to weaker lattice vibration and reduced electron-phonon coupling [10] and in SiC at 2.2-3.9 at.% boron doping with a T c ≈ 1.4 K [11]. The discovery of superconductivity in these prominent semiconductors has prompted further explorations to advance fundamental understanding and potential innovative electronics applications [12,13].
Our recent studies show that diamond can be metallized under compression-shear (CS) deformation where a compressive stress prevents graphitization usually occurring in severely deformed diamond crystal [14]. This discovery raises intriguing prospects of intrinsic superconductivity in deformed diamond without external carrier doping. Here we report on a computational study of the evolution of electronic band structure, lattice vibration, and electronphonon coupling along a typical deformation path where metallic states emerge and develop. We find that CS strains progressively enhance conduction electronic density of states near the Fermi energy and induce phonon softening in deformed diamond crystal. These effects combine to generate increasingly stronger electron-phonon coupling, driving diamond into a superconducting state. This phenomenon presents a distinct route toward intrinsic superconductivity in undoped diamond, expanding fundamental benchmarks of this prominent material. Our findings show that superconductivity may develop under complex strains in materials that are normally nonsuperconductive at ambient or simple pressure and stress conditions, opening a new area of research of hitherto largely unexplored undoped semiconductor-based superconductors. Such select deformation driven superconductivity is usually not accompanied by a conventional structural phase transition; instead, strain induced changes of bonding states and charge distribution in the deformed crystal generate concurrent electronic conduction and phonon softening that produce increasingly robust superconductivity. Recent experimental results show that strain modifies phonon modes in SrTiO3 films, resulting in enhanced superconducting transition temperatures, which suggests a promising approach to generating or enhancing superconductivity [15].
We have employed QUANTUM ESPRESSO (QE) code [16] for structural relaxation, stress-strain relation, and electronic band structure calculations based on the density functional theory (DFT) in the local density approximation (LDA) [17,18]. We also have performed lattice dynamics and electron-phonon coupling (EPC) calculations using the density-functional perturbation theory (DFPT) in linear response as implemented in QE. Stress responses from QE calculations are in excellent agreement (less than 2% difference) with the results obtained from calculations using VASP code [19,20] with an energy convergence around 1 meV per atom and residual forces and stresses less than 0.005 eV Å −1 and 0.1 GPa, respectively. Electronic calculations adopted a kinetic energy cutoff of 100 Ry and a 16 × 16 × 24 k mesh, and lattice dynamics calculations used a 4 × 4 × 6 q mesh. LDA calculations tend to underestimate electronic band gap in semiconductors, but such calculations adequately describe metallic states and produce fairly accurate T c in metallized semiconductors, e.g., in borondoped diamond and silicon [3][4][5][6][7][8][9][10] and in MgB 2 [21][22][23][24][25]. Further computational details on consistency, convergence, and reliability tests are provided in the Supplemental Material [26].
From the Eliashberg theory of superconductivity [31,32], McMillan derived [33], later modified by Allen and Dynes [34], an analytic expression for transition temperature, where ω log is a logarithmically averaged characteristic phonon frequency, and μ Ã is the Coulomb pseudopotential that describes the effective electron-electron repulsion [35].
The ω-dependent EPC parameter λðωÞ is obtained from the Eliashberg spectral function α 2 FðωÞ, with λð∞Þ giving the total EPC parameter λ in the T c equation, and Here, the integration is done over the Brillouin zone (BZ) of volume Ω BZ , and EPC parameter for mode j at wave vector q is Þ is the linewidth for mode j at wave vector q, arising from electron-phonon coupling, ε kn is the band energy at k, and the electron-phonon interaction matrix element is where M is the ionic mass, ∇V q is the gradient of crystal potential with respect to ionic displacements, and e qj is the polarization vector of phonon mode j at wave vector q.
The Allen-Dynes modified McMillan T c equation is widely adopted and generally accurate for materials with λ < 1.5 [36][37][38][39], whereas a numerical solution to the Eliashberg equation [31] is more adequate at larger λ as in, e.g., high-T c superconducting superhydrides [40,41]. The interaction between electrons is commonly described by the Coulomb pseudopotential μ Ã , which can be obtained from an inversion process by fitting experimental singleparticle tunneling data [36][37][38], and is often treated as an adjustable parameter. The values of μ Ã mostly stay within a very narrow range around 0.1 for most materials, making this formulism highly robust [34][35][36][37][38][39]. Pertinent examples include works on MgB 2 and boron-doped diamond and silicon that all adopted μ Ã ∼ 0.10 (0.08-0.13, mostly set to 0.10), producing T c in good agreement with measured data [4,5,10,[21][22][23][24][25]. The robustness of this approach is also verified by the latest ab initio Eliashberg theory calculations that map out effective μ Ã by fitting Eq. (1) to the fully solved Eliashberg T c [39]. Smaller values of μ Ã also occur, like μ Ã ¼ 0.06 for a high-pressure phase of Si [42]. In this work, we select a relatively broad range of μ Ã ¼ 0.05-0.15 for deformed diamond, which hosts strain-dependent electronic states thus changing electronic interactions, making the ranged μ Ã a well-suited choice.
We explore mechanical deformation induced superconductivity, focusing on diamond under [111] CS strain, which exhibits representative structural and electronic behaviors under CS strains. At increasing strains, we closely examine (i) evolutions of electronic bands and the associated variations in bonding and conduction charge distribution and (ii) phonon softening patterns and rising electron-phonon coupling in the deformed diamond crystal. We present in Fig. 1 our main findings on the emergence of superconductivity in progressively deformed diamond, which goes through three distinct stages under CS strains. In the first stage, the electronic band gap gradually decreases with increasing deformation and comes to a complete closure at CS strain ϵ ≈ 0.161 [43]. The second stage develops at further increasing strains, with rising accumulation of electronic states near the Fermi energy, but electronic density of states (EDOS) and EPC are too weak to produce an appreciable superconducting state. Superconductivity occurs in the third stage well inside the metallic regime at CS strain ϵ ≈ 0.349 where a sufficiently high EDOS has been reached and, perhaps more significant, the deformed diamond lattice softens considerably to generate a strong EPC (see below). This process leads to a rising CS-strain driven T c that reaches 2.4-12.4 K for Coulomb pseudopotential μ Ã ¼ 0.15-0.05. These surprisingly high T c values under realistic μ Ã place deformed diamond among the most prominent elemental superconductors [44].
To decipher key factors driving CS deformed diamond into a superconducting state, we first examine evolution of electronic states, focusing on changes in bonding charge and emergence of conduction charge. The electronic band structures and the corresponding EDOS of deformed diamond at selected strains are shown in Fig. 2. It is seen that band states start to cross the Fermi level around shear strain 0.161, and an increasing number of bands and amount of states pass the Fermi level at higher strains, generating a rising EDOS (N E F ) at the Fermi energy. At shear strain 0.349, N E F reaches a level comparable to those in boron-doped diamond and Si [4,5], initiating an appreciable superconducting state [ Figs. 1(a) and 1(b)]. Accumulated electronic states distribute smoothly around the Fermi level, conforming to near-constant EDOS close to E F required for the derivation of many equations [37,38], including Eqs. (1)-(4). In Fig. 2(e), we show total charge in minimally (ϵ ¼ 0.000) and maximally (ϵ ¼ 0.566) strained stable diamond structures. A significant depletion of bonding charge is seen in the latter case, which weakens C-C bonds, especially those aligned in the [111] and  directions. To track strain induced charge redistribution, particularly their contribution toward conduction, we integrated EDOS in the ½−0.3; 0.3 eV energy window around the Fermi level, and the resulting conduction charge patterns are plotted in Fig. 2(f), where both a local view (left panel) and extended view (right panel) are shown. Conduction charges center on carbon atoms between bonds that span a large angular space driven by CS strains, and point away from all C-C bonds in the crystal. The extended view further highlights that conduction charges locate in buckled (111) and  planes separated by the deformation weakened [111] and  bonds. This pattern of nonbonding conduction charge is similar to that in graphite, where nonbonding π states in the graphitic sheets contribute to electrical conduction. It is noted, however, the bulk bonding network remains intact without graphitization in CS deformed diamond. The EDOS values at large CS strains, e.g., 0.520 and 0.566, are considerably higher than in borondoped diamond and silicon [4,5], which should produce better screening thus smaller μ Ã in CS strained diamond than previously reported 0.10 [4,5], suggesting T c of deformed diamond should reach the upper range shown in Fig. 1(b).
We next explore lattice dynamics of deformed diamond and assess the evolution of phonon spectra and the associated EPC with rising strains. Plotted in Figs. 3(a)-3(d) are calculated phonon dispersion curves and the corresponding spectral function α 2 FðωÞ=ω and EPC parameter λðωÞ at selected strains. Several spectral features are noteworthy. There is an overall spectral softening as increasing strains, reflecting a general weakening of deformed C-C bonds under CS strains, which explains the decrease of logarithmically averaged phonon frequency ω log in the superconducting regime as shown in Fig. 1(c). There is an especially pronounced phonon softening at A in the Brillouin zone as seen in Figs. 3(a)-3(d), where a large drop of frequency occurs concomitantly with a significant increase of EPC, resulting in a steep rise of λðωÞ in the corresponding low-frequency range. To elucidate the underlying mechanism, we show in Fig. 3(e) the vibrational modes of the two lowest-frequency phonon branches at A, labeled mode A1 and mode A2, which exhibit the most profound CS strain induced softening. These phonon modes correspond to relative shear oscillations of the (111) and (11-1) crystal planes connected by the charge-depletion weakened bonds, which explains the pronounced reduction of the associated phonon frequency. Moreover, the directions of the polarization e qj of these phonon modes, as indicated by the arrows, are aligned with the charge-depleted [111] and  bonds, respectively. Since charge depletion occurs mostly in the middle of these weakened bonds, the resulting charge disparities along these bonds produce large magnitudes for the gradients of the crystal potential ∇V q aligned in the same directions as those of e qj . As such, the ∇V q · e qj term in the matrix element in Eq. (4) is maximally optimized, accounting for the greatly enhanced EPC. These results showcase an effective strain altered charge redistribution in the deformed crystal lattice that plays the dual role of softening phonon modes and enhancing EPC, and this may be a robust mechanism in a broad range of materials, offering crucial guiding principles in search of strain engineered nontraditional superconductors.
Further studies of aspects not considered in this work, e.g., the anharmonic phonon effect and EPC anisotropy, may offer insights for more accurate understanding of superconducting diamond; also of interest are thermodynamic properties that can be derived from numerically solving the full Eliashberg equation. Anisotropic EPC has been shown to enhance T c considerably in MgB 2 [25], and such an effect would strengthen our reported results. A major complication in studying boron-doped diamond, Si, and SiC is structural disorder introduced by the doping process [12]. In contrast, undoped CS deformed diamond provides a platform for exploring intrinsic superconductivity in a metallized semiconductor, and this phenomenon may be prevalent among many semiconductors, suggesting a promising new research area of fundamental and practical significance. Our results also hold major implications for high-pressure experiments using diamond anvil cells (DAC), whose culet regions hosting four-probe leads for transport measurements commonly experience large pressure gradients with considerable CS strains [45]. At extreme loadings these regions may become superconducting and impede measurements probing superconductivity in specimens inside the DAC chamber. Meticulous measurements should verify this intriguing phenomenon. It is noted that the predicted superconductivity in deformed diamond occurs in a strain range near ductile failure. The newly identified compression-shear mode [14], however, offers an effective channel for releasing stress and energy thus sustaining the structural integrity in the deformation regime of a flat stress-strain curve. This new mechanism results in a significantly reduced overall bond elongation under CS, leading to a much-reduced rate of stress and energy increase with rising strains, in stark contrast to the uniaxial case. Moreover, loaded DAC contains small regions with high CS stresses surrounded by much larger less-stressed volumes [46]. This setting helps contain the severely CS deformed region, preventing any runaway structural failure. Similar situations exist in rotational DAC [47] and manipulated nanoscale diamond [48,49], where the largest CS deformed region is surrounded and supported by less-deformed crystal structure or other confining device components, thereby making the CS deformed superconducting region viable and accessible to experimental probes.
In summary, we have uncovered a new route to superconductivity in diamond by mechanical deformation. Compression-shear strained diamond crystal hosts deformed bonds with significant charge depletion, facilitating a buildup of conduction charge in crystal planes separated by the depletion-weakened bonds. At large strains well inside the metallic regime, superconductivity emerges at increasing accumulation of conduction states at the Fermi energy and progressively more pronounced softening of phonon modes, especially the shear modes associated with the most severely deformation-weakened bonds. These effects together produce increasingly stronger electron-phonon coupling, leading to rising superconducting transition temperature T c that reaches 2.4-12.4 K for a broad range of Coulomb pseudopotentials μ Ã ¼ 0.15-0.05. These results place deformed diamond among prominently select elemental solids that exhibit the highest T c values. The present work raises exciting prospects of turning normally nonsuperconductive materials into intrinsic superconductors by a hitherto little explored strain engineering approach, which holds promise for finding a distinct class of deformation driven nontraditional superconductors.