Theoretical Predictions of Intrinsic Superconductivity in a Novel Two-Dimensional Cr₃B₅ Material

Abstract

The rapid development of electronic, energy, and quantum technologies has elevated the demand for high-performance superconducting materials, making the design of new 2D superconductors a key research hotspot. In this study, by means of structure prediction method combined with first-principles calculations, we successfully identified an unprecedented two-dimensional­(2D) Cr₃B₅ material. The 2D Cr₃B₅ features a unique layered structure with high stability, as validated by cohesive energy calculations (7.54 eV/atom), elastic constant analysis, and phonon dispersion studies. Electronic structure analysis reveals a high density of states at the Fermi level, dominated by Cr-3d orbitals, supporting strong electron–phonon coupling (EPC). Low-frequency in-plane vibrations of Cr atoms contribute significantly to the EPC, yielding a computed constant (λ) of 0.896 and a superconducting transition temperature (T _c) of 11.678 K, making it practical and viable under liquid helium conditions. These findings establish Cr₃B₅ as an intrinsic phonon-mediated transition-metal boride superconductor with significant potential for experimental realization, paving the way for designing and understanding new 2D superconducting materials.

1. Introduction

The discovery of graphene heralded a new epoch in materials science, with its extraordinary propertiesincluding ultrahigh carrier mobility, , exceptional mechanical strength, , and outstanding thermal conductivityigniting intensive research interest in two-dimensional (2D) materials. This fervor has propelled investigations into diverse 2D systems beyond graphene, driving the development of distinctive materials tailored for electronic, optoelectronics, and energy storage applications. , The discovery of 2D superconducting materials holds immense significance for advancing next-generation technologies. Among these, 2D superconducting materials have emerged as pivotal for next-generation technologies, owing to their indispensable roles in lossless electronic devices, low-dissipation integrated circuits, and compact, highly accurate measurement, and control or imaging equipment, , coupled with their capacity to host unique quantum phenomena such as enhanced quantum coherence and reduced dimensional constraints. In contrast to their bulk counterparts, 2D superconductors further exhibit superior integration capabilities, , enabling seamless incorporation into nanoscale devices and thereby opening avenues for miniaturized and high-precision applications.

Despite these advantages, achieving superconductivity in 2D materials remains a challenging endeavor due to stringent requirements, including a nonzero density of states at the Fermi level and sufficient electron–phonon coupling. Efforts to induce superconductivity in graphenedespite its groundbreaking discovery, inherently lacking superconductivity due to its vanishing density of states at the Fermi levelsuch as extreme doping, strain engineering, and substrate modifications, have seen limited success. This limitation highlights the critical need for new 2D materials with intrinsic superconducting properties, that can overcome these challenges without external modifications. Intrinsic 2D superconductors not only simplify discovering and designing material processes between theoretical potential and practical application but also offer unique opportunities to study fundamental quantum phenomena under reduced dimensional constraints, further emphasizing their significance in advancing both theoretical and experimental research.

Boron occupies a unique position in the periodic table, straddling the divide next to metals and insulators, with chemical properties just after carbon, having only three valence electrons but occupying four electron orbitals, which facilitates metal richness and also results in significant chemical reactivity. Therefore, transition-metal atoms with d orbitals are frequently introduced to construct stable borides as these metal elements have different stable valence states, i.e., they can provide a different number of electrons when forming compounds with boron, resulting in various common and uncommon metal boride configurations. − Transition-metal borides (TMBs) have emerged as a promising class of materials for addressing this challenge. These compounds, characterized by their intrinsic metallicity, structural versatility, and high stability, are fertile grounds for exploring 2D superconductivity. Some structures have been proven to possess superconductive electrical properties, such as the monolayer VB₂, NbB₂, and TaB₂, which are similar in structure to the two-dimensional multilayer superconducting AlB₂ and are forecasted as superconductors, boasting superconductive transition temperatures (T _c) of 8.1 K, 35.5 K, and 7.1 K. Tetragonal and hexagonal phases of W₂B₂ are both fundamentally phonon-mediated superconductors, with superconducting transition temperatures (T _c) of 7.8 and 1.5 K. The monolayer h-MnB₃, composed of two boron kagome layers, possesses intrinsic superconductivity with a superconducting transition temperature of 24.9 K. Recently, ZrB₆, known as hexaboride zirconium, has emerged as a novel 2D transition-metal compound and is shining as a native superconductor. Its superconducting critical temperature (T _c) marks an impressive 15.79 K. Among transition-metal elements, the atomic mass of Cr is relatively small but not the smallest. The realization of superconductivity depends not only on the phonon vibration frequency (related to atomic mass) but also closely on the electron–phonon coupling strength. However, in the electronic structure of Cr, the d-orbital electrons have a higher density of states and stronger electron correlation effects compared with those of other transition-metal elements with smaller atomic masses, which can enhance the electron–phonon coupling effect. In addition, in low-dimensional systems (such as single-layer CrB₂), the magnetic ordering characteristics of Cr may be effectively suppressed. Taking into account key factors such as electron–phonon coupling strength, theoretical performance predictions, and structural stability, this study selects the Cr–B compound system as the core research direction. By screening Cr–B compounds with different stoichiometric ratios through structural prediction, we ultimately identified the structurally stable Cr₃B₅ as the ideal research object.

In this context, we have forecast a novel 2D TMB, Cr₃B₅ sheet materials, through structural prediction and first-principles calculations. Cr₃B₅ not only exhibits exceptional stability but also possesses intrinsic metallicity and a high density of states at the Fermi level, which are critical for superconductivity. Our study explores the structural, electronic, and superconducting properties of Cr₃B₅, demonstrating its potential as an intrinsic phonon-mediated superconductor with a T _c of 11.678 K and an EPC strength of 0.896. In the opening segment of our exposition, we embarked on an exploration into the atomic architecture of the novel 2D material, Cr₃B₅. In the subsequent part of our exposition, based on the predicted structure, we have carried out a diverse range of studies, focusing on the structural stability of the novel two-dimensional structure Cr₃B₅; in the third section of our study, we delved into an exploration of the electronic characteristics inherent to the newly predicted two-dimensional structure Cr₃B₅ and found it to exhibit intrinsic metallic behavior, which sparked our interest in its superconducting properties; in the fourth part, we partially discussed the superconducting coupling mechanism of the structure. Our study helps deepen the understanding of the physical origin of superconductivity and provides a new perspective for research into two-dimensional superconductors.

2. Computational Details

In this study, CALYPSO structure search software was employed to combine the particle swarm optimization algorithm with the random search algorithm for an in-depth exploration of the Cr–B system. The upper limit of the number of atomic systems was set to 12 to generate diverse atomic ratios and configurations, thereby determining the corresponding atomic structures. Over 3000 structures were generated and optimized. When implementing two-dimensional constraints, the thickness of the vacuum layer was controlled to be less than 15 Å to meet the structural property requirements of two-dimensional materials, ensuring the scientificity and accuracy of the search results. All calculations related to in this article are performed in the framework of density functional theory, employing a dual-software strategy utilizing both the Vienna Ab Initio Simulation Package (VASP) and the Quantum Espresso (QE) program , to leverage their respective strengths in structural optimization and phonon calculations. VASP was primarily used for crystal structure relaxation, electronic structure analysis, and mechanical property evaluations, whereas QE was specifically applied for computing phonon dispersion spectra and electron–phonon coupling strengths. This integrated approach ensures comprehensive and accurate investigation of both structural properties and phonon-mediated superconducting mechanisms. The interaction between electrons and ions is realized by the Projector–Augmented–Wave. The Perdew–Burke–Ernzerhof (PBE) parametrized generalized gradient approximation is chosen to describe the exchange–correlation energy. , The ionic positions in the cell were fully relaxed to their minimum-energy configuration using the Broyden–Fletcher–Goldfarb–Shanno algorithm. The convergence criteria for energy and force were set as 10^–6 and 10^–2 Ry a.u.^–1

Ultrasoft pseudopotentials are employed in the QE package for calculating the phonon dispersion and the electron–phonon interaction. To ensure that the structures optimized by VASP are completely relaxed, the convergence criterion for each atomic force is set to 10^–3 Ry/Bohr. For an enhanced level of computational accuracy in our research, we utilized the Methfessel–Paxton smearing method, meticulously setting its width parameter at 0.02 Ry. This meticulous adjustment was instrumental in ensuring high reliability of our data. In the quest for calculating self-consistent electron density in Cr₃B₅, two distinct k-point grid schemes were meticulously crafted: a foundational 24 × 24 × 1 grid and a more refined 48 × 48 × 1 grid, significantly enhancing precision and granularity in our computations. The EPC matrix elements are calculated on a 6 × 6 × 1 q-point grid of Cr₃B₅. An in-depth exploration of phononic characteristics and electron–phonon coupling (EPC) in Cr₃B₅ was undertaken, leveraging the computational prowess of Density-Functional Perturbation Theory coupled with Eliashberg theory.

The magnitude of the EPC λ_qν is calculated by the Migdal–Eliashberg equation

$${\lambda }_{\mathrm{q}\mathit{\nu }}=\frac{{\gamma }_{\mathrm{q}\nu }}{\mathit{\pi }\mathit{h}\mathit{N}(E_{\mathrm{F}}){{\omega }_{\mathrm{q}\nu }}^2}$$ 1

In the equation, γ_qv signifies the phonon line width, , ω_qv denotes the phonon frequency, and N­(E _F), a cornerstone parameter, represents the density of electronic states at the Fermi energy level. The expression for γ_qv can be gracefully captured by the following sophisticated formula

$${\gamma }_{\mathrm{q}\nu }=\frac{2\pi {\omega }_{\mathrm{q}\nu }}{{\mathrm{\Omega }}_{\mathrm{B}\mathrm{Z}}}\sum _{k,n,m}|g_{kn,k+qm}^{\nu }{|}^2\delta ({ϵ}_{kn}-{ϵ}_{\mathrm{F}})\delta ({ϵ}_{k+qm}-{ϵ}_{\mathrm{F}})$$ 2

within the aforementioned equation, Ω_BZ stands as the volume of the first Brillouin zone (BZ), ϵ_kn and ϵ_k+qm denote the Kohn–Sham energies, and $g_{kn,k+qm}^{\nu }$ represents the EPC matrix element. The Eliashberg spectral function, − denoted as α² F(ω), unfolds its definition thus

$${\alpha }^{2}F(\omega )=\frac{1}{2\pi N(E_{\mathrm{F}})}\sum _{\mathrm{q}\nu }\frac{{\gamma }_{\mathrm{q}\nu }}{{\omega }_{\mathrm{q}\nu }}\delta (\omega -{\omega }_{\mathrm{q}\nu })$$ 3

The EPC constant λ­(ω) and the logarithmic average frequency ω_log can be obtained from

$$\lambda (\omega )=2{\int }_0^{\omega }\frac{{\alpha }^{2}F(\omega )}{\omega }\mathrm{d}\omega$$ 4

and

$${\omega }_{\log }=\exp [\frac{2}{\lambda }{\int }_0^{\infty }\frac{\mathrm{d}\omega }{\omega }{\alpha }^{2}F(\omega )\log \omega ]$$ 5

Then, the superconducting transition temperature T _c can be calculated from

$$T_{\mathrm{c}}=\frac{{\omega }_{\log }}{1.2}\exp [-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{*}(1+0.62\lambda )}]$$ 6

Within our computational paradigm, we opt to employ the effective screened Coulombic repulsion constant, μ*, set at a value of 0.1. −

3. Results and Discussion

3.1. Lattice Structure and Stability

Figure shows the optimized crystal structure of Cr₃B₅ determined by calculation. Chromium atoms are represented by larger green spheres, while boron atoms are represented by smaller pink spheres. The overall structure is approximately a network structure with a certain regularity. The green Cr atoms are distributed on the outer layer of the structure, and the pink B atoms are embedded between the Cr atoms. The atomic structure of Cr₃B₅ is composed of three layers: upper, middle, and lower. In the top view of Figure a, each Cr atom is connected to five B atoms. As shown in the side view of Figure b, the layered characteristics can be more clearly observed with atoms arranged in layers. The Cr atoms in the middle layer are connected to one B atom directly above and below them. The unit cell of the structure is represented by black solid lines, which define the repeating unit of the crystal. After optimization, the structural parameters of Cr₃B₅ are obtained: the lattice parameters are a = 2.86 Å, b = 3.97 Å, and γ = 111.09°, the distance between the upper and lower layers is approximately 3.32 Å. The structure includes two types of B–B bond lengths, namely, 1.652 Å and 1.88 Å, and three types of B–Cr bond lengths, namely, 3.63 Å, 2.12 Å, and 2.09 Å, with the space group being Amm2. The lattice constant along the c-axis is intentionally set to 15 Å to eliminate the interaction between the layers.

1.

Top (a) and left side (b) views of optimized Cr₃B₅. The larger green spheres denote the Cr atoms, while the notably smaller pinkish-red spheres represent the B atoms. Top (c) and left side (d) views of difference charge density for Cr₃B₅. Yellow regions emblemize the gain in electron cloud, whereas teal outlines the contours where electron density recedes. The black solid line represents the unit cell.

To analyze the electron transfer and bonding characteristics of this crystal structure, a differential charge density analysis, Bader charge analysis, and the calculation of the electron localization function (ELF) for the novel Cr₃B₅ structure are performed. According to the differential charge calculation, which is the difference between the charge density after bonding and the corresponding atomic charge density before bonding, as detailed in Figure c,d, most areas around the B atoms are bright yellow, indicating that they gain electrons, while the transition-metal element Cr is mainly surrounded by blue-green, indicating that it loses electrons. The actual charge quantities of each atom obtained through the Bader charge calculation are shown in Table . The results indicate that Cr atoms lose electrons while B atoms gain electrons, leading to positively charged Cr ions and negatively charged B ions. Moreover, B atoms acquire electrons from Cr atoms, which is consistent with the differential charge results calculated earlier.

1. Actual Charge Number of Each Atom in Cr₃B₅ after Completing the Bader Charge Calculation (Quantified in Units of |e|).

Cr3B5	Cr1	Cr2	Cr3	B1	B2	B3	B4	B5
charge	11.03	11.03	11.18	3.54	3.53	3.49	3.49	3.71

The values of the ELF are shown in Figure . The ELF value between adjacent boron (B) atoms is 0.79 (greater than 0.5), and the red area is mainly distributed around the boron atoms with a large coverage, indicating the presence of covalent bonds between them. The region near the Cr atoms is blue-green, and the ELF values in this region are generally less than 0.5, indicating strong electron delocalization. The ELF values here accurately reflect the characteristics of the chromium–boron (Cr–B) ionic bond.

2.

ELF graphical representations for Cr₃B₅ furnish visual insights into its Miller indices, specifically displaying orientations at (a) 100 and (b,c) 001 surface views.

The stability of a new material is one of the key factors determining its application prospects. For the optimized structure based on Cr₃B₅, a series of calculations have been performed to evaluate its stability and other physical properties, including cohesive energy calculations, phonon dispersion along high-symmetry directions, and elastic constants and Young’s modulus related to this structure. The absence of imaginary phonon modes throughout the BZ serves as conclusive evidence confirming the dynamic stability of Cr₃B₅. This stability indicates a high likelihood that the material can be synthesized under experimental conditions, and a comprehensive analysis of the phonon dispersion will be conducted in subsequent chapters. Cohesive energy is a key parameter for measuring the stability of materials, reflecting the degree of combination of internal atoms or molecules. The cohesive energy is typically determined by using a series of formulas that calculate the energy required to separate a solid into individual atoms or molecules. These equations often involve the total energy of the system and the energies associated with the isolated components. In the newly predicted Cr₃B₅ structure that we have forecast, the formula for calculating the cohesion energy is as follows

$$E_{\mathrm{c}\mathrm{o}\mathrm{h}}=\frac{{\mathbf{3}\mathit{E}}_{\mathrm{C}\mathrm{r}}+\mathbf{5}{E}_{\mathrm{B}}-E_{{\mathrm{C}\mathrm{r}}_3{B}_5}}{8}$$ 7

In the formula for calculating the cohesive energy related to the Cr₃B₅ structure, $E_{{\mathrm{C}\mathrm{r}}_3{\mathrm{B}}_5}$ is the total energy of Cr₃B₅, and E _Cr and E _B represent the energies of isolated metal atoms (Cr atoms) and isolated B atoms, respectively. Such energy figures serve as cardinal metrics in the assessment of the material’s holistic energetic condition, playing a pivotal role in determining its stability, potential applications, and response to various environmental conditions. Given this definition, a higher cohesive energy value means greater system stability, and it is more likely that such a structure can be synthesized experimentally. The computed cohesive energy for this structure amounts to 7.54 eV/atom, which is larger than the 2D materials that were previously predicted through computational methods and subsequently fabricated experimentally, such as Cu₂Si, , Cu₂Ge, black phosphorus, and silicene.

The mechanical properties of Cr₃B₅ are analyzed from three aspects: the elastic modulus is an index to measure the ability of the material to resist shape changes under stress, and the elastic constants, as fundamental physical quantities, are listed in Table . According to the elastic stability criterion: the elastic constants C₁₁ and det (C_ij) for Cr₃B₅ are found to be positive, and crucially, they satisfy the criterion where the multiplication of C₁₁ and C₂₂ is greater than the square of C₁₁*C₂₂ > C₁₂*C₁₂, C₁₁C₂₂C₆₆ + 2C₁₂C₁₆C₂₆–C₁₁C₂₆ ²–C₂₂C₁₆ ²–C₆₆C₁₂ ² > 0. Taken together, these conditions confirm the mechanical stability of the Cr₃B₅ structure. This result indicates that the elastic properties of Cr₃B₅ meet the stability requirements, demonstrating its unwavering resistance to deformation in the field of mechanics. In Table , Young’s modulus and Poisson’s ratio are listed in detail, providing precise numerical basis for the mechanical properties of this material and laying a foundation for further analysis and understanding of its performance under various conditions. For Cr₃B₅, both Y _y and Y _x exceed the corresponding values of graphene, which are 295 N/m, indicating that it has good mechanical properties. The value of σ _x /σ _y (0.190–0.210) is similar to that of graphene (σ = 0.173).

2. For Cr₃B₅, the Values of the Elastic Stiffness Constants C_ij (in Units of Newtons per Meter), the Young’s Modulus Y (Again, in Newtons per Meter), and the Poisson’s Ratios σ _x and σ _y Were Calculated.

structure	C11	C12	C22	C16	C26	C66	Y x	Y y	σ x	σ y
Cr3B5	341.20	73.85	436.66	–1.68	0.87	184.95	328.72	420.67	0.17	0.22

3.2. Electronic Structure

To delve into the electronic structural properties of Cr₃B₅, computations were executed to ascertain band structures segregated by orbitals, the density of electronic states (DOS), orbital-projected densities of states (PDOS), and the crystal orbital Hamilton population (COHP)which quantifies the bonding and antibonding characteristics between adjacent atomsand the Fermi surface (FS) as shown in Figure . In the orbitally resolved band structure diagram, each band is distinguished according to its corresponding atomic orbital, thereby revealing the electronic characteristics that determine the unique properties of the compound. In the corresponding graphic Figure a, the spectacle of two prominent bands slicing through the Fermi energy level along the Γ–X, H_1–C, C–H, and Y–Γ trajectories is most directly observed. This distinct phenomenon conclusively confirms the metallic nature of Cr₃B₅, as evidenced by the intersection of its energy bands at the Fermi point, demonstrating its inherent electrical conductivity. More specifically, in the atomic-orbital-projected band diagrams of Cr₃B₅, namely Figure a,b, the projected orbital that mainly dominates at the Fermi level (E _F) is Cr-4d, followed by B-2p. Displayed in Figure d,f, the comprehensive view of total electronic density and the decomposed orbital densities mirrors the insights gleaned from the band structure analysis. Most notably, a pronounced peak adjacent to the Fermi energy level catches the eye, echoing the high degree of electronic engagement inherent to Cr₃B₅. This observation fortifies the compound’s electronic landscape, highlighting its rich metal-like characteristics. Furthermore, we evaluated the influence of spin–orbit coupling (SOC) on the electronic structure, as shown in Figure e. The results show that the inclusion of SOC induces a minor change of approximately 2.8% in the density of states at the Fermi level (increasing from 4.839 states/eV to 4.976 states/eV), indicating a relatively weak SOC effect in the present system.

3.

(a) Orbital-projected electronic band structure of Cr₃B₅ along high-symmetry lines. (b,c) Orbital-resolved energy bands of individual Cr atoms and individual B atoms, respectively. (d) The total DOS without SOC of Cr₃B₅ and the total DOS of Cr and B atoms, respectively. (e) The total DOS with SOC of Cr₃B₅. (f) The PDOS of Cr₃B₅. The Fermi level indicated by the dotted line is set to 0 eV. (g) COHP analysis on the B–B, B–Cr, and Cr–Cr bonding interactions in Cr₃B₅. (h) The FS of Cr₃B₅ with the color drawn proportional to the magnitude of the Fermi velocity υ_F.

According to the core of the BCS theory of superconductivity, the superconducting transition temperature is often directly proportional to the electron density of states around the Fermi level. , In an environment with a rich electron density of states near the Fermi level, electron pairing becomes easier, thus promoting the formation of Cooper pairsthe cornerstone of the superconducting phenomenonwhich, in turn, paves a broad path for the increase of the superconducting transition temperature. In the complex electronic structure system of Cr₃B₅, the d orbitals of Cr atoms play a dominant role, forming the basis for its metallic properties. The p orbitals of B serve an auxiliary function, enhancing its metallicity without overshadowing the dominant role of the Cr atoms. In contrast, the s orbitals of both Cr and B can be regarded as background elements, making extremely minor contributions to the Fermi level that are almost negligible. Conversely, the s orbitals from both chromium and boron can be regarded as background extras, their contributions to the Fermi energy level so minuscule as to be virtually nonexistent. This observation is consistent with the previous band-projection results and is also in line with the previous calculation results of superconductivity in most two-dimensional transition-metal borides. − Similar to the density of states, the COHP curve can also be projected onto two atoms whose interactions are of interest, allowing researchers to determine whether the two atoms are bonding or antibonding and the strength of the bonding. We can have a look at Figure g, which presents the COHP diagrams for the interactions among Cr–Cr, B–Cr, and B–B atoms. In Cr₃B₅, the COHP clearly indicates that the B–B bonding is the major source of structural stability, followed by the B–C interaction, and finally the Cr–Cr interaction.

Figure h unveils the FS traced along the high-symmetry path Γ–X–H_1–C–H–Y–Γ. The corresponding Fermi velocity (υ_F) is indicated by color, reflecting the different slopes constituting the FS. Together with the further interpretation of the band structure distinguished by orbitals, this provides a comprehensive perspective for an in-depth understanding of the electronic properties of the compound. For Cr₃B₅, near the Fermi level, the peak of the Fermi velocity (υ_F) mainly originates from the band configuration along the H_1–C–H path. Specifically, the electron distribution interval provided by the Cr-d orbitals is particularly evident throughout the high-symmetry path; this phenomenon is consistent with the orbital-decomposed band structure shown in Figure , demonstrating the electronic structural characteristics of the compound.

3.3. Electron–Phonon Coupling and Possible Superconductivity

Our exploration of the phonon-mediated superconductivity potential in Cr₃B₅ involves in-depth theoretical investigations into the phonon dispersion spectrum phonon density of states (PhDOS) and the spectral function α² F(ω) characterizing the evolution of the superconducting gap obtained from the Eliashberg equation. From these, we derived the temperature-dependent electron–phonon coupling function λ­(ω) through integration. These studies provide a comprehensive perspective on our research findings, as illustrated in Figure . These detailed inspection offers a panoramic view of the compound’s phonon dynamics, essential for discerning its superconductivity mechanisms. The original structure of Cr₃B₅ contains eight atoms, which gives rise to 24 phonon modes, including twenty-one optical phonon branches and three acoustic phonon branches. The absence of imaginary vibrational modes signifies the thermodynamic stability of this structurea testament to its soundness under thermal dynamics.

4.

(a) The phonon dispersion is colored based on the intensity of the electron–phonon coupling λ_qν, with its size represented uniformly throughout. (b,c) The orbital-resolved band structure along the high-symmetry line Γ–X–H_1–C–H–Y–Γ, incorporating the contributions from Cr and B atoms. (d) Total DOS and atom-projected phonon DOS. (e) Eliashberg spectral function α ² F(ω) and cumulative frequency-dependent EPC function λ_(ω).

The phonon spectrum of this system exhibits a wide frequency domain distribution, with a frequency span exceeding 1000 cm^–1. This phenomenon originates from the high-energy phonon excited states induced by the strong coupling between light atoms. Strong lattice dynamic interactions cause the system to support higher-energy quantized lattice vibration modes, whose physical essence is reflected in the significantly broadened high-frequency characteristic peaks in the phonon density of states function. According to the magnitude of the frequency, the phonon dispersion is roughly divided into three regions: high-frequency, medium-frequency, and low-frequency. Due to the significantly larger atomic mass of Cr compared to boron, in the low-frequency region of 0 to 400 cm^–1, Cr and B atoms vibrate cooperatively, with the influence generated by Cr atomic vibrations being slightly greater than that of boron atomic vibrations. When entering the medium-frequency region (400–700 cm^–1), the contribution of B atomic vibrations becomes extremely significant compared to that of Cr atomic vibrations. Upon entering the high-frequency zone (700–1200 cm^–1), the stage is entirely commandeered by the vibrational dynamics of Boron atoms, rendering their impact singularly significant.

Illustrated in the phonon dispersion diagram, meticulously weighted by phonon line width as depicted in Figure a, phonon softening phenomena of varying degrees are clearly observed near the high-symmetry points Γ and H, as well as in the vicinity of H_1. These regions exhibit notably large λ_qv values and precisely fall within the low-frequency range of 0 to 200 cm^–1. Correspondingly, the electron–phonon coupling strength λ­(ω) demonstrates a significant increase in this frequency region. The softening of phonon modes in this frequency interval typically coincides with a stronger electron–phonon coupling. In the phonon dispersion diagrams of Cr and B atom vibration projections shown in Figure b,c, in the low-frequency range from 0 to 400 cm^–1, the in-plane and out-of-plane vibrations of Cr atoms are dominant; while in the high-frequency region above 400 cm^–1, the in-plane and out-of-plane vibrations of B atoms become the dominant factors. As shown in Figure a, in the low-frequency region of the phonon dispersion diagram, atomic vibration scenarios at three representative phonon softening locations were selected based on the magnitude of λ_qv. Mode I is in the frequency range of 0 to 100 cm^–1, mainly along the acoustic branches pointing in the Γ–X and Γ–Y directionsthere are two distinct Kohn anomalies, which primarily originate from the in-plane vibrations of Cr atoms. There are two tiny softened modes in the frequency range of (0–100 cm^–1) along the H₁–C and C–H directions for Mode II, which is mainly attributed to the in-plane and out-of-plane vibrations of Cr atoms. Mode III is characterized by the emergence of a softened phonon mode near the Γ point within the frequency range of 100–200 cm^–1. These three vibration modes in the low-frequency region contribute significantly to electron–phonon coupling (EPC), accounting for 79.62% of the total electron–phonon coupling.

As is well-known, the Coulombic pseudopotential μ* is a key empirical parameter affecting the superconducting transition temperature T _c. When calculating the T _c of Cr₃B₅, a clear correlation between T _c and μ* was found, as shown in Figure . The value range of μ* is from 0.05 to 0.15. Within this range, as μ* increases, the superconducting transition temperature T _c of Cr₃B₅ exhibits a monotonically decreasing trend. Specifically, when the Coulombic pseudopotential μ* increases from 0.05 to 0.15, the superconducting transition temperature T _c of Cr₃B₅ decreases from 15.042 to 8.543 K. For the widely used μ* value of 0.10, the superconducting transition temperature T _c of Cr₃B₅ is determined to be 11.678 K. The superconductivity exhibited in this new structure fundamentally depends on the coupling between the in-plane phonon motion in Cr atoms and the electrons in the Cr-d orbitals. As shown in Table , the T _c of Cr₃B₅ exceeds that of two-dimensional TMBs obtained theoretically in recent years, such as Li₂B₇, Mo₂B₂, and t-MnB₃. Although the superconducting transition temperature of Cr₃B₅ is lower than that of two-dimensional ZrB₆ (15.79 K), it still lies within the liquid helium temperature range, facilitating experimental implementation. Its electron–phonon coupling strength λ = 0.896, which is close to that of ZrB₆ (λ = 0.93), showing the characteristic of strong interaction. This material satisfies the Born mechanical stability criterion, and its cohesive energy of 7.54 eV/atom is significantly superior to that of ZrB₆ (whose elastic parameters have not been reported), demonstrating better structural stability. Compared with the MnB₃ monolayer, although the superconducting temperature of Cr₃B₅ is lower than that of MnB₃ (24.9 K), its λ value (0.896) is higher than that of MnB₃(0.79), and its cohesive energy (7.54 eV/atom) is far greater than that of h-MnB₃ (−0.25 eV/atom). This indicates that its atomic bonding is tighter, and it has higher synthesis feasibility and application potential.

5.

(a) Vibration modes at the marked I, II, and III positions in Figure a, which exhibit large EPC and contribute most to the EPC. The arrows represent the directions of the atomic vibrations. (b) The superconducting transition temperature Tc of Cr₃B₅ obtained through calculation is a function of the Coulombic pseudopotential μ*.

3. EPC Constants (λ) and Superconducting Critical Temperatures T _c (K) of Several Two-Dimensional TMB Superconductors.

2D materials	λ	T c	refs.
Li2B2	0.56	6.2
Mo2B2	0.49	3.9
ZrB6	0.93	15.79
t-MnB3	0.37	2.9
h-MnB3	0.79	24.9

4. Conclusion

In summary, we successfully identified a novel 2D Cr₃B₅ material through advanced structural prediction and first-principles calculations. This material is not only intrinsically metallic but also exhibits high thermodynamic stabilitya trait shared with many TMBsalong with strong phonon-coupling mechanisms. First-principles calculations reveal a remarkable cohesive energy of 7.54 eV/atom, surpassing most reported two-dimensional TMBs, along with robust mechanical stability demonstrated by elastic constants satisfying the Born criteria. Remarkably, its superconducting transition temperature T _c reaches 11.678 K through a phonon-mediated mechanism, enabled by strong coupling between low-frequency in-plane chromium vibrations and chromium d-orbitals. This unique interaction promotes the emergence of superconductivity. The calculated coupling constant λ = 0.896 confirms significant electron–phonon interactions, while ab initio phonon spectra validate dynamical stability across the BZ. This high stability + strong coupling synergy positions Cr₃B₅ as an experimentally accessible platform for superconductivity research. Moreover, this material system exhibits excellent potential for property modulation: strategies such as strain engineering, carrier doping, or heterostructure construction can further optimize its electronic structure near the Fermi level and phonon behavior, thereby effectively enhancing the superconducting transition temperature. This study not only expands the chemical space of two-dimensional functional materials but also provides new design concepts and material foundations for the development of high-performance superconducting systems.

All data needed to evaluate the conclusions in this study are included in the article itself. No additional Supporting Information are available.

The authors declare no competing financial interest.